OptiStruct 13.0 User Guide

HyperWorks 13.0 OptiStruct User's Guide

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OptiStruct 13.0 User's Guide

User's Guide ........................................................................................................................................... 1 Overview............................................................................................................................................... 2 Features ................................................................................................................................... 5 Capabilities ................................................................................................................................... 12 Formats ................................................................................................................................... 13 Enhancing ................................................................................................................................... the Design Process 14 Pre-processing and Post-processing in HyperWorks ................................................................................................................................... 17 Running............................................................................................................................................... OptiStruct 21 Run Options for OptiStruct ................................................................................................................................... 25 OptiStruct ................................................................................................................................... GPU 39 OptiStruct ................................................................................................................................... SPMD 41 Platforms and Hardware Recommendations ................................................................................................................................... 60 OptiStruct ................................................................................................................................... Configuration File 62 Expanded ................................................................................................................................... Error Message File 67 Memory Limitations ................................................................................................................................... 69 Restarting ................................................................................................................................... OptiStruct 71 OptiStruct ................................................................................................................................... Compression Run 72 Structural Analysis ............................................................................................................................................... 74 Linear Static Analysis ................................................................................................................................... 75 Linear Buckling Analysis ................................................................................................................................... 76 Nonlinear Analysis ................................................................................................................................... 78 Normal Modes Analysis ................................................................................................................................... 103 Frequency................................................................................................................................... Response Analysis 107 Complex ................................................................................................................................... Eigenvalue Analysis 113 Random Response Analysis ................................................................................................................................... 115 Response................................................................................................................................... Spectrum Analysis 119 Transient................................................................................................................................... Response Analysis 123 Thermal Analysis ............................................................................................................................................... 129 Linear Steady-State Heat Transfer Analysis ................................................................................................................................... 130 Linear Transient Heat Transfer Analysis ................................................................................................................................... 133 Nonlinear................................................................................................................................... Steady-State Heat Transfer Analysis 135 Contact-based Thermal Analysis ................................................................................................................................... 137 Acoustic Analysis ............................................................................................................................................... 140 Coupled Frequency Response Analysis of Fluid-Structure Models ................................................................................................................................... 141 Radiated ................................................................................................................................... Sound Analysis 258 Fatigue............................................................................................................................................... Analysis 266 Multi-body Dynamics Simulation ............................................................................................................................................... 282 i

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Transient................................................................................................................................... Analysis for MBD 284 Static Analysis for MBD ................................................................................................................................... 286 Quasi-static Analysis for MBD ................................................................................................................................... 287 Linear Analysis for MBD ................................................................................................................................... 288 Bodies

................................................................................................................................... 289 Markers ................................................................................................................................... 290 Constraints ................................................................................................................................... 291 Contact ................................................................................................................................... 293 Compliant................................................................................................................................... Elements 295 Applied Forces and Motions ................................................................................................................................... 296 Initial Velocity ................................................................................................................................... 297 Function Expressions ................................................................................................................................... 298 Results of................................................................................................................................... a Multi-body Dynamics Analysis 299 Rotor Dynamics ............................................................................................................................................... 300 NVH Applications and Techniques ............................................................................................................................................... 309 Transfer Path Analysis on an Automobile ................................................................................................................................... 310 Residual Runs using Super Elements ................................................................................................................................... 316 Basic OptiStruct NVH Output Files ................................................................................................................................... 319 Global Search Option ................................................................................................................................... 322 Create Door and Deck Lid Seals ................................................................................................................................... 325 Create a HyperGraph Template for Reading in Multiple Files ................................................................................................................................... 328 Using AMSES (Automatic Multi-Level Sub-Structuring Eigensolver Solution) ................................................................................................................................... 329 Modeling Techniques ............................................................................................................................................... 331 Parts and................................................................................................................................... Instances 332 Subcase Specific Modeling ................................................................................................................................... 341 Direct Matrix Input (Superelements) ................................................................................................................................... 345 Flexible Body Generation ................................................................................................................................... 364 Poroelastic Materials (Biot theory) ................................................................................................................................... 369 Elements ................................................................................................................................... and Materials 371 Loads and................................................................................................................................... Boundary Conditions 385 Modeling ................................................................................................................................... Errors 404 Results............................................................................................................................................... 407 Coupling OptiStruct with Third Party Software ............................................................................................................................................... 417 Design ............................................................................................................................................... Optimization 425 Optimization Problem ................................................................................................................................... 426 Responses ................................................................................................................................... 429 Topology ................................................................................................................................... Optimization 446 Free-size ................................................................................................................................... Optimization 460 Topography Optimization ................................................................................................................................... 467 Size Optimization ................................................................................................................................... 471 Shape Optimization ................................................................................................................................... 473

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Free-shape Optimization ................................................................................................................................... 475 Manufacturing Constraints ................................................................................................................................... 493 Reliability-based Design Optimization (Beta) ................................................................................................................................... 558 Optimization of Arbitrary Beam Sections ................................................................................................................................... 564 Optimization of Composite Structures ................................................................................................................................... 565 Equivalent Static Load Method (ESLM) ................................................................................................................................... 573 Gradient-based Optimization Method ................................................................................................................................... 587 Global Search Option ................................................................................................................................... 596 Design ............................................................................................................................................... Interpretation - OSSmooth 598 OSSmooth Parameter File ................................................................................................................................... 601 Running OSSmooth ................................................................................................................................... 606 Interpretation of Topology Optimization Results ................................................................................................................................... 607 Laplacian................................................................................................................................... Smoothing 608 Interpretation of Topography Optimization Results ................................................................................................................................... 610 FEA Topology for Reanalysis ................................................................................................................................... 613 FEA Topography for Reanalysis ................................................................................................................................... 615 OptiStruct References ............................................................................................................................................... 617

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User's Guide Overview Running OptiStruct Structural Analysis Thermal Analysis Acoustic Analysis Fatigue Analysis Multi-body Dynamics Simulation Rotor Dynamics NVH Applications and Techniques Modeling Techniques Results Coupling OptiStruct with Third Party Software Design Optimization Design Interpretation - OSSmooth OptiStruct References

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Overview Altair® OptiStruct® is an industry proven, modern structural analysis solver for linear and non-linear structural problems under static and dynamic loadings. It is the market-leading solution for structural design and optimization. Based on finite element and multi-body dynamics technology, and through advanced analysis and optimization algorithms, OptiStruct helps designers and engineers rapidly develop innovative, lightweight and structurally efficient designs. OptiStruct is used by thousands of companies worldwide to analyze and Optimize structures for their strength, durability and NVH (noise, vibration and harshness) characteristics. Refer to the Features page for a list of solutions available in OptiStruct. Finite element solutions via OptiStruct include: Linear static analysis Nonlinear implicit quasi-static analysis Linear buckling analysis Normal modes analysis Complex eigenvalue analysis Frequency response analysis Random response analysis Linear transient response analysis Geometric non-linear explicit and implicit analysis Linear fluid-structure coupled (acoustic) analysis Linear steady-state heat transfer analysis Coupled thermal-structural analysis Nonlinear steady-state heat transfer analysis Linear transient heat transfer analysis Contact-based thermal analysis Inertia relief analysis with static, non-linear contact, modal frequency response, and modal transient response analyses Component Mode Synthesis (CMS) for the generation of flexible bodies for multi-body dynamics analysis Reduced matrix generation One-step (inverse) sheet metal stamping analysis Fatigue analysis A typical set of finite elements including shell, solid, bar, scalar, and rigid elements as well as loads and materials are available for modeling complex events. Multi-body dynamics solutions integrated via OptiStruct for rigid and flexible bodies include: Kinematics analysis Dynamics analysis

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Static and quasi-static analysis Linearization All typical types of constraints like joints, gears, couplers, user-defined constraints, and highpair joints can be defined. High pair joints include point-to-curve, point-to-surface, curve-tocurve, curve-to-surface, and surface-to-surface constraints. They can connect rigid bodies, flexible bodies, or rigid and flexible bodies. For this multi-body dynamics solution, the power of Altair MotionSolve has been integrated with OptiStruct.

Structural Design and Optimization Structural design tools include topology, topography, and free-size optimization. Sizing, shape and free-shape optimization are available for structural optimization. In the formulation of design and optimization problems, the following responses can be applied as the objective or as constraints: compliance, frequency, volume, mass, moment of inertia, center of gravity, displacement, velocity, acceleration, buckling factor, stress, strain, composite failure, force, synthetic response, and external (user-defined) functions. Static, inertia relief, nonlinear quasi-static (contact), normal modes, buckling, and frequency response solutions can be included in a multi-disciplinary optimization setup. Topology, topography, size, and shape optimization can be combined in a general problem formulation.

Topology Optimization Topology optimization generates an optimized material distribution for a set of loads and constraints within a given design space. The design space can be defined using shell or solid elements, or both. The classical topology optimization set up solving the minimum compliance problem, as well as the dual formulation with multiple constraints are available. Constraints on von Mises stress and buckling factor are available with limitations. Manufacturing constraints can be imposed using a minimum member size constraint, draw direction constraints, extrusion constraints, symmetry planes, pattern grouping, and pattern repetition. A conceptual design can be imported in a CAD system using an iso-surface generated with OSSmooth, which is part of the OptiStruct package. Free-size optimization is available for shell design spaces. The shell thickness or composite ply-thickness of each element is the design variable.

Topography Optimization Topography optimization generates an optimized distribution of shape based reinforcements such as stamped beads in shell structures. The problem set up is simply done by defining the design region, the maximum bead depth and the draw angle. OptiStruct automatically provides the design variable creation and optimization control. Manufacturing constraints can be imposed using symmetry planes, pattern grouping, and pattern repetition.

Size and Shape Optimization General size and shape optimization problems can be solved. Variables can be assigned to perturbation vectors, which control the shape of the model. Variables can also be assigned to

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properties, which control the thickness, area, moments of inertia, stiffness, and nonstructural mass of elements in the model. All of the variables supported by OptiStruct can be assigned using HyperMesh. Shape perturbation vectors can be created using HyperMorph. The reduction of local stress can be accomplished easily using free-shape optimization. Shape perturbations are automatically determined by OptiStruct (based on the stress levels in the design) when using this technique. The layout of laminated shells can be improved by modifying the ply thickness and ply angle of these materials.

Multi-body Dynamics Analysis Different solution sequences for the analysis of mechanical systems are available; these include Kinematics, Dynamics, Static, and Quasi-static solutions. Flexible bodies can be derived from any finite element model defined in OptiStruct.

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Features Finite Element Analysis using OptiStruct Structural Analysis - Linear Static Analysis - Linear Buckling Analysis - Nonlinear Quasi-Static Analysis - Large Displacement Nonlinear Static Analysis - Geometric Nonlinear Analysis (RADIOSS Integration) - Normal Modes Analysis - Frequency Response Analysis - Complex Eigenvalue Analysis - Random Response Analysis - Response Spectrum Analysis - Transient Response Analysis Thermal Analysis - Linear Steady-State Heat Transfer Analysis - Linear Transient Heat Transfer Analysis - Nonlinear Steady-State Heat Transfer Analysis - Contact-based Thermal Analysis Acoustic Analysis - Coupled Frequency Response Analysis of Fluid-Structure Models - Radiated Sound Analysis Fatigue Analysis - Stress-Life method - Strain-Life method Rotor Dynamics Fast equation solver - Sparse matrix solver - Iterative PCG solver - Lanczos eigensolver - SMP parallelization - SPMD parallelization - DMIG input - AMLS Interface - FastFRS Interface

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Advanced element formulations - Triangular, quadrilateral, first and second order shells - Laminated shells - Hexahedron, pyramid, tetrahedron first and second order solids - Bar, beam, bushing, and rod elements - Spring, mass, and damping scalar elements - Mesh independent gap and weld elements - Rigid elements - Concentrated and non-structural mass - Direct matrix input Geometric element quality check Local coordinate systems Multi-point constraints Contact, tie interfaces Prestressed analysis Linear-elastic materials - Isotropic - Anisotropic - Orthotropic Nonlinear materials - Elastoplastic - Hyperelastic - Viscoelastic Material consistency checks Ground check for unintentionally constrained rigid body modes.

Modeling Techniques Parts and Instances Subcase Specific Modeling Direct Matrix Input (Superelements) - Direct Matrix Input - Creating Superelements - Component Dynamic Analysis Flexible Body Generation Poroelastic Materials

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Multi-body Dynamics using OptiStruct Solution sequences - Kinematics - Dynamics - Static - Quasi-static - Linearization Bodies - Rigid - Flexible - Flexible body generation in using the CMS modeling technique, integrated with multi-body analysis if the model is set up in OptiStruct. Constraints (between any body, flexible, or rigid) - Joints: Ball (spherical), free, fixed, revolute, translational, cylindrical, universal, planar, at-point, in-plane, parallel-axes, orient, perpendicular-axes, constant velocity, and in-line. - Gear - Couplers - Higher-pair joints: point-to-curve, point-to-surface, curve-to-curve, curve-tosurface, and surface-to-surface constraints. Loads - Forces - Gravity - Motions (Joint and Marker) - Initial velocities (Body and Joint) Function Expressions

Optimization General optimization problem formulation for all optimization types - Response based - Equation utility - Interface to external user-defined routines - Minmax (maxmin) problems - System identification - Continuous and discrete design variables Solution sequences for optimization - Linear static - Normal modes

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- Linear buckling - Quasi-static nonlinear (gap/contact) - Frequency response (modal method with residual vectors) - Acoustic response - Random response - Linear steady-state heat transfer - Coupled thermo-mechanical - Multi-body Dynamics - Fatigue Responses for optimization - All optimization types: - Compliance - Frequency - Compliance index - Volume - Mass - Volume fraction - Mass fraction - Center of gravity - Moments of inertia - Displacement - Velocity - Acceleration - Temperature - Pressure - Stress (global von Mises stress in topology/free-size optimization) - Buckling factor (with limitations in topology/free-size optimization) - Fatigue life/damage - User-defined responses - Size, shape, free-shape, and topography optimization: (In problems with topology/free-size design domains, these responses can be used in the non-design domain) - Strain - Force - Composite stress, strain, and failure (linear static analysis only) Automatic selection of best optimization algorithm - Optimality criteria method - Convex approximation method - Method of feasible directions - Sequential quadratic programming - Advanced approximations Automatic selection of best method for design sensitivity analysis - Direct method - Adjoint variable method Topology, free-size, topography, size, shape, and free-shape optimization problems can

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be solved simultaneously Multi-disciplinary optimization using combinations of the supported solution sequences Mode tracking

Topology Optimization Generalized optimization problem formulation Multiple load cases with different solution sequences in combination Global von Mises stress constraint for static loads Density method 1-D, 2-D, and 3-D elements in the design space Non-design space can contain any element type and response Extensive manufacturing control: - Minimum member size control to avoid mesh dependent results - Maximum member size control to avoid large material concentrations - Draw direction constraints - Extrusion constraints - Pattern grouping - Pattern repetition - Multiple symmetry planes Checkerboard control Discreteness control Smoothing and geometry generation for 3-D results

Free-Size Optimization Generalized optimization problem formulation Multiple load cases with different solution sequences in combination Global von Mises stress constraint for static loads Shell element thickness and composite ply-thickness design variables Non-design space can contain any element type and response Extensive manufacturing control: - Minimum member size control to avoid mesh dependent results - Maximum member size control to avoid large material concentrations - Draw direction constraints - Extrusion constraints - Pattern grouping

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- Pattern repetition - Multiple symmetry planes

Topography Optimization Shape optimization for shells with automated design variable definition Easy set up with one DTPG card Extensive bead pattern control to allow for manufacturing constraints - Pattern grouping - Pattern repetition - Multiple symmetry planes - Discreteness control

Size Optimization Shell, rod, and beam properties can be designed Spring and concentrated mass properties can be designed Composite ply thickness and ply angle can be designed Material properties can be designed Continuous and discrete design variables

Shape Optimization Perturbation vector approach Shape functions are defined through DVGRID cards Continuous and discrete design variables

Free-shape Optimization Perturbation vector approach Automatic generation of perturbation vectors Reduction of stress concentrations

Structural Optimization in Multi-body Dynamics Systems Equivalent Static Load (ESL) method Size, shape, free-shape, topology, topography, free-size, and material optimization of flexible bodies in multi-body dynamics systems

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Generalized optimization problem definition Large number of design variables and constraints

Pre-processing Fully supported in HyperMesh and MotionView Nastran type input format

Post-processing HyperView - Direct output of H3D format for model and results - Direct output for iteration history - Export of iso-density surface in STL format HyperGraph - Iteration history graphs - Sensitivity bar charts - Complex frequency response displacement, velocity, and acceleration plots for up to 500 nodes - Random response PSD and auto/cross correlation of displacement, velocity, and acceleration - Transient response displacement, velocity, and acceleration time history plots for up to 500 nodes - Bar chart for effective mass HTML report - Model summary - Model and result displayed using HyperView Player HyperMesh - Direct binary result file output Microsoft Excel - Design sensitivities for size and shape variable approximations Support of Nastran Punch and OP2 output formats

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Capabilities OptiStruct can be used to solve and optimize a wide variety of design problems in which the structural and system behavior can be simulated using finite element and multi-body dynamics analysis. The design and optimization capabilities of OptiStruct allow for the development of preliminary design concepts and for the improvement of existing designs based on finite element analyses. Some types of optimization problems are listed below: Two-dimensional truss structure optimization Ribbed reinforcement patterns for 3-D shell structures Ribbed reinforcements for solid structures Spotweld reduction Lightening holes for existing 2-D planar and 3-D bending shell problems Discrete optimized structures for problems modeled using 3 dimensional solid element problems Bead (Swages) reinforcements in 3-D shell structures Shape modifications for volume parts Gage optimization of 3-D shell structures Beam cross-section optimization of structures modeled with beam elements Layout of laminated shell by modifying ply thickness and ply angle Reduction of stress concentrations Optimization of mechanisms and mechanical systems to minimize weight and reduce stress

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Formats OptiStruct supports the following input/output formats: Formats Input

Nastran Bulk Data Format

Output

HyperMesh Result File (Results) H3D Binary File (Results) Patran ASCII (Results) Nastran Output2 (Results) Nastran Punch File (Results) OptiStruct 2.0 (Results) HyperView Format (Iteration history, sensitivities, effective mass) Microsoft Excel (Sensitivities) From Bulk Data Format input: HyperMesh Result File Nastran Output2 File Nastran Punch File OptiStruct 2.0 File Patran ASCII File

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Enhancing the Design Process OptiStruct enhances the design process by: Accelerating the design process Shortening the number of design cycles Increasing the design performance Providing fast and accurate finite element analysis Generating optimal design concepts using topology and topography optimization Providing traditional size and shape optimization to maximize the design performance

The design process can be viewed as an optimization process to find structures, mechanical systems, and structural parts that fulfill certain expectations towards their economy, functionality, and appearance. Generally, the design process is an iterative procedure consisting of the following components: Conceptual design Design Testing Optimization

Today’s testing ground is usually the computer. Finite element analysis (FEA) and Multi-body dynamics analysis (MBD) are the most used tools for computational design testing. The results of computational analyses are used to determine design improvements. Changes to the design are introduced in all phases of the process. At a certain stage of this process, changes to the concept become prohibitive. The concept phase plays a fundamental role concerning overall efficiency of the design and the cost of the overall development process. In the concept phase of a design process, the freedom of the designer is limited only by the specifications of the design (Figure 1). Today, the decision on how a new design should look is based largely upon a benchmark design or on previous designs. The decision making is based on the experience of those involved in the design process. Conceptual design tools such as topology and topography optimization can be introduced to enhance the process. The concept can be based on results of a computational optimization rather than on estimations. Using topology and topography optimization, the initial design step is already based on input generated using computational analysis. Topology and topography optimization redefine the role of computational analysis and simulation in the design process. Finite element analysis has matured from a testing tool to a design tool.

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Figure 1: Decision making in the design process.

Figure 2 compares the design process using topology optimization with the conventional method of leaving the concept entirely to experience and intuition. The overall cost of design development can be reduced substantially by avoiding concept changes introduced in the testing phase of the design. This is the major benefit of modifying the design process by introducing topology and topography optimization. In the real world, the design process is not as straightforward as described above. The design is not just driven by one performance measure -- it has to be viewed as a multidisciplinary task. Today, the different disciplines work more or less independently. Analysis and optimization is performed for single phenomena such as linear static behavior or noise, vibration and harshness. Still, the idea persists that if one performance measure improves, the whole performance improves. A simple example shows that this is not quite true. Take the design of a car -- a high stiffness is necessary for good driving and handling, and high deformability is important for the crashworthiness of the design. This shows that improving one measure may result in degrading another. Therefore, compromises must go into the formulation of the optimization problem. The definition of the design problem and of the design target is most important. The solution can be left to computational means. Multidisciplinary considerations, especially in the conceptual design, are, in many ways, still active research topics and are being covered by future developments of topology optimization. However, the inclusion of manufacturing constraints into topology and topography optimization is already implemented in OptiStruct.

Figure 2: The design process without and with the use of topology optimization.

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OptiStruct also provides size and shape optimization to completely support the design process with finite element based structural optimization. Using the advanced interfacing with HyperMesh, the generation of input data for structural optimization becomes an easy task. This allows structural optimization to be integrated into the design process seamlessly.

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Pre-processing and Post-processing in HyperWorks Pre-processing Pre-processing tools must be used to prepare models for OptiStruct, RADIOSS, and MotionSolve. HyperWorks provides specialized pre-processors interfacing with the solvers. HyperMesh can be used to mesh and set up finite element simulations for OptiStruct and RADIOSS. Two user profiles are provided: OptiStruct RADIOSS (with sub-profiles for the different input formats)

HyperCrash is useful to set up finite element models for automotive crash simulation in RADIOSS. It provides a number of useful tools for dummy positioning and model interrogation that are not available in HyperMesh. Translation of models from OptiStruct to RADIOSS and vice versa can be performed efficiently in HyperCrash. HyperForm is used to set up and execute sheet metal stamping simulations. Two user profiles are provided to run RADIOSS: One_Step Incremental_Radioss

MotionView is used to set up multi-body dynamics models for MotionSolve. The respective SolverMode has to be chosen.

Figure 1. HyperMesh

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Figure 2. HyperCrash

Figure 3. HyperForm

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Figure 4. MotionView

Post-processing Graphical tools must be used to visualize and evaluate the results of OptiStruct, RADIOSS, and MotionSolve. HyperWorks provides HyperView, a specialized post-processor, for this. HyperView allows animation, 2D and 3D plotting, video and text processing to work with the solver results and to generate reports. It can be used for all post-processing purposes in finite element and multi-body dynamics analysis. Direct readers are provided for the animation and time history file written by OptiStruct, RADIOSS, and MotionSolve.

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Figure 1. HyperView

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Running OptiStruct Note: Your system administrator may need to modify the script to make it compatible with your system. This section describes the execution of OptiStruct. There are several ways to run OptiStruct: From the script. From the HyperWorks Solver Run Manager. From inside the preprocessors HyperMesh. From inside HyperView and HyperGraph. In all the above cases, HyperWorks will initialize $PATH and other environment variables required to run the selected solver, however you are responsible for initializing environment variables for third party products. In particular, MPI and AMLS/FFRS external solvers (if needed) may require PATH and LD_LIBRARY_PATH.

Running OptiStruct from the Script To run on UNIX from the command line, type the following: /altair/scripts/optistruct "filename" –option argument To run OptiStruct from a Windows DOS prompt, type the following: \hwsolvers\bin\win64\optistruct.bat "filename" –option argument The options and arguments are described under Run Options for OptiStruct. OptiStruct looks for "filename" in the following manner ("filename" may contain a file path that is either absolute or relative to the run directory): First, it checks to see if "filename" exists exactly as input. If "filename" does not exist exactly as input, and if "filename" does not contain an extension (that is, if the actual file name without the path does not contain a period), then it checks for "filename".parm and then for "filename".fem. If none of these checks results in a match, OptiStruct reports an error and terminates.

Running OptiStruct from HyperWorks Solver Run Manager On Windows, a utility to start each solver is provided through Start > Programs > Altair HyperWorks 13.0 > OptiStruct. This utility allows you to start multiple solver runs, select options from the menu, and maintains a history of solutions. On UNIX platforms, this utility can be started from command line as: /altair/scripts/ -gui

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Running OptiStruct from HyperMesh If you set up a finite element model in HyperMesh, you can run the simulation directly out of HyperMesh by going to the OptiStruct panel in the respective user profiles. The panels can be accessed through the Analysis page, from the Utility menu, or through the Applications pull down. The panels ask for the file name. After clicking the solver button, the model is exported using the given export options. Then the solver runs the script that is provided locally on the machine. After solver execution, the results can be viewed in HyperView. You can bring up HyperView with the results loaded by clicking HyperView. Note: When running OptiStruct from HyperMesh on UNIX and Linux, a shell is spawned with the DISPLAY setting :0.0. If this is different from the DISPLAY setting for HyperMesh, 50 HyperWorks units (in addition to the 21 HyperWorks units being used for HyperMesh) will be checked out. To avoid the checking out of additional units, be sure that the DISPLAY is set to :0.0 before starting HyperMesh.

Running OptiStruct from HyperView or HyperGraph If you are in HyperView or HyperGraph, OptiStruct can be run from the Applications pulldown. After selecting OptiStruct, the HyperWorks Solver Run Manager main form will appear, which will allow you to select a file, enter run options, and run the simulation.

The OptiStruct Configuration File The configuration file optistruct.cfg may be used to establish default settings for OptiStruct either system wide, for a particular user, or for a local directory. A full description of the settings allowed and the usage of the configuration file is provided on the OptiStruct Configuration File page.

Environment Variables The following environment variable is optional and may be set on either UNIX or PC platforms; however, the preferred way is to define them using the OptiStruct Configuration File. OS_TMP_DIR = path

Path – Path name to directory for scratch file storage (Default = directory where the solver is started – can be overwritten by the definition in the script or input deck).

The following environment variable is optional and may only be set on UNIX platforms; however, the preferred way is to define this using the OptiStruct Configuration File.

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DOS_DRIVE_$ = path

This environment variable allows drive letters to be assigned to UNIX paths. This facilitates copying files which contain INCLUDE, TMPDIR, INFILE or OUTFILE definitions containing drive letters from PC to UNIX on hybrid networks. $ - Drive letter to be defined (case sensitive). Path - UNIX path with which you want to replace the drive letter. Note that after such expansion, the paths are always interpreted as if there were a ‘\’ immediately after the drive letter in the original PC path.

Memory Allocation Memory is dynamically allocated for a run. The allocation starts with the initial memory. The default setting for the memory limit is 1GB for 64-bit solver version (PC and Linux). This setting can be changed by using the SYSSETTING option OS_RAM, or by defining the –len option in the run script. The script overwrites the environment variable. OptiStruct will always attempt to assign enough memory for a minimum core solution. The initial memory is 10% of the memory limit by default. This setting can be changed by using the SYSSETTING option OS_RAM_INIT. A check run can be very helpful in estimating the memory and disk space usage. In a check run, the memory necessary is automatically allocated. The solver automatically chooses an in-core, out-of-core, or minimum core solution based on the memory allocated. A solution type can be forced by defining the –core option in the run script; the memory necessary for the specified solution type is then assigned. Refer to the Memory Limitations section for detailed information on the following topics: 32bit versus 64-bit computations, virtual versus physical memory, and automatic memory allocation versus fixed memory runs.

Summary Information OptiStruct always creates an .out file which contains summary information for the job. This information can be echoed to the screen through the inclusion of the SCREEN I/O option in the input data or through the use of the -out command line option (see Run Options for OptiStruct). This file also contains memory and disk space estimates. The disk space estimates for eigenvalue analyses (normal modes, linear buckling, modal methods of frequency, transient response, and fluid-structure coupling (acoustics)) are sometimes very conservative and can be three times as much as is truly used. This is because it is not fully predictable how much data needs to be saved to scratch files. The true usage of memory and disk space is reported at the bottom of the file after the solver has finished.

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Should the job be re-run in the same location, the .out file is not overwritten, but is instead moved to _#.out, where # is the lowest available three digit number that creates a unique file name. For example, if filename.fem were run in a directory already containing filename.out, the existing filename.out would be moved to filename_001.out, and the summary information for the new job would be written to filename.out. Should the job be repeated again, the existing filename.out would be moved to filename_002.out, and the summary information for the latest job would be written to filename.out. filename.out is the only file that is saved in this manner. All other results files will be overwritten.

Recommendations 1. Try running OptiStruct with the default setting first (without specification of the –len or – core options). 2. Do a check run before submitting large jobs (>500,000 dof) to NQS to make sure sufficient NQS memory is being provided. The –lM option can be used to change the NQS memory. Be sure to include at least 12Mb for the executable in addition to the memory necessary to solve the problem. A check run can also assist in debugging input data without having to wait in a queue.

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Run Options for OptiStruct Option

Argument

Description

Available on

-acf

N/A

Option to specify that the input file is an ACF file for a multi-body dynamics solution sequence.

All Platforms

-amls

YES/NO

Invokes the external AMLS eigenvalue solver. Linux The AMLS_EXE environment variable needs to point to the AMLS executable for this setting to work. Overrides the PARAM, AMLS setting in the input file. (Example: optistruct infile.fem –amls yes)

-amlsncpu

1, 2, or 4

Defines the number of CPUs to be used by the Linux external AMLS eigenvalue solver. This parameter will set the environment variable OMP_NUM_THREADS. The default value is the current value of OMP_NUM_THREADS. Note that this value can be set by the command line arguments – nproc or –ncpu. OptiStruct and AMLS can be run with different allocations of processors. For example, OptiStruct can be run with 1 processor and AMLS with 4 processors in the same run. Only valid with –amls run option or when PARAM, AMLS is set to YES. Overrides the PARAM, AMLSNCPU setting in the input file. Default: Number of processors used by OptiStruct. (Example: optistruct infile.fem –amls yes –amlsncpu 4)

-amlsmem

Memory in GB

Defines the amount of memory in Gigabytes to be used by the external AMLS eigenvalue solver. This run option is only supported for AMLS versions 5 and later.

Linux

Note:

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Option

Argument

Description

Available on

1. This run option will override the memory value set by PARAM, AMLSMEM in the input file and the environment variable AMLS_MEM. 2. This run option is valid only if –amls or PARAM, AMLS is set to YES. -analysis

N/A

Submit an analysis run. This option will also check the optimization data; the job will be terminated if any errors exist.

All Platforms

-optskip will skip checking the optimization data and the analysis will be performed. Cannot be used with -check or -restart (Example: optistruct infile.fem – analysis) -buildinfo N/A

Displays build information for selected solver executables.

OptiStruct

-check

Submit a check job through the command line.

All Platforms

N/A

The memory needed is automatically allocated. Cannot be used with –analysis, -optskip or -restart (Example: optistruct infile.fem –check) -checkel

yes, no, full Note: An argument for – checkel is optional. If an argument is not specified, the default argument

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If NO, element quality checks are not performed, but mathematical validity checks are performed.

All Platforms

If YES, or if no argument is given, the geometric quality of each element is checked. Any violation of the error limits is counted as a fatal error and the run will stop. Any violation of warning limits is non-fatal. Error or warning messages are printed for elements violating the limits along with the offending property values. The amount of output is limited to the first 3 occurrences for each individual case, plus a summary table of all errors. If FULL, the same checks are performed as for YES, but the error or warning messages are


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Option

Argument

Description

Available on

(yes) is assigned.

printed for all of the elements violating the error or warning limits. Default is YES. (Example: optistruct infile.fem –checkel full) (Example: optistruct infile.fem –checkel) Note: An argument for –checkel is optional. If an argument is not specified, the default argument (yes) is assigned.

-compress

N/A

Submits a compression run.

All Platforms

Reduces out matching material and property definitions. Property definitions referencing deleted material definitions are updated with the retained matching material definition (reduction of property definition occurs after this process). Element definitions referencing deleted property definitions are updated with the retained matching property definition. The resulting bulk data file will be written to a file named .echo. It is assumed that there is no optimization, nonlinear or thermal-material data in the bulk data. If such data are present in the input file, the resulting file (.echo) may not be valid. The –compress run option cannot be used in combination with any other option as OptiStruct terminates the run after the .echo file is generated. (Example: optistruct infile.fem – compress) See OptiStruct Compression Run for more information. -core

in, out, min

in – in-core solution is forced

All Platforms

out – out-of-core solution is forced

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Option

Argument

Description

Available on

min – minimum core solution is forced The solver assigns the appropriate memory required. If there is not enough memory available, OptiStruct will error out. Overwrites the –len option. (Example: optistruct infile.fem –core in) -cpu or -proc or -nproc or -ncpu or -nt

Number of cores

-ddm

N/A

Runs MPI based OptiStruct SPMD in Domain Decomposition Mode.

Not all platforms are supported. Refer to the OptiStruct SPMD User's Guide for the list of supported platforms.

-delay

Number of seconds

Delays the start of an OptiStruct run for the specified number of seconds. This functionality does not use licenses, computer memory or CPU before the start of the run (the delay expires).

All Platforms

Number of cores to be used for SMP solution. (See comment 2).

All Platforms

(Example: optistruct infile.fem -ncpu 2)

Note: The –delay option can only be used for a single job. Delays cannot be scheduled for multiple jobs in a queue. If the run is started using the HWSolver Run Manager (GUI), the Schedule delay option should be used. -dir

N/A

Change directory to the location of input file before starting the solver.

-ffrs

YES/NO

Invokes the external FastFRS (Fast Frequency Linux Response Solver) solver. The FASTFRS_EXE

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All Platforms

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Option

Argument

Description

Available on

environment variable should point to the FastFRS executable for this setting to work. Overrides the PARAM, FFRS setting in the input file. (Example: optistruct infile.fem –ffrs yes) -ffrsncpu

1, 2, or 4

Defines the number of CPUs to be used by the Linux external FastFRS eigenvalue solver. This parameter will set the environment variable OMP_NUM_THREADS. The default value is the current value of OMP_NUM_THREADS. Note that this value can be set by the command line arguments – nproc or –ncpu. OptiStruct and FastFRS can be run with different allocations of processors. For example, OptiStruct can be run with 1 processor and FastFRS with 4 processors in the same run. Valid only when the –ffrs run option or PARAM, FFRS is set to YES. Overrides the PARAM, FFRSNCPU setting in the input file. Default: Number of processors used by OptiStruct. (Example: optistruct infile.fem –ffrs yes –ffrsncpu 4)

-ffrsmem

Memory in GB

Defines the amount of memory in Gigabytes to be used by the external FastFRS eigenvalue solver. This run option is only supported for FastFRS versions 2 and later.

Linux

Note: 1. This run option will override the memory value set by PARAM, FFRSMEM in the input file and the environment variable FFRS_MEM. 2. This run option is valid only when the – ffrs run option or PARAM, FFRS is set to YES.

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Option

Argument

Description

Available on

-fixlen

RAM in MBytes

Disables dynamic memory allocation.

All Platforms

OptiStruct will allocate the given amount of memory and use it throughout the run. If this memory is not available, or if the allocated amount is not sufficient for the solution process, OptiStruct will terminate with an error. To avoid over specifying the memory when using this option, it is suggested first to run OptiStruct with the -check option and use the results of that run to properly define the memory size for the -fixlen option. This option allows, on certain platforms, to avoid memory fragmentation and allocate more memory than is possible with dynamic memory allocation. Overwritten by -len and -core options. (Example: optistruct infile.fem fixlen 500) -gpu

N/A

Activates GPU Computing

All Platforms

-gpuid

N/A

N: Integer, Optional, Selects the GPU Card. Default = 1.

All Platforms

-h

N/A

Displays script usage.

All Platforms

-len

RAM in MBytes

Preferred upper bound on dynamic memory allocation.

All Platforms

When different algorithms can be chosen, the solver will try to use the fastest algorithm which can run within the specified amount of memory. If no such algorithm is available, then the algorithm with minimum memory requirement will be used. For example, the sparse linear solver, which can run in-core, out-of-core or min-core will be selected. The – core option will override the –len option. The default for –len is 1000MB, this means that all except for very small models, OptiStruct will use only the minimum memory needed to run the job. If –len value is larger than the amount of available physical RAM, it may cause excessive swapping during

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Option

Argument

Description

Available on

computations, and significantly slow down the solution process. Default = 1000 MB. (Example: optistruct infile.fem –len 32) Best practices for –len specification: For proper memory allocation while using – len in an OptiStruct run, avoid using the exact reported memory estimate value (for eg. Using Check). The –len value should be provided based on the actual memory of the system. This would be the recommended memory limit to run the job, it may not necessarily represent the memory utilized by the job or the actual memory limit. This way, the job is more likely to run with the best possible performance. If the same system is shared by multiple jobs, then the memory allocation should follow the same procedure as above; except, that the individual maximum memory should be used in place of the total system memory. (If a job runs outof-core instead of in-core (it exceeded the memory allocation) it will still run very efficiently. However, make sure that the job does not exceed the actual memory of the system itself as this will slow the run down by a large factor. The recommended method to deal with this is to specify –maxlen as the actual memory of the system to limit the maximum memory that can be used on the system. -lic

FEA, OPT

FEA -

FE analysis only (OptiStructFEA).

OPT -

Optimization (OptiStruct or OptiStructMulti).

All Platforms

The solver checks out a license of the specified type before reading the input data. Once the input data is read, the solver verifies that the requested license is of the correct type. If this is not the case, OptiStruct will terminate with an error.

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Option

Argument

Description

Available on

No default (Example: OptiStruct infile.fem -lic FEA) -licwait

Hours to wait for a license to become available Note: An argument for – licwait is optional. If the argument is not specified, the default argument (12) is assigned.

If present and there are not 50 HyperWorks Units available, OptiStruct will wait for up to the number of hours specified (default=12) for licenses to become available and then start to run. The maximum wait period that can be specified to wait is 168 hours (a week). OptiStruct will check for available HyperWorks Units every two minutes.

All Platforms

-manual

N/A

Launches the online OptiStruct User’s manual.

All Platforms

-maxlen

RAM in Mbytes

Hard limit on the upper bound of dynamic memory allocation.

All Platforms

OptiStruct will not exceed this limit. No default (Example: optistruct infile.fem –maxlen 1000) -mmo

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N/A

The –mmo option can be used to run multiple optimization models in a single run.


Not all platforms are supported. Refer to the OptiStruct SPMD User's Guide for the list of supported

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Option

Argument

Description

Available on platforms.

-monitor

N/A

Monitor convergence from an optimization or nonlinear run. Equivalent to SCREEN, LOG in the input deck.

-mpi

i (Intel MPI),

Initiate an MPI-based SPMD run on supported Not all platforms are supported. platforms. Refer to the (Example: optistruct infile.fem –mpi – OptiStruct SPMD np 4) User's Guide for the list of supported platforms.

pl (IBM PlatformMPI (formerly HP-MPI)),

All Platforms

ms (MSMPI), pl8 (for versions 8 and newer of IBM PlatformMPI) Note: An argument for –mpi is optional. If an argument is not specified, the default argument is assigned. -mpipath

path

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Not all platforms are supported. Refer to the Note: This option is useful if MPI OptiStruct SPMD environments from multiple MPI vendors are installed on the system. Valid for an User's Guide for the list of MPI run only. supported (Example: optistruct infile.fem –mpi – platforms. np 4 –mpipath /apps/hpmpi/bin) Specify the directory containing HP-MPI’s mpirun executable.


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Option

Argument

Description

Available on

-nlrestart Subcase ID Restart a geometric nonlinear solution sequence from specified subcase ID.

All Platforms

If Subcase ID is not specified, it will restart from the first geometric nonlinear subcase ending with error in previous run. Note: The geometric nonlinear solution sequence is a series of geometric nonlinear subcases (ANALYSIS = NLGEOM, IMPDYN or EXPDYN) linked by CNTNLSUB. -np

Number of Number of processors to be used in SPMD processors analysis.

All Platforms

(Example: optistruct infile.fem –mpi – np 4) -optskip

N/A

Submit an analysis run without performing check on optimization data (skip reading all optimization related cards).

All Platforms

Cannot be used with –check or –restart. (Example: optistruct infile.fem optskip) -out

N/A

Echos the output file to the screen. This takes precedence over the I/O option SCREEN.

All Platforms

(Example: optistruct infile.fem -out) -outfile

Prefix for Option to direct the output files to a directory All Platforms output different from the one in which the input file filenames exists. If such a directory does not exist, the last part of the path is assumed to be the prefix of the output files. This takes precedence over the I/O option OUTFILE. (Example: optistruct infile.fem outfile results); here OptiStruct will output results.out, etc.

-rad

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Run RADIOSS optimizati on in OptiStruct

Option to run RADIOSS optimization in All Platforms OptiStruct. A RADIOSS optimization file .rad should be input to OptiStruct and the –rad run option should be specified to request an optimization run for a RADIOSS


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Option

Argument

Description

Available on

input deck. Note: The RADIOSS Starter and input files supporting the optimization input should be available in the same directory as the .rad file. Refer to RADIOSS Optimization in the User’s Guide for more information. -ramdisk

Size of virtual disk (in MB)

Option to specify area in RAM allocated to store information which otherwise would be stored in scratch files on the hard drive.

All Platforms

(Example: optistruct infile.fem – ramdisk 800) For a more detailed description, see the RAMDISK setting on I/O option SYSSETTING.

-reanal

Density threshold

This option can only be used in combination with -restart.

All Platforms

Inclusion of this option on a restart run will cause the last iteration to be reanalyzed without penalization. If the "density threshold" given is less than the value of MINDENS (default = 0.01) used in the optimization, all elements will be assigned the densities they had during the final iteration of the optimization. As there is no penalization, stiffness will now be proportional to density. If the "density threshold" given is greater than the value of MINDENS, those elements whose density is less than the given value will have density equal to MINDENS, all others will have a density of 1.0. (Example: optistruct infile.fem restart -reanal 0.3) -restart

filename.s Specify a restart run. If no argument is All Platforms h provided, OptiStruct will look for the restart file, which will have the same root as the input file with the extension .sh. If you enter an argument On PC, you will need to provide the full path to the restart file including the file name.

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Option

Argument

Description

Available on

Cannot be used with –check, -analysis or – optskip. (Example: optistruct infile.fem restart); here OptiStruct looks for the restart file infile.sh. (Example: optistruct infile.fem – restart C:\oldrun\old_infile.sh); here OptiStruct looks for the restart file old_infile.sh. -rnp

Number of Number of processors to be used in processors OptiStruct SPMD for IMPDYN, EXPDYN, and NLGEOM analysis types.

All Platforms

(Example: optistruct infile.fem –mpi – rnp 4) -rnt

Number of cores

Number of cores to be used for OptiStruct SMP for IMPDYN, EXPDYN, and NLGEOM analysis types.

All Platforms

(Example: optistruct infile.fem -rnt 2) -rsf

Safety factor

Specify a safety factor over the limit of allocated memory.

All Platforms

Not applicable when -maxlen is used. (Example: optistruct infile.fem –rsf 1.2) (Example: optistruct infile.fem –len 32 –rsf 1.2) (Example: optistruct infile.fem –core out –rsf 1.2) -scr or -tmpdir

Path, Option to choose directories in which the filesize=n scratch files are to be written. filesize=n , slow=1 and slow=1 arguments are optional. Multiple arguments may be comma separated.

All Platforms

path ; give the path to the directory for scratch file storage. filesize=n ; defines the maximum file size (in GB) that may be written to that location. slow=1 ; indicates a network drive.

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Option

Argument

Description

Available on

(Example: optistruct infile.fem –scr filesize=2,slow=1,/network_dir/tmp) Multiple scratch directories may be defined through repeated instances of –tmpdir or – scr. (Example: optistruct infile.fem –tmpdir C:\tmp –tmpdir filesize=2,slow=1,Z: \network_drive\tmp) This overwrites the environment variable OS_TMP_DIR, and the TMPDIR definition in the I/O section of the input deck. For a more detailed description, see the I/O Option TMPDIR. -scrfmode

basic, buffered, unbuffer, smbuffer, stripe, mixfcio

Option to select different mode of storing scratch files for linear solver (especially for out-of-core and minimum-core solution modes). Multiple arguments may be comma separated.

All Platforms

(Example: optistruct infile.fem – scrfmode buffered, stripe – tmpdir C: \tmp) For a description of the arguments, see the SCRFMODE setting on I/O option SYSSETTING. -testmpi

N/A

Check if MPI is configured properly and if the All Platforms SPMD version of the OptiStruct executables is available for this system. (Example: optistruct infile.fem –mpi – np 4 –mpipath /apps/hpmpi/bin -testmpi)

-uselen

RAM in MBytes

Suggested dynamic memory usage limit. All Platforms OptiStruct will use more than the minimum memory required up to this limit, but only when it improves the speed of the solution. This value is used only for some solution sequences, which can profit from additional memory available (for example, to use bigger buffers to store intermediate results). This value is automatically limited by the value specified by –len, so –uselen can be

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Option

Argument

Description

Available on

set safely to a very large value. -version

N/A

Checks version and build time information from OptiStruct.

All Platforms

-xml

N/A

Option to specify that the input file is an XML file for a multi-body dynamics solution sequence.

All Platforms

Comments 1. Any arguments containing spaces or special characters must be quoted in {} , for example: -mpipath {C:\Program Files\MPI}. File paths on Windows may use backward "\" or forward slash "/" but must be within quotes when using a backslash "\". 2. Currently, the solver executable (OptiStruct) does not have a specific limit on the number of processors/cores assigned to the SMP part of the run ( -nt/-nthread ). However, practical tests indicate that there is little advantage in increasing this value beyond 4, and if the value for this option is set too high, it may actually increase the run time. Therefore the solver script is programmed to error out if the value of -nt exceeds 16. Users interested in testing this limitation may edit the hwsolver.tcl script (text file) located at: {ALTAIR_HOME}/hwsolvers/scripts/ To do so, increase '16' in the following lines: add_arg nthread

"-nproc="

range { 1 16 }

(Or) add_arg nthread

"-nt="

range { 1 16 }

This line appears several times in the script, each appearance is clearly commented to indicate the specific solver executable it applies to. 3. The above arguments are processed by solver script(s) and not by the actual executable. If you are developing internal scripts which use the executable directly, then you may get specific information about command line arguments that are accepted by the executable by looking at the content of the .stat file, where these arguments are listed for each run, or you can contact [email protected] for more information. 4. The order of the above options is arbitrary. However, options for which arguments are optional should not be followed immediately by the INPUT_FILE_NAME argument.

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OptiStruct GPU Introduction A Graphics Processing Unit (GPU) is a system which can be used to improve the performance of computationally intensive engineering applications. GPU Computing is a process which uses the GPU to execute the time consuming sections of the application and the rest of the code runs on the CPU.

Implementation Starting from OptiStruct version 12.0, the GPU can be used to accelerate the sparse direct equation solver through the NVIDIA CUDA programming model. GPU computing is implemented by off-loading most of the computation intensive work to the GPU and concurrently overlapping the communication and data transfer between the CPU cores and the GPU.

Speedup A speedup in the equation solver of up to 4 times, and up to 3 times overall when compared to a Quad-core Intel Nehalem Xeon run, can be achieved. This heterogeneous computing model is particularly suitable for jobs dominated by the equation solver. For example: nonlinear static analysis on power train structures, topology optimization on blocky structures and so on.

Compatibility 1. GPU computing is available for static analysis/optimization. 2. GPU computing is available in 64-bit Linux platform only. 3. GPU computing is NOT supported in the SPMD module. 4. NVIDIA Fermi and Kepler architecture based Tesla and Quadro graphic cards are supported. Tesla C2050/C2070/M2090/K10/K20, Quadro 6000/K5000/K6000 cards are recommended for computing by NVIDIA.

Activating OptiStruct GPU Command option “-gpu” is used to activate OptiStruct GPU. Currently, only one graphics card is supported, and “–gpuid” can be used to pick the desired graphic card for computation when multiple cards are present. Compatible drivers for the graphics card needs to be installed by the user prior to launching OptiStruct GPU using the option “-gpu”.

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Command Line Option

Value

Action Activates GPU computing

-gpu -gpuid

N

Integer: Optional, selects the GPU card Default = 1

Note: I/O usually accounts for an appreciable percentage of the total solution time in OptiStruct for an out-of-core or min-core run. This cannot be addressed or improved through GPU computing. Therefore, -core in (at least –core out) is recommended when the memory in system is large enough. OptiStruct, currently only supports one graphics card of a GPU in a specific solution. Each GPU card may typically consist of a multitude of small cores (not comparable to a CPU core). Each GPU graphics card is considered equivalent to 1 CPU core for licensing purposes. Refer to the Altair HyperWorks 13.0 Product Licensing Unit Draw page for OptiStruct GPU licensing information.

Recommended Tesla GPU Computing Processor List for OptiStruct The following table lists the recommended Tesla graphic boards for use with the Altair HyperWorks Solver suite of applications for high-powered GPU computing.

Manufacturer

Adaptor Type

Driver Version (minimum or higher)

NVIDIA (Tesla C-CLASS series)

C2070 C2075

Linux (64-bit): 295.59

NVIDIA (Tesla M-CLASS series)

M2090

Linux (64-bit): 295.59

NVIDIA (Tesla Kepler)

K20

Linux (64-bit)

Note: The most recent vendor/manufacturer drivers should be used and all driver support for these cards should be addressed to the appropriate manufacturer of the graphic board.

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OptiStruct SPMD Single Program, Multiple Data (SPMD) is a parallelization technique in computing that is employed to achieve faster results by splitting the program into multiple subsets and running them simultaneously on multiple processors/machines. SPMD typically implies running the same process or program on different machines (Nodes) with different input data for each individual task.

Supported Platforms Supported platforms and MPI versions for subcase based parallelization are listed in Table 1: Application

Version

Supported Platforms

Linux (64-bit)

OptiStruct SPMD

13.0

Windows (64-bit)

MPI Requires IBM Platform MPI (formerly HP-MPI) (Version 7.1); (or) Intel MPI (Version 3.2.011 (or) Version 4.1) Requires IBM Platform MPI (formerly HP-MPI) (Version 7.1); (or) Intel MPI (Version 3.2.011 (or) Version 4.1) (or) Microsoft MPI (Version 3.04.4169)

Table 1: Supported Platforms for OptiStruct SPMD

However, SPMD can sometimes be implemented on a single machine with multiple processors depending upon the program and hardware limitations/requirements. SPMD in OptiStruct is implemented by the following three MPI-based functionalities: Task-based parallelization (TBP) Domain Decomposition Method (DDM) Multi-model optimization (MMO)

Task-based parallelization Task-based parallelization (TBP) in OptiStruct can be used when a run is distributed into parallel tasks, as shown in Figure 1. The schematic shown in Figure 1 is applicable to an SPMD run on multiple machines. The entire model is divided into parallelizable subcases, Table 2 lists the various supported solution sequences and parallelizable steps.

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Figure 1: Overview of Task-based Parallelization in OptiStruct

In Task-based parallelization, the model (analysis/optimization) is split into several tasks, as shown in Figure 1. The tasks are assigned to various nodes that run in parallel. Ideally, if the model is split into N parallel tasks, then (N+1) nodes/machines would be required for maximum efficiency. (This is dependent on various other factors like: type of tasks, processing power of the nodes, memory allocation at each node and so on. During a TBP run, using more than (N+1) nodes for N parallelizable tasks would not increase efficiency). The extra node is known as the Manager Node. The manager node decides the nature of data assigned to each node and the identity of the Master Node. The manager node also distributes multiple input decks and tasks to various nodes. It does not require a machine with high processing power, as no analysis or optimization is run on the manager node. The Master Node, however, requires a higher amount of memory, since it contains the main input deck and it also collects all results and performs all processes that cannot be parallelized. Optimization is run on the Master Node. The platform dependent Message Passing Interface (MPI) helps in the communication between various nodes and also between the Master Node and the Slave Nodes.

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Note: 1. A Task is a minimum distribution unit used in parallelization. Each buckling analysis subcase is one task. Each Left-Hand Side (LHS) of the static analysis subcases is one task. Typically, the static analysis subcases sharing the same SPC (Single Point Constraint) belong to one task. Not all tasks can be run in parallel at the same time (For example: A buckling subcase can not start before the execution of its STATSUB subcase). 2. The manager can also be included within the master node by specifying np = N+1 for N nodes and repeating the first node in the appfile/hostfile in a cluster setup (-np option, appfile/ hostfile are explained in the following sections).

Supported Solution Sequences OptiStruct can handle a variety of solution sequences as listed in the overview. However, all solution sequences do not lend themselves to parallelization. In general, many steps in a program execution are not parallelized. Steps like Pre-processing and Matrix Assembly are repeated on all nodes, while response recovery, screening, approximation, optimization and output of results are all executed on the Master Node. Solution Sequences that Support Parallelization Static Analysis

Parallelizable Steps

Two or more static Boundary Conditions are parallelized (Matrix Factorization is the step that is parallelized since it is computationally intensive.)

Non-Parallelizable Steps

Iterative Solution is not parallelized. (Only Direct Solution is parallelized).

Sensitivities are parallelized (Even for a single Boundary Condition as analysis is repeated on all slave nodes). Buckling Analysis

Two or more Buckling Subcases are parallelized. Sensitivities are parallelized.

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Solution Sequences that Support Parallelization Direct Frequency Response Analysis

Parallelizable Steps

Non-Parallelizable Steps

Loading Frequencies are parallelized. No Optimization.

Modal Frequency Response Analysis

Loading Frequencies are parallelized.

Modal FRF preprocessing is not parallelized. Sensitivities are not parallelized.

Table 2: Task-based parallelization - Parallelizable Steps for various solution sequences

As of HyperWorks 11.0, the presence of non-parallelizable subcases WILL NOT make the entire program non-parallelizable. The program execution will continue in parallel and the non-parallelizable subcase will be executed as a serial run.

Number and Type of Nodes available for Parallelization The types and functions of the nodes that are used in Task-based Parallelization are indicated in Table 3. The first node is automatically selected as the manager, the second node is the master node and the rest are slave nodes. Node Type

Functions

Master Node

Runs all non-parallelizable tasks

(1 Node)

Optimization is run here

Slave Node

Runs all parallelizable tasks

(N-2 Nodes)

Input deck copies are provided

Manager Node

No tasks are run on this node, it manages the way nodes are assigned tasks.

(1 Node)

Manager makes multiple copies of the input deck and sends them to the slave nodes. Table 3: Types and functions of the Nodes

This assignment is based on the sequence of nodes that you specify in the appfile. The appfile is a text file which contains process counts and the list of programs. Nodes can be repeated in the appfile, multiple cores of the repeated nodes will be assigned parallel jobs in the same sequence discussed here.

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Frequently Asked Questions How many nodes should I use? Parallelization is based on task distribution. If the maximum number of tasks which can be run at the same time is N, then using (N+1) nodes is ideal (the extra one node for the manager distributing tasks). Using more than (N+1) nodes will not improve the performance. The .out file suggests the number of nodes you can use, based on your model. When there are only M physical nodes available (M < N), then the correct way to start the job is to use M+1 nodes in the hostfile/appfile. The manager node requires only a small amount of resources and can be safely shared with the master: The way to assign such a distribution is repeating the first physical host in the hostfile/appfile. For example: Hostfile for Intel MPI can be: Node1 Node1 Node2 Node3 ……. Note: For Frequency Response Analysis any number of nodes may be used. (up to the number of loading frequencies in the model.) How to run OptiStruct SPMD on a dual/quad CPU’s/Cores machine? Follow the instructions to run OptiStruct SPMD on a single machine. The ideal number of nodes is min(N+1, M), where N is the maximum number of tasks that can be run at the same time, and M is the number of CPU’s/Cores. Note: For dual/quad code machines it may be more efficient to run OptiStruct in serial + SMP mode. (that is, use –nt argument in the solver script).

How to run OptiStruct SPMD over LAN? It is possible to run OptiStruct SPMD over LAN. Follow the HP-MPI manual to setup different working directories of each node the OptiStruct SPMD is launched.

Is it better to run on cluster of separate machines or on shared memory machine(s) with multiple CPU’s? There is no easy answer to this question. If the computer has enough memory to run all tasks in-core, then we can expect faster solution times as MPI communication is not slowed down by the network speed. But if the tasks have to run out-of-core, then computations are slowed down by disk read/write delay. Multiple tasks on the same machine may compete for disk access, and (in extreme situations) even result in wall clock time slower than that for serial (non-MPI) runs.

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Will OptiStruct SPMD use less memory on each node than in the serial run? No, Memory estimates for serial runs and parallel runs on each node are the same. They are based on the solution of a single (most demanding) subcase.

Will OptiStruct SPMD use less disk space on each node than in the serial run? Yes. Disk space usage on each node will be smaller, because only temporary files related to task(s) solved on this node will be stored. But the total amount of disk space will be larger than that in the serial run and this can be noticed, especially in parallel runs on a sharedmemory machine.

I have a cluster with N nodes each with M cores. What is the most efficient way I can use the resources that I possess? 1. When each host has sufficient RAM to execute only a single serial OptiStruct run, then use multiple cores to activate SMP on each node. (using more than four cores is usually not effective). For example: on a 4 host cluster, each with 8 cores, you can run: optistruct -mpi -np 5 –nt 4 –hostfile… 2. When each host has sufficient RAM to efficiently execute more than one serial run, then you can assign multiple MPI nodes to each host. For example: optistruct -mpi -np 9 –nt 4 –hostfile…

Domain Decomposition Method

In addition to Task-based parallelization (TBP), OptiStruct SPMD includes another approach for parallelization called Domain Decomposition Method (DDM) for static analysis and optimization. DDM allows you to run a single subcase of static analysis and/or optimization with multiple processors in either Shared Memory Processing (SMP) or Distributed Memory Processing (DMP) cluster computers. The solution time will be significantly reduced in DDM mode and the scalability is much higher compared to the legacy shared memory processing parallelization approach, especially on machines with a high number of processors (for example, greater than 8).

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Figure 2: Example illustrating Graph Partitioning for the DDM implementation in OptiStruct

The DDM process utilizes graph partition algorithms to automatically partition the geometric structure into multiple domains (equal to the number of MPI nodes). During FEA analysis/optimization, an individual domain only processes its domain related calculations. Such procedures include element matrix assembly, linear solution, stress calculations, sensitivity calculations, and so on. The necessary communication across domains is accomplished by OptiStruct and is required to guarantee the accuracy of the final solution. When the solution is complete, result data is collected and output to a single copy of the .out file. From the user’s perspective, there will be no difference between DDM and serial runs in this aspect. Supported Solution Sequences Linear and nonlinear static analysis/optimization solution sequences are generally supported. The following solutions, however, are currently not supported. 1. Static analysis (iterative solver) 2. Level set method (Static optimization) 3. Preloading (static analysis) Note: 1. The –ddm run option can be used to activate DDM. Refer to the Setting up OptiStruct SPMD and Launching OptiStruct SPMD for information on setting up and launching Domain Decomposition in OptiStruct. 2. The installation steps and supported platforms for DDM are the same as that of the Task-based parallelization (TBP) mode. 3. In DDM mode, there is no distinction between node types (for example, manager node, master node, slave node, and so on). All nodes are considered as working nodes. If –np n is specified, OptiStruct partitions n geometric domains and assigns each domain to one MPU node.

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4. Hybrid computation is supported. –nt can be used to specify the number of threads in an SMP run. Sometimes, hybrid performance may be better than pure MPI or pure SMP mode, especially for blocky structures. It is also recommended that the total number of cores (n x m) should not exceed the physical cores of the machine.

Multi-Model Optimization In addition to Task-based parallelization (TBP) and Domain Decomposition Method (DDM), OptiStruct SPMD includes another approach for MPI-based parallelization called Multi-Model Optimization (MMO) for optimization of multiple structures with common design variables in a single optimization run. ASSIGN, MMO can be used to include multiple solver decks in a single run. Common design variables are identified by common user identification numbers in multiple models. Design variables with identical user identification numbers are linked across the models. Responses in multiple models can be referenced via the DRESPM continuation lines on DRESP2/DRESP3 entries. Common responses in different models can be qualified by using the name of the model on the DRESPM continuation line. The model names can be specified via ASSIGN, MMO for each model.

Figure 3: Example usecase for Multi-Model Optimization

Multi-model optimization is a MPI based parallelization method, requiring OptiStruct MPI executables for it to run. Existing solver decks do not need any additional input, can be easily included, and are fully compatible with the MMO mode. MMO allows greater flexibility to optimize components across structures. The –mmo run option can be used to activate Multi-Model Optimization in OptiStruct. Supported Solution Sequences 1. All optimization types are currently supported. 2. Multi-body Dynamics (OS-MBD) and Geometric Nonlinear Analysis (RADIOSS Integration) are currently not supported. 3. MMO currently cannot be used in conjunction with the Domain Decomposition Method (DDM). 4. The DTPG and DSHAPE entries are supported; however linking of design variables is not. For example, it makes no difference to the solution if multiple DSHAPE entries in different slave files contain the same ID’s or not.

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Note: 1. The number of processes should be equal to one more than the number of models. 2. Refer to the Setting up OptiStruct SPMD and Launching OptiStruct SPMD sections for information on setting up and launching Multi-Model Optimization in OptiStruct. 3. The installation steps and supported platforms for MMO are the same as that of the Task-based parallelization (TBP) and Domain Decomposition (DDM) modes. 4. If multiple objective functions are defined across different models in the master/slaves, then OptiStruct always uses minmax [Objective(i)] (where, i is the number of objective functions) to define the overall objective for the solution. 5. The following entries are allowed in the Master deck: Control cards: SCREEN, DIAG/OSDIAG, DEBUG/OSDEBUG, TITLE, ASSIGN, RESPRINT, DESOBJ, DESGLB, REPGLB, MINMAX, MAXMIN, ANALYSIS, LOADLIB Bulk data cards: DSCREEN, DOPTPRM (see section below), DRESP3, DRESP3, DOBJREF, DCONSTR, DCONADD, DREPORT, DREPADD, DEQATN, DTABLE, PARAM DOPTPRM parameters (these work from within the master deck – all other DOPTPRM’s should be specified in the slave): CHECKER, DDVOPT, DELSHP, DELSIZ, DELTOP, DESMAX, DISCRETE, OBJTOL, OPTMETH, SHAPEOPT

Setting up OptiStruct SPMD Linux Machines Below are detailed instructions on installing and launching OptiStruct SPM on Linux machines.

Installing OptiStruct SPMD System Requirements Operating system: Linux64 The MPI library: IBM Platform-MPI (Formerly HP-MPI) or Intel MPI must be installed and accessible from every machine in the cluster. Installing Software and activating the License 1. OptiStruct SPMD is included with the OptiStruct solver package and the SPMD executables are included in the installation. 2. Test if OptiStruct SPMD is able to run in serial mode.

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Configuring the machines 1. The account used to run OptiStruct SPMD must exist on every node in the cluster. 2. OptiStruct SPMD executables must be accessible to all cluster nodes. 3. The local scratch directories must be the same (same path). 4. It is highly recommended to have all the computation (working) directories at the same location on all cluster nodes. SSH installation 1. IBM Platform MPI (formerly HP-MPI) default installation uses ssh to launch OptiStruct SPMD on different nodes. 2. ssh should be configured to permit the connection on all hosts of the cluster without the need to type passwords. 3. Refer to ssh man pages to generate and install rsa keys (ssh-keygen tool). To check the functionality of ssh, the following test can be performed on the different nodes: [optistruct@host1] ssh host1 ls [optistruct@host1] ssh host2 ls ... [optistruct@host1] ssh host[n] ls RSH (An alternative to SSH) 1. It is also possible to use rsh instead of ssh to launch OptiStruct SPMD. 2. Computation nodes need to be accessible to all the other nodes without need for a password. 3. Refer to the rsh manpages for installation instructions. Check with IBM Platform MPI (formerly HP-MPI) manual for instructions on how to use RSH instead of SSH. 4. To check the functionality of rsh, the following test can be performed on the different nodes: [optistruct@host1] rsh host1 ls [optistruct@host1] rsh host2 ls ... [optistruct@host1] rsh host[n] ls IBM Platform MPI (formerly HP-MPI) installation 1. IBM Platform MPI (formerly HP-MPI) should be accessible to all computation nodes on which OptiStruct SPMD will be launched. 2. Download the IBM Platform MPI images for the platform you desire to use from the vendor (Contact IBM for help with procuring IBM Platform MPI). 3. Install the IBM Platform MPI package on each node.

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4. No license is required for Platform-MPI to run OptiStruct SPMD. 5. Add $MPI_ROOT/platform_mpi/lib/[platform] into LD_LIBRARY_PATH, if needed. Intel-MPI installation 1. Intel-MPI must be accessible from all computation nodes on which OptiStruct SPMD will be launched. 2. Users can download (purchase/trial) the Intel-MPI images for their respective platforms from the following Intel MPI download site: http://software.intel.com/en-us/intel-mpi-library 3. Install the Intel-MPI package on each node. [root@host[i]] ./install.sh A license is required to install Intel-MPI libraries. (However, OptiStruct SPMD does not require a separate license). 4. Add $MPI_ROOT/intel/impi/3.2.2.006/lib into LD_LIBRARY_PATH, if needed.

Launching OptiStruct SPMD There are several ways to launch parallel programs with OptiStruct SPMD. Remember to propagate environment variables when launching OptiStruct SPMD, if needed. Refer to the respective MPI vendor’s manual for more details. Note: 1. A minimum of three processes are required to launch OptiStruct SPMD. 2. OptiStruct SPMD must match the MPI implementation you use.

Using Solver Scripts On a single host (for IBM Platform MPI (Formerly HP-MPI) using solver script Task-based Parallelization (TBP) [optistruct@host1~]$ $ALTAIR_HOME/scripts/optistruct –mpi [MPI_TYPE] –np [n] [INPUTDECK] [OS_ARGS] Domain Decomposition Method (DDM) [optistruct@host1~]$ $ALTAIR_HOME/scripts/optistruct –ddm [MPI_TYPE] –np [n] [INPUTDECK] [OS_ARGS] Multi Model Optimization (MMO) [optistruct@host1~]$ $ALTAIR_HOME/scripts/optistruct –mmo [MPI_TYPE] –np [n] [INPUTDECK] [OS_ARGS] Where, [MPI_TYPE]: is the MPI implementation used: pl for IBM Platform-MPI (Formerly HP-MPI) i for Intel MPI

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-- ( [MPI_TYPE] is optional, default MPI implementation on Linux machines is pl Refer to the Run Options page for further information). [n]: is the number of processors [INPUTDECK]: is the input deck file name [OS_ARGS]: lists the arguments to OptiStruct -- ( [OS_ARGS] is optional. Refer to the Run Options page for further information). Note: 1. Adding the command line option “-testmpi”, runs a small program which verifies whether your MPI installation, setup, library paths and so on are accurate. 2. OptiStruct SPMD can also be launched using the Run Manager GUI. (Refer to HyperWorks Solver Run Manager) 3. It is also possible to launch OptiStruct SPMD without the GUI/ Solver Scripts. (Refer to the Appendix) 4. Adding the optional command line option “–mpipath PATH” helps you find the MPI installation if it is not included in the current search path or when multiple MPI’s are installed.

Windows Machines Below are detailed instructions on installing and launching OptiStruct SPM on Windows machines.

Installing OptiStruct SPMD System Requirements Operating system: Windows XP/Vista/7: 64-bit only The MPI library: IBM Platform-MPI (Formerly HP-MPI), Intel MPI or MS-MPI must be installed and accessible to each machine in the cluster. Software Installation and License Activation 1. OptiStruct SPMD is included with the OptiStruct solver package and the SPMD executables are included in the installation. 2. Test if the OptiStruct SPMD be able to run in serial mode. Machine Configuration 1. The account used to run OptiStruct SPMD must exist on every node in a cluster. 2. OptiStruct SPMD executables must be accessible to all cluster nodes. The local scratch directories must be the same (same path). It is highly recommended that the computation (working) directories also remain the same on all nodes in a cluster.

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IBM Platform-MPI, MS-MPI installation On windows it is quite straightforward to install the MPI implementation. Refer to their respective Installation Guides for further assistance. Below are the instructions for MPI download: 1. Download the IBM Platform MPI images for the platform you desire to use from the vendor (Contact IBM for help with procuring IBM Platform MPI). 2. Users can download (purchase/trial) the Intel-MPI images for their respective platforms from the following Intel MPI download site: http://software.intel.com/en-us/intel-mpi-library 3. Users can download (purchase/trial) the Microsoft-MPI images for their respective platforms from the following Microsoft MPI (MS-MPI) download site: http://www.microsoft.com/en-us/download/details.aspx?id=14737

Launching OptiStruct SPMD There are several ways to launch parallel programs with each MPI. Below are some typical ways to launch OptiStruct SPMD. Remember to propagate environment variables when launching OptiStruct SPMD, if needed. Refer to corresponding MPI’s manual for more details. Note: 1. A minimum of three processes are required to launch OptiStruct SPMD. 2. OptiStruct SPMD must match the MPI implementation you use.

Using Solver Scripts On a single host using solver script (for HP-MPI, Platform-MPI, Intel-MPI and MSMPI) Task-based Parallelization (TBP) [optistruct@host1~]$ $ALTAIR_HOME/hwsolvers/scripts/optistruct.bat –mpi [MPI_TYPE] –np [n] [INPUTDECK] [OS_ARGS] Domain Decomposition Method (DDM) [optistruct@host1~]$ $ALTAIR_HOME/hwsolvers/scripts/optistruct.bat –ddm [MPI_TYPE] –np [n] [INPUTDECK] [OS_ARGS] Multi Model Optimization (MMO) [optistruct@host1~]$ $ALTAIR_HOME/hwsolvers/scripts/optistruct.bat –mmo [MPI_TYPE] –np [n] [INPUTDECK] [OS_ARGS] Where, [MPI_TYPE]: is the MPI implementation used: pl for versions 7 and older of IBM Platform-MPI (Formerly HP-MPI).

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pl8 for versions 8 and newer of IBM Platform-MPI i for Intel MPI ms for MS-MPI -- ( [MPI_TYPE] is optional, default MPI implementation on Windows machines is pl Refer to the Run Options page for further information) [n]: is the number of processors [INPUTDECK]: is the input deck file name [OS_ARGS]: lists the arguments to OptiStruct -- ( [OS_ARGS] is optional. Refer to the Run Options page for further information) Notes: 1. Adding the command line option “-testmpi”, runs a small program which verifies whether your MPI installation, setup, library paths and so on are accurate. 2. OptiStruct SPMD can also be launched using the Run Manager GUI. (Refer to HyperWorks Solver Run Manager) 3. It is also possible to launch OptiStruct SPMD without the GUI/ Solver Scripts. (Refer to the Appendix) 4. Adding the optional command line option “–mpipath PATH” helps you find the MPI installation if it is not included in the current search path or when multiple MPI’s are installed.

Appendix Launching OptiStruct SPMD on Linux Machines using Direct calls to Executable On a Single Host (for IBM Platform-MPI and Intel MPI) Task-based Parallelization (TBP) [optistruct@host1~]$ mpirun -np [n] $ALTAIR_HOME/hwsolvers/optistruct/bin/ linux64/optistruct_spmd [INPUTDECK] [OS_ARGS] -mpimode Domain Decomposition Method (DDM) [optistruct@host1~]$ mpirun -np [n] $ALTAIR_HOME/hwsolvers/optistruct/bin/ linux64/optistruct_spmd [INPUTDECK] [OS_ARGS] -ddmmode Multi-Model Optimization (MMO) [optistruct@host1~]$ mpirun -np [n] $ALTAIR_HOME/hwsolvers/optistruct/bin/ linux64/optistruct_spmd [INPUTDECK] [OS_ARGS] -mmomode Where,

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optistruct_spmd is the OptiStruct SPMD binary [n]: is the number of processors [INPUTDECK]: is the input deck file name [OS_ARGS]: lists the arguments to OptiStruct other than –mpimode/-ddmmode/-mmomode -- ( [OS_ARGS] is optional. Refer to the Run Options page for further information). Note: Running OptiStruct SPMD, using direct calls to the executable, requires an additional command-line option – mpimode/-ddmmode/-mmomode (as shown above). If one of these run options is not used, there will be no parallelization and the entire program will be run on each node.

On a Linux cluster (for IBM Platform-MPI) Task-based Parallelization (TBP) [optistruct@host1~]$ mpirun –f [appfile] -h [host i] -np [n] $ALTAIR_HOME/hwsolvers/optistruct/bin/linux64/ optistruct_spmd [INPUTDECK] -mpimode Domain Decomposition Method (DDM) [optistruct@host1~]$ mpirun –f [appfile] -h [host i] -np [n] $ALTAIR_HOME/hwsolvers/optistruct/bin/linux64/ optistruct_spmd [INPUTDECK] -ddmmode Multi-Model Optimization (MMO) [optistruct@host1~]$ mpirun –f [appfile] -h [host i] -np [n] $ALTAIR_HOME/hwsolvers/optistruct/bin/linux64/ optistruct_spmd [INPUTDECK] -mmomode Where, [appfile]: is a text file which contains process counts and a list of programs. Note: Running OptiStruct SPMD, using direct calls to the executable, requires an additional command-line option – mpimode/-ddmmode/-mmomode (as shown above). If one of these options is not used, there will be no parallelization and the entire program will be run on each node. Example: 4 CPU job on 2 dual-CPU hosts (the two machines are named: c1 and c2) [optistruct@host1~]$ cat appfile -h c1 –np 2 $ALTAIR_HOME/hwsolvers/optistruct/bin/linux64/optistruct_spmd [INPUTDECK] -mpimode -h c2 –np 2 $ALTAIR_HOME/hwsolvers/optistruct/bin/linux64/optistruct_spmd [INPUTDECK] –mpimode

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On a Linux cluster (for Intel-MPI) Task-based Parallelization (TBP) [optistruct@host1~]$ mpirun –f [hostfile] -np [n] $ALTAIR_HOME/hwsolvers/ optistruct/bin/linux64/optistruct_spmd [INPUTDECK] [OS_ARGS] -mpimode Domain Decomposition Method (DDM) [optistruct@host1~]$ mpirun –f [hostfile] -np [n] $ALTAIR_HOME/hwsolvers/ optistruct/bin/linux64/optistruct_spmd [INPUTDECK] [OS_ARGS] -ddmmode Multi-Model Optimization (MMO) [optistruct@host1~]$ mpirun –f [hostfile] -np [n] $ALTAIR_HOME/hwsolvers/ optistruct/bin/linux64/optistruct_spmd [INPUTDECK] [OS_ARGS] -mmomode Where, [hostfile]: is a text file which contains the host names. Line format is as follows: [host i] Note: 1. One host requires only one line. 2. Running OptiStruct SPMD, using direct calls to the executable, requires an additional command-line option –mpimode/ddmmode/-mmomode (as shown above). If one of these options is not used, there will be no parallelization and the entire program will be run on each node. Example: 4 CPU job on 2 dual-CPU hosts (the two machines are named: c1 and c2) [optistruct@host1~]$ cat hostfile c1 c2

Launching OptiStruct SPMD on Windows Machines using Direct calls to Executable On a Single Host (for IBM Platform-MPI) Task-based Parallelization (TBP) [optistruct@host1~]$ mpirun -np [n] $ALTAIR_HOME/hwsolvers/optistruct/bin/win64/optistruct_spmd [OS_ARGS] -mpimode

[INPUTDECK]

Domain Decomposition Method (DDM) [optistruct@host1~]$ mpirun -np [n] $ALTAIR_HOME/hwsolvers/optistruct/bin/win64/optistruct_spmd [OS_ARGS] -ddmmode

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Multi-Model Optimization (MMO) [optistruct@host1~]$ mpirun -np [n] $ALTAIR_HOME/hwsolvers/optistruct/bin/win64/optistruct_spmd [OS_ARGS] -mmomode

[INPUTDECK]

Where, optistruct_spmd is the OptiStruct SPMD binary [n]: is the number of processors [INPUTDECK]: is the input deck file name [OS_ARGS]: lists the arguments to OptiStruct other than –mpimode/-ddmmode/-mmomode -- ( [OS_ARGS] is optional. Refer to the Run Options page for further information) Note: Running OptiStruct SPMD, using direct calls to the executable, requires an additional command-line option -mpimode/-ddmmode/-mmomode (as shown above). If one of these options is not used, there will be no parallelization and the entire program will be run on each node.

On a Single Host (for Intel-MPI and MS-MPI) Task-based Parallelization (TBP) [optistruct@host1~]$ mpiexec -np [n] $ALTAIR_HOME/hwsolvers/optistruct/bin/win64/optistruct_spmd [OS_ARGS] -mpimode

[INPUTDECK]

Domain Decomposition Method (DDM) [optistruct@host1~]$ mpiexec -np [n] $ALTAIR_HOME/hwsolvers/optistruct/bin/win64/optistruct_spmd [OS_ARGS] -ddmmode

[INPUTDECK]

Multi-Model Optimization (MMO) [optistruct@host1~]$ mpiexec -np [n] $ALTAIR_HOME/hwsolvers/optistruct/bin/win64/optistruct_spmd [OS_ARGS] -mmomode

[INPUTDECK]

Where, optistruct_spmd is the OS SPMD binary [n]: is the number of processors [INPUTDECK]: is the input deck file name

[OS_ARGS]: lists the arguments to OptiStruct SPMD other than –mpimode -- ( [OS_ARGS] is optional. Refer to the Run Options page for further information)

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Note: Running OptiStruct SPMD, using direct calls to the executable, requires an additional command-line option –mpimode/-ddmmode/-mmomode (as shown above). If one of these options is not used, there will be no parallelization and the entire program will be run on each node.

On Multiple Windows Hosts (for IBM Platform-MPI) Task-based Parallelization (TBP) [optistruct@host1~]$ mpirun –f [appfile] -h [host i] -np [n] $ALTAIR_HOME\optistruct

[INPUTDECK] -mpimode

Domain Decomposition Method (DDM) [optistruct@host1~]$ mpirun –f [appfile] -h [host i] -np [n] $ALTAIR_HOME\optistruct

[INPUTDECK] -ddmmode

Multi-Model Optimization (MMO) [optistruct@host1~]$ mpirun –f [appfile] -h [host i] -np [n] $ALTAIR_HOME\optistruct

[INPUTDECK] -mmomode

Where, [appfile]: is a text file which contains process counts and a list of programs. Note: Running OptiStruct SPMD, using direct calls to the executable, requires an additional command-line option –mpimode/-ddmmode/-mmomode (as shown above). If one of these options is not used, there will be no parallelization and the entire program will be run on each node. Example: 4 CPU job on 2 dual-CPU hosts (the two machines are named: c1 and c2) [optistruct@host1~]$ cat appfile -h c1 –np 2 $ALTAIR_HOME/hwsolvers/optistruct/bin/win64/optistruct_spmd [INPUTDECK] -mpimode -h c2 –np 2 $ALTAIR_HOME/hwsolvers/optistruct/bin/win64/optistruct_spmd [INPUTDECK] –mpimode

On Multiple Windows Hosts (for Intel-MPI and MS-MPI) Task-based Parallelization (TBP) [optistruct@host1~]$ mpiexec –configfile [config_file] -host [host i] –n [np] $ALTAIR_HOME/hwsolvers/optistruct/bin/win64/ optistruct_spmd [INPUTDECK] -mpimode

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Domain Decomposition Mode (DDM) [optistruct@host1~]$ mpiexec –configfile [config_file] -host [host i] –n [np] $ALTAIR_HOME/hwsolvers/optistruct/bin/win64/ optistruct_spmd [INPUTDECK] –ddmmode Multi-Model Optimization (MMO) [optistruct@host1~]$ mpiexec –configfile [config_file] -host [host i] –n [np] $ALTAIR_HOME/hwsolvers/optistruct/bin/win64/ optistruct_spmd [INPUTDECK] -mmomode Where, [config_file]: is a text file which contains the command for each host. Note: 1. One host needs only one line. 2. Running OptiStruct SPMD, using direct calls to the executable, requires an additional command-line option – mpimode/-ddmmode/-mmomode (as shown above). If one of these options is not used, there will be no parallelization and the entire program will be run on each node. Example: 4 CPU job on 2 dual-CPU hosts (the two machines are named: c1 and c2) [optistruct@host1~]$ cat hostfile -host c1 –n 2 $ALTAIR_HOME/hwsolvers/optistruct/bin/win64/optistruct_spmd [INPUTDECK] -mpimode -host c2 –n 2 $ALTAIR_HOME/hwsolvers/optistruct/bin/win64/optistruct_spmd [INPUTDECK] –mpimode

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Platforms and Hardware Recommendations Platforms OptiStruct runs on the following platforms: Operatin Architecture g System Linux

64-bit

Windows

64-bit

Mac OS X

Version

SMP

SPMD

RHEL 5.9 RHEL 6.2 SLES 11 SP2

Yes

Yes

Windows/Vista/7/8.1

Yes*

Yes

Server 2008 (R2/HPC)

Yes

Yes

10.8

Yes

No

64-bit

SMP

Symmetric Multiprocessing (Multiple processors, single memory).

SPMD

Single Process Multiple Data (Massive parallel processing, Multiple processors each having its own memory).

RHEL

Red Hat Enterprise Linux

SLES

SUSE Linux Enterprise Server

*

Performance gain for SMP runs on Windows platforms is poor, therefore using more than one processor on these platforms is not recommended.

Hardware Recommendations Altair does not recommend any particular brand of hardware. All hardware purchases are going to balance the cost versus performance. The following are some items which can affect the performance with OptiStruct. CPU – The faster the clock speed of the processor, along with the speed at which data is exchanged between CPU cores of processor the better the performance. Memory – The amount of memory required by an analysis depends on the solution type, types of elements in the model, and model size. Large OptiStruct solutions can require large amounts of memory. Also, memory that is not used by OptiStruct is still available for I/O caching. So the amount of free memory can dramatically effect the wall clock time of the run. The more free memory, the less I/O wait time and the faster the job will run. Even if an analysis is too large to run in-core, having extra memory available will increase the speed of the analysis because unused RAM will be used by the operating system to buffer disk

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requests. Disk drives – OptiStruct solutions often require the writing of large temporary scratch files to the hard drive. Therefore, it is important to have fast hard drives. The best solution is to use two or more fast hard drives in RAID 0 (striped) as a dedicated place for scratch files during the solution. A typical configuration is to have one drive for the operating system and software, and then 2-15 drives striped together as the scratch space for the runs. Interconnect – The parallel SPMD versions of OptiStruct can run on multiple processors and/ or on multiple nodes in the cluster. To run parallel jobs on a cluster, each should have enough RAM to run a full job in non-parallel mode. And, each node in a cluster should have its own disk space that is sufficient to store all the scratch files on that node. Cluster architecture with separate disks for each node will achieve better performance than single shared RAID array of disks. A fast interconnect is important, but anything over Gigabit Ethernet will not speed the solution visibly. When nodes use a shared scratch disk area, the interconnect speed is a critical factor for all out-of-core jobs. For a large NVH analysis, it is recommended to have at least 8 GB per CPU with at least 4 disks in RAID 0 for temporary scratch files.

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OptiStruct Configuration File OptiStruct configuration files may be used to establish default settings for the OptiStruct. Configuration files must be named optistruct.cfg. Any errors or warnings caused by the content of these files will be echoed to the screen only. Any default setting will be ignored if it is defined by an environment variable (see Running OptiStruct), bulk data input deck entry (see The Input File), or command line argument (see Run Options for OptiStruct).

File Location The configuration file allows default settings to be established on five different levels: 1. System level If a configuration file is located in the ${ALTAIR_HOME}/hwsolvers directory, the default settings defined in this file apply to all solver runs. A configuration file is included in the installation at this location. This configuration file contains all of the configuration file options in a comment format. Uncommenting an option will activate it. This file may be used as a template for all configuration files. If the ${ALTAIR_HOME} environment variable is not set, then the configuration file at this location will not be used. This variable is automatically set when the recommended HyperWorks installation and execution procedures are followed. 2. Corporate level If a configuration file is located in the ${HW_CORPORATE_CUSTOMIZATION_DIR} directory, the default settings defined in this file are added to the system defaults. If a default setting, which was defined at the system level, is redefined at this level, the redefined setting is used. If the ${HW_CORPORATE_CUSTOMIZATION_DIR} environment variable is not set, then the configuration file at this location will not be used. 3. Group level If a configuration file is located in the ${HW_GROUP_CUSTOMIZATION_DIR} directory, the default settings defined in this file are added to the system and corporate defaults. If a default setting, which was defined at the system or corporate level, is redefined at this level, the redefined setting is used. If the ${HW_GROUP_CUSTOMIZATION_DIR} environment variable is not set, then the configuration file at this location will not be used. 4. User level If a configuration file is located in the ${HOME} directory, the default settings defined in this file are added to the system, corporate and group defaults. If a default setting, which was defined at the system, corporate or group level, is redefined at this level, the redefined setting is used.

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If the ${HOME} environment variable is not set, then the configuration file at this location will not be used. This variable is normally set for UNIX and Linux operating systems to point to a user's home directory, but it may vary for different versions of the Windows operating system. 5. Local level If a configuration file is located in the current directory, the default settings defined in this file are added to the system, corporate, group and user defaults. If a default setting, which was defined at the system, corporate, group or user level, is redefined at this level, the redefined setting is used.

Entries and Format Below is a list of entries recognized in the configuration file. As stated above, the configuration file contained in the installation may be used as a template for other configuration files. The format of the entries (with the exception of ELEMQUAL) is similar to the format of the I/ O Options in the input deck, namely: ENTRY = Argument Comments may be inserted using the $ character; which indicates that everything which follows on that line is a comment. ELEMQUAL is a recognized entry in the configuration file. It is used as described in the bulk data entry description ELEMQUAL with the condition that it must be written in free format (see Guidelines for Bulk Data Entries). Entry

Argument

Description

DOS_DRIVE_$

Path

Same as DOS_DRIVE_$ environment variable (see Running OptiStruct).

SYNTAX

Same as SYNTAX setting on the I/O option SYSSETTING.

SPSYNTAX

Same as SPSYNTAX setting on the I/O option SYSSETTING.

CORE

Same as the –core run option in the Run Options for OptiStruct section.

SAVEFILE

Same as SAVEFILE setting on the I/O option SYSSETTING.

RAMDISK

Integer

Same as RAMDISK setting on the I/O option SYSSETTING.

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Entry

Argument

Description

SKIP10FIELD

Same as SKIP10FIELD setting on the I/O option SYSSETTING.

CARDLENGTH

Integer

Same as CARDLENGTH setting on the I/ O option SYSSETTING.

TABSTOPS

Integer

Used to change the default tab length from 8 to another value.

MAXLEN

Integer

Used to define the maximum allowable amount of memory to be used in MB. There is no default.

MINLEN

Integer

Used to define the initial memory allocation in MB. The default is 10% of OS_RAM.

BUFFSIZE

Integer Default = 16832

The maximum size in 8 byte words of the records of data written to the .op2 file. Use -1 to turn off buffering.

MSGLMT

Various

See MSGLMT setting on the I/O options section MSGLMT.

ASSIGN, UPDATE, filename

Various

See ASSIGN in the I/O options section.

LOADTEMP

Same as LOADTEMP setting on the I/O option SYSSETTING.

OS_RAM

RAM in Mbytes

Same as SYSSETTING option OS_RAM.

PLOTELID

Same as SYSSETTING option PLOTELID.

RAM_SAFETY_FAC TOR

Multiplier

Same as -rsf option for running from the script (see Run Options for OptiStruct).

FORMAT

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Entry

Argument

Description

SCREEN

Same as I/O option SCREEN.

TMPDIR

Path

Same as I/O option TMPDIR.

SCRFMODE

CHECKEL

Same as CHECKEL option for bulk data entry PARAM.

OutputDefault

The OutputDefault entry allows default outputs to be disabled. This entry controls output for subcases for which there is no output requested. AUTO: Output is automatically generated for certain solution sequences. NONE: No output that is not specifically requested is output.

CHECKMAT

Same as CHECKMAT option for bulk data entry PARAM.

COUPMASS

<-1, 0, 1, YES, NO>

Same as COUPMASS option for bulk data entry PARAM.

EFFMASS

Integer

Same as EFFMASS option for bulk data entry PARAM.

PRGPST

Same as PRGPST option for bulk data entry PARAM.

KGRGD

Same as KGRGD option for bulk data entry PARAM.

WTMASS

Real > 0.0

Same as WTMASS option for bulk data entry PARAM.

MBDH3D

Same as MBDH3D option for bulk data entry PARAM.

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Entry

Argument

Description

FLEXH3D

Same as FLEXH3D option for bulk data entry PARAM.

USERAM

RAM in Mbytes

Same as SYSSETTING option USERAM.


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Expanded Error Message File OptiStruct Expanded Error Message files may be used to expand the error and warning messages for the OptiStruct. In this way, you can customize the error and warning messages so that they are more meaningful and informative for you and others in your organization. The Expanded Error Message file must be named optistruct_err.msg. The file contains an expanded message for each message you wish to expand. If the same message is expanded in multiple files, then all expansions will be printed. The expanded error message is printed only for the first instance of the error. The format of the file is to have four asterisks left justified on a line followed by the error number to be expanded. The next lines contain the expanded error message. An example for error 9009 and 9008 are below. Note that the order of the error message numbers does not matter. **** 9009 This error usually happens when there is not enough room on the disk, but it can happen also when one of the output files does not have write permissions for the current user, or when the directory assigned for output or temporary files does not exist, does not have write permission or is located on a read-only filesystem (e.g. on a CD). Please check following cards: OUTFILE, TMPDIR, EIGVSAVE, ASSIGN, or the command line arguments. Note that TMPDIR may be located in any of config files, and (on Unix) the filenames may be affected by Dos_drive conversion (e.g. DOS_DRIVE_n environment variable). **** 9008 This error usually happens when the input file name is mistyped, either on a command line or on any of following cards: INFILE, INCLUDE, EIGVNAE, RESTART, LOADLIB, ASSIGN. It can also happen if the user does not have read permission to an input file, or to any directory on a path leading to the input file.

File Location The expanded error message file allows default settings to be established on four different levels: 1. System level If an expanded error message file is located in the ${ALTAIR_HOME}/hwsolvers

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directory, the expanded error messages defined in this file are added to all solver runs. An expanded error message file is included in the installation at this location. This file contains samples of expanded messages. This file may be used as a template for all expanded error message files. If the ${ALTAIR_HOME} environment variable is not set, then the expanded error message file at this location will not be used. 2. Corporate level If an expanded error message file is located in the ${HW_CORPORATE_CUSTOMIZATION_DIR} directory, the expanded messages defined in this file are added to the system expanded messages. If the ${HW_CORPORATE_CUSTOMIZATION_DIR} environment variable is not set, then the expanded error message file at this location will not be used. 3. Group level If an expanded error message file is located in the ${HW_GROUP_CUSTOMIZATION_DIR} directory, the expanded error message defined in this file are added to the system and corporate messages. If the ${HW_GROUP_CUSTOMIZATION_DIR} environment variable is not set, then the expanded error message file at this location will not be used. 4. User level If an expanded error message file is located in the ${HOME} directory, the expanded error messages defined in this file are added to the system, corporate and group expanded error messages. If the ${HOME} environment variable is not set, then the expanded error message file at this location will not be used. This variable is normally set for UNIX and Linux operating systems to point to a user's home directory, but it may vary for different versions of the Windows operating system.

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Memory Limitations 32-bit Versus 64-bit Computations On 64-bit machines, when using a 64-bit compiled version of OptiStruct and a 64-bit operating system, OptiStruct can use all memory available in the system (real RAM and virtual memory). There are still some size limitations as a result of the size of standard integer variables, but they should only occur in very rare cases. Currently, OptiStruct is built with two versions of linear solvers: one using 32-bit integers, and the other one using 64-bit integers. By default, the 32-bit solver is used (as it requires less memory/disk and runs visibly faster), but the 64-bit solver is automatically selected when the size of a problem requires it.

Virtual Versus Physical Memory OptiStruct can use more memory than is actually installed on a given system (i.e. more than the installed RAM). This is what the virtual memory (swap space) is for. OptiStruct is more efficient, however, if it uses only actual RAM (remember to allow some RAM to be used by the operating system and other codes running at the same time). When more memory is requested than actual available RAM, OptiStruct will run much slower due to swapping. You will hear disks working constantly with little CPU being used, and there will be a significant difference between the elapsed time and the CPU time. Memory specification for OptiStruct (using –len command line option) is actually only giving OptiStruct a hint about the amount of physical RAM available for the run (i.e. it should specify the amount of physical memory not used by the operating system and other running programs, and as explained above, always less than the total amount of RAM in the computer). Based on this information, OptiStruct will try to use the fastest algorithm which can run within the specified amount of memory. If no such algorithm is available, then the algorithm with minimum memory requirement will be used. Specifying a larger value for – len than the amount of physical RAM may cause excessive swapping during computations, and will significantly slow down the solution process. On most machines OptiStruct asks operating systems for information about available memory. This information is printed in the header of the .out file, and can be used to issue a warning, when it is possible that the run may fail because of lack of this resource. This information is dynamic (changes with other programs running at the machine) and therefore is never used inside OptiStruct – user supplied information (example: with –len argument or from the config file) is used instead.

Automatic Memory Allocation Versus Fixed Memory Runs In standard modes of operation, OptiStruct automatically estimates the amount of memory required, and this memory is requested in successive steps from the operating system. Sometimes the memory could be used more efficiently if requested at once and not in increments. This can be done using the "-fixlen" command line option (see Run Options). When using the "-fixlen" option, OptiStruct may start to run, but fail after some time with a memory allocation error. This can happen when almost all available memory is requested by

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the "-fixlen" argument, because in addition to the memory required by the OptiStruct solver, OptiStruct launches a bandwidth minimizer, which uses an additional small amount of memory. Requesting slightly less memory with the "-fixlen" option is a possible solution. Again, you see that it is incorrect to assign too much memory to OptiStruct.

Additional Control of Used Memory OptiStruct has new command line argument "-maxlen" which can be used in automatic mode. This switch can be useful in some batch scheduling installations, as it will not allow OptiStruct to use more than a given amount of memory. Note that for models which require more memory than allowed by this argument, OptiStruct will abort during the solution, potentially after spending some time in computations. OptiStruct has a new command line argument “-uselen” which can be used in automatic mode. –uselen is used to specify an increased dynamic memory usage limit. If –uselen is not defined, then the algorithms which may use variable amount of memory try to use as minimal an amount as possible. When this option is used, OptiStruct will use more than the minimum memory required, up to this limit, but only when it improves the speed of the solution. This value is used only for some solution sequences, which can profit from additional memory available (for example, to use bigger buffers to store intermediate results). This value is automatically limited by the value specified by –len, so –uselen can be set safely to a very large value. (Example: optistruct infile.fem –uselen 32) Best practices for –uselen specification: The speed gain is usually modest, and is limited to certain solution sequences, therefore, this run option should not be used unless solution speed is critical and excess memory is available. For single-user hosts, it is useful to set this value to the same as –len (or higher). This maximizes the use of available memory to achieve possible better performance. Different values of this run option may be used on systems shared by multiple jobs (for example HyperMesh or other solvers). In such scenarios, using a lower value of this run option (or not using it at all), will result in a lower use of memory and may improve overall speed an response time.

OptiStruct Configuration File All options for memory control can be specified in the OptiStruct Configuration File, however, this is not advisable if the configuration file is shared on the common file server. The configuration files should be tuned for specific hardware independently, and should be placed in the configuration file local to each machine.

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Restarting OptiStruct It is possible to restart an OptiStruct optimization by using the command line option restart (see Run Options), by adding the I/O option RESTART to the input file, or from the OptiStruct panel in HyperMesh. To restart an optimization, you will need information about the final iteration of the previous optimization run. This information is stored in the .sh file. The DESMAX entry on the DOPTPRM card, in the .fem file, specifies the maximum number of additional iterations. To perform an analysis on the optimized structure, restart with DESMAX set to 0. If DESMAX is not defined, then the default value of DESMAX is assumed (30 iterations is the default value for DESMAX unless topology manufacturing constraints are used, in which case the default is 80 iterations). There are a number of conditions that must be observed when restarting an optimization: The number of design variables or design elements cannot be changed. It is invalid to restart with minimum member size control removed if it was present in the original run. It is invalid to restart with checkerboard control turned on if it was not activated in the original run. It is, however, acceptable to deactivate checkerboard control in the restart if it was activated in the original run. It is invalid to restart with manufacturing constraints that differ from those of the original run. The purpose of the restart functionality is for restarting with unconverged optimization runs or optimization runs that were terminated before completion (due to a power outage, etc.). Output files from a restart run are appended with the extension _rst#, where # is a 3 digit number indicating the starting iteration for the restart run. For example, filename_rst030.out is the .out file created when restarting filename.fem from iteration 30. Iterations for the restart are numbered starting with the iteration number in the .sh file (the last iteration from the previous run). You may manually append new .dens, .disp, and .strs files to old ones and post-process the combined files.

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OptiStruct Compression Run The OptiStruct solver can be used as a simple input deck preprocessor, intended to reduce out matching material and property definitions. It is useful for models (produced on some simpleminded mesh generator tools) which sometimes have unique property IDs and unique material IDs assigned to each element, even in cases where actual data is identical. Such models (although technically correct) will be expensive to solve, or slow to process in HyperMesh or HyperView. In this special run mode, OptiStruct will simply read the model, compare all material and property data, and remove redundant data.

Example of Run optistruct infile.fem -compress will produce a new bulk data file named: infile.echo which will contain a new model with all duplicate materials and properties deleted. All references to removed data will be replaced with the remaining ones, so for all practical purposes the model should yield identical results. The additional argument to -compress represents the tolerance value in percent. All floating point values in material and property data are compared using that tolerance. Using tolerance may increase significantly run time.

Restrictions 1. Comparison is performed exactly (meaning all data are compared without allowing for any tolerance or round-off). If optional tolerance value is specified, then the run is performed in two passes: exact matches are removed first, then all remaining materials and properties are compared with each other using following formula: (2 * abs(value1-value2)) / (abs(value1)+abs(value2)) < tolerance *0.01. 2. Optimization data, nonlinear data, and thermal materials are not processed. If such data are present they may reference removed entities, but a compress run will not adjust references. The resulting file (.echo) may not be valid. 3. Cards which extend or modify Materials or Properties (such as MATT1, MATX02, MATS1, or PSHELLX) are not used in comparison, and can also be left orphaned as a result of a compress run. 4. SETs referencing Materials or Properties are not processed. This will not result in a bad deck because SETs are allowed to reference non-existent IDs, however SETs in the output file may be different from the input file. 5. After the .echo file is produced, OptiStruct terminates the run, therefore -compress cannot be combined with any other option.

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The following cards are processed: 1. Properties: PBEAM, PBAR, PBEAML, PBARL, PSOLID (also fluid), PCOMP, PSHELL, PROD, PWELD, and PSHEAR as well as properties not referring to materials: PELAS, PBUSH, PVISC, PDAMP, PGAP, PCONT, PAABSF, and PACABS. 2. Materials: MAT1, MAT2, MAT4, MAT5, MAT8, MAT9, and MAT10. 3. All elements referencing properties or materials, including PRBODY and PFBODY.

Any other cards present in the deck are allowed only if they do not reference materials or properties; however OptiStruct does not verify this assumption. If such a card is present in the deck (for example, DTPG referring to a list of properties), it may be printed with negative IDs for removed entities. The resulting file (.echo) is produced using the same routines which produce ECHO; all restrictions present for ECHO will affect a -compress run; in particular: Some optimization cards are currently known to produce incorrect ECHO, meaning an ECHO of these cards cannot be read back into OptiStruct. Results are formatted in fixed format, irrespective of the format used in the input file. This limits the accuracy of most coefficients because of 8 character fields. Current formatting preserves as many decimal places as possible within 8 characters, but for values which require an exponential form, it is sometimes possible to retain accuracy to only 3-4 decimal places. Exceptions: GRID and DMIG cards are printed in free format with accuracy to at least 10 decimal places. Only bulk data is printed to .echo file (no i/o or control sections). Some cards are not printed: in particular, PARAM and DOPTRM do not appear in ECHO files.

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Structural Analysis The Structural Analysis section provides an overview of the following analyses: Linear Static Analysis Linear Buckling Analysis Nonlinear Analysis Normal Modes Analysis Frequency Response Analysis Complex Eigenvalue Analysis Random Response Analysis Response Spectrum Analysis Transient Response Analysis

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Linear Static Analysis The basic finite element equation to be solved for structures experiencing static loads can be expressed as:

Ku

P

Where, K is the stiffness matrix of the structure (an assemblage of individual element stiffness matrices). The vector u is the displacement vector, and P is the vector of loads applied to the structure. The above equation is the equilibrium of external and internal forces. The stiffness matrix is singular, unless displacement boundary conditions are applied to fix the rigid body degrees of freedom of the model. The equilibrium equation is solved either by a direct or an iterative solver. By default, the direct solver is invoked, whereby the unknown displacements are simultaneously solved using a Gauss elimination method that exploits the sparseness and symmetry of the stiffness matrix, K, for computational efficiency. Alternatively, an iterative solver using the preconditioning conjugate gradient method may be used. While the direct solver is very robust, accurate and efficient, the iterative solver is sometimes superior, in terms of speed, for thick-walled solid structures. The iterative solver is selected through the SOLVTYP subcase information entry, which in turn references a SOLVTYP bulk data entry. Once the unknown displacements at the nodal points of the elements are calculated, the stresses can be calculated by using the constitutive relations for the material. For linear static analysis where the deformations are in the elastic range, ta=hat is the stresses, , are assumed to be linear functions of the strains, , Hooke’s law can be used to calculate the stresses. Hooke’s law can be stated as:

C with the elasticity matrix C of the material. The strains displacements.

are a function of the

The static loads and boundary conditions are defined in the bulk data section of the input deck. They need to be referenced in the subcase information section using an SPC and LOAD statement in a SUBCASE. Each SUBCASE defines a load vector. Thermal loading is defined by referencing bulk data entries with the TEMPERATURE statement in a SUBCASE. Unconstrained models can be solved using inertia relief. SUPORT1 subcase statements can then reference the boundary conditions that restrain the rigid body motions. Up to six degrees of freedom can be restrained. These restraints can also be defined without subcase reference using the SUPORT bulk data entry or automated using PARAM, INREL, -2.

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Linear Buckling Analysis The problem of linear buckling in finite element analysis is solved by first applying a reference level of loading, PRef , to the structure. A standard linear static analysis is then carried out to obtain stresses which are needed to form the geometric stiffness matrix KG. The buckling loads are then calculated by solving an eigenvalue problem:

K

KG x 0

Where, K is the stiffness matrix of the structure and is the multiplier to the reference load. The solution of the eigenvalue problem generally yields n eigenvalues , where n is the number of degrees of freedom (in practice, only a subset of eigenvalues is usually calculated). The vector x is the eigenvector corresponding to the eigenvalue. The eigenvalue problem is solved using a matrix method called the Lanczos method. Not all eigenvalues are required. Only a small number of the lowest eigenvalues are normally calculated for buckling analysis. The lowest eigenvalue

PCr

Cr

is associated with buckling. The critical or buckling load is:

Cr PRe f

In order to run a linear buckling analysis, an EIGRL bulk data entry needs to be given because it defines the number of modes to be extracted. The EIGRL card needs to be referenced by a METHOD statement in a SUBCASE in the subcase information section. In addition, it is necessary to use a STATSUB card to reference the appropriate referential static loading, fre , SUBCASE. STATSUB cannot refer to a subcase that uses inertia relief. The buckling analysis will ignore zero-dimensional elements, MPC, RBE3, and CBUSH elements. These elements can be used in buckling analysis, but they do not contribute to the geometric stiffness matrix, KG. By default, the contribution from the rigid elements to the geometric stiffness matrix is not included. You have to add PARAM,KGRGD,YES to the bulk data section to include the contribution of rigid elements to the geometric stiffness matrix. In addition, through the EXCLUDE subcase information entry, you may decide to omit the contribution of other elements to the geometric stiffness matrix, effectively allowing you to control which parts of the structure are analyzed for buckling. The excluded properties are only removed from the geometric stiffness matrix, resulting in a buckling analysis with elastic boundary conditions. This means that the excluded properties may still be showing movement in the buckling mode. Buckling analysis cannot be performed if the referential static loading subcase uses inertia relief. In such cases, the stiffness matrix is positive semi-definite and the buckling eigenvalue solution ends in singularity.

Linear Buckling and Offset Elements Some one-dimensional and shell elements can use offset to “shift” the element stiffness relative to the location determined by element’s nodes. For example, shell elements can be offset from the plane defined by element nodes by means of ZOFFS. In this case all other

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information, such as material matrices or fiber locations for the calculation of stresses, are given relative to the offset reference plane. Similarly, shell results, such as shell element forces, are output on the offset reference plane. Offset is applied to all element matrices (stiffness, mass, and geometric stiffness), and to respective element loads (such as gravity). Hence, in principle offset can be used in all types of analysis and optimization, including linear buckling. However, caution is advised when interpreting the results. Without offset, a typical simple structure will bifurcate and loose stability “instantly” at the critical load. With offset, though, the loss of stability is gradual and asymptotically reaches a limit load, as shown below in figure (b):

In practice then, the structure with offset can reach excessive deformation before the limit load is reached. (Note that more complex structures, such as frames or structures experiencing bending moments, buckle via limit load, even in absence of ZOFFS on the element card). Furthermore, in a fully nonlinear approach, additional instability points may be present on the limit load path.

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Nonlinear Analysis Nonlinear Quasi-Static Analysis Large Displacement Nonlinear Static Analysis Geometric Nonlinear Analysis

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Nonlinear Quasi-Static Analysis This solution sequence performs quasi-static nonlinear analysis. Presently, the sources of nonlinearity include CONTACT interfaces, GAP elements, and MATS1 elastic-plastic material. Small deformation theory is used in the solution of nonlinear problems, similar to the way it is used with Linear Static Analysis. Inertia relief is also possible. Small deformation theory means that strains should be within linear elasticity range (some 5 percent strain), and rotations within small rotation range (some 5 degrees rotation). This also means that there is no update of gap/contact element locations or orientation due to the deformations – they remain the same throughout the nonlinear computations. The orientation may change, however, due to geometry changes in optimization runs.

Nonlinear Solution Method The basic Newton method is used for the solution of nonlinear problems. The principle of this method is illustrated for a one-dimensional problem in the figure below and can be formulated as follows:

Consider a nonlinear problem:

L(u ) P Where, u is the displacement vector, P is the global load vector, and L(u) is the nonlinear response of the system (nodal reactions). Note that for a linear problem, L(u) would simply be Ku (as described in the Linear Static Analysis section). Application of Newton's method to this equation leads to an iterative solution procedure:

K n un un 1 un

Rn un

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Where,

Kn Rn

L(u ) / u at un P L un

In the above formulas, Kn represents a "slope" matrix, defined as a tangent to the L(u) curve at a point un , and Rn is the nonlinear residual. Repeating this procedure iteratively, under certain convergence conditions, leads to systematic reduction of residual Rn and hence, convergence. Note that the above scheme is somewhat modified to an equivalent format wherein, instead of calculating

K n un 1

u , the new solution u is directly obtained: n+1

Rn

K n un

This form is readily produced by adding Kn un to both sides of Newton's equation, and has certain advantages in practical implementations.

Incremental Loading For a large class of problems satisfying certain stability and smoothness conditions, the Newton's iterative method is proven to converge, provided that the initial guess is sufficiently close to the true force-displacement path L(u). Hence, to improve convergence for strongly nonlinear problems, the total loading P is often applied in smaller increments, as shown in the figure below. At each of the intermediate loads, P1, P2, etc., the standard Newton iterations are performed.

This procedure, known as incremental loading, helps to keep the consecutive iterations closer to the true load path, thereby improving the chances of obtaining a final, converged solution (though usually at the expense of an increased total number of iterations).

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Nonlinear Convergence Criteria In order to assess whether the nonlinear process has converged, a number of convergence criteria are available. These criteria and respective tolerances can be selected on the NLPARM bulk data card. The basic principle in assessing nonlinear convergence is to compare an error measure of the solution with a pre-determined tolerance level. When the error falls below the prescribed tolerance, the problem is considered converged. In a case of multiple, simultaneous convergence criteria, all criteria need to be satisfied for the solution to be converged. The relative error in displacements (printed in the convergence summary as EUI) is calculated as:

A

q

EU

1 q

u

A u

Here, A is a normalizing vector consisting of square roots of diagonal elements of stiffness matrix

K A1

Kii

Au

and the vector norm II. II is calculated as:

Ai ui

i

Furthermore, q is a contraction factor that corrects the increment of solution represent the actual error in the nonlinear solution. It is expressed as:

q

un to better

un un 1

In order to stabilize the behavior of q in practical computations, it is updated iteratively according to the formula:

qn

2 3

un un 1

1 q 3 n 1

starting from initial value q1 = 0.99. Note that the contraction factor is meaningful when the solution is close to having converged – it then reasonably well estimates the actual error remaining in the nonlinear solution. The relative error in terms of loads (printed in convergence summary as EPI) measures the relative strength of the residual R, and is calculated as:

EP

R u P u

The load vector P in this formula includes nodal reactions due to prescribed displacements.

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The relative error in terms of work (printed in convergence summary as EWI) measures the relative change in solution energy, and is calculated as:

R

EW

u

P u

Note that the above norms only measure the error of the nonlinear iterative process. Their values do not represent the accuracy of the finite element solution, only the fact that the nonlinear process has converged properly.

Nonlinear Problem Setup The setup for the nonlinear solution is very straightforward. The static loads and boundary conditions are defined in the bulk data section of the input deck. They need to be referenced in the subcase information section using an SPC and LOAD statement in a SUBCASE. Each SUBCASE defines a load vector. Loads or enforced displacements are not mandatory for nonlinear quasi-static solutions, if GAP or CONTACT elements are present in the model. Unconstrained models can be solved using inertia relief. SUPORT1 subcase statements can then reference the boundary conditions that restrain the rigid body motions. Up to six degrees of freedom can be restrained. These restraints can also be defined without subcase reference using the SUPORT bulk data entry or automated using PARAM, INREL, -2. To indicate that a nonlinear solution is required for any subcase, a subcase information command NLPARM needs to be present for the subcase. This command, in turn, points to the bulk data NLPARM card that contains the convergence tolerances and other nonlinear parameters. Example: SUBCASE 10 SPC = 1 LOAD = 2 NLPARM = 99 . . BEGIN BULK NLPARM 99

12

UPW+1.1e-5

. Note that nonlinear gap and contact analysis are also supported in optimization.

Nonlinear Convergence Considerations The Newton's method is a reliable tool for the solution of nonlinear problems and can provide a fast quadratic convergence rate. However, convergence is not guaranteed under all circumstances. Contact problems, especially those with friction, often cause convergence difficulties. In order to improve the chances of a successfully converged solution, methods have been built in to help problems converge that would otherwise oscillate back-and-forth and never

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converge. One method involves a "sticky gap," wherein a residual stickiness is introduced to prevent the "undecided" nodes from bouncing in and out of contact. Another method is gap/ contact status freezing where, after a number of oscillating iterations, gap/contact elements are not allowed to change their open/closed status. Note that these methods are activated only for near-converging yet stagnated problems, and do not interfere with converging (or radically diverging) cases.

User's Considerations for Nonlinear Convergence There are a number of precautions you can take to increase the chances of a successful convergence. Realistic Problem Setup Make sure that the nonlinear problem represents a realistic physical situation for which a feasible solution exists. In particular, special care needs to be taken in selecting the proper orientation of gap elements. This is especially important when using a prescribed gap coordinate system. See the description of the CGAP and CGAPG elements for more details. Sufficient Support Since gap/contact elements only provide one-way support, it is possible to formulate the problem in such a way that the individual components will have rigid body freedom under certain loading conditions. This will manifest as zero pivot in the solution process. To avoid such situations, it is advisable to provide sufficient support to all components so that, even without gap/contact elements, there are no rigid body modes. If "solid" supports are not feasible for all parts (the part needs to move), a very weak set of springs can be used to prevent the part from "flying away" when gap/contact elements are not engaged. The stiffness of such auxiliary springs can be selected so as to allow for large motion of the part, compatible with the overall size of the model. If the gap elements and contact interfaces are properly set up, such weak springs will exert virtually no effect when the solution has converged. Reasonable Gap Stiffness The gap stiffness values KA and KT essentially represent penalty springs that are hard enough to prevent perceptible penetration of contacting nodes. While, theoretically, higher stiffness values enforce the contact conditions more precisely, excessively high values may cause difficulties in convergence or poor conditioning of the stiffness matrix (this is especially true for KT). If any such symptoms are observed, it may be beneficial to reduce the value of gap stiffness. As a baseline recommendation, a reasonable range of gap stiffness is of the order of:

103 to 106 * E * h Where, E is the typical value of elastic modulus and h is the typical element size in the area surrounding the gap elements. Such range will generally keep the gap penetration below one thousandth / one millionth of the element size, respectively. A good value for KT is of the order of 0.1*KA. To facilitate reasonable values of KA and KT, OptiStruct supports the automatic calculation of these parameters, specifically:

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Option KA=AUTO determines the value of KA for each gap element using the stiffness of surrounding elements. Additional options SOFT and HARD create respectively softer or harder penalties. SOFT can be used in cases of convergence difficulties and HARD can be used if undesirable penetration is detected in the solution. Option KT=AUTO automatically calculates the value of KT. If MU1>0, the result here is the same as with blank KT -- its value is calculated as MU1*KA. However, if MU1=0 or blank, KT=AUTO produces a non-zero value of KT, calculated as KT=0.1*KA. Therefore, KT=AUTO can be used to prescribe enforced stick conditions. Friction The presence of friction, due to its strongly nonlinear, non-conservative nature, may cause difficulties in nonlinear convergence, especially when sliding is present. Therefore, solving the problem without friction can often provide convergence in otherwise failing problems. Or, in cases when presence of frictional resistance is necessary and minimal sliding is expected, enforcing a stick condition may be a viable solution, and will often lead to a better convergence than Coulomb friction (see the PGAP and PCONT bulk data card for details). Note that in cases of larger sliding motions, the stick condition may lead to divergence through a "tumbling" mode. Gap Offset In order to provide theoretical correctness, friction produces bending moments in gap/contact elements of non-zero length (this results from the transfer of frictional force from the contact surface to the end nodes). This offset operation can, however, cause convergence problems and counter-intuitive results. In problems with friction, it may be advisable to turn off the offset operation via a parameter: GAPPRM,GAPOFFS,NO This will produce more intuitive results in the presence of friction. However, it may violate the rigid body balance of the body, and should therefore be used with caution, especially for problems without full SPC support. See the PGAP and PCONT bulk data card for details. Incremental Loading If the nonlinear procedure diverges in spite of taking the measures described above, the incremental loading procedure (applying the total load in a number of increments) can be used to achieve convergence. See the description of the NLPARM bulk data card for details. Note, however, that if the problem is incorrectly formulated (the solution exhibits excessive deformations, free rigid body motions, an ill-conditioned stiffness matrix, extremely high nonlinear error, etc.), then incremental loading cannot be counted on to provide a converged solution. Nonlinear Expert System In some difficult to converge cases an expert system can be used to achieve convergence: PARAM,EXPERTNL,YES The expert system will try to adjust the load increment and other nonlinear parameters to achieve convergence. Note, however, that if the problem is incorrectly formulated (the solution exhibits excessive deformations, free rigid body motions, an ill-conditioned stiffness matrix, extremely high nonlinear error, etc.), then expert system cannot be counted on to provide a converged solution. Moreover, in some cases it can lead to long computational times without success. This may

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be due to using very small load increments or re-running the solution with modified nonlinear parameters.

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Large Displacement Nonlinear Static Analysis Large displacement nonlinear static analysis is used for the solution of problems wherein the load-response relationship is nonlinear and structural large displacements are involved. The source of this nonlinearity can be attributed to multiple system properties, for example, materials, geometry, nonlinear loading and constraint. Currently, in OptiStruct the following large displacement nonlinear capabilities are available, including large strain elasto-plasticity, hyperelasticity of polynomial form, contact with small tangential motion, and rigid body constraints. Geometric Nonlinearity In analyses involving geometric nonlinearity, changes in geometry as the structure deforms are considered in formulating the constitutive and equilibrium equations. Many engineering applications require the use of large deformation analysis based on geometric nonlinearity. Applications such as metal forming, tire analysis, and medical device analysis. Small deformation analysis based on geometric nonlinearity is required for some applications, like analysis involving cables, arches and shells. Such applications involve small deformation, except finite displacement or rotation. Material Nonlinearity Material nonlinearity involves the nonlinear behavior of a material based on current deformation, deformation history, rate of deformation, temperature, pressure, and so on. Constraint and Contact Nonlinearity Constraint nonlinearity in a system can occur if kinematic constraints are present in the model. The kinematic degrees of freedom of a model can be constrained by imposing restrictions on its movement. In OptiStruct, constraints are enforced with Lagrange multipliers. In the case of contact, the constraint condition is based on inequalities and such a constraint generally does not allow penetration between any two bodies in contact.

Nonlinear Solution Method Nonlinear problems are generally history dependent. In order to achieve a certain level of accuracy, the solution must be obtained in a series of small increments. For this purpose we need to solve the equilibrium equation at each increment and a corresponding increment size is selected. Newton’s method is used to solve the nonlinear equilibrium equation in OptiStruct. If the solution is smooth, quadratic of rate of convergence may be achieved when compared with other methods. This method is also very robust in highly nonlinear situations. Choosing a suitable time increment is very important. In OptiStruct, an automatic time increment control is available. It should be suitable for a wide range of nonlinear problems and, in general, is a very reliable approach. The automatic time increment control functionality measures the difficulty of convergence at the current increment. If the calculated number of iterations is equal to optimal number of iterations for convergence, OptiStruct will proceed with the same increment size. If a lesser number of iterations is required to achiever convergence, the increment size will be increased for next increment. Similarly if it is determined that too many iterations are required, the current increment will be attempted again with a smaller increment size.

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Nonlinear Large Displacement Analysis Problem Setup The setup for the large displacement nonlinear static solution is straightforward. 1. PARAM, LGDISP,1 is used to activate large displacement analysis. 2. To indicate that a nonlinear solution is required for any subcase, the NLPARM subcase information entry should be included in the corresponding subcase. 3. This subcase entry, in turn, references a NLPARM bulk data entry that contains the convergence tolerances and other nonlinear parameters. 4. If constraints or contacts are defined in the model, the matrix profile may be updated over time, so it is recommended that hash assembly is used for nonlinear analysis. This is activated using PARAM, HASHASSM,1. 5. The material MATS1 (TYPE=PLASTIC) is required in conjunction with PARAM, LGDISP,1 to activate large strain elasto-plasticity analysis. 6. The material MATHE can be used in conjunction with PARAM, LGDISP, 1 to activate large displacement analysis with hyperelastic materials. Example PARAM,LGDISP,1 PARAM,HASHASSM,YES SUBCASE 10 SPC = 1 LOAD = 2 NLPARM = 99 . . BEGIN BULK NLPARM 99

12

UPW+1.1e-5

Note: 1. Large displacement nonlinear analysis is supported only for solid elements, RROD, RBAR, and RBE2 entries. (see Note 8 for further details). 2. 1D/3D Bolt Pretensioning and RBE3 rigid elements are not currently supported in large displacement nonlinear analysis. 3. Direct Matrix Input (using the DMIG entry) is currently not supported in large displacement nonlinear analysis. 4. Linear Buckling Analysis and Preloaded Analysis are not supported with large displacement nonlinear analysis. However, you can use PARAM,PRESUBNL,YES to force OptiStruct to run in such models. Linear Buckling Analysis or Preloaded Analysis is not recommended in models with nonlinear materials or in large displacement nonlinear analysis. It is the user’s responsibility to interpret the results with caution. 5. Currently, MATS1 (TYPE=PLASTIC) should be specified to conduct a large displacement nonlinear analysis. However, if linear material properties are required, then a very large value can be specified for the LIMIT1 field on the MATS1 entry. 6. The expert system (PARAM, EXPERTNL) is currently not supported with large displacement nonlinear analysis.

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7. Nonlinear Heat Transfer Analysis is currently not supported with Large Displacement Nonlinear Analysis. 8. Large Displacement Nonlinear Analysis is not supported in conjunction with the following elements: (a) The following elements can exist in the model, but they will be resolved using small displacement theory: SHELL, GASKET, BUSHING, RROD, RBAR, RBE2, CROD, CELAS, CONM (b) The following elements are not allowed and OptiStruct will error out if they are present: CBAR, CBEAM, CGAP, CGAPG, CWELD, CSEAM, CFAST, RBE1, RBE3

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Geometric Nonlinear Analysis (RADIOSS Integration) Geometric nonlinear analysis in OptiStruct is provided thru an integration of the RADIOSS Starter and RADIOSS Engine via a translator. Implicit (static and dynamic), as well as explicit integration schemes, are available. Transparent to you, OptiStruct input data is directly translated into RADIOSS input data. The RADIOSS Starter and RADIOSS Engine are then executed and the results are brought back into the OptiStruct output module to export the different output formats.

Solution Method This section discusses the basic concepts of the solution methods to highlight the characteristics of the solution methods and to identify the use of certain parameters to control convergence. The geometric nonlinear solution utilizes a general Newmark integration scheme. The following equation of motion shall be solved.

Mu&& Cu& Ku P The matrix M is the mass matrix, C is the damping matrix and K is the stiffness matrix. These matrices are derived using finite elements. The vector P describes the external loads and u is the displacement vector. The dots describe the derivatives with respect to time. The equation of motion can be solved using a general Newmark integration scheme. Newmark is a one-step time integration method. All solutions can be derived from it and are formulated in terms of a time history (Figure 1).

Figure 1: Time History

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In general Newmark, the state vector is computed as follows:

u&t 1 u&t ut 1 ut

u&&t

t 1 tu&t

1

u&&t 1

t 2 u&&t

2

t 2 u&&t 1

Then the equation of motion yields:

M

t 2 K u&&t 1

tC

P C u&t

1

tu&&t

K ut

tu&t

1

2

t 2 u&&t

This can be rewritten into:

1 M t2

t

ut 1 ut

ut

C K

ut

P%t

using:

In short:

A u

P%

The matrix A is the dynamic stiffness. In nonlinear time-dependent problems, this system becomes nonlinear and its solution requires an additional iteration loop at each time step using a Newton-type method. An implicit (quasi-)static analysis scheme follows directly when omitting mass and damping terms. Therefore:

K (u ) ut

Pt

The linear static case reduces to the systems equation:

Ku

P

Normal modes analysis is a linear analysis that solves the eigenvalues problem.

K

M x 0

For implicit dynamic analysis, an extension of Newmark method, known as a-HHT, is the default time integrator. This method is named after Hilber, Hughes, and Taylor – it allows for effective algorithmic damping of high-frequency spurious vibrations. This method introduces additional parameter α and assumes:

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2

1

1 2 , 2

with

4

1 M t2

1 t

C

1

K

1 3

ut

0

P%t

The smaller the value of a, the more algorithmic damping is included in the numerical solution. With a = 0.0 there is no numerical damping and the method is the trapezoidal method. The second method available is the general Newmark with user-defined defaults are typically:

1 , 2

1

and

. The

4

which is equivalent to HHT method with a = 0.0. This is an unconditionally stable implicit integration scheme with:

u&t 1 u&t

ut 1 ut

1

&& && 2 t ut ut 1

tu&t

1

2 && && 4 t ut ut 1

And from the equation of motion:

4 M t2

2 C K t

ut

P%t

By default a Modified Newton method is employed to solve the implicit problems stated above (Figure 2).

K (ut ) ui

Ri

ui 1 ui

ui

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Figure 2: Implicit scheme

Convergence for implicit (quasi-)static and dynamic analysis is controlled via NLPARM, TSTEPNL bulk data entries, respectively. Modified Newton requires that the stiffness matrix is kept constant thru a number of iterations (KSTEP) before it is rebuilt. This saves computation time in terms of matrix factorizations, but may increase the number of iterations. Full Newton can be achieved by using KSTEP = 1. Convergence is defined by a change in results less than a specified tolerance. Relative residual force (EPSP), relative displacement (EPSU), or relative residual energy (EPSW) can be chosen as convergence criteria (CONV). In implicit analysis the time step is controlled via NLPARMX, TSTEPNX bulk data entries, respectively. Time step control includes a minimum (DTMIN) which terminates the solution, a maximum (DTMAX) time step, as well as a maximum number of time steps which cannot be exceeded. Using convergence acceleration methods (SACC), more control can be asserted. If the number of iterations within a time step reaches a specified limit (LDTN), then the iteration is repeated with a smaller time step. The time step is also reduced should the iteration diverge. If the number of iterations is below a certain limit (ITW), then the time step is increased. A BFGS Quasi-Newton method is also available to solve the implicit equations. It works similarly to Modified Newton. However, in addition to the tangential stiffness, it uses an approximate Hessian to improve convergence. A conditionally stable explicit integration scheme can be derived from the Newmark scheme by setting:

1

2,

u&t 1 u&t

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0 1

&& && 2 t ut ut 1


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tu&t

ut 1 ut

1

2 && 2 t ut

From these relationships the central differences explicit integration scheme can be derived.

u&t

1/2

u&t

t2 u&1 1/2

ut 1 ut Mu&&t 1

t12 u&&t

1/2

P Cu&1 1/2 Kut 1

Figure 3 illustrates the relationships.

Figure 3: Explicit integration

Assuming that

Cu& t 1

Cu&t 1/2

The equation of motion for the central differences scheme simplifies to:

Mu&&t 1

P Cu&t 1 Kut 1

Mu&&t 1

P

Btt 1 t 1dV

This central differences scheme is used if explicit analysis is selected. The time step must always be smaller than the critical time step to ensure stability of the solution. The critical time step depends on the highest frequency in the system and is computed from the corresponding angular frequency max as:

tcr

2 max

For a discrete system, the time step must be small enough to excite all frequencies in the finite element mesh. This requires such a short time step that a shock wave does not miss any node when traveling the mesh. Therefore,

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t

lc c

with lc being the critical length of an element and c is the speed of sound in the given material. Different ways of time step control are available. The default method is the nodal time step which is computed from the nodal mass m and the equivalent nodal stiffness k such that:

tcrn

min n

2m k

The element time step based on the critical length of each element is also available. The choice can be made on the XSTEP bulk data entry.

Problem Setup Geometric nonlinear analysis is defined thru a SUBCASE. For implicit (quasi-)static analysis an NLPARM statement as well as ANALYSIS = NLGEOM must be present in the subcase. To define the termination time a TTERM subcase entry can be used. TTERM is mandatory if a nonlinear load NLOAD is used. NLPARM references an NLPARM bulk data entry. Additional parameters to control the geometric nonlinear solution can be defined on the optional NLPARMX bulk data entry. These include convergence acceleration methods. In the case of post-buckling analysis Riks method can be selected. Linear static analysis is provided as a debugging option. It is defined thru NLPARMX, ILIN. Such analysis can help investigate the model for modeling errors. In linear static analysis the load vector is determined at the termination time. Normal modes analysis requires a METHOD subcase statement in addition. For implicit dynamic analysis a TSTEPNL statement as well as ANALYSIS = IMPDYN must be present in the subcase. To define the termination time a TTERM subcase entry is mandatory. TSTEPNL references a TSTEPNL bulk data entry. Additional parameters to control the geometric nonlinear solution can be defined on the optional TSTEPNX bulk data entry. For explicit dynamic analysis an XSTEP statement as well as ANALYSIS = EXPDYN must be present in the subcase. To define the termination time a TTERM subcase entry is mandatory. XSTEP references an XSTEP bulk data entry. Time step control can be defined on the XSTEP bulk data entry. The implicit schemes require the solution of linear systems equations. By default, the direct solver is invoked, whereby the unknowns are simultaneously solved using a Gauss elimination method that exploits the sparseness and symmetry of the stiffness matrix, K, for computational efficiency. Alternatively, an iterative solver using the preconditioning conjugate gradient method may be used. While the direct solver is very robust, accurate and efficient, the iterative solver is sometimes superior in terms of speed, for example for bulky solid structures. The iterative solver is selected through the SOLVTYP subcase information entry, which in turn references a SOLVTYP bulk data entry. The definition of a unit system thru the DTI, UNITS or UNITS bulk data statement is required. The geometric nonlinear analysis loads and boundary conditions are defined in the bulk data

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section of the input deck. They need to be referenced under the SUBCASE using an SPC, NLOAD, LOAD, IC and RWALL statements in a SUBCASE. Each SUBCASE defines one loading condition that is executed separately. Subcase continuation is available thru the use of CNTNLSUB. Any number of explicit and implicit analyses can be linked. However, geometric nonlinear (ANALYSIS = NLGEOM, EXPDYN, or IMPDYN) analysis subcases cannot yet be linked with small displacement quasistatic nonlinear (ANALYSIS = NLSTAT) analysis subcases and vice versa.

Example for implicit (quasi-)static analysis SUBCASE 1 ANALYSIS = NLGEOM SPC = 1 NLOAD = 2 NLPARM = 3 TTERM = 1.0 DISP = ALL STRESS = ALL BEGIN BULK NLPARM,3 NLOAD1,2,2,,L,88 TABLED1,88, +,0.0,0.0,1.0,1.0,ENDT DTI,UNITS,1,kg,N,m,s

Alternative example for implicit (quasi-)static analysis SUBCASE 1 ANALYSIS = NLGEOM SPC = 1 LOAD = 4 NLPARM = 3 DISP = ALL STRESS = ALL BEGIN BULK NLPARM,3 FORCE,4,233,,1.0,0.0,0.0,1.0 DTI,UNITS,1,kg,N,m,s

Example for implicit dynamic analysis SUBCASE 2 ANALYSIS = IMPDYN SPC = 1 IC = 5 TSTEPNL = 3 TTERM = 0.2 DISP = ALL STRESS = ALL BEGIN BULK TSTEPNL,3 TIC,5,123,1,,13.88 DTI,UNITS,1,kg,N,m,s

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Example for explicit analysis SUBCASE 3 ANALYSIS = EXPDYN SPC = 1 NLOAD = 2 XSTEP = 3 TTERM = 1.0 DISP = ALL STRESS = ALL BEGIN BULK XSTEP,3 NLOAD1,2,2,,L,88 TABLED1,88, +,0.0,0.0,1.0,1.0,ENDT DTI,UNITS,1,kg,N,m,s

Example for subcase continuation DISP = ALL STRESS = ALL SUBCASE 1 ANALYSIS = NLGEOM SPC = 1 NLOAD = 2 NLPARM = 3 TTERM = 1.0 SUBCASE 2 ANALYSIS = EXPDYN IC = 5 XSTEP = 4 TTERM = 1.1 CNTNLSUB = 1 BEGIN BULK NLPARM,3 XSTEP,4 NLOAD1,2,2,,L,88 GRAV,2,,9.81,0.0,0.0,1.0 TABLED1,88, +,0.0,0.0,1.0,1.0ENDT TIC,5,123,1,,13.88 DTI,UNITS,1,kg,N,m,s

Alternative example for subcase continuation for implicit (quasi-) static analysis DISP = ALL STRESS = ALL ANALYSIS = NLGEOM CNTNLSUB, YES SUBCASE 1 SPC = 1 LOAD = 2 NLPARM = 3 SUBCASE 2

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SPC = 1 LOAD = 4 NLPARM = 3 BEGIN BULK NLPARM,3 GRAV,2,,9.81,0.0,0.0,1.0 FORCE,4,233,,1.0,0.0,0.0,1.0 UNITS = SI

User's Considerations Geometric Nonlinear Analysis Properties and Materials Special element types and nonlinear materials are available for geometric nonlinear analysis. As a general rule property and material definitions that are only applicable in geometric nonlinear analysis are defined on extensions to the original property and to a MAT1 material, respectively. The extensions are grouped with the base entry by sharing the same PID or MID. In the case of a subcase that is not a geometric nonlinear analysis, these extensions are ignored. Property defaults can be set for shells (XSHLPRM) and solids (XSOLPRM) that may replace the use of property extensions. Property example: PSHELL, 3, 7, 1.0, 7, , 7 PSHELLX, 3, 24, , , 5 Material example: MAT1, 102, 60.4, , 0.33, 2.70e-6 MATX02, 102, 0.09026, 0.22313, 0.3746, 100.0, 0.175

Coordinate Systems In geometric nonlinear analysis there are moving and fixed coordinate systems. Rectangular coordinate systems that are defined thru grid points (CORD1R, CORD3R) are moving with the deformations of the model. Systems defined in terms of point coordinates (CORD2R, CORD4R) are fixed. The behavior of loads depends on the coordinate system referenced. If the loads FORCE, MOMENT are desired to be follower forces, a CID that references a moving coordinate system (CORD1R, CORD3R) must be defined. Otherwise these loads are not following the deformation. PLOAD always follows the deformations.

Difference Between Geometric Linear and Geometric Nonlinear Analysis In geometric linear analysis all deformations and rotations are small (infinitesimal). As a general rule, displacements of say 5% of the model dimension and rotations up to 5 degrees can be treated as small. Rotations are trickier. A rotating body seems to get bigger linearly under deformation even if defined as rigid. Nonlinearities can only come from contact or materials. This type of analysis is supported in Nonlinear Quasi-Static Analysis with ANALYSIS = NLSTAT. Loads stay in the undeformed coordinates and simply move along the

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axis they are defined in. In geometric nonlinear analysis, displacements and rotations are large (finite). The magnitude of a force actually matters. Changes in magnitude of a load can change convergence behavior considerably. Also the direction of a force needs to be controlled. Forces may follow the deformation or keep their direction. This can be controlled thru the choice of coordinate systems (see above). The images below display two examples of these differences. Figure 4 shows a cantilever beam solved with small displacements, large displacements with a follower force, and large displacements without a follower force. Figure 5 shows a simple rigid rotated by an angle solved with small and finite rotations.

Figure 4: Cantilever beam with small (GLIN) and large (GNL) displacements

Figure 5: Small (GLIN) vs. finite (GNL) rotations

Difference Between Implicit and Explicit Analysis Implicit static analysis has the following characteristics: Involves matrix factorization Stiffness matrix must be positive definite

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- Model must be sufficiently constraint - No unattached parts Iteration is needed to reach equilibrium Equilibrium is achieved within iteration tolerances Larger time steps Long-term, (quasi-)static events

Implicit dynamic analysis has the following characteristics: Involves matrix factorization Dynamic stiffness matrix must be positive definite Iteration is needed to reach equilibrium Equilibrium is achieved within iteration tolerances Larger time steps Long-term events

Explicit (dynamic) analysis has the following characteristics: In general a diagonal mass matrix is used No matrix factorization necessary Equilibrium is always guaranteed Maximum stable time step needs to be respected Small time steps Short-term events Implicit Contact In an implicit contact analysis, you need to take care of the following two concerns: First, there should be no initial penetrations in the mesh. Sometimes, initial penetration is necessary to begin the simulation then only a small (< 0.01*GAP) value is recommended to not change reality too much. With high initial penetrations, the solution will progress but may lead to incorrect results. You will be warned about initial penetration during the check run. Secondly, in quasi-static analysis the model needs to be sufficiently constrained. For example, having two blocks on top of each other (Figure 6) the top part is not constrained. It is recommended to have the meshes completely depenetrated and to define a very small GAP. This would create small springs constraining the upper body in vertical direction. Of course, the other rigid body motions of the part have to be constrained too. More information can be found in CONTACT, CONTPRM, and PCONTX bulk data definitions.

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Figure 6: Initial GAP

Implicit Snap-thru, Post-buckling Analysis Some nonlinear problems with large deformations encounter bifurcations. The solution becomes instable and the structure buckles or snaps from one state to another (Figure 7). The load vs. displacement does not simply increase but may reduce until another stable point is reached from which the load then can continue to increase (Figure 8). In the implicit solution procedure it is clear that a simple load increment may not be sufficient to determine the point where the force starts reducing. A special method needs to be employed to find the proper search direction s for the solution to stay on its path. This solution is called Riks method and can be defined via NLPARMX, SACC. The search direction is defined by satisfying certain constraints of which two methods can be selected via NLPARMX, CTYP. There are currently some limitations in the way the results are written. Internal forces cannot be plotted yet.

Figure 7: Snap-thru

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Figure 8: Snap-thru - Load vs. displacement

Not Sufficiently Constrained Model in Implicit (Quasi-)static Analysis For models that are not sufficiently constrained, inertia stiffness can be used to overcome a singular stiffness matrix (NLPARMX, KINER = ON). The inertia stiffness, Kinertia = 1 / (DTSCQ * dt)2M, is added to the stiffness matrix K in a (quasi-)static analysis. Care needs to be taken in the selection of DTSCQ. An added mass that is too large may lead to incorrect results. This function is similar to inertia relief in other analysis types.

Implicit Convergence Issues Sometimes the iteration process stops with TIME STEP LIMIT ERROR. This means the time step reached DTMIN. In this case, the following measures can be taken to remedy the situation: Implicit (Quasi-)static Check if there is any rigid motion by launching a linear run or eigenvalue analysis. Double check the values and units of input parameters (material properties, loads, etc). It is usually helpful to view the results of intermediate animation outputs. Increase the number of load increments (NLPARM, NINC), decrease minimum time step (NLPARMX, DTMIN), and/or decrease maximum time step (NLPARMX, DTMAX). If post-buckling happens, activate RIKS method (NLPARMX, TSCTRL = RIKS). Check the displacement, force and energy residual values during the iteration in the .out file, find out which convergence criteria is causing the divergence, and then modify convergence control criteria (NLPARM, CONV) and relax the tolerances (NLPARM, EPSU, EPSW, and EPSP). It must be understood that reducing the convergence tolerance may lead to inaccurate results.

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Use small displacement formulation (PARAM, SMDISP, 1) if the displacements and rotations are small. Implicit Dynamic Double check the values and units of input parameters (material properties, loads, etc). It is usually helpful to view the results of intermediate animation outputs. Decrease the initial time step (TSTEPNL, NX), decrease minimum time step (TSTEPNX, DTMIN), and/or decrease maximum time step (TSTEPNX, DTMAX). Check the displacement, force and energy residual values during the iteration in the .out file, find out which convergence criteria is causing the divergence, and then modify convergence control criteria (NLPARM, CONV) and relax the tolerances (NLPARM, EPSU, EPSW, and EPSP). It must be understood that reducing the convergence tolerance may lead to inaccurate results. Use small displacement formulation (PARAM, SMDISP, 1) if the displacements and rotations are small. Limitations The solution will be terminated if unsupported Bulk Data entries are encountered. The following Bulk Data properties and elements are currently not translated: - PBUSHT (partially, KN is translated) - PDAMP, CDAMPi - PGAP, CGAP, CGAPG (partially, friction is not allowed) - PMASS, CMASSi - PSHEAR, CSHEAR - PVISC, CVISC Additional relevant Bulk Data entries (except loads) that are currently not translated: - CORD1C, CORD1S, CORD2S - DMIG - MAT2, MAT4, MAT5, MAT8, MAT9, MAT10 - MATTi, TABLEST - MPC, MPCADD - RBE1, RROD Relevant loads that are currently not translated: - PLOAD1, PLOAD2 - PLOAD4 (partially, N1, N2, N3 cannot be used) - RFORCE (partially, RACC is not supported) - TLOAD1, TLOAD2

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Normal Modes Analysis Normal Modes Analysis, also called eigenvalue analysis or eigenvalue extraction, is a technique used to calculate the vibration shapes and associated frequencies that a structure will exhibit. It is important to know these frequencies because if cyclic loads are applied at these frequencies, the structure can go into a resonance condition that will lead to catastrophic failure. It is also important to know the shapes in order to make sure that loads are not applied at points that will cause the resonance condition. Normal modes analysis is also required for modal frequency response and modal transient analysis. In these analyses the problem is transformed from the direct mesh coordinates, where the number of degrees of freedom can be in the millions, to the modal coordinates where the number of degrees of freedom is just the number of modes used. Typically, the upper bound frequency in this case is 1.5 times the highest loading frequency or response frequency of interest. In OptiStruct, normal modes analysis can be performed using one of two algorithms: Lanczos or the automated multi-level sub-structuring eigenvalue solution (AMSES). The eigenvalue extraction data for Lanczos is specified on the EIGRL data and for the automated multi-level sub-structuring eigenvalue solution method, the EIGRA data is used. In addition, OptiStruct has an interface to the AMLS software developed at the University of Texas. AMLS uses the automated multi-level sub-structuring method for eigenvalue extraction. The use of AMLS is triggered by using the PARAM, AMLS set to YES input data in conjunction with the EIGRL card (only). The Lanczos Method The Lanczos method has the advantage that the eigenvalues and associated mode shapes are calculated exactly. This method is efficient for calculations in which the number of modes is small and the full shape of each mode is required. The disadvantage of the Lanczos method is that it is slow for large problems with millions of degrees of freedom for which hundreds of modes are required. The run times for these types of problems can easily stretch into days. In these cases, the AMSES or AMLS method must be used. The Automated Multi-level Sub-structuring Eigenvalue Solution Method (AMSES) The AMSES method has the advantage that only a portion of the eigenvector need be calculated. Since only a portion of the eigenvector is calculated, the disk space and disk I/O is greatly reduced. This leads to much shorter run times. For typical NVH frequency response analysis there is only about 100 degrees of freedom of interest. In these cases, solutions of thousands of modes for meshes of millions of degrees of freedom can be solved in just a few hours. The disadvantage of the AMSES method is that the calculations are not exact. However, the modal frequencies are still accurate to a few digits. Also, for NVH analysis it is important that the mode shapes form modal space that covers all possible deformation patterns, but not so important that each individual mode shape is accurate. AMSES Usage Guidelines The following guidelines list the factors affecting AMSES usage: 1. The AMSES solution is, generally, much faster than Lanczos, but the results are approximate. Accuracy of the lower modes is very high; therefore, AMSES is a good candidate for solutions with a large number of modes (greater than a few hundred) where an approximated eigen-space is sufficient (as in Modal Frequency Response and Modal Transient Response Analysis). Although approximate, the large number of modes used for

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modal analysis will encompass the modal space and the resulting motion will match very closely with the Lanczos results. Lanczos is recommended in solutions where accurate mode shapes of a small number of modes are required. 2. AMSES is also recommended in cases where: 1) A low number of eigenvalues are requested but the model consists of more than a million degrees of freedom, and/or; 2) The upper bound (V2) is specified or the number of modes (ND) is greater than 50 on the EIGRL entry. In such cases, it is likely that Lanczos runs are slower than AMSES runs. 3. For optimization runs, if accuracy of the eigenvector is important, normal modes analysis with AMSES can be run first and then Lanczos can be run with precise lower and upper bounds to check the AMSES run for accuracy. The AMSES upper bound can then be adjusted to achieve acceptable accuracy of the desired eigenvectors. Now, AMSES can be used for all optimization runs in this analysis. 4. The AMSES solution is much faster for flexible body generation and modal solutions with many residual vectors. 5. AMSES should be used cautiously in situations with very large RBE3’s (if the RBE3 is connected to 1/4th of the structure). It may be better to eliminate such RBE3’s. 6. AMSES solution speeds depend on the number of eigenvector degrees of freedom (DOF) to be calculated. DISP=ALL will cause the entire eigenvector to be calculated and the speedup will not be large. However, if results for only a few DOF are required (typical for NVH analysis), AMSES can be up to 100 times faster than Lanczos. To improve AMSES run times, it is recommended to request results only for the required DOF. 7. For an AMSES run with V1, V2 and ND specified on the EIGRA entry, AMSES calculates all the modes up to the specified V2 (upper bound) regardless of the value of ND. Then “ND” number of requested modes is output. Therefore, reducing ND by keeping the upper bound (V2) the same will not significantly improve the AMSES run times, the upper bound must also be correspondingly reduced to prevent the extraction of extra modes. 8. AMSES is also useful in checking for model irregularities. AMSES can be used to print the list of grids associated with a massless mechanism or a singularity.

The Governing Equations Normal Modes Analysis The equilibrium equation for a structure performing free vibration appears as the eigenvalue problem:

K

M x 0

Where, K is the stiffness matrix of the structure and M is the mass matrix. Damping is neglected. The solution of the eigenvalue problem yields n eigenvalues , where n is the number of degrees of freedom. The vector { i } is the eigenvector corresponding to the eigenvalue.

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The eigenvalue problem is solved using Lanczos, AMSES, or AMLS. The natural frequency fi follows directly from the eigenvalue .

fi

i

2

Input Specification In order to run a normal modes analysis, an EIGRL or EIGRA bulk data entry needs to be given to define the number of modes to be extracted. EIGRL or EIGRA data needs to be referenced by a METHOD statement in a SUBCASE in the subcase information section. It is not necessary to define boundary conditions using an SPC statement. If no boundary conditions are applied, a zero eigenvalue is computed for each rigid body degree of freedom of the model. It is possible to request the computation of residual vectors in conjunction with a normal modes analysis. Residual vectors are static displacements ortho-normalized with the eigenvectors to be used in an external frequency response analysis. In order to get this output, users have to define degrees of freedom using USET, USET1. The degrees of freedom are then used to define loads in the unit load method to compute the residual vectors. RESVEC = YES needs to be defined in the normal modes subcase, if the Lanczos eigensolver is used. Residual vectors associated with USET and USET1 data are always created, if the AMSES or AMLS eigensolvers are used. Boundary conditions defined using SPC or inertia relief must be applied to create residual vectors.

Subcase Definition A normal modes subcase may be explicitly identified by setting ANALYSIS=MODES, but it is also implicitly chosen for any subcase containing the METHOD data selector (when the ANALYSIS entry is not present). The following data selectors are recognized for an normal modes subcase definition. 1. METHOD – references an eigenvalue extraction bulk data definition (EIGRL or EIGRA). This reference is required. 2. SPC – references single point constraint bulk data entries (SPCADD, SPC or SPC1). 3. MPC – references multi-point constraint bulk data entries (MPCADD or MPC).

Bulk Data Bulk data entries which have particular significance for normal modes analysis include: 1. EIGRL – specifies the modes to be calculated and solution parameters for the Lanczos eigenvalue extraction method. 2. EIGRA – specifies the modes to be calculated and solution parameters for the AMSES eigenvalue extraction method. 3. PARAM,AMLS,YES – specifies that the AMLS software will be used for eigenvalue extraction based on the modal parameters on the EIGRL or EIGRA data.

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4. SPC, SPC1, and SPCADD - specify the base where excitation is applied and other constraints. 5. MPC and MPCADD - specify multi-point constraints.

Sample Input SUBCASE 100 SPC = 5 METHOD = 24 $ BEGIN BULK $ EIGRL, 24, 0.0, 1000. ENDDATA $

Output Results of interest from eigenvalue extraction include maximum displacement, modal stresses, energies and multi-point constraint forces. These are requested via the I/O Options DISPLACEMENT, EKE, ESE, STRESS, GPSTRESS and MPCFORCE respectively.

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Frequency Response Analysis Frequency response analysis is used to calculate the response of a structure to steady state oscillatory excitation. Typical applications are noise, vibration and harshness (NVH) analysis of vehicles, rotating machinery, transmissions, and powertrain systems. Frequency response analysis is used to compute the response of the structure, which is actually transient, in a static frequency domain. The loading is sinusoidal. A simple case is a load of given amplitude at a specified frequency. The response occurs at the same frequency, and damping would lead to a phase shift (Figure 1). The loads can be forces, displacements, velocity, and acceleration. They are dependent on the excitation frequency . The results from a frequency response analysis are displacements, velocities, accelerations, forces, stresses, and strains. The responses are usually complex numbers that are either given as magnitude and phase angle or as real and imaginary part. OptiStruct supports Direct and Modal frequency response analysis.

Figure 1: Excitation and response of a frequency response analysis.

Direct Frequency Response Analysis Direct frequency response analysis can be used to compute the structural responses directly at discrete excitation frequencies by solving a set of complex matrix equations.

Mu&& Bu& Ku

Pei

t

The quantity is the angular loading frequency. The applied harmonic excitation can be assumed to generate a harmonic response.

u

dei

t

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The vector u is the displacement vector. Substituting the assumed harmonic displacement response into the equation of motion and rewriting the damping matrix (B), you get:

K

2

M

iGK iK E

i B1 dei

t

Pei

t

The matrix K is the stiffness matrix and the matrix M is the mass matrix. There are three ways to define damping in the system. 1. Using a uniform structural damping coefficient G. 2. Structural element damping using the damping coefficients GE on the materials as well as GE on bushing and spring element property definitions. These form the matrix KE. 3. Viscous damping generated by damper elements. These form the matrix B1. The equation of motion is solved directly using complex algebra.

Running Direct Frequency Response Analysis using OptiStruct The frequency response loads and boundary conditions are defined in the bulk data section of the input deck. They need to be referenced in the subcase information section using an SPC and DLOAD statement in a SUBCASE. OptiStruct does not support inertia relief for direct frequency response analysis. The solver will error out if it is attempted. A frequency set must be referenced using a FREQUENCY statement. In addition to the various damping elements and material damping, uniform structural damping G can be applied using PARAM, G.

Modal Frequency Response Analysis The modal method first performs a normal modes analysis to obtain the eigenvalues

X

i

2 i

X

i and the corresponding eigenvectors of the system. The response can be expressed as a scalar product of the eigenvectors X and the modal responses d.

u

Xdei

t

The equation of motion without damping is then transformed into modal coordinates using the eigenvectors.

X T MX

X T KX dei

t

X T Pei

T

t

T

The modal mass matrix X MX and the modal stiffness matrix X KX are diagonal. If the eigenvectors are normalized with respect to the mass matrix, the modal mass matrix is the

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unity matrix and the modal stiffness matrix is a diagonal matrix holding the eigenvalues of the system. This way, the system equation is reduced to a set of uncoupled equations for the components of d that can be solved easily. The inclusion of damping, as discussed in the direct method, yields:

X T KX

2

X T MX

iGX T KX

XTK X

iX T K E X

i X T B1 X dei

t

X T Pei

t

XTB X

E 1 Here, the matrices and are generally non-diagonal. Then the coupled problem is similar to the system solved in the direct method, but of much lesser degree of freedom. It is solved using the direct method.

The evaluation of the equation of motion is much faster if the equations can be kept decoupled. This can be achieved if the damping is applied to each mode separately. This is done through a damping table TABDMP1 that lists damping values g versus natural

i

frequency f . If this approach is used, no structural element or viscous damping should be

i

defined. The decoupled equation is: 2

mi

bi / (2mi

i

Where,

ki di ei

i bi

i

)

t

pi ei

t

is the modal damping ratio, and

2 i

is the modal eigenvalue.

g (f )

Three types of modal damping values i i can be defined: G – Structural damping, CRIT – Critical damping, and Q – Quality factor. They are related through the following three equations at resonance:

G:

i

bi bcr

gi 2

CRIT : bcr

2mi

1 2 i

Q : Qi

i

1 gi

Modal damping is entered in to the complex stiffness matrix as structural damping if PARAM, KDAMP, -1 is used. Then the uncoupled equation becomes: 2

mi

(1 ig ( )) * ki di ei

t

pi ei

t

A METHOD statement is required for the modal method to control the normal modes analysis. The METHOD statement can refer to either EIGRL or EIGRA data.

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Residual Vector Generation (Increases accuracy) The accuracy of the modal method can be vastly improved by adding the displacement vectors of a static analysis based on the dynamic loading to the matrix of eigenvectors X. These vectors are frequently referred to as residual vectors, the method as the modal acceleration. There are two ways this is implemented. The unit load method generates residual vectors based on static loads which are unit vectors at the dynamic load degrees of freedom. That is, the static loads for the residual vector generation are unit vectors at the degrees of freedom where the dynamic load is applied. The number of residual vectors is equal to the number of loaded degrees of freedom. This is the default method since it is generally more accurate. The applied load method generates a maximum of two residual vectors which are the dynamic load vector at a loading frequency of zero. If the real and the imaginary parts of the dynamic load are the same, or if one of them is zero, only one of them is used. In the case of excited displacements, the residual vectors are obtained by solving static load cases with unit displacements at the same degrees of freedom as the dynamic excited displacement degrees of freedom. The following image illustrates the effect that the use of the residual vectors has on the result accuracy of the modal frequency response analysis (FRA) compared to the accurate direct method.

Running Modal Frequency Response Analysis using OptiStruct The frequency response loads and boundary conditions are defined in the bulk data section of the input deck. They need to be referenced in the subcase information section using an SPC and DLOAD statement in a SUBCASE. Residual vectors are relevant for modal FRF/acoustics/transient analysis. They enhance the accuracy of these analyses and, hence, are computed by default. You can control RESVEC calculations using the case control statement: RESVEC(APPLOD/UNITLOD,DAMPLOD/NODAMP)=Value

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Where, Value can be Yes or No. The keyword(s) within parentheses are ignored if the Value specified is No – in this case all RESVEC calculations are turned off. The keyword APPLOD generates RESVECs based on the dynamic loading of the modal FRF/acoustics/transient analysis. The keyword UNITLOAD generates RESVECs based on unit loads at the dynamic loading’s degrees of freedom. The keyword DAMPLOD generates viscous damping RESVECs based on unit loads at the viscous damping degrees of freedom. The keyword NODAMP turns off the generation of the viscous damping RESVECs that are otherwise generated by default. Even though DAMPLOD and NODAMP are options in the case control, they are global switches that will be applied to all the modal FRF/acoustics/transient subcases in the model. When the underlying eigenvalue analysis is done using the Lanczos method, the default RESVECs are generated based on the applied loading and viscous damping degrees of freedom. If the underlying eigenvalue analysis is done using AMSES or AMLS, the default RESVECs are generated based on unit loading at the load degrees of freedom and viscous damping degrees of freedom. Residual vectors are always generated if enforced displacements, velocities or accelerations are defined. In addition, if there is USET U6 data, residual vectors will be calculated if the AMSES or AMLS eigensolver is used. USET U6 residual vectors will not be calculated if the Lanczos eigensolver is used. When residual vectors are included, inertia relief will be applied by default to unconstrained models. If inertia relief is not desired for RESVECs, it has to be turned off using PARAM, INREL, 0. When residual vectors are included, the eigenmodes from the underlying eigenvalue analysis of the FRF/transient subcase are used in inertia relief. All modes with eigenvalues below a limit value (FZERO) are used as rigid body modes in the inertia relief analysis. If there are no eigenmodes below FZERO, up to 6 global rigid body modes are internally generated based on the geometry of the model and used in the inertia relief. You can set FZERO using PARAM, FZERO, Value. The default value for FZERO is 0.1 A frequency set must be referenced using a FREQUENCY statement. A METHOD statement is required for the modal method to control the normal modes analysis. In order to save computational effort, previously saved eigenvectors can be retrieved using the EIGVRETRIEVE subcase statement. In addition to the various damping elements and material damping, uniform structural damping G can be applied using PARAM, G. Modal damping is being applied using the SDAMPING reference of a damping table TABDMP1. The parameter PARAM, KDAMP is to define the method of applying the damping table. Frequency-dependent materials (MATFi bulk data entries) can be used in Direct and Modal Frequency Response Analysis, via TABLEDi entries for corresponding fields on the MATi entries. MATF1, MATF2, MATF3, MATF8, MATF9 and MATF10 bulk data entries can be used to define the currently available frequency-dependent materials. Frequency-dependent properties (PBUSHT bulk data entry) can also be used in Frequency Response Analysis, via TABLEDi entries for the corresponding fields on the PBUSHT entry.

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Output The results of a frequency response analysis are displacements, velocities, accelerations, forces, stresses, and strains. The usual output entries like STRESS, STRAIN, DISPLACEMENT, etc. can be used to request corresponding output values. PARAM, ENFMOTN, REL can be used to generate displacement, velocity and acceleration output relative to the specified enforced motion. In such cases, subsequently calculated outputs like stresses and forces are also generated relative to the specified enforced motion. PARAM, ENFMOTN, TOTAL/ABS can be used to generate the total output values including the specified enforced motion (TOTAL/ABS is the default).

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Complex Eigenvalue Analysis Real eigenvalue analysis is used to compute the normal modes of a structure. Complex eigenvalue analysis computes the complex modes of the structure. The complex modes contain the imaginary part, which represents the cyclic frequency, and the real part which represents the damping of the mode. If the real part is negative, then the mode is said to be stable. If the real part is positive, then the mode is unstable. Complex eigenvalue analysis is usually used to determine the stability of a structure when unsymmetric matrices are presented due to special physical behavior. It is also used to determine the modes of a damped structure. The complex eigenvalue analysis is formulated in the following manner:

p2 M

pB

K igK i GE

f

Kf

0

Where,

K is the stiffness matrix of the structure M is the mass matrix GE is the element structural damping matrix B is viscous damping matrix g is global structural damping coefficient Kf is the extra stiffness matrix by direct matrix input f is the coefficient of extra stiffness matrix

p

i , and The solution of the complex eigenvalue problem yields complex eigenvalue, complex mode shape, . Complex modes with positive real parts are considered unstable modes. Unstable modes are often in pairs.

The circular frequency f is then calculated through the relationship:

f

2

The damping coefficient is also computed from the equation:

g

2

This corresponds to the real part of a complex eigenvalue; modes with negative damping coefficients have positive real parts and are unstable modes. The extraction of complex modes directly from the above formulation is usually quite computationally expensive, especially if the problem size is not small. Instead, a modal method is used to solve the complex eigenvalue problem. First, the real modes are calculated via a normal modes analysis. Then, a complex eigenvalue problem is formed on

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the projected subspace spanned by the real modes and thus much smaller than the real space. Finally, the complex modes extraction of the reduced problem follows the well known Hessenberg reduction method. In order to run a complex eigenvalue analysis, both the EIGRL bulk data and the EIGC bulk data entry need to be given. They define the number of the real modes and the number of complex modes to be extracted, respectively. The EIGRL card has to be referenced by a METHOD statement in a SUBCASE definition. The EIGC card is referenced by a CMETHOD statement in the same SUBCASE definition. The complex eigen value analysis usually involves an unsymmetric matrix which represents the source of the physical instability. The external matrix should be provided as a DMIG bulk data entry, and then referenced by a K2PP statement in the SUBCASE definition. You can define a specific coefficient for the external matrix by PARAM, FRIC. Otherwise, the default value of the coefficient is 1.0.

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Random Response Analysis Random response analysis is used when a structure is subjected to a nondeterministic, continuous excitation. Cases likely to involve nondeterministic loads are those linked to conditions such as turbulence on an airplane structure, road surface imperfections on a car structure, noise loads on a given structure, etc. The complex frequency response can be achieved by direct and modal frequency response. If Hxa( f ) and Hxb( f ) are the complex frequency responses (displacement, velocity or acceleration) of x th degree of freedom, due to load cases a and b respectively, the power spectral density of the response of x th degree of freedom Rx( f ) is as follows:

Rx ( f ) H xa ( f )Sab ( f ) H xb ( f ) Sab( f ) is the power spectral density of the two sources, where the individual source a is the excited load case and b is the applied load case. If Sa( f ) is the spectral density of the individual source (a th load case), the power spectral density of the response of x th degree of freedom due to a th load case will be:

Rx ( f )

2

H xa ( f ) Sa ( f )

b The cross spectral density Sab( f ) with two different sources could possibly be a complex number. The power spectral density of the response of x th degree of freedom due to a th and b th load cases will be:

Rx ( f ) H xa ( f )Sab ( f ) H xb ( f ) The total power spectral density of the response will be the summation of the power spectral density of all individual load cases as well as all cross load cases. The autocorrelation

of a variable x(t) can be defined by the following equation:

T /2

Ax ( )

x (t ) x (t

lim

T

) dt

T /2

The variance of the x(t) will be equal to Ax(0). The variance function of power spectral density Sx( f ) as follows:

Ax (0)

2 (x)

can be expressed as a

S x ( f ) df

The root mean square value of the response x(t) can also be written in the following equation:

xRMS

S x ( f ) df

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The autocorrelation function and the power spectral density are Fourier transforms of each other. Therefore, the auto correlation can be described as follows:

Ax ( )

S x ( f ) exp(i 2 f ) df

There could be fatigue failure due to random vibration. The number of fatigue cycles of random vibration is evaluated by multiplying the vibration duration and another parameter called maximum number of positive zero crossing. The maximum number of positive zero crossing is defined in the following equation:

Pc

f 2 S x ( f ) df S x ( f ) df

Whenever there is a request for XYPLOT, XYPEAK or XYPUNCH, the root mean square value and the maximum number of positive crossing will be exported to the *.peak file. Setup for Random Response Analysis Random response analysis is activated, for all subcases, through the inclusion of the RANDOM data selector in the Subcase Information section of the input. This selector identifies RANDPS and RANDT1 bulk data entries to be used for random response analysis. The input spectral density is described by the RANDPS bulk data entry. The RANDPS data refers to a TABRND1 bulk data entry, which contains the power spectral density of the loading versus frequency. The RANDT1 bulk data entry describes the time span for the autocorrelation. The RCROSS bulk data is used to request the output of the cross-power spectral density function for random response analysis and is referred to by the RCROSS I/O section selector. Loading for each frequency response subcase may be distinct, but all frequency response subcases must reference the same frequency data. Results Output from Random Response Analysis The random response Power Spectral Density Function (PSDF) can be written to the .h3d file for DISP, VELO, and ACCE using the PSDF output option on these I/O option data selectors. At the end of the output for all the frequencies is the RMS over frequencies output selector in HyperView as shown below. The random response Power Spectral Density Function (PSDF) can be written to the .h3d file for CBUSH element forces using the FORCE I/O option by specifying the PSDF output option. At the end of the output for each frequency is the RMS over frequencies output selector for the .h3d file in HyperView as shown below. The random response Power Spectral Density Function (PSDF) can be written to the .h3d and .op2 files for solid and shell elements for stress and strain with the STRESS and STRAIN I/O options using the PSDF output option. At the end of the output for each frequency is the RMS over frequencies output selector for the .h3d file and the Simulation selector for the .op2 file in HyperView as shown below. Additionally, PSDF and RMS von Mises stress and strain results based on the Segalman Method are also written to the .h3d file for Random Response Analysis (only available in the H3D format).

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RMS output selection in .h3d file in HyperView

RMS output selection in .op2 file in HyperView

Plotting Output from Random Response Analysis Three plotting output requests may be used for random response analysis results. These output requests are placed in the I/O Options section of the input data. The three plotting controllers are: XYPEAK

Generates a .peak file containing a summary of the requested output.

XYPLOT

Generates a HyperGraph session file (_rand.mvw file) and related data file (.rand file) for the requested output. Also generates the .peak file.

XYPUNCH

Generates a .pch file for the requested output. Also generates the .peak file.

These output requests are different from most other OptiStruct output requests in that they may be combined on the same line.

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The requests are formatted as follows: Operation, Curve-type, Plot-type or / Grid (Component) list. "Operation" can be any combination of XYPLOT, XYPUNCH, and XYPEAK. "Curve-type" can be FORCE, STRESS, STRAIN, DISP, VELO, or ACCE to request force, stress, strain, displacement, velocity or acceleration, respectively. "Plot-type" can be either PSDF or AUTO to request power spectral density function or autocorrelation, respectively. "Grid (Component) list" must come after a slash "/". Each entry in the list is comma separated. Each entry consists of a GRID or SPOINT ID followed by a component of motion (T1, T2, T3, R1, R2, or R3) in parentheses. For SPOINTs the component must be T1. In addition, plot titles and axis labels may be controlled using TCURVE (plot title), XTITLE (xaxis label), and YTITLE (y-axis label). Default titles and labels are generated when these controls are not used.

Example 1 Requesting random response results in a HyperGraph session file for the velocity PSDF for GRIDs 3 and 6 for component T2: XYPLOT, VELO, PSDF / 3(T2), 6(T2)

Example 2 Requesting random response summary results to be written to the .peak file for the autocorrelation of displacement for GRID 223 for component R3: XYPEAK, DISP, AUTO / 223(T3)

Example 3 Requesting random response results output, in all formats, for the acceleration PSDF for GRIDS 8 and 9 for components T1 and T2: XYPEAK, XYPLOT, XYPUNCH, ACCE, PSDF / 8(T1), 9(T1), 8(T2), 9(T2) Here the XYPEAK request is valid, but redundant as it is always created when XYPLOT or XYPUNCH is present.

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Response Spectrum Analysis Response Spectrum Analysis (RSA) is a technique used to estimate the maximum response of a structure for a transient event. Maximum displacement, stresses, and/or forces may be determined in this manner. The technique combines response spectra for a prescribed dynamic loading with results of a normal modes analysis. The time-history of the responses are not available. Response spectra describes the maximum response versus natural frequency of a 1-DOF system for a prescribed dynamic loading. They are employed to calculate the maximum modal response for each structural mode. These modal maxima may then be combined using various methods, such as the Absolute Sum (ABS) method or the Complete Quadratic Combination (CQC) method, to obtain an estimate of the peak structural response. RSA is a simple and computationally inexpensive method to provide an approximation of peak response, compared to conventional transient analysis. The major computational effort is to obtain a sufficient number of normal modes in order to represent the entire frequency range of input excitation and resulting response. Response spectra are usually provided by design specifications; given these, peak responses under various dynamic excitations can be quickly calculated. Therefore, it is widely used as a design tool in areas such as seismic analysis of buildings.

The Governing Equations Normal Modes Analysis The equilibrium equation for a structure performing free vibration appears as the eigenvalue problem:

[K

M]

0

Where, K is the stiffness matrix of the structure and M is the mass matrix. Damping is neglected. The solution of the eigenvalue problem yields n eigenvalues degrees of freedom. The vector

{

i

}

i

, where n is the number of

is the eigenvector corresponding to the eigenvalue.

The eigenvalue problem is solved using the Lanczos or the AMLS method. Not all eigenvalues are required and only a small number of the lowest eigenvalues are normally calculated. The results of eigenvalue analysis are the fundamentals of response spectrum analysis. Response spectrum analysis can be performed together with normal modes analysis in a single run, or eigenvalue analysis with Lanczos solver can be performed first to save eigenvalues and eigenvectors by using EIGVSAVE, which can be retrieved later by using EIGVRETRIEVE for response spectrum analysis.

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Modal Combination It is assumed each individual mode behaves like a single degree-of-freedom system. The transient response at a degree of freedom is:

uk

ik

i

i

Where, is the eigenvector, spectrum

is modal participation factor, and X is the response

For loading due to base acceleration, the modal participation factor can be expressed as:

[ i ]T [ M ]{T }

i

Where, is the eigenvector, M is the mass matrix, and T is rigid body motion due to excitation In ABS modal combination, the peak response is estimated by:

uk

ik

i

i

In CQC modal combination, the peak response is estimated by:

uk

vm m

mn n

n

v

Where, m is the modal response associated with mode m, and coefficient.

mn

is the cross-modal

Directional combination In order to estimate peak response due to dynamic excitations in different directions, the peak response in each direction must be combined to obtain total peak response. Methods such as ALG (algebraic) and SRSS (square root of sum of squares) can be used.

Input Specification Subcase Definition An RSA subcase may be explicitly identified by setting ANALYSIS=RSPEC, but it is also implicitly chosen for any subcase containing the RSPEC data selector (when the ANALYSIS entry is not present). The following data selectors are recognized for an RSA subcase definition. METHOD – references an eigenvalue extraction bulk data definition (EIGRL). Only METHOD(STRUCTURE) is supported. This reference is required. RSPEC – references an RSPEC bulk data entry where the combination rules, excitation DOF, and the input spectra are identified. This reference is required.

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SDAMPING – references damping table bulk data entries (TABDMP1) to specify modal damping. This reference is required. SPC – references single point constraint bulk data entries (SPCADD, SPC and SPC1). For RSA analysis, these entries define the base degrees of freedom where excitation is applied. MPC – references multi-point constraint bulk data entries (MPCADD or MPC).

Bulk Data Bulk data entries which have particular significance for RSA include: RSPEC – specifies combination rules, excitation DOF, and references the input spectra. DTI,SPECSEL – defines response spectra. EIGRL – defines parameters for eigenvalue extraction. PARAM, LFREQ and PARAM, HFREQ – define the range of modes used in modal combinations. TABDMP1 – specifies modal damping SPC, SPC1, and SPCADD - specifies base where excitation is applied and other constraints.

Sample input SUBCASE 100 RSPEC = 2 SPC = 5 SDAMPING = 12 METHOD = 24 $ BEGIN BULK $ PARAM, LFREQ, 0.1 PARAM, HFREQ, 1000. EIGRL, 24, 0.0, 1000. RSPEC, 2, ABS, CQC, 0.1 , 99, 2.0, 1.0, 0.0, 0.0 DTI, SPECSEL, 99, , A, 2, 0., 3, 0.02, , 4, 0.04, ENDREC TABDMP1, 12, … TABLED1, 2 +,… TABLED1, 3 +,… TABLED1, 4 +,…

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ENDDATA $

Output Results of interest from RSA include maximum displacement, stress and force. These are requested via the I/O Options DISPLACEMENT, STRESS and FORCE respectively.

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Transient Response Analysis Transient response analysis is used to calculate the response of a structure to timedependent loads. Typical applications are structures subject to earthquakes, wind, explosions, or a vehicle going through a pothole. The loads are time-dependent forces and displacements. Initial conditions define the initial displacement and initial velocities in grid points. The results of a transient response analysis are displacements, velocities, accelerations, forces, stresses, and strains. The responses are usually time-dependent. The transient response analysis computes the structural responses solving the following equation of motion with initial conditions in matrix form.

Mu&& Bu&

Ku

u (t

0)

u0

u& (t

0)

v0

P(t )

The matrix K is the global stiffness matrix, the matrix M the mass matrix, and the matrix B is the damping matrix formed by the damping elements. The initial conditions are part of the problem formulation and are applicable for the direct transient response only. The equation of motion is integrated over time using the Newmark beta method. A time step and an end time need to be defined. Direct and modal transient response analysis methods are implemented as follows.

Direct Transient Response The equation of motion is solved directly using the Newmark Beta method. The use of complex coefficients for damping is not allowed in transient response analysis. Therefore, structural damping is included using equivalent viscous damping. The damping matrix B is composed of several contributions as follows:

B

B1

G

K

3

1

KE

4

Where, B1 is the matrix of the viscous damper elements, plus the external damping matrices input through the DMIG bulk data entry; G is the overall structural damping (PARAM, G); is the frequency of interest for the conversion of the overall structural damping into

3

equivalent viscous damping (PARAM, W3); 4 is the frequency of interest for the conversion of the element structural damping into equivalent viscous damping (PARAM, W4); and KE is the contribution from structural element damping coefficients GE.

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Running Direct Transient Response Analysis using OptiStruct The transient response loads and boundary conditions are defined in the bulk data section of the input deck. They need to be referenced in the subcase information section using an SPC statement and a DLOAD statement in a SUBCASE. Inertia relief is not supported for direct transient response analysis. OptiStruct will error out if this is attempted. Only one transient subcase can be defined. Initial conditions need to be referenced through the IC subcase statement. The analysis time step and termination time need to be defined through a TSTEP(TIME) subcase reference. In addition to the various damping elements and material damping, uniform structural damping G can be applied using PARAM, G.

Modal Transient Response In the modal method, a normal modes analysis to obtain the eigenvalues

X

i

2 i

and the

X

i corresponding eigenvectors of the system is performed first. The state vector u can be expressed as a scalar product of the eigenvectors X and the modal responses d.

u

Xd

The equation of motion without damping is then transformed into modal coordinates using the eigenvectors:

X T MXd&&

X T KXd

XTP T

T

The modal mass matrix X MX and the modal stiffness matrix X KX are diagonal. This way the system equation is reduced to a set of uncoupled equations for the components of d that can be solved easily. The inclusion of damping yields:

X T MXd&&

X T BXd&

X T KXd

XTP

T

Here, the matrices X BX are generally non-diagonal. Then coupled problem is similar to the system solved in the direct method, but of a much lesser degree of freedom. The solution of the reduced equation of motion is performed using the Newmark Beta method. The decoupling of the equations can be maintained if the damping is applied to each mode separately. This is done through a damping table TABDMP1 that lists damping values versus natural frequency

fi

gi

.

The decoupled equation is:

mi d&&i (t ) bi d&i (t ) ki di (t )

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or

d&&i (t ) 2 i

Where,

i

i

d&i (t )

2 i

bi / (2mi

i

)

di (t )

1 pi (t ) mi

is the modal damping ratio, and

2 i

is the modal eigenvalue.

g (f )

Three types of modal damping values i i can be defined: G – Structural damping, CRIT – Critical damping, and Q – Quality factor. They are related through the following three equations at resonance:

G:

i

bi bcr

CRIT : bcr

Q : Qi

1 2 i

gi 2 2mi

i

1 gi

Residual Vector Generation (Increases accuracy) The accuracy of the modal method can be vastly improved by adding the displacement vectors of a static analysis based on the dynamic loading to the matrix of eigenvectors X. These vectors are frequently referred to as residual vectors, the method as modal acceleration. There are two ways this is implemented. The unit load method generates residual vectors based on static loads, which are unit vectors at the dynamic load degrees of freedom. That is, the static loads for the residual vector generation are unit vectors at the degrees of freedom, where the dynamic load is applied. The number of residual vectors is equal to the number of loaded degrees of freedom. The applied load method generates a maximum of two residual vectors which are the dynamic load vector at loading frequency of zero. If the real and the imaginary parts of the dynamic load are the same, or if one of them is zero, only one of them is used. This is the default method since it is generally more efficient. In the case of excited displacements, the residual vectors are obtained by solving static load cases with unit displacements at the same degrees of freedom as the dynamic excited displacement degrees of freedom. Running Modal Transient Response Analysis using OptiStruct Transient response loads and boundary conditions are defined in the bulk data section of the input deck. They need to be referenced in the subcase information section using an SPC statement and a DLOAD statement in a SUBCASE. Residual vectors can be activated using the subcase statement RESVEC with the options APPLOD or UNITLOD. They are computed by default. Residual vectors are always generated if enforced displacements, velocities or accelerations are defined. Residual vectors are also calculated for viscous damping DOF. These are created by default and can be turned off with

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the RESVEC option NODAMP. In addition, if there is USET U6 data, residual vectors will be calculated if the AMSES or AMLS eigensolver is used. USET U6 residual vectors will not be calculated if the Lanczos eigensolver is used. When residual vectors are included, inertia relief can be applied to unconstrained models. A SUPORT1 subcase entry references the boundary conditions that restrain the rigid body motions. These restraints can also be defined without subcase reference using the SUPORT bulk data entry or automated using PARAM, INREL, -2. Only one transient subcase can be defined. Initial conditions cannot be defined if the modal method is used. A METHOD statement is required for the modal method to control the normal modes analysis. The METHOD statement can refer to either EIGRL or EIGRA data. The analysis time step and termination time need to be defined through a TSTEP(TIME) subcase reference. In order to save computational effort, previously saved eigenvectors can be retrieved using the EIGVRETRIEVE subcase statement. In addition to the various damping elements and material damping, uniform structural damping G is applied using PARAM, G. Modal damping can be applied using the SDAMPING reference of a damping table TABDMP1. Output The results of a transient response analysis are displacements, velocities, accelerations, forces, stresses, and strains. The responses are usually time-dependent. The usual output entries like STRESS, STRAIN, DISPLACEMENT, etc. can be used to request corresponding output values. PARAM, ENFMOTN, REL can be used to generate displacement, velocity and acceleration output relative to the specified enforced motion. In such cases, subsequently calculated outputs like stresses and forces are also generated relative to the specified enforced motion. PARAM, ENFMOTN, TOTAL/ABS can be used to generate the total output values including the specified enforced motion (TOTAL/ABS is the default).

Transient Response Analysis by Fourier Transformation Using the Fourier transformation method, frequency response analysis can be used for the transient analysis. The Fourier transformation method may be used to solve for the transient response of structural models under periodic loads. A typical application for this method is a vehicle on a bumpy road. Time-dependent applied loads are transformed into the frequency domain and all frequency dependent matrix calculations are completed. The frequency response results are then transformed back into the time domain. The results are displacements, velocities, accelerations, forces, stresses, and strains. The responses are usually time-dependent. The following equation of motion with initial conditions in matrix form is solved.

Mu&& Bu& Ku

P(t )

The matrix K is the stiffness matrix, the matrix M is the mass matrix, and the matrix B is the damping matrix formed by the damping elements. Initial conditions cannot be defined.

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The load vector is transformed from the time domain into the frequency domain using:

The response in given by:

u%

h

P%

Where, h( ) is the frequency response due to unit load. After the frequency response analysis, the time-dependent response can be recovered using:

For the results to be accurate it is important to note that: 1. The system has to be reasonably well damped. Too little damping may lead to incorrect results. 2. The forcing function should be zero for some time interval to allow decay. 3. The frequency interval should follow: . The direct and modal methods are implemented.

Direct Method Direct frequency response analysis is applied (Frequency Response Analysis). Transient response loads and boundary conditions are defined in the bulk data section of the input deck. They need to be referenced in the subcase information section using an SPC and DLOAD statement in a SUBCASE. Inertia relief is not implemented for direct frequency response. The solver will error out if it is attempted. A frequency set must be referenced using a FREQUENCY statement. Initial conditions cannot be applied. The analysis time step and termination time need to be defined through a TSTEP(FOURIER) subcase reference. In addition to the various damping elements and material damping, uniform structural damping G can be applied using PARAM, G.

Modal Method Modal frequency response analysis is applied (Frequency Response Analysis).

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Transient response loads and boundary conditions are defined in the bulk data section of the input deck. They need to be referenced in the subcase information section using an SPC statement and a DLOAD statement in a SUBCASE. Residual vectors can be activated using the subcase statement RESVEC with the options APPLOD or UNITLOD. They are computed by default. Residual vectors are always generated if enforced displacements, velocities or accelerations are defined. When residual vectors are included, inertia relief can be applied to unconstrained models. A SUPORT1 subcase entry references the boundary conditions that restrain the rigid body motions. These restraints can also be defined without subcase reference using the SUPORT bulk data entry or automated using PARAM, INREL, -2. A frequency set must be referenced using a FREQUENCY statement. Initial conditions cannot be defined. A METHOD statement is required for the modal method to control the normal modes analysis. The analysis time step and termination time need to be defined through a TSTEP(FOURIER) subcase reference. In order to save computational effort, previously saved eigenvectors can be retrieved using the EIGVRETRIEVE subcase statement. In addition to the various damping elements and material damping, uniform structural damping G can be applied using PARAM, G. Modal damping can be applied using the SDAMPING reference of a damping table TABDMP1. The parameter PARAM, KDAMP is to define the method of applying the damping table.

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Thermal Analysis The Thermal Analysis section provides an overview of the following analyses: Linear Steady-State Heat Transfer Analysis Linear Transient Heat Transfer Analysis Nonlinear Steady-State Heat Transfer Analysis Contact-based Thermal Analysis

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Linear Steady-State Heat Transfer Analysis Heat transfer analysis solves for unknown temperatures and fluxes under thermal loading. Temperature represents the amount of thermal energy available, and fluxes represent the flow of thermal energy. Conduction deals with thermal energy exchange by molecular motion. Free convection deals with thermal energy exchange between solids and surrounding fluids. Thermal loading is defined as energy flows into and out of the system. In linear steady state analysis, material properties such as conductivity and convection coefficient are linear. Temperature and fluxes at the final thermal equilibrium state are of interest. The basic finite element equation is:

Kc

H

T

p

(1)

Where, [Kc] is the conductivity matrix, [H] is the boundary convection matrix due to free convection, {T} is an unknown nodal temperature, {p} is the thermal loading vector. The system of linear equation is solved to find nodal temperature {T}. Thermal load vector can be expressed as:

p

PB

PH

PQ

(2)

Where, {PB} is the power due to heat flux at boundary specified by QBDY1 card, {PH} is the boundary convection vector due to convection specified by CONV card, and {PQ} is the power vector due to internal heat generation specified by QVOL card. The matrix on the left hand side of equation (1) is singular unless temperature boundary conditions are specified. The equilibrium equation is solved simultaneously for the unknown temperatures using a Gauss elimination method that exploits the sparseness and symmetry for computational efficiency. Once the unknown temperatures at the nodal points of the elements are calculated, temperature gradient { } can be calculated according to element shape functions. Element fluxes can be calculated by using:

f

k

T

(3)

Where, [k] is the conductivity of the material. An analogy of heat transfer analysis and structural analysis is shown in Table 1. Heat Transfer

Structural

Temperature

Displacement

Temperature gradient

Strain

Flux

Stress

[Kc]

Conductivity matrix

Stiffness matrix

[H]

Boundary convection matrix

Elastic foundation stiffness matrix

{p}

Heat flux vector

Load vector

Unknown

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{PQp}

Heat Transfer

Structural

Element volumetric

Gravity load

Table 1. Analogy of heat transfer and structural analysis

The thermal loads and boundary conditions are defined in the bulk data section of the input deck. They need to be referenced in the SUBCASE information section using an SPC or MPC and LOAD statement in a SUBCASE.

Input Data for Thermal Structural Analysis Both GRID and SPOINT can be used to specify a thermal point. Fixed temperatures are specified with SPC/SPC1/SPCD data with the component ID blank or zero. MPC data can be used to specify the relationship between temperatures of different points using component ID blank or zero. If you want to use component ID 1, then SPSYNTAX=mixed must be specified in the input deck. Rigid elements are ignored in heat transfer analysis. For all elements that have property data that reference material data (CROD, CONROD, CBAR, CBEAM, CQUAD4, CTRIA3, CQUAD8, CTRIA6, CHEXA, CHEXA20, CPENTA, CPENTA15, CPYRA, CPYRA13, CTETRA, and CTET10) can be used as conduction elements. The property data references MAT4 data for the isotropic conduction coefficient and MAT5 data for anisotropic conduction coefficients. Note that the thermal material property data has the same ID as the structural property data for any element. For CELAS1-4, the value of K is treated as the conduction coefficient. Elements that generate heat are listed in QVOL data. The heat generated by an element is equal to the element volume * QVOL * HGEN, where HGEN is a scale factor (default=1.0) listed on the material (MAT4 or MAT5) data. Heat flux load QBDY1 and convective heat transfer CONV are applied to the structure through surfaces identified by the CHBDYE card. The CHBDYE elements associate heat exchange surfaces with conduction elements. A 1D element can have heat flux applied at each end and along its length. A 2D element can have heat flux on its surface and along any edge. A 3D element can have head flux applied on any face. Fixed values of heat flux are specified using the QBDY1 card. This data lists the CHBDYE element ID and the heat flux value (Q0). The power exchanged through a CHBDYE element is equal to Q0 multiplied by the effective area of the CHBDYE element. For a 1D element, the area at the end is the cross-sectional area of the element. For flux into the side of a 1D element, the effective area is the length times the circumference of the element which is calculated from the cross-sectional area, assuming that the cross-section is circular. For 2D elements, the effective area for the surface of the element is its area and the effective area of a side is equal to the length of the side multiplied by the thickness of the element. For 3D elements, the effective area is just the area of the face. Free convection heat flux is specified for CHBDYE elements using the CONV data which lists the CHBDYE element ID, the ambient temperature (TAMB), and the ID of the PCONV data which lists the MAT4 material ID. The MAT4 data contains the convection coefficient H. The heat flux per unit area from convection is H*(T-TAMB), where T is the grid temperature.

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Coupled Thermal Structural Analysis and Optimization Each heat transfer SUBCASE defines a temperature set, which can be referred by a structural SUBCASE by TEMP(LOAD) to perform thermal-structural analysis. The temperature set identification is the same as heat transfer SUBCASE identification by default. It can be changed by using TSTRU card. If the temperature set identification is the same as a bulk data temperature set identification, the temperatures from heat transfer analysis override bulk data temperatures. Coupled thermal structural analysis is done in the following fashion. Heat transfer analysis is performed first to determine the temperature field of the structure. The temperature field is used as part of the loading for structural analysis. A single finite element mesh is usually used for both thermal and structural analysis. The finite element governing equation for static structural analysis is:

K D

f

fT

(4)

Where, [K] is the global stiffness matrix, {D} is the unknown displacement vector, {fT} is the temperature loading, and {f} is the structural loading such as forces, pressures, etc. Displacement vector {D} is solved by the linear equation solver. In coupled thermal structural optimization, {fT} sensitivities due to design changes are calculated. Besides the usual responses such as displacement, stress, mass, etc., temperature can also be a response in optimization. The coupling in thermal structural analysis is sequential, i.e. the thermal analysis affects the subsequent structural analysis. On the other hand, in coupled thermal structural optimization, the coupling works both ways, that is the thermal influence on structural and the structural influence on thermal. In other words, the optimizer modifies the structural design to satisfy constraints, which in turn affects the thermal analysis. Temperature responses are supported in Sizing, Shape, Topography, and Topology Optimization, but the CHBDYE element cannot be used in the Design Domain of Topology Optimization.

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Linear Transient Heat Transfer Analysis Linear transient heat transfer analysis can be used to calculate the temperature distribution in a system with respect to time. The applied thermal loads can either be time-dependent or time-invariant; transient thermal analysis is used to capture the thermal behavior of a system over a specific period in time. The basic finite element equation for transient heat transfer analysis is given by:

C T&

K

H

T

p

(1)

Where,

[C] is the heat capacity matrix [K] is the conductivity matrix [H] is the boundary convection matrix due to free convection is the temperature derivative with respect to time

{T} is the unknown nodal temperature {p} is the thermal loading vector. The differential equation (1) is solved to find nodal temperature {T} at the specified time steps. When equation (1) is compared to the steady-state heat transfer equation, you see that there is an additional term

that captures the transient nature of the analysis.

Guide to Request a Linear Transient Heat Transfer Analysis The following steps can be considered as a guide to define the linear transient heat transfer subcase. 1. Use the solution sequence identifier (ANALYSIS) in the subcase information section to select the linear transient heat transfer analysis using: ANALYSIS=HEAT. 2. Define the time step intervals at which the solutions will be calculated for transient analysis using the TSTEP bulk data entry. This is referenced in the subcase information section by the TSTEP subcase information entry which is used to select the integration type (TSTEP=SID) for transient analysis. 3. The initial conditions for transient heat transfer analysis are selected by the use of the IC subcase information entry. This entry can be used in the subcase information section to specify the set identification number of the temperature field defined by TEMP or TEMPD bulk data entries. 4.

Use the single point constraint (SPC) data entry to specify the fixed boundary conditions for this analysis.

5.

Use the DLOAD subcase information entry to reference the set ID’s of DLOAD, TLOAD1 and TLOAD2 bulk data entries Use the TLOAD1 and TLOAD2 bulk data entries to specify: (a) Time dependent thermal loading The EXCITEID field of the TLOAD1 and TLOAD2 bulk data entries should point to the ID’s of QVOL, QBDY1 bulk data entries or a combination of them using LOADADD.

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(b) Temperature boundary condition The EXCITEID field of the TLOAD1 and TLOAD2 bulk data entries should point to the ID of the SPCD bulk data entry. Also, the TYPE field in the TLOAD1 and TLOAD2 entries should be set to 1. 6. The MAT4 and MAT5 bulk data entries can be used to define thermal material properties such as thermal conductivity [K], heat capacity [C], density [RHO], convection heat transfer coefficient [H] and heat generation capability [HGEN] used in the QVOL data entry. 7. The THERMAL I/O option can be used to request nodal temperature output {T} for transient heat transfer analysis subcases. The FLUX I/O options entry can be used to request temperature gradient and flux output for transient heat transfer analysis subcases.

Applying Heat Flux Loads In Step 5(a) of the guide above, the ability to use QBDY1 data to apply heat flux loading is illustrated. This is accomplished as explained in the following steps: 1. The value of the heat flux load is input in the Q0 field of a QBDY1 data entry. 2. The EID# field in the QBDY1 data entry requires the identification number of CHBDYE surface elements. These surface elements should be created on the surfaces of the model to which heat flux loads are to be applied. 3. This is done in HyperMesh by creating an interface of type CONDUCTION, selecting all the relevant surfaces and then adding CHBDYE surface elements to those surfaces. 4. These newly created surface elements via the interface group can then be referenced in the EID# field of the QBDY1 data entry. Refer to the OS-1090 tutorial for detailed information on setting up heat flux loads and free convection for transient heat transfer analysis.

Coupled Thermal-structural Analysis The temperature results from the final time step of a linear transient heat transfer analysis can be applied to a structural subcase. Both TEMPERATURE(LOAD) and TEMPERATURE(MATERIAL) are allowed to reference the subcase ID or temperature result sets from the linear transient heat transfer analysis for use in either material property calculations or thermal loading. Note: Non-zero SPC will be considered as zero SPC for transient thermal analysis, except when non-zero SPC are used to specify ambient points for convection. When an ambient point is controlled by TLOAD1/TLOAD2 via SPCD, the corresponding SPC should be zero.

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Nonlinear Steady-State Heat Transfer Analysis Nonlinear steady-state heat transfer analysis can be used to calculate the temperature distribution in a system, in which material properties are a function of temperature. The basic finite element equation for nonlinear heat transfer can be written as:

L(T) = P

(1)

Where, {T} is unknown temperature, P is the global power vector, and L(T) is the global response (nodal power). The system of equations (1) is solved using the Newton’s method. Solution control is provided by defining parameters on the NLPARM bulk data entry. TEMPERATURE (INITIAL) can be used to provide a likely initial temperature distribution. The temperature results from the nonlinear heat transfer analysis can be used in subsequent structural analysis.

Nonlinear Steady-State Heat Transfer Analysis Setup The following steps can be considered as a guide to setup a nonlinear steady-state heat transfer analysis: 1. Use the solution sequence identifier (ANALYSIS) in the subcase information section to select the nonlinear steady-state heat transfer analysis using: ANALYSIS=NLHEAT. 2. The likely initial temperature distribution can be defined using the TEMPERATURE subcase information entry (type=INITIAL). A good initial temperature estimate improves the convergence of the solver. 3. The MATT4 bulk data entry can be used to define temperature dependent thermal material properties. 4. To indicate that a nonlinear solution is required for any subcase, a NLPARM subcase information entry is required. This subcase entry points to a NLPARM bulk data entry that specifies convergence tolerances and other nonlinear parameters. 5. Loads and boundary conditions are defined in the bulk data section of the input deck. These should be referenced in the subcase information section using SPC and LOAD entries in a subcase. Each subcase defines a load vector.

Example An example solver deck section showing the usage of ANALYSIS and NLPARM is shown below: SUBCASE 5 ANALYSIS=NLHEAT SPC=10 LOAD=20 NLPARM=30 … BEGIN BULK … NLPARM, 30 …

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ENDDATA

Note: Optimization based on nonlinear heat transfer analysis is currently not supported.

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Contact-based Thermal Analysis Introduction In OptiStruct structural models involving contact are solved by using nonlinear quasi-static analysis. The analysis involves finding the contact status, such as contact clearance and pressure. Contact clearance spans the distance between the master and slave, while contact pressure is developed between two surfaces in contact. Motivation The traditional thermal structural analysis is one-way coupling, in the sense that thermal analysis influences structural analysis by providing temperature, but structural problem does not affect the thermal problem.

Figure 1: Traditional thermal-structural analysis – Thermal results affect the structural problem.

When contact problems are involved, thermal structural analysis becomes fully coupled since contact status changes thermal conductivity.

Figure 2: Contact based (Coupled) Thermal-Structural Analysis – contact status affects the thermal problem

In Figure 1, you can see that a change in contact status does not affect the thermal problem. This may lead to inaccurate solutions if thermal conductivity depends on the contact status. In Figure 2, the contact clearance and/or pressure changes during the course of the quasistatic nonlinear analysis, the corresponding change in the thermal conductivity will affect the solution of the thermal problem. Implementation Thermal analysis is performed first using initial contact status. Nonlinear structural analysis is employed to find contact status. Thermal conductivity at the contact interface is calculated based on contact clearance or pressure or based on user-defined values. Coupling is essential because the contact status is used to determine thermal conductivity. An iterative solution process is developed to solve fully coupled nonlinear thermal structural problem, as shown in Figure 3. Temperature results from thermal analysis are used as convergence criteria.

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Thermal conductivity across contact interface can be based on user-defined values, clearance, or pressure. The PGAPHT and PCONTHT entries can be used to define the thermal conductivity values. Note: 1. For thermal contact problems with CGAP/CGAPG, PGAPHT is required. The PGAPHT entry should have the same PID as PGAP. For problems with CONTACT and PCONT, the PCONTHT entry should be used and it requires the same PID as PCONT. 2. For problems without PCONT, PCONTHT is not required. 3. Thermal conductivity based on the AUTO option (KC/KAHT fields on PCONTHT/PGAPHT entries) can be used in thermal analysis to allow OptiStruct to automatically determine the conductivity values based on the conductivity of surrounding elements.

Figure 3: Fully coupled contact-based thermal-structural analysis.

Theoretically, while higher conductivity values enforce a perfect conductor, excessively high values may cause poor conditioning of the conductivity matrix. If such effects are observed, it may be beneficial to reduce the value of conductivity, or use conductivity based contact clearance and pressure.

Clearance based thermal conductivity (TCID on PCONTHT/PGAPHT, via TABLED#) The clearance based conductivity values can be specified by you via TABLED# entries. The typical conductivity values vary as follows:

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Figure 4: Thermal conductivity based on contact clearance.

Pressure based thermal conductivity (TPID on PCONTHT, via TABLED#) The pressure based conductivity values can be specified by you via TABLED# entries. The typical conductivity values vary as follows:

Figure 5: Thermal conductivity based on contact pressure.

Clearance and pressure based thermal conductivity (TCID and TPID on PCONTHT via TABLED#) The clearance and pressure based conductivity values can be specified by you via TABLED# entries. The typical conductivity values vary as follows:

Figure 6: Thermal conductivity based on contact clearance and pressure.

Typical thermal conductivity values increase as the clearance between the master and slave decreases. In the case of contact pressure, the thermal conductivity increases with a corresponding increase in pressure.

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Acoustic Analysis The Acoustic Analysis section provides an overview of the following analyses: Coupled Frequency Response Analysis of Fluid-Structure Models Radiated Sound Analysis

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Coupled Frequency Response Analysis of Fluid-Structure Models Coupled frequency response analysis of fluid-structure models, commonly termed acoustic analysis, is generally performed to model sound propagation within a structural cavity, such as the interior of a vehicle or a musical instrument. OptiStruct allows both direct and modal frequency response analysis for fluid-structure models. The responses of both the structural and fluid domains are computed; for the structural domain, these responses are the displacements and rotations of the structural grids, and for the fluid domain, these responses are the pressures at the fluid grid points. The accelerations of structural grids at the fluid-structure interface excite the fluid domain and conversely, the pressures on the fluid grids at the fluid-structure interface excite the structural domain. Hence the problem is coupled and the motions of structural and fluid degrees-of-freedom are solved simultaneously. Loading is sinusoidal with excitation frequency , and can be in the form of forces, acoustic sources enforced displacements, enforced velocities, and/or enforced accelerations. Frequency response loads and boundary conditions are defined in the bulk data section of the input deck. These are then referenced in a subcase definition through an SPC or DLOAD data selector. Damping may be defined for both the structural and fluid domains. For the structural domain damping may be defined through structural damping elements, material damping, structural damping (PARAM,G), or modal damping (SDAMPING referenced by a SDAMPING(STRUCT) subcase data selector). For the fluid domain damping may be defined through material damping, fluid damping (PARAM,GFL), or modal damping (SDAMPING referenced by a SDAMPING(FLUID) subcase data selector). In addition, the normalized admittance coefficient for porous materials can be specified by ALPHA on the MAT10 data. Frequency dependent fluid acoustic absorber elements can be specified on the fluid faces of the fluid-structure boundary using the CAABSF elements. The absorber elements can be point, line, triangular, or quadrilateral in shape. The CAABSF data references the PAABSF data which is used to specify the frequency dependent resistance (real part of the impendence) and reactance (imaginary part of the impendence) as well as the area factors for point and line elements. Frequency dependent structural acoustic absorber elements can be specified on the fluidstructure boundary using the CHACAB elements. These absorbers are solid elements between the fluid and structural meshes. The CHACAB data references the PACABS data which is used to specify the frequency dependent resistance (real part of the impendence) and reactance (imaginary part of the impendence). Another option is to calculate these values mass, stiffness, and damping values per unit area specified in this data. PARAM, LFREQFL and PARAM, HFREQFL can be used to exclude modes from a coupled modal frequency response analysis (Acoustic analysis). The acoustic analysis is based on inviscid flow with linear pressure-density relation as:

1

P u&& 0

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and the continuity equation is:

P

.u

0

Where,

P and u are the pressure of the fluid domain and displacement of the structural domain respectively, and are the compressibility of the fluid domain and density of the structural domain respectively. Combining the above equations, the governing equation of the fluid domain is:

&& P

1

2P

0

Effect of the structure on the fluid domain at the interface After finite element discretization, the assembly of equations for the fluid domain is:

&& B p P& K p P Au&& S p M pP Where, Mp, Bp, Kp and Sp are the mass matrix, damping matrix, stiffness matrix and source vector respectively, of the fluid domain. The matrix A represents the interface matrix and is the acceleration of the structural grids at the fluid-structure interface. (The pressure gradient at the interface will be influenced by the acceleration of the structural grids).

Effect of fluid on the structural domain at the interface The structural equation assembly can be written as:

M S u&& BS u&& K S u AT P

SS

Where, MS, BS, KS and SS are the mass matrix, damping matrix, stiffness matrix and source vector respectively, of the structural domain. The matrix A represents the transpose of the interface matrix and is the pressure at the interface fluid grids at the fluid-structure interface (The displacement, velocity and acceleration of the structural grids at the interface will be influenced by the pressure at the interface fluid grids).

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Coupled Fluid-Structure interface equation Therefore the combined fluid structure interface equation is:

The above equations are solved simultaneously for unknowns in the structural and the fluid domains, either by direct frequency response or modal frequency response. For modal frequency response, OptiStruct will calculate the eigenspace for both structure and fluid domain automatically.

Loads on the Fluid Mesh The fluid grid points can be loaded by specifying the load magnitude and GRID using the SLOAD data. The SLOAD data is referenced by the ACSRCE data which defines the dynamic characteristics of the load (DELAY, DPHASE, and a tabular listing of the load scale factor vs. frequency). In addition, the density and bulk modulus of the loaded fluid are specified on the ASRCE data. The material characteristics of the fluid must be specified in the ASRCE in case the same fluid GRID is shared by two different fluid meshes.

Fluid-Structure Interface Visualization and Refinement OptiStruct has support for both grid-to-grid matching and non-matching interfaces. The interface is specified through the ACMODL card. If an ACMODL card is not specified in the input deck, the fluid-structure interface is automatically defined by OptiStruct based on default values for the ACMODL parameters. Based on a search box specified on the ACMODL card, OptiStruct outputs an *.interface file, containing information about the fluid-structure interface. With the model loaded in HyperMesh, you can import the *.interface file to visualize the fluid-structure interface (ensure that the “FE overwrite” option is activated on import). The “^Fluid Faces at Interface” component is created, which allows you to view the interface found between the structural and fluid domains. If a component “Âcoustically Rigid Fluid Faces” is created, that means at those fluid surfaces, there are no structural grids found. A structural grid set “^Structural grids at Interface” is also created to display the structural grids found at the interface. There are several steps you can take to improve the interface: 1. Perform an OptiStruct check run. This will create the *.interface file, allowing you to visualize the interface. 2. If the interface is not as desired, you may create a new SET containing those fluid grids that describe the fluid boundary. 3. You can then specify the newly created set on the ACMODL card under FSET. 4. Perform another OptiStruct check run, and review the new interface.

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Using an External Fluid Structure Coupling Definition File OptiStruct can use the binary ftn.70 coupling file generated by AKUSMOD instead of internally calculating the coupling. To use this option, add PARAM,AKUSMOD,YES to the input deck.

Modal and Panel Participation Modal participation is a measure of how much each mode participates at a given frequency in a modal frequency response calculation. Output of modal participation may be requested for structural degrees of freedom as well as for fluid degrees of freedom. Calculation and output of modal participation can be requested, for any number of degrees of freedom, using the PFMODE I/O Option. Panel participation is a measure of the influence of sets of specified structural grids, defined by PANEL bulk data entries. The response of a fluid grid is influenced through each panel or each grid at the acoustic interface. Calculation and output of the contribution from each panel at specific loading frequencies can be requested through the PFPANEL I/O Option, for modal frequency response. Also, the calculation and output of the contribution from each grid at the interface can be requested through the PFGRID I/O Option. A file *.pfmode.pch is generated based on the definition of the I/O Options PFMODE and PFPANEL. The output for PFGRID would be in a H3D file. The results for PFMODE and PFPANEL are best plotted in HyperGraph, whereas the contour results for PFGRID are best visualized in HyperView.

Non-Reflecting Boundary To create a non-reflecting boundary, set the values of the TABLEDi entry referenced by the TZREID field (Resistance-real part of Impedance) in the PAABSF data entry to be equal to * C fluid

fluid

for all frequencies. This will allow the acoustic wave to propagate normally through the boundary, without reflection. This condition is called the Sommerfeld boundary condition. Where,

is the density of the fluid, and

is the speed of sound in the fluid.

Input File - chacab.fem $$------------------------------------------------------------------------------$ $$ $ $$ NASTRAN Input Deck Generated by HyperMesh Version : 8.0SR1 $ $$ Generated using HyperMesh-Nastran Template Version : 8.0sr1 $$ $ $$ Template: general $ $$ $ $$------------------------------------------------------------------------------$ $$------------------------------------------------------------------------------$ $$ Executive Control Cards $ $$------------------------------------------------------------------------------$ SOL 111

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CEND $$------------------------------------------------------------------------------$ $$ Case Control Cards $ $$------------------------------------------------------------------------------$ SET 1 = 1734 DISPLACEMENT = 1 $ $HMNAME LOADSTEP 1"Load2" SUBCASE 1 LABEL= Load2 SPC = 4 FREQUENCY = 5 DLOAD = 2 $$------------------------------------------------------------------------------$ $$ Bulk Data Cards $ $$------------------------------------------------------------------------------$ BEGIN BULK $CHEXA 1056 2 1650 1661 1662 $+ 1683 1672 CHACAB 1056 100 1650 1645 1657 + 1671 1672 PACABS,100,YES,1,2,3,1.5,10.0,2.0 PARAM,G,0.001 PARAM,COUPMASS,-1 PARAM,POST,-1 $ACMODL DIFF 0.1 $$ EIGRL,20,,,300 EIGRL,21,,,300 $$ GRID Data $$ GRID 1 2.0 2.0 0.0 GRID 2 2.0 1.5 0.0 GRID 3 2.0 1.0 0.0 GRID 4 2.0 0.5 0.0 GRID 5 2.0 0.0 0.0 GRID 6 2.0 -0.5 0.0 GRID 7 2.0 -1.0 0.0 GRID 8 2.0 -1.5 0.0 GRID 9 2.0 -2.0 0.0 GRID 10 1.5 2.0 0.0 GRID 11 1.5 1.5 0.0 GRID 12 1.5 1.0 0.0 GRID 13 1.5 0.5 0.0 GRID 14 1.5 0.0 0.0 GRID 15 1.5 -0.5 0.0 GRID 16 1.5 -1.0 0.0 GRID 17 1.5 -1.5 0.0 GRID 18 1.5 -2.0 0.0 GRID 19 1.0 2.0 0.0 GRID 20 1.0 1.5 0.0 GRID 21 1.0 1.0 0.0 GRID 22 1.0 0.5 0.0 GRID 23 1.0 0.0 0.0 GRID 24 1.0 -0.5 0.0 GRID 25 1.0 -1.0 0.0 GRID 26 1.0 -1.5 0.0 GRID 27 1.0 -2.0 0.0 GRID 28 0.5 2.0 0.0 GRID 29 0.5 1.5 0.0 GRID 30 0.5 1.0 0.0 GRID 31 0.5 0.5 0.0 GRID 32 0.5 0.0 0.0 GRID 33 0.5 -0.5 0.0 GRID 34 0.5 -1.0 0.0 GRID 35 0.5 -1.5 0.0 GRID 36 0.5 -2.0 0.0 GRID 37 0.0 2.0 0.0

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1651 1658

1671 1676

1682+ 1675+

-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1


145

GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID

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38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0 -1.5 -1.5 -1.5 -1.5 -1.5 -1.5 -1.5 -1.5 -1.5 -2.0 -2.0 -2.0 -2.0 -2.0 -2.0 -2.0 -2.0 -2.0 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 -2.5 -2.5 -2.5 -2.5 -2.5 -2.5 -2.5 -2.5 -2.5 2.5 2.5 2.0 2.0 1.5 1.5 1.0 1.0

1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0 2.5 -2.5 2.5 -2.5 2.5 -2.5 2.5 -2.5

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1


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108 109 110 111 112 113 114 115 116 117 118 119 120 121 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782

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0.5 0.5 0.0 0.0 -0.5 -0.5 -1.0 -1.0 -1.5 -1.5 -2.0 -2.0 -2.5 -2.5 2.5 2.5 2.0 2.0 2.5 2.0 2.5 2.0 2.5 2.0 2.5 2.0 2.5 2.0 2.5 2.0 2.5 2.0 2.5 2.0 2.5 2.0 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.0

2.5 0.0 -2.5 0.0 2.5 0.0 -2.5 0.0 2.5 0.0 -2.5 0.0 2.5 0.0 -2.5 0.0 2.5 0.0 -2.5 0.0 2.5 0.0 -2.5 0.0 2.5 0.0 -2.5 0.0 2.5 1.0 2.0 1.0 2.0 1.0 2.5 1.0 1.5 1.0 1.5 1.0 1.0 1.0 1.0 1.0 0.5 1.0 0.5 1.0 -4.2E-191.0 -6.5E-201.0 -0.5 1.0 -0.5 1.0 -1.0 1.0 -1.0 1.0 -1.5 1.0 -1.5 1.0 -2.0 1.0 -2.0 1.0 -2.5 1.0 -2.5 1.0 2.0 1.0 2.5 1.0 1.5 1.0 1.0 1.0 0.5 1.0 -9.8E-211.0 -0.5 1.0 -1.0 1.0 -1.5 1.0 -2.0 1.0 -2.5 1.0 2.0 1.0 2.5 1.0 1.5 1.0 1.0 1.0 0.5 1.0 -1.5E-211.0 -0.5 1.0 -1.0 1.0 -1.5 1.0 -2.0 1.0 -2.5 1.0 2.0 1.0 2.5 1.0 1.5 1.0 1.0 1.0 0.5 1.0 -2.3E-221.0 -0.5 1.0 -1.0 1.0 -1.5 1.0 -2.0 1.0 -2.5 1.0 2.0 1.0

-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1


147


148

783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0 -1.5 -1.5 -1.5 -1.5 -1.5 -1.5 -1.5 -1.5 -1.5 -1.5 -1.5 -2.0 -2.0 -2.0 -2.0 -2.0 -2.0 -2.0 -2.0 -2.0 -2.0 -2.0 -2.5 -2.5 -2.5 -2.5 -2.5 -2.5 -2.5 -2.5 -2.5 -2.5 -2.5 2.5 2.5 2.0 2.0 2.5

2.5 1.0 1.5 1.0 1.0 1.0 0.5 1.0 -3.5E-231.0 -0.5 1.0 -1.0 1.0 -1.5 1.0 -2.0 1.0 -2.5 1.0 2.0 1.0 2.5 1.0 1.5 1.0 1.0 1.0 0.5 1.0 -5.3E-241.0 -0.5 1.0 -1.0 1.0 -1.5 1.0 -2.0 1.0 -2.5 1.0 2.0 1.0 2.5 1.0 1.5 1.0 1.0 1.0 0.5 1.0 -8.1E-251.0 -0.5 1.0 -1.0 1.0 -1.5 1.0 -2.0 1.0 -2.5 1.0 2.0 1.0 2.5 1.0 1.5 1.0 1.0 1.0 0.5 1.0 -9.3E-181.0 -0.5 1.0 -1.0 1.0 -1.5 1.0 -2.0 1.0 -2.5 1.0 2.0 1.0 2.5 1.0 1.5 1.0 1.0 1.0 0.5 1.0 -2.0E-181.0 -0.5 1.0 -1.0 1.0 -1.5 1.0 -2.0 1.0 -2.5 1.0 2.0 1.0 2.5 1.0 1.5 1.0 1.0 1.0 0.5 1.0 -1.0E-181.0 -0.5 1.0 -1.0 1.0 -1.5 1.0 -2.0 1.0 -2.5 1.0 2.5 2.0 2.0 2.0 2.0 2.0 2.5 2.0 1.5 2.0

-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1


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853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922

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2.0 2.5 2.0 2.5 2.0 2.5 2.0 2.5 2.0 2.5 2.0 2.5 2.0 2.5 2.0 2.5 2.0 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5

1.5 2.0 1.0 2.0 1.0 2.0 0.5 2.0 0.5 2.0 -6.0E-192.0 -1.2E-192.0 -0.5 2.0 -0.5 2.0 -1.0 2.0 -1.0 2.0 -1.5 2.0 -1.5 2.0 -2.0 2.0 -2.0 2.0 -2.5 2.0 -2.5 2.0 2.0 2.0 2.5 2.0 1.5 2.0 1.0 2.0 0.5 2.0 -2.1E-202.0 -0.5 2.0 -1.0 2.0 -1.5 2.0 -2.0 2.0 -2.5 2.0 2.0 2.0 2.5 2.0 1.5 2.0 1.0 2.0 0.5 2.0 -3.8E-212.0 -0.5 2.0 -1.0 2.0 -1.5 2.0 -2.0 2.0 -2.5 2.0 2.0 2.0 2.5 2.0 1.5 2.0 1.0 2.0 0.5 2.0 -6.7E-222.0 -0.5 2.0 -1.0 2.0 -1.5 2.0 -2.0 2.0 -2.5 2.0 2.0 2.0 2.5 2.0 1.5 2.0 1.0 2.0 0.5 2.0 -1.2E-222.0 -0.5 2.0 -1.0 2.0 -1.5 2.0 -2.0 2.0 -2.5 2.0 2.0 2.0 2.5 2.0 1.5 2.0 1.0 2.0 0.5 2.0 -2.0E-232.0 -0.5 2.0 -1.0 2.0 -1.5 2.0

-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1


149


150

923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992

-0.5 -0.5 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0 -1.5 -1.5 -1.5 -1.5 -1.5 -1.5 -1.5 -1.5 -1.5 -1.5 -1.5 -2.0 -2.0 -2.0 -2.0 -2.0 -2.0 -2.0 -2.0 -2.0 -2.0 -2.0 -2.5 -2.5 -2.5 -2.5 -2.5 -2.5 -2.5 -2.5 -2.5 -2.5 -2.5 2.5 2.5 2.0 2.0 2.5 2.0 2.5 2.0 2.5 2.0 2.5 2.0 2.5 2.0 2.5 2.0 2.5 2.0 2.5 2.0 2.5 2.0 1.5 1.5

-2.0 2.0 -2.5 2.0 2.0 2.0 2.5 2.0 1.5 2.0 1.0 2.0 0.5 2.0 -1.4E-182.0 -0.5 2.0 -1.0 2.0 -1.5 2.0 -2.0 2.0 -2.5 2.0 2.0 2.0 2.5 2.0 1.5 2.0 1.0 2.0 0.5 2.0 -1.3E-172.0 -0.5 2.0 -1.0 2.0 -1.5 2.0 -2.0 2.0 -2.5 2.0 2.0 2.0 2.5 2.0 1.5 2.0 1.0 2.0 0.5 2.0 -4.1E-182.0 -0.5 2.0 -1.0 2.0 -1.5 2.0 -2.0 2.0 -2.5 2.0 2.0 2.0 2.5 2.0 1.5 2.0 1.0 2.0 0.5 2.0 -2.5E-182.0 -0.5 2.0 -1.0 2.0 -1.5 2.0 -2.0 2.0 -2.5 2.0 2.5 3.0 2.0 3.0 2.0 3.0 2.5 3.0 1.5 3.0 1.5 3.0 1.0 3.0 1.0 3.0 0.5 3.0 0.5 3.0 -6.7E-193.0 -1.5E-193.0 -0.5 3.0 -0.5 3.0 -1.0 3.0 -1.0 3.0 -1.5 3.0 -1.5 3.0 -2.0 3.0 -2.0 3.0 -2.5 3.0 -2.5 3.0 2.0 3.0 2.5 3.0

-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1


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993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062

Altair Engineering

1.5 1.5 3.0 1.5 1.0 3.0 1.5 0.5 3.0 1.5 -3.1E-203.0 1.5 -0.5 3.0 1.5 -1.0 3.0 1.5 -1.5 3.0 1.5 -2.0 3.0 1.5 -2.5 3.0 1.0 2.0 3.0 1.0 2.5 3.0 1.0 1.5 3.0 1.0 1.0 3.0 1.0 0.5 3.0 1.0 -6.2E-213.0 1.0 -0.5 3.0 1.0 -1.0 3.0 1.0 -1.5 3.0 1.0 -2.0 3.0 1.0 -2.5 3.0 0.5 2.0 3.0 0.5 2.5 3.0 0.5 1.5 3.0 0.5 1.0 3.0 0.5 0.5 3.0 0.5 -1.2E-213.0 0.5 -0.5 3.0 0.5 -1.0 3.0 0.5 -1.5 3.0 0.5 -2.0 3.0 0.5 -2.5 3.0 1.50E-322.0 3.0 3.80E-332.5 3.0 2.67E-331.5 3.0 4.07E-341.0 3.0 6.20E-350.5 3.0 6.35E-32-2.3E-223.0 1.29E-31-0.5 3.0 -4.0E-32-1.0 3.0 -5.5E-32-1.5 3.0 -3.2E-32-2.0 3.0 -4.8E-33-2.5 3.0 -0.5 2.0 3.0 -0.5 2.5 3.0 -0.5 1.5 3.0 -0.5 1.0 3.0 -0.5 0.5 3.0 -0.5 -2.2E-193.0 -0.5 -0.5 3.0 -0.5 -1.0 3.0 -0.5 -1.5 3.0 -0.5 -2.0 3.0 -0.5 -2.5 3.0 -1.0 2.0 3.0 -1.0 2.5 3.0 -1.0 1.5 3.0 -1.0 1.0 3.0 -1.0 0.5 3.0 -1.0 -2.6E-183.0 -1.0 -0.5 3.0 -1.0 -1.0 3.0 -1.0 -1.5 3.0 -1.0 -2.0 3.0 -1.0 -2.5 3.0 -1.5 2.0 3.0 -1.5 2.5 3.0 -1.5 1.5 3.0 -1.5 1.0 3.0 -1.5 0.5 3.0 -1.5 -1.5E-173.0

-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1


151


152

1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100 1101 1102 1103 1104 1105 1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120 1121 1122 1123 1124 1125 1126 1127 1128 1129 1130 1131 1132

-1.5 -1.5 -1.5 -1.5 -1.5 -2.0 -2.0 -2.0 -2.0 -2.0 -2.0 -2.0 -2.0 -2.0 -2.0 -2.0 -2.5 -2.5 -2.5 -2.5 -2.5 -2.5 -2.5 -2.5 -2.5 -2.5 -2.5 2.5 2.5 2.0 2.0 2.5 2.0 2.5 2.0 2.5 2.0 2.5 2.0 2.5 2.0 2.5 2.0 2.5 2.0 2.5 2.0 2.5 2.0 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

-0.5 3.0 -1.0 3.0 -1.5 3.0 -2.0 3.0 -2.5 3.0 2.0 3.0 2.5 3.0 1.5 3.0 1.0 3.0 0.5 3.0 -5.3E-183.0 -0.5 3.0 -1.0 3.0 -1.5 3.0 -2.0 3.0 -2.5 3.0 2.0 3.0 2.5 3.0 1.5 3.0 1.0 3.0 0.5 3.0 -3.3E-183.0 -0.5 3.0 -1.0 3.0 -1.5 3.0 -2.0 3.0 -2.5 3.0 2.5 4.0 2.0 4.0 2.0 4.0 2.5 4.0 1.5 4.0 1.5 4.0 1.0 4.0 1.0 4.0 0.5 4.0 0.5 4.0 -1.0E-164.0 -1.8E-164.0 -0.5 4.0 -0.5 4.0 -1.0 4.0 -1.0 4.0 -1.5 4.0 -1.5 4.0 -2.0 4.0 -2.0 4.0 -2.5 4.0 -2.5 4.0 2.0 4.0 2.5 4.0 1.5 4.0 1.0 4.0 0.5 4.0 -3.1E-164.0 -0.5 4.0 -1.0 4.0 -1.5 4.0 -2.0 4.0 -2.5 4.0 2.0 4.0 2.5 4.0 1.5 4.0 1.0 4.0 0.5 4.0 -3.6E-164.0 -0.5 4.0 -1.0 4.0 -1.5 4.0 -2.0 4.0

-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1


Altair Engineering


1133 1134 1135 1136 1137 1138 1139 1140 1141 1142 1143 1144 1145 1146 1147 1148 1149 1150 1151 1152 1153 1154 1155 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 1169 1170 1171 1172 1173 1174 1175 1176 1177 1178 1179 1180 1181 1182 1183 1184 1185 1186 1187 1188 1189 1190 1191 1192 1193 1194 1195 1196 1197 1198 1199 1200 1201 1202

Altair Engineering

1.0 -2.5 4.0 0.5 2.0 4.0 0.5 2.5 4.0 0.5 1.5 4.0 0.5 1.0 4.0 0.5 0.5 4.0 0.5 -2.8E-164.0 0.5 -0.5 4.0 0.5 -1.0 4.0 0.5 -1.5 4.0 0.5 -2.0 4.0 0.5 -2.5 4.0 -1.7E-162.0 4.0 -2.3E-162.5 4.0 -2.7E-171.5 4.0 -3.0E-171.0 4.0 -7.0E-170.5 4.0 2.16E-17-1.4E-164.0 1.65E-16-0.5 4.0 3.53E-16-1.0 4.0 2.86E-16-1.5 4.0 -5.5E-17-2.0 4.0 -2.5E-16-2.5 4.0 -0.5 2.0 4.0 -0.5 2.5 4.0 -0.5 1.5 4.0 -0.5 1.0 4.0 -0.5 0.5 4.0 -0.5 -2.5E-174.0 -0.5 -0.5 4.0 -0.5 -1.0 4.0 -0.5 -1.5 4.0 -0.5 -2.0 4.0 -0.5 -2.5 4.0 -1.0 2.0 4.0 -1.0 2.5 4.0 -1.0 1.5 4.0 -1.0 1.0 4.0 -1.0 0.5 4.0 -1.0 9.63E-174.0 -1.0 -0.5 4.0 -1.0 -1.0 4.0 -1.0 -1.5 4.0 -1.0 -2.0 4.0 -1.0 -2.5 4.0 -1.5 2.0 4.0 -1.5 2.5 4.0 -1.5 1.5 4.0 -1.5 1.0 4.0 -1.5 0.5 4.0 -1.5 2.14E-164.0 -1.5 -0.5 4.0 -1.5 -1.0 4.0 -1.5 -1.5 4.0 -1.5 -2.0 4.0 -1.5 -2.5 4.0 -2.0 2.0 4.0 -2.0 2.5 4.0 -2.0 1.5 4.0 -2.0 1.0 4.0 -2.0 0.5 4.0 -2.0 1.84E-164.0 -2.0 -0.5 4.0 -2.0 -1.0 4.0 -2.0 -1.5 4.0 -2.0 -2.0 4.0 -2.0 -2.5 4.0 -2.5 2.0 4.0 -2.5 2.5 4.0 -2.5 1.5 4.0

-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1


153


154

1203 1204 1205 1206 1207 1208 1209 1210 1211 1212 1213 1214 1215 1216 1217 1218 1219 1220 1221 1222 1223 1224 1225 1226 1227 1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264 1265 1266 1267 1268 1269 1270 1271 1272

-2.5 1.0 4.0 -2.5 0.5 4.0 -2.5 1.10E-164.0 -2.5 -0.5 4.0 -2.5 -1.0 4.0 -2.5 -1.5 4.0 -2.5 -2.0 4.0 -2.5 -2.5 4.0 2.5 2.5 5.0 2.5 2.0 5.0 2.0 2.0 5.0 2.0 2.5 5.0 2.5 1.5 5.0 2.0 1.5 5.0 2.5 1.0 5.0 2.0 1.0 5.0 2.5 0.5 5.0 2.0 0.5 5.0 2.5 -1.8E-165.0 2.0 -2.4E-165.0 2.5 -0.5 5.0 2.0 -0.5 5.0 2.5 -1.0 5.0 2.0 -1.0 5.0 2.5 -1.5 5.0 2.0 -1.5 5.0 2.5 -2.0 5.0 2.0 -2.0 5.0 2.5 -2.5 5.0 2.0 -2.5 5.0 1.5 2.0 5.0 1.5 2.5 5.0 1.5 1.5 5.0 1.5 1.0 5.0 1.5 0.5 5.0 1.5 -2.8E-165.0 1.5 -0.5 5.0 1.5 -1.0 5.0 1.5 -1.5 5.0 1.5 -2.0 5.0 1.5 -2.5 5.0 1.0 2.0 5.0 1.0 2.5 5.0 1.0 1.5 5.0 1.0 1.0 5.0 1.0 0.5 5.0 1.0 -3.0E-165.0 1.0 -0.5 5.0 1.0 -1.0 5.0 1.0 -1.5 5.0 1.0 -2.0 5.0 1.0 -2.5 5.0 0.5 2.0 5.0 0.5 2.5 5.0 0.5 1.5 5.0 0.5 1.0 5.0 0.5 0.5 5.0 0.5 -2.4E-165.0 0.5 -0.5 5.0 0.5 -1.0 5.0 0.5 -1.5 5.0 0.5 -2.0 5.0 0.5 -2.5 5.0 -2.4E-162.0 5.0 -2.4E-162.5 5.0 -1.4E-161.5 5.0 -1.1E-161.0 5.0 -6.8E-170.5 5.0 -2.1E-17-1.4E-165.0 1.13E-16-0.5 5.0

-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1


Altair Engineering


1273 1274 1275 1276 1277 1278 1279 1280 1281 1282 1283 1284 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294 1295 1296 1297 1298 1299 1300 1301 1302 1303 1304 1305 1306 1307 1308 1309 1310 1311 1312 1313 1314 1315 1316 1317 1318 1319 1320 1321 1322 1323 1324 1325 1326 1327 1328 1329 1330 1331 1332 1333 1334 1335 1336 1337 1338 1339 1340 1341 1342

Altair Engineering

2.64E-16-1.0 5.0 2.07E-16-1.5 5.0 -8.9E-18-2.0 5.0 -1.0E-16-2.5 5.0 -0.5 2.0 5.0 -0.5 2.5 5.0 -0.5 1.5 5.0 -0.5 1.0 5.0 -0.5 0.5 5.0 -0.5 -2.9E-175.0 -0.5 -0.5 5.0 -0.5 -1.0 5.0 -0.5 -1.5 5.0 -0.5 -2.0 5.0 -0.5 -2.5 5.0 -1.0 2.0 5.0 -1.0 2.5 5.0 -1.0 1.5 5.0 -1.0 1.0 5.0 -1.0 0.5 5.0 -1.0 8.84E-175.0 -1.0 -0.5 5.0 -1.0 -1.0 5.0 -1.0 -1.5 5.0 -1.0 -2.0 5.0 -1.0 -2.5 5.0 -1.5 2.0 5.0 -1.5 2.5 5.0 -1.5 1.5 5.0 -1.5 1.0 5.0 -1.5 0.5 5.0 -1.5 1.34E-165.0 -1.5 -0.5 5.0 -1.5 -1.0 5.0 -1.5 -1.5 5.0 -1.5 -2.0 5.0 -1.5 -2.5 5.0 -2.0 2.0 5.0 -2.0 2.5 5.0 -2.0 1.5 5.0 -2.0 1.0 5.0 -2.0 0.5 5.0 -2.0 1.09E-165.0 -2.0 -0.5 5.0 -2.0 -1.0 5.0 -2.0 -1.5 5.0 -2.0 -2.0 5.0 -2.0 -2.5 5.0 -2.5 2.0 5.0 -2.5 2.5 5.0 -2.5 1.5 5.0 -2.5 1.0 5.0 -2.5 0.5 5.0 -2.5 5.75E-175.0 -2.5 -0.5 5.0 -2.5 -1.0 5.0 -2.5 -1.5 5.0 -2.5 -2.0 5.0 -2.5 -2.5 5.0 2.0 2.0 0.0 2.0 1.5 0.0 2.0 1.0 0.0 2.0 0.5 0.0 2.0 -2.2E-180.0 2.0 -0.5 0.0 2.0 -1.0 0.0 2.0 -1.5 0.0 2.0 -2.0 0.0 1.5 2.0 0.0 1.5 1.5 0.0

-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1


155


156

1343 1344 1345 1346 1347 1348 1349 1350 1351 1352 1353 1354 1355 1356 1357 1358 1359 1360 1361 1362 1363 1364 1365 1366 1367 1368 1369 1370 1371 1372 1373 1374 1375 1376 1377 1378 1379 1380 1381 1382 1383 1384 1385 1386 1387 1388 1389 1390 1391 1392 1393 1394 1395 1396 1397 1398 1399 1400 1401 1402 1403 1404 1405 1406 1407 1408 1409 1410 1411 1412

1.5 1.0 0.0 1.5 0.5 0.0 1.5 -1.9E-180.0 1.5 -0.5 0.0 1.5 -1.0 0.0 1.5 -1.5 0.0 1.5 -2.0 0.0 1.0 2.0 0.0 1.0 1.5 0.0 1.0 1.0 0.0 1.0 0.5 0.0 1.0 -1.9E-180.0 1.0 -0.5 0.0 1.0 -1.0 0.0 1.0 -1.5 0.0 1.0 -2.0 0.0 0.5 2.0 0.0 0.5 1.5 0.0 0.5 1.0 0.0 0.5 0.5 0.0 0.5 -1.9E-180.0 0.5 -0.5 0.0 0.5 -1.0 0.0 0.5 -1.5 0.0 0.5 -2.0 0.0 -2.8E-182.0 0.0 -2.8E-181.5 0.0 -2.5E-181.0 0.0 -2.8E-180.5 0.0 -3.1E-18-1.7E-180.0 -2.8E-18-0.5 0.0 -3.1E-18-1.0 0.0 -1.9E-18-1.5 0.0 -2.8E-18-2.0 0.0 -0.5 2.0 0.0 -0.5 1.5 0.0 -0.5 1.0 0.0 -0.5 0.5 0.0 -0.5 -1.7E-180.0 -0.5 -0.5 0.0 -0.5 -1.0 0.0 -0.5 -1.5 0.0 -0.5 -2.0 0.0 -1.0 2.0 0.0 -1.0 1.5 0.0 -1.0 1.0 0.0 -1.0 0.5 0.0 -1.0 -1.9E-180.0 -1.0 -0.5 0.0 -1.0 -1.0 0.0 -1.0 -1.5 0.0 -1.0 -2.0 0.0 -1.5 2.0 0.0 -1.5 1.5 0.0 -1.5 1.0 0.0 -1.5 0.5 0.0 -1.5 -1.7E-180.0 -1.5 -0.5 0.0 -1.5 -1.0 0.0 -1.5 -1.5 0.0 -1.5 -2.0 0.0 -2.0 2.0 0.0 -2.0 1.5 0.0 -2.0 1.0 0.0 -2.0 0.5 0.0 -2.0 -2.2E-180.0 -2.0 -0.5 0.0 -2.0 -1.0 0.0 -2.0 -1.5 0.0 -2.0 -2.0 0.0


Altair Engineering


1413 1414 1415 1416 1417 1418 1419 1420 1421 1422 1423 1424 1425 1426 1427 1428 1429 1430 1431 1432 1433 1434 1435 1436 1437 1438 1439 1440 1441 1442 1443 1444 1445 1446 1447 1448 1449 1450 1451 1452 1453 1454 1455 1456 1457 1458 1459 1460 1461 1462 1463 1464 1465 1466 1467 1468 1469 1470 1471 1472 1473 1474 1475 1476 1477 1478 1479 1480 1481 1482

Altair Engineering

2.4964642.0 0.004472 2.4964641.5 0.004472 2.4964641.0 0.004472 2.4964640.5 0.004472 2.496464-2.6E-180.004472 2.496464-0.5 0.004472 2.496464-1.0 0.004472 2.496464-1.5 0.004472 2.496464-2.0 0.004472 -2.496462.0 0.004472 -2.496461.5 0.004472 -2.496461.0 0.004472 -2.496460.5 0.004472 -2.49646-2.6E-180.004472 -2.49646-0.5 0.004472 -2.49646-1.0 0.004472 -2.49646-1.5 0.004472 -2.49646-2.0 0.004472 2.4961522.4961520.005963 2.496152-2.496150.005963 2.0 2.4964640.004472 2.0 -2.496460.004472 1.5 2.4964640.004472 1.5 -2.496460.004472 1.0 2.4964640.004472 1.0 -2.496460.004472 0.5 2.4964640.004472 0.5 -2.496460.004472 -2.6E-182.4964640.004472 -2.6E-18-2.496460.004472 -0.5 2.4964640.004472 -0.5 -2.496460.004472 -1.0 2.4964640.004472 -1.0 -2.496460.004472 -1.5 2.4964640.004472 -1.5 -2.496460.004472 -2.0 2.4964640.004472 -2.0 -2.496460.004472 -2.496152.4961520.005963 -2.49615-2.496150.005963 -2.49615-2.496154.994037 -2.49646-2.0 4.995528 -2.49646-1.5 4.995528 -2.49646-1.0 4.995528 -2.49646-0.5 4.995528 -2.496465.58E-174.995528 -2.496460.5 4.995528 -2.496461.0 4.995528 -2.496461.5 4.995528 -2.496152.4961524.994037 -2.496462.0 4.995528 -2.0 -2.496464.995528 -2.0 -2.0 5.0 -2.0 -1.5 5.0 -2.0 -1.0 5.0 -2.0 -0.5 5.0 -2.0 1.05E-165.0 -2.0 0.5 5.0 -2.0 1.0 5.0 -2.0 1.5 5.0 -2.0 2.4964644.995528 -2.0 2.0 5.0 -1.5 -2.496464.995528 -1.5 -2.0 5.0 -1.5 -1.5 5.0 -1.5 -1.0 5.0 -1.5 -0.5 5.0 -1.5 1.32E-165.0 -1.5 0.5 5.0 -1.5 1.0 5.0


157


158

1483 1484 1485 1486 1487 1488 1489 1493 1494 1495 1496 1497 1498 1499 1500 1504 1505 1506 1507 1508 1509 1510 1511 1515 1516 1517 1518 1519 1520 1521 1522 1526 1527 1528 1529 1530 1531 1532 1533 1534 1535 1536 1537 1538 1539 1540 1541 1542 1543 1544 1545 1546 1547 1548 1549 1550 1551 1552 1553 1554 1555 1556 1557 1558 1559 1560 1561 1562 1563 1564

-1.5 1.5 5.0 -1.5 2.4964644.995528 -1.5 2.0 5.0 -1.0 -2.496464.995528 -1.0 -2.0 5.0 -1.0 -1.5 5.0 -1.0 -1.0 5.0 -1.0 1.0 5.0 -1.0 1.5 5.0 -1.0 2.4964644.995528 -1.0 2.0 5.0 -0.5 -2.496464.995528 -0.5 -2.0 5.0 -0.5 -1.5 5.0 -0.5 -1.0 5.0 -0.5 1.0 5.0 -0.5 1.5 5.0 -0.5 2.4964644.995528 -0.5 2.0 5.0 -1.0E-16-2.496464.995528 -1.1E-17-2.0 5.0 2.04E-16-1.5 5.0 2.61E-16-1.0 5.0 -1.2E-161.0 5.0 -1.4E-161.5 5.0 -2.4E-162.4964644.995528 -2.4E-162.0 5.0 0.5 -2.496464.995528 0.5 -2.0 5.0 0.5 -1.5 5.0 0.5 -1.0 5.0 0.5 1.0 5.0 0.5 1.5 5.0 0.5 2.4964644.995528 0.5 2.0 5.0 1.0 -2.496464.995528 1.0 -2.0 5.0 1.0 -1.5 5.0 1.0 -1.0 5.0 1.0 -0.5 5.0 1.0 -3.0E-165.0 1.0 0.5 5.0 1.0 1.0 5.0 1.0 1.5 5.0 1.0 2.4964644.995528 1.0 2.0 5.0 1.5 -2.496464.995528 1.5 -2.0 5.0 1.5 -1.5 5.0 1.5 -1.0 5.0 1.5 -0.5 5.0 1.5 -2.8E-165.0 1.5 0.5 5.0 1.5 1.0 5.0 1.5 1.5 5.0 1.5 2.4964644.995528 1.5 2.0 5.0 2.0 -2.496464.995528 2.496152-2.496154.994037 2.0 -2.0 5.0 2.496464-2.0 4.995528 2.0 -1.5 5.0 2.496464-1.5 4.995528 2.0 -1.0 5.0 2.496464-1.0 4.995528 2.0 -0.5 5.0 2.496464-0.5 4.995528 2.0 -2.4E-165.0 2.496464-1.8E-164.995528 2.0 0.5 5.0


Altair Engineering


1565 1566 1567 1568 1569 1570 1571 1572 1573 1574 1575 1576 1577 1578 1579 1580 1581 1582 1583 1584 1585 1586 1587 1588 1589 1590 1591 1592 1593 1594 1595 1596 1597 1598 1599 1600 1601 1602 1603 1604 1605 1606 1607 1608 1609 1610 1611 1612 1613 1614 1615 1616 1617 1618 1619 1620 1621 1622 1623 1624 1625 1626 1627 1628 1629 1630 1631 1632 1633 1634

Altair Engineering

2.4964640.5 4.995528 2.0 1.0 5.0 2.4964641.0 4.995528 2.0 1.5 5.0 2.4964641.5 4.995528 2.0 2.4964644.995528 2.0 2.0 5.0 2.4964642.0 4.995528 2.4961522.4961524.994037 -2.49776-2.497764.0 -2.5 -2.0 4.0 -2.5 -1.5 4.0 -2.5 -1.0 4.0 -2.5 -0.5 4.0 -2.5 1.07E-164.0 -2.5 0.5 4.0 -2.5 1.0 4.0 -2.5 1.5 4.0 -2.497762.4977644.0 -2.5 2.0 4.0 -2.0 -2.5 4.0 -2.0 2.5 4.0 -1.5 -2.5 4.0 -1.5 2.5 4.0 -1.0 -2.5 4.0 -1.0 2.5 4.0 -0.5 -2.5 4.0 -0.5 2.5 4.0 -2.5E-16-2.5 4.0 -2.3E-162.5 4.0 0.5 -2.5 4.0 0.5 2.5 4.0 1.0 -2.5 4.0 1.0 2.5 4.0 1.5 -2.5 4.0 1.5 2.5 4.0 2.0 -2.5 4.0 2.497764-2.497764.0 2.5 -2.0 4.0 2.5 -1.5 4.0 2.5 -1.0 4.0 2.5 -0.5 4.0 2.5 -1.0E-164.0 2.5 0.5 4.0 2.5 1.0 4.0 2.5 1.5 4.0 2.0 2.5 4.0 2.5 2.0 4.0 2.4977642.4977644.0 -2.49776-2.497763.0 -2.5 -2.0 3.0 -2.5 -1.5 3.0 -2.5 -1.0 3.0 -2.5 -0.5 3.0 -2.5 -5.4E-183.0 -2.5 0.5 3.0 -2.5 1.0 3.0 -2.5 1.5 3.0 -2.497762.4977643.0 -2.5 2.0 3.0 -2.0 -2.5 3.0 -2.0 2.5 3.0 -1.5 -2.5 3.0 -1.5 2.5 3.0 -1.0 -2.5 3.0 -1.0 2.5 3.0 -0.5 -2.5 3.0 -0.5 2.5 3.0 -2.9E-18-2.5 3.0 -3.1E-182.5 3.0


159


160

1635 1636 1637 1638 1639 1640 1641 1642 1643 1644 1645 1646 1647 1648 1649 1650 1651 1652 1653 1654 1655 1656 1657 1658 1659 1660 1661 1662 1663 1664 1665 1666 1667 1668 1669 1670 1671 1672 1673 1674 1675 1676 1677 1678 1679 1680 1681 1682 1683 1684 1685 1686 1687 1688 1689 1690 1691 1692 1693 1694 1695 1696 1697 1698 1699 1700 1701 1702 1703 1704

0.5 -2.5 3.0 0.5 2.5 3.0 1.0 -2.5 3.0 1.0 2.5 3.0 1.5 -2.5 3.0 1.5 2.5 3.0 2.0 -2.5 3.0 2.497764-2.497763.0 2.5 -2.0 3.0 2.5 -1.5 3.0 2.5 -1.0 3.0 2.5 -0.5 3.0 2.5 -3.4E-183.0 2.5 0.5 3.0 2.5 1.0 3.0 2.5 1.5 3.0 2.0 2.5 3.0 2.5 2.0 3.0 2.4977642.4977643.0 -2.49776-2.497762.0 -2.5 -2.0 2.0 -2.5 -1.5 2.0 -2.5 -1.0 2.0 -2.5 -0.5 2.0 -2.5 -5.0E-182.0 -2.5 0.5 2.0 -2.5 1.0 2.0 -2.5 1.5 2.0 -2.497762.4977642.0 -2.5 2.0 2.0 -2.0 -2.5 2.0 -2.0 2.5 2.0 -1.5 -2.5 2.0 -1.5 2.5 2.0 -1.0 -2.5 2.0 -1.0 2.5 2.0 -0.5 -2.5 2.0 -0.5 2.5 2.0 -2.5E-18-2.5 2.0 -2.5E-182.5 2.0 0.5 -2.5 2.0 0.5 2.5 2.0 1.0 -2.5 2.0 1.0 2.5 2.0 1.5 -2.5 2.0 1.5 2.5 2.0 2.0 -2.5 2.0 2.497764-2.497762.0 2.5 -2.0 2.0 2.5 -1.5 2.0 2.5 -1.0 2.0 2.5 -0.5 2.0 2.5 -3.1E-182.0 2.5 0.5 2.0 2.5 1.0 2.0 2.5 1.5 2.0 2.0 2.5 2.0 2.5 2.0 2.0 2.4977642.4977642.0 -2.49776-2.497761.0 -2.5 -2.0 1.0 -2.5 -1.5 1.0 -2.5 -1.0 1.0 -2.5 -0.5 1.0 -2.5 -3.5E-181.0 -2.5 0.5 1.0 -2.5 1.0 1.0 -2.5 1.5 1.0 -2.497762.4977641.0 -2.5 2.0 1.0


Altair Engineering

GRID 1705 -2.0 -2.5 1.0 GRID 1706 -2.0 2.5 1.0 GRID 1707 -1.5 -2.5 1.0 GRID 1708 -1.5 2.5 1.0 GRID 1709 -1.0 -2.5 1.0 GRID 1710 -1.0 2.5 1.0 GRID 1711 -0.5 -2.5 1.0 GRID 1712 -0.5 2.5 1.0 GRID 1713 -2.5E-18-2.5 1.0 GRID 1714 -2.5E-182.5 1.0 GRID 1715 0.5 -2.5 1.0 GRID 1716 0.5 2.5 1.0 GRID 1717 1.0 -2.5 1.0 GRID 1718 1.0 2.5 1.0 GRID 1719 1.5 -2.5 1.0 GRID 1720 1.5 2.5 1.0 GRID 1721 2.0 -2.5 1.0 GRID 1722 2.497764-2.497761.0 GRID 1723 2.5 -2.0 1.0 GRID 1724 2.5 -1.5 1.0 GRID 1725 2.5 -1.0 1.0 GRID 1726 2.5 -0.5 1.0 GRID 1727 2.5 -2.9E-181.0 GRID 1728 2.5 0.5 1.0 GRID 1729 2.5 1.0 1.0 GRID 1730 2.5 1.5 1.0 GRID 1731 2.0 2.5 1.0 GRID 1732 2.5 2.0 1.0 GRID 1733 2.4977642.4977641.0 GRID 1734 -0.25 3.33E-165.0 $$ $$ SPOINT Data $$ $$ $$------------------------------------------------------------------------------$ $$ Group Definitions $ $$------------------------------------------------------------------------------$ $$ $$ RBE2 Elements - Multiple dependent nodes $$ RBE2 1553 1734 123456 1478 1479 1480 1481 1482+ + 1489 1493 1500 1504 1511 1515 1522 1526+ + 1533 1534 1535 1536 1537 $ $HMMOVE 6 $ 1553 $ $ CQUAD4 Elements $ CQUAD4 1101 4 1332 1341 1342 1333 CQUAD4 1102 4 1333 1342 1343 1334 CQUAD4 1103 4 1334 1343 1344 1335 CQUAD4 1104 4 1335 1344 1345 1336 CQUAD4 1105 4 1336 1345 1346 1337 CQUAD4 1106 4 1337 1346 1347 1338 CQUAD4 1107 4 1338 1347 1348 1339 CQUAD4 1108 4 1339 1348 1349 1340 CQUAD4 1109 4 1341 1350 1351 1342 CQUAD4 1110 4 1342 1351 1352 1343 CQUAD4 1111 4 1343 1352 1353 1344 CQUAD4 1112 4 1344 1353 1354 1345 CQUAD4 1113 4 1345 1354 1355 1346 CQUAD4 1114 4 1346 1355 1356 1347 CQUAD4 1115 4 1347 1356 1357 1348 CQUAD4 1116 4 1348 1357 1358 1349 CQUAD4 1117 4 1350 1359 1360 1351 CQUAD4 1118 4 1351 1360 1361 1352 CQUAD4 1119 4 1352 1361 1362 1353 CQUAD4 1120 4 1353 1362 1363 1354 CQUAD4 1121 4 1354 1363 1364 1355

Altair Engineering


161

CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4

162

1122 1123 1124 1125 1126 1127 1128 1129 1130 1131 1132 1133 1134 1135 1136 1137 1138 1139 1140 1141 1142 1143 1144 1145 1146 1147 1148 1149 1150 1151 1152 1153 1154 1155 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 1169 1170 1171 1172 1173 1174 1175 1176 1177 1178 1179 1180 1181 1182 1183 1184 1185 1186 1187 1188 1189 1190 1191

4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

1355 1356 1357 1359 1360 1361 1362 1363 1364 1365 1366 1368 1369 1370 1371 1372 1373 1374 1375 1377 1378 1379 1380 1381 1382 1383 1384 1386 1387 1388 1389 1390 1391 1392 1393 1395 1396 1397 1398 1399 1400 1401 1402 1413 1414 1415 1416 1417 1418 1419 1420 1404 1405 1406 1407 1408 1409 1410 1411 1431 1433 1421 1340 1435 1349 1437 1358 1439 1367 1441

1364 1365 1366 1368 1369 1370 1371 1372 1373 1374 1375 1377 1378 1379 1380 1381 1382 1383 1384 1386 1387 1388 1389 1390 1391 1392 1393 1395 1396 1397 1398 1399 1400 1401 1402 1404 1405 1406 1407 1408 1409 1410 1411 1332 1333 1334 1335 1336 1337 1338 1339 1422 1423 1424 1425 1426 1427 1428 1429 1433 1435 1340 1349 1437 1358 1439 1367 1441 1376 1443

1365 1366 1367 1369 1370 1371 1372 1373 1374 1375 1376 1378 1379 1380 1381 1382 1383 1384 1385 1387 1388 1389 1390 1391 1392 1393 1394 1396 1397 1398 1399 1400 1401 1402 1403 1405 1406 1407 1408 1409 1410 1411 1412 1333 1334 1335 1336 1337 1338 1339 1340 1423 1424 1425 1426 1427 1428 1429 1430 1332 1341 1434 1436 1350 1438 1359 1440 1368 1442 1377

1356 1357 1358 1360 1361 1362 1363 1364 1365 1366 1367 1369 1370 1371 1372 1373 1374 1375 1376 1378 1379 1380 1381 1382 1383 1384 1385 1387 1388 1389 1390 1391 1392 1393 1394 1396 1397 1398 1399 1400 1401 1402 1403 1414 1415 1416 1417 1418 1419 1420 1421 1405 1406 1407 1408 1409 1410 1411 1412 1413 1332 1432 1434 1341 1436 1350 1438 1359 1440 1368


Altair Engineering


1192 1193 1194 1195 1196 1197 1198 1199 1200 1201 1202 1203 1204 1205 1206 1207 1208 1209 1210 1211 1212 1213 1214 1215 1216 1217 1218 1219 1220 1221 1222 1223 1224 1225 1226 1227 1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261

Altair Engineering

4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

1376 1443 1385 1445 1394 1447 1403 1449 1412 1431 1413 1414 1433 1415 1416 1417 1418 1419 1420 1421 1432 1434 1436 1435 1438 1437 1440 1439 1442 1441 1444 1443 1446 1445 1448 1447 1450 1449 1452 1451 1422 1423 1424 1425 1426 1427 1428 1429 1430 1732 1733 1730 1731 1729 1728 1727 1726 1725 1724 1723 1722 1721 1719 1720 1717 1718 1715 1716 1713 1714

1385 1445 1394 1447 1403 1449 1412 1451 1430 1733 1732 1730 1731 1729 1728 1727 1726 1725 1724 1723 1722 1721 1719 1720 1717 1718 1715 1716 1713 1714 1711 1712 1709 1710 1707 1708 1705 1706 1694 1703 1704 1702 1701 1700 1699 1698 1697 1696 1695 1692 1693 1690 1691 1689 1688 1687 1686 1685 1684 1683 1682 1681 1679 1680 1677 1678 1675 1676 1673 1674

1444 1386 1446 1395 1448 1404 1450 1422 1452 1731 1733 1732 1720 1730 1729 1728 1727 1726 1725 1724 1723 1722 1721 1718 1719 1716 1717 1714 1715 1712 1713 1710 1711 1708 1709 1706 1707 1703 1705 1704 1702 1701 1700 1699 1698 1697 1696 1695 1694 1693 1691 1692 1680 1690 1689 1688 1687 1686 1685 1684 1683 1682 1681 1678 1679 1676 1677 1674 1675 1672

1442 1377 1444 1386 1446 1395 1448 1404 1450 1433 1431 1413 1435 1414 1415 1416 1417 1418 1419 1420 1421 1432 1434 1437 1436 1439 1438 1441 1440 1443 1442 1445 1444 1447 1446 1449 1448 1451 1450 1422 1423 1424 1425 1426 1427 1428 1429 1430 1452 1733 1731 1732 1720 1730 1729 1728 1727 1726 1725 1724 1723 1722 1721 1718 1719 1716 1717 1714 1715 1712


163


164

1262 1263 1264 1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280 1281 1282 1283 1284 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294 1295 1296 1297 1298 1299 1300 1301 1302 1303 1304 1305 1306 1307 1308 1309 1310 1311 1312 1313 1314 1315 1316 1317 1318 1319 1320 1321 1322 1323 1324 1325 1326 1327 1328 1329 1330 1331

4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

1711 1712 1709 1710 1707 1708 1705 1706 1694 1703 1704 1702 1701 1700 1699 1698 1697 1696 1695 1692 1693 1690 1691 1689 1688 1687 1686 1685 1684 1683 1682 1681 1679 1680 1677 1678 1675 1676 1673 1674 1671 1672 1669 1670 1667 1668 1665 1666 1654 1663 1664 1662 1661 1660 1659 1658 1657 1656 1655 1652 1653 1650 1651 1649 1648 1647 1646 1645 1644 1643

1671 1672 1669 1670 1667 1668 1665 1666 1654 1663 1664 1662 1661 1660 1659 1658 1657 1656 1655 1652 1653 1650 1651 1649 1648 1647 1646 1645 1644 1643 1642 1641 1639 1640 1637 1638 1635 1636 1633 1634 1631 1632 1629 1630 1627 1628 1625 1626 1614 1623 1624 1622 1621 1620 1619 1618 1617 1616 1615 1612 1613 1610 1611 1609 1608 1607 1606 1605 1604 1603

1673 1670 1671 1668 1669 1666 1667 1663 1665 1664 1662 1661 1660 1659 1658 1657 1656 1655 1654 1653 1651 1652 1640 1650 1649 1648 1647 1646 1645 1644 1643 1642 1641 1638 1639 1636 1637 1634 1635 1632 1633 1630 1631 1628 1629 1626 1627 1623 1625 1624 1622 1621 1620 1619 1618 1617 1616 1615 1614 1613 1611 1612 1600 1610 1609 1608 1607 1606 1605 1604

1713 1710 1711 1708 1709 1706 1707 1703 1705 1704 1702 1701 1700 1699 1698 1697 1696 1695 1694 1693 1691 1692 1680 1690 1689 1688 1687 1686 1685 1684 1683 1682 1681 1678 1679 1676 1677 1674 1675 1672 1673 1670 1671 1668 1669 1666 1667 1663 1665 1664 1662 1661 1660 1659 1658 1657 1656 1655 1654 1653 1651 1652 1640 1650 1649 1648 1647 1646 1645 1644


Altair Engineering


1332 1333 1334 1335 1336 1337 1338 1339 1340 1341 1342 1343 1344 1345 1346 1347 1348 1349 1350 1351 1352 1353 1354 1355 1356 1357 1358 1359 1360 1361 1362 1363 1364 1365 1366 1367 1368 1369 1370 1371 1372 1373 1374 1375 1376 1377 1378 1379 1380 1381 1382 1383 1384 1385 1386 1387 1388 1389 1390 1391 1392 1393 1394 1395 1396 1397 1398 1399 1400 1401

Altair Engineering

4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

1642 1641 1639 1640 1637 1638 1635 1636 1633 1634 1631 1632 1629 1630 1627 1628 1625 1626 1614 1623 1624 1622 1621 1620 1619 1618 1617 1616 1615 1612 1613 1610 1572 1611 1569 1609 1567 1608 1565 1607 1563 1606 1561 1605 1559 1604 1557 1603 1555 1602 1601 1553 1599 1571 1568 1600 1566 1564 1562 1560 1558 1556 1554 1552 1597 1551 1549 1598 1548 1547

1602 1601 1599 1600 1597 1598 1595 1596 1593 1594 1591 1592 1589 1590 1587 1588 1585 1586 1574 1583 1584 1582 1581 1580 1579 1578 1577 1576 1575 1572 1573 1569 1571 1570 1568 1567 1566 1565 1564 1563 1562 1561 1560 1559 1558 1557 1556 1555 1554 1553 1552 1552 1541 1551 1549 1550 1548 1547 1546 1545 1544 1543 1542 1541 1530 1540 1538 1539 1537 1536

1603 1602 1601 1598 1599 1596 1597 1594 1595 1592 1593 1590 1591 1588 1589 1586 1587 1583 1585 1584 1582 1581 1580 1579 1578 1577 1576 1575 1574 1573 1570 1572 1570 1550 1571 1569 1568 1567 1566 1565 1564 1563 1562 1561 1560 1559 1558 1557 1556 1555 1553 1554 1552 1550 1551 1539 1549 1548 1547 1546 1545 1544 1543 1542 1541 1539 1540 1528 1538 1537

1643 1642 1641 1638 1639 1636 1637 1634 1635 1632 1633 1630 1631 1628 1629 1626 1627 1623 1625 1624 1622 1621 1620 1619 1618 1617 1616 1615 1614 1613 1611 1612 1573 1600 1572 1610 1569 1609 1567 1608 1565 1607 1563 1606 1561 1605 1559 1604 1557 1603 1602 1555 1601 1570 1571 1598 1568 1566 1564 1562 1560 1558 1556 1554 1599 1550 1551 1596 1549 1548


165


166

1402 1403 1404 1405 1406 1407 1408 1409 1410 1411 1412 1417 1418 1419 1420 1421 1422 1423 1424 1429 1430 1431 1432 1433 1434 1435 1436 1441 1442 1443 1444 1445 1446 1447 1448 1453 1454 1455 1456 1457 1458 1459 1460 1465 1466 1467 1468 1469 1470 1471 1472 1473 1474 1475 1476 1477 1478 1479 1480 1481 1482 1483 1484 1485 1486 1487 1488 1489 1490 1491

4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

1546 1545 1544 1543 1542 1541 1595 1540 1538 1596 1537 1532 1531 1530 1593 1529 1527 1594 1526 1521 1520 1519 1591 1518 1516 1592 1515 1510 1509 1508 1589 1507 1505 1590 1504 1499 1498 1497 1587 1496 1494 1588 1493 1488 1487 1486 1585 1485 1483 1586 1482 1481 1480 1479 1478 1477 1476 1475 1574 1583 1474 1584 1472 1582 1471 1581 1470 1580 1469 1579

1535 1534 1533 1532 1531 1530 1519 1529 1527 1528 1526 1521 1520 1519 1508 1518 1516 1517 1515 1510 1509 1508 1497 1507 1505 1506 1504 1499 1498 1497 1486 1496 1494 1495 1493 1488 1487 1486 1475 1485 1483 1484 1482 1477 1476 1475 1464 1474 1472 1473 1471 1470 1469 1468 1467 1466 1465 1464 1453 1462 1463 1463 1461 1461 1460 1460 1459 1459 1458 1458

1536 1535 1534 1533 1532 1531 1530 1528 1529 1517 1527 1522 1521 1520 1519 1517 1518 1506 1516 1511 1510 1509 1508 1506 1507 1495 1505 1500 1499 1498 1497 1495 1496 1484 1494 1489 1488 1487 1486 1484 1485 1473 1483 1478 1477 1476 1475 1473 1474 1462 1472 1471 1470 1469 1468 1467 1466 1465 1464 1463 1462 1461 1463 1460 1461 1459 1460 1458 1459 1457

1547 1546 1545 1544 1543 1542 1597 1539 1540 1594 1538 1533 1532 1531 1595 1528 1529 1592 1527 1522 1521 1520 1593 1517 1518 1590 1516 1511 1510 1509 1591 1506 1507 1588 1505 1500 1499 1498 1589 1495 1496 1586 1494 1489 1488 1487 1587 1484 1485 1583 1483 1482 1481 1480 1479 1478 1477 1476 1585 1584 1473 1582 1474 1581 1472 1580 1471 1579 1470 1578


Altair Engineering

CQUAD4 1492 4 1468 CQUAD4 1493 4 1578 CQUAD4 1494 4 1467 CQUAD4 1495 4 1577 CQUAD4 1496 4 1466 CQUAD4 1497 4 1576 CQUAD4 1498 4 1465 CQUAD4 1499 4 1575 CQUAD4 1500 4 1464 $ $ CHEXA Elements: First Order $ CHEXA 601 1 100 + 729 728 CHEXA 602 1 82 + 732 731 CHEXA 603 1 83 + 734 733 CHEXA 604 1 84 + 736 735 CHEXA 605 1 85 + 738 737 CHEXA 606 1 86 + 740 739 CHEXA 607 1 87 + 742 741 CHEXA 608 1 88 + 744 743 CHEXA 609 1 89 + 746 745 CHEXA 610 1 90 + 748 747 CHEXA 611 1 102 + 749 729 CHEXA 612 1 1 + 751 732 CHEXA 613 1 2 + 752 734 CHEXA 614 1 3 + 753 736 CHEXA 615 1 4 + 754 738 CHEXA 616 1 5 + 755 740 CHEXA 617 1 6 + 756 742 CHEXA 618 1 7 + 757 744 CHEXA 619 1 8 + 758 746 CHEXA 620 1 9 + 759 748 CHEXA 621 1 104 + 760 749 CHEXA 622 1 10 + 762 751 CHEXA 623 1 11 + 763 752 CHEXA 624 1 12 + 764 753 CHEXA 625 1 13 + 765 754 CHEXA 626 1 14 + 766 755 CHEXA 627 1 15 + 767 756 CHEXA 628 1 16 + 768 757 CHEXA 629 1 17 + 769 758

Altair Engineering

1457 1457 1456 1456 1455 1455 1454 1454 1453

1458 1456 1457 1455 1456 1454 1455 1453 1454

1469 1577 1468 1576 1467 1575 1466 1574 1465

102

1

82

727

730+

1

2

83

728

729+

2

3

84

731

732+

3

4

85

733

734+

4

5

86

735

736+

5

6

87

737

738+

6

7

88

739

740+

7

8

89

741

742+

8

9

90

743

744+

9

103

101

745

746+

104

10

1

730

750+

10

11

2

729

749+

11

12

3

732

751+

12

13

4

734

752+

13

14

5

736

753+

14

15

6

738

754+

15

16

7

740

755+

16

17

8

742

756+

17

18

9

744

757+

18

105

103

746

758+

106

19

10

750

761+

19

20

11

749

760+

20

21

12

751

762+

21

22

13

752

763+

22

23

14

753

764+

23

24

15

754

765+

24

25

16

755

766+

25

26

17

756

767+

26

27

18

757

768+


167

CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA +

168

630 770 631 771 632 773 633 774 634 775 635 776 636 777 637 778 638 779 639 780 640 781 641 782 642 784 643 785 644 786 645 787 646 788 647 789 648 790 649 791 650 792 651 793 652 795 653 796 654 797 655 798 656 799 657 800 658 801 659 802 660 803 661 804 662 806 663 807 664 808

1 759 1 760 1 762 1 763 1 764 1 765 1 766 1 767 1 768 1 769 1 770 1 771 1 773 1 774 1 775 1 776 1 777 1 778 1 779 1 780 1 781 1 782 1 784 1 785 1 786 1 787 1 788 1 789 1 790 1 791 1 792 1 793 1 795 1 796 1 797

18

27

107

105

758

769+

106

108

28

19

761

772+

19

28

29

20

760

771+

20

29

30

21

762

773+

21

30

31

22

763

774+

22

31

32

23

764

775+

23

32

33

24

765

776+

24

33

34

25

766

777+

25

34

35

26

767

778+

26

35

36

27

768

779+

27

36

109

107

769

780+

108

110

37

28

772

783+

28

37

38

29

771

782+

29

38

39

30

773

784+

30

39

40

31

774

785+

31

40

41

32

775

786+

32

41

42

33

776

787+

33

42

43

34

777

788+

34

43

44

35

778

789+

35

44

45

36

779

790+

36

45

111

109

780

791+

110

112

46

37

783

794+

37

46

47

38

782

793+

38

47

48

39

784

795+

39

48

49

40

785

796+

40

49

50

41

786

797+

41

50

51

42

787

798+

42

51

52

43

788

799+

43

52

53

44

789

800+

44

53

54

45

790

801+

45

54

113

111

791

802+

112

114

55

46

794

805+

46

55

56

47

793

804+

47

56

57

48

795

806+

48

57

58

49

796

807+


Altair Engineering


665 809 666 810 667 811 668 812 669 813 670 814 671 815 672 817 673 818 674 819 675 820 676 821 677 822 678 823 679 824 680 825 681 826 682 828 683 829 684 830 685 831 686 832 687 833 688 834 689 835 690 836 691 837 692 839 693 840 694 841 695 842 696 843 697 844 698 845 699 846

Altair Engineering

1 798 1 799 1 800 1 801 1 802 1 803 1 804 1 806 1 807 1 808 1 809 1 810 1 811 1 812 1 813 1 814 1 815 1 817 1 818 1 819 1 820 1 821 1 822 1 823 1 824 1 825 1 826 1 828 1 829 1 830 1 831 1 832 1 833 1 834 1 835

49

58

59

50

797

808+

50

59

60

51

798

809+

51

60

61

52

799

810+

52

61

62

53

800

811+

53

62

63

54

801

812+

54

63

115

113

802

813+

114

116

64

55

805

816+

55

64

65

56

804

815+

56

65

66

57

806

817+

57

66

67

58

807

818+

58

67

68

59

808

819+

59

68

69

60

809

820+

60

69

70

61

810

821+

61

70

71

62

811

822+

62

71

72

63

812

823+

63

72

117

115

813

824+

116

118

73

64

816

827+

64

73

74

65

815

826+

65

74

75

66

817

828+

66

75

76

67

818

829+

67

76

77

68

819

830+

68

77

78

69

820

831+

69

78

79

70

821

832+

70

79

80

71

822

833+

71

80

81

72

823

834+

72

81

119

117

824

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118

120

91

73

827

838+

73

91

92

74

826

837+

74

92

93

75

828

839+

75

93

94

76

829

840+

76

94

95

77

830

841+

77

95

96

78

831

842+

78

96

97

79

832

843+

79

97

98

80

833

844+

80

98

99

81

834

845+


169


170

700 847 701 850 702 853 703 855 704 857 705 859 706 861 707 863 708 865 709 867 710 869 711 870 712 872 713 873 714 874 715 875 716 876 717 877 718 878 719 879 720 880 721 881 722 883 723 884 724 885 725 886 726 887 727 888 728 889 729 890 730 891 731 892 732 894 733 895 734 896

1 836 1 849 1 852 1 854 1 856 1 858 1 860 1 862 1 864 1 866 1 868 1 850 1 853 1 855 1 857 1 859 1 861 1 863 1 865 1 867 1 869 1 870 1 872 1 873 1 874 1 875 1 876 1 877 1 878 1 879 1 880 1 881 1 883 1 884 1 885

81

99

121

119

835

846+

727

730

729

728

848

851+

728

729

732

731

849

850+

731

732

734

733

852

853+

733

734

736

735

854

855+

735

736

738

737

856

857+

737

738

740

739

858

859+

739

740

742

741

860

861+

741

742

744

743

862

863+

743

744

746

745

864

865+

745

746

748

747

866

867+

730

750

749

729

851

871+

729

749

751

732

850

870+

732

751

752

734

853

872+

734

752

753

736

855

873+

736

753

754

738

857

874+

738

754

755

740

859

875+

740

755

756

742

861

876+

742

756

757

744

863

877+

744

757

758

746

865

878+

746

758

759

748

867

879+

750

761

760

749

871

882+

749

760

762

751

870

881+

751

762

763

752

872

883+

752

763

764

753

873

884+

753

764

765

754

874

885+

754

765

766

755

875

886+

755

766

767

756

876

887+

756

767

768

757

877

888+

757

768

769

758

878

889+

758

769

770

759

879

890+

761

772

771

760

882

893+

760

771

773

762

881

892+

762

773

774

763

883

894+

763

774

775

764

884

895+


Altair Engineering


735 897 736 898 737 899 738 900 739 901 740 902 741 903 742 905 743 906 744 907 745 908 746 909 747 910 748 911 749 912 750 913 751 914 752 916 753 917 754 918 755 919 756 920 757 921 758 922 759 923 760 924 761 925 762 927 763 928 764 929 765 930 766 931 767 932 768 933 769 934

Altair Engineering

1 886 1 887 1 888 1 889 1 890 1 891 1 892 1 894 1 895 1 896 1 897 1 898 1 899 1 900 1 901 1 902 1 903 1 905 1 906 1 907 1 908 1 909 1 910 1 911 1 912 1 913 1 914 1 916 1 917 1 918 1 919 1 920 1 921 1 922 1 923

764

775

776

765

885

896+

765

776

777

766

886

897+

766

777

778

767

887

898+

767

778

779

768

888

899+

768

779

780

769

889

900+

769

780

781

770

890

901+

772

783

782

771

893

904+

771

782

784

773

892

903+

773

784

785

774

894

905+

774

785

786

775

895

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775

786

787

776

896

907+

776

787

788

777

897

908+

777

788

789

778

898

909+

778

789

790

779

899

910+

779

790

791

780

900

911+

780

791

792

781

901

912+

783

794

793

782

904

915+

782

793

795

784

903

914+

784

795

796

785

905

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785

796

797

786

906

917+

786

797

798

787

907

918+

787

798

799

788

908

919+

788

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800

789

909

920+

789

800

801

790

910

921+

790

801

802

791

911

922+

791

802

803

792

912

923+

794

805

804

793

915

926+

793

804

806

795

914

925+

795

806

807

796

916

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796

807

808

797

917

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797

808

809

798

918

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798

809

810

799

919

930+

799

810

811

800

920

931+

800

811

812

801

921

932+

801

812

813

802

922

933+


171


172

770 935 771 936 772 938 773 939 774 940 775 941 776 942 777 943 778 944 779 945 780 946 781 947 782 949 783 950 784 951 785 952 786 953 787 954 788 955 789 956 790 957 791 958 792 960 793 961 794 962 795 963 796 964 797 965 798 966 799 967 800 968 801 971 802 974 803 976 804 978

1 924 1 925 1 927 1 928 1 929 1 930 1 931 1 932 1 933 1 934 1 935 1 936 1 938 1 939 1 940 1 941 1 942 1 943 1 944 1 945 1 946 1 947 1 949 1 950 1 951 1 952 1 953 1 954 1 955 1 956 1 957 1 970 1 973 1 975 1 977

802

813

814

803

923

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805

816

815

804

926

937+

804

815

817

806

925

936+

806

817

818

807

927

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807

818

819

808

928

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808

819

820

809

929

940+

809

820

821

810

930

941+

810

821

822

811

931

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811

822

823

812

932

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812

823

824

813

933

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813

824

825

814

934

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816

827

826

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937

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815

826

828

817

936

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817

828

829

818

938

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818

829

830

819

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819

830

831

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940

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820

831

832

821

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821

832

833

822

942

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822

833

834

823

943

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823

834

835

824

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955+

824

835

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825

945

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827

838

837

826

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959+

826

837

839

828

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958+

828

839

840

829

949

960+

829

840

841

830

950

961+

830

841

842

831

951

962+

831

842

843

832

952

963+

832

843

844

833

953

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833

844

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834

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834

845

846

835

955

966+

835

846

847

836

956

967+

848

851

850

849

969

972+

849

850

853

852

970

971+

852

853

855

854

973

974+

854

855

857

856

975

976+


Altair Engineering


805 980 806 982 807 984 808 986 809 988 810 990 811 991 812 993 813 994 814 995 815 996 816 997 817 998 818 999 819 1000 820 1001 821 1002 822 1004 823 1005 824 1006 825 1007 826 1008 827 1009 828 1010 829 1011 830 1012 831 1013 832 1015 833 1016 834 1017 835 1018 836 1019 837 1020 838 1021 839 1022

Altair Engineering

1 979 1 981 1 983 1 985 1 987 1 989 1 971 1 974 1 976 1 978 1 980 1 982 1 984 1 986 1 988 1 990 1 991 1 993 1 994 1 995 1 996 1 997 1 998 1 999 1 1000 1 1001 1 1002 1 1004 1 1005 1 1006 1 1007 1 1008 1 1009 1 1010 1 1011

856

857

859

858

977

978+

858

859

861

860

979

980+

860

861

863

862

981

982+

862

863

865

864

983

984+

864

865

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866

985

986+

866

867

869

868

987

988+

851

871

870

850

972

992+

850

870

872

853

971

991+

853

872

873

855

974

993+

855

873

874

857

976

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857

874

875

859

978

995+

859

875

876

861

980

996+

861

876

877

863

982

997+

863

877

878

865

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998+

865

878

879

867

986

999+

867

879

880

869

988

1000+

871

882

881

870

992

1003+

870

881

883

872

991

1002+

872

883

884

873

993

1004+

873

884

885

874

994

1005+

874

885

886

875

995

1006+

875

886

887

876

996

1007+

876

887

888

877

997

1008+

877

888

889

878

998

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878

889

890

879

999

1010+

879

890

891

880

1000

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882

893

892

881

1003

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881

892

894

883

1002

1013+

883

894

895

884

1004

1015+

884

895

896

885

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1016+

885

896

897

886

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886

897

898

887

1007

1018+

887

898

899

888

1008

1019+

888

899

900

889

1009

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889

900

901

890

1010

1021+


173


174

840 1023 841 1024 842 1026 843 1027 844 1028 845 1029 846 1030 847 1031 848 1032 849 1033 850 1034 851 1035 852 1037 853 1038 854 1039 855 1040 856 1041 857 1042 858 1043 859 1044 860 1045 861 1046 862 1048 863 1049 864 1050 865 1051 866 1052 867 1053 868 1054 869 1055 870 1056 871 1057 872 1059 873 1060 874 1061

1 1012 1 1013 1 1015 1 1016 1 1017 1 1018 1 1019 1 1020 1 1021 1 1022 1 1023 1 1024 1 1026 1 1027 1 1028 1 1029 1 1030 1 1031 1 1032 1 1033 1 1034 1 1035 1 1037 1 1038 1 1039 1 1040 1 1041 1 1042 1 1043 1 1044 1 1045 1 1046 1 1048 1 1049 1 1050

890

901

902

891

1011

1022+

893

904

903

892

1014

1025+

892

903

905

894

1013

1024+

894

905

906

895

1015

1026+

895

906

907

896

1016

1027+

896

907

908

897

1017

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897

908

909

898

1018

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898

909

910

899

1019

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899

910

911

900

1020

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900

911

912

901

1021

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901

912

913

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1022

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904

915

914

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1025

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903

914

916

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1024

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905

916

917

906

1026

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906

917

918

907

1027

1038+

907

918

919

908

1028

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908

919

920

909

1029

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909

920

921

910

1030

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910

921

922

911

1031

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911

922

923

912

1032

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912

923

924

913

1033

1044+

915

926

925

914

1036

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914

925

927

916

1035

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916

927

928

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1037

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917

928

929

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1038

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918

929

930

919

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919

930

931

920

1040

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920

931

932

921

1041

1052+

921

932

933

922

1042

1053+

922

933

934

923

1043

1054+

923

934

935

924

1044

1055+

926

937

936

925

1047

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925

936

938

927

1046

1057+

927

938

939

928

1048

1059+

928

939

940

929

1049

1060+


Altair Engineering


875 1062 876 1063 877 1064 878 1065 879 1066 880 1067 881 1068 882 1070 883 1071 884 1072 885 1073 886 1074 887 1075 888 1076 889 1077 890 1078 891 1079 892 1081 893 1082 894 1083 895 1084 896 1085 897 1086 898 1087 899 1088 900 1089 901 1092 902 1095 903 1097 904 1099 905 1101 906 1103 907 1105 908 1107 909 1109

Altair Engineering

1 1051 1 1052 1 1053 1 1054 1 1055 1 1056 1 1057 1 1059 1 1060 1 1061 1 1062 1 1063 1 1064 1 1065 1 1066 1 1067 1 1068 1 1070 1 1071 1 1072 1 1073 1 1074 1 1075 1 1076 1 1077 1 1078 1 1091 1 1094 1 1096 1 1098 1 1100 1 1102 1 1104 1 1106 1 1108

929

940

941

930

1050

1061+

930

941

942

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1051

1062+

931

942

943

932

1052

1063+

932

943

944

933

1053

1064+

933

944

945

934

1054

1065+

934

945

946

935

1055

1066+

937

948

947

936

1058

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936

947

949

938

1057

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938

949

950

939

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939

950

951

940

1060

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940

951

952

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1061

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941

952

953

942

1062

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942

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954

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943

954

955

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1064

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944

955

956

945

1065

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945

956

957

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1066

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948

959

958

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1069

1080+

947

958

960

949

1068

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949

960

961

950

1070

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950

961

962

951

1071

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951

962

963

952

1072

1083+

952

963

964

953

1073

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953

964

965

954

1074

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954

965

966

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1075

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955

966

967

956

1076

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956

967

968

957

1077

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969

972

971

970

1090

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970

971

974

973

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1092+

973

974

976

975

1094

1095+

975

976

978

977

1096

1097+

977

978

980

979

1098

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979

980

982

981

1100

1101+

981

982

984

983

1102

1103+

983

984

986

985

1104

1105+

985

986

988

987

1106

1107+


175


176

910 1111 911 1112 912 1114 913 1115 914 1116 915 1117 916 1118 917 1119 918 1120 919 1121 920 1122 921 1123 922 1125 923 1126 924 1127 925 1128 926 1129 927 1130 928 1131 929 1132 930 1133 931 1134 932 1136 933 1137 934 1138 935 1139 936 1140 937 1141 938 1142 939 1143 940 1144 941 1145 942 1147 943 1148 944 1149

1 1110 1 1092 1 1095 1 1097 1 1099 1 1101 1 1103 1 1105 1 1107 1 1109 1 1111 1 1112 1 1114 1 1115 1 1116 1 1117 1 1118 1 1119 1 1120 1 1121 1 1122 1 1123 1 1125 1 1126 1 1127 1 1128 1 1129 1 1130 1 1131 1 1132 1 1133 1 1134 1 1136 1 1137 1 1138

987

988

990

989

1108

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991

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971

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976

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980

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986

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1000

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1000

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998

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999

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1000

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1000

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1003

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1124

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1002

1013

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1004

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1005

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1006

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1127

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1007

1018

1019

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1008

1019

1020

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1129

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1009

1020

1021

1010

1130

1141+

1010

1021

1022

1011

1131

1142+

1011

1022

1023

1012

1132

1143+

1014

1025

1024

1013

1135

1146+

1013

1024

1026

1015

1134

1145+

1015

1026

1027

1016

1136

1147+

1016

1027

1028

1017

1137

1148+


Altair Engineering


945 1150 946 1151 947 1152 948 1153 949 1154 950 1155 951 1156 952 1158 953 1159 954 1160 955 1161 956 1162 957 1163 958 1164 959 1165 960 1166 961 1167 962 1169 963 1170 964 1171 965 1172 966 1173 967 1174 968 1175 969 1176 970 1177 971 1178 972 1180 973 1181 974 1182 975 1183 976 1184 977 1185 978 1186 979 1187

Altair Engineering

1 1139 1 1140 1 1141 1 1142 1 1143 1 1144 1 1145 1 1147 1 1148 1 1149 1 1150 1 1151 1 1152 1 1153 1 1154 1 1155 1 1156 1 1158 1 1159 1 1160 1 1161 1 1162 1 1163 1 1164 1 1165 1 1166 1 1167 1 1169 1 1170 1 1171 1 1172 1 1173 1 1174 1 1175 1 1176

1017

1028

1029

1018

1138

1149+

1018

1029

1030

1019

1139

1150+

1019

1030

1031

1020

1140

1151+

1020

1031

1032

1021

1141

1152+

1021

1032

1033

1022

1142

1153+

1022

1033

1034

1023

1143

1154+

1025

1036

1035

1024

1146

1157+

1024

1035

1037

1026

1145

1156+

1026

1037

1038

1027

1147

1158+

1027

1038

1039

1028

1148

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1028

1039

1040

1029

1149

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1029

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1041

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1042

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1033

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1036

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1035

1157

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1035

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1037

1156

1167+

1037

1048

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1038

1158

1169+

1038

1049

1050

1039

1159

1170+

1039

1050

1051

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1040

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1052

1041

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1041

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1053

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1042

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1054

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1043

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1055

1044

1164

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1044

1055

1056

1045

1165

1176+

1047

1058

1057

1046

1168

1179+

1046

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1059

1048

1167

1178+

1048

1059

1060

1049

1169

1180+

1049

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1061

1050

1170

1181+

1050

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1062

1051

1171

1182+

1051

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1063

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1172

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1052

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1064

1053

1173

1184+

1053

1064

1065

1054

1174

1185+

1054

1065

1066

1055

1175

1186+


177


178

980 1188 981 1189 982 1191 983 1192 984 1193 985 1194 986 1195 987 1196 988 1197 989 1198 990 1199 991 1200 992 1202 993 1203 994 1204 995 1205 996 1206 997 1207 998 1208 999 1209 1000 1210 1001 1213 1002 1216 1003 1218 1004 1220 1005 1222 1006 1224 1007 1226 1008 1228 1009 1230 1010 1232 1011 1233 1012 1235 1013 1236 1014 1237

1 1177 1 1178 1 1180 1 1181 1 1182 1 1183 1 1184 1 1185 1 1186 1 1187 1 1188 1 1189 1 1191 1 1192 1 1193 1 1194 1 1195 1 1196 1 1197 1 1198 1 1199 1 1212 1 1215 1 1217 1 1219 1 1221 1 1223 1 1225 1 1227 1 1229 1 1231 1 1213 1 1216 1 1218 1 1220

1055

1066

1067

1056

1176

1187+

1058

1069

1068

1057

1179

1190+

1057

1068

1070

1059

1178

1189+

1059

1070

1071

1060

1180

1191+

1060

1071

1072

1061

1181

1192+

1061

1072

1073

1062

1182

1193+

1062

1073

1074

1063

1183

1194+

1063

1074

1075

1064

1184

1195+

1064

1075

1076

1065

1185

1196+

1065

1076

1077

1066

1186

1197+

1066

1077

1078

1067

1187

1198+

1069

1080

1079

1068

1190

1201+

1068

1079

1081

1070

1189

1200+

1070

1081

1082

1071

1191

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1083

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1072

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1193

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1073

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1194

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1074

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1075

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1076

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1077

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1078

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1090

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1092

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1211

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1091

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1095

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1212

1213+

1094

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1097

1096

1215

1216+

1096

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1099

1098

1217

1218+

1098

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1101

1100

1219

1220+

1100

1101

1103

1102

1221

1222+

1102

1103

1105

1104

1223

1224+

1104

1105

1107

1106

1225

1226+

1106

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1109

1108

1227

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1108

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1111

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1093

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1214

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1112

1114

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1213

1233+

1095

1114

1115

1097

1216

1235+

1097

1115

1116

1099

1218

1236+


Altair Engineering


1015 1238 1016 1239 1017 1240 1018 1241 1019 1242 1020 1243 1021 1244 1022 1246 1023 1247 1024 1248 1025 1249 1026 1250 1027 1251 1028 1252 1029 1253 1030 1254 1031 1255 1032 1257 1033 1258 1034 1259 1035 1260 1036 1261 1037 1262 1038 1263 1039 1264 1040 1265 1041 1266 1042 1268 1043 1269 1044 1270 1045 1271 1046 1272 1047 1273 1048 1274 1049 1275

Altair Engineering

1 1222 1 1224 1 1226 1 1228 1 1230 1 1232 1 1233 1 1235 1 1236 1 1237 1 1238 1 1239 1 1240 1 1241 1 1242 1 1243 1 1244 1 1246 1 1247 1 1248 1 1249 1 1250 1 1251 1 1252 1 1253 1 1254 1 1255 1 1257 1 1258 1 1259 1 1260 1 1261 1 1262 1 1263 1 1264

1099

1116

1117

1101

1220

1237+

1101

1117

1118

1103

1222

1238+

1103

1118

1119

1105

1224

1239+

1105

1119

1120

1107

1226

1240+

1107

1120

1121

1109

1228

1241+

1109

1121

1122

1111

1230

1242+

1113

1124

1123

1112

1234

1245+

1112

1123

1125

1114

1233

1244+

1114

1125

1126

1115

1235

1246+

1115

1126

1127

1116

1236

1247+

1116

1127

1128

1117

1237

1248+

1117

1128

1129

1118

1238

1249+

1118

1129

1130

1119

1239

1250+

1119

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1131

1120

1240

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1132

1121

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1121

1132

1133

1122

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1124

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1134

1123

1245

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1123

1134

1136

1125

1244

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1125

1136

1137

1126

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1126

1137

1138

1127

1247

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1127

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1139

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1140

1129

1249

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1129

1140

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1130

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1130

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1131

1251

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1131

1142

1143

1132

1252

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1132

1143

1144

1133

1253

1264+

1135

1146

1145

1134

1256

1267+

1134

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1147

1136

1255

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1136

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1148

1137

1257

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1137

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1149

1138

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1139

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1141

1261

1272+

1141

1152

1153

1142

1262

1273+

1142

1153

1154

1143

1263

1274+


179

CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA +

1050 1276 1051 1277 1052 1279 1053 1280 1054 1281 1055 1282

1 1265 1 1266 1 1268 1 1269 1 1270 1 1271

1143

1154

1155

1144

1264

1275+

1146

1157

1156

1145

1267

1278+

1145

1156

1158

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1147

1158

1159

1148

1268

1279+

1148

1159

1160

1149

1269

1280+

1149

1160

1161

1150

1270

1281+

CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA +

1057 1284 1058 1285 1059 1286 1060 1287 1061 1288 1062 1290 1063 1291 1064 1292 1065 1293 1066 1294 1067 1295 1068 1296 1069 1297 1070 1298 1071 1299 1072 1301 1073 1302 1074 1303 1075 1304 1076 1305 1077 1306 1078 1307 1079 1308 1080 1309 1081 1310 1082 1312 1083 1313 1084 1314

1 1273 1 1274 1 1275 1 1276 1 1277 1 1279 1 1280 1 1281 1 1282 1 1283 1 1284 1 1285 1 1286 1 1287 1 1288 1 1290 1 1291 1 1292 1 1293 1 1294 1 1295 1 1296 1 1297 1 1298 1 1299 1 1301 1 1302 1 1303

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1162

1163

1152

1272

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1152

1163

1164

1153

1273

1284+

1153

1164

1165

1154

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1167

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1167

1169

1158

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1170

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1171

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1310+

1180

1191

1192

1181

1301

1312+

1181

1192

1193

1182

1302

1313+

180


Altair Engineering

CHEXA 1085 1 1182 1193 1194 1183 1303 1314+ + 1315 1304 CHEXA 1086 1 1183 1194 1195 1184 1304 1315+ + 1316 1305 CHEXA 1087 1 1184 1195 1196 1185 1305 1316+ + 1317 1306 CHEXA 1088 1 1185 1196 1197 1186 1306 1317+ + 1318 1307 CHEXA 1089 1 1186 1197 1198 1187 1307 1318+ + 1319 1308 CHEXA 1090 1 1187 1198 1199 1188 1308 1319+ + 1320 1309 CHEXA 1091 1 1190 1201 1200 1189 1311 1322+ + 1321 1310 CHEXA 1092 1 1189 1200 1202 1191 1310 1321+ + 1323 1312 CHEXA 1093 1 1191 1202 1203 1192 1312 1323+ + 1324 1313 CHEXA 1094 1 1192 1203 1204 1193 1313 1324+ + 1325 1314 CHEXA 1095 1 1193 1204 1205 1194 1314 1325+ + 1326 1315 CHEXA 1096 1 1194 1205 1206 1195 1315 1326+ + 1327 1316 CHEXA 1097 1 1195 1206 1207 1196 1316 1327+ + 1328 1317 CHEXA 1098 1 1196 1207 1208 1197 1317 1328+ + 1329 1318 CHEXA 1099 1 1197 1208 1209 1198 1318 1329+ + 1330 1319 CHEXA 1100 1 1198 1209 1210 1199 1319 1330+ + 1331 1320 $$ $$------------------------------------------------------------------------------$ $$ HyperMesh name information for generic property collectors $ $$------------------------------------------------------------------------------$ $$ $$------------------------------------------------------------------------------$ $$ Property Definition for 1-D Elements $ $$------------------------------------------------------------------------------$ $$ $$------------------------------------------------------------------------------$ $$ HyperMesh name and color information for generic components $ $$------------------------------------------------------------------------------$ $HMNAME COMP 6"auto1" $HWCOLOR COMP 6 3 $ $$ $$------------------------------------------------------------------------------$ $$ Property Definition for Surface and Volume Elements $ $$------------------------------------------------------------------------------$ $$ $$ PSHELL Data $ $HMNAME COMP 4"shells" $HWCOLOR COMP 4 7 PSHELL 4 20.2 2 2 $$ $$ PSOLID Data $ $HMNAME COMP 1"solids" $HWCOLOR COMP 1 26 PSOLID 1 1 PFLUID PSOLID 2 2 $$ $$------------------------------------------------------------------------------$ $$ Material Definition Cards $ $$------------------------------------------------------------------------------$ $$-------------------------------------------------------------$$ HYPERMESH TAGS

Altair Engineering


181

$$-------------------------------------------------------------$$BEGIN TAGS $$END TAGS $$ $$ MAT1 Data $ $HMNAME MAT 2"MAT1" $HWCOLOR MAT 2 18 MAT1 2200000.0 0.3 0.9e-5 $$ $$ $$ MAT10 Data $HMNAME MAT 1"MAT10_1" $HWCOLOR MAT 1 3 MAT10 11.0 0.01 $$ $$ $$------------------------------------------------------------------------------$ $$ HyperMesh name information for generic materials $ $$------------------------------------------------------------------------------$ $$ $$------------------------------------------------------------------------------$ $$ Material Definition Cards $ $$------------------------------------------------------------------------------$ $$ $$------------------------------------------------------------------------------$ $$ Loads and Boundary Conditions $ $$------------------------------------------------------------------------------$ $$ $$HyperMesh name and color information for generic loadcollectors $$ $HMNAME LOADCOL 4"SPC" $HWCOLOR LOADCOL 4 3 $ $HMNAME LOADCOL 6"spcd" $HWCOLOR LOADCOL 6 4 $ $$ $$ $$ $$ $$ FREQ1 cards $$ $HMNAME LOADCOL 5"freq" $HWCOLOR LOADCOL 5 4 FREQ1 50.1 10.0 5 $$ $$ $$ $$ $$ $$ RLOAD2 cards $$ $HMNAME LOADCOL 2"rload2" $HWCOLOR LOADCOL 2 5 RLOAD2 2 6 1 0 ACCE $$ $HMNAME LOADCOL 3"darea" $HWCOLOR LOADCOL 3 5 RLOAD2 3 3 1 0 LOAD $$ $$ $$ $$ TABLED1 cards $$ $HMNAME LOADCOL 1"tab" $HWCOLOR LOADCOL 1 41 TABLED1 1 LINEAR LINEAR + 0.0 0.0 1000.0 1.0ENDT $$

182


Altair Engineering

TABLED1 2 LINEAR + 0.0 0.0 $$ TABLED1 3 LINEAR + 0.0 5.0 $$ DLOAD cards $$ $HMNAME LOADCOL $HWCOLOR LOADCOL DLOAD 111.0 $$ $$ $$ $$ $$ $$ $$ $$ SPC Data $$ SPC 4 1431 SPC 4 1432 SPC 4 1451 SPC 4 1452 SPC 4 1734 $$ $$ SPCD Data $$ SPCD 6 1734 $ $ DAREA Data $ $$ $$ DAREA Data $$ DAREA 3 1734 ENDDATA

LINEAR 1000.0

1.0ENDT

LINEAR 1000.0

5.0ENDT

11"DLOAD11" 11 3 1.0 2

1.0

3

1234560.0 1234560.0 1234560.0 1234560.0 3 0.0

3

3.0

3-10.0

ALTDOCTAG "HqTD_ARNMI\S\pMpN13G;5oANN]l[enE7fmSbTJro20LOpNriZFOQfUk] _`5hfS5ATf6pT7RXMjA3e@k_r^K?GP;?OeEbD0" ADI0.1.0 2011-05-13T19:57:45 0of1 OSQA ENDDOCTAG

Input File - mdcaabsf.parm $$ $$ Optistruct Input Deck Generated by HyperMesh Version : 10.0build60 $$ Generated using HyperMesh-Optistruct Template Version : 10.0-SA1-120 $$ $$ Template: optistruct $$ $$ $ DISPLACEMENT(PHASE) = 1 OUTPUT,HGFREQ,ALL OUTPUT,OPTI,ALL OUTPUT,H3D,ALL OUTPUT,PUNCH,ALL $$------------------------------------------------------------------------------$ $$ Case Control Cards $ $$------------------------------------------------------------------------------$

Altair Engineering


183

$ $HMNAME LOADSTEP 1"Piston_Load" $ SUBCASE 1 LABEL Piston_Load SPC = 12 METHOD(STRUCTURE) = 4 METHOD(FLUID) = 5 FREQUENCY = 3 DLOAD = 9 XYPUNCH DISP 1/ 11(T1) XYPUNCH DISP 1/ 43(T1) XYPUNCH DISP 1/ 55(T1) XYPUNCH DISP 1/ 67(T1) XYPUNCH DISP 1/ 79(T1) XYPUNCH DISP 1/ 91(T1) XYPUNCH DISP 1/ 103(T1) XYPUNCH DISP 1/ 115(T1) XYPUNCH DISP 1/ 127(T1) XYPUNCH DISP 1/ 139(T1) XYPUNCH DISP 1/ 151(T1) XYPUNCH DISP 1/ 163(T1) XYPUNCH DISP 1/ 175(T1) XYPUNCH DISP 1/ 187(T1) XYPUNCH DISP 1/ 199(T1) XYPUNCH DISP 1/ 531(T1) XYPUNCH DISP 1/ 543(T1) XYPUNCH DISP 1/ 555(T1) XYPUNCH DISP 1/ 567(T1) XYPUNCH DISP 1/ 579(T1) XYPUNCH DISP 1/ 591(T1) XYPUNCH DISP 1/ 603(T1) XYPUNCH DISP 1/ 615(T1) XYPUNCH DISP 1/ 627(T1) XYPUNCH DISP 1/ 639(T1) XYPUNCH DISP 1/ 651(T1) XYPUNCH DISP 1/ 663(T1) XYPUNCH DISP 1/ 675(T1) XYPUNCH DISP 1/ 687(T1) $ $HMSET 1 1 "pressure" SET 1 = 43,55,67,79,91,103,115, 127,139,151,163,175,187,199, 531,543,555,567,579,591,603, 615,627,639,651,663,675,687, 6798

6

$ $$-------------------------------------------------------------$$ HYPERMESH TAGS $$-------------------------------------------------------------$$BEGIN TAGS $$END TAGS $ BEGIN BULK ACMODL $$ $$ Stacking Information for Ply-Based Composite Definition $$ PARAM,AUTOSPC,YES PARAM,POST,-1 $$ $$ DESVARG Data $$ $$ $$ GRID Data $$ GRID 9

184

0.492

0.0

-1.72-15

-1


Altair Engineering


10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79

Altair Engineering

0.246 0.0 -8.59-16 0.0 0.0 0.0 -0.246 0.0 8.589-16 -0.492 0.0 1.718-15 -0.492 0.246 1.718-15 -0.492 0.492 1.718-15 -0.246 0.492 8.589-16 0.0 0.492 0.0 0.246 0.492 -8.59-16 0.492 0.492 -1.72-15 0.492 0.246 -1.72-15 0.0 0.246 0.0 -0.246 0.246 8.589-16 0.246 0.246 -8.59-16 0.492 -0.246 -1.72-15 0.492 -0.492 -1.72-15 0.246 -0.492 -8.59-16 0.0 -0.492 0.0 -0.246 -0.492 8.589-16 -0.492 -0.492 1.718-15 -0.492 -0.246 1.718-15 0.0 -0.246 0.0 0.246 -0.246 -8.59-16 -0.246 -0.246 8.589-16 0.246 5.049-29-.300073 -5.99-130.0 -.300073 -5.62-130.246 -.300073 0.246 0.246 -.300073 0.246 2.524-29-.600146 -1.2-12 0.0 -.600146 -1.12-120.246 -.600146 0.246 0.246 -.600146 0.246 2.919-29-0.90022 -1.79-120.0 -.900219 -1.68-120.246 -.900219 0.246 0.246 -0.90022 0.246 3.787-29-1.20029 -2.39-120.0 -1.20029 -2.24-120.246 -1.20029 0.246 0.246 -1.20029 0.246 4.733-29-1.50037 -3.0-12 0.0 -1.50037 -2.81-120.246 -1.50037 0.246 0.246 -1.50037 0.246 5.364-29-1.80044 -3.6-12 0.0 -1.80044 -3.37-120.246 -1.80044 0.246 0.246 -1.80044 0.246 6.311-29-2.10051 -4.2-12 0.0 -2.10051 -3.93-120.246 -2.10051 0.246 0.246 -2.10051 0.246 7.258-29-2.40059 -4.79-120.0 -2.40059 -4.49-120.246 -2.40059 0.246 0.246 -2.40059 0.246 8.204-29-2.70066 -5.39-120.0 -2.70066 -5.06-120.246 -2.70066 0.246 0.246 -2.70066 0.246 8.835-29-3.00073 -5.99-120.0 -3.00073 -5.62-120.246 -3.00073 0.246 0.246 -3.00073 0.246 9.782-29-3.30081 -6.59-120.0 -3.30081 -6.18-120.246 -3.30081 0.246 0.246 -3.30081 0.246 1.073-28-3.60088 -7.19-120.0 -3.60088

-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1


185


186

80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149

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-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1


Altair Engineering


150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539

Altair Engineering

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-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1


187


188

540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609

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-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1


Altair Engineering


610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679

Altair Engineering

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-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1


189


190

680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749

-9.09-110.246 -24.6059 0.246 0.246 -24.6059 0.246 9.466-30-24.9059 -9.76-110.0 -24.9059 -9.15-110.246 -24.9059 0.246 0.246 -24.9059 0.246 4.151-12-25.206 -9.82-112.767-12-25.206 -9.16-110.246 -25.206 0.246 0.246 -25.206 0.492 4.323-13-.300073 0.492 0.246 -.300073 0.492 1.621-13-.600146 0.492 0.246 -.600146 0.492 1.592-13-0.90022 0.492 0.246 -0.90022 0.492 2.009-13-1.20029 0.492 0.246 -1.20029 0.492 2.5-13 -1.50037 0.492 0.246 -1.50037 0.492 3.004-13-1.80044 0.492 0.246 -1.80044 0.492 3.51-13 -2.10051 0.492 0.246 -2.10051 0.492 4.016-13-2.40059 0.492 0.246 -2.40059 0.492 4.522-13-2.70066 0.492 0.246 -2.70066 0.492 5.028-13-3.00073 0.492 0.246 -3.00073 0.492 5.534-13-3.30081 0.492 0.246 -3.30081 0.492 6.041-13-3.60088 0.492 0.246 -3.60088 0.492 6.547-13-3.90095 0.492 0.246 -3.90095 0.492 7.053-13-4.20102 0.492 0.246 -4.20102 0.492 7.559-13-4.5011 0.492 0.246 -4.5011 0.492 8.066-13-4.80117 0.492 0.246 -4.80117 0.492 8.572-13-5.10124 0.492 0.246 -5.10124 0.492 9.078-13-5.40132 0.492 0.246 -5.40132 0.492 9.584-13-5.70139 0.492 0.246 -5.70139 0.492 1.009-12-6.00146 0.492 0.246 -6.00146 0.492 1.06-12 -6.30154 0.492 0.246 -6.30154 0.492 1.11-12 -6.60161 0.492 0.246 -6.60161 0.492 1.161-12-6.90168 0.492 0.246 -6.90168 0.492 1.212-12-7.20176 0.492 0.246 -7.20176 0.492 1.262-12-7.50183 0.492 0.246 -7.50183 0.492 1.313-12-7.8019 0.492 0.246 -7.8019 0.492 1.363-12-8.10198 0.492 0.246 -8.10198 0.492 1.414-12-8.40205 0.492 0.246 -8.40205 0.492 1.465-12-8.70212 0.492 0.246 -8.70212 0.492 1.515-12-9.0022 0.492 0.246 -9.0022

-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1


Altair Engineering


750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979

Altair Engineering

0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492

1.566-12-9.30227 0.246 -9.30227 1.616-12-9.60234 0.246 -9.60234 1.667-12-9.90242 0.246 -9.90242 1.718-12-10.2025 0.246 -10.2025 1.768-12-10.5026 0.246 -10.5026 1.819-12-10.8026 0.246 -10.8026 1.87-12 -11.1027 0.246 -11.1027 1.92-12 -11.4028 0.246 -11.4028 1.971-12-11.7029 0.246 -11.7029 2.021-12-12.0029 0.246 -12.0029 2.072-12-12.303 0.246 -12.303 2.123-12-12.6031 0.246 -12.6031 6.223-12-12.903 0.246 -12.903 6.274-12-13.2031 0.246 -13.2031 6.324-12-13.5032 0.246 -13.5032 6.375-12-13.8032 0.246 -13.8032 6.425-12-14.1033 0.246 -14.1033 6.476-12-14.4034 0.246 -14.4034 6.527-12-14.7034 0.246 -14.7034 6.577-12-15.0035 0.246 -15.0035 6.628-12-15.3036 0.246 -15.3036 6.679-12-15.6037 0.246 -15.6037 6.729-12-15.9037 0.246 -15.9037 6.78-12 -16.2038 0.246 -16.2038 6.83-12 -16.5039 0.246 -16.5039 6.881-12-16.804 0.246 -16.804 6.932-12-17.104 0.246 -17.104 6.982-12-17.4041 0.246 -17.4041 7.033-12-17.7042 0.246 -17.7042 7.083-12-18.0042 0.246 -18.0042 7.134-12-18.3043 0.246 -18.3043 7.185-12-18.6044 0.246 -18.6044 7.235-12-18.9045 0.246 -18.9045 7.286-12-19.2045 0.246 -19.2045 7.337-12-19.5046 0.246 -19.5046

-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1


191


192

980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049

0.492 7.387-12-19.8047 0.492 0.246 -19.8047 0.492 7.438-12-20.1048 0.492 0.246 -20.1048 0.492 7.488-12-20.4048 0.492 0.246 -20.4048 0.492 7.539-12-20.7049 0.492 0.246 -20.7049 0.492 7.59-12 -21.005 0.492 0.246 -21.005 0.492 7.64-12 -21.3051 0.492 0.246 -21.3051 0.492 7.691-12-21.6051 0.492 0.246 -21.6051 0.492 7.742-12-21.9052 0.492 0.246 -21.9052 0.492 7.792-12-22.2053 0.492 0.246 -22.2053 0.492 7.843-12-22.5053 0.492 0.246 -22.5053 0.492 7.893-12-22.8054 0.492 0.246 -22.8054 0.492 7.944-12-23.1055 0.492 0.246 -23.1055 0.492 7.995-12-23.4056 0.492 0.246 -23.4056 0.492 8.045-12-23.7056 0.492 0.246 -23.7056 0.492 8.096-12-24.0057 0.492 0.246 -24.0057 0.492 8.146-12-24.3058 0.492 0.246 -24.3058 0.492 8.197-12-24.6059 0.492 0.246 -24.6059 0.492 8.248-12-24.9059 0.492 0.246 -24.9059 0.492 5.534-12-25.206 0.492 0.246 -25.206 -5.24-130.492 -.300073 0.246 0.492 -.300073 -1.05-120.492 -.600146 0.246 0.492 -.600146 -1.57-120.492 -.900219 0.246 0.492 -0.90022 -2.09-120.492 -1.20029 0.246 0.492 -1.20029 -2.63-120.492 -1.50037 0.246 0.492 -1.50037 -3.15-120.492 -1.80044 0.246 0.492 -1.80044 -3.67-120.492 -2.10051 0.246 0.492 -2.10051 -4.19-120.492 -2.40059 0.246 0.492 -2.40059 -4.72-120.492 -2.70066 0.246 0.492 -2.70066 -5.24-120.492 -3.00073 0.246 0.492 -3.00073 -5.76-120.492 -3.30081 0.246 0.492 -3.30081 -6.29-120.492 -3.60088 0.246 0.492 -3.60088 -6.81-120.492 -3.90095 0.246 0.492 -3.90095 -7.33-120.492 -4.20102 0.246 0.492 -4.20102 -7.87-120.492 -4.5011 0.246 0.492 -4.5011 -8.39-120.492 -4.80117 0.246 0.492 -4.80117

-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1


Altair Engineering


1050 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100 1101 1262 1263 1264 1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279

Altair Engineering

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193


194

1280 1281 1282 1283 1284 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294 1295 1296 1297 1298 1299 1300 1301 1302 1303 1304 1305 1306 1307 1308 1309 1310 1311 1312 1313 1314 1315 1316 1317 1318 1319 1320 1321 1322 1323 1324 1325 1326 1327 1328 1329 1330 1331 1332 1333 1334 1335 1336 1337 1338 1339 1340 1341 1342 1343 1344 1345 1346 1347 1348 1349

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Altair Engineering


1350 1351 1352 1353 1354 1355 1356 1357 1358 1359 1360 1361 1362 1363 1364 1365 1366 1367 1368 1369 1370 1371 1372 1373 1374 1375 1376 1377 1378 1379 1380 1381 1382 1383 1384 1385 1386 1387 1468 1469 1470 1471 1472 1473 1474 1475 1476 1477 1478 1479 1480 1481 1482 1483 1484 1485 1486 1487 1488 1489 1490 1491 1492 1493 1494 1495 1496 1497 1498 1499

Altair Engineering

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195


196

1500 1501 1502 1503 1504 1505 1506 1507 1508 1509 1510 1511 1512 1513 1514 1515 1516 1517 1518 1519 1520 1521 1522 1523 1524 1525 1526 1527 1528 1529 1530 1531 1532 1533 1534 1535 1536 1537 1538 1539 1540 1541 1542 1543 1544 1545 1546 1547 1548 1549 1550 1551 1552 1553 1554 1555 1556 1557 1558 1559 1560 1561 1562 1563 1564 1565 1566 1567 1568 1569

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-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1


Altair Engineering


1570 1571 1572 1573 1574 1575 1576 1577 1578 1579 1580 1581 1582 1583 1584 1585 1586 1587 1588 1589 1590 1591 1592 1593 1594 1595 1596 1597 1598 1599 1600 1601 1602 1603 1604 1605 1606 1607 1608 1609 1610 1611 1612 1613 1614 1615 1616 1617 1618 1619 1620 1621 1622 1623 1624 1625 1626 1627 1628 1629 1630 1631 1632 1633 1634 1635 1636 1637 1638 1639

Altair Engineering

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197


198

1640 1641 1642 1643 1644 1645 1646 1647 1648 1649 1650 1651 1652 1653 1654 1655 1656 1657 1658 1659 1660 1661 1662 1663 1664 1665 1666 1667 1668 1669 1670 1671 1672 1673 1674 1675 1676 1677 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 2023 2024 2025 2026 2027 2028 2029

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-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1


Altair Engineering


2030 2031 2032 2033 2034 2035 2036 2037 2038 2039 2040 2041 2042 2043 2044 2045 2046 2047 2048 2049 2050 2051 2052 2053 2054 2055 2056 2057 2058 2059 2060 2061 2062 2063 2064 2065 2066 2067 2068 2069 2070 2071 2072 2073 2074 2075 2076 2077 2078 2079 2080 2081 2082 2083 2084 2085 2086 2087 2088 2089 2090 2091 2092 2093 2094 2095 2096 2097 2098 2099

Altair Engineering

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199


200

2100 2101 2102 2103 2104 2105 2106 2107 2108 2109 2110 2111 2112 2113 2114 2115 2116 2117 2118 2119 2120 2121 2122 2123 2124 2125 2126 2127 2128 2129 2130 2131 2132 2133 2134 2135 2136 2137 2138 2139 2140 2141 2142 2143 2144 2145 2146 2147 2148 2149 2150 2151 2152 2153 2154 2155 2156 2157 2158 2159 2160 2161 2162 2163 2164 2165 2166 2167 2168 2169

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Altair Engineering


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Altair Engineering

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201


202

2240 2241 2242 2243 2244 2245 2246 2247 2248 2249 2410 2411 2412 2413 2414 2415 2416 2417 2418 2419 2420 2421 2422 2423 2424 2425 2426 2427 2428 2429 2430 2431 2432 2433 2434 2435 2436 2437 2438 2439 2440 2441 2442 2443 2444 2445 2446 2447 2448 2449 2450 2451 2452 2453 2454 2455 2456 2457 2458 2459 2460 2461 2462 2463 2464 2465 2466 2467 2468 2469

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Altair Engineering

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203


204

2540 2541 2542 2543 2544 2545 2546 2547 2548 2549 2550 2551 2552 2553 2554 2555 2556 2557 2558 2559 2560 2561 2562 2563 2564 2565 2566 2567 2568 2569 2570 2571 2572 2573 2574 2575 2576 2577 2738 2739 2740 2741 2742 2743 2744 2745 2746 2747 2748 2749 2750 2751 2752 2753 2754 2755 2756 2757 2758 2759 2760 2761 2762 2763 2764 2765 2766 2767 2768 2769

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Altair Engineering


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Altair Engineering

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205


206

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207


208

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Altair Engineering


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209


210

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211


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-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1


Altair Engineering


3810 3811 3812 3813 3814 3815 3816 3817 3818 3819 3820 3821 3822 3823 3824 3825 3826 3827 3828 3829 3830 3831 3832 3833 3834 3835 3836 3837 3838 3839 3840 3841 3842 3843 3844 3845 3846 3847 3848 3849 3850 3851 3852 3853 3854 3855 3856 3857 3858 3859 3860 3861 3862 3863 3864 3865 3866 3867 3868 3869 3870 3871 3872 3873 3874 3875 3876 3877 3878 3879

Altair Engineering

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-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1


213


214

3880 3881 3882 3883 3884 3885 3886 3887 3888 3889 4050 4051 4052 4053 4054 4055 4056 4057 4058 4059 4060 4061 4062 4063 4064 4065 4066 4067 4068 4069 4070 4071 4072 4073 4074 4075 4076 4077 4078 4079 4080 4081 4082 4083 4084 4085 4086 4087 4088 4089 4090 4091 4092 4093 4094 4095 4096 4097 4098 4099 4100 4101 4102 4103 4104 4105 4106 4107 4108 4109

0.246 0.492 0.246 0.492 0.246 0.492 0.246 0.492 0.246 0.492 0.246 0.492 0.246 0.492 0.246 0.492 0.246 0.492 0.246 0.492 0.246 0.492 0.246 0.492 0.246 0.492 0.246 0.492 0.246 0.492 0.246 0.492 0.246 0.492 0.246 0.492 0.246 0.492 0.246 0.492 0.246 0.492 0.246 0.492 0.246 0.492 0.246 0.492 0.246 0.492 0.246 0.492 0.246 0.492 0.246 0.492 0.246 0.492 0.246 0.492 0.246 0.492 0.246 0.492 0.246 0.492 0.246 0.492 0.246 0.492

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-11.4028 -11.4028 -11.7029 -11.7029 -12.0029 -12.0029 -12.303 -12.303 -12.6031 -12.6031 -12.903 -12.903 -13.2031 -13.2031 -13.5032 -13.5032 -13.8032 -13.8032 -14.1033 -14.1033 -14.4034 -14.4034 -14.7034 -14.7034 -15.0035 -15.0035 -15.3036 -15.3036 -15.6037 -15.6037 -15.9037 -15.9037 -16.2038 -16.2038 -16.5039 -16.5039 -16.804 -16.804 -17.104 -17.104 -17.4041 -17.4041 -17.7042 -17.7042 -18.0042 -18.0042 -18.3043 -18.3043 -18.6044 -18.6044 -18.9045 -18.9045 -19.2045 -19.2045 -19.5046 -19.5046 -19.8047 -19.8047 -20.1048 -20.1048 -20.4048 -20.4048 -20.7049 -20.7049 -21.005 -21.005 -21.3051 -21.3051 -21.6051 -21.6051

-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1


Altair Engineering

GRID 4110 GRID 4111 GRID 4112 GRID 4113 GRID 4114 GRID 4115 GRID 4116 GRID 4117 GRID 4118 GRID 4119 GRID 4120 GRID 4121 GRID 4122 GRID 4123 GRID 4124 GRID 4125 GRID 4126 GRID 4127 GRID 4128 GRID 4129 GRID 4130 GRID 4131 GRID 4132 GRID 4133 GRID 6776 GRID 6777 GRID 6778 GRID 6779 GRID 6780 GRID 6781 GRID 6782 GRID 6783 GRID 6784 GRID 6785 GRID 6786 GRID 6787 GRID 6788 GRID 6789 GRID 6790 GRID 6791 GRID 6792 GRID 6793 GRID 6794 GRID 6795 GRID 6796 GRID 6797 GRID 6798 GRID 6799 GRID 6800 $$ $$ SPOINT Data $$ $ $ CQUAD4 Elements $ CQUAD4 5627 CQUAD4 5629 CQUAD4 6116 CQUAD4 6122 CQUAD4 6125 CQUAD4 6520 CQUAD4 6521 CQUAD4 6523 CQUAD4 6528 CQUAD4 6954 CQUAD4 7220 CQUAD4 7647 CQUAD4 7652 CQUAD4 7945 CQUAD4 7948

Altair Engineering

0.246 0.492 0.246 0.492 0.246 0.492 0.246 0.492 0.246 0.492 0.246 0.492 0.246 0.492 0.246 0.492 0.246 0.492 0.246 0.492 0.246 0.492 0.246 0.492 -0.246 0.246 0.0 -0.492 -0.492 -0.246 0.0 0.246 0.492 0.492 0.246 -0.246 0.0 0.492 0.492 0.246 0.0 -0.246 -0.492 -0.492 -0.492 -0.246 0.0 0.246 0.492

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

-0.492 -0.492 -0.492 -0.492 -0.492 -0.492 -0.492 -0.492 -0.492 -0.492 -0.492 -0.492 -0.492 -0.492 -0.492 -0.492 -0.492 -0.492 -0.492 -0.492 -0.492 -0.492 -0.492 -0.492 -0.246 -0.246 -0.246 -0.246 -0.492 -0.492 -0.492 -0.492 -0.492 -0.246 0.246 0.246 0.246 0.246 0.492 0.492 0.492 0.492 0.492 0.246 0.0 0.0 0.0 0.0 0.0

6778 6782 6777 6783 6799 6779 6776 6780 6781 6797 6788 6787 6798 6786 6789

-21.9052 -21.9052 -22.2053 -22.2053 -22.5053 -22.5053 -22.8054 -22.8054 -23.1055 -23.1055 -23.4056 -23.4056 -23.7056 -23.7056 -24.0057 -24.0057 -24.3058 -24.3058 -24.6059 -24.6059 -24.9059 -24.9059 -25.206 -25.206 8.589-16 -8.59-16 0.0 1.718-15 1.718-15 8.589-16 0.0 -8.59-16 -1.72-15 -1.72-15 -8.59-16 8.589-16 0.0 -1.72-15 -1.72-15 -8.59-16 0.0 8.589-16 1.718-15 1.718-15 1.718-15 8.589-16 0.0 -8.59-16 -1.72-15

6798 6778 6799 6777 6798 6796 6797 6779 6776 6796 6787 6795 6797 6788 6786

6799 6777 6800 6785 6788 6797 6798 6776 6778 6795 6793 6794 6787 6792 6791

-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1

6777 6783 6785 6784 6786 6776 6778 6781 6782 6787 6792 6793 6788 6791 6790


215

CQUAD4 7955 1 6800 $ $HMMOVE 5 $ 5627 5629 6116 $ 6528 6954 7220 $ $ $ CHEXA Elements: First Order $ CHEXA 17 2 10 + 36 37 CHEXA 18 2 34 + 40 41 CHEXA 19 2 38 + 44 45 CHEXA 20 2 42 + 48 49 CHEXA 21 2 46 + 52 53 CHEXA 22 2 50 + 56 57 CHEXA 23 2 54 + 60 61 CHEXA 24 2 58 + 64 65 CHEXA 25 2 62 + 68 69 CHEXA 26 2 66 + 72 73 CHEXA 27 2 70 + 76 77 CHEXA 28 2 74 + 80 81 CHEXA 29 2 78 + 84 85 CHEXA 30 2 82 + 88 89 CHEXA 31 2 86 + 92 93 CHEXA 32 2 90 + 96 97 CHEXA 33 2 94 + 100 101 CHEXA 34 2 98 + 104 105 CHEXA 35 2 102 + 108 109 CHEXA 36 2 106 + 112 113 CHEXA 37 2 110 + 116 117 CHEXA 38 2 114 + 120 121 CHEXA 39 2 118 + 124 125 CHEXA 40 2 122 + 128 129 CHEXA 41 2 126 + 132 133 CHEXA 42 2 130 + 136 137 CHEXA 43 2 134 + 140 141 CHEXA 44 2 138 + 144 145 CHEXA 45 2 142 + 148 149 CHEXA 46 2 146 + 152 153 CHEXA 47 2 150

216

6799

6786

6789

6122 7647

6125 7652

6520THRU 7945 7948

11

21

23

34

35

35

36

37

38

39

39

40

41

42

43

43

44

45

46

47

47

48

49

50

51

51

52

53

54

55

55

56

57

58

59

59

60

61

62

63

63

64

65

66

67

67

68

69

70

71

71

72

73

74

75

75

76

77

78

79

79

80

81

82

83

83

84

85

86

87

87

88

89

90

91

91

92

93

94

95

95

96

97

98

99

99

100

101

102

103

103

104

105

106

107

107

108

109

110

111

111

112

113

114

115

115

116

117

118

119

119

120

121

122

123

123

124

125

126

127

127

128

129

130

131

131

132

133

134

135

135

136

137

138

139

139

140

141

142

143

143

144

145

146

147

147

148

149

150

151

151

152

153

154

155


6521 7955

6523

Altair Engineering

+ CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA + CHEXA

156 48 160 49 164 50 168 51 172 52 176 53 180 54 184 55 188 56 192 57 196 58 200 139 524 140 528 141 532 142 536 143 540 144 544 145 548 146 552 147 556 148 560 149 564 150 568 151 572 152 576 153 580 154 584 155 588 156 592 157 596 158 600 159 604 160 608 161 612 162

Altair Engineering

157 2 161 2 165 2 169 2 173 2 177 2 181 2 185 2 189 2 193 2 197 2 201 2 525 2 529 2 533 2 537 2 541 2 545 2 549 2 553 2 557 2 561 2 565 2 569 2 573 2 577 2 581 2 585 2 589 2 593 2 597 2 601 2 605 2 609 2 613 2

154

155

156

157

158

159

158

159

160

161

162

163

162

163

164

165

166

167

166

167

168

169

170

171

170

171

172

173

174

175

174

175

176

177

178

179

178

179

180

181

182

183

182

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184

185

186

187

186

187

188

189

190

191

190

191

192

193

194

195

194

195

196

197

198

199

198

199

200

201

522

523

522

523

524

525

526

527

526

527

528

529

530

531

530

531

532

533

534

535

534

535

536

537

538

539

538

539

540

541

542

543

542

543

544

545

546

547

546

547

548

549

550

551

550

551

552

553

554

555

554

555

556

557

558

559

558

559

560

561

562

563

562

563

564

565

566

567

566

567

568

569

570

571

570

571

572

573

574

575

574

575

576

577

578

579

578

579

580

581

582

583

582

583

584

585

586

587

586

587

588

589

590

591

590

591

592

593

594

595

594

595

596

597

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599

598

599

600

601

602

603

602

603

604

605

606

607

606

607

608

609

610

611

610

611

612

613

614

615


217


218

616 163 620 164 624 165 628 166 632 167 636 168 640 169 644 170 648 171 652 172 656 173 660 174 664 175 668 176 672 177 676 178 680 179 684 180 688 181 37 182 41 183 45 184 49 185 53 186 57 187 61 188 65 189 69 190 73 191 77 192 81 193 85 194 89 195 93 196 97 197

617 2 621 2 625 2 629 2 633 2 637 2 641 2 645 2 649 2 653 2 657 2 661 2 665 2 669 2 673 2 677 2 681 2 685 2 689 2 691 2 693 2 695 2 697 2 699 2 701 2 703 2 705 2 707 2 709 2 711 2 713 2 715 2 717 2 719 2 721 2

614

615

616

617

618

619

618

619

620

621

622

623

622

623

624

625

626

627

626

627

628

629

630

631

630

631

632

633

634

635

634

635

636

637

638

639

638

639

640

641

642

643

642

643

644

645

646

647

646

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648

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650

651

650

651

652

653

654

655

654

655

656

657

658

659

658

659

660

661

662

663

662

663

664

665

666

667

666

667

668

669

670

671

670

671

672

673

674

675

674

675

676

677

678

679

678

679

680

681

682

683

682

683

684

685

686

687

9

10

23

20

690

34

690

34

37

691

692

38

692

38

41

693

694

42

694

42

45

695

696

46

696

46

49

697

698

50

698

50

53

699

700

54

700

54

57

701

702

58

702

58

61

703

704

62

704

62

65

705

706

66

706

66

69

707

708

70

708

70

73

709

710

74

710

74

77

711

712

78

712

78

81

713

714

82

714

82

85

715

716

86

716

86

89

717

718

90

718

90

93

719

720

94

720

94

97

721

722

98


Altair Engineering


101 198 105 199 109 200 113 201 117 202 121 203 125 204 129 205 133 206 137 207 141 208 145 209 149 210 153 211 157 212 161 213 165 214 169 215 173 216 177 217 181 218 185 219 189 220 193 221 197 222 201 303 525 304 529 305 533 306 537 307 541 308 545 309 549 310 553 311 557 312

Altair Engineering

723 2 725 2 727 2 729 2 731 2 733 2 735 2 737 2 739 2 741 2 743 2 745 2 747 2 749 2 751 2 753 2 755 2 757 2 759 2 761 2 763 2 765 2 767 2 769 2 771 2 773 2 935 2 937 2 939 2 941 2 943 2 945 2 947 2 949 2 951 2

722

98

101

723

724

102

724

102

105

725

726

106

726

106

109

727

728

110

728

110

113

729

730

114

730

114

117

731

732

118

732

118

121

733

734

122

734

122

125

735

736

126

736

126

129

737

738

130

738

130

133

739

740

134

740

134

137

741

742

138

742

138

141

743

744

142

744

142

145

745

746

146

746

146

149

747

748

150

748

150

153

749

750

154

750

154

157

751

752

158

752

158

161

753

754

162

754

162

165

755

756

166

756

166

169

757

758

170

758

170

173

759

760

174

760

174

177

761

762

178

762

178

181

763

764

182

764

182

185

765

766

186

766

186

189

767

768

190

768

190

193

769

770

194

770

194

197

771

772

198

772

198

201

773

934

522

934

522

525

935

936

526

936

526

529

937

938

530

938

530

533

939

940

534

940

534

537

941

942

538

942

538

541

943

944

542

944

542

545

945

946

546

946

546

549

947

948

550

948

550

553

949

950

554

950

554

557

951

952

558


219


220

561 313 565 314 569 315 573 316 577 317 581 318 585 319 589 320 593 321 597 322 601 323 605 324 609 325 613 326 617 327 621 328 625 329 629 330 633 331 637 332 641 333 645 334 649 335 653 336 657 337 661 338 665 339 669 340 673 341 677 342 681 343 685 344 689 345 1018 346 1020 347

953 2 955 2 957 2 959 2 961 2 963 2 965 2 967 2 969 2 971 2 973 2 975 2 977 2 979 2 981 2 983 2 985 2 987 2 989 2 991 2 993 2 995 2 997 2 999 2 1001 2 1003 2 1005 2 1007 2 1009 2 1011 2 1013 2 1015 2 1017 2 1019 2 1021 2

952

558

561

953

954

562

954

562

565

955

956

566

956

566

569

957

958

570

958

570

573

959

960

574

960

574

577

961

962

578

962

578

581

963

964

582

964

582

585

965

966

586

966

586

589

967

968

590

968

590

593

969

970

594

970

594

597

971

972

598

972

598

601

973

974

602

974

602

605

975

976

606

976

606

609

977

978

610

978

610

613

979

980

614

980

614

617

981

982

618

982

618

621

983

984

622

984

622

625

985

986

626

986

626

629

987

988

630

988

630

633

989

990

634

990

634

637

991

992

638

992

638

641

993

994

642

994

642

645

995

996

646

996

646

649

997

998

650

998

650

653

999

1000

654

1000

654

657

1001

1002

658

1002

658

661

1003

1004

662

1004

662

665

1005

1006

666

1006

666

669

1007

1008

670

1008

670

673

1009

1010

674

1010

674

677

1011

1012

678

1012

678

681

1013

1014

682

1014

682

685

1015

1016

686

23

21

17

18

37

36

37

36

1018

1019

41

40

41

40

1020

1021

45

44


Altair Engineering


1022 348 1024 349 1026 350 1028 351 1030 352 1032 353 1034 354 1036 355 1038 356 1040 357 1042 358 1044 359 1046 360 1048 361 1050 362 1052 363 1054 364 1056 365 1058 366 1060 367 1062 368 1064 369 1066 370 1068 371 1070 372 1072 373 1074 374 1076 375 1078 376 1080 377 1082 378 1084 379 1086 380 1088 381 1090 382

Altair Engineering

1023 2 1025 2 1027 2 1029 2 1031 2 1033 2 1035 2 1037 2 1039 2 1041 2 1043 2 1045 2 1047 2 1049 2 1051 2 1053 2 1055 2 1057 2 1059 2 1061 2 1063 2 1065 2 1067 2 1069 2 1071 2 1073 2 1075 2 1077 2 1079 2 1081 2 1083 2 1085 2 1087 2 1089 2 1091 2

45

44

1022

1023

49

48

49

48

1024

1025

53

52

53

52

1026

1027

57

56

57

56

1028

1029

61

60

61

60

1030

1031

65

64

65

64

1032

1033

69

68

69

68

1034

1035

73

72

73

72

1036

1037

77

76

77

76

1038

1039

81

80

81

80

1040

1041

85

84

85

84

1042

1043

89

88

89

88

1044

1045

93

92

93

92

1046

1047

97

96

97

96

1048

1049

101

100

101

100

1050

1051

105

104

105

104

1052

1053

109

108

109

108

1054

1055

113

112

113

112

1056

1057

117

116

117

116

1058

1059

121

120

121

120

1060

1061

125

124

125

124

1062

1063

129

128

129

128

1064

1065

133

132

133

132

1066

1067

137

136

137

136

1068

1069

141

140

141

140

1070

1071

145

144

145

144

1072

1073

149

148

149

148

1074

1075

153

152

153

152

1076

1077

157

156

157

156

1078

1079

161

160

161

160

1080

1081

165

164

165

164

1082

1083

169

168

169

168

1084

1085

173

172

173

172

1086

1087

177

176

177

176

1088

1089

181

180

181

180

1090

1091

185

184


221


222

1092 383 1094 384 1096 385 1098 386 1100 467 1262 468 1264 469 1266 470 1268 471 1270 472 1272 473 1274 474 1276 475 1278 476 1280 477 1282 478 1284 479 1286 480 1288 481 1290 482 1292 483 1294 484 1296 485 1298 486 1300 487 1302 488 1304 489 1306 490 1308 491 1310 492 1312 493 1314 494 1316 495 1318 496 1320 497

1093 2 1095 2 1097 2 1099 2 1101 2 1263 2 1265 2 1267 2 1269 2 1271 2 1273 2 1275 2 1277 2 1279 2 1281 2 1283 2 1285 2 1287 2 1289 2 1291 2 1293 2 1295 2 1297 2 1299 2 1301 2 1303 2 1305 2 1307 2 1309 2 1311 2 1313 2 1315 2 1317 2 1319 2 1321 2

185

184

1092

1093

189

188

189

188

1094

1095

193

192

193

192

1096

1097

197

196

197

196

1098

1099

201

200

201

200

1100

1101

525

524

525

524

1262

1263

529

528

529

528

1264

1265

533

532

533

532

1266

1267

537

536

537

536

1268

1269

541

540

541

540

1270

1271

545

544

545

544

1272

1273

549

548

549

548

1274

1275

553

552

553

552

1276

1277

557

556

557

556

1278

1279

561

560

561

560

1280

1281

565

564

565

564

1282

1283

569

568

569

568

1284

1285

573

572

573

572

1286

1287

577

576

577

576

1288

1289

581

580

581

580

1290

1291

585

584

585

584

1292

1293

589

588

589

588

1294

1295

593

592

593

592

1296

1297

597

596

597

596

1298

1299

601

600

601

600

1300

1301

605

604

605

604

1302

1303

609

608

609

608

1304

1305

613

612

613

612

1306

1307

617

616

617

616

1308

1309

621

620

621

620

1310

1311

625

624

625

624

1312

1313

629

628

629

628

1314

1315

633

632

633

632

1316

1317

637

636

637

636

1318

1319

641

640

641

640

1320

1321

645

644


Altair Engineering


1322 498 1324 499 1326 500 1328 501 1330 502 1332 503 1334 504 1336 505 1338 506 1340 507 1342 508 1344 509 1019 510 1021 511 1023 512 1025 513 1027 514 1029 515 1031 516 1033 517 1035 518 1037 519 1039 520 1041 521 1043 522 1045 523 1047 524 1049 525 1051 526 1053 527 1055 528 1057 529 1059 530 1061 531 1063 532

Altair Engineering

1323 2 1325 2 1327 2 1329 2 1331 2 1333 2 1335 2 1337 2 1339 2 1341 2 1343 2 1345 2 1346 2 1347 2 1348 2 1349 2 1350 2 1351 2 1352 2 1353 2 1354 2 1355 2 1356 2 1357 2 1358 2 1359 2 1360 2 1361 2 1362 2 1363 2 1364 2 1365 2 1366 2 1367 2 1368 2

645

644

1322

1323

649

648

649

648

1324

1325

653

652

653

652

1326

1327

657

656

657

656

1328

1329

661

660

661

660

1330

1331

665

664

665

664

1332

1333

669

668

669

668

1334

1335

673

672

673

672

1336

1337

677

676

677

676

1338

1339

681

680

681

680

1340

1341

685

684

685

684

1342

1343

689

688

20

23

18

19

691

37

691

37

1019

1346

693

41

693

41

1021

1347

695

45

695

45

1023

1348

697

49

697

49

1025

1349

699

53

699

53

1027

1350

701

57

701

57

1029

1351

703

61

703

61

1031

1352

705

65

705

65

1033

1353

707

69

707

69

1035

1354

709

73

709

73

1037

1355

711

77

711

77

1039

1356

713

81

713

81

1041

1357

715

85

715

85

1043

1358

717

89

717

89

1045

1359

719

93

719

93

1047

1360

721

97

721

97

1049

1361

723

101

723

101

1051

1362

725

105

725

105

1053

1363

727

109

727

109

1055

1364

729

113

729

113

1057

1365

731

117

731

117

1059

1366

733

121

733

121

1061

1367

735

125

735

125

1063

1368

737

129


223


224

1065 533 1067 534 1069 535 1071 536 1073 537 1075 538 1077 539 1079 540 1081 541 1083 542 1085 543 1087 544 1089 545 1091 546 1093 547 1095 548 1097 549 1099 550 1101 631 1263 632 1265 633 1267 634 1269 635 1271 636 1273 637 1275 638 1277 639 1279 640 1281 641 1283 642 1285 643 1287 644 1289 645 1291 646 1293 647

1369 2 1370 2 1371 2 1372 2 1373 2 1374 2 1375 2 1376 2 1377 2 1378 2 1379 2 1380 2 1381 2 1382 2 1383 2 1384 2 1385 2 1386 2 1387 2 1468 2 1469 2 1470 2 1471 2 1472 2 1473 2 1474 2 1475 2 1476 2 1477 2 1478 2 1479 2 1480 2 1481 2 1482 2 1483 2

737

129

1065

1369

739

133

739

133

1067

1370

741

137

741

137

1069

1371

743

141

743

141

1071

1372

745

145

745

145

1073

1373

747

149

747

149

1075

1374

749

153

749

153

1077

1375

751

157

751

157

1079

1376

753

161

753

161

1081

1377

755

165

755

165

1083

1378

757

169

757

169

1085

1379

759

173

759

173

1087

1380

761

177

761

177

1089

1381

763

181

763

181

1091

1382

765

185

765

185

1093

1383

767

189

767

189

1095

1384

769

193

769

193

1097

1385

771

197

771

197

1099

1386

773

201

773

201

1101

1387

935

525

935

525

1263

1468

937

529

937

529

1265

1469

939

533

939

533

1267

1470

941

537

941

537

1269

1471

943

541

943

541

1271

1472

945

545

945

545

1273

1473

947

549

947

549

1275

1474

949

553

949

553

1277

1475

951

557

951

557

1279

1476

953

561

953

561

1281

1477

955

565

955

565

1283

1478

957

569

957

569

1285

1479

959

573

959

573

1287

1480

961

577

961

577

1289

1481

963

581

963

581

1291

1482

965

585

965

585

1293

1483

967

589


Altair Engineering


1295 648 1297 649 1299 650 1301 651 1303 652 1305 653 1307 654 1309 655 1311 656 1313 657 1315 658 1317 659 1319 660 1321 661 1323 662 1325 663 1327 664 1329 665 1331 666 1333 667 1335 668 1337 669 1339 670 1341 671 1343 672 1345 673 1512 674 1516 675 1520 676 1524 677 1528 678 1532 679 1536 680 1540 681 1544 682

Altair Engineering

1484 2 1485 2 1486 2 1487 2 1488 2 1489 2 1490 2 1491 2 1492 2 1493 2 1494 2 1495 2 1496 2 1497 2 1498 2 1499 2 1500 2 1501 2 1502 2 1503 2 1504 2 1505 2 1506 2 1507 2 1508 2 1509 2 1513 2 1517 2 1521 2 1525 2 1529 2 1533 2 1537 2 1541 2 1545 2

967

589

1295

1484

969

593

969

593

1297

1485

971

597

971

597

1299

1486

973

601

973

601

1301

1487

975

605

975

605

1303

1488

977

609

977

609

1305

1489

979

613

979

613

1307

1490

981

617

981

617

1309

1491

983

621

983

621

1311

1492

985

625

985

625

1313

1493

987

629

987

629

1315

1494

989

633

989

633

1317

1495

991

637

991

637

1319

1496

993

641

993

641

1321

1497

995

645

995

645

1323

1498

997

649

997

649

1325

1499

999

653

999

653

1327

1500

1001

657

1001

657

1329

1501

1003

661

1003

661

1331

1502

1005

665

1005

665

1333

1503

1007

669

1007

669

1335

1504

1009

673

1009

673

1337

1505

1011

677

1011

677

1339

1506

1013

681

1013

681

1341

1507

1015

685

1015

685

1343

1508

1017

689

12

13

14

22

1510

1511

1510

1511

1512

1513

1514

1515

1514

1515

1516

1517

1518

1519

1518

1519

1520

1521

1522

1523

1522

1523

1524

1525

1526

1527

1526

1527

1528

1529

1530

1531

1530

1531

1532

1533

1534

1535

1534

1535

1536

1537

1538

1539

1538

1539

1540

1541

1542

1543

1542

1543

1544

1545

1546

1547


225


226

1548 683 1552 684 1556 685 1560 686 1564 687 1568 688 1572 689 1576 690 1580 691 1584 692 1588 693 1592 694 1596 695 1600 696 1604 697 1608 698 1612 699 1616 700 1620 701 1624 702 1628 703 1632 704 1636 705 1640 706 1644 707 1648 708 1652 709 1656 710 1660 711 1664 712 1668 713 1672 714 1676 795 2000 796 2004 797

1549 2 1553 2 1557 2 1561 2 1565 2 1569 2 1573 2 1577 2 1581 2 1585 2 1589 2 1593 2 1597 2 1601 2 1605 2 1609 2 1613 2 1617 2 1621 2 1625 2 1629 2 1633 2 1637 2 1641 2 1645 2 1649 2 1653 2 1657 2 1661 2 1665 2 1669 2 1673 2 1677 2 2001 2 2005 2

1546

1547

1548

1549

1550

1551

1550

1551

1552

1553

1554

1555

1554

1555

1556

1557

1558

1559

1558

1559

1560

1561

1562

1563

1562

1563

1564

1565

1566

1567

1566

1567

1568

1569

1570

1571

1570

1571

1572

1573

1574

1575

1574

1575

1576

1577

1578

1579

1578

1579

1580

1581

1582

1583

1582

1583

1584

1585

1586

1587

1586

1587

1588

1589

1590

1591

1590

1591

1592

1593

1594

1595

1594

1595

1596

1597

1598

1599

1598

1599

1600

1601

1602

1603

1602

1603

1604

1605

1606

1607

1606

1607

1608

1609

1610

1611

1610

1611

1612

1613

1614

1615

1614

1615

1616

1617

1618

1619

1618

1619

1620

1621

1622

1623

1622

1623

1624

1625

1626

1627

1626

1627

1628

1629

1630

1631

1630

1631

1632

1633

1634

1635

1634

1635

1636

1637

1638

1639

1638

1639

1640

1641

1642

1643

1642

1643

1644

1645

1646

1647

1646

1647

1648

1649

1650

1651

1650

1651

1652

1653

1654

1655

1654

1655

1656

1657

1658

1659

1658

1659

1660

1661

1662

1663

1662

1663

1664

1665

1666

1667

1666

1667

1668

1669

1670

1671

1670

1671

1672

1673

1674

1675

1674

1675

1676

1677

1998

1999

1998

1999

2000

2001

2002

2003

2002

2003

2004

2005

2006

2007


Altair Engineering


2008 798 2012 799 2016 800 2020 801 2024 802 2028 803 2032 804 2036 805 2040 806 2044 807 2048 808 2052 809 2056 810 2060 811 2064 812 2068 813 2072 814 2076 815 2080 816 2084 817 2088 818 2092 819 2096 820 2100 821 2104 822 2108 823 2112 824 2116 825 2120 826 2124 827 2128 828 2132 829 2136 830 2140 831 2144 832

Altair Engineering

2009 2 2013 2 2017 2 2021 2 2025 2 2029 2 2033 2 2037 2 2041 2 2045 2 2049 2 2053 2 2057 2 2061 2 2065 2 2069 2 2073 2 2077 2 2081 2 2085 2 2089 2 2093 2 2097 2 2101 2 2105 2 2109 2 2113 2 2117 2 2121 2 2125 2 2129 2 2133 2 2137 2 2141 2 2145 2

2006

2007

2008

2009

2010

2011

2010

2011

2012

2013

2014

2015

2014

2015

2016

2017

2018

2019

2018

2019

2020

2021

2022

2023

2022

2023

2024

2025

2026

2027

2026

2027

2028

2029

2030

2031

2030

2031

2032

2033

2034

2035

2034

2035

2036

2037

2038

2039

2038

2039

2040

2041

2042

2043

2042

2043

2044

2045

2046

2047

2046

2047

2048

2049

2050

2051

2050

2051

2052

2053

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2055

2054

2055

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2058

2059

2060

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2068

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2070

2071

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2078

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2086

2087

2088

2089

2090

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2090

2091

2092

2093

2094

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2095

2096

2097

2098

2099

2098

2099

2100

2101

2102

2103

2102

2103

2104

2105

2106

2107

2106

2107

2108

2109

2110

2111

2110

2111

2112

2113

2114

2115

2114

2115

2116

2117

2118

2119

2118

2119

2120

2121

2122

2123

2122

2123

2124

2125

2126

2127

2126

2127

2128

2129

2130

2131

2130

2131

2132

2133

2134

2135

2134

2135

2136

2137

2138

2139

2138

2139

2140

2141

2142

2143

2142

2143

2144

2145

2146

2147


227


228

2148 833 2152 834 2156 835 2160 836 2164 837 1513 838 1517 839 1521 840 1525 841 1529 842 1533 843 1537 844 1541 845 1545 846 1549 847 1553 848 1557 849 1561 850 1565 851 1569 852 1573 853 1577 854 1581 855 1585 856 1589 857 1593 858 1597 859 1601 860 1605 861 1609 862 1613 863 1617 864 1621 865 1625 866 1629 867

2149 2 2153 2 2157 2 2161 2 2165 2 36 2 40 2 44 2 48 2 52 2 56 2 60 2 64 2 68 2 72 2 76 2 80 2 84 2 88 2 92 2 96 2 100 2 104 2 108 2 112 2 116 2 120 2 124 2 128 2 132 2 136 2 140 2 144 2 148 2 152 2

2146

2147

2148

2149

2150

2151

2150

2151

2152

2153

2154

2155

2154

2155

2156

2157

2158

2159

2158

2159

2160

2161

2162

2163

11

12

22

21

35

1510

35

1510

1513

36

39

1514

39

1514

1517

40

43

1518

43

1518

1521

44

47

1522

47

1522

1525

48

51

1526

51

1526

1529

52

55

1530

55

1530

1533

56

59

1534

59

1534

1537

60

63

1538

63

1538

1541

64

67

1542

67

1542

1545

68

71

1546

71

1546

1549

72

75

1550

75

1550

1553

76

79

1554

79

1554

1557

80

83

1558

83

1558

1561

84

87

1562

87

1562

1565

88

91

1566

91

1566

1569

92

95

1570

95

1570

1573

96

99

1574

99

1574

1577

100

103

1578

103

1578

1581

104

107

1582

107

1582

1585

108

111

1586

111

1586

1589

112

115

1590

115

1590

1593

116

119

1594

119

1594

1597

120

123

1598

123

1598

1601

124

127

1602

127

1602

1605

128

131

1606

131

1606

1609

132

135

1610

135

1610

1613

136

139

1614

139

1614

1617

140

143

1618

143

1618

1621

144

147

1622

147

1622

1625

148

151

1626

151

1626

1629

152

155

1630


Altair Engineering


1633 868 1637 869 1641 870 1645 871 1649 872 1653 873 1657 874 1661 875 1665 876 1669 877 1673 878 1677 959 2001 960 2005 961 2009 962 2013 963 2017 964 2021 965 2025 966 2029 967 2033 968 2037 969 2041 970 2045 971 2049 972 2053 973 2057 974 2061 975 2065 976 2069 977 2073 978 2077 979 2081 980 2085 981 2089 982

Altair Engineering

156 2 160 2 164 2 168 2 172 2 176 2 180 2 184 2 188 2 192 2 196 2 200 2 524 2 528 2 532 2 536 2 540 2 544 2 548 2 552 2 556 2 560 2 564 2 568 2 572 2 576 2 580 2 584 2 588 2 592 2 596 2 600 2 604 2 608 2 612 2

155

1630

1633

156

159

1634

159

1634

1637

160

163

1638

163

1638

1641

164

167

1642

167

1642

1645

168

171

1646

171

1646

1649

172

175

1650

175

1650

1653

176

179

1654

179

1654

1657

180

183

1658

183

1658

1661

184

187

1662

187

1662

1665

188

191

1666

191

1666

1669

192

195

1670

195

1670

1673

196

199

1674

199

1674

1677

200

523

1998

523

1998

2001

524

527

2002

527

2002

2005

528

531

2006

531

2006

2009

532

535

2010

535

2010

2013

536

539

2014

539

2014

2017

540

543

2018

543

2018

2021

544

547

2022

547

2022

2025

548

551

2026

551

2026

2029

552

555

2030

555

2030

2033

556

559

2034

559

2034

2037

560

563

2038

563

2038

2041

564

567

2042

567

2042

2045

568

571

2046

571

2046

2049

572

575

2050

575

2050

2053

576

579

2054

579

2054

2057

580

583

2058

583

2058

2061

584

587

2062

587

2062

2065

588

591

2066

591

2066

2069

592

595

2070

595

2070

2073

596

599

2074

599

2074

2077

600

603

2078

603

2078

2081

604

607

2082

607

2082

2085

608

611

2086

611

2086

2089

612

615

2090


229


230

2093 983 2097 984 2101 985 2105 986 2109 987 2113 988 2117 989 2121 990 2125 991 2129 992 2133 993 2137 994 2141 995 2145 996 2149 997 2153 998 2157 999 2161 1000 2165 1001 2166 1002 2168 1003 2170 1004 2172 1005 2174 1006 2176 1007 2178 1008 2180 1009 2182 1010 2184 1011 2186 1012 2188 1013 2190 1014 2192 1015 2194 1016 2196 1017

616 2 620 2 624 2 628 2 632 2 636 2 640 2 644 2 648 2 652 2 656 2 660 2 664 2 668 2 672 2 676 2 680 2 684 2 688 2 2167 2 2169 2 2171 2 2173 2 2175 2 2177 2 2179 2 2181 2 2183 2 2185 2 2187 2 2189 2 2191 2 2193 2 2195 2 2197 2

615

2090

2093

616

619

2094

619

2094

2097

620

623

2098

623

2098

2101

624

627

2102

627

2102

2105

628

631

2106

631

2106

2109

632

635

2110

635

2110

2113

636

639

2114

639

2114

2117

640

643

2118

643

2118

2121

644

647

2122

647

2122

2125

648

651

2126

651

2126

2129

652

655

2130

655

2130

2133

656

659

2134

659

2134

2137

660

663

2138

663

2138

2141

664

667

2142

667

2142

2145

668

671

2146

671

2146

2149

672

675

2150

675

2150

2153

676

679

2154

679

2154

2157

680

683

2158

683

2158

2161

684

687

2162

22

14

15

16

1513

1512

1513

1512

2166

2167

1517

1516

1517

1516

2168

2169

1521

1520

1521

1520

2170

2171

1525

1524

1525

1524

2172

2173

1529

1528

1529

1528

2174

2175

1533

1532

1533

1532

2176

2177

1537

1536

1537

1536

2178

2179

1541

1540

1541

1540

2180

2181

1545

1544

1545

1544

2182

2183

1549

1548

1549

1548

2184

2185

1553

1552

1553

1552

2186

2187

1557

1556

1557

1556

2188

2189

1561

1560

1561

1560

2190

2191

1565

1564

1565

1564

2192

2193

1569

1568

1569

1568

2194

2195

1573

1572

1573

1572

2196

2197

1577

1576


Altair Engineering


2198 1018 2200 1019 2202 1020 2204 1021 2206 1022 2208 1023 2210 1024 2212 1025 2214 1026 2216 1027 2218 1028 2220 1029 2222 1030 2224 1031 2226 1032 2228 1033 2230 1034 2232 1035 2234 1036 2236 1037 2238 1038 2240 1039 2242 1040 2244 1041 2246 1042 2248 1123 2410 1124 2412 1125 2414 1126 2416 1127 2418 1128 2420 1129 2422 1130 2424 1131 2426 1132

Altair Engineering

2199 2 2201 2 2203 2 2205 2 2207 2 2209 2 2211 2 2213 2 2215 2 2217 2 2219 2 2221 2 2223 2 2225 2 2227 2 2229 2 2231 2 2233 2 2235 2 2237 2 2239 2 2241 2 2243 2 2245 2 2247 2 2249 2 2411 2 2413 2 2415 2 2417 2 2419 2 2421 2 2423 2 2425 2 2427 2

1577

1576

2198

2199

1581

1580

1581

1580

2200

2201

1585

1584

1585

1584

2202

2203

1589

1588

1589

1588

2204

2205

1593

1592

1593

1592

2206

2207

1597

1596

1597

1596

2208

2209

1601

1600

1601

1600

2210

2211

1605

1604

1605

1604

2212

2213

1609

1608

1609

1608

2214

2215

1613

1612

1613

1612

2216

2217

1617

1616

1617

1616

2218

2219

1621

1620

1621

1620

2220

2221

1625

1624

1625

1624

2222

2223

1629

1628

1629

1628

2224

2225

1633

1632

1633

1632

2226

2227

1637

1636

1637

1636

2228

2229

1641

1640

1641

1640

2230

2231

1645

1644

1645

1644

2232

2233

1649

1648

1649

1648

2234

2235

1653

1652

1653

1652

2236

2237

1657

1656

1657

1656

2238

2239

1661

1660

1661

1660

2240

2241

1665

1664

1665

1664

2242

2243

1669

1668

1669

1668

2244

2245

1673

1672

1673

1672

2246

2247

1677

1676

1677

1676

2248

2249

2001

2000

2001

2000

2410

2411

2005

2004

2005

2004

2412

2413

2009

2008

2009

2008

2414

2415

2013

2012

2013

2012

2416

2417

2017

2016

2017

2016

2418

2419

2021

2020

2021

2020

2420

2421

2025

2024

2025

2024

2422

2423

2029

2028

2029

2028

2424

2425

2033

2032

2033

2032

2426

2427

2037

2036


231


232

2428 1133 2430 1134 2432 1135 2434 1136 2436 1137 2438 1138 2440 1139 2442 1140 2444 1141 2446 1142 2448 1143 2450 1144 2452 1145 2454 1146 2456 1147 2458 1148 2460 1149 2462 1150 2464 1151 2466 1152 2468 1153 2470 1154 2472 1155 2474 1156 2476 1157 2478 1158 2480 1159 2482 1160 2484 1161 2486 1162 2488 1163 2490 1164 2492 1165 2167 1166 2169 1167

2429 2 2431 2 2433 2 2435 2 2437 2 2439 2 2441 2 2443 2 2445 2 2447 2 2449 2 2451 2 2453 2 2455 2 2457 2 2459 2 2461 2 2463 2 2465 2 2467 2 2469 2 2471 2 2473 2 2475 2 2477 2 2479 2 2481 2 2483 2 2485 2 2487 2 2489 2 2491 2 2493 2 1018 2 1020 2

2037

2036

2428

2429

2041

2040

2041

2040

2430

2431

2045

2044

2045

2044

2432

2433

2049

2048

2049

2048

2434

2435

2053

2052

2053

2052

2436

2437

2057

2056

2057

2056

2438

2439

2061

2060

2061

2060

2440

2441

2065

2064

2065

2064

2442

2443

2069

2068

2069

2068

2444

2445

2073

2072

2073

2072

2446

2447

2077

2076

2077

2076

2448

2449

2081

2080

2081

2080

2450

2451

2085

2084

2085

2084

2452

2453

2089

2088

2089

2088

2454

2455

2093

2092

2093

2092

2456

2457

2097

2096

2097

2096

2458

2459

2101

2100

2101

2100

2460

2461

2105

2104

2105

2104

2462

2463

2109

2108

2109

2108

2464

2465

2113

2112

2113

2112

2466

2467

2117

2116

2117

2116

2468

2469

2121

2120

2121

2120

2470

2471

2125

2124

2125

2124

2472

2473

2129

2128

2129

2128

2474

2475

2133

2132

2133

2132

2476

2477

2137

2136

2137

2136

2478

2479

2141

2140

2141

2140

2480

2481

2145

2144

2145

2144

2482

2483

2149

2148

2149

2148

2484

2485

2153

2152

2153

2152

2486

2487

2157

2156

2157

2156

2488

2489

2161

2160

2161

2160

2490

2491

2165

2164

21

22

16

17

36

1513

36

1513

2167

1018

40

1517

40

1517

2169

1020

44

1521


Altair Engineering


2171 1168 2173 1169 2175 1170 2177 1171 2179 1172 2181 1173 2183 1174 2185 1175 2187 1176 2189 1177 2191 1178 2193 1179 2195 1180 2197 1181 2199 1182 2201 1183 2203 1184 2205 1185 2207 1186 2209 1187 2211 1188 2213 1189 2215 1190 2217 1191 2219 1192 2221 1193 2223 1194 2225 1195 2227 1196 2229 1197 2231 1198 2233 1199 2235 1200 2237 1201 2239 1202

Altair Engineering

1022 2 1024 2 1026 2 1028 2 1030 2 1032 2 1034 2 1036 2 1038 2 1040 2 1042 2 1044 2 1046 2 1048 2 1050 2 1052 2 1054 2 1056 2 1058 2 1060 2 1062 2 1064 2 1066 2 1068 2 1070 2 1072 2 1074 2 1076 2 1078 2 1080 2 1082 2 1084 2 1086 2 1088 2 1090 2

44

1521

2171

1022

48

1525

48

1525

2173

1024

52

1529

52

1529

2175

1026

56

1533

56

1533

2177

1028

60

1537

60

1537

2179

1030

64

1541

64

1541

2181

1032

68

1545

68

1545

2183

1034

72

1549

72

1549

2185

1036

76

1553

76

1553

2187

1038

80

1557

80

1557

2189

1040

84

1561

84

1561

2191

1042

88

1565

88

1565

2193

1044

92

1569

92

1569

2195

1046

96

1573

96

1573

2197

1048

100

1577

100

1577

2199

1050

104

1581

104

1581

2201

1052

108

1585

108

1585

2203

1054

112

1589

112

1589

2205

1056

116

1593

116

1593

2207

1058

120

1597

120

1597

2209

1060

124

1601

124

1601

2211

1062

128

1605

128

1605

2213

1064

132

1609

132

1609

2215

1066

136

1613

136

1613

2217

1068

140

1617

140

1617

2219

1070

144

1621

144

1621

2221

1072

148

1625

148

1625

2223

1074

152

1629

152

1629

2225

1076

156

1633

156

1633

2227

1078

160

1637

160

1637

2229

1080

164

1641

164

1641

2231

1082

168

1645

168

1645

2233

1084

172

1649

172

1649

2235

1086

176

1653

176

1653

2237

1088

180

1657

180

1657

2239

1090

184

1661


233


234

2241 1203 2243 1204 2245 1205 2247 1206 2249 1287 2411 1288 2413 1289 2415 1290 2417 1291 2419 1292 2421 1293 2423 1294 2425 1295 2427 1296 2429 1297 2431 1298 2433 1299 2435 1300 2437 1301 2439 1302 2441 1303 2443 1304 2445 1305 2447 1306 2449 1307 2451 1308 2453 1309 2455 1310 2457 1311 2459 1312 2461 1313 2463 1314 2465 1315 2467 1316 2469 1317

1092 2 1094 2 1096 2 1098 2 1100 2 1262 2 1264 2 1266 2 1268 2 1270 2 1272 2 1274 2 1276 2 1278 2 1280 2 1282 2 1284 2 1286 2 1288 2 1290 2 1292 2 1294 2 1296 2 1298 2 1300 2 1302 2 1304 2 1306 2 1308 2 1310 2 1312 2 1314 2 1316 2 1318 2 1320 2

184

1661

2241

1092

188

1665

188

1665

2243

1094

192

1669

192

1669

2245

1096

196

1673

196

1673

2247

1098

200

1677

200

1677

2249

1100

524

2001

524

2001

2411

1262

528

2005

528

2005

2413

1264

532

2009

532

2009

2415

1266

536

2013

536

2013

2417

1268

540

2017

540

2017

2419

1270

544

2021

544

2021

2421

1272

548

2025

548

2025

2423

1274

552

2029

552

2029

2425

1276

556

2033

556

2033

2427

1278

560

2037

560

2037

2429

1280

564

2041

564

2041

2431

1282

568

2045

568

2045

2433

1284

572

2049

572

2049

2435

1286

576

2053

576

2053

2437

1288

580

2057

580

2057

2439

1290

584

2061

584

2061

2441

1292

588

2065

588

2065

2443

1294

592

2069

592

2069

2445

1296

596

2073

596

2073

2447

1298

600

2077

600

2077

2449

1300

604

2081

604

2081

2451

1302

608

2085

608

2085

2453

1304

612

2089

612

2089

2455

1306

616

2093

616

2093

2457

1308

620

2097

620

2097

2459

1310

624

2101

624

2101

2461

1312

628

2105

628

2105

2463

1314

632

2109

632

2109

2465

1316

636

2113

636

2113

2467

1318

640

2117

640

2117

2469

1320

644

2121


Altair Engineering


2471 1318 2473 1319 2475 1320 2477 1321 2479 1322 2481 1323 2483 1324 2485 1325 2487 1326 2489 1327 2491 1328 2493 1329 35 1330 39 1331 43 1332 47 1333 51 1334 55 1335 59 1336 63 1337 67 1338 71 1339 75 1340 79 1341 83 1342 87 1343 91 1344 95 1345 99 1346 103 1347 107 1348 111 1349 115 1350 119 1351 123 1352

Altair Engineering

1322 2 1324 2 1326 2 1328 2 1330 2 1332 2 1334 2 1336 2 1338 2 1340 2 1342 2 1344 2 2495 2 2497 2 2499 2 2501 2 2503 2 2505 2 2507 2 2509 2 2511 2 2513 2 2515 2 2517 2 2519 2 2521 2 2523 2 2525 2 2527 2 2529 2 2531 2 2533 2 2535 2 2537 2 2539 2

644

2121

2471

1322

648

2125

648

2125

2473

1324

652

2129

652

2129

2475

1326

656

2133

656

2133

2477

1328

660

2137

660

2137

2479

1330

664

2141

664

2141

2481

1332

668

2145

668

2145

2483

1334

672

2149

672

2149

2485

1336

676

2153

676

2153

2487

1338

680

2157

680

2157

2489

1340

684

2161

684

2161

2491

1342

688

2165

33

12

11

31

2494

1510

2494

1510

35

2495

2496

1514

2496

1514

39

2497

2498

1518

2498

1518

43

2499

2500

1522

2500

1522

47

2501

2502

1526

2502

1526

51

2503

2504

1530

2504

1530

55

2505

2506

1534

2506

1534

59

2507

2508

1538

2508

1538

63

2509

2510

1542

2510

1542

67

2511

2512

1546

2512

1546

71

2513

2514

1550

2514

1550

75

2515

2516

1554

2516

1554

79

2517

2518

1558

2518

1558

83

2519

2520

1562

2520

1562

87

2521

2522

1566

2522

1566

91

2523

2524

1570

2524

1570

95

2525

2526

1574

2526

1574

99

2527

2528

1578

2528

1578

103

2529

2530

1582

2530

1582

107

2531

2532

1586

2532

1586

111

2533

2534

1590

2534

1590

115

2535

2536

1594

2536

1594

119

2537

2538

1598

2538

1598

123

2539

2540

1602


235


236

127 1353 131 1354 135 1355 139 1356 143 1357 147 1358 151 1359 155 1360 159 1361 163 1362 167 1363 171 1364 175 1365 179 1366 183 1367 187 1368 191 1369 195 1370 199 1451 523 1452 527 1453 531 1454 535 1455 539 1456 543 1457 547 1458 551 1459 555 1460 559 1461 563 1462 567 1463 571 1464 575 1465 579 1466 583 1467

2541 2 2543 2 2545 2 2547 2 2549 2 2551 2 2553 2 2555 2 2557 2 2559 2 2561 2 2563 2 2565 2 2567 2 2569 2 2571 2 2573 2 2575 2 2577 2 2739 2 2741 2 2743 2 2745 2 2747 2 2749 2 2751 2 2753 2 2755 2 2757 2 2759 2 2761 2 2763 2 2765 2 2767 2 2769 2

2540

1602

127

2541

2542

1606

2542

1606

131

2543

2544

1610

2544

1610

135

2545

2546

1614

2546

1614

139

2547

2548

1618

2548

1618

143

2549

2550

1622

2550

1622

147

2551

2552

1626

2552

1626

151

2553

2554

1630

2554

1630

155

2555

2556

1634

2556

1634

159

2557

2558

1638

2558

1638

163

2559

2560

1642

2560

1642

167

2561

2562

1646

2562

1646

171

2563

2564

1650

2564

1650

175

2565

2566

1654

2566

1654

179

2567

2568

1658

2568

1658

183

2569

2570

1662

2570

1662

187

2571

2572

1666

2572

1666

191

2573

2574

1670

2574

1670

195

2575

2576

1674

2576

1674

199

2577

2738

1998

2738

1998

523

2739

2740

2002

2740

2002

527

2741

2742

2006

2742

2006

531

2743

2744

2010

2744

2010

535

2745

2746

2014

2746

2014

539

2747

2748

2018

2748

2018

543

2749

2750

2022

2750

2022

547

2751

2752

2026

2752

2026

551

2753

2754

2030

2754

2030

555

2755

2756

2034

2756

2034

559

2757

2758

2038

2758

2038

563

2759

2760

2042

2760

2042

567

2761

2762

2046

2762

2046

571

2763

2764

2050

2764

2050

575

2765

2766

2054

2766

2054

579

2767

2768

2058

2768

2058

583

2769

2770

2062


Altair Engineering


587 1468 591 1469 595 1470 599 1471 603 1472 607 1473 611 1474 615 1475 619 1476 623 1477 627 1478 631 1479 635 1480 639 1481 643 1482 647 1483 651 1484 655 1485 659 1486 663 1487 667 1488 671 1489 675 1490 679 1491 683 1492 687 1493 1510 1494 1514 1495 1518 1496 1522 1497 1526 1498 1530 1499 1534 1500 1538 1501 1542 1502

Altair Engineering

2771 2 2773 2 2775 2 2777 2 2779 2 2781 2 2783 2 2785 2 2787 2 2789 2 2791 2 2793 2 2795 2 2797 2 2799 2 2801 2 2803 2 2805 2 2807 2 2809 2 2811 2 2813 2 2815 2 2817 2 2819 2 2821 2 2494 2 2496 2 2498 2 2500 2 2502 2 2504 2 2506 2 2508 2 2510 2

2770

2062

587

2771

2772

2066

2772

2066

591

2773

2774

2070

2774

2070

595

2775

2776

2074

2776

2074

599

2777

2778

2078

2778

2078

603

2779

2780

2082

2780

2082

607

2781

2782

2086

2782

2086

611

2783

2784

2090

2784

2090

615

2785

2786

2094

2786

2094

619

2787

2788

2098

2788

2098

623

2789

2790

2102

2790

2102

627

2791

2792

2106

2792

2106

631

2793

2794

2110

2794

2110

635

2795

2796

2114

2796

2114

639

2797

2798

2118

2798

2118

643

2799

2800

2122

2800

2122

647

2801

2802

2126

2802

2126

651

2803

2804

2130

2804

2130

655

2805

2806

2134

2806

2134

659

2807

2808

2138

2808

2138

663

2809

2810

2142

2810

2142

667

2811

2812

2146

2812

2146

671

2813

2814

2150

2814

2150

675

2815

2816

2154

2816

2154

679

2817

2818

2158

2818

2158

683

2819

2820

2162

30

13

12

33

2822

1511

2822

1511

1510

2494

2823

1515

2823

1515

1514

2496

2824

1519

2824

1519

1518

2498

2825

1523

2825

1523

1522

2500

2826

1527

2826

1527

1526

2502

2827

1531

2827

1531

1530

2504

2828

1535

2828

1535

1534

2506

2829

1539

2829

1539

1538

2508

2830

1543

2830

1543

1542

2510

2831

1547


237


238

1546 1503 1550 1504 1554 1505 1558 1506 1562 1507 1566 1508 1570 1509 1574 1510 1578 1511 1582 1512 1586 1513 1590 1514 1594 1515 1598 1516 1602 1517 1606 1518 1610 1519 1614 1520 1618 1521 1622 1522 1626 1523 1630 1524 1634 1525 1638 1526 1642 1527 1646 1528 1650 1529 1654 1530 1658 1531 1662 1532 1666 1533 1670 1534 1674 1615 1998 1616 2002 1617

2512 2 2514 2 2516 2 2518 2 2520 2 2522 2 2524 2 2526 2 2528 2 2530 2 2532 2 2534 2 2536 2 2538 2 2540 2 2542 2 2544 2 2546 2 2548 2 2550 2 2552 2 2554 2 2556 2 2558 2 2560 2 2562 2 2564 2 2566 2 2568 2 2570 2 2572 2 2574 2 2576 2 2738 2 2740 2

2831

1547

1546

2512

2832

1551

2832

1551

1550

2514

2833

1555

2833

1555

1554

2516

2834

1559

2834

1559

1558

2518

2835

1563

2835

1563

1562

2520

2836

1567

2836

1567

1566

2522

2837

1571

2837

1571

1570

2524

2838

1575

2838

1575

1574

2526

2839

1579

2839

1579

1578

2528

2840

1583

2840

1583

1582

2530

2841

1587

2841

1587

1586

2532

2842

1591

2842

1591

1590

2534

2843

1595

2843

1595

1594

2536

2844

1599

2844

1599

1598

2538

2845

1603

2845

1603

1602

2540

2846

1607

2846

1607

1606

2542

2847

1611

2847

1611

1610

2544

2848

1615

2848

1615

1614

2546

2849

1619

2849

1619

1618

2548

2850

1623

2850

1623

1622

2550

2851

1627

2851

1627

1626

2552

2852

1631

2852

1631

1630

2554

2853

1635

2853

1635

1634

2556

2854

1639

2854

1639

1638

2558

2855

1643

2855

1643

1642

2560

2856

1647

2856

1647

1646

2562

2857

1651

2857

1651

1650

2564

2858

1655

2858

1655

1654

2566

2859

1659

2859

1659

1658

2568

2860

1663

2860

1663

1662

2570

2861

1667

2861

1667

1666

2572

2862

1671

2862

1671

1670

2574

2863

1675

2863

1675

1674

2576

2944

1999

2944

1999

1998

2738

2945

2003

2945

2003

2002

2740

2946

2007


Altair Engineering


2006 1618 2010 1619 2014 1620 2018 1621 2022 1622 2026 1623 2030 1624 2034 1625 2038 1626 2042 1627 2046 1628 2050 1629 2054 1630 2058 1631 2062 1632 2066 1633 2070 1634 2074 1635 2078 1636 2082 1637 2086 1638 2090 1639 2094 1640 2098 1641 2102 1642 2106 1643 2110 1644 2114 1645 2118 1646 2122 1647 2126 1648 2130 1649 2134 1650 2138 1651 2142 1652

Altair Engineering

2742 2 2744 2 2746 2 2748 2 2750 2 2752 2 2754 2 2756 2 2758 2 2760 2 2762 2 2764 2 2766 2 2768 2 2770 2 2772 2 2774 2 2776 2 2778 2 2780 2 2782 2 2784 2 2786 2 2788 2 2790 2 2792 2 2794 2 2796 2 2798 2 2800 2 2802 2 2804 2 2806 2 2808 2 2810 2

2946

2007

2006

2742

2947

2011

2947

2011

2010

2744

2948

2015

2948

2015

2014

2746

2949

2019

2949

2019

2018

2748

2950

2023

2950

2023

2022

2750

2951

2027

2951

2027

2026

2752

2952

2031

2952

2031

2030

2754

2953

2035

2953

2035

2034

2756

2954

2039

2954

2039

2038

2758

2955

2043

2955

2043

2042

2760

2956

2047

2956

2047

2046

2762

2957

2051

2957

2051

2050

2764

2958

2055

2958

2055

2054

2766

2959

2059

2959

2059

2058

2768

2960

2063

2960

2063

2062

2770

2961

2067

2961

2067

2066

2772

2962

2071

2962

2071

2070

2774

2963

2075

2963

2075

2074

2776

2964

2079

2964

2079

2078

2778

2965

2083

2965

2083

2082

2780

2966

2087

2966

2087

2086

2782

2967

2091

2967

2091

2090

2784

2968

2095

2968

2095

2094

2786

2969

2099

2969

2099

2098

2788

2970

2103

2970

2103

2102

2790

2971

2107

2971

2107

2106

2792

2972

2111

2972

2111

2110

2794

2973

2115

2973

2115

2114

2796

2974

2119

2974

2119

2118

2798

2975

2123

2975

2123

2122

2800

2976

2127

2976

2127

2126

2802

2977

2131

2977

2131

2130

2804

2978

2135

2978

2135

2134

2806

2979

2139

2979

2139

2138

2808

2980

2143

2980

2143

2142

2810

2981

2147


239


240

2146 1653 2150 1654 2154 1655 2158 1656 2162 1657 2495 1658 2497 1659 2499 1660 2501 1661 2503 1662 2505 1663 2507 1664 2509 1665 2511 1666 2513 1667 2515 1668 2517 1669 2519 1670 2521 1671 2523 1672 2525 1673 2527 1674 2529 1675 2531 1676 2533 1677 2535 1678 2537 1679 2539 1680 2541 1681 2543 1682 2545 1683 2547 1684 2549 1685 2551 1686 2553 1687

2812 2 2814 2 2816 2 2818 2 2820 2 2987 2 2989 2 2991 2 2993 2 2995 2 2997 2 2999 2 3001 2 3003 2 3005 2 3007 2 3009 2 3011 2 3013 2 3015 2 3017 2 3019 2 3021 2 3023 2 3025 2 3027 2 3029 2 3031 2 3033 2 3035 2 3037 2 3039 2 3041 2 3043 2 3045 2

2981

2147

2146

2812

2982

2151

2982

2151

2150

2814

2983

2155

2983

2155

2154

2816

2984

2159

2984

2159

2158

2818

2985

2163

28

33

31

27

2986

2494

2986

2494

2495

2987

2988

2496

2988

2496

2497

2989

2990

2498

2990

2498

2499

2991

2992

2500

2992

2500

2501

2993

2994

2502

2994

2502

2503

2995

2996

2504

2996

2504

2505

2997

2998

2506

2998

2506

2507

2999

3000

2508

3000

2508

2509

3001

3002

2510

3002

2510

2511

3003

3004

2512

3004

2512

2513

3005

3006

2514

3006

2514

2515

3007

3008

2516

3008

2516

2517

3009

3010

2518

3010

2518

2519

3011

3012

2520

3012

2520

2521

3013

3014

2522

3014

2522

2523

3015

3016

2524

3016

2524

2525

3017

3018

2526

3018

2526

2527

3019

3020

2528

3020

2528

2529

3021

3022

2530

3022

2530

2531

3023

3024

2532

3024

2532

2533

3025

3026

2534

3026

2534

2535

3027

3028

2536

3028

2536

2537

3029

3030

2538

3030

2538

2539

3031

3032

2540

3032

2540

2541

3033

3034

2542

3034

2542

2543

3035

3036

2544

3036

2544

2545

3037

3038

2546

3038

2546

2547

3039

3040

2548

3040

2548

2549

3041

3042

2550

3042

2550

2551

3043

3044

2552

3044

2552

2553

3045

3046

2554


Altair Engineering


2555 1688 2557 1689 2559 1690 2561 1691 2563 1692 2565 1693 2567 1694 2569 1695 2571 1696 2573 1697 2575 1698 2577 1779 2739 1780 2741 1781 2743 1782 2745 1783 2747 1784 2749 1785 2751 1786 2753 1787 2755 1788 2757 1789 2759 1790 2761 1791 2763 1792 2765 1793 2767 1794 2769 1795 2771 1796 2773 1797 2775 1798 2777 1799 2779 1800 2781 1801 2783 1802

Altair Engineering

3047 2 3049 2 3051 2 3053 2 3055 2 3057 2 3059 2 3061 2 3063 2 3065 2 3067 2 3069 2 3231 2 3233 2 3235 2 3237 2 3239 2 3241 2 3243 2 3245 2 3247 2 3249 2 3251 2 3253 2 3255 2 3257 2 3259 2 3261 2 3263 2 3265 2 3267 2 3269 2 3271 2 3273 2 3275 2

3046

2554

2555

3047

3048

2556

3048

2556

2557

3049

3050

2558

3050

2558

2559

3051

3052

2560

3052

2560

2561

3053

3054

2562

3054

2562

2563

3055

3056

2564

3056

2564

2565

3057

3058

2566

3058

2566

2567

3059

3060

2568

3060

2568

2569

3061

3062

2570

3062

2570

2571

3063

3064

2572

3064

2572

2573

3065

3066

2574

3066

2574

2575

3067

3068

2576

3068

2576

2577

3069

3230

2738

3230

2738

2739

3231

3232

2740

3232

2740

2741

3233

3234

2742

3234

2742

2743

3235

3236

2744

3236

2744

2745

3237

3238

2746

3238

2746

2747

3239

3240

2748

3240

2748

2749

3241

3242

2750

3242

2750

2751

3243

3244

2752

3244

2752

2753

3245

3246

2754

3246

2754

2755

3247

3248

2756

3248

2756

2757

3249

3250

2758

3250

2758

2759

3251

3252

2760

3252

2760

2761

3253

3254

2762

3254

2762

2763

3255

3256

2764

3256

2764

2765

3257

3258

2766

3258

2766

2767

3259

3260

2768

3260

2768

2769

3261

3262

2770

3262

2770

2771

3263

3264

2772

3264

2772

2773

3265

3266

2774

3266

2774

2775

3267

3268

2776

3268

2776

2777

3269

3270

2778

3270

2778

2779

3271

3272

2780

3272

2780

2781

3273

3274

2782

3274

2782

2783

3275

3276

2784


241


242

2785 1803 2787 1804 2789 1805 2791 1806 2793 1807 2795 1808 2797 1809 2799 1810 2801 1811 2803 1812 2805 1813 2807 1814 2809 1815 2811 1816 2813 1817 2815 1818 2817 1819 2819 1820 2821 1821 2494 1822 2496 1823 2498 1824 2500 1825 2502 1826 2504 1827 2506 1828 2508 1829 2510 1830 2512 1831 2514 1832 2516 1833 2518 1834 2520 1835 2522 1836 2524 1837

3277 2 3279 2 3281 2 3283 2 3285 2 3287 2 3289 2 3291 2 3293 2 3295 2 3297 2 3299 2 3301 2 3303 2 3305 2 3307 2 3309 2 3311 2 3313 2 2986 2 2988 2 2990 2 2992 2 2994 2 2996 2 2998 2 3000 2 3002 2 3004 2 3006 2 3008 2 3010 2 3012 2 3014 2 3016 2

3276

2784

2785

3277

3278

2786

3278

2786

2787

3279

3280

2788

3280

2788

2789

3281

3282

2790

3282

2790

2791

3283

3284

2792

3284

2792

2793

3285

3286

2794

3286

2794

2795

3287

3288

2796

3288

2796

2797

3289

3290

2798

3290

2798

2799

3291

3292

2800

3292

2800

2801

3293

3294

2802

3294

2802

2803

3295

3296

2804

3296

2804

2805

3297

3298

2806

3298

2806

2807

3299

3300

2808

3300

2808

2809

3301

3302

2810

3302

2810

2811

3303

3304

2812

3304

2812

2813

3305

3306

2814

3306

2814

2815

3307

3308

2816

3308

2816

2817

3309

3310

2818

3310

2818

2819

3311

3312

2820

29

30

33

28

3314

2822

3314

2822

2494

2986

3315

2823

3315

2823

2496

2988

3316

2824

3316

2824

2498

2990

3317

2825

3317

2825

2500

2992

3318

2826

3318

2826

2502

2994

3319

2827

3319

2827

2504

2996

3320

2828

3320

2828

2506

2998

3321

2829

3321

2829

2508

3000

3322

2830

3322

2830

2510

3002

3323

2831

3323

2831

2512

3004

3324

2832

3324

2832

2514

3006

3325

2833

3325

2833

2516

3008

3326

2834

3326

2834

2518

3010

3327

2835

3327

2835

2520

3012

3328

2836

3328

2836

2522

3014

3329

2837

3329

2837

2524

3016

3330

2838


Altair Engineering


2526 1838 2528 1839 2530 1840 2532 1841 2534 1842 2536 1843 2538 1844 2540 1845 2542 1846 2544 1847 2546 1848 2548 1849 2550 1850 2552 1851 2554 1852 2556 1853 2558 1854 2560 1855 2562 1856 2564 1857 2566 1858 2568 1859 2570 1860 2572 1861 2574 1862 2576 1943 2738 1944 2740 1945 2742 1946 2744 1947 2746 1948 2748 1949 2750 1950 2752 1951 2754 1952

Altair Engineering

3018 2 3020 2 3022 2 3024 2 3026 2 3028 2 3030 2 3032 2 3034 2 3036 2 3038 2 3040 2 3042 2 3044 2 3046 2 3048 2 3050 2 3052 2 3054 2 3056 2 3058 2 3060 2 3062 2 3064 2 3066 2 3068 2 3230 2 3232 2 3234 2 3236 2 3238 2 3240 2 3242 2 3244 2 3246 2

3330

2838

2526

3018

3331

2839

3331

2839

2528

3020

3332

2840

3332

2840

2530

3022

3333

2841

3333

2841

2532

3024

3334

2842

3334

2842

2534

3026

3335

2843

3335

2843

2536

3028

3336

2844

3336

2844

2538

3030

3337

2845

3337

2845

2540

3032

3338

2846

3338

2846

2542

3034

3339

2847

3339

2847

2544

3036

3340

2848

3340

2848

2546

3038

3341

2849

3341

2849

2548

3040

3342

2850

3342

2850

2550

3042

3343

2851

3343

2851

2552

3044

3344

2852

3344

2852

2554

3046

3345

2853

3345

2853

2556

3048

3346

2854

3346

2854

2558

3050

3347

2855

3347

2855

2560

3052

3348

2856

3348

2856

2562

3054

3349

2857

3349

2857

2564

3056

3350

2858

3350

2858

2566

3058

3351

2859

3351

2859

2568

3060

3352

2860

3352

2860

2570

3062

3353

2861

3353

2861

2572

3064

3354

2862

3354

2862

2574

3066

3355

2863

3355

2863

2576

3068

3436

2944

3436

2944

2738

3230

3437

2945

3437

2945

2740

3232

3438

2946

3438

2946

2742

3234

3439

2947

3439

2947

2744

3236

3440

2948

3440

2948

2746

3238

3441

2949

3441

2949

2748

3240

3442

2950

3442

2950

2750

3242

3443

2951

3443

2951

2752

3244

3444

2952

3444

2952

2754

3246

3445

2953


243


244

2756 1953 2758 1954 2760 1955 2762 1956 2764 1957 2766 1958 2768 1959 2770 1960 2772 1961 2774 1962 2776 1963 2778 1964 2780 1965 2782 1966 2784 1967 2786 1968 2788 1969 2790 1970 2792 1971 2794 1972 2796 1973 2798 1974 2800 1975 2802 1976 2804 1977 2806 1978 2808 1979 2810 1980 2812 1981 2814 1982 2816 1983 2818 1984 2820 1985 690 1986 692 1987

3248 2 3250 2 3252 2 3254 2 3256 2 3258 2 3260 2 3262 2 3264 2 3266 2 3268 2 3270 2 3272 2 3274 2 3276 2 3278 2 3280 2 3282 2 3284 2 3286 2 3288 2 3290 2 3292 2 3294 2 3296 2 3298 2 3300 2 3302 2 3304 2 3306 2 3308 2 3310 2 3312 2 3479 2 3481 2

3445

2953

2756

3248

3446

2954

3446

2954

2758

3250

3447

2955

3447

2955

2760

3252

3448

2956

3448

2956

2762

3254

3449

2957

3449

2957

2764

3256

3450

2958

3450

2958

2766

3258

3451

2959

3451

2959

2768

3260

3452

2960

3452

2960

2770

3262

3453

2961

3453

2961

2772

3264

3454

2962

3454

2962

2774

3266

3455

2963

3455

2963

2776

3268

3456

2964

3456

2964

2778

3270

3457

2965

3457

2965

2780

3272

3458

2966

3458

2966

2782

3274

3459

2967

3459

2967

2784

3276

3460

2968

3460

2968

2786

3278

3461

2969

3461

2969

2788

3280

3462

2970

3462

2970

2790

3282

3463

2971

3463

2971

2792

3284

3464

2972

3464

2972

2794

3286

3465

2973

3465

2973

2796

3288

3466

2974

3466

2974

2798

3290

3467

2975

3467

2975

2800

3292

3468

2976

3468

2976

2802

3294

3469

2977

3469

2977

2804

3296

3470

2978

3470

2978

2806

3298

3471

2979

3471

2979

2808

3300

3472

2980

3472

2980

2810

3302

3473

2981

3473

2981

2812

3304

3474

2982

3474

2982

2814

3306

3475

2983

3475

2983

2816

3308

3476

2984

3476

2984

2818

3310

3477

2985

32

10

9

24

3478

34

3478

34

690

3479

3480

38

3480

38

692

3481

3482

42


Altair Engineering


694 1988 696 1989 698 1990 700 1991 702 1992 704 1993 706 1994 708 1995 710 1996 712 1997 714 1998 716 1999 718 2000 720 2001 722 2002 724 2003 726 2004 728 2005 730 2006 732 2007 734 2008 736 2009 738 2010 740 2011 742 2012 744 2013 746 2014 748 2015 750 2016 752 2017 754 2018 756 2019 758 2020 760 2021 762 2022

Altair Engineering

3483 2 3485 2 3487 2 3489 2 3491 2 3493 2 3495 2 3497 2 3499 2 3501 2 3503 2 3505 2 3507 2 3509 2 3511 2 3513 2 3515 2 3517 2 3519 2 3521 2 3523 2 3525 2 3527 2 3529 2 3531 2 3533 2 3535 2 3537 2 3539 2 3541 2 3543 2 3545 2 3547 2 3549 2 3551 2

3482

42

694

3483

3484

46

3484

46

696

3485

3486

50

3486

50

698

3487

3488

54

3488

54

700

3489

3490

58

3490

58

702

3491

3492

62

3492

62

704

3493

3494

66

3494

66

706

3495

3496

70

3496

70

708

3497

3498

74

3498

74

710

3499

3500

78

3500

78

712

3501

3502

82

3502

82

714

3503

3504

86

3504

86

716

3505

3506

90

3506

90

718

3507

3508

94

3508

94

720

3509

3510

98

3510

98

722

3511

3512

102

3512

102

724

3513

3514

106

3514

106

726

3515

3516

110

3516

110

728

3517

3518

114

3518

114

730

3519

3520

118

3520

118

732

3521

3522

122

3522

122

734

3523

3524

126

3524

126

736

3525

3526

130

3526

130

738

3527

3528

134

3528

134

740

3529

3530

138

3530

138

742

3531

3532

142

3532

142

744

3533

3534

146

3534

146

746

3535

3536

150

3536

150

748

3537

3538

154

3538

154

750

3539

3540

158

3540

158

752

3541

3542

162

3542

162

754

3543

3544

166

3544

166

756

3545

3546

170

3546

170

758

3547

3548

174

3548

174

760

3549

3550

178

3550

178

762

3551

3552

182


245


246

764 2023 766 2024 768 2025 770 2026 772 2107 934 2108 936 2109 938 2110 940 2111 942 2112 944 2113 946 2114 948 2115 950 2116 952 2117 954 2118 956 2119 958 2120 960 2121 962 2122 964 2123 966 2124 968 2125 970 2126 972 2127 974 2128 976 2129 978 2130 980 2131 982 2132 984 2133 986 2134 988 2135 990 2136 992 2137

3553 2 3555 2 3557 2 3559 2 3561 2 3723 2 3725 2 3727 2 3729 2 3731 2 3733 2 3735 2 3737 2 3739 2 3741 2 3743 2 3745 2 3747 2 3749 2 3751 2 3753 2 3755 2 3757 2 3759 2 3761 2 3763 2 3765 2 3767 2 3769 2 3771 2 3773 2 3775 2 3777 2 3779 2 3781 2

3552

182

764

3553

3554

186

3554

186

766

3555

3556

190

3556

190

768

3557

3558

194

3558

194

770

3559

3560

198

3560

198

772

3561

3722

522

3722

522

934

3723

3724

526

3724

526

936

3725

3726

530

3726

530

938

3727

3728

534

3728

534

940

3729

3730

538

3730

538

942

3731

3732

542

3732

542

944

3733

3734

546

3734

546

946

3735

3736

550

3736

550

948

3737

3738

554

3738

554

950

3739

3740

558

3740

558

952

3741

3742

562

3742

562

954

3743

3744

566

3744

566

956

3745

3746

570

3746

570

958

3747

3748

574

3748

574

960

3749

3750

578

3750

578

962

3751

3752

582

3752

582

964

3753

3754

586

3754

586

966

3755

3756

590

3756

590

968

3757

3758

594

3758

594

970

3759

3760

598

3760

598

972

3761

3762

602

3762

602

974

3763

3764

606

3764

606

976

3765

3766

610

3766

610

978

3767

3768

614

3768

614

980

3769

3770

618

3770

618

982

3771

3772

622

3772

622

984

3773

3774

626

3774

626

986

3775

3776

630

3776

630

988

3777

3778

634

3778

634

990

3779

3780

638

3780

638

992

3781

3782

642


Altair Engineering


994 2138 996 2139 998 2140 1000 2141 1002 2142 1004 2143 1006 2144 1008 2145 1010 2146 1012 2147 1014 2148 1016 2149 34 2150 38 2151 42 2152 46 2153 50 2154 54 2155 58 2156 62 2157 66 2158 70 2159 74 2160 78 2161 82 2162 86 2163 90 2164 94 2165 98 2166 102 2167 106 2168 110 2169 114 2170 118 2171 122 2172

Altair Engineering

3783 2 3785 2 3787 2 3789 2 3791 2 3793 2 3795 2 3797 2 3799 2 3801 2 3803 2 3805 2 3478 2 3480 2 3482 2 3484 2 3486 2 3488 2 3490 2 3492 2 3494 2 3496 2 3498 2 3500 2 3502 2 3504 2 3506 2 3508 2 3510 2 3512 2 3514 2 3516 2 3518 2 3520 2 3522 2

3782

642

994

3783

3784

646

3784

646

996

3785

3786

650

3786

650

998

3787

3788

654

3788

654

1000

3789

3790

658

3790

658

1002

3791

3792

662

3792

662

1004

3793

3794

666

3794

666

1006

3795

3796

670

3796

670

1008

3797

3798

674

3798

674

1010

3799

3800

678

3800

678

1012

3801

3802

682

3802

682

1014

3803

3804

686

31

11

10

32

2495

35

2495

35

34

3478

2497

39

2497

39

38

3480

2499

43

2499

43

42

3482

2501

47

2501

47

46

3484

2503

51

2503

51

50

3486

2505

55

2505

55

54

3488

2507

59

2507

59

58

3490

2509

63

2509

63

62

3492

2511

67

2511

67

66

3494

2513

71

2513

71

70

3496

2515

75

2515

75

74

3498

2517

79

2517

79

78

3500

2519

83

2519

83

82

3502

2521

87

2521

87

86

3504

2523

91

2523

91

90

3506

2525

95

2525

95

94

3508

2527

99

2527

99

98

3510

2529

103

2529

103

102

3512

2531

107

2531

107

106

3514

2533

111

2533

111

110

3516

2535

115

2535

115

114

3518

2537

119

2537

119

118

3520

2539

123

2539

123

122

3522

2541

127


247


248

126 2173 130 2174 134 2175 138 2176 142 2177 146 2178 150 2179 154 2180 158 2181 162 2182 166 2183 170 2184 174 2185 178 2186 182 2187 186 2188 190 2189 194 2190 198 2271 522 2272 526 2273 530 2274 534 2275 538 2276 542 2277 546 2278 550 2279 554 2280 558 2281 562 2282 566 2283 570 2284 574 2285 578 2286 582 2287

3524 2 3526 2 3528 2 3530 2 3532 2 3534 2 3536 2 3538 2 3540 2 3542 2 3544 2 3546 2 3548 2 3550 2 3552 2 3554 2 3556 2 3558 2 3560 2 3722 2 3724 2 3726 2 3728 2 3730 2 3732 2 3734 2 3736 2 3738 2 3740 2 3742 2 3744 2 3746 2 3748 2 3750 2 3752 2

2541

127

126

3524

2543

131

2543

131

130

3526

2545

135

2545

135

134

3528

2547

139

2547

139

138

3530

2549

143

2549

143

142

3532

2551

147

2551

147

146

3534

2553

151

2553

151

150

3536

2555

155

2555

155

154

3538

2557

159

2557

159

158

3540

2559

163

2559

163

162

3542

2561

167

2561

167

166

3544

2563

171

2563

171

170

3546

2565

175

2565

175

174

3548

2567

179

2567

179

178

3550

2569

183

2569

183

182

3552

2571

187

2571

187

186

3554

2573

191

2573

191

190

3556

2575

195

2575

195

194

3558

2577

199

2577

199

198

3560

2739

523

2739

523

522

3722

2741

527

2741

527

526

3724

2743

531

2743

531

530

3726

2745

535

2745

535

534

3728

2747

539

2747

539

538

3730

2749

543

2749

543

542

3732

2751

547

2751

547

546

3734

2753

551

2753

551

550

3736

2755

555

2755

555

554

3738

2757

559

2757

559

558

3740

2759

563

2759

563

562

3742

2761

567

2761

567

566

3744

2763

571

2763

571

570

3746

2765

575

2765

575

574

3748

2767

579

2767

579

578

3750

2769

583

2769

583

582

3752

2771

587


Altair Engineering


586 2288 590 2289 594 2290 598 2291 602 2292 606 2293 610 2294 614 2295 618 2296 622 2297 626 2298 630 2299 634 2300 638 2301 642 2302 646 2303 650 2304 654 2305 658 2306 662 2307 666 2308 670 2309 674 2310 678 2311 682 2312 686 2313 3479 2314 3481 2315 3483 2316 3485 2317 3487 2318 3489 2319 3491 2320 3493 2321 3495 2322

Altair Engineering

3754 2 3756 2 3758 2 3760 2 3762 2 3764 2 3766 2 3768 2 3770 2 3772 2 3774 2 3776 2 3778 2 3780 2 3782 2 3784 2 3786 2 3788 2 3790 2 3792 2 3794 2 3796 2 3798 2 3800 2 3802 2 3804 2 3807 2 3809 2 3811 2 3813 2 3815 2 3817 2 3819 2 3821 2 3823 2

2771

587

586

3754

2773

591

2773

591

590

3756

2775

595

2775

595

594

3758

2777

599

2777

599

598

3760

2779

603

2779

603

602

3762

2781

607

2781

607

606

3764

2783

611

2783

611

610

3766

2785

615

2785

615

614

3768

2787

619

2787

619

618

3770

2789

623

2789

623

622

3772

2791

627

2791

627

626

3774

2793

631

2793

631

630

3776

2795

635

2795

635

634

3778

2797

639

2797

639

638

3780

2799

643

2799

643

642

3782

2801

647

2801

647

646

3784

2803

651

2803

651

650

3786

2805

655

2805

655

654

3788

2807

659

2807

659

658

3790

2809

663

2809

663

662

3792

2811

667

2811

667

666

3794

2813

671

2813

671

670

3796

2815

675

2815

675

674

3798

2817

679

2817

679

678

3800

2819

683

2819

683

682

3802

2821

687

26

32

24

25

3806

3478

3806

3478

3479

3807

3808

3480

3808

3480

3481

3809

3810

3482

3810

3482

3483

3811

3812

3484

3812

3484

3485

3813

3814

3486

3814

3486

3487

3815

3816

3488

3816

3488

3489

3817

3818

3490

3818

3490

3491

3819

3820

3492

3820

3492

3493

3821

3822

3494

3822

3494

3495

3823

3824

3496


249


250

3497 2323 3499 2324 3501 2325 3503 2326 3505 2327 3507 2328 3509 2329 3511 2330 3513 2331 3515 2332 3517 2333 3519 2334 3521 2335 3523 2336 3525 2337 3527 2338 3529 2339 3531 2340 3533 2341 3535 2342 3537 2343 3539 2344 3541 2345 3543 2346 3545 2347 3547 2348 3549 2349 3551 2350 3553 2351 3555 2352 3557 2353 3559 2354 3561 2435 3723 2436 3725 2437

3825 2 3827 2 3829 2 3831 2 3833 2 3835 2 3837 2 3839 2 3841 2 3843 2 3845 2 3847 2 3849 2 3851 2 3853 2 3855 2 3857 2 3859 2 3861 2 3863 2 3865 2 3867 2 3869 2 3871 2 3873 2 3875 2 3877 2 3879 2 3881 2 3883 2 3885 2 3887 2 3889 2 4051 2 4053 2

3824

3496

3497

3825

3826

3498

3826

3498

3499

3827

3828

3500

3828

3500

3501

3829

3830

3502

3830

3502

3503

3831

3832

3504

3832

3504

3505

3833

3834

3506

3834

3506

3507

3835

3836

3508

3836

3508

3509

3837

3838

3510

3838

3510

3511

3839

3840

3512

3840

3512

3513

3841

3842

3514

3842

3514

3515

3843

3844

3516

3844

3516

3517

3845

3846

3518

3846

3518

3519

3847

3848

3520

3848

3520

3521

3849

3850

3522

3850

3522

3523

3851

3852

3524

3852

3524

3525

3853

3854

3526

3854

3526

3527

3855

3856

3528

3856

3528

3529

3857

3858

3530

3858

3530

3531

3859

3860

3532

3860

3532

3533

3861

3862

3534

3862

3534

3535

3863

3864

3536

3864

3536

3537

3865

3866

3538

3866

3538

3539

3867

3868

3540

3868

3540

3541

3869

3870

3542

3870

3542

3543

3871

3872

3544

3872

3544

3545

3873

3874

3546

3874

3546

3547

3875

3876

3548

3876

3548

3549

3877

3878

3550

3878

3550

3551

3879

3880

3552

3880

3552

3553

3881

3882

3554

3882

3554

3555

3883

3884

3556

3884

3556

3557

3885

3886

3558

3886

3558

3559

3887

3888

3560

3888

3560

3561

3889

4050

3722

4050

3722

3723

4051

4052

3724

4052

3724

3725

4053

4054

3726


Altair Engineering


3727 2438 3729 2439 3731 2440 3733 2441 3735 2442 3737 2443 3739 2444 3741 2445 3743 2446 3745 2447 3747 2448 3749 2449 3751 2450 3753 2451 3755 2452 3757 2453 3759 2454 3761 2455 3763 2456 3765 2457 3767 2458 3769 2459 3771 2460 3773 2461 3775 2462 3777 2463 3779 2464 3781 2465 3783 2466 3785 2467 3787 2468 3789 2469 3791 2470 3793 2471 3795 2472

Altair Engineering

4055 2 4057 2 4059 2 4061 2 4063 2 4065 2 4067 2 4069 2 4071 2 4073 2 4075 2 4077 2 4079 2 4081 2 4083 2 4085 2 4087 2 4089 2 4091 2 4093 2 4095 2 4097 2 4099 2 4101 2 4103 2 4105 2 4107 2 4109 2 4111 2 4113 2 4115 2 4117 2 4119 2 4121 2 4123 2

4054

3726

3727

4055

4056

3728

4056

3728

3729

4057

4058

3730

4058

3730

3731

4059

4060

3732

4060

3732

3733

4061

4062

3734

4062

3734

3735

4063

4064

3736

4064

3736

3737

4065

4066

3738

4066

3738

3739

4067

4068

3740

4068

3740

3741

4069

4070

3742

4070

3742

3743

4071

4072

3744

4072

3744

3745

4073

4074

3746

4074

3746

3747

4075

4076

3748

4076

3748

3749

4077

4078

3750

4078

3750

3751

4079

4080

3752

4080

3752

3753

4081

4082

3754

4082

3754

3755

4083

4084

3756

4084

3756

3757

4085

4086

3758

4086

3758

3759

4087

4088

3760

4088

3760

3761

4089

4090

3762

4090

3762

3763

4091

4092

3764

4092

3764

3765

4093

4094

3766

4094

3766

3767

4095

4096

3768

4096

3768

3769

4097

4098

3770

4098

3770

3771

4099

4100

3772

4100

3772

3773

4101

4102

3774

4102

3774

3775

4103

4104

3776

4104

3776

3777

4105

4106

3778

4106

3778

3779

4107

4108

3780

4108

3780

3781

4109

4110

3782

4110

3782

3783

4111

4112

3784

4112

3784

3785

4113

4114

3786

4114

3786

3787

4115

4116

3788

4116

3788

3789

4117

4118

3790

4118

3790

3791

4119

4120

3792

4120

3792

3793

4121

4122

3794

4122

3794

3795

4123

4124

3796


251


252

3797 2473 3799 2474 3801 2475 3803 2476 3805 2477 3478 2478 3480 2479 3482 2480 3484 2481 3486 2482 3488 2483 3490 2484 3492 2485 3494 2486 3496 2487 3498 2488 3500 2489 3502 2490 3504 2491 3506 2492 3508 2493 3510 2494 3512 2495 3514 2496 3516 2497 3518 2498 3520 2499 3522 2500 3524 2501 3526 2502 3528 2503 3530 2504 3532 2505 3534 2506 3536 2507

4125 2 4127 2 4129 2 4131 2 4133 2 3806 2 3808 2 3810 2 3812 2 3814 2 3816 2 3818 2 3820 2 3822 2 3824 2 3826 2 3828 2 3830 2 3832 2 3834 2 3836 2 3838 2 3840 2 3842 2 3844 2 3846 2 3848 2 3850 2 3852 2 3854 2 3856 2 3858 2 3860 2 3862 2 3864 2

4124

3796

3797

4125

4126

3798

4126

3798

3799

4127

4128

3800

4128

3800

3801

4129

4130

3802

4130

3802

3803

4131

4132

3804

27

31

32

26

2987

2495

2987

2495

3478

3806

2989

2497

2989

2497

3480

3808

2991

2499

2991

2499

3482

3810

2993

2501

2993

2501

3484

3812

2995

2503

2995

2503

3486

3814

2997

2505

2997

2505

3488

3816

2999

2507

2999

2507

3490

3818

3001

2509

3001

2509

3492

3820

3003

2511

3003

2511

3494

3822

3005

2513

3005

2513

3496

3824

3007

2515

3007

2515

3498

3826

3009

2517

3009

2517

3500

3828

3011

2519

3011

2519

3502

3830

3013

2521

3013

2521

3504

3832

3015

2523

3015

2523

3506

3834

3017

2525

3017

2525

3508

3836

3019

2527

3019

2527

3510

3838

3021

2529

3021

2529

3512

3840

3023

2531

3023

2531

3514

3842

3025

2533

3025

2533

3516

3844

3027

2535

3027

2535

3518

3846

3029

2537

3029

2537

3520

3848

3031

2539

3031

2539

3522

3850

3033

2541

3033

2541

3524

3852

3035

2543

3035

2543

3526

3854

3037

2545

3037

2545

3528

3856

3039

2547

3039

2547

3530

3858

3041

2549

3041

2549

3532

3860

3043

2551

3043

2551

3534

3862

3045

2553

3045

2553

3536

3864

3047

2555


Altair Engineering


3538 2508 3540 2509 3542 2510 3544 2511 3546 2512 3548 2513 3550 2514 3552 2515 3554 2516 3556 2517 3558 2518 3560 2599 3722 2600 3724 2601 3726 2602 3728 2603 3730 2604 3732 2605 3734 2606 3736 2607 3738 2608 3740 2609 3742 2610 3744 2611 3746 2612 3748 2613 3750 2614 3752 2615 3754 2616 3756 2617 3758 2618 3760 2619 3762 2620 3764 2621 3766 2622

Altair Engineering

3866 2 3868 2 3870 2 3872 2 3874 2 3876 2 3878 2 3880 2 3882 2 3884 2 3886 2 3888 2 4050 2 4052 2 4054 2 4056 2 4058 2 4060 2 4062 2 4064 2 4066 2 4068 2 4070 2 4072 2 4074 2 4076 2 4078 2 4080 2 4082 2 4084 2 4086 2 4088 2 4090 2 4092 2 4094 2

3047

2555

3538

3866

3049

2557

3049

2557

3540

3868

3051

2559

3051

2559

3542

3870

3053

2561

3053

2561

3544

3872

3055

2563

3055

2563

3546

3874

3057

2565

3057

2565

3548

3876

3059

2567

3059

2567

3550

3878

3061

2569

3061

2569

3552

3880

3063

2571

3063

2571

3554

3882

3065

2573

3065

2573

3556

3884

3067

2575

3067

2575

3558

3886

3069

2577

3069

2577

3560

3888

3231

2739

3231

2739

3722

4050

3233

2741

3233

2741

3724

4052

3235

2743

3235

2743

3726

4054

3237

2745

3237

2745

3728

4056

3239

2747

3239

2747

3730

4058

3241

2749

3241

2749

3732

4060

3243

2751

3243

2751

3734

4062

3245

2753

3245

2753

3736

4064

3247

2755

3247

2755

3738

4066

3249

2757

3249

2757

3740

4068

3251

2759

3251

2759

3742

4070

3253

2761

3253

2761

3744

4072

3255

2763

3255

2763

3746

4074

3257

2765

3257

2765

3748

4076

3259

2767

3259

2767

3750

4078

3261

2769

3261

2769

3752

4080

3263

2771

3263

2771

3754

4082

3265

2773

3265

2773

3756

4084

3267

2775

3267

2775

3758

4086

3269

2777

3269

2777

3760

4088

3271

2779

3271

2779

3762

4090

3273

2781

3273

2781

3764

4092

3275

2783

3275

2783

3766

4094

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2785


253

+ 3768 4096 CHEXA 2623 2 3277 2785 3768 4096 3279 2787 + 3770 4098 CHEXA 2624 2 3279 2787 3770 4098 3281 2789 + 3772 4100 CHEXA 2625 2 3281 2789 3772 4100 3283 2791 + 3774 4102 CHEXA 2626 2 3283 2791 3774 4102 3285 2793 + 3776 4104 CHEXA 2627 2 3285 2793 3776 4104 3287 2795 + 3778 4106 CHEXA 2628 2 3287 2795 3778 4106 3289 2797 + 3780 4108 CHEXA 2629 2 3289 2797 3780 4108 3291 2799 + 3782 4110 CHEXA 2630 2 3291 2799 3782 4110 3293 2801 + 3784 4112 CHEXA 2631 2 3293 2801 3784 4112 3295 2803 + 3786 4114 CHEXA 2632 2 3295 2803 3786 4114 3297 2805 + 3788 4116 CHEXA 2633 2 3297 2805 3788 4116 3299 2807 + 3790 4118 CHEXA 2634 2 3299 2807 3790 4118 3301 2809 + 3792 4120 CHEXA 2635 2 3301 2809 3792 4120 3303 2811 + 3794 4122 CHEXA 2636 2 3303 2811 3794 4122 3305 2813 + 3796 4124 CHEXA 2637 2 3305 2813 3796 4124 3307 2815 + 3798 4126 CHEXA 2638 2 3307 2815 3798 4126 3309 2817 + 3800 4128 CHEXA 2639 2 3309 2817 3800 4128 3311 2819 + 3802 4130 CHEXA 2640 2 3311 2819 3802 4130 3313 2821 + 3804 4132 $ $HMMOVE 2 $ 17THRU 58 139THRU 222 303THRU 386 $ 467THRU 550 631THRU 714 795THRU 878 $ 959THRU 1042 1123THRU 1206 1287THRU 1370 $ 1451THRU 1534 1615THRU 1698 1779THRU 1862 $ 1943THRU 2026 2107THRU 2190 2271THRU 2354 $ 2435THRU 2518 2599THRU 2640 $ $$ $$------------------------------------------------------------------------------$ $$ HyperMesh name and color information for generic components $ $$------------------------------------------------------------------------------$ $HMNAME COMP 2"Air" 2 "Air" 5 $HWCOLOR COMP 2 5 $ $HMNAME COMP 5"Piston" $HWCOLOR COMP 5 8 $ $HMNAME COMP 6"absorber" $HWCOLOR COMP 6 3 $ $ $HMDPRP $ 17THRU 58 139THRU 222 303THRU 386 $ 467THRU 550 631THRU 714 795THRU 878 $ 959THRU 1042 1123THRU 1206 1287THRU 1370 $ 1451THRU 1534 1615THRU 1698 1779THRU 1862 $ 1943THRU 2026 2107THRU 2190 2271THRU 2354 $ 2435THRU 2518 2599THRU 2640 5627 5629 6116 $ 6122 6125 6520THRU 6521 6523 6528 6954 7220 $ 7647 7652 7945 7948 7955 $

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$ $$ $$ PSHELL Data $$ $ $ $ $ $ $ $ $HMNAME PROP 1"tube" 4 $HWCOLOR PROP 1 52 PSHELL 1 20.1 2 2 0.0 $$ $$ PSOLID Data $$ $HMNAME PROP 2"Air" 5 $HWCOLOR PROP 2 4 PSOLID 2 1 PFLUID $$ $$ MAT1 Data $$ $HMNAME MAT 2"alum" "MAT1" $HWCOLOR MAT 2 3 MAT1 21.0+7 0.3 0.000254 $$ $$ MAT10 Data $HMNAME MAT 1"Air" "MAT10" $HWCOLOR MAT 1 3 MAT10 1 1.21-7 13000.0 $$ $$------------------------------------------------------------------------------$ $$ HyperMesh Commands for loadcollectors name and color information $ $$------------------------------------------------------------------------------$ $HMNAME LOADCOL 2"spc" $HWCOLOR LOADCOL 2 6 $$ $HMNAME LOADCOL 8"Force" $HWCOLOR LOADCOL 8 7 $$ $HMNAME LOADCOL 12"SPC" $HWCOLOR LOADCOL 12 5 $$ $$ $$ FREQi cards $$ $HMNAME LOADCOL 3"Freq" $HWCOLOR LOADCOL 3 6 $FREQ1 3 0.0 5.0 600 FREQ 3480. $ $$ $$ RLOAD1 cards $$ $HMNAME LOADCOL 6"Rload" $HWCOLOR LOADCOL 6 6 RLOAD1 6 8 7 0 VELO $$ $$ $$ TABLED1 cards $$ $HMNAME LOADCOL 7"Table" $HWCOLOR LOADCOL 7 6 TABLED1 7 LINEAR LINEAR + 0.0 1.0 3000.0 1.0ENDT $$ $HMNAME LOADCOL 10"reactance"

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$HWCOLOR LOADCOL 10 5 TABLED1 10 LINEAR LINEAR + 0.0 0.00154 3000.0 0.00154ENDT $$ $HMNAME LOADCOL 11"Impedance" $HWCOLOR LOADCOL 11 5 TABLED1 11 LINEAR LINEAR + 0.0 0.0 3000.0 0.0ENDT $$ $$ $$ DLOAD cards $$ $HMNAME LOADCOL 9"Dload" $HWCOLOR LOADCOL 9 6 DLOAD 91.0 1.0 6 $$ $$ EIGRL cards $$ $HMNAME LOADCOL 4"EigrlTube" $HWCOLOR LOADCOL 4 6 EIGRL 4 5 $HMNAME LOADCOL 5"EigrlAir" $HWCOLOR LOADCOL 5 6 EIGRL 5 30 $$ $$ SPC Data $$ SPC1 12123456 6776 thru 6800 spcd 86776 3 1.0 spcd 86777 3 1.0 spcd 86778 3 1.0 spcd 86779 3 1.0 spcd 86780 3 1.0 spcd 86781 3 1.0 spcd 86782 3 1.0 spcd 86783 3 1.0 spcd 86784 3 1.0 spcd 86785 3 1.0 spcd 86786 3 1.0 spcd 86788 3 1.0 spcd 86789 3 1.0 spcd 86790 3 1.0 spcd 86791 3 1.0 spcd 86792 3 1.0 spcd 86793 3 1.0 spcd 86794 3 1.0 spcd 86795 3 1.0 spcd 86796 3 1.0 spcd 86797 3 1.0 spcd 86798 3 1.0 spcd 86799 3 1.0 spcd 86800 3 1.0 $ $ DAREA Data $ $$ $$ DAREA Data $$ DAREA 8 6798 3-15.0 $$ $$ CAABSF 7957 5 689 688 687 CAABSF 7960 5 1017 689 686 CAABSF 7964 5 1345 1344 688 CAABSF 7969 5 1509 1345 689 CAABSF 7972 5 2165 2164 2163 CAABSF 7977 5 688 2165 2162 CAABSF 7978 5 4133 3805 3804 CAABSF 7980 5 2493 2492 2164 CAABSF 7984 5 1344 2493 2165

256

MASS MASS

686 1016 689 1017 2162 687 4132 2165 688


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CAABSF 7985 5 2821 687 2162 2820 CAABSF 7988 5 2820 2162 2163 2985 CAABSF 7990 5 3313 2821 2820 3312 CAABSF 7994 5 3312 2820 2985 3477 CAABSF 7996 5 3805 1016 686 3804 CAABSF 7998 5 3804 686 687 2821 CAABSF 8003 5 4132 3804 2821 3313 PAABSF 5 11 10 ENDDATA $$ $$------------------------------------------------------------------------------$$ $$ Data Definition for AutoDV $$ $$------------------------------------------------------------------------------$$ $$ $$-----------------------------------------------------------------------------$$ $$ Design Variables Card for Control Perturbations $$ $$-----------------------------------------------------------------------------$$ $ $------------------------------------------------------------------------------$ $ Domain Element Definitions $ $------------------------------------------------------------------------------$ $$ $$------------------------------------------------------------------------------$$ $$ Nodeset Definitions $$ $$------------------------------------------------------------------------------$$ $$ Design domain node sets $$ $$------------------------------------------------------------------------------$$ $$ Control Perturbation $$ $$------------------------------------------------------------------------------$$ $$ $$ $$ CONTROL PERTURBATION Data $$

ALTDOCTAG "0mjpRI@DXd^3_0ASnbi`;l;q6A23R@9_67hgW8R?OiZ] Eq:PeN`À;WXh3ITgJeq5NZRd5jSHQK3X@:à12;n4qD_I^RYMo" ADI0.1.0 2011-02-11T20:16:20 0of1 OSQA ENDDOCTAG

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Radiated Sound Analysis Radiated Sound Output Radiated Sound Output can be requested for grid points on the structural surface and in the exterior acoustic field. Grid points are used to represent microphones to record the radiated sound, sound power, and sound intensity.

Guide for Requesting Radiated Sound Output The following procedure can be considered as a guide for requesting radiated sound output: 1. Microphones that record sound levels in the acoustic field can be defined as grid point sets using the RADSND (MSET field) bulk data entry. 2. PANELG (TYPE=SOUND/Blank) can be used to define the sound generating panel(s) which are to be considered for radiated sound output calculations. 3. The PANEL continuation line in the RADSND bulk data entry can be used to list the panel ID’s of the panels defined using PANELG (TYPE=SOUND/Blank). This allows the definition of the sound generating panels that contribute to the calculation of radiated sound output at the microphones (Grid points) listed in the MSET field of the RADSND bulk data entry. 4. The value of the speed of sound “c” required to define the wave number and the complex particle velocity vector is input using PARAM, SPLC. The density of the acoustic medium “e” used in the calculation of the complex acoustic sound pressure and the complex particle velocity vector is defined using PARAM, SPLRHO. An additional scale factor “q” can be specified using PARAM, SPLFAC in the Sound Pressure Level calculation. 5. Various outputs can be requested for this analysis. SPOWER output request can be used to request sound power, SINTENS can be used to request sound intensity and SPL can be used to request sound pressure.

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Figure 1: Radiated sound output from a panel.

The set up guide for radiated sound output calculation is described in the previous section. The procedure is based on the following set of equations for the calculation of each output type.

Analytical Background for Radiated Sound Output The sound radiated from the sound generating panel is reduced to sound generation from discrete point sources. The grid points of the finite element mesh on the surface of the panel are considered as sound sources. Sound power and sound intensity can be requested for both the source grids and the microphone grids.

At the Microphone Location Wave Number The wave number, k is defined as follows:

k

2 f c

Where,

c is the speed of sound defined by PARAM, SPLC. f is the frequency of the sound wave in the medium.

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Velocity Flux of the Source Grid The velocity flux of the source grid is the rate at which panel material in an infinitesimal area surrounding the grid point moves through the medium.

Figure 2: Defining the Velocity Flux

For each frequency, it is calculated as follows:

V flux f

V f

uuur A

Where, is the velocity vector of the source grid. is the area vector associated with the source grid defined as follows:

uuur A

r AX n s

Where,

A is the area associated with the source grid. is the unit normal to the panel surface at the source grid (see Figure 2).

Complex Acoustic Sound Pressure (Requested using SPL) The complex acoustic sound pressure is the deviation from the ambient atmospheric pressure caused by a sound wave. This is denoted by and is defined as the sound pressure deviation, due to a single sound panel grid j at the microphone location for each frequency as follows:

Total Complex Acoustic Sound Pressure requested by SPL is:

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Where, is the frequency of the sound wave in the medium. is the density of the acoustic medium defined by PARAM, SPLRHO.

rj is the distance from the acoustic source grid j on the panel to the microphone location grid (see Figure 1). is the velocity flux of the source grid.

k is the wave number as defined in Wave Number. i is the square root of -1 np is the number of source grids (see Figure 1). q is the value of the scale factor specified using the parameter PARAM, SPLFAC. The Sound Pressure Level in decibels (SPLdB - Also requested using SPL) can be calculated using the following equation:

SPLdB

20.0 * log10 (

SPL SPLREFDB

)

Where,

SPLdB is the Sound Pressure Level in decibels.

SPL

is the magnitude of the acoustic sound pressure.

SPLREFDB is the reference sound pressure value specified using the parameter PARAM, SPLREFDB

Complex Particle Velocity Vector The complex particle velocity vector is the velocity of a particle in a medium measured as a wave passes through it. The particle velocity is not the velocity of the wave itself; rather it is the velocity of a particle as it oscillates about a mean position, due to the passage of the wave. It is denoted by at the location of the microphone, due to the source grid (see Figure 1) and is defined for each frequency as follows:

uuur ( pv) j ( f )

ˆ pj ( f ) X j

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c

1

j

i krj


261

Where, is the complex acoustic pressure, due to source grid, is the unit vector from the source grid

r Xj r Xj

ˆ X j

j at the microphone location.

j to the microphone grid (see Figure 1).

r Xj rj

is the density of the acoustic medium defined by PARAM, SPLRHO.

c is the speed of sound defined by PARAM, SPLC. k is the wave number as defined in Wave Number. rj is the distance from the acoustic source grid j on the panel to the microphone grid (see Figure 1).

i is the square root of -1 Total Sound Power (Requested using SPOWER) The total sound power is the rate of change of sound energy with time in the domain of reference. The total sound power , due to all the source grids can be calculated at a microphone location for each frequency as follows: np

sp( f )

real p j ( f ). p j ( f ) j 1

Where, is the acoustic pressure at a microphone location, due to the source grid "j". is the complex conjugate of

.

np is the number of source grids (see Figure 1). Total Complex Intensity Vector (Requested using SINTENS) The total complex intensity vector is the sound power per unit area. The sound intensity can be defined as a product of sound pressure and the particle velocity vector. For multiple source grids, the total sound intensity at a microphone location for each frequency is given as follows:

uur

iv( f )

262

1 2

np

uuuur

real p j ( f ).( pv) j ( f ) j 1


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Where, is the acoustic pressure at the microphone location due to the sound generated at the source grid "j". is the complex conjugate of , which is the complex particle velocity vector at the microphone location, due to the sound generated at the source grid "j".

At the Source Grid Location Wave Number The wave number is independent of the location of the grid points. Now define a set of displacement vectors that relate source grids to one another. To do this, each source grid is considered to be associated with an area (A) on the panel.

Figure 3: Displacement vectors at the source grids.

The vector addition operation for displacement vectors from Figure 3 is as follows:

uuur

Xs

uur uuur

X

Xr

Where, is the vector from a source grid (1) to the source grid (2) of interest. is defined as:

uuur

Xr

1 Ar Xn s 2

Where,

A is the area associated with a source grid. is the unit normal to the area,

A associated with a source grid.

Complex Acoustic Sound Pressure [at the source grid] The complex acoustic sound pressure is the deviation from the ambient atmospheric

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pressure caused by a sound wave. This is denoted using source grid for each frequency as follows:

f V f (rs ) j flux

( ps ) j f

j

ie

and is defined at the

ik ( rs ) j

Total Complex Acoustic Sound Pressure at a source grid requested by SPL is: ( np 1)

( ps )total f j 1

f V f (rs ) j flux

j

ie

ik ( rs ) j

Where, is the frequency of the sound wave in the medium. is the density of the acoustic medium defined by PARAM, SPLRHO. is equal to , for each grid, j (j=1 to np), the magnitude (length) of in At the Source Grid Location (see Figure 1). is the velocity flux of the source grid,

as defined

j (see Velocity Flux of the Source Grid)

k is the wave number as defined in Wave Number. i is the square root of -1 np is the number of source grids (see Figure 1). q is the value of the scale factor specified using the parameter PARAM, SPLFAC.

Total Sound Power (Requested using SPOWER) [at the source grid] The total sound power is the rate of change of sound energy with time in the domain of reference. The total sound power , due to all the source grids can be calculated at a source grid of interest for each frequency as follows: ( np 1)

real ( ps ) j f .( ps )* j f

( sp) s ( f ) j 1

Where, is the acoustic pressure at a source grid, due to the source grid "j". is the complex conjugate of

.

np is the number of source grids (see Figure 1).

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Total Complex Intensity Vector (Requested using SINTENS) [at the source grid] The total complex intensity vector is the sound power per unit area. The sound intensity can be defined as a product of sound pressure and the normal velocity vector. For multiple source grids, the sound intensity

uuuuur

(iv) s ( f )

for each frequency is given as follows:

uuuur 1 ( np 1) real ( ps ) j f . ( pv) s ( f ) 2 j1

j

Where, is the acoustic pressure at the source grid location of interest, due to the sound generated at the source grid "j".

is the complex conjugate of the normal velocity vector source grid of interest.

of the

Where the normal velocity vector of the source grid of interest is given as:

Vn

f

r r V f .X n s X n s

j

Refer to at the source grid location and velocity flux of the source grid sections for a description of the terms.

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Fatigue Analysis Fatigue analysis, using S-N (stress-life), E-N (strain-life), and Dang Van Criterion (Factor of Safety) approaches for predicting the life (number of loading cycles) of a structure under cyclical loading may be performed by using OptiStruct. The stress-life method works well in predicting fatigue life when the stress level in the structure falls mostly in the elastic range. Under such cyclical loading conditions, the structure typically can withstand a large number of loading cycles; this is known as highcycle fatigue. When the cyclical strains extend into plastic strain range, the fatigue endurance of the structure typically decreases significantly; this is characterized as low-cycle fatigue. The generally accepted transition point between high-cycle and low-cycle fatigue is around 10,000 loading cycles. For low-cycle fatigue prediction, the strain-life (E-N) method is applied, with plastic strains being considered as an important factor in the damage calculation. Sections of a model on which fatigue analysis is to be performed must be identified on a FATDEF bulk data entry. The appropriate FATDEF bulk data entry may be referenced from a fatigue subcase definition through the FATDEF Subcase Information entry. The Dang Van criterion is used to predict if a component will fail in its entire load history. The conventional fatigue result that specifies the minimum fatigue cycles to failure is not applicable in such cases. It is necessary to consider if any fatigue damage will occur during the entire load history of the component. If damage does occur, the component cannot experience infinite life.

The Stress-Life (S-N) Approach S-N Curve The S-N curve, first developed by Wöhler, defines a relationship between stress and number of cycles to failure. Typically, the S-N curve (and other fatigue properties) of a material is obtained from experiment; through fully reversed rotating bending tests. Due to the large amount of scatter that usually accompanies test results, statistical characterization of the data should also be provided (certainty of survival is used to modify the S-N curve according to the standard error of the curve and a higher reliability level requires a larger certainty of survival).

Figure 1: S-N data from testing

When S-N testing data is presented in a log-log plot of alternating nominal stress amplitude

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Sa or range SR versus cycles to failure N, the relationship between S and N can be described by straight line segments. Normally, a one or two segment idealization is used.

Figure 2: One segment S-N curves in log-log scale

S

S1 N f

b1

for segment 1 (1)

Where,

S is the nominal stress range Nf are the fatigue cycles to failure b1 is the first fatigue strength exponent S1 is the fatigue strength coefficient The S-N approach is based on elastic cyclic loading, inferring that the S-N curve should be confined, on the life axis, to numbers greater than 1000 cycles. This ensures that no significant plasticity is occurring. This is commonly referred to as high-cycle fatigue. S-N curve data is provided for a given material on a MATFAT bulk data entry. It is referenced through a Material ID (MID) which is shared by a structural material definition.

Damage Model Palmgren-Miner's linear damage summation rule is used. Failure is predicted when:

Di

ni Nif

1.0 (2)

Where, Nif is the material’s fatigue life (number of cycles to failure) from its S-N curve at a combination of stress amplitude and means stress level i

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ni is the number of stress cycles at load level i Di is the cumulative damage under ni load cycle The linear damage summation rule does not take into account the effect of the load sequence on the accumulation of damage due to cyclic fatigue loading. However, it has been proved to work well for many applications.

Cycle Counting Cycle counting is used to extract discrete simple "equivalent" constant amplitude cycles from a random loading sequence. One way to understand “cycle counting” is as a changing stressstrain versus time signal. Cycle counting will count the number of stress-strain hysteresis loops and keep track of their range/mean or maximum/minimum values. Rainflow cycle counting is the most widely used cycle counting method. It requires that the stress time history be rearranged so that it contains only the peaks and valleys and it starts either with the highest peak or the lowest valley (whichever is greater in absolute magnitude). Then, three consecutive stress points (S1, S2, and S3) will define two consecutive ranges as

S1 = |S1 - S2| and S2 = |S2 - S3| . A cycle from S1 to S2 is only extracted if S1 S2. Once a cycle is extracted, the two points forming the cycle are discarded and the remaining points are connected to each other. This procedure is repeated until the remaining data points are exhausted.

Figure 3: Determine cycles using rainflow cycle counting method

Parameters affecting rainflow cycle counting may be defined on a FATPARM bulk data entry. The appropriate FATPARM bulk data entry may be referenced from a fatigue subcase definition through the FATPARM Subcase Information entry.

Equivalent Nominal Stress Since S-N theory deals with uniaxial stress, the stress components need to be resolved into one combined value for each calculation point, at each time step, and then used as equivalent nominal stress applied on the S-N curve.

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Various stress combination types are available with the default being “Absolute maximum principle stress”. In general “Absolute maximum principle stress” is recommended for brittle materials, while “Signed von Mises stress” is recommended for ductile material. The sign on the signed parameters is taken from the sign of the Maximum Absolute Principal value. Parameters affecting stress combination may be defined on a FATPARM bulk data entry. The appropriate FATPARM bulk data entry may be referenced from a fatigue subcase definition through the FATPARM Subcase Information entry.

Mean Stress Influence Generally S-N curves are obtained from standard experiments with fully reversed cyclic loading. However, the real fatigue loading could not be fully reversed and the normal mean stresses have significant effect on fatigue performance of components. Tensile normal mean stresses are detrimental and compressive normal mean stresses are beneficial, in terms of fatigue strength. Mean stress correction is used to take into account the effect of non-zero mean stresses. The Gerber parabola and the Goodman line in Haigh's coordinates are widely used when considering mean stress influence, and can be expressed as:

Sa

Se

Sm Su

1

2

Gerber:

(3)

Sa

Se

Sm Su

1 Goodman:

(4)

Where, Mean stress S

m = (Smax + Smin) / 2

Stress amplitude S = (S

a

max - Smin) / 2

Se is the stress range for fully reversed loading that is equivalent to the load case with a stress range SR and a mean stress Sm

Su is ultimate strength The Gerber method treats positive and negative mean stress correction in the same way that mean stress always accelerates fatigue failure, while the Goodman method ignores the negative means stress. Both methods give conservative result for compressive means stress. The Goodman method is recommended for brittle material while the Gerber method is recommended for ductile material. For the Goodman method, if the tensile means stress is greater than UTS, the damage will be greater than 1.0. For Gerber method, if the mean stress is greater than UTS, no matter tensile or compressive, the damage will be greater than 1.0.

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A Haigh diagram characterizes different combinations of stress amplitude and mean stress for a given number of cycles to failure.

Figure 4: Haigh diagram and mean stress correction methods

Parameters affecting mean stress influence may be defined on a FATPARM bulk data entry. The appropriate FATPARM bulk data entry may be referenced from a fatigue subcase definition through the FATPARM Subcase Information entry.

The Strain-Life (E-N) Approach Monotonic Stress-Strain Behavior Relative to the current configuration, the true stress and strain relationship can be defined as:

P/ A

(5)

(6) Where, A is the current cross-section area, l is the current specimen length, l0 is the initial specimen length, and and are the true stress and strain, respectively, Figure 5 shows the monotonic stress-strain curve in true stress-strain space. In the whole process, the stress continues increasing to a large value until the specimen fails at C.

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Figure 5: Monotonic stress-strain curve

The curve in Figure 5 is comprised of two typical segments, namely the elastic segment OA and plastic segment AC. The segment OA keeps the linear relationship between stress and elastic strain following Hooke Law:

E e

(7)

Where, E is elastic modulus and e is elastic strain. The formula can also be rewritten as:

/E

e

(8)

by expressing elastic strain in terms of stress. For most of materials, the relationship between the plastic strain and the stress can be represented by a simple power law of the form: n

K

p

(9)

Where, p is plastic strain, K is strength coefficient, and n is work hardening coefficient. Similarly, the plastic strain can be expressed in terms of stress as:

1n p

K

(10)

The total strain induced by loading the specimen up to point B or D is the sum of plastic strain and elastic strain:

1n e

p

E

K

(11)

Cyclic Stress-Strain Curve Material exhibits different behavior under cyclic load compared with that of monotonic load. Generally, there are four kinds of response.

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stable state cyclically hardening cyclically softening softening or hardening depending on strain range Which response will occur depends on its nature and initial condition of heat treatment. Figure 6 illustrates the effect of cyclic hardening and cyclic softening where the first two hysteresis loops of two different materials are plotted. In both cases, the strain is constrained to change in fixed range, while the stress is allowed to change arbitrarily. If the stress range increases relative to the former cycle under fixed strain range, as shown in the upper part of Figure 6, it is called cyclic hardening; otherwise, it is called cyclic softening, as shown in the lower part of Figure 6. Cyclic response of material can also be described by specifying the stress range and leaving strain unconstrained. If the strain range increases relative to the former cycle under fixed stress range, it is called cyclic softening; otherwise, it is called cyclic hardening. In fact, the cyclic behavior of material will reach a steady state after a short time which generally occupies less than 10 percent of the material total life. Through specifying different strain ranges, a series of hysteresis loops at steady state can be obtained. By placing these hysteresis loops in one coordinate system, as shown in Figure 7, the line connecting all the vertices of these hysteresis loops determine cyclic stress-strain curve which can be expressed in the similar form with monotonic stress-strain curve as:

Figure 6: Material cyclic response (a) Cyclic hardening; (b) Cyclic softening

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Figure 7: Definition of stable stress-strain curve

1 n' e

p

E

K'

(12)

Where, K' is cyclic strength coefficient, n' is strain cyclic hardening exponent.

Hysteresis Loop Shape Bauschinger observed that after the initial load had caused plastic strain, load reversal caused materials to exhibit anisotropic behavior. Based on experiment evidence, Massing put forward the hypothesis that a stress-strain hysteresis loop is geometrically similar to the cyclic stress strain curve, but with twice the magnitude. This implies that when the quantity

is two times of , the stress-strain cycle will lie on the hysteresis loop. This can be expressed with formulas:

2

(13)

2

(14)

Expressing in terms of , in terms of hysteresis loop formula can be deduced as:

, and substituting it into Eq. 12, the

1 n'

E

2

2K '

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(15)


273

Strain-Life Approach Almost a century ago, Basquin observed the linear relationship between stress and fatigue life in log scale when the stress is limited. He put forward the following fatigue formula controlled by stress: ' f

2N f

b

(16)

Where, a is stress amplitude, fatigue strength coefficient, b fatigue strength exponent. Later in the 1950s, Coffin and Manson independently proposed that plastic strain may also be related with fatigue life by a simple power law:

pa

c

' 2N f f

(17)

Where, is plastic strain amplitude, fatigue ductility coefficient, c fatigue ductility exponent. Morrow combined the work of Basquin, Coffin and Manson to consider both elastic strain and plastic strain contribution to the fatigue life. He found out that the total strain has more direct correlation with fatigue life. By applying Hooke Law, Basquin rule can be rewritten as:

ea

a

' f

E

E

2N f

b (18)

Where, is elastic strain amplitude. Total strain amplitude, which is the sum of the elastic strain and plastic stain, therefore, can be described by applying Basquin formula and CoffinManson formula:

' f a

ea

pa

E

2N f

b

' 2N f f

c (19)

Where, is the total strain amplitude, the other variable is the same with above. Figure 8 illustrates three methods in log scale in stress-life space. Two straight lines, which represent Basquin formula and Coffin-Manson rule respectively, intersect at a point where elastic strain is equal to the plastic strain and the fatigue life predicted by the two methods is the same. The fatigue life at the intersection point is called transition life and can be calculated as: ' E/ f

2 Nt

' f

1 b c

(20)

by combining Eq.17 and Eq.18, at the same time, applying the conditions: ea

Nt

pa

(21)

Nf

(22)

Where, Nt is the transition life. When fatigue life is less than the transition life, plastic strain plays the controlling role in life prediction; otherwise, elastic strain plays the key role.

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Figure 8: Strain-life curve in log scale

Damage Accumulation Model In the E-N approach, use the same damage accumulation model as the S-N approach, which is Palmgren-Miner's linear damage summation rule.

Mean Stress Influence The fatigue experiments carried out in the laboratory are always fully reversed, whereas in practice, the mean stress is inevitable, thus the fatigue law established by the fully reversed experiments must be corrected before applied to engineering problems. Morrow is the first to consider the effect of mean stress through introducing the mean stress 0 in fatigue strength coefficient by: ' f ea

0

2N f

E

b

(23)

Thus the entire fatigue life formula becomes: ' f a

0

E

2N f

b

' f

2N f

c

(24)

Morrow's equation is consistent with the observation that mean stress effects are significant at low value of plastic strain and of little effect at high plastic strain. Smith, Watson and Topper proposed a different method to account for the effect of mean stress by considering the maximum stress during one cycle (for convenience, this method is called SWT in the following). In this case, the damage parameter is modified as the product of the maximum stress and strain range in one cycle. For a fully reversed cycle, the maximum stress is given by: max

' f

2N f

b

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(25)


275

By multiplying Eq.19 with Eq.25, it can be rewritten as:

' f max a

E

2N f

2b

' ' 2N f f f

b c (26)

The SWT method will predict that no damage will occur when the maximum stress is zero or negative, which is not consistent with the reality. When comparing the two methods, the SWT method predicted conservative life for loads predominantly tensile, whereas, the Morrow approach provides more realistic results when the load is predominantly compressive.

Neuber Correction Strain-life analysis is based on the fact that many critical locations such as notch roots have stress concentration, which will have obvious plastic deformation during the cyclic loading before fatigue failure. Thus, the elastic-plastic strain results are essential for performing strain-life analysis. Neuber correction is the most popular practice to correct elastic analysis results into elastic-plastic results. In order to derive the local stress from the nominal stress that is easier to obtain, the concentration factors are introduced such as the local stress concentration factor the local strain concentration factor

K , and

K .

K

/S

(27)

K

/e

(28)

Where, is the local stress, is the local strain, S is the nominal stress, and e is the nominal strain. If nominal stress and local stress are both elastic, the local stress concentration factor is equal to the local strain concentration factor. However, if the plastic strain is present, the

K

K

relationship between and no long holds. Thereafter, focusing on this situation, Neuber introduced a theoretically elastic stress concentration factor Kt defined as:

Kt2

K K

(29)

Substitute Eq.27 and Eq.28 into Eq.29, the theoretical stress concentration factor Kt can be rewritten as:

Kt2

S

e

(30)

Through linear static FEA, the local stress instead of nominal stress is provided, which implies the effect of the geometry in Eq.30 is removed, thus you can set Kt as 1 and rewrite Eq.30 as: e e

Where, ,

276

e

,

(31) is locally elastic stress and locally elastic strain obtained from elastic analysis,

the stress and strain at the presence of plastic strain. Both

and


can be calculated

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from Eq.31 together with the equations for the cyclic stress-strain curve and hysteresis loop.

Dang Van Criterion (Factor of Safety) The Dang Van criterion is used to predict if a component will fail in its entire load history. In certain physical systems, components may be required to last infinitely long. For example, automobile components which undergo multiaxial cyclic loading at high rotational velocities (like propeller shafts) reach their high cycle fatigue limit within a short operating life. The conventional fatigue result that specifies the minimum fatigue cycles to failure is not applicable in such cases. It is not necessary to quantify the amount of fatigue damage, but just to consider if any fatigue damage will occur during the entire load history of the component. If damage does occur, the component cannot experience infinite life. Fatigue analysis based on the Dang Van criterion is designed for this purpose. Fatigue crack initiation usually occurs at zones of stress concentration such as geometric discontinuities, fillets, notches and so on. This phenomenon takes place in the microscopic level and is localized to certain regions like grains which have undergone local plastic deformation in characteristic intra-crystalline bands. The Dang Van approach postulates a fatigue criterion using microscopic variables in the apparent stabilization state; this is a state of elastic shakedown if no damage occurs. The main principle of the criterion is that the usual characterization of the fatigue cycle is replaced by the local loading path and so damaging loads can be distinguished. The general procedure of Dang Van fatigue analysis is: 1. Evaluate the macroscopic stresses 2. Split the macroscopic stress

ij

(t )

ij

(t )

, for each location at a different point in time.

into a hydrostatic part

3. Calculate the stabilized microscopic residual stress equation:

dev

*

dev

p (t ) and a deviatoric part Sij (t ) .

*

based on the following

Min( Max( J 2 ( Sij (t ) dev )))

The expression is minimized with respect to

and maximized with respect to t.

4. Calculate the deviatoric part of microscopic stress.

sij (t )

Sij (t ) dev

*

5. Calculate factor of safety (FOS):

FOS

(t )

Min

b (t ) ap (t )

0.5Tresca ( sij (t ))

Where, b and a are material constants. If FOS is less than 1, the component cannot experience infinite life.

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OptiStruct Factor of Safety setup 1. The torsion fatigue limit and hydrostatic stress sensitivity values required for an FOS analysis can be set in the optional FOS continuation line on the MATFAT bulk data entry. 2. The Dang Van criterion type can be selected on the FATPARM bulk data entry. 3. Factor of Safety output can be requested using the FOS I/O options entry.

Other Factors Affecting Fatigue Surface Condition (Finish and Treatment) Surface condition is an extremely important factor influencing fatigue strength, as fatigue failures nucleate at the surface. Surface finish and treatment factors are considered to correct the fatigue analysis results. Surface finish correction factor Cfinish is used to characterize the roughness of the surface. It's presented on diagrams that categorize finish by means of qualitative terms such as polished, machined or forged.

Figure 9*: Surface finish correction factor for steels (* Source: Yung-Li Lee, Jwo. Pan, Richard B. Hathaway and Mark E. Barekey. Fatigue testing and analysis: Theory and practice, Elsevier, 2005)

Surface treatment can improve the fatigue strength of components. NITRIDED, SHOTPEENED, COLD-ROLLED are considered for surface treatment correction. It is also possible to input a value to specify the surface treatment factor Ctreat. In general cases, the total correction factor is Csur = Ctreat * Cfinish. If treatment type is NITRIDED, then the total correction is Csur = 2.0 * Cfinish (Ctreat = 2.0). If treatment type is SHOT-PEENED or COLD-ROLLED, then the total correction is Csur = 1.0. It means you will ignore the effect of surface finish. The fatigue endurance limit FL will be modified by Csur as: FL' = FL * Csur. For two segment S-N curve, the stress at the transition point is also modified by multiplying by Csur.

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Surface conditions may be defined on a PFAT bulk data entry. Surface conditions are then associated with sections of the model through the FATDEF bulk data entry.

Fatigue Strength Reduction Factor In addition to the factors mentioned above, there are various others factors that could affect the fatigue strength of a structure, e.g., notch effect, size effect, loading type. Fatigue strength reduction factor Kf is introduced to account for the combined effect of all such corrections. The fatigue endurance limit FL will be modified by Kf as: FL' = FL / Kf The fatigue strength reduction factor may be defined on a PFAT bulk data entry. It may then be associated with sections of the model through the FATDEF bulk data entry. If both Csur and Kf are specified, the fatigue endurance limit FL will be modified as: FL' = FL * Csur / Kf. Csur and Kf have similar influences on the E-N formula through its elastic part as on the S-N formula. In the elastic part of the E-N formula, a nominal fatigue endurance limit FL is calculated internally from the reversal limit of endurance Nc. FL will be corrected if Csur and Kf are presented. The elastic part will be modified as well with the updated nominal fatigue limit.

Setting Up a Fatigue Analysis Linear Superposition of Multiple FEA/Load Time History Load Cases When there are several load cases at the same time, all of which vary independently of one another, the principle of linear superposition will be used to combine all load cases together to determine the stress variation at each calculation point due to the combination of all loads. The formula is: n ij (t )

ij ,k

Pk (t ) k 1 PFEA,k

(32)

Where, n is the total number of load cases Pk(t) and

are, respectively, the time variation of the k-th load time history and the

total stress tensor PFEA,k and

are, respectively, the k-th load magnitude and stress tensor from FE

analysis

Load Time History Compression This option is used to save calculation time. It will remove small cycles (defined by a gate value) and intermediate points.

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Figure 10: Sample showing removal of small cycles

When removing small cycles, adjacent turning points, where the difference is less than the maximum range multiplied by relative gate value, will be removed from each channel. However, phase relationship will be maintained, when peaks and valleys occur on different channels at different times. This is shown by the sample above. In the first channel (top), the points at time 4 and 5 will be removed when the absolute gate equals one, while in the second channel (bottom), the points at time 1 and 2 will not be removed in order to keep the phase relationship between channels.

Figure 11: Sample showing removal of intermediate points

Removing intermediate points is another important mechanism to save computation time. If any point on the load-time history is neither a peak nor valley point, it will not contribute in determining any stress cycle. Such points could be screened out in the fatigue computation without losing the accuracy, but the computation time could be saved significantly. For example, the left column in Fig 11 shows three load-time histories of three super-positioned

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loadcases, respectively. After removing the intermediate points, the three load-time histories are obtained as in the right column, which can produce the same fatigue results as the left column, but use much less time. This mechanism is built in OptiStruct and is effective automatically.

Fatigue Loads, Events and Sequences Fatigue loading is defined by scaling a static subcase with a load-time history. A fatigue event consists of one or more static loadcases applied simultaneously in the same time duration scaled by load-time histories. For fatigue events with more than one static loadcase stress, linear superposition is used. A fatigue sequence consists of a number of fatigue events and repeated instances of these events. A fatigue sequence can be made up of other sub fatigue sequences and/or fatigue events. In this way, you can define very complex events and sequences for fatigue analysis. In OptiStruct, fatigue sequences defined in fatigue subcases (referred by FATSEQ) are the basic loading blocks. The fatigue life results of these fatigue subcases are calculated as the number of repeats of the loading block. Below is an example of a "tree-like" fatigue sequence, which can be defined in OptiStruct, with FSEQ# identifying fatigue sequences and FEVN# identifying fatigue events:

Figure 12: Example of a "tree-like" fatigue sequence

Fatigue loading is defined by a FATLOAD bulk data entry, where a static subcase and a loadtime history are associated. A fatigue loading event is defined by a FATEVNT bulk data entry, where one or more fatigue loads (FATLOAD) are selected. A fatigue loading sequence is defined by a FATSEQ bulk data entry, where a sequence of one or more fatigue loading events or other fatigue loading sequences is given. The appropriate FATSEQ bulk data entry may be referenced from a fatigue subcase definition through the FATSEQ Subcase Information entry.

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Multi-body Dynamics Simulation A multi-body system is defined to be an assembly of sub-systems (bodies, components, or sub-structures). The motion of the sub-systems may be kinematically constrained and each sub-system or component may undergo large translational and rotational displacements. Bodies can be considered rigid or flexible. Rigid bodies do not undergo deformations. Rigid body motion can be described completely by using six generalized coordinates. The resulting mathematical model is highly nonlinear. Neglecting the body deformations can lead to inaccurate results. Therefore, some of the bodies are considered flexible, that is they can undergo deformations. Modal reduction procedures are used to include flexible bodies in multi-body dynamics simulations. Joints, force elements, and controls connect the bodies. Initial velocities, forces, and motions may be applied to the system. Different types of analysis can be performed on a multi-body system to determine its behavior under certain loading, applied motion, and initial velocity. Transient (kinematic, dynamic) analysis determines the response under time dependent loading. Static and quasistatic analyses determine the static equilibrium of a system. The multi-body solution is based on an extended absolute coordinates formulation.

Multi-body system

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This implementation is targeted at the typical finite-element user who wants to solve multibody dynamics problems in the context of a finite element model, and is still somewhat limited. HyperMesh is used for modeling. All geometry entities are defined in terms of a finite element mesh. Flexible body modeling is fully integrated. HyperStudy can be used for optimization. Shape optimization of rigid and flexible bodies is available through the use of HyperMorph.

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Transient Analysis for MBD Transient analysis is used to calculate the response of a multi-body system to timedependent loads and motions. Forces and motions are time-dependent. Body initial conditions define the initial body velocities, while joint initial conditions define the initial displacement of a particular joint. The results of a transient analysis are displacements, velocities, accelerations, forces, as well as modal contributions to stresses and strains in flexible bodies. The responses are usually time-dependent. The equation of motion is given in the following form:

M q&& P(t ) q& (t 0) v0 The matrix M is the mass matrix, the vector P is the vector of external forces, and the vector q represents the generalized coordinates. Stiffness, damping, constraint forces, external loads, and gravity are all included in the external force vector P. An initial and maximum integration time step, an end time, and integrator tolerance need to be defined. Two analyses, kinematic and dynamic, are defined depending on the degree of freedom of the system analysis. A kinematic simulation is performed if all degrees of freedom are constrained through appropriate joints and/or motions, making it a zero degree of freedom model. A kinematics simulation finds a system configuration that satisfies all kinematic constraints and motion equations at any given time. The configuration is obtained by solving a system of nonlinear algebraic equations representing constraints. During a kinematic simulation, there is no need to integrate the differential equations of motion because the system configuration is fully determined by solving the constraint and motion equations alone. Even though forces are not used to compute the kinematics solution, joint reaction forces can be computed at any given time. The mass and inertia properties of bodies involved, and external forces acting on them, do not affect the resultant system configuration, but they do affect the joint reaction forces requested as outputs. A dynamic simulation is employed whenever the model has one or more degrees of freedom. A dynamic simulation involves integrating the differential equations of motion subject to nonlinear algebraic equations representing kinematics constraints. In other words, the solution is obtained by solving a mixed system of differential-algebraic equations. The resultant solution takes into account various dynamic effects and is dependent upon mass and inertia properties of bodies, damping within the system, and applied forces and motions. Additional simulation parameters, such as the integration scheme, integration time step, convergence tolerance, etc. could also affect the solution and; therefore, need to be specified appropriately. If a simulation type of transient is requested, the solver automatically determines whether to run a kinematic or dynamic solution from the degree of freedom. The equation of motion is solved using one of the three different integrators that are available. The choice is based on the stiffness of the problem. A problem is stiff if the numerical solution has its step size limited more severely by the stability of the numerical technique than by the accuracy of the technique. These are systems with high damping and

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low transience. VSTIFF (Default) – Implicit integrator that utilizes the Variable Coefficient Differential Equation Solver (VODE). It is suited for stiff and non-stiff problems. MSTIFF – Implicit integrator that utilizes the Modified Extended Backward Differentiation Formula (MEBDF) to solve the nonlinear equations of motion. It is suited for stiff problems. ABAM (Adams-Bashforth-Adams-Moulton) – Explicit integrator that uses a finite differences scheme to solve the nonlinear equations of motion. This integrator is suitable for systems that are non-stiff. A multi-body subcase needs to be defined in the input deck. Only one such subcase can be used in a model. The simulation type "transient" is defined on an MBSIM bulk data entry which must be referenced through a subcase statement MBSIM. The MBSIM bulk data entity also defines the integrator, end time, and time step. A sequence of several simulations of different types can be defined by referring to an MBSEQ bulk data statement instead. Loads and motions are referenced on MLOAD and MOTION subcase entries, respectively. Initial velocity is referenced through INVEL. SPC type constraints in Multi-body Dynamics analysis are allowed only for MBD-ESL optimization of a flexible body if displacements are used as constraints. Further information on loads and boundary conditions can be obtained from the sections Applied Forces and Motions and Initial Velocity. The unit system for the simulation can be defined using a DTI, UNITS bulk data entry.

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Static Analysis for MBD A static simulation is also called an equilibrium simulation. The system must have at least one degree of freedom to undergo a static simulation and an initial configuration must be specified. A zero degree of freedom system is always considered to be in static equilibrium once all of the kinematic constraints are satisfied. Starting with the user-specified initial configuration, a final configuration is arrived at iteration-by-iteration, such that there are no unbalanced forces or torques on any of the bodies in the system and all of the kinematics constraints are satisfied. All of the velocities and accelerations are set to zero. A multi-body subcase needs to be defined in the input deck. Only one such subcase can be used in a model. The simulation type "static" is defined on an MBSIM bulk data entry which must be referenced through a subcase statement MBSIM. The MBSIM bulk data entity also defines the integrator, end time, and time step. A sequence of several simulations of different types can be defined by referring to an MBSEQ bulk data statement instead. Loads and motions are referenced on MLOAD and MOTION subcase entries, respectively. Initial velocities do not apply here. SPC type constraints in Multi-body Dynamics analysis are allowed only for MBD-ESL optimization of a flexible body if displacements are used as constraints. Further information on loads and boundary conditions can be obtained from the sections Applied Forces and Motions and Initial Velocity. The unit system for the simulation can be defined using a DTI, UNITS bulk data entry.

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Quasi-static Analysis for MBD A quasi-static simulation is a sequence of static simulation steps applied to a model over a given duration at specified intervals. A quasi-static simulation is employed when you have time-dependent forces or motions in the model and you want a static equilibrium configuration at every time step. The system must have at least one degree of freedom to undergo a quasi-static simulation and you must specify the initial configuration. For the first time step, the user-specified initial configuration is used as a starting point, whereas for all other time-steps, a configuration from the previous time step is used as the starting point for equilibrium simulation at that time step. A final configuration is arrived at iteration-by-iteration, such that there are no unbalanced forces or torques on any of the bodies in the system and all of the kinematic constraints are satisfied at that time step. For every step, all velocities and accelerations are set to zero. A multi-body subcase needs to be defined in the input deck. Only one such subcase can be used in a model. The simulation type "quasi-static" is defined on an MBSIM bulk data entry which must be referenced through a subcase statement MBSIM. The MBSIM bulk data entity also defines the integrator, end time, and time step. A sequence of several simulations of different types can be defined by referring to a MBSEQ bulk data statement instead. Loads and motions are referenced on MLOAD and MOTION subcase entries, respectively. Initial velocities do not apply here. SPC type constraints in Multi-body Dynamics analysis are allowed only for MBD-ESL optimization of a flexible body if displacements are used as constraints. Further information on loads and boundary conditions can be obtained from the sections Applied Forces and Motions and Initial Velocity. The unit system for the simulation can be defined using a DTI, UNITS bulk data entry.

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Linear Analysis for MBD In linearization analysis, the nonlinear representations of force, motion, stiffness, or damping are linearized. A linearization can be performed on these models to prepare a model for use with Matlab or to obtain eigenmodes characteristics of the model. In the case of performing a linear analysis on the mechanical system, optional files of type eig_info, Simulink MDL, and Matlab ABCD matrices are available to be exported. A multi-body subcase needs to be defined in the input deck. Only one such subcase can be used in a model. A linear simulation is defined by referring an MBSIM subcase entry to a MBLIN bulk data statement. The MBLIN bulk data entity also selects linear analysis types, EIGEN or STMAT. A sequence of several simulations of different types can be defined by referring to an MBSEQ bulk data statement instead. Loads and motions are referenced on MLOAD and MOTION subcase entries, respectively. SPC type constraints in Multi-body Dynamics analysis are allowed only for MBD-ESL optimization of a flexible body if displacements are used as constraints. Initial velocities do not apply here. Further information on loads and boundary conditions can be obtained from the sections Applied Forces and Motions and Initial Velocity. The unit system for the simulation can be defined using a DTI, UNITS bulk data entry.

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Bodies Bodies are the model elements that have mass and inertia. Bodies can be rigid or flexible. A rigid body has only mass and inertia, and does not deform during the simulation. An initial velocity can be assigned. Mass and inertia information can be omitted for kinematic, static, and quasi-static simulations. It does not affect displacement, velocity, and acceleration results of kinematic simulation or displacement results of a static or a quasi-static simulation. Mass and inertia information must be correctly specified if joint-reaction forces are of interest in kinematic, static, or quasi-static simulations. A flexible body deforms during the simulation. Mass and inertia are determined by the geometry and material of the structure defining the body. An initial velocity and damping can be assigned. Flexible bodies are formulated using an orthogonal set of modes that represent the displacements u of the flexible body such that

q where, q are the modal coordinates which are to be determined by the multi-body dynamics analysis. The set of orthogonal modes is determined in a Component Mode Synthesis (CMS). Depending on the model, CMS can be performed as a pre-processing step using a special simulation (see Direct Matrix Approach). Besides displacements, velocities, and accelerations, stresses and strains can also be computed for flexible bodies. One special rigid body is the ground body. It describes the reference environment, and does not add any degrees of freedom to the system. It is at absolute rest. Any grounded body is merged into one. Bodies are defined in terms of a finite element model. A body is formed by a group of properties, elastic, rigid, and mass elements as well as grid points. Rigid bodies are defined on a PRBODY entry. Mass and inertia are either determined from the geometric entities or can be entered on PRBODY. The ground body is defined using the GROUND bulk data entry. Flexible bodies are defined using the PFBODY bulk data entry. The interface grid points are automatically determined or are defined on PFBODY using the FLXNODE flag. The procedure, as described in Direct Matrix Approach, is applied to each PFBODY definition. The procedure is fully integrated in the multi-body dynamics solution sequence, where flexh3d files are generated for multiple flexible bodies in the same model. The parameter PARAM, FLEXH3D may be used to control the regeneration of flexh3d files for subsequent runs.

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Markers A marker is a coordinate system attached to a body at a geometric point. Markers are used as a reference for joints, compliant elements, applied loads, and output requests. Markers are defined using a grid point and a coordinate system. The MARKER bulk data entry is used.

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Constraints The multi-body system must be sufficiently constrained. Typical types of constraints like joints, couplers, and high-pair joints can be defined. Higherpair joints include point-to-curve, point-to-surface, and curve-to-curve constraints. They can connect rigid bodies, flexible bodies, or a rigid and flexbody. Before running the solver, any redundant constraints in the model are removed. Since the constraint forces associated with redundant constraints are set to zero, it is important to review all of the constraints in the model to make sure they are physically meaningful and that there are no unintended redundant constraints. Joints connect two grid points that belong to a body. They constrain the motion between the bodies. They are defined using the JOINT or JOINTM bulk data entries. SPC type constraints in Multi-body Dynamics analysis are allowed only for MBD-ESL optimization of a flexible body if displacements are used as constraints.

Joint Type

Constrained Degrees of Freedom

Number of GRIDs

Translation

Rotation

Fixed

3

3

2

Revolute

3

2

3

Translational

2

3

3

Cylindrical

2

2

3

Universal

3

1

4

Planar

1

2

3

Ball

3

0

2

Perpendicular

0

1

4

Parallel axes

0

2

4

Orientation

0

3

2

In-plane

1

0

4

Inline

2

0

3

Constant velocity

3

1

4

(development source only)

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Couplers are constraints between the translational and/or the rotational motion of two or three joints. They are defined using the COUPLER bulk data entries. Higher-pair joints are connecting points, curves and surfaces. They can be rigid or deformable. Higher-pair Joint Type

Bulk Data Entry

Point-to-curve

MBPTCV

Point-to-deformable MBPTDCV curve Point-to-deformable MBPTDSF surface Curve-to-curve

MBCVCV

These entries refer to the parametric curve definition (MBPCRV), deformable curve definition (MBPTDCV), and deformable surface definition (MBDSRF).

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Contact The contact modeling capability for multi-body dynamics can handle complex contact scenarios between rigid bodies and rigid and flexible bodies. For the definition you have to identify geometries on one body that can contact a different set of geometries on a second body. You also specify the contact material properties such as coefficient of restitution and friction. The solver monitors the proximity of the specified geometries to each other. When contact between the two sets of geometry occurs, a force based on the defined physical properties is generated. This represents the contact force. Both normal and frictional forces are modeled. When the bodies separate, the force becomes zero. There are four key features to the contact capability: Modeling the geometry of the bodies that are in contact Detecting the onset of contact Applying the contact force Detecting the end of a contact "incident" and removing the contact force Two contact types are available: Rigid body to rigid body (MBCNTR) which is defined as the contact of two element sets (SET) and rigid to flexible body (MBCNTDS) which is defined as the contact between a node set (SET) and a deformable surface. The deformable surface must be defined by the MBDSRF bulk data entry. When the onset of a collision is detected, the collision detection algorithm returns a set of interfering polygons. From those the solver computes the following: The point of contact and surface normal vector The magnitude and direction of the normal and friction forces Once the point of contact and surface normal vector are known, the normal and friction force magnitudes are computed using a penalty-based Poisson contact normal force model. The two primary inputs to this model are the penalty and the coefficient of restitution (COR). COR is defined as the ratio of relative speed of separation to the relative speed of approach of the colliding bodies. A COR of 1.0 implies a perfectly elastic collision and a COR of 0.0 represents a perfectly plastic collision. One may think of the COR as damping and penalty as stiffness. Too high of a penalty value may cause numerical difficulties, while too small of a value may lead to excessive penetration. Some fine-tuning of these two parameters is usually required to reach stable and accurate results. The frictional force is modeled as a viscous force according to the following law:

In the above equations: is the current slip speed at the point of contact. is the coefficient of static friction. is the coefficient of dynamic friction. is the stiction transition slip speed at which the full value of coefficient of friction.

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is used for the


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is the dynamic friction slip speed at which the full value of coefficient of friction.

is used for the

is the friction force that is to be applied. The friction force opposes the direction of the slip velocity.

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Compliant Elements Compliant elements are bushings, spring-dampers, and beams. For each, the relevant information such as stiffness, damping, preload, attachment markers, etc. needs to be defined. Stiffness and damping value numbers should correctly represent the actual system and must be physically meaningful. Otherwise, the system may inadvertently turn out to be numerically stiff, even though the physical system may not be. Compliant elements can be defined with respect to grid points or with respect to markers. CMBEAM and CMBEAMM bulk data entries define beam elements. CMBUSH and CMBUSHM bulk data entries define bushing elements. A bushing element has linear stiffness and damping properties. CMSPDP and CMSPDPM bulk data entries define spring-damper elements.

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Applied Forces and Motions Forces and moments can be present in the system. There are action-only forces which are applied to one point, and action-reaction forces which are applied to two points. Force components can be a constant value, a curve, an expression, or a user-written subroutine. A special force is gravity. Acceleration is applied to a body and from mass and acceleration, the gravitational force is computed. Motion is a scalar constraint to the system. Displacement, velocity, and acceleration-type motions are possible. The motion must depend only on time and not on any other measures in the model that could change during the simulation. In other words, at every time step with only time as the independent parameter, the solver should be able to evaluate the expression completely without using any other information about the model. For example, the motion cannot depend upon displacement or velocity or acceleration between two points in the model. Motion can be specified as a constant value, a curve, an expression, or user-written subroutine. Motion is either defined as motion between two points or joint motion. When a motion is applied on a joint, one joint degree of freedom is controlled as a function of time. When a motion is applied between two points, movement along a user-specified direction is controlled as a function of time. Forces are always defined at grid points, and can be applied to one grid point (action-only) or two (action-reaction). The bulk data entry MBFRC defines constant force; the entry MBFRCC defines force by a curve; and the entry MBFRCE defines a force by equation. Moments are always defined at grid points, and can be applied to one grid point (action-only) or two (action-reaction). The bulk data entry MBMNT defines constant moment; the entry MBMNTC defines moment by a curve; and the entry MBMNTE defines a moment by equation. GRAV defines the gravity acceleration. The entry MLOAD can be used to derive force and moment set combinations. Motion can be defined as grid point motion or as joint motion. Grid point motion can be applied to one grid point or two (relative motion). The bulk data entry MOTNG defines constant motion; the entry MOTNGC defines motion by a curve, and MOTNGE defines motion by an equation. Joint motion can be applied to translational or revolute joints only. The bulk data entry MOTNJ defines constant motion; the entry MOTNJC defines motion by a curve, and MOTNJE defines motion by an equation. The entry MOTION can be used to derive motion set combinations.

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Initial Velocity Initial velocity is part of the problem formulation of the equation of motion. It can be applied to bodies or to cylindrical, translational, and rotational joints. The INVELB bulk data entry defines body initial velocity. The entry INVEL can be used to derive initial velocity set combinations.

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Function Expressions Expressions can be used in many places. They formulate relationships as functions of time, displacements, velocities, acceleration, forces, etc. If geometric points are used in an equation, they are always related to markers. The bulk data entry MBVAR is used to define equations.

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Results of a Multi-body Dynamics Analysis The primary results in a multi-body dynamics analysis are the motions of the bodies. They are written as nodal displacements, velocities, and accelerations. For flexible bodies, element results such as deformations, stresses, and strains are derived from those results. These results can be displayed in an animation of the entire system in a graphical toll such as Altair HyperView (see figure). Aside from the full nodal results, the solver provides a more compressed form of animation data only for multi-body dynamics analysis that can only be displayed in Altair HyperView, where HyperView does many of the typical transformations and final computations. See the Results of a Finite Element Analysis section to find more information on how to postprocess nodal and elemental results. Measures like body displacements, velocities, accelerations, joint forces, and user-defined expressions are being written as time history. Marker time history can be written upon request. These results can be plotted in a graphical plotting tool such as Altair HyperGraph. The definitions of the output options can be found in the I/O Options Section. An overview of the result files can be found in the Results Output by OptiStruct section.

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Rotor Dynamics Introduction Rotor dynamics is the analysis of structures containing rotating components. The dynamic behavior of such structures is influenced by the type and angular velocity of rotating components and their locations within the model. Rotor dynamics is available in OptiStruct for modal frequency response and complex eigenvalue analyses. Motivation When a component within the structure rotates, additional forces like the gyroscopic force and circular damping force act on it. It is important to determine the effects of rotating components on the system as a whole. The natural frequencies of a system usually change, if gyroscopic forces act on the model due to a rotating component. Circulating damping forces due to rotating components can lead to system instability. These forces are a function of the frequency of rotating component. In OptiStruct, they are included in the calculation of the response of the structure of interest when required in applicable subcases.

Figure 1: Example illustration depicting an application of Rotor Dynamics analysis

In Figure 1, the rotating components of the structure are the shafts on which gears are mounted. The design of the rotors and their angular frequencies can affect the dynamic response of the structure. Any design will most likely lead to asymmetrical mass distribution about the rotor axes. This unbalanced mass, even if it isn’t significant, can result in deflection of the rotor depending on various factors. The magnitude of these deflections will be augmented when the rotating speed of the shafts equals the natural frequency of the structure (Resonance), and can lead to catastrophic failure of the system. Implementation The Rotor Dynamics functionality is activated in OptiStruct with the use of the RGYRO subcase information entry (RGYRO = ID). This RGYRO entry references the identification number of a RGYRO bulk data entry. Related bulk data entries, RSPINR, UNBALNC, ROTORG and RSPEED are defined in the model for Rotor Dynamics. Parameters PARAM, GYROAVG, PARAM, WR3, and PARAM, WR4 are also used.

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Whirl A rotor is a structure that rotates about its own axis at a specific angular velocity. If a lateral force is applied to the rotor, it will deform in the lateral direction. This deformation is dependent on various factors, such as, magnitude of the applied force, rotor material properties, stator stiffness, and damping within the system. Due to rotor rotation, the deformed rotor will also whirl about an axis. Synchronous and Asynchronous Analysis The whirling speed can either be the same as rotor speed or it can be different from it. The type of analysis performed if the whirling speed and the rotor speed match is known as synchronous analysis. If the speeds don’t match, then asynchronous analysis is used to determine the dynamic response of the model. In OptiStruct, the RGYRO bulk data entry can be used to select synchronous/asynchronous analysis.

Figure 2: Illustration depicting the types of Whirl and the two analysis types that are dependent on the angular frequency of a rotor.

Forward Whirl and Backward Whirl The type of whirl depends on the spin direction of a rotor. If the rotor spin direction is the same as that of its whirl direction, then it is termed as forward whirl. If the rotor spin direction is opposite to the whirl direction, it is termed as backward whirl. In complex eigenvalue analysis, you can determine and differentiate between the modes of a structure undergoing backward whirl and forward whirl.

Supported solution sequences OptiStruct supports the Rotor dynamics functionality in the following solution sequences: Frequency Response Analysis The response of a structure with rotating components to a specified external excitation can be determined using the rotor dynamics functionality in frequency response analysis. Asynchronous analysis (RGYRO = ASYNC) If ASYNC is specified in the RGYRO bulk data entry, the rotors within the structure have userdefined spin rates. The excitation frequency (FREQi entries) is independent of the reference rotor speed defined in the RGYRO entry. Synchronous analysis (RGYRO = SYNC)

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If SYNC is specified in the RGYRO bulk data entry, the reference rotor spin rate is equal to (or synchronous with) the excitation frequency. The reference rotor speed is not input via the RGYRO entry and the FREQi entry values are used in this analysis. Complex Eigenvalue Analysis The eigenvalues and critical speeds of a structure with rotating components can be determined using the rotor dynamics functionality in complex eigenvalue analysis. Asynchronous analysis (RGYRO = ASYNC) If ASYNC is specified in the RGYRO bulk data entry, the rotors within the structure have userdefined spin rates via the RSPEED entry and the Campbell Diagram can be plotted to find the critical speeds. Additionally, since the calculated eigenvalues are complex, you can determine unstable modes by studying the real parts of the calculated eigenvalues. If the real part of a complex eigenvalue is positive, then the corresponding system mode is unstable. Synchronous analysis (RGYRO = SYNC) If SYNC is specified in the RGYRO bulk data entry, only the critical speeds are calculated as the rotor speeds are equal to the whirl frequencies. These critical speeds can lead to structural resonance and the design should be modified to change its whirl frequencies or the operating rotor spin rate should be limited to avoid reaching the critical speeds. Note: In a frequency response analysis, the synchronous analysis (SYNC) option is generally used to model rotors with an inherent unbalance. The rotor unbalance can be specified as a force or via the UNBALNC entry. The analysis is synchronous because the unbalanced load vibrates at the whirl frequency of the system which is equal to the rotor spin speed. Implementation - Frequency Response Analysis (ASYNC) Asynchronous analysis is activated using the RGYRO=ASYNC option. Frequency response analysis in rotor dynamics involves defining the excitation either as an external varying load as a function of frequency or as a rotor unbalance via the UNBALNC entry (or as a force that simulates the effect of the rotor unbalance). Asynchronous frequency response analysis in OptiStruct is designed for an external varying force at a specific set of frequencies. The following equation implements the external loading functionality in OptiStruct. The rotor speeds should be specified by you for Asynchronous frequency response analysis. 2

[ M ] i ([ BS ] [ BR ] (1 i GR

N

j 1

Rj

(

ref

j

R1

[M R ]

R2

[ K R ]) (1 i G )[ K S ] i [ K 4 S ]

)[ K Rj ] i [ K 4 Rj ])

G C ) i [ BRj ] [ BRj ]

u( ) R1

C [ M Rj ]

R2

C [ K Rj ]

GR

C [ K Rj ]

1

[ K 4CRj ]

The response of a system with rotating components to an external load in the frequency domain is calculated based on the above equation.

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F( )

Frequency Response Analysis (SYNC) Synchronous analysis is activated using the RGYRO=SYNC option. Frequency response analysis in rotor dynamics involves defining the excitation either as an external varying load as a function of frequency or as a rotor unbalance via the UNBALNC entry (or as a force that simulates the effect of the rotor unbalance). Synchronous frequency response analysis in OptiStruct is designed to calculate the response of a system with a rotor unbalance. The following equation implements the rotor unbalance functionality in OptiStruct. The rotor speeds are determined from the FREQi entries for Synchronous frequency response analysis. 2 ref

[M ] i

ref

([ BS ] [ BR ]

R1

[M R ]

R2

[ K R ]) (1 iG )[ K S ] i[ K 4 S ]

(1 i GR j )[ K Rj ] i[ K 4 Rj ])

N j 1

Rj

(

) i

ref

ref

u(

[ BRjG ] [ BRjC ]

R1

[ M RjC ]

R2

GR

[ K RjC ]

1

[ K RjC ]

ref

ref

)

F(

[ K 4CRj ]

ref

The response of a system with rotating components to a rotor imbalance which is considered as a force acting in the frequency domain is calculated based on the above equation. Frequency Response Analysis with WR3 and WR4 (ASYNC) Parameters PARAM, WR3 and PARAM, WR4 can be used to avoid frequency dependent calculation of the rotor speeds in systems with multiple rotors. The frequency values in the circulation damping terms are replaced with the values of the parameters as shown in the equation below. 2

[ M ] i ([ BS ] [ BR ]

R1

[M R ]

R2

[ K R ]) (1 iG )[ K S ] i[ K 4 S ]

(1 i GR j )[ K Rj ] i[ K 4 Rj ])

N j 1

Rj

(

ref

u( )

) i [ BRjG ] [ BRjC ]

R1

[ M RjC ]

R2

[ K RjC ]

GR [ K RjC ] WR3

F( )

1 [ K 4CRj ] WR 4

Frequency Response Analysis with WR3 and WR4 (SYNC) Parameters PARAM, WR3 and PARAM, WR4 can be used to avoid frequency dependent calculation of the rotor speeds in systems with multiple rotors. The rotor speeds can be calculated as a linear function of the reference rotor spin rate (see description of terms below). The reference rotor spin rate values in the circulation damping terms are replaced with the values of the parameters as shown in the equation below. 2 ref

N j 1

[M ] i

ref

([ BS ] [ BR ]

R1

[M R ]

R2

[ K R ]) (1 iG )[ K S ] i[ K 4 S ]

(1 i GR j )[ K Rj ] i[ K 4 Rj ]) Rj

(

ref

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) i

ref

[ BRjG ] [ BRjC ]

u( R1

[ M RjC ]

R2

[ K RjC ]

GR [ K RjC ] WR3

1 [ K 4CRj ] WR 4


303

ref

)

F(

Complex Eigenvalue Analysis with WR3 and WR4 (ASYNC) The eigenvalues and critical speeds of a structure with rotating components can be determined using the rotor dynamics functionality in complex eigenvalue analysis. In asynchronous analysis the critical speeds can also be determined by plotting the Campbell diagram for frequencies specified using the RSPEED entry. The parameters PARAM, WR3 and PARAM, WR4 can be used to replace the values of WR3 and WR4 in the equation below. 2

[ M ] i ([ BS ] [ BR ]

R1

[M R ]

R2

[ K R ]) (1 iG )[ K S ] i[ K 4 S ]

(1 i GR j )[ K Rj ] i[ K 4 Rj ])

N j 1

Rj

(

ref

u( )

) i [ BRjG ] [ BRjC ]

R1

[ M RjC ]

R2

[ K RjC ]

GR [ K RjC ] WR3

0

1 [ K 4CRj ] WR 4

Complex Eigenvalue Analysis with WR3 and WR4 (SYNC) Only the rotor speeds are required to perform the synchronous complex eigenvalue analysis as the whirl frequencies are equal to the reference rotor spin rates. Only the critical speeds are output as a result of this analysis. The parameters PARAM, WR3 and PARAM, WR4 can be used to replace the values of WR3 and WR4 in the equation below. 2 ref

[M ] i

j 1

Rj

ref

Where,

(

ref

([ BS ] [ BR ]

R1

[M R ]

R2

[ K R ]) (1 iG )[ K S ] i[ K 4 S ]

(1 i GR j )[ K Rj ] i[ K 4 Rj ])

N

Rj

ref

(

ref

) i

ref

[ BRjG ] [ BRjC ]

u( R1

[ M RjC ]

R2

[ K RjC ]

GR [ K RjC ] WR3

1 [ K 4CRj ] WR 4

is the reference rotor spin rate

)

(

)

Rj ref is the spin rate of rotor “j” as a function of the reference rotor spin rate. can be determined for each excitation frequency or it can be calculated as a linear function of the reference rotor spin rate:

Rj

(

ref

)

j

j

ref

j

and j are scaling factors calculated from the relative spin rates defined in the RSPINR bulk data entry. [M] is the structural mass

[ BS ]

is the viscous damping of the support

[ BR ]

is the rotor viscous damping

[M R ]

is the rotor mass

[K R ]

is the rotor stiffness

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ref

)

0

[ K 4R ] C R

[B ]

is the rotor material damping

is the circulation due to rotor viscous damping

[ M RC ] C R

[K ]

is the circulation due to rotor ‘mass’ is the circulation due to rotor structural ‘stiffness’

C R

[K 4 ] [K S ]

is the circulation due to rotor material damping

is the stiffness of the support

[ K 4S ]

is the material damping of the support

N is the number of rotors in the model

u ( ) is the displacement as a function of frequency u(

ref

)

is the displacement as a function of reference rotor spin rate

F ( ) is the external excitation as a function of frequency F(

ref

)

is the unbalanced load as a function of reference rotor spin rate (via DAREA or UNBALNC entries) G is the structural damping value of the support defined using PARAM, G GR is the structural damping value of the rotor defined using PARAM, G R1

and

R2

[ BR ]Rayleigh

are used to define the Rayleigh viscous damping as follows: R1

[M R ]

R2

[K R ]

and

[ BRC ]Rayleigh

R1

[ M RC ]

R2

[K CR ]

R and R are used to define the scale factors of the linear fit (between SPDLOW and SPDHIGH on the ROTORG entry) of the rotor speed to the reference rotor speed.

WR3 and WR4 are defined by the parameters PARAM, WR3, PARAM, WR4, respectively. The general form of a circulation damping term is given as:

[ DC ]

1 [T ][ D] [ D][T ] 2

Where, [D] is the regular damping matrix and [T] is a rotation matrix defined as follows:

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[T ]

[ BRG ]

0 1 0 0

1 0 0 0

0 0 0 0

0 0 0 0

0 0

0 0

0 1 0 0

0 0 0 1

0 0 0 0

0 0

0 0

is the gyroscopic matrix defined in a rotor coordinate system as follows:

[ BRG ]

0 0 0 0

0

0

0 0 0 0

0 0 0 0

0 0 0 I 33

I 33

0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

Model Restrictions 1D Rotor model The OptiStruct rotor dynamics feature currently supports only 1D rotors. Rotor shafts modeled with 1D elements like CBEAM, CBAR, or CBUSH only can be used. CONM1 or CONM2 entries should be used to define the mass and inertia of the rotors. Grid points are necessary for the definition of mass and inertia via CONM1 or CONM2. All grid points that belong to rotors should be listed in the ROTORG entries and only grids listed in the ROTORG entries are included in the calculation of gyroscopic terms. The I33 field on CONM1/CONM2 entries should contain meaningful values as only the inertia about the local Z axis plays a role in the gyroscopic forces (see above description). Detached Rotor model The rotor should be detached from the rest of the structure. Only rigid elements (RBEi) can be used to attach rotors to the ground or to flexible bearings. If any connection exists between the rotor and other parts of the structure using elements other than RBEi, then the program will error out. Symmetric rotor in a fixed reference frame Rotor dynamics analysis in OptiStruct is performed based on assumption that the rotor is symmetric. Therefore, the rotor model is required to be symmetric about the rotation axis. The implementation is based on equations of motion formulated in a fixed reference frame. Asymmetric rotors in a rotating reference frame is planned to be implemented in future versions of OptiStruct.

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Multiple Rotors During synchronous analysis, the calculations are performed with respect to the reference rotor. In synchronous frequency response analysis, the reference rotor is rotating at the frequency of the unbalanced load and in synchronous complex eigenvalue analysis, the reference rotor is rotating at the whirl frequency of the system. The interpretation of results in a multiple rotor system should always be done with respect to the reference rotor. Any deduction of results from the behavior of rotors other than the reference rotor will be inaccurate and can lead to incorrect results. If the behavior of a rotor other than the reference rotor is to be studied, a different analysis should be run with the rotor of interest as the reference rotor.

Campbell Diagram The critical speeds of a rotating structure should be calculated and the design parameters can then be altered if necessary to restrict the operating speeds of the structure from attaining those resonant speeds. The structure may undergo excessive amplitude and phase changes if its operating speeds reach critical speeds. The calculation of critical speeds in OptiStruct can be undertaken in two ways: 1. Synchronous Complex Eigenvalue Analysis The RGYRO=SYNC option in Complex Eigenvalue Analysis can be used to determine the exact critical speeds of the rotating structure. During a synchronous analysis, the rotor speed is equal to the whirl frequency of the structure, which by definition, are the critical speeds of the structure that should be avoided during its operation.

Figure 3: An example Campbell Diagram to calculate the critical speeds.

2. Asynchronous Complex Eigenvalue Analysis The RGYRO=ASYNC option and the RSPEED bulk data entry in Complex Eigenvalue Analysis can be used to determine the whirl frequencies (backward whirl and forward whirl) of the structure. These Whirl frequencies can be calculated for a sequence of rotor spin rates. Forward Whirl and Backward Whirl frequencies can then be plotted against the range of rotor spin rates (Figure 3). The critical speeds can be calculated by superimposing the “Rotor Spin Rate = Whirl Frequencies” line on the plot. The points of intersection are the critical speeds.

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Note: The rotor speeds specified on the RSPEED entry should be input with sufficiently fine resolution to be able to capture the critical speeds. If the specified rotor speeds are too far apart, the critical speeds may be missed.

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NVH Applications and Techniques The NVH Applications and Techniques section provides an overview of the following: Transfer Path Analysis on an Automobile Residual Runs using Super Elements Basic OptiStruct NVH Output Files Global Search Option Create Door and Deck Lid Seals Create a HyperGraph Template for Reading in Multiple Files Using AMSES (Automatic Multi-Level Sub-Structuring Eigensolver Solution) Poroelastic Materials (Biot theory)

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Transfer Path Analysis on an Automobile Transfer Function Body Analysis The function of the transfer path analysis is to determine which body interface dominates the critical NVH response in the interior of the body for a given type of vehicle loading. The first step is to determine all the transfer functions at all the body interfaces, such as the front and rear cradle mounts, front and rear suspension attachments, powertrain mounts, exhaust hangers and steering system. The major component file in this first run is the fully trimmed body. One issue that needs to be determined is whether or not the steering column and steering wheel are part of the body model to start with. This would determine to which component these attachments belong. If one of the critical response points is the steering wheel response, then both the steering column and steering wheel must be included with the body model. The front cradle can also be included in the model and the paths from the front suspension to the front cradle can also be evaluated. Another requirement is that the body model is in its fully trimmed state and that it contains all bolt-on components that belong to the body, such as the doors, deck lid, hood, seats, instrument panel, etc. Also the body model will need to include the air cavities, if the transfer path can determine the critical paths causing interior noise problems in the vehicle. Below is an example of how to set up the first deck to obtain the needed attachment results. The results from this run are used later to determine the transfer load paths for a full vehicle model subjected to a powertrain loading.

Example OUTPUT, H3D OUTPUT, MASSPROP PARAM, AMLS, YES PARAM, AMLSNCPU, 4 PARAM, AUTOSPC, YES PARAM, CHECKEL, NO TITLE = TRIM BODY MOBILITY ANALYSIS SUBTITLE = WITH CAVITY RESPONSE METHOD(FLUID) = 2 METHOD(STRUCTURE) = 3 FREQUENCY= 1 $ DRIVER'S EAR ACOUSTIC RESPONSE SET 2 = 80000000 ACCELERATION(PUNCH,PHASE) = 1 $ $ UNIT INPUT LOAD AT EACH ATTACHEMENT POINT IN ALL 6 DOF'S $ SUBCASE 2 LABEL = 4003003:+X<>3003:+X<>Frt Susp.:LCA - Frt Bush:LHS:+X DLOAD = 101 DISPLACEMENT (PUNCH,PHASE) = 2 SET 3 = 4003003

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VELOCITY (PUNCH,PHASE) = 3 $ SUBCASE 3 LABEL = 4003003:+Y<>3003:+Y<>Frt Susp.:LCA - Frt Bush:LHS:+Y DLOAD = 102 DISPLACEMENT (PUNCH,PHASE) = 2 SET 4 = 4003003 VELOCITY (PUNCH,PHASE) = 4 $ SUBCASE 4 LABEL = 4003003:+Z<>3003:+Z<>Frt Susp.:LCA - Frt Bush:LHS:+Z DLOAD = 103 DISPLACEMENT (PUNCH,PHASE) = 2 SET 5 = 4003003 VELOCITY (PUNCH,PHASE) = 5 $ SUBCASE 5 LABEL = 4003003:+RX<>3003:+RX<>Frt Susp.:LCA - Frt Bush:LHS:+RX DLOAD = 104 DISPLACEMENT (PUNCH,PHASE) = 2 SET 6 = 4003003 VELOCITY (PUNCH,PHASE) = 6 $ SUBCASE 6 LABEL = 4003003:+RY<>3003:+RY<>Frt Susp.:LCA - Frt Bush:LHS:+RY DLOAD = 105 DISPLACEMENT (PUNCH,PHASE) = 2 SET 7 = 4003003 VELOCITY (PUNCH,PHASE) = 7 $ SUBCASE 7 LABEL = 4003003:+RZ<>3003:+RZ<>Frt Susp.:LCA - Frt Bush:LHS:+RZ DLOAD = 106 DISPLACEMENT (PUNCH,PHASE) = 2 SET 8 = 4003003 VELOCITY (PUNCH,PHASE) = 8 $ $ Not all subcases are shown in this example ------------------------------------------------------------------SUBCASE 273 LABEL = 9005852:+RX<>9011999:+RX<>Frt Susp.:Int. Shaft to Col.::+RX DLOAD = 372 DISPLACEMENT (PUNCH,PHASE) = 2 SET 274 = 9005852 VELOCITY (PUNCH,PHASE) = 274 $ SUBCASE 274 LABEL = 9005852:+RY<>9011999:+RY<>Frt Susp.:Int. Shaft to Col.::+RY DLOAD = 373 DISPLACEMENT (PUNCH,PHASE) = 2 SET 275 = 9005852 VELOCITY (PUNCH,PHASE) = 275 $ SUBCASE 275 LABEL = 9005852:+RZ<>9011999:+RZ<>Frt Susp.:Int. Shaft to Col.::+RZ DLOAD = 374 DISPLACEMENT (PUNCH,PHASE) = 2 SET 276 = 9005852 VELOCITY (PUNCH,PHASE) = 276

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$ $ BEGIN BULK $ $ PARAM CARDS FOR ANALYSIS PARAM WTMASS 1. $==01==><==02==><==03==><==04==><==05==><==06==><==07==><==08==><==09==><==10 ==> $ FREQ1 1 5.0 1.0 195 $==01==><==02==><==03==><==04==><==05==><==06==><==07==><==08==><==09==><==10 ==> EIGRL 2 600. EIGRL 3 300. ACMODL 4.0 1.0 1.0 1.0 $==01==><==02==><==03==><==04==><==05==><==06==><==07==><==08==><==09==><==10 ==> $ $ 4003003 +X DLOAD 101 1.0 1.0 401 RLOAD1 401 1001 0 0 400 0 DAREA 1001 4003003 1 1.0 $ 4003003 +Y DLOAD 102 1.0 1.0 402 RLOAD1 402 1002 0 0 400 0 DAREA 1002 4003003 2 1.0 $ 4003003 +Z DLOAD 103 1.0 1.0 403 RLOAD1 403 1003 0 0 400 0 DAREA 1003 4003003 3 1.0 $ 4003003 +RX DLOAD 104 1.0 1.0 404 RLOAD1 404 1004 0 0 400 0 DAREA 1004 4003003 4 1.0 $ 4003003 +RY DLOAD 105 1.0 1.0 405 RLOAD1 405 1005 0 0 400 0 DAREA 1005 4003003 5 1.0 $ 4003003 +RZ DLOAD 106 1.0 1.0 406 RLOAD1 406 1006 0 0 400 0 DAREA 1006 4003003 6 1.0 $ $ Not all load cards are shown in this example ---------------------------------------------------------------------------$ $==01==><==02==><==03==><==04==><==05==><==06==><==07==><==08==><==09==><==10 ==> TABLED1 400 +400A +400A 20.0 1.0 400.0 1.0 ENDT $ INCLUDE '/ANALYSIS/TRIM_BODY_CONNECTIONS.dat' INCLUDE '/MODELS/CAVITY/CAVITY.dat' INCLUDE '/ANALYSIS/TRIM_BODY_FILES.dat' INCLUDE '/MODELS/FRONT_CRADLE/FRONT_CRADLE.dat' INCLUDE '/MODELS/STEERING/STEERING_COLUMN.dat' INCLUDE '/MODELS/STEERING/STEERING_WHEEL.dat' ENDDATA

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Note: Around 275 subcases were needed to define all the interface point unit loads in all six degrees of freedom for this example. The label card: LABEL = 4003003:+X<>3003:+X<>Frt Susp.:LCA - Frt Bush:LHS:+X The first parameter defines the input attachment point and its loading direction. The second parameter defines a shortened version of this input attachment point. The third parameter defines the attachment by its name and also includes the loading direction. The creation of the subcases and this labeling information will be automated in a future release of NVH Director. The major output from this analysis is the displacement and velocity output in the .pch file, which can be around 40 MB in size.

Full Vehicle Load Case The second run is a full vehicle model analysis with a particular critical loading on one of the non-body components. Below is an example of a P/T type of analysis. A torque loading is applied to the crankshaft and the acoustic response at the driver’s ear is captured.

Example OUTPUT, H3D OUTPUT, MASSPROP PARAM, AMLS, YES PARAM, AMLSNCPU, 4 PARAM, AUTOSPC, YES PARAM, CHECKEL, NO $MODEL,100 $ TITLE = P/T FULL VEHICLE ANALYSIS SUBTITLE = BASELINE COMPONENTS MPC = 406 SPC = 1 $ Acoustic response output set SET 1 = 80000000, 80000002, 80000004, 80000006 $ Structural response output set SET 2 = 1006001,9106012 $ Body attachment forces SET 3 = 1002001,1002001,1002002,1002003,1002004,1003015,1003016, 1003521,1004503,1004507,1004515,1004523,1005003,1005004, 1005011,1005012,1005013,1005014,1005015,1005016,1005017, 1005018,2005807,2005809,2005810,4003003,4003004,4003005, 4003006,4003007,4003008,4003501,4003511,4003541,4005811, 4005812,9005852 INCLUDE 'display_set.dat' $ This file contains set 200 that has the full vehicle plotel grid points identified. $ SUBCASE 1 $ MODAL DEFLECTION SHAPE LABEL = P over T Modal

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METHOD(FLUID) = 2 METHOD(STRUCTURE) = 3 DISPLACEMENT(H3D)= 200 $ SUBCASE 2 $ FREQUENCY RESPONSE ANALYSIS LABEL = P over T Baseline METHOD(FLUID) = 2 METHOD(STRUCTURE) = 3 DLOAD = 110 FREQUENCY= 1 GPFORCE (PUNCH,PHASE) = 3 DISPLACEMENT(PUNCH,PHASE)= 1 DISPLACEMENT(H3D,PHASE)= 1 ACCELERATION (PUNCH,PHASE) = 2 ACCELERATION (H3D,PHASE) = 2 $ SUBCASE 3 $ OPERATING DEFLECTION MODE SHAPE LABEL = P over T Post METHOD(FLUID) = 2 METHOD(STRUCTURE) = 3 $ Critical Frequencies specified in set 300 SET 300 = 54.0,64.0,80.0,92.0,104.0,114.0,146.0 OFREQ = 300 DLOAD = 110 FREQUENCY= 1 DISPLACEMENT(H3D)= 200 $ BEGIN BULK $ $ PARAM CARDS FOR ANALYSIS PARAM WTMASS 1. $ $==01==><==02==><==03==><==04==><==05==><==06==><==07==><==08==><==09==><==10 ==> FREQ1 1 5.0 1.0 195 EIGRL 2 600. EIGRL 3 300. ACMODL 4.0 1.0 1.0 1.0 $ INCLUDE '/ANALYSIS/P_OVER_T/PT_LOADS.dat' $ INCLUDE '/ANALYSIS/P_OVER_T/PT_CONNECTIONS_FULL.dat' $ INCLUDE '/ANALYSIS/FULL_VEHICLE_FILES_W_CAVITY.dat' ENDDATA This run also puts out a large .pch file that includes the response and the body attachment forces. Once these two runs are completed, a transfer patch analysis can be performed in HyperView.

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Transfer Path Analysis To perform a transfer path analysis on this model, open up HyperView. 1. From the File menu, select Load > Preference File. 2. From the Preference dialog, select NVH Utilities and click Load. 3. From the NVH menu, select Transfer Path Analysis. 4. Click on the file browser icon to select a Transfer Function file. This is the PCH from the first run. 5. Click on the file browser icon to select a Force file. 6. Select Load.

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Residual Runs using Super Elements Super Elements Only with No Structure in the Residual Run To run an analysis on just a super element model with no residual structure, you must include a dummy grid point or include the .seplot file generated with the super element. OUTPUT, H3D ASSIGN,H3DDMIG,BODY,/H3D/TRIMMED_BODY23_H3D.h3d TITLE = TRIM BODY MOBILITY ANALYSIS METHOD=1 FREQUENCY=1 $ SET 2 = 1006001 ACCELERATION (PUNCH,SORT1,PHASE) = 2 $ SUBCASE 2 LABEL = 1002001: +X DLOAD = 101 DISPLACEMENT = NONE SET 3 = 1002001 VELOCITY (PUNCH,SORT1,PHASE) = 3 $ SUBCASE 3 LABEL = 1002001: +Y DLOAD = 102 DISPLACEMENT = NONE SET 4 = 1002001 VELOCITY (PUNCH,SORT1,PHASE) = 4 $ BEGIN BULK GRID, 1 $ FREQ1 1 10.0 1.0 490 EIGRL 1 450.0 $ $ 1002001 +X DLOAD 101 1.0 1.0 401 RLOAD1 401 1001 0 0 400 0 DAREA 1001 1002001 1 1.0 $ 1002001 +Y DLOAD 102 1.0 1.0 402 RLOAD1 402 1002 0 0 400 0 DAREA 1002 1002001 2 1.0 TABLED1 400 +400A 10.0 1.0 500.0 1.0 ENDT $ INCLUDE '/MODELS/SEPLOTS/TRIMMED_BODY23_H3DFF.seplot' ENDDATA

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Residual Run Containing Several Super Elements An example to combine super elements in a residual run is shown in the following example. OUTPUT, H3D OUTPUT, MASSCOMP PARAM, AUTOSPC, YES PARAM, CHECKEL, NO INCLUDE '/ANALYSIS/H3D_FILES_W_CAVITY.dat' TITLE = SUBTITLE = SPC = 1 SUBCASE 1 LABEL = UNIT TORQUE INPUT DLOAD = 1000 METHOD(FLUID) = 2 METHOD(STRUCTURE) = 3 FREQUENCY=1 SET 1 = 80000000, 80000002, 80000004, 80000006 DISPLACEMENT(PUNCH,SORT2,PHASE)= 1 SET 2 = 1006001,9106012 ACCELERATION (PUNCH,SORT2,PHASE) = 2 $ OUTPUT(XYPLOT) TCURVE = DRIVER'S ACOUSTIC RESPONSE XYPUNCH DISP RESPONSE / 80000000(T1RM) TCURVE = FRONT PASSENGER ACOUSTIC RESPONSE XYPUNCH DISP RESPONSE / 80000002(T1RM) TCURVE = RIGHT REAR PASSENGER ACOUSTIC RESPONSE XYPUNCH DISP RESPONSE / 80000004(T1RM) TCURVE = LEFT REAR PASSENGER ACOUSTIC RESPONSE XYPUNCH DISP RESPONSE / 80000006(T1RM) BEGIN BULK $==01==><==02==><==03==><==04==><==05==><==06==><==07==><==08==><==09==><==10= => FREQ1 1 50.0 2.0 49 EIGRL 2 600. EIGRL 3 300. $==01==><==02==><==03==><==04==><==05==><==06==><==07==><==08==><==09==><==10= => ACMODL $ INCLUDE '/LOADS.dat' $ INCLUDE '/CONNECTIONS_BETWEEN_COMPONENTS.dat' $ INCLUDE '/NON_H3D_FILES.dat' ENDDATA Where the '/ANALYSIS/H3D_FILES_W_CAVITY.dat' looks like this. ASSIGN,H3DDMIG,BODY,/H3D/TRIMMED_BODY23_W_CAVITY_H3DFF.h3d ASSIGN,H3DDMIG,EXHAUS,/H3D/EXHAUST_H3DFF.h3d ASSIGN,H3DDMIG,FRCALLS,/H3D/FRONT_CALIPHER_LS_H3DFF.h3d ASSIGN,H3DDMIG,FRCALRS,/H3D/FRONT_CALIPHER_RS_H3DFF.h3d ASSIGN,H3DDMIG,FRCRAD,/H3D/FRONT_CRADLE_H3DFF.h3d ASSIGN,H3DDMIG,FRDILS,/H3D/FRONT_DISC_LS_H3DFF.h3d ASSIGN,H3DDMIG,FRDIRS,/H3D/FRONT_DISC_RS_H3DFF.h3d

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-------------------------------------------------ASSIGN,H3DDMIG,RRKNRS,/H3D/REAR_KNUCKLE_RS_H3DFF.h3d ASSIGN,H3DDMIG,RRRLLS,/H3D/REAR_LAT_LINK_LS_H3DFF.h3d ASSIGN,H3DDMIG,RRRLRS,/H3D/REAR_LAT_LINK_RS_H3DFF.h3d ASSIGN,H3DDMIG,RRSTAB,/H3D/REAR_STAB_BAR_H3DFF.h3d ASSIGN,H3DDMIG,RRSTLS,/H3D/REAR_STRUT_LS_H3DFF.h3d ASSIGN,H3DDMIG,RRSTRS,/H3D/REAR_STRUT_RS_H3DFF.h3d A unique name, six or less characters long, must be entered in the third field for each component. Note: A residual run with a large Craig-Chang super element should be run in either Lanczos or the Direct method. It will be extremely slow with AMLS. AMLS is not needed since the residual run is small in size. Note that to get results for interior points in the super element response points that were not included in the list with the interface points, output using the PUNCH command; you must also include the DEBUG,SETDMIG,1 line in the file. It is the only output command that requires this extra line.

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Basic OptiStruct NVH Output Files For more detail information, refer to Files Created by OptiStruct.

From a Standard Modal or Frequency Response Analysis *.h3d

This file contains the super element information from a super element creation run, the modal information from a modal run, the response output from a frequency response run, or the output from an optimization run. Information in this file can come from various types of analysis. This is a binary file that is used by HyperView.

*.html

This file contains a problem summary and results summary of the run. This file is created by default, but can be turned off using OUTPUT=HTML,NO. Open up this file in an internet browser like Internet Explorer or Firefox. It is very useful in debugging modal problems.

*.interface

This file contains the coupling between the cavity model and the structural model. This file can be viewed by reading in both the cavity and structure file first and then reading in the file through the standard file input selection in HyperMesh. This file is created if there are both structural and fluid meshes in the input data file.

*.mvw

This file contains the information to quickly load in the requested output information into HyperGraph. It references the .pch file information.

*.op2

Duplicates the standard Nastran .op2 file information. File is created by the OUTPUT2 output format command. Binary file.

*.out

This file contains the run information such as warning and error messages, mass information, memory requirements, and AMLS information. This file is always created.

*.pch

This file can contains output in both the XYPUNCH or PUNCH Nastran format. This file is created when XYPEAK, XYPLOT, XYPUNCH or PUNCH output is requested. ASCII format.

*.peak

This file contains the peak response information from a random response run. It contains RMS value, the number of positive crossings, and the peak power spectral density and responses. ASCII format.

*.res

The .res file is a HyperMesh binary results file. Output results can be view in HyperMesh Post capabilities. This file is created when the OUTPUT,HM,YES is turned on.

*.stat

This file contains the module timing information. This file is created by default, but can be turned off with OUTPUT=STAT,NO. ASCII format.

*_frames.html This file is used by the *.html file to view the H3D results in the HyperView Player browser plug-in. *_menu.html

This file is used by the *.html file to view the H3D results in the HyperView Player browser plug-in.

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From an Optimization Run *.desvar

Updated design variables at final iteration. ASCII format.

*.prop

Update property values at final iteration. Output is for all property values even those not being optimized. Creation of this file is controlled by the PROPERTY I/O option. ASCII format.

*.slk

This file contains the sensitivity information for the selected DESVARS. This output can be viewed in Excel. This file is created when the SENSITIVITY command is used.

*_noengl.slk

This file is always generated if sensitivity information is requested. NonEnglish version of the .slk file.

*.hgdata

This file contains the iteration history of the objective function, constraint functions, design variables, and response functions. Contents of this file are controlled by the I/O option HISOUT. This file can be read into HyperGraph to display its contents. ASCII format.

*.hist

This file contains the iteration history of the objective function, maximum constraint violation, design variables, DRESP1 type responses, and DRESP2 type responses. Contents of this file are controlled by the I/O option HISOUT. ASCII format.

*.sh

This file is created when an optimization is performed. Contains information necessary to restart the optimization from a given iteration. Output of this file is controlled by the I/O Option SHRES.

*_s#.h3d

This file is a compressed binary file, containing both model and result data. It can be used to post-process results in HyperView or using the HyperView Player. The _s#.h3d file is created when the H3D format is chosen.

*_des.h3d

This file is a compressed binary file, containing both model and result data. It can be used to post-process results in HyperView or when using the HyperView Player. The _des.h3d file is created when the H3D format is chosen (see I/O option FORMAT), and an optimization run is performed.

*_gauge.0.h3d This file is a compressed binary file containing both model and result data. It can be used to post-process shell thickness (gauge) sensitivity in HyperView. The *_gauge.0.h3d file is created when the H3D format is chosen (see I/O option FORMAT), and an optimization run is performed. *_hist.mvw

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This file is a HyperView session file and may be opened from the File menu in HyperView or HyperGraph. The file automatically creates individual plots for each of the results contained in the .hist file. Each plot occupies its own page within HyperView (HyperGraph). This file is created when an optimization is performed. Creation of this file is controlled by the I/O option DESHIS.


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From PFMODE, PFGRID Analysis *.pfmode.pch

This file contains the output from PFMODE and PFPANEL requests. This information can be viewed in the HyperView NVH PFMODE-PFGRID module. In version 12.0, this file is only available when output=punch is requested in PFMODE and PFPANEL. It is recommended to export the modal and panel participation data into a H3D file, due to the large volume of data.

*.h3d

This file contains the output from PFMODE, PFPANEL, and PFGRID requests. Results of PFGRID is only available in H3D file.

From a Super Element Creation Analysis *.seplot

This file contains the exterior and interior grids, plotels and plate plotels, retained in the super element creation run. This file is created when the PARAM, SEPLOT,YES command is included in the run. ASCII format.

*.h3d

This file contains the super element binary information to be included in a future residual run. Always created for a super element run. This file is created by default, but can be turned off with OUTPUT,H3D, NO.

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Global Search Option The global search option is incorporated directly in OptiStruct. It does not require an external program to run with OptiStruct. Below is an example on how it can be used to optimize engine mount locations in a full vehicle model for a simple rough road shake input. Initially here are the main cards for the global search option. Everything is controlled by the DGLOBAL card. Now, this card might seem daunting and overloaded with parameters, but try running with default values first. Most parameters were implemented for advanced usage and for futureproofing the feature. Here's all you need for basic usage: DGLOBAL = 10 ... BEGIN BULK ... DGLOBAL 10

Engine Mount Optimization Example PARAM, MASSPROP DGLOBAL = 10 SENSITIVITY = ALL SENSOUT = FL $ INCLUDE '/ANALYSIS/H3D_FILES.dat' TITLE = ENGINE MOUNT LOCATION OPTIMIZATION $ ENGINE MOUNT LOCATIONS SET 400 = 6966 6967 6968 6998 6999 7000 DESVAR = 400 DESOBJ = 1 RANDOM = 2400 SET 2 =1006001,9006002 ACCE(SORT1,PHASE,PLOT,PSDF) = 2 SUBCASE 10 $RIGHT SIDE INPUT DLOAD=10 ANALYSIS = MFREQ FREQUENCY = 100 SPC = 1 MPC = 400 METHOD = 1 SUBCASE 20 $LEFT SIDE INPUT DLOAD=20 FREQUENCY = 100 ANALYSIS = MFREQ SPC = 1 MPC = 400 METHOD = 1 $ OUTPUT(XYPLOT) XYPUNCH ACCE PSDF / 1006001(T1) XYPUNCH ACCE PSDF / 1006001(T2) XYPUNCH ACCE PSDF / 1006001(T3)

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XYPUNCH ACCE PSDF / 9006002(T1) XYPUNCH ACCE PSDF / 9006002(T2) XYPUNCH ACCE PSDF / 9006002(T3) BEGIN BULK $-----------------------------------------------------------------------------$ PARAM CARDS FOR ANALYSIS PARAM WTMASS 1. $==01==><==02==><==03==><==04==><==05==><==06==><==07==><==08==><==09==><==10 ==> $ FREQ1 1 1.0 0.2 95 EIGRL 1 45.0 $ $-----------------------------------------------------------------------------$==01==><==02==><==03==><==04==><==05==><==06==><==07==><==08==><==09==><==10 ==> DOPTPRM DESMAX 50 DGLOBAL 10 $-----------------------------------------------------------------------------$==01==><==02==><==03==><==04==><==05==><==06==><==07==><==08==><==09==><==10 ==> $ $-----------------------------------------------------------------------$ $ Left Engine Mount Point of Action $ GRID 4500 1250.0 -325.0 747.0 GRID 4505 1250.0 -325.0 747.0 GRID 4501 1260.0 -325.0 747.0 123456 GRID 4503 1250.0 -325.0 757.0 123456 CBUSH 5955 5964 4505 4500 5901 CORD1R 5901 4500 4503 4501 $--1---|---2---|---3---|---4---|---5---|---6---|---7---|---8---|--9---|-------| RBE2 5961 4004501 123456 4500 RBE2 5962 6004501 123456 4505 CONM2 5956 6004501 00.0035 0.0 0.0 0.0 CONM2 5957 4004501 00.0035 0.0 0.0 0.0 DESVAR 6966 EM4501X 10.0 -70.00 80.00 0.2 DVGRID 6966 4505 1.0 1.0 0.0 0.0 DVGRID 6966 4500 1.0 1.0 0.0 0.0 DESVAR 6967 EM4501Y 10.0 -60.00 30.00 0.2 DVGRID 6967 4505 1.0 0.0 1.0 0.0 DVGRID 6967 4500 1.0 0.0 1.0 0.0 DESVAR 6968 EM4501Z 10.0 -90.00 70.00 0.2 DVGRID 6968 4505 1.0 0.0 0.0 1.0 DVGRID 6968 4500 1.0 0.0 0.0 1.0 $--1---|---2---|---3---|---4---|---5---|---6---|---7---|---8---|--9---|-------| PLOTEL 5977 6004501 4501 PLOTEL 5979 6004501 4503 $ PBUSH 5964 K 450.0 300.0 500.0 0.0 0.0 0.0 B 0.0 0.0 0.0 0.0 0.0 0.0 GE 0.040 0.040 0.040 0.0 0.0 0.0 $-----------------------------------------------------------------------$ $ Right Engine Mount Point of Action

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$ $ GRID 4510 1250.0 325.0 747.0 GRID 4515 1250.0 325.0 747.0 GRID 4511 1260.0 325.0 747.0 123456 GRID 4517 1250.0 325.0 757.0 123456 CBUSH 5964 5964 4515 4510 6001 CORD1R 6001 4510 4517 4511 $--1---|---2---|---3---|---4---|---5---|---6---|---7---|---8---|--9---|-------| RBE2 5994 4004511 123456 4510 RBE2 5995 6004511 123456 4515 CONM2 5996 6004511 00.0035 0.0 0.0 0.0 CONM2 5997 4004511 00.0035 0.0 0.0 0.0 DESVAR 6998 EM4511X 10.0 -75.00 80.00 0.2 DVGRID 5998 4515 1.0 1.0 0.0 0.0 DVGRID 5998 4510 1.0 1.0 0.0 0.0 DESVAR 6999 EM4511Y 10.0 -20.00 80.00 0.2 DVGRID 6999 4515 1.0 0.0 1.0 0.0 DVGRID 6999 4510 1.0 0.0 1.0 0.0 DESVAR 7000 EM4511Z 10.0 -65.00 60.00 0.2 DVGRID 7000 4515 1.0 0.0 0.0 1.0 DVGRID 7000 4510 1.0 0.0 0.0 1.0 $--1---|---2---|---3---|---4---|---5---|---6---|---7---|---8---|--9---|-------| PLOTEL 6011 6004511 4511 PLOTEL 6013 6004511 4517 PLOTEL 24511 6004511 4004511 INCLUDE '/ANALYSIS/OPTIMIZATION_CARDS.dat' INCLUDE '/ANALYSIS/SIMPLE_ROAD_INPUT.dat' INCLUDE '/ANALYSIS/CONNECTIONS_WO_ENGING_MOUNTS.dat' INCLUDE '/ANALYSIS/NON_H3D_FILES.dat' ENDDATA In this model, the left and right mount locations are being optimized for improving the driver’s seat track for a simple rough road shake input. Most of the component files are in the CMS super element format. The simple component files are in OptiStruct. The super elements are required in order to make each optimization run faster. This run will make several optimization runs from different starting points. Each optimization output will be put into a separate directory. The .pch files from each directory can be viewed in HyperGraph and the best results can be chosen. The resulting .grid file for the best results can be included in the basic model file by the ASSIGN UPDATE card. This will automatically update the engine mount locations for you.

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Create Door and Deck Lid Seals 1. The first thing that needs to be done is to save this HyperMesh macro in one of your directories. The best directory to use is the Work_dir_hw11. This is the default directory for the HyperMesh command window. It orients the seal CBUSH elements properly. 2. The easiest way to create the door seals is to isolate both side of the structure that the seal is attached to. To do this, open up the body model in HyperMesh and save the outer door frame in a separate file. Do the same thing with the door inner panel to which the door seal contacts. 3. Open up both the body side frame and the door inner panel in HyperMesh. 4. Display only the body side frame. 5. Go to the Geom panel and select lines. 6. In the Lines panel, select node list and smooth as the two options displayed. 7. Start at one point on the door side frame where the seal is attached to the body. Do not select the edge point if the door seal is perpendicular to the door edge. For the deck lid seal the deck lid edge is parallel to the seal and the edge points can be picked. Do not pick nodes that are shared by two components. 8. Continue around the perimeter selecting points. The points can be spaced far apart if they are in a straight line. Around the curve areas, select enough points to adequately follow the curve surface. Select almost each point around a curve. If you select a wrong point, right-click on the last point to remove it. The right-click option can be performed several times to remove the last few points. Do not close the line. Select the last point close to the first point. 9. Click create. A line will be created along the path chosen. 10. Click return. 11. In the Geom panel, select Nodes. 12. Select on line. Number of nodes =: around 200 bias style: linear bias intensity: 0.0 13. Click the lines button and select the line just created. 14. Click create. The temporary nodes will be created equidistant from each other. 15. Display just these new nodes using the Model panel on the left side. Turn off the body side frame. Note the distance between nodes using the distance option in the Geom panel and selecting two sequential points. Record this distance for future use. 16. Select the 1D panel at the bottom. Then click connectors > spot. 17. Select the spot selection. Location: nodes

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connect with: comps type = sealing fe file; < return. 29. Go to the Properties selection on the top menu line. Select Edit > Prop and select the PBUSH0 property that represent the property for the spot weld CBUSH elements. Note that there may be multiple PBUSH properties for the door seals. The Spot command creates a new property card when one of the components that the door seal attaches to changes. The following needs to be performed for each door seal property card. 30. Click Select > update/edit. This will bring up the PBUSH card. Enter your values for the door seal properties. The seal stiffness properties will normally be given in stiffness per unit length for each of the three translation directions. Multiply these numbers by the length distance recorded above and enter the values into the K1, K2 and K3 locations. K1 will be the compressive direction; K2 the direction along the seal length; and K3 the direction perpendicular to the seal length.

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31. For K4, K5 and K6, enter 0.0. 32. Click Return twice. 33. Go to the Model panel on the left hand side and turn on the spot welds auto1. 34. Select the Tool > renumber panel. 35. Select elems. Click the elems button and select displayed. 36. Enter a start with value that moves the element IDs above all the rest of the elements in the full vehicle model. 37. Click the renumber button. 38. Go to Mask by Config in the left hand panel. Select the Springs/Gaps to isolate just the spot welds. Just the CBUSH elements of the spot welds will appear. 39. Now select lines and click the “+” symbol. This will bring up the line created initially. 40. Now use the macro to align the CBUSH elements properly. Go to the top line in the HyperMesh window and select View; select command window, if it is not already present. 41. Enter the macro name source orient…tbc. in the white command window. 42. Click enter. 43. In the bottom window there is a button named nodes list, which is highlighted. Expand the CBUSH elements. Select the node closest to the line on each CBUSH element. This is the GA point for the CBUSH element. Click proceed at any time and the macro will align just those CBUSH’s that were selected. Vectors that are created pointing along the line direction are shown. This defines the “XY” plane for the CBUSH elements. You can restart the macro by moving the curser into the command window and hitting the up arrow button. 44. Click enter. Now proceed to do all the seal CBUSH elements. 45. Go to the Model panel on the left hand side and turn on the spot welds auto1. 46. Either keep the welds with the body or the door, or the seals can be put into a separate file by going to the file output panel entering a new name for the file and set export to Displayed. 47. Click Export. 48. Presently in HyperMesh 10.0, the PBUSH ID cannot be renumbered initially. If the seal model is put in a separate file, close HyperMesh and reopen it and read this seal model back in. 49. The PBUSH property can now be renumbered by clicking Tool > Renumber. Select prop and click the prop button. Now select the PBUSH associated with the door seal CBUSHs. 50. Enter a start with value that moves the property ID above all the rest of the properties in the full vehicle model. 51. Click the renumber button. You will now need to resave the door seal file. 52. If you save the door seals in a separate file for use with your trim body model, the body and door seal grid points from the deck will need to be removed. You will also need to remove the control cards at the top of the deck and the enddata card at the end of the deck, so that when it is included with the body and doors there will be no duplicate ID’s.

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Create a HyperGraph Template for Reading in Multiple Files All the .pch files should be in the same directory and the $label values should be the same. 1. Open HyperGraph. 2. Read in file number 1. 3. To read in multiple curves at one time, select one Y Type. Select multiple Y Request. Select multiple Y Components. 4. From the Layout menu, select One curve per plot. 5. Click Apply. Multiple plots will be created. 6. From the File menu, select Save as > Report Template. same directory as where the .pch files are located.

Save file as a .tpl in the

7. From the File menu, select New > Session. This will restart HyperGraph. 8. From the Tools toolbar, select Open Reports Panel. 9. In the Report definition list, the .tpl file will be listed. 10. Click Apply. The original curves will be shown. 11. Click Add to open another file to compare the results. 12. Select Overlay. 13. Once you have finished adding the files, Save the .tpl. 14. You can now open the .tpl file in HyperGraph and enter new file names for different comparisons.

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Using AMSES (Automatic Multi-Level Sub-Structuring Eigensolver Solution) For the solution of large eigenvalue problems, the AMSES solution can be used instead of the Lanczos eigensolver. The resulting eigenvalues and eigenvectors are used in eigenvalue analysis, CMS Super Element creation, Modal Frequency Response, and Modal Transient Analysis. In addition, the AMSES solver can be used during topology, topography, shape, and sizing optimizations. The AMSES solver can be 2-100 times faster than Lanczos. AMSES is a multi-threaded application and can use any number of processors. AMSES will use the same number of processors that OptiStruct is using. Activating AMSES To use AMSES, one of the following must be defined: 1. Use of the EIGRA data, instead of EIGRL data. 2. Use of AMSES solver keyword on the CMSMETH data

AMSES Usage Guidelines The following guidelines list the factors affecting AMSES usage: 1. The AMSES solution is, generally, much faster than Lanczos, but the results are approximate. Accuracy of the lower modes is very high; therefore, AMSES is a good candidate for solutions with a large number of modes (greater than a few hundred) where an approximated eigen-space is sufficient (as in Modal Frequency Response and Modal Transient Response Analysis). Although approximate, the large number of modes used for modal analysis will encompass the modal space and the resulting motion will match very closely with the Lanczos results. Lanczos is recommended in solutions where accurate mode shapes of a small number of modes are required. 2. AMSES is also recommended in cases where: 1) A low number of eigenvalues are requested but the model consists of more than a million degrees of freedom, and/or; 2) The upper bound (V2) is specified or the number of modes (ND) is greater than 50 on the EIGRL entry. In such cases, it is likely that Lanczos runs are slower than AMSES runs. 3. For optimization runs, if accuracy of the eigenvector is important, normal modes analysis with AMSES can be run first and then Lanczos can be run with precise lower and upper bounds to check the AMSES run for accuracy. The AMSES upper bound can then be adjusted to achieve acceptable accuracy of the desired eigenvectors. Now, AMSES can be used for all optimization runs in this analysis. 4. The AMSES solution is much faster for flexible body generation and modal solutions with many residual vectors. 5. AMSES should be used cautiously in situations with very large RBE3’s (i.e., if the RBE3 is connected to 1/4th of the structure). It may be better to eliminate such RBE3’s. 6. AMSES solution speeds depend on the number of eigenvector degrees of freedom (DOF) to be calculated. DISP=ALL will cause the entire eigenvector to be calculated and the speedup will not be large. However, if results for only a few DOF are required (typical for NVH analysis), AMSES can be up to 100 times faster than Lanczos. To improve AMSES run times, it is recommended to request results only for the required DOF.

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7. For an AMSES run with V1, V2 and ND specified on the EIGRA entry, AMSES calculates all the modes up to the specified V2 (upper bound) regardless of the value of ND. Then “ND” number of requested modes is output. Therefore, reducing ND by keeping the upper bound (V2) the same will not significantly improve the AMSES run times, the upper bound must also be correspondingly reduced to prevent the extraction of extra modes. 8. AMSES is also useful in checking for model irregularities. AMSES can be used to print the list of grids associated with a massless mechanism or a singularity.

Parameters Affecting AMSES AMSES controls the accuracy and the cost of a solution with the parameter AMPFFACT. The “optimal” value of AMPFFACT for typical NVH analysis, 5.0, has been established through extensive testing. AMPFFACT is set on the EIGRA and CMSMETH data. In case of predominately solid models, such as engine blocks, AMPFACT should be set to 10.0. PARAM,RBMEIG can be used to adjust the upper limit on eigenvalues associated with rigid body modes. The default upper limit is 1.0 (equivalent to a natural frequency of 0.16 Hz) if PARAM,RBMEIG is not included in the deck.

Residual Vector Calculations When the AMSES eigensolver is used, residual vectors for each of the following are calculated: USET U6 data Frequency Response Dynamic Loads Transient Response Dynamic Loads Damping DOF from CBUSH, CDAMPi or CVISC data One Residual Vector is calculated for each USET U6 degree of freedom, each DAREA degree of freedom, and each damping degree of freedom associated with the CBUSH, CDAMPi and CVISC data. The Residual Vector calculations are controlled by the Solution Control data RESVEC. To control Residual Vector calculations with AMSES, the following commands can be used: Use RESVEC=NO to turn off Residual Vector calculations with AMSES Use RESVEC(NODAMP)=YES to turn off Residual Vectors associated with Damping DOF. If the center of a large RBE3 is loaded, a residual vector will be created that includes terms for each of the independent DOF. If this number is large, say over 500, the AMSES run time will increase dramatically. For large loaded RBE3 it is recommended to use the RBE3 UM data to make the center GRID independent.

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Modeling Techniques The Modeling Techniques section provides an overview of the following: Parts and Instances Subcase Specific Modeling Direct Matrix Input Flexible Body Generation Poroelastic Materials Elements and Materials Loads and Boundary Conditions Virtual Fluid Mass Modeling Errors

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Parts and Instances Introduction The Parts and Instances functionality can be used to combine independently created substructures (or, parts) into a single model. This feature allows you greater flexibility in the creation of a finite element model. By following a few simple numbering requirements, you can define independent substructures which in turn can be easily combined into a final bigger model using simple translational and rotational transformations. Explicit instancing of parts can be achieved, as explained in the Instances section. This functionality is available for all analysis solution sequences and is currently not supported for optimization runs. OptiStructMulti-body Dynamics (OS-MBD) and Geometric Nonlinear Analysis are not supported. Motivation There are various advantages to defining a large finite element model as a combination of substructures or parts: 1. Model complexity is reduced as the structure is segregated into manageable substructures which are interconnected using simple transformations. 2. Individual part modules can be locally updated without having to make cascading edits to the entire structure.

Figure 1: Illustration depicting a model created as a combination of multiple parts (substructures)

3. Each substructure can be independently developed in a modular environment and later assembled into a single structure. This allows various departments working on a project to focus on independent modules while following a few simple numbering requirements. Numbering Requirements In this implementation, specific ID control for grid points and elements (including rigid elements) is not required. Refer to ID Resolution Guidelines for a detailed explanation of part numbering and how this influences the various other data entries in the model. The following simple numbering requirements should be enforced for any solver deck containing multiple parts:

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1. Entities which are part specific, like grid points and elements, are numbered by part based local numbering. 2. Global entities such as properties, materials, loads and boundary conditions are defined in the global numbering system. 3. The individual parts forming the total structure can be combined without any changes in format if the numbering requirements (1) and (2) are met.

Parts A part can be visualized as an independent entity which is connected to a global structure during assembly (The global structure is also considered to be a part). Each part can be defined as a section of the entire finite element model that is used for a specific purpose. For example, a door of an automobile can be defined as a part and multiple instances of this part can be instantiated (refer to Instances for a detailed explanation of part instancing) to save time during modeling. Definition A part is included in the global structure between BEGIN and END bulk data entries potentially using the INCLUDE entry. The INCLUDE entry accepts a string that references the file name (filename.fem) of a specific part. Parts are defined as separate solver decks and could be included within the same working directory. Multiple INCLUDE entries can be used within a single set of BEGIN and END entries to add multiple sections of a single part (refer to Instances for a detailed explanation of part instancing). A part can also be split between separate BEGIN and END entries with the same part name. Format BEGIN and END bulk data entries mark the start and end of the definition of a part in the global structure. A part is defined as a separate file which is included in the global structure using the INCLUDE bulk data entry (The entire solver deck of the part can also be inserted between the BEGIN and END entries, instead of using the INCLUDE entry). The format used to include a part in a global structure is as follows: BEGIN, FEMODEL, name INCLUDE “filename.fem” END, FEMODEL, name The second field of the BEGIN entry should be set to FEMODEL and the third field should contain the name of the part. This part name will be used to define fully qualified references to local entries within the part from anywhere in the model. All part names in the model should be unique. Part names are not case sensitive and should start with a letter. They can contain letters, digits, and underscores, only. filename.fem follows standard requirements for INCLUDE, that is, it can refer to a local file or contain the full path. No other BEGIN or END entries are allowed between BEGIN and END. All bulk data entries should be located between BEGIN, FEMODEL and END, FEMODEL. A full model consists of several parts. One part is designated as global. The other parts can be moved to arbitrary locations using the INSTNCE bulk data entry and connected to each other using connector elements or CONNECT bulk data entries.

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Local and Global Entries Local entries, as their name suggests, are data entries that can be defined locally within a specific part. Local entries are currently limited to the following (refer to the Appendix for a list of all local entries). Grid Point Definition GRID Bulk Data Entry Rigid Elements RROD Bulk Data Entry RBAR Bulk Data Entry RBE1 Bulk Data Entry RBE2 Bulk Data Entry Elements CHEXA Bulk Data Entry CQUAD Bulk Data Entry CBUSH Bulk Data Entry and so on All other data entries are considered global and should be defined in the global structure. Even if they are located within BEGIN – END entries in a different part, they are still interpreted as if they are in the global part. Some global data entries can reference local data entries (see above list) using fully qualified references.

Fully Qualified References The same ID number can be used for non-unique local data entries defined in multiple parts. Such ID’s cannot be referenced by entries in the global structure without the use of fully qualified references. Fully qualified references contain information about both the local data entry and its corresponding part. Numeric Reference

Fully Qualified Reference

A numeric reference on a local entry References a local entry defined references another local entity within within any part. the same part, or a global entity. A numeric reference can be used to reference a global entry only if another entry of the same type and with the same ID does not exist within the current part. A numeric reference on a global entry A fully qualified reference can be references an entity within the global used to reference an entry when part only. another entry of the same type and with the same ID exists within the model.

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Numeric Reference

Fully Qualified Reference

Format: A equal to the set ID of a non-local entry that is to be referenced is input in the corresponding field(s) of the referencing entry.

Format: This is similar to the format of a numeric reference. The difference is that is input in the corresponding field(s) of the referencing entry. The PartName is the name specified on the BEGIN entry for the part that contains the referenced entry.

Example:

Example:

RBE2, 16, 9, 123, 10

RBE2, 16, door.9, 123, bpillar.1

This connects grid points 9 and 10 located in the same part as the RBE2 This connects grid points 9 and 10 located in part “door” and part entry (RBE2 is a local entry). “bpillar”, respectively. CHEXA, 5, 9, … This RBE2 entry can be located in This references global material 9 any part. irrespective of where the MAT#, 9 entry is located (Currently, all Material entries are global entries).

ID Resolution Guidelines The following guidelines can be used to implement proper ID resolution in a model containing multiple parts and instances. 1. Each part can be included only once within a specific set of BEGIN and END bulk data entries. Inclusion of multiple copies of a single part is known as instancing (refer to Instances). 2. All references to properties and materials are resolved in a standard way. These entities are global and should be defined only once anywhere in the model. 3. Subcase information and I/O options entries are also handled similarly. These entries refer only to numeric ID of entries in the global part (for example, SPC = 5 will expect SPCADD, 5 or SPC, 5 within the global part). SPC’s, MPC’s, SPCADD and MPCADD are local entries and they allow fully qualified referencing of local entries anywhere in the model. SPCADD, MPCADD entries in parts are allowed but will not be used in the solution as they cannot be activated by subcase selectors. 4. Fully qualified references are allowed in some data entries (refer to the latest OptiStruct Reference Guide to check if a data entry accepts fully qualified references). Not all entries and not all fields within these entries allow fully qualified references. 5. This generalized syntax is allowed in all four bulk formats – fixed small field, fixed large field, free and free large field. As fully qualified references are usually longer than 8 characters, free formats are more useful for this purpose.

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6. As a general practice, all data entries using fully qualified references are placed inside the global part. This is not mandatory. 7. The own ID of each data entry (usually the second field after the card name) cannot be a fully qualified reference. 8. As explained previously, for local entries, their own ID’s within a particular part cannot be equal. Whereas, similar data entries can be the same own ID if they are defined in different parts. In such cases they are completely independent entities and their ID’s are resolved by using fully qualified references. The same rules apply to set ID’s, for example, in SPC’s or MPC’s – the same SID in different parts represent completely independent entities. 9. Any reference to a global entry must be a numeric reference regardless of whether it is being referred to from a global or a local part. 10. A fully qualified reference (if allowed) is resolved to a specific instance defined by part name and ID within that part. 11. If a local entry contains a numeric reference (instead of a fully qualified reference), OptiStruct resolves the reference to a local entry within the same part. If the part does not contain an entry (of the required type) with the ID equal to the numeric reference, OptiStruct looks in the global part for a possible match. If the entry is not available in the global part also, then the program errors out, regardless of whether the required entry (with same ID) is available locally in a different part. For example: This entry is located in local part “grip”: RBE2, 15, 5, 123, 7, 8 Grid points 1,3,5 are included in part “grip” Grid points 1,3,5,7,8 are included in local part “frame” Grid points 3,5,7 are included in global part “racquet” In the above example, on the RBE2 element, grid point 5 refers to grid grip.5, grid point 7 refers to racquet.8 and grid point results in an error. 12. OptiStruct allows repetition of some global data entries, even if only unique ID’s are allowed, only if the content of such cards is identical (for example, material and property entries).

Logical Sets The SET bulk data entry can be used in the global part to reference SET’s defined within different parts. These SET entries in the global part can contain fully qualified references to part-specific SET data only if logical operators (OPERATOR field on the SET entry) are used. For example: The following SET entry exists in part “A”: BEGIN, FEMODEL, A SET, 29, ELEM, LIST 15 THRU 30 … END, FEMODEL, A

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Referencing SET, 29 in the global part “G”: BEGIN, FEMODEL, G SET, 78, ELEM, OR A.29 … END, FEMODEL, G This process can be used to reference local sets in the global part on entries which do not support fully-qualified referencing of local sets (like, output entries). For example: The following SET entry exists in part “A”: BEGIN, FEMODEL, A SET, 3, GRID, LIST 15 THRU 30 … END, FEMODEL, A In the global part “G”: Incorrect BEGIN, FEMODEL, G DISPLACEMENT(H3D)=part.3 or DISPLACEMENT(H3D)=3 … END, FEMODEL, G Correct BEGIN, FEMODEL, G DISPLACEMENT(H3D)=3 SET, 3, GRID, OR A.3 … END, FEMODEL, G

Relocation A full model in an OptiStruct Parts and Instances consists of a collection of parts. These parts are inserted into the global structure automatically (without relocation) or by using entries which allow relocation (RELOC and INSTNCE). Relocation involves the positioning of parts within the model relative to each other or the global structure. Currently relocation of parts included in the full structure can be accomplished using the INSTNCE and RELOC bulk data entries. Relocation involves translational and rotational movement of parts relative to their initial position. INSTNCE Bulk Data Entry The full model consists of several parts. One part is designated as global and the rest of the included parts are relatively positioned with respect to the global part with the help of the INSTNCE entry. The INSTNCE entry references the RELOC bulk data entry that defines the relocation or mapping of grid points from one position to another.

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Format A INSTNCE bulk data entry can be specified only within the global part. No INSTNCE entry can reference the global part. Each INSTNCE entry should reference a unique part name, however, it is not required that every part is positioned using an INSTNCE entry. Parts which are not specified on the INSTNCE entry are included in the full model without any relocation. INSTNCE, SID, name, NN The third field “name” specifies the name of the part defined via the BEGIN, FEMODEL, name entry and NN references the ID of the RELOC bulk data entry. The RELOC bulk data entry defines the actual location of the part in the final model. Refer to the OptiStruct Reference Guide for detailed information. RELOC Bulk Data Entry The RELOC bulk data entry defines relocation or mapping of grid points from one position to another. It can be used in conjunction with the INSTNCE entry to relocate parts relative to each other or to the global structure. Format RELOC entries are used to relocate grid points within the full model. RELOC defines independent transformation of each part, not the transformation of one part with respect to another. In particular, it does define that two parts should be connected together and then relocated as an assembly. It can be specified in the global part and is referred to by part specific entries like INSTNCE. RELOC, ID, type, GID# … The third field “type” specifies the type of relocation: MOVE, MATCH, ROTATE, or MIRROR. There are multiple formats of the RELOC entry depending on the specified type. Refer to the OptiStruct Reference Guide for detailed information.

Connectivity The final step in the part assembly process is connection. Connectivity between parts can be achieved in two different ways, the first involves using the CONNECT bulk data entry and the other is using connectors like rigid or bush elements that can explicitly reference grids in any part. Connectors can contain a mix of regular and fully qualified ID’s. Regular ID’s reference: In a Local entry: An entry with that ID in the same part, or to an entry with the same ID in the global part. In a Global entry: An entry with that ID in the global part. Refer to comment 11 in the ID Resolution Guidelines section for information on how regular ID references in localized entries are resolved. The Appendix contains a list of supported Local entries which can be defined within a part. CONNECT Bulk Data Entry The CONNECT bulk data entry defines equivalence for degrees of freedom between two parts. Using two forms of the entry, you can equivalence all degrees of freedom for grids of both parts within the specified tolerance distance from each other.

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CONNECT always equivalences grid points located within two parts at the same location (X, Y, and Z) after all parts are moved to their final location using INSTNCE entries. If multiple grid points exist in both parts at the same location (within the specified tolerance), then they will be equivalenced together. Format The CONNECT entry can be used to define equivalence between two parts (name_a and name_b) by searching for all grid points within a specified tolerance (tol). The alternate form involves specifying a set of grid points at which to search for neighboring grids of either part “name_a” or part “name_b” within a specific tolerance (tol). CONNECT, name_a, name_b, tol The tolerance (tol) is a numeric value defining the maximum distance between two grid points to allow equivalence. All grids in “part_a” are considered, if the search in “part_b” finds a grid point matching the location (within the specified tolerance), then these two grids are equivalenced. Refer to the OptiStruct Reference Guide for detailed information.

Instances Instances are multiple copies of a part that are exactly the same as the part itself in all respects. Currently, part instancing in OptiStruct is available as a logical extension to the part inclusion process. A direct approach to part instancing will be available in a future release. Creating instances of a part To create an additional instance of an existing part, the part inclusion process can be repeated. For example, to create an additional instance of the part “CrankShaft”, the BEGIN FEMODEL and END FEMODEL statements are repeated. The same set of include files (using the INCLUDE entry) are repeated inside multiple part definitions. BEGIN, FEMODEL, CrankShaft_1 include “CrankS_a.fem” include “CrankS_b.fem” END, FEMODEL, CrankShaft BEGIN, FEMODEL, CrankShaft_2 include “CrankS_a.fem” include “CrankS_b.fem” END, FEMODEL, CrankShaft

Appendix Local Data Entries A list of data entries which can currently be defined as local entries in OptiStruct are listed below: Miscellaneous GRID, SPC, MPC, SET

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Elements CAABSF, CBAR, CBUSH1D, CBUSH, CDUM1, CELAS1, CELAS2, CELAS3, CELAS4, CFAST, CGAP, CGAPG, CGASK12, CGASK16, CGASK6, CGASK8, CHACAB, CHEXA, CONM1, CONM2, CONROD, CONV, CPENTA, CPYRA, CQUAD4, CQUAD8, CROD, CSEAM, CSHEAR, CTETRA, CTRIA3, CTRIA6, CTRIAX6, CTUBE, CVISC, CWELD, and PLOTEL Rigid Elements RBAR, RROD, RBE1, and RBE2

Entries which allow fully qualified references TO GRID POINTS: A list of data entries which currently allow fully qualified references to grid points in OptiStruct are listed below: Local entries CBUSH1D, CBUSH, RBAR, RROD, RBE1, RBE2, and RBE3 Global entries SPC, MPC, FORCE, MOMENT, SPCD, and RELOC TO OTHER ENTITIES: A list of data entries which currently allow fully qualified references to other entities in OptiStruct are listed below: SPCADD and MPCADD (allow fully qualified references to SPC and MPC, respectively) SET (allow fully qualified references to other local SET entries only if logical operators are used) Refer to the Release Notes for the latest list of all global/local entries and for entries which allow fully qualified references to other entries. Analysis entries are available for Parts and Instances entries specific to optimization (for example, DVGRID, DESVAR) are not allowed.

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Subcase Specific Modeling Introduction Subcase specific modeling in OptiStruct allows you to analyze multiple structures in a single solver run. The structures can be completely independent; can represent different regions of the same model, or different assemblies sharing common parts. Subcase specific modeling is also known as submodeling. In a traditional modeling scenario without submodeling capabilities, the entire model will be solved for each solver run, regardless of the boundary conditions and multiple models will have to be created to solve structures with variable sections. Submodeling allows specific sections of the model to be solved independently without affecting the rest of the structure. Motivation There are various advantages to subcase specific modeling for both analysis and optimization. The primary motivation for submodeling is the ability to solve independent structures with common parts. For example, if you consider the case of a pickup truck, various cabin shapes can be solved for, by allowing the bed and the chassis to remain unchanged. This is accomplished by assigning elements belonging to different cabin types to different sets. The constant chassis and bed can be a defined as another element set. These element sets can now be combined to allow various structural submodels to be modeled without having to repeat the common parts of the structure for each solution.

Figure 1: Example illustration depicting an application of subcase specific modeling.

In Figure 1, each cabin body can be defined as a specific element set and the common parts will make up one element set. These can be independently combined under three different subcases with different boundary conditions and solved in one single solver run. Implementation The subcase specific modeling functionality is realized with the use of element sets and the subcase selector entry SUBMODEL. The SUBMODEL entry can be used within a specific subcase in the subcase information section to select a certain element set for the solution. SUBMODEL subcase information entry The full model consists of several common or shared sections which remain constant while other sections are changed to find the best fit for a particular application. In such cases, the

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SUBMODEL entry can be used to choose a set of elements for selective solution. The element set identification number is the only input required for this entry. Format SUBMODEL, SID, SID_r The fields “SID” and “SID_r” specify the identification numbers of the element and rigid element sets, respectively defining the submodel that is solved within this subcase. The “SID_r” field is optional. Refer to the OptiStruct Reference Guide for detailed information. Submodeling - Example OptiStruct input deck $ Subcase Information Section SUBCASE 1 $ Submodel specific SPC’s and LOAD’s can be defined here. SUBMODEL, 11 SUBCASE 2 $ Submodel specific SPC’s and LOAD’s can be defined here. SUBMODEL, 12 $ Bulk Data Section SET, 1, ELEM $ defines the shared/common part, for example, the chassis, bed and wheels in Figure 1 SET, 2, ELEM

$ defines the individual part, for example, cabin body 1 in Figure 1

SET, 3, ELEM

$ defines the individual part, for example, cabin body 2 in Figure 2

SET, 11, ELEM, OR, 1, 2 common parts

$ defines the full truck model with cabin body 1 and the

SET, 12, ELEM, OR, 1, 3 common parts

$ defines the full truck model with cabin body 2 and the

Comments 1. Single Point Constraints (SPC), Loads (LOAD), Multi Point Constraints (MPC) and other similar subcase selectors should define attributes only applicable to the specific submodel or substructure. These attributes should apply exclusively to the subcase-specific model defined via SUBMODEL. The SUBMODEL entry does not trim the specified attributes (loads, constraints and so on) to the defined subcase. 2. This functionality is currently available for linear static analysis only. All optimization types with responses from Linear Static analysis are supported (except SPCFORCE/ residual force responses).

Global-Local Modeling Global-local analysis is a technique in which a full model is solved using two (or more) submodels; one submodel represents the full structure but at a lower accuracy (for example, a larger mesh size) and the second submodel represents only a part of the structure (for example, using a smaller mesh size). The global structure is solved first and the displacements from the selected zone are interpolated and applied to the local structure.

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Global-local analysis is implemented with the use of the subcase specific modeling technique defined above. Note that Global-local modeling is only an approximation, and its use depends on the assumption that a more accurate local model will not significantly affect the displacements of the global structure. This process should not be used whenever small stiffness changes in the local submodel may have a large impact on the solution outside of it. Motivation Global-local analysis may help improve results in models with local stress concentrations. Parts of the structure with small details which require relatively higher accuracy can be modeled as local submodels with a fine mesh and the full structure can be modeled using a coarse mesh. This will allow for faster solution times as only a part of the structure is being solved with a fine mesh.

Figure 2: Example illustration depicting an application of global-local analysis.

In Figure 2, an example building is illustrated wherein sections containing the pillar-roof joint are modeled as separate submodels with a refined mesh. The finer mesh allows for better accuracy at regions with high stress concentrations. Using the global-local analysis capability, the results from the coarser global model are interpolated and applied to the finer mesh of the local model at the transfer region. This allows for the local model to be driven by the results of the global model. Implementation The global-local modeling functionality is realized with the help of the subcase specific modeling feature and the subcase selector entry GLOBSUB. This entry is defined in the subcase which contains the local model definition. The GLOBSUB entry identifies the global model which is used to drive the specific local model. GLOBSUB subcase information entry The full model consists of several sections with areas of high stress concentration or regions of interest which require a higher accuracy. In such cases, the entire model can be solved with a coarser mesh and in each local subcase defining the submodel of interest, the global structure can be referenced using the GLOBSUB entry. The set of grids within the local structure at the transfer zone is also specified.

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Format GLOBSUB, SUBID, SID The field “SUBID” specifies the identification number of the subcase that contains the global structure definition (via SUBMODEL) and the field “SID” specifies the set of grid points in the local structure that defines the transfer zone. The displacements from the global structure are applied to this set of grid points. Refer to the OptiStruct Reference Guide for detailed information. Global-Local Analysis - Example OptiStruct input deck $ Subcase Information Section SUBCASE 1 $ Submodel specific SPC’s and LOAD’s can be defined here. SUBMODEL, 11 SUBCASE 2 $ Submodel specific SPC’s and LOAD’s can be defined here. SUBMODEL, 12 GLOBSUB, 1, 15 $ Bulk Data Section SET, 11, ELEM

$ defines the global structure, for example, the full building in Figure 2

SET, 12, ELEM

$ defines the local structure, for example, the pillar-roof joint in Figure 2

SET, 15, GRID $ defines the transfer zone, for example, the interface grids of the pillarroof joint in Figure 2 at which the displacements are interpolated. Comments 1. The transfer zone should contain only 3-dimensional elements in both the local and global structures. Second order elements (for example, CHEXA20) are allowed. There is no further restriction on element types elsewhere in the structure. 2. The transfer zone may represent single or multiple cuts (sections) through the structure. Multiple cuts should be separated from each other, that is, they should not exist closer than the element size of the global model. 3. The GLOBSUB entry should always reference the subcase ID of a global subcase that is defined above its corresponding local subcase. 4. This functionality is currently available for linear static analysis only. All optimization types with responses from Linear Static analysis are supported. Exceptions: SPCFORCE/residual force responses are not supported. Topology must lie outside the local part(s) and, any design variables affecting the local submodel should also be mapped to the global model.

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Direct Matrix Input (Superelements) Direct Matrix Input Creating Superelements Component Dynamic Analysis Super Element Generation

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Direct Matrix Input In the course of a finite element solution, matrix representations of a structure's stiffness, mass, damping, and loading are generated. The matrices are based on the information provided in the Bulk Data section of the input file. Depending on the analysis type, system equations using these matrices are solved to simulate the structure’s behavior. In a linear static analysis, for example, a system of linear equations Ku=f is solved. Here, K is the stiffness matrix, f is the loading vector, and u is the vector of the unknown displacements. The time taken for these matrix solutions is about proportional to the square of the number of degrees-of-freedom of the structure. The solution speed can be improved by representing sections of the structure (super element assemblies) with a smaller subset of degrees of freedom of those sections (boundary degrees of freedom of the super element assemblies) and a representative set of reduced matrices. In an optimization, for example, the solution speed can be improved dramatically by removing the non-design portion of the structure and keeping only the design portion of the model. For the purpose of deriving the matrices, the displacement vector may be partitioned into displacements of inner (OSET) and outer (ASET, interface) degrees of freedom.

(1) Here, the subscript ‘o’ denotes the inner degrees of freedom, and ‘a’ the interface degrees of freedom. The static equilibrium is given as:

(2)

The eigenvalue problem for a normal modes analysis of the body using a diagonal mass matrix represents itself as:

(3)

Note: PARAM, WTMASS cannot be applied to superelements (.h3d or .pch) that are read into the model. If the unit of mass is incorrect on the MAT# entries and PARAM, WTMASS is required to update the structural mass matrix; then this should be done in the creation run. There are two ways of obtaining the reduced matrices: 1. Static Condensation (or Guyan Reduction) reduces the linear matrix equation to the interface degrees of freedom of the substructure through algebraic substitution. The result can be used as external matrices, representing a super element assembly, in a finite element analysis. This method is accurate for the stiffness matrix and approximate for the mass matrix.

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In addition, the load vectors are reduced to the ASET DOF. This includes the load vectors from point and pressure loads as well as distributed loads due to acceleration (GRAV and RLOAD). 2. Component Mode Synthesis (CMS) reduces a finite element model of an elastic body to the interface degrees of freedom and a set of normal modes. The result can either be used for inclusion as a flexible body in a multi-body dynamics analysis (see Flexible Body Generation) or as external matrices, representing a super element assembly, in a finite element analysis. It is always an approximation; however, it is the preferred method for dynamic analysis as it captures the mass matrix correctly. Load vectors are not reduced during CMS Super Element creation.

Static Condensation The first line of equation (2) reads as:

Koo uo Koa ua

Po

The displacements of the inner degrees of freedom, therefore are:

uo

(4)

Koo1 Po Koa ua

Also from equation (2):

K ao uo K aa ua

Pa

Substituting equation (4) into this:

K ao Koo1 Po Koa ua

K aa ua

Pa

or

K aa

K ao Koo1 K oa ua

Pa

K ao Koo1 Po

This is interpreted as:

K reduced ua

Preduced

So the reduced stiffness is:

K reduced

K ao Koo1 Koa

K aa

And the reduced loading is:

Preduced

Pa

K ao Koo1 Po

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The reduced matrix can also be written as:

K reduced

S T KS

K aa

K ao Koo1 Koa

Then, using the same transformation, you can obtain the reduced mass matrix:

M reduced

M aa

K ao Koo1M oa Koo1 Koa

K ao Koo1M oa M ao Koo1 Koa

The solution of the reduced linear static problem provides an exact solution, whereas the

K u M reduced ua , only provides solution of the reduced eigen problem, reduced a approximations to the solution of the full eigenvalue problem as only vectors that satisfy the constraint

uo

Koo1 Koa ua will be included in the solution.

Component Mode Synthesis (CMS) The interface or boundary degrees of freedom (ASET) that are used in the construction of mode shapes should be representative of the set of force-bearing degrees of freedom in the subsequent analysis. For a finite element analysis, this refers to those nodes that connect to either the residual structure or other super element assemblies. The purpose of specifying the interface or boundary degrees of freedom for CMS is mainly to account for the static deformation due to constraint or applied forces acting on the interface degrees of freedom. A large number of eigenmodes is required if these static modes are omitted. The flexible deformations due to constraint forces, compared to the deformation due to the body inertia forces, are often dominant in most constrained models. The inclusion of all force-bearing degrees of freedom as interface degrees of freedom is therefore an essential step to get accurate results from subsequent analyses. The task of the component mode synthesis is to find a set of orthogonal modes represent the displacements u of the reduced structure such that:

that

q Where, q is the matrix of modal participation factors or modal coordinates which are to be determined by the analysis. For creating external matrix representations of super element assemblies for use in subsequent finite element analyses, only the Craig-Bamption method of component mode synthesis is currently available.

Craig-Bampton Method This method uses a system constrained in the interface degrees of freedom. Normal modes analysis of the system yields the diagonal matrix of eigenvalue and the matrix of eigenmodes . In this normal analysis, you can select the cut-off frequency or the number of modes to be solved. This determines the column dimension of . In addition, a static analysis is performed with a unit displacement in each interface degree

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of freedom while all other interface degrees of freedom are fixed. The number of subcases in this static analysis is six times the number of interface nodes. Note that it is important to constrain each interface node with its neighboring nodes, if necessary, to ensure that it has non-zero stiffness along the direction of all six degree of freedom. This yields the displacement matrix and the interface forces . Reduced modal stiffness

and mass matrices

are now generated using,

which yields:

. It follows an othogonalization step that transforms the original shapes X into a set of orthogonal modes .

Eigenvector Normalization First, a new eigenvalue problem using the reduced matrices above is solved.

The resulting diagonal matrix of eigenvalues D and the normal modes N are used to transform the set of original shapes into the set of orthogonal modes .

It can be shown that the resulting modes are orthogonal with respect to the system stiffness matrix K and mass matrix M.

If the orthogonal modes are normalized with respect to the mass matrix M, the reduced matrices for the subsequent analysis appear as:

Generating the Matrices Through the inclusion of certain bulk data and I/O options entries described here, static condensation and component mode synthesis may be performed on a structure and the reduced matrices written to a file for use in subsequent analyses. For static condensation, only stiffness, mass, and load matrices can be generated. If the SUBCASE is for static analysis, by default the reduced mass matrix will not be created. However, PARAM,DMIGMASS,YES can be used to create the reduces mass matrix for static analysis. For eigenvalue analysis, the mass and stiffness matrix will be reduced, but there is

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no reduced load vector. For component mode synthesis, stiffness, mass, structural and viscous damping, and fluidstructure coupling matrices can all be generated. No reduced load vector is created. Matrix reduction is activated by the presence of ASET or ASET1 bulk data entries. These bulk data entries indicate the interface or boundary degrees of freedom of a super element assembly, i.e. the set of degrees-of-freedom where the component, being replaced by direct matrix input, connects to the modeled structure. If ASET or ASET1 bulk data entries are present, and there is no CMSMETH I/O Option, then the static condensation method is used to generate the reduced matrices. Component mode synthesis is activated by the presence of the I/O Option CMSMETH. The I/ O Option references a CMSMETH bulk data entry, which defines the method of matrix reduction to be used (CBN or GUYAN methods apply to external super element generation). When CBN is the selected method, the frequency range or number of modes to be calculated and the starting SPOINT ID for storing the modal data are also defined on CMSMETH. For GUYAN, this additional information is ignored. With component mode synthesis, no loads or SPC boundary conditions can be applied directly to the portion of the structure that is being removed. For the definition of loads or SPCs, however, an ASET can be defined at the grid point of a load or SPC. Then, loads or SPCs can be applied to that grid point in the assembled model. Reduced matrices are automatically written to the .h3d output format, unless OUTPUT,H3D,NONE is defined. The matrices can also be saved in the Nastran punch format (.pch file) or in a binary format (.dmg file) by using the PARAM, EXTOUT bulk data entry. The matrices are written to the .pch file in the DMIG bulk data entry format. They are defined by a single header entry and one or more column entries. The I/O option entry DMIGNAME provides you with control of the name of the matrices written to the .pch and .dmg files. This is an optional entry and if not used matrices are given the suffix “AX.” With static condensation, the stiffness matrix is always output when PARAM, EXTOUT is present in the input file; however, the load matrix is only output if a linear static subcase is present, and the mass matrix is only output if an eigenvalue subcase is present. All subcases must use the same boundary conditions (SPC set) and multi-point constraints (MPC set) or an error termination will occur. With component mode synthesis, the orthogonal modes and the corresponding eigenvalues D are exported to the .h3d file by default. The export of the model to this file can be controlled using the MODEL output statement. This allows for only a small portion of the model or a “display model” (a coarse representation of the structure consisting of PLOTELs) to be exported. For component modal synthesis super elements written to the .h3d file, there is no interior grid or element data stored by default. However, the MODEL data can be used to specify interior grids for displacement, velocity, or acceleration results, and interior elements for stress and strain results in the residual run. The SEINTPNT data can be used to convert the interior points to exterior points in the residual run, so that they can be used as connection points, loading points, or response points in optimization.

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Using the Matrices in a Finite Element Analysis To use the reduced matrices that were written to an .h3d file, the ASSIGN I/O option must be used to assign names to such matrices. Unlike matrices stored in .pch or .dmg formats, those stored in the .h3d file are named on retrieval, rather than on creation. The ASSIGN, H3DDMIG command provides a suffix “matrixname” for the matrices retrieved from that file. Once matrices in an .h3d file have been assigned a name they may be referenced using one of the subcase information entries listed below. All of the matrices in the .h3d file are used in the analysis by default. If only some of the matrices are to be used, then use the K2GG, M2GG, K42GG, and B2GG data to specify which matrices are to be used. The unreferenced matrices will not be used in this case. The SEINTPNT can be used to convert the interior points specified by the MODEL data in the creation run to exterior points in the residual run, so that they can be used as connection points, loading points, or response points in optimization. Reduced matrices that are written to .pch files are stored as DMIG bulk data input. As these reduced matrices are already in a recognized input format, the files simply need to be included in the bulk data section by an INCLUDE statement. Reduced matrices that are written to a .dmg file are stored in a binary format. Similar to the .pch file, these reduced matrices simply need to be included in the bulk data section by an INCLUDE statement. As matrices are referred to by name, it is important to ensure that when multiple reduced matrices are used that they have unique names. Matrices may be chosen through one of the following subcase information entries as either stiffness, mass, damping, or load matrices. The K2GG subcase information entry references a matrix by name, indicating that it is a stiffness matrix. The stiffness matrix must be symmetric and it applies to all subcases. The M2GG subcase information entry references a matrix by name, indicating that it is a mass matrix. The mass matrix must be symmetric and it applies to all subcases. Gravity and centrifugal loads are not considered on the external mass matrix (M2GG). Gravity and centrifugal loads must be included in generating the reduced loads (P2G). The P2G can then be used in DMIG input to get the exact static results. The B2GG subcase information entry references a matrix by name, indicating that it is a viscous damping matrix. The viscous damping matrix must be symmetric and terms are added to it before any constraints are applied. The K42GG subcase information entry references a matrix by name, indicating that it is a structural element damping matrix. The structural element damping matrix must be symmetric and terms are added to it before any constraints are applied. The P2G and P2GSUB subcase information entries reference a matrix by name, indicating that it is a load matrix. The load matrix must be columnar and terms are added to it before any constraints are applied. Gravity and centrifugal loads are not considered on the external mass matrix (M2GG). Gravity and centrifugal loads must be included in generating the reduced loads. The P2G and P2GSUB can then be used in DMIG input to get the exact static results. P2G must be used above the first SUBCASE. If there are multiple load vectors in the DMIG, then they will be applied in successive static SUBCASE until they are all used or until every static SUBCASE has a load vector. P2GSUB is used within a specific SUBCASE. If there is more than one load vector in the DMIG, then P2GSUB can be used to specify which load vector is to be used in that SUBCASE.

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The A2GG subcase information entry references a matrix by name, indicating that it is a fluid-structure coupling matrix. Only one instance of the fluid-structure coupling matrix is allowed.

Example Figure 1 shows a finite element model composed of two components: a design component and a non-design component. The structure is fully clamped along the left-hand edge and has a downward vertical force applied along its right-hand edge. This example will demonstrate how to replace the non-design component with a set of reduced matrices at the interface nodes (the nodes which are common to both components) for both an analysis problem and an optimization problem.

Figure 1: Example model indicating design and non-design components and boundary nodes.

First of all, a linear static analysis is performed on the complete structure. The displacement and von Mises stress results from this analysis are shown in Figure 3. Next, a reduced stiffness matrix and a load vector are generated for the non-design component. This is achieved by creating a new finite element model containing just the nondesign component and the loads and boundary conditions directly applied to that component. This is shown in Figure 2.

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Figure 2: Reducing out the non-design component.

All of the degrees of freedom at the interface nodes (see Figure 1) are selected as boundary degrees of freedom. ASET or ASET1 bulk data entries are used to indicate this. The reduced matrices output is requested through the inclusion of the PARAM, EXTOUT, DMIGPCH bulk data entry. The model is submitted to OptiStruct, resulting in the creation of the file filename_AX.pch which contains the reduced matrices in ASCII format. (Had PARAM,EXTOUT,DMIGBIN been used, the reduced matrices would be written in binary form to the file filename_AX.dmg). The non-design component can now be replaced in the original model by its reduced matrix representation. This is done by manipulating the original model as follows: 1. Delete the bulk data entries for the nodes and elements of the non-design component. 2. Delete all loads and boundary conditions that were only applied to the non-design component. 3. Include the file containing the reduced matrices in the bulk data section. 4. Select the reduced stiffness matrix with the subcase information entry K2GG. (In this example, the reduced stiffness matrix will have the default name KAAX). 5. Select the reduced load vector with the subcase information entry P2G. (In this example, the reduced load vector will have the default name PAX). Figure 4 shows the displacement and von Mises Stress for the design component when the non-design component was replaced by a reduced matrix representation. It can clearly be seen, they match perfectly with the original model's results.

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(a) Displacements

(b) von Mises Stress

Figure 3: Displacement and von Mises stress results for a linear static analysis on the complete structure.

(a) Displacements

(b) von Mises Stress

Figure 4: Displacement and von Mises stress results for a linear static analysis with reduced matrix substitution.

Finally, for both the complete structural model and the reduced matrix model, a topology optimization was performed. For both models, the design component was identified as designable, the objective was to minimize the global compliance, and an upper limit of 50% was put on the volume fraction. The optimization results are shown in Figure 5.

(a) Complete Structure

(b) Reduced Matrix Substitution

Figure 5: Density results for a topology optimization.

Compliance results not matching for reduced models. If external forces are applied to the degrees of freedom that are reduced out of the model, then the strain energy or compliance values for the complete model will not match the strain energy or compliance values for the reduced model.

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The reduced loading is:

Therefore, strain energy for the reduced model is:

The strain energy for the complete model is:

So the compliance for the reduced model is missing the term:

But when no external forces are applied to the degrees of freedom that are reduced out of the model, then [f0] = 0, and the strain energy or compliance values for the reduced model match the complete model.

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Creating Superelements There are three types of super elements that can be created in OptiStruct. The first is the Craig-Bampton in which the interface DOFs are fixed. These fixed interface DOF are specified using ASET, BSET, or BNDFIX data. The second is the Craig-Chang in which the interface DOF is free. These free interface DOFs are specified using the CSET or BNDFREE data. The third is a combination of both, Craig-Bampton and Craig-Chang. This third version would be used for an automobile part such as the exhaust system where the fixed DOFs are coupling the exhaust system to the powertrain; and the free DOFs are coupling the exhaust hangers to the body. The other version that uses either form is a combination of a structure and fluid such as an automobile body and its interior cavity. A super element can be a combination of a structure and fluid grids and elements to model a trimmed automobile body and its interior cavity. The fluid grids where the sound pressure response is to be calculated (microphone points) must be in the ASET list. The interface points are where the component is attached to another component either through a rigid connection or a stiffness connection. These interface points must be independent in all connection degrees of freedom. If they are the dependent DOF in an RBE3, the dependencies must be transferred to one of the independent grids referenced by the RBE3 element. In the Bulk Data section on the RBE3 element, the UM parameter shows how to redefine the dependency on the initial grid. The RBE3 can also be changed to an RBE2 but this might stiffen up the local area as a result. Interior points used in the super element to define the rough shape of the component using PLOTEL elements do not need to be independent. A component model has to be complete in order to be made into a super element. All grids referenced in the super element must be in the component model file, including local coordinate systems grids. All properties and materials referenced in the components must also be included in the component. The component model must be able to be run successfully in a modal analysis run. Note: PARAM, WTMASS cannot be applied to superelements (.h3d or .pch) that are read into the model. If the unit of mass is incorrect on the MAT# entries and PARAM, WTMASS is required to update the structural mass matrix; then this should be done in the creation run.

Craig-Bampton Approach Initially, look at the Craig-Bampton input deck for an automobile body model without a bolt on components referred to as a BODY-IN_WHITE (BIW). MODEL = PLOTEL,,NORIGID TITLE = BODY-IN-WHITE CMS MODEL SUBTITLE = DMIGNAME=BIW $-----------------------------------------------------------------------------SUBCASE 1 CMSMETH = 1

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BEGIN BULK $==01==><==02==><==03==><==04==><==05==><==06==><==07==><==08==><==09==><==10= => CMSMETH 1 CBN 300.0 1008000 AMSES ASET1 123456 1000000 THRU 1006004 INCLUDE '/MODELS/BODY/BODY.dat' ENDDATA First, the Craig-Bampton approach can be run in either AMSES or in Lanczos. The results should be the same. Use AMSES for large component models and Lanczos for small component models. Second, the MODEL card will include all PLOTEL and the associated interior grids in the super element. An alternate method to include extra grids can be accomplished by listing the grids in a set and referencing the set in the MODEL card between the ,, in the MODEL card. The PLOTEL generally are used to define a rough outline of the component for viewing its mode shapes. The PLOTEL value will also now include the new plate PLOTEL that are available in OptiStruct. Third, the PARAM, SEPLOT, YES will create a file including all the grids referenced in the MODEL card, plus the interface grids and the PLOTEL referenced in the MODEL card. This file can be used later in residual runs, if desired, along with the super element H3D file, although it is not required. If a residual run on just this single component is to be made, such as a modal or mobility run, then the .seplot file should be created and used in the residual run. Otherwise the runs will fail without having any other input besides the H3D file because no GRID data is in the input data. The DMIGNAME is optional and can be specified later in the residual run. It creates a unique name to the stiffness, mass and damping matrix and can only be a maximum of six characters. The CMSMETH case control card calls the new OptiStruct super element eigenvalue card CMSMETH. The CMSMETH bulk data card referenced in this case the CBN (Craig-Bampton Method). The next field is the upper frequency bound for the modes. The sixth field references the starting number for where the SPOINTS for the modes will be stored. This range of numbers must be unique for the full vehicle model and each super element will need its own numbering range. The seventh field is used to specify the eigensolver (Lanczos or AMSES). The ASET1 lists the interface points of the super element model in the example above. Only the connection and specific response points should be included in the ASET list. The other points in the super element are called interior points. The interior points should not be in this ASET list, since the model size and run time will increase with the number of points included in the ASET list. If you have special response points in the component, they should be included in the ASET list. Generally there are only a few critical response points. The critical response points must be independent points. The calculated responses will show up in the output files. If any type of random analysis is being performed, it is essential that the response points be included in the ASET list with the exterior points. This file will generate an H3D binary file that includes all the information that is needed for this super element. It can also create the optional .seplot ASCII file, if requested. The H3D file will include matrices for the mass and stiffness and it will also include the matrices for structural damping and viscous damping if these elements exist in the model. Look at the bottom of the .out file to see what matrices are created.

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Below is an example. There were 274 modes found for this model. There were 166 connection grids with six DOF each for a total of 996 connections DOF. The total number of modes (connection plus normal modes) is 1270. OUTPUT DMIG MATRIX IN H3D FORMAT OUTPUT DMIG MATRIX: KA

NCOL =

1270

OUTPUT DMIG MATRIX: MA

NCOL =

1270

OUTPUT DMIG MATRIX: K4

NCOL =

1270

The .out file will show the modes for the fixed boundary condition. In general, it is desirable to have at least 30 modes for each component model. For small items like a control arm, an upper limit of around 5,000 Hz may be needed.

Craig-Chang Approach The following cards show the Craig-Chang approach. PARAM, SEPLOT, YES PARAM, AUTOSPC, YES MODEL = PLOTEL,,NORIGID TITLE = BODY-IN-WHITE CMS MODEL SUBTITLE = DMIGNAME=BIW $-----------------------------------------------------------------------------SUBCASE 1 CMSMETH = 1 BEGIN BULK $==01==><==02==><==03==><==04==><==05==><==06==><==07==><==08==><==09==><==10= => CMSMETH 1 GM 300.0 1008000 AMSES BNDFREE1 123456 1000000 THRU 1006016 INCLUDE '/MODELS/BODY/BODY.dat' ENDDATA For the Craig-Chang Approach, use AMSES to generate the super element. With AMSES you will get the appropriate static modes. Static modes are additional modes created by perturbing each connection DOF. The number of static modes is the number of BNDFREE DOF. Since Lanczos does not generate the static modes for the Craig-Chang approach, the AMSES eigensolver must be used. The only difference in the input data for creating Craig-Chang super elements from CraigBampton super elements is in the CMSMETH and the BNDFREE1 cards. The CMSMETH card now references the GM (General Method). This method is used for the Craig-Chang or the combination of both methods. The BNDFIX or ASET data should be changed to BNDFREE or CSET data. The BNDFREE1 card replaces the ASET1 card and has the same meaning for the inclusions of the interface points.

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For this method, verify that the components have six good rigid body modes (that is below 1.0E-02 Hz). This insures that the body model does not have any modeling problems. Having more than six rigid body modes or rigid body modes greater than 1.0E-2 indicates that the component has modeling problems that need to be identified and fixed. To create a combination of both fixed and free modes, just add the ASET1 list to the GM and include in the list the interface points that you want to be fixed. The superelements created by using the Craig-Chang method should not be used for subsequent static analysis.

Craig-Bampton & Craig-Chang Approach The following cards show an example of the combination approach for an automobile exhaust system. MODEL = PLOTEL,,NORIGID $ $--1---|---2---|---3---|---4---|---5---|---6---|---7---|---8---|--9---|-------| TITLE = EXHAUST H3D MODEL CREATION SUBTITLE = FIXED AND FREE GM APPROACH $ DMIGNAME=EXHAUST $-----------------------------------------------------------------------------SUBCASE 1 CMSMETH = 1 $-----------------------------------------------------------------------------BEGIN BULK $==01==><==02==><==03==><==04==><==05==><==06==><==07==><==08==><==09==><==10= => CMSMETH 1 GM 400.0 7308000 $ EXHAUST ATTACHMENTS TO POWERTRAIN ASET1 123456 7304501 7304502 $ EXHAUST HANGERS TO BODY BNDFREE1 123456 7304503 7304507 7304515 7304523 $ INCLUDE '/MODELS/EXHAUST/EXHAUST.dat' ENDDATA The ASET1 list includes the two attachments of the exhaust system which are rigidly connected to the powertrain. The BNDFREE1 list includes the exhaust hangers which are rubber mounted to the body. In this case there will not be any rigid body modes. Note that the Craig-Bampton approach can be used in either the CBN or GM process on the CMSMETH card. However, in the GM the underlying eigen space has additional rotation; which will make the residual run times with super elements longer if the Craig-Bampton approach is created with the GM method. GM with BNDFIX will be useful if you want to do more component level analysis in the residual run.

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In summary: 1. CSET and BNDFREE are equivalent. 2. ASET, BSET/BNDFIX and CSET/BNDFREE; all three are not allowed in the same deck. 3. If CSET/BNDFREE and ASET are present, the DOFs associated with ASET would be in BSET; except the DOF assigned to CSET/BNDFREE. 4. If BSET/BNDFIX and ASET are present, the DOFs associated with ASET would be in CSET; except the DOF assigned to BSET/BNDFIX.

Automobile Body – Cavity Approach Another special super element is an automobile body - cavity super element. Below is an example of how this super element is created. ANALYSIS PARAM, CHECKEL NO PARAM, SEPLOT, YES MODEL = PLOTEL,,NORIGID TITLE = FULLY TRIMMED BODY CMS MODEL WITH CAVITY MODEL SUBTITLE = DMIGNAME=TRIMBDY $-----------------------------------------------------------------------------SUBCASE 1 CMSMETH = 1 $-----------------------------------------------------------------------------BEGIN BULK $==01==><==02==><==03==><==04==><==05==><==06==><==07==><==08==><==09==><==10= => ACMODL CMSMETH 1 GM 300.0 1008000 AMSES 600.0 90000000 BNDFREE1 123456 1002000 THRU 1006100 BNDFREE1 123456 2005807 2005809 2005810 BNDFREE1 1 80000000 THRU 80000007 $==01==><==02==><==03==><==04==><==05==><==06==><==07==><==08==><==09==><==10= => INCLUDE '/TRIM_BODY/TRIM_BODY_CONNECTIONS.dat' INCLUDE '/TRIM_BODY/TRIM_BODY_FILES.dat' INCLUDE '/MODELS/CAVITY/CAVITY.dat' ENDDATA In this file, the BNDFREE1 connections contain the interface points for the BODY-IN-WHITE and the interface points for the instrument panel for the steering column connections. It also includes the acoustic response points in the cavity model. Note only the one degree of freedom is referenced for the acoustic response points. The CMSMETH card also contains a second line that references the upper frequency limit for the cavity modes and the numbering starting point for the cavity modes. The file also contains the ACMODL card that references the connections of the cavity to the trim body parts.

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The .out file shows the inclusion of the acoustic modes in the A matrix. OUTPUT DMIG MATRIX IN H3D FORMAT OUTPUT DMIG MATRIX: KA

NCOL =

1687

OUTPUT DMIG MATRIX: MA

NCOL =

1687

OUTPUT DMIG MATRIX: K4

NCOL =

1402

OUTPUT DMIG MATRIX: A

NCOL =

285

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Component Dynamic Analysis Super Element Generation The CDS super element should be used when it's anticipated that a large number of residual runs will be made on a very large model at the higher end of the frequency range of study. For example, this approach should be used when studying a variety of inputs on an automobile model in the frequency range of 400 to 800 Hz. For the residual analysis to run as fast as possible, all components, except very small ones, should be converted into CDS super elements. The major limitation of this approach is that it takes longer to generate the CDS super elements than with the other super element creation methods. Also, the analysis must be performed at the fixed set of frequencies specified when the CDS supe element is formed. The major benefit of the CDS super element is that the residual run will be much faster than with super elements created by other methods. For an example of the body CDS super element generation, see the input data for a body-inwhite below. The special data for this input are the case control data: CDSMETH = 1; the FREQ card which restricts the residual analyses to just those frequencies; the CDSMETH data (see the CDSMETH card definition); and the BNDFREE data which defines the exterior points on the component. CDS Body Super Element Example MODEL = NONE TITLE = BODY-IN-WHITE CDS MODEL $-----------------------------------------------------------------------------SUBCASE 1 FREQUENCY = 1 METHOD = 2 CDSMETH = 1 BEGIN BULK $==01==><==02==><==03==><==04==><==05==><==06==><==07==><==08==><==09==><==10= => FREQ1 1 1.0 0.2 94 FREQ1 1 20.0 0.5 159 FREQ1 1 100.0 1.0 99 FREQ1 1 200.0 2.0 200 EIGRA 2 1000.0 CDSMETH 1 CMSOUT 1008000 BNDFREE1 123456 1000000 THRU 1006001 INCLUDE '/MODELS/BODY/BODYNEW.dat' ENDDATA This run will generate not only the CDS super element but also the GM method super element because the CMSOUT keyword was present. The GM method super element is saved in the file: XXX.h3d; while the CDS super element is saved in the file: XXX_CDS.h3d. The interface points (exterior points) are where the component is attached to another super element directly or to the residual structure. These interface points must be independent in all six degrees of freedom. If they are the dependent point of an RBE3, they must be made independent by transferring the dependencies to one of the independent grids referenced by the RBE3 element using UM data on the RBE3 definition. In the Bulk Data section on the RBE3 element, the UM parameter shows how to redefine the dependency. The RBE3 can also be changed to an RBE2, but this could stiffen up the local area as a result.

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In order to be formed into a super element, a component FEA model has to be complete. All grids referenced in the super element must be in the component model file. This includes local coordinate systems grids. All properties and material referenced in the components must also be included in the component. The component model must be able to be run successfully by itself in a modal analysis run. The MODEL card includes all PLOTEL elements and the associated interior grids in the super element. An alternate method of specifying interior GRID points is to use the GRID SET ID on the MODEL data. The PLOTEL elements generally are used to define a rough outline of the component for viewing its mode shapes. The PLOTEL set can include the 3 and 4 node PLOTEL elements. Residual Run Using the CDS Super Elements The residual run on the full model must be run with the direct analysis approach. Also, the same or a subset of the frequencies specified in the CDS super element generation run must be used in the residual run. $ THIS FILE CONTAIN ALL THE H3D FILES FOR THE CDS SUPER ELEMENTS INCLUDE '/CDS_FILES.dat' $ TITLE = P/T FULL VEHICLE ANALYSIS SUBTITLE = FVM DIR H3D SPC = 1 MPC = 406 SUBCASE 1 LABEL = UNIT LOAD INPUT INTO CDS MODEL DLOAD = 110 FREQUENCY=1 SET 2 = 1006001,9106012 ACCELERATION (PUNCH,SORT2,PHASE) = 2 $ BEGIN BULK $ $==01==><==02==><==03==><==04==><==05==><==06==><==07==><==08==><==09==><==10= => FREQ1 1 1.0 0.2 94 FREQ1 1 20.0 0.5 159 FREQ1 1 100.0 1.0 99 FREQ1 1 200.0 2.0 200 EIGRL 1 800.0 $==01==><==02==><==03==><==04==><==05==><==06==><==07==><==08==><==09==><==10= => $ INCLUDE '/LOADS.dat' $ INCLUDE '/CONNECTIONS_BETWEEN_COMPONENTS.dat' $ INCLUDE '/NON_CDS_COMPONENTS.dat' ENDDATA

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Flexible Body Generation Component Mode Synthesis (CMS) is used to reduce a finite element model of an elastic body to the interface degrees of freedom and a set of normal modes for inclusion as a flexible body in a multi-body dynamics analysis. For the purpose of deriving the matrices, the displacement vector may be partitioned into displacements of inner (OSET) and outer (ASET, interface) degrees of freedom. (1) Here, the subscript ‘o’ denotes the inner degrees of freedom, and ‘a’ the interface degrees of freedom. The interface nodes that are used in the component mode synthesis process for the construction of mode shapes should be coincidental to the set of force-bearing nodes in the subsequent multi-body dynamics analysis. In a multi-body dynamics model, the flexible body interacts with other components of the model through joints, constraints, or force elements, which are connected or applied on the nodes of the flexible body. Except for body forces due to gravity or acceleration of the flexible body, all nodes that are subject to constraint or applied forces in the multi-body dynamics analysis are denoted as force-bearing nodes. The purpose of specifying the interface nodes for CMS is mainly to account for the static deformation due to constraints or applied forces acting on the interface nodes. A huge number of eigenmodes is required if these static modes are omitted. The flexible deformations due to constraint forces, compared to the deformation due to the body inertia forces, are often dominant in most constrained models; therefore, the inclusion of all forcebearing nodes as interface nodes is an essential step to get accurate results from subsequent flexible multi-body dynamics analysis. The static equilibrium is given as: (2) The eigenvalue problem for a normal modes analysis of the body using a diagonal mass matrix represents itself as: (3) The task of the component mode synthesis is to find a set of orthogonal modes represent the displacements u of the reduced structure such that:

that

Where, q is the matrix of modal participation factors or modal coordinates which are to be determined by the analysis. The Craig-Bampton and Craig-Chang methods of component mode synthesis are available.

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Craig-Bampton Method Normal Modes Analysis with constrained Interface degrees of freedom This method uses a system constrained in the interface degrees of freedom. Normal modes analysis of the system yields the diagonal matrix of eigenvalue and the matrix of eigenmodes . In this normal analysis, you can select the cut-off frequency or the number of modes to be solved. This determines the column dimension of . Static Analysis with alternately fixed Interface degrees of freedom In addition, a static analysis is performed with a unit displacement in each interface degree of freedom while all other interface degrees of freedom are fixed. The number of subcases in this static analysis is six times the number of interface nodes. Note that it is important to constrain each interface node with its neighboring nodes, if necessary, to ensure that it has non-zero stiffness along all six degrees of freedom directions. This process yields the displacement matrix and the interface forces . Reduced modal stiffness

and mass matrices

are now generated using:

Which yields:

And

It follows an othogonalization step that transforms the original shapes orthogonal modes .

X into a set of

Craig-Chang Method This method uses a system that is unconstrained (free-free) and therefore has six rigid body modes. Normal modes analysis of the system yields the diagonal matrix of eigenvalue and the matrix of eigenmodes . In this normal analysis, you can select the cut-off frequency or the number of modes to be solved. This determines the column dimension of . The eigenmodes associated with the rigid body modes will be normalized with respect to the mass matrix such that:

In addition, a static analysis is performed using an equilibrated load matrix

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The equilibrated load matrix is applied in an inertia relief static analysis without any SPC constraints, but with proper support to remove the six rigid modes. The vector is a collection of the attachment force vectors which otherwise have all zero entries, except a unit force along each degree of freedom of the interface nodes. The resulting modes are called the inertia relieve attachment modes. Reduced modal stiffness

and mass matrices

are now generated using:

Which yields:

And

It follows an othogonalization step that transforms the original shapes X into a set of orthogonal modes .

Eigenvector Normalization First, a new eigenvalue problem using the reduced matrices above is solved.

The resulting diagonal matrix of eigenvalues D and the normal modes N are used to transform the set of original shapes into the set of orthogonal modes .

It can be shown that the resulting modes are orthogonal with respect to the system stiffness matrix K and mass matrix M.

If the orthogonal modes are normalized with respect to the mass matrix M, the reduced matrices for the subsequent analysis appear as:

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Distributed Loads In MotionSolve, any arbitrary displacement field of the flexible body is approximated by:

x

q

Where, q is the m by 1 modal participation vector. Suppose pi is the i-th distributed load (pi is an n by 1 vector) defined in the FE model, xi is the corresponding displacement field, and x is any arbitrary displacement field of the flexible body. The virtual work due to pi is:

Taking the variation on both sides of the arbitrary displacement field equation and plugging the results into the above equation, results in:

Define modal distributed load as

i, i=1,…,l , as:

Each modal distributed load is an m by 1 vector. The above modal distributed loads, along with their load ID in the FE model, should be precomputed in OS-Flexprep and written as new modal distributed loads blocks in the h3d file. In addition, the corresponding displacement fields xi, i=1,…, l, should enter the mode shapes X in the CMS process.

Input and Output Component mode synthesis is activated by the presence of the I/O Option CMSMETH. The I/ O Option references a CMSMETH bulk data entry which defines the method (only CC and CB methods apply to flexible body generation), frequency range or number of modes to be calculated. Only one subcase is allowed per model. The interface degrees of freedom are defined using ASET or ASET1 bulk data statements. An MPC reference is allowed. While subcases are irrelevant for this run mode, if the model has multiple subcases, the MPC reference must match for all subcases. Subcase information entries other than MPC are ignored. In general, offsets should not be used in flexible body generation. However, PARAM, CMSOFST may be used to allow small offsets for shell elements. The orthogonal modes and the corresponding eigenvalues D are exported to a flexh3d file by default. The modal stresses and strains can be output optionally using the STRESS and STRAIN output statements. Sets can be applied to reduce the amount of data. The export of the model to this file can be controlled using the MODEL output statement.

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Recovering MBD Analysis results in OptiStruct After MotionSolve is run, it is possible to recover displacements, velocities, accelerations, stresses, and strains for the flexbody in OptiStruct in order to create .op2 and .h3d results files for fatigue analysis. The procedure is explained below. After running MotionSolve, a residual run can be made to recover displacement, velocity, acceleration, stress, and strain results for interior grids and elements in the Flex Body based on the modal participation results from MotionSolve. After running MotionSolve, a resulting .mrf file is created that contains MotionSolve results including the modal participation factors at each time step for the Flex Body for transient analysis. In the residual run, the Flex Body _recov.h3d file and the .mrf results file are specified using the ASSIGN data: ASSIGN,H3DMBD,30101,’pfbody_1_recov.h3d' ASSIGN,H3DMBD,30102,’pfbody_2_recov.h3d' ASSIGN,MBDINP,10,’pfbody.mrf' Where the 10 in the ASSIGN,MBDINP data references the SUBCASE for which the MotionSolve results will be used. In SUBCASE 10, instead of performing a transient analysis, OptiStruct will just use the results from MotionSolve. The 30101 and 30102 in the ASSIGN,H3DMBD data refer to the Flex Body ID’s in the .mrf file. For transient analysis, the number of time steps in the transient analysis run must match the number of time steps used in the MotionSolve analysis. While the transient analysis data is ignored, there must still be some dummy loading data (TLOAD, DAREA, and TABLED data). A sample of input data for a transient analysis run is shown below: OUTPUT OP2 OUTPUT H3D ASSIGN,H3DMBD,30103,MBD_pfbody_BODY_2_PROP_6_recov.h3d ASSIGN,H3DMBD,30102,MBD_pfbody_BODY_1_PROP_9_recov.h3d ASSIGN,H3DMBD,30104,MBD_pfbody_BODY_3_PROP_10_recov.h3d ASSIGN,MBDINP,1,MBD_pfbody_mbd.mrf SUBCASE 1 TSTEP(TIME) = 4 DLOAD = 3 DISPLACEMENT = ALL STRESS = ALL SPC = 1 $ BEGIN BULK GRID 9999999 $ Dummy GRID since at least one GRID is required $------+-------+-------+-------+-------+-------+-------+-------+------TSTEP 4 300 .0003333 1 TLOAD1 3 3 DISP 2 TABLED1 2 LINEAR LINEAR + 0.0 1.0 10.0 1.0ENDT $ $ Dummy load on the dummy grid SPC 1 9999999 1 SPCD 3 9999999 1 -200.0 ENDDATA A dummy grid and a dummy load have been added for the OptiStruct analysis run.

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Poroelastic Materials (Biot theory) Introduction Poroelastic materials can be used to model coupled fluid-structure systems where the fluid exists within the interstitial spaces of a porous solid. The mechanical response of a porous material varies, depending on the amount of fluid present within the pores, the type of fluid, the pressure of the interstitial fluid, and the structure of the porous material. In OptiStruct, a poroelastic material based on the Biot theory of poroelasticity can be modeled for use in relevant applications. Motivation A porous material containing fluid within interstitial spaces cannot be accurately modeled without accounting for the influence of the fluid on the mechanical response of the structure. There are various physical applications for poroelastic material models; for example, trimmings of an automobile are porous and the cabin cavity is filled with air. The fluid (air) enters the interstitial spaces within these trimmings and the dynamic behavior of the system is altered. This difference in dynamic behavior should be accurately accounted for in Noise Vibration and Harshness (NVH) studies of the automobile.

Figure 1: Difference in mechanical response between porous and non-porous materials.

For a possible application, the Biot poroelastic material implementation in OptiStruct can be used to model automobile trim components in frequency response analyses to generate a more accurate solution. Trim materials, carpets, foam pads, and other porous materials can be modeled.

Implementation Poroelastic materials in OptiStruct are implemented using the Biot poroelastic theory. They can be modeled using CTETRA, CHEXA, or CPENTA solid elements. The Biot u-p (displacement-pressure) formulation is used wherein each grid consists of three displacement degrees of freedom (DOF) and one pressure component. The required material properties are listed in detail on the MATPE1 entry in the Reference Guide. The MATPE1 entry can be selected on the PSOLID property entry with FCTN=PORO.

Supported Solution Sequences The Biot poroelastic material is frequency-dependent. The frequency-dependent element matrices are calculated at each frequency for each element. Direct and Modal Frequency

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Response Analyses are supported. The frequency-dependent matrices are reduced to modal spaces for modal frequency response analysis. Panel participation calculation for the Biot material is available similar to that of other panels. The fluid-structural grid participation is also available for detailed interpretation of panel contributions. Note 1. Coupling between the acoustic cavity (FLUID) and trim component (BIOT): (a) A GRID to GRID match or Multi-point constraints (MPCs) are required to connect the acoustic cavity to fluid dof of the BIOT material. (b)The ACMODL entry will be used to couple the fluid dof and the structural dof of the BIOT material. (c) If a GRID to GRID match is used to couple the acoustic cavity to the BIOT material, the grids on the fluid that are shared with the BIOT material should not have their CD fields set to -1. However, if Multi-point constraints (MPCs) are used, then the CD fields should be set equal to -1. 2. Coupling between the trim component (BIOT) and the body structure: (a) Coincident nodes (GRID to GRID match) can be used to achieve displacement continuity. However, if the nodes between the trim component and the body structure are not shared, TIE/CONTACT entries with the FREEZE option, rigids, MPC’s, or other structural elements can be used to ensure continuity. Limitations The pressure output for the trim component (poroelastic Biot material) is not currently supported. However, the pressure of the closest acoustic domain is available.

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Elements and Materials The following features can be found in this section

Elastic, Damper, and Mass Elements Zero-dimensional Elements Elements in this group only connect to grid points having a single degree of freedom at each end. Elements also included in this group are those that connect to scalar points at one end and ground at the other, like the following: CELAS1, CELAS2, CELAS3, and CELAS4 that are used to model elastic springs. The properties for CELAS1 and CELAS3 are defined on PELAS. CELAS2 and CELAS4 define spring properties. CDAMP1, CDAMP2, CDAMP3, and CDAMP4 that are used to model scalar dampers. The properties for CDAMP1 and CDAMP3 are defined on PDAMP. CDAMP2 and CDAMP4 define scalar damper properties. CMASS1, CMASS2, CMASS3, and CMASS4 that are used to model point masses. The properties for CMASS1 and CMASS3 are defined on PMASS. CMASS2 and CMASS4 define the mass. CONM1 and CONM2, which are concentrated mass elements. CONM1 defines a 6x6 mass matrix at a grid point. CONM2 defines mass and inertia properties at a grid point. CVISC is used to model viscous dampers. The properties for CVISC are defined on PVISC.

One-dimensional Elements Elements in this group are represented by a line connecting grid points at each end. The following actions involving forces (and displacements) at each end are possible: Forces and displacements along the axis of the element. Transverse shear forces (and displacements) in the two lateral directions. Bending moments (and rotations) in two perpendicular, bending planes. Torsional moments (and resulting rotations). Twisting of the cross-section (or cross-sectional warping). The elements in this category are: CBEAM - a general beam element that supports all types of action listed above. CBAR - a simple, prismatic beam element that supports all of the above types of actions except cross-sectional warping. CBUSH - a general spring-damper element that supports forces, moments, and displacements along the axis of the element.

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CBUSH1D - a rod-type spring-damper element. CGAP - a gap element that supports axial and friction forces. CGAPG - a gap element that supports axial and friction forces. It does not have to be placed between grid points. It can also connect surface patches. CROD - a simple, axial bar element that supports only axial forces and torsional moments. CWELD - a simple, axial bar element that supports forces, moments, and torsional moments. It does not have to be placed between grid points. It can also connect surface patches. The properties for these elements are defined on PBEAM, PBAR, PBUSH, PBUSH1D, PGAP, PROD, and PWELD, respectively. CONROD - a simple, axial bar element that supports only axial forces and torsional moments. This element does not reference a property definition; the property information is provided with the element definition.

Two-dimensional Elements Two-dimensional elements are used to model thin-shell behavior, which incorporates inplane or membrane actions, plane strain, and bending action (including transverse shear characteristics and membrane-bending coupling actions). Reissner-Mindlin shell theory is used to model bending. A plane strain option is available for pure 2D applications. The element shapes may be triangular (CTRIA3) or quadrilateral (CQUAD4). Second order triangular (CTRIA6) and quadrilateral (CQUAD8) shell elements are also available. The shell properties, including the behavior, are defined on the PSHELL entry. The first order shell element formulation for CQUAD4 and CTRIA3 has the special characteristic of using six degrees of freedom per grid. Hence, there is stiffness associated to each degree of freedom. In some finite element codes, shell elements do not have a drilling stiffness normal to the mid-plane, which may cause singular stiffness matrix. Then, a user-defined artificial stiffness value is assigned to this degree of freedom to avoid the singularity. The second order shell elements (CTRIA6 and CQUAD8) have five degrees of freedom per grid. Rotational degrees of freedom without stiffness are removed through SPC. Another form of two-dimensional elements may also be used to model thin buckled plates. These elements support shear stress in their interior and extensional forces between their adjacent grid points. These elements are used in situations where the bending stiffness and axial membrane stiffness of a plate is negligible. The elements are quadrilateral and are defined as CSHEAR. Their properties are defined on the PSHEAR entry.

Three-dimensional Elements The three-dimensional solid elements are used to model thick plates, solid structures. In general, structures in which the lateral dimensions are of the same order of magnitude as the longitudinal dimensions can support the use of three-dimensional solid elements in modeling. The elements in this category are the CHEXA, CPENTA, CPYRA, and CTETRA elements. The property definition associated with these elements is called PSOLID.

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Offset for One-dimensional and Two-dimensional Elements Some one-dimensional and two-dimensional elements can use offset to “shift” the element stiffness relative to the location determined by the element’s nodes. For example, shell elements can be offset from the plane defined by element nodes by means of ZOFFS. In this case, all other information, such as material matrices or fiber locations for the calculation of stresses, are given relative to the offset reference plane. Similarly, the results, such as shell element forces, are output on the offset reference plane. Offset is applied to all element matrices (stiffness, mass, and geometric stiffness), and to respective element loads (such as gravity). Hence, in principle, offset can be used in all types of analysis and optimization. However, caution is advised when interpreting the results, especially in linear buckling analysis. Without offset, a typical simple structure will bifurcate and loose stability “instantly” at the critical load. With offset, though, the loss of stability is gradual and asymptotically reaches a limit load, as shown below in figure (b):

In practice then, the structure with offset can reach excessive deformation before the limit load is reached. (Note that more complex structures, such as frames or structures experiencing bending moments, buckle via limit load even in absence of ZOFFS on the element card). Furthermore, in a fully nonlinear approach, additional instability points may be present on the limit load path.

Elements for Geometric Nonlinear Analysis Special element formulations are available for geometric nonlinear analysis. As a general rule, property definitions that are only applicable in geometric nonlinear analysis are defined on extensions to the original property. The extensions are grouped with the base entry by sharing the same PID. In the case of a subcase that is not a geometric nonlinear analysis, these extensions are ignored. The following extensions are available: PSHELLX, PSOLIDX, PBARX, and PBEAMX. Property defaults can be set for shells (XSHLPRM) and solids (XSOLPRM) that may replace the use of property extensions. All other properties are used as they are.

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Example: PSHELL, 3, 7, 1.0, 7, , 7 PSHELLX, 3, 24, , , 5

Non-structural Mass Non-structural mass may be specified in two different ways. 1. Many property definitions (PSHELL, PCOMP, PBAR, PBARL, PBEAM, PBEAML, PROD, CONROD, PSHEAR, and PTUBE), have an NSM data field that allows a value of nonstructural mass per unit area or non-structural mass per unit length to be defined. When non-structural mass is defined in this way, it is considered in all analyses. 2. Non-structural mass may be defined via a number of non-structural mass bulk data entries (NSM, NSM1, NSML, NSML1, and NSMADD) for a list of elements or properties. In the case of a list of properties, non-structural mass is applied to the elements referencing the properties in the list. These non-structural mass definitions must be selected for use in an analysis through the NSM subcase information entry. Only one NSM subcase information entry can exist in a model and it must occur before the first SUBCASE statement. The bulk data entry NSM and its alternate form NSM1 allow you to define a value of non-structural mass per unit area or non-structural mass per unit length to be applied to a selected list of elements. The bulk data entry NSML and its alternate form NSML1 allow you to allocate and smear a lumped non-structural mass value to be evenly distributed over a list of elements. The non-structural mass value per unit area or per unit length to be applied to the elements is computed as:

NSM per unit area

VALUE n

area of element i i 1

NSM per unit length

VALUE n

length of element i i 1

Where, n is the number of elements in the set and VALUE is the value of the lumped mass. The NSML and NSML1 entries cannot mix "area" and "line" elements on the same entry. The "area" elements are: CQUAD4, CQUAD8, CTRIA3, CTRIA6, and CSHEAR; "line" elements are: CBAR, CBEAM, CTUBE, CROD, and CONROD. The bulk data entry NSMADD allows you to form combinations of NSM, NSM1, NSML, and NSML1. An element can have more than one non-structural mass value specified for it. The actual non-structural mass value will be the sum of all of the individual non-structural mass values.

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Virtual Fluid Mass Virtual Fluid Mass mimics the mass effect of an incompressible inviscid fluid in contact with a structure. It does not represent the actual mass of the fluid. There is no mesh needed for the fluid domain. The Virtual Fluid Mass represents the full coupling between acceleration and pressure at the fluid-structure interface. A dense mass matrix is generated among damp grids at the fluid-structure interface. This simulation is applicable to automobile containers, such as a fuel tank, which hold non-pressurized fluids. Assumptions 1. The fluid is inviscid and incompressible. The fluid flow is a potential flow. 2. Because the fluid is nearly incompressible, the structural modes are below the compressible fluid modes. 3. There is no gravity effect or sloshing effect. 4. There is no acoustic effect involved. The modes from the structural side do not couple with the modes of the nearly incompressible fluid modes.

MFLUID Interface If a fish can swim to every point inside fluid domain without leaving the fluid, the fluid domain can be represented by a single MFLUID card in the bulk data section. Each MFLUID card in the bulk section can only be referred to by a single MFLUID card in the control section. Multiple bulk data MFLUID cards can be referred by a single MFLUID card in the control section. Symmetry and anti-symmetry options can be applied to a MFLUID card. If PARAM,VMOPT,1 is used (default), the virtual mass is included in the regular mass matrix and it can be applied to both direct and modal dynamic subcases. Because the virtual mass matrix is dense for the damp grids, the computational time increases significantly. However, you have the option to use PARAM,VMOPT,2. Although, PARAM,VMOPT,2 can only be applied to modal dynamic subcases. In this case the virtual mass is added after the eigen solution, and the computational time is not increased significantly. When PARAM,VMOPT,2 is used, the dry modes are computed without adding virtual mass in the computation. Then the modes are modified based on the virtual mass matrix.

Theory The elemental pressure and acceleration are calculated with respect to the source potential of the element. The pressure is calculated based on displacement potential as:

If the source potential of element j is

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, then the pressure can be represented as:


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An additional area integration is done to convert pressure into force. Similarly, the acceleration vector

can be represented as:

Using the force and acceleration, the effective mass matrix can be calculated.

Arbitrary Beam Section Definition In addition to using predefined beam cross-sections selected by the TYPE field on the PBARL and PBEAML bulk data entries, defining arbitrary beam cross-sections. This is referred to here as section definitions. To define an Arbitrary Beam Section, HYPRBEAM should be entered into the GROUP field on the PBARL and PBEAML bulk data entries. Also, the ND field should specify the number of dimensions input during the definition of the arbitrary beam section in the DIMi fields of the PBARL and PBEAML bulk data entries. Section definitions are contained within the bulk data section of the input file. A section definition begins with the statement BEGIN and ends with the statement END. Section definitions are referenced from a PBARL or PBEAML definition through the NAME field. The NAME entered on the PBARL or PBEAML definition must match the NAME following the BEGIN, statement. The section is defined by a 2D finite element mesh. The finite element mesh is composed of nodes (denoted by GRIDS entries), which are connected by 2-node, 3-node, 4-node, 6node or 8-node elements (denoted by CSEC2, CSEC3, CSEC4, CSEC6, or CSEC8 entries, respectively). These elements reference PSEC entries; these provide a material reference for all elements and thickness information for the 2-noded CSEC2 elements. The following is an example of a simple thin-walled section definition named SQUARE: $ BEGIN,HYPRBEAM,SQUARE $ GRIDS,1,0.0,0.0 GRIDS,2,1.0,0.0 GRIDS,3,1.0,1.0 GRIDS,4,0.0,1.0 $ CSEC2,10,100,1,2 CSEC2,20,100,2,3 CSEC2,30,100,3,4 CSEC2,40,100,4,1 $ PSEC,100,1000,0.1 $ END,HYPRBEAM $

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The following is an example of a solid section definition named CUTOUT: $ BEGIN,HYPRBEAM,CUTOUT $ GRIDS,1,0.0,0.0 GRIDS,2,0.05,0.0 ... ... GRIDS,895,0.35,1.18 GRIDS,896,0.38,1.19 $ CSEC3,806,100,887,873,872 CSEC3,809,100,868,820,885 CSEC3,812,100,813,803,817 $ CSEC4,1,100,147,148,149,157 CSEC4,2,100,157,149,150,158 ... ... CSEC4,813,100,648,712,895,896 CSEC4,814,100,647,646,896,895 $ PSEC,100,1000 $ END,HYPRBEAM $

Rigid Elements and Multi-Point Constraints Rigid elements and multi-point constraints are used to constrain one or more degrees of freedom to be equal to linear combinations of the values of other degrees of freedom. Rigid elements are equations generated internally. You provide the connection data only. Rigid elements function as rigid bodies; therefore they are also known as rigid bodies or constraint elements. Internally, they are treated the same way as multi-point constraints. The RROD element can be used to model a pin-ended rod which is rigid in extension. One equation of constraint will be generated for this element. The RBAR element can be used to model a rigid bar with six degrees of freedom at each end. From one to six (depending on your input) equations of constraint will be generated for this element. The RBE1 and RBE2 elements are rigid bodies connected to an arbitrary number of grid points. The number of equations of constraint generated is equal to or greater than one, depending on the dependent degrees of freedom selected by you. For the RBE1 element, the independent degrees of freedom are six components of motion that must be jointly capable of representing any general rigid body motion of the element; whereas for the RBE2 element, the independent degrees of freedom are the six components of motion at a single grid point. The RBE3 element provides for specification from one to six equations of constraint developed from the relation that the motion at a "reference grid point" is the least square

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weighted average of the motion at other grid points. This element is generally used to "beam" loads and masses from a reference point to a set of grid points. Multi-point constraints are equations in which you explicitly provide the coefficients of the equations. Each multi-point constraint is described by a single equation that specifies a linear relationship for two or more degrees of freedom. Multiple sets of multi-point constraints can be provided in the bulk data section. In the subcase information section, the multipoint constraints are assigned to the specific load case using the MPC statement. The bulk data entry MPC is the statement for defining multi-point constraints. The first coordinate mentioned on the card is taken as the dependent degree of freedom (that is, the degree of freedom that is removed from the equations of motion). Dependent degrees of freedom may appear as independent terms in other equations of the set; however, they may appear as dependent terms in only a single equation. Some uses of multi-point constraints are: To enforce zero motion in directions other than those corresponding to components of the global coordinate system. In this case, the multi-point constraint will involve only the degrees of freedom at a single grid point. The constraint equation relates the displacement in the direction of zero motion to the displacement components in the global system at the grid point. To describe rigid elements and mechanisms such as levers, pulleys, and gear trains. In this application, the degrees of freedom associated with the rigid element that are in excess of those needed to describe rigid body motion are eliminated with multipoint constraint equations. Treatment of very stiff members as being rigid elements eliminates the ill-conditioning associated with their treatment as ordinary elastic elements. To be used with scalar elements to generate non-standard structural elements and other special effects. When using rigid elements or multi-point constraints, you must make sure that the following requirements are satisfied: A dependent degree of freedom cannot be in the SPC. A dependent degree of freedom in any rigid element or multi-point constraint cannot be defined as a dependent degree of freedom in any other rigid element or multipoint constraint.

Materials The different elastic material types provided by OptiStruct are: isotropic, orthotropic, and an-isotropic materials. The material property definition cards are used to define the properties for each of the materials used in a structural model. The MAT1 bulk data entry is used to define the properties for isotropic elastic materials. It can be referenced by any of the structural elements, and can also be referenced by any property card. The MAT2 entry is used to define the properties for an-isotropic materials. It applies only to triangular or quadrilateral membrane and bending elements, and can only be referenced by PSHELL, PCOMP, and PCOMPG property cards. This material type specifies the relationship between the in-plane stresses and strains. The angle between the material coordinate system and the element coordinate system is specified on the connection cards.

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The MAT4 entry is used to define the properties for isotropic elastic materials. It can be referenced by any of the structural elements, and can also be referenced by any property card. The MAT5 entry is used to define the properties for an-isotropic elastic materials. It can be referenced by any of the structural elements, and can also be referenced by any property card. The MAT8 card is used to define the properties for planar orthotropic elastic materials in two dimensions. Individual plys of a layered composite lay-up typically possess such orthotropic properties. Since layered composite laminates are modeled using shell elements, MAT8 property data can only be referenced by PSHELL, PCOMP, and PCOMPG property cards. The MAT9 bulk data entry can be used to define the properties for an-isotropic elastic materials for three dimensional solid elements. The general an-isotropic stress-strain relationship linking the six independent stress components of the stress tensor at a point and the six independent strain components of the tensor at the point contain 21 independent constants in the elasticity matrix. These values are supplied using the MAT9 bulk data card. The MAT9 bulk data card is used with the CHEXA, CPENTA, CPYRA, and CTETRA solid elements, and can only be referenced on the PSOLID property card. The optional coordinate system in which MAT9 data are specified is supplied via the PSOLID bulk data entry. The MAT10 bulk data entry is used to define material properties for fluid elements in coupled fluid-structural (acoustic) analysis. It may only be referenced on PSOLID entries with FCTN=’PFLUID’. Temperature dependent material properties are defined using MATT1, MATT2, MATT8, and MATT9. All four have the same characteristics as described above. The temperature dependency of each property is defined through TABLEM1, TABLEM2, TABLEM3, or TABLEM4 table entries. Composite laminates are defined using the PCOMP and PCOMPG properties. They are not material types; each ply in the laminate lay-up can reference a different material. Nonlinear material properties are defined using MATS1. The nonlinear material characteristics may need the table input TABLES1. MATS1 is defined as an extension to a MAT1 with the same MID. MATS1 is applicable to all nonlinear solutions. For geometric nonlinear subcases, more nonlinear material laws are available. As a general rule, material definitions that are only applicable in geometric nonlinear analysis are defined on extensions to a MAT1 material that defines the basic elastic properties. The extensions are grouped with the base entry by sharing the same MID. The table below lists the MATXyz extensions available. If a law requires material curves, TABLES1 entries are used. Example MAT1, 102, 60.4, , 0.33, 2.70e-6 MATX02, 102, 0.09026, 0.22313, 0.3746, 100.0, 0.175

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MATXy

Description

MATX0

Void material

MATX02

Johnson-Cooke elastic-plastic material

MATX13

Rigid material

MATX21

Rock-Concrete material

MATX25

Composite shell material, TSAI-WU and CRASURV formulations

MATX27

Brittle elastic-plastic material

MATX28

Honeycomb material

MATX33

Visco-elastic foam material

MATX36

Piece-wise linear elastic-plastic material

MATX42

Ogden-Mooney and Rivlin material

MATX43

Hill orthotropic material

MATX44

Cowper-Symonds elastic-plastic material

MATX60

Piece-wise nonlinear elastic-plastic material

MATX62

Hyper-visco-elastic material

MATX65

Tabulated strain-rate dependent elastic-plastic material

MATX68

Honeycomb material

MATX70

Tabulated visco-elastic foam material

MATX82

Ogden material

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Composite Laminates Overview Plates and shells can be made of layered composites in which several layers of different materials (plies) are bonded together to form a cohesive structure. Typically, the plies are made of unidirectional fibers or of woven fabrics and are joined together by a bonding medium (matrix). In OptiStruct composite shells, the plies are assumed to be laid in layers parallel to the middle plane of the shell. Each layer may have a different thickness and different orientation of fiber directions.

Four-layer composite with ply angle shown.

Classical lamination theory is used to calculate effective stiffness and mass density of the composite shell. This is done automatically within the code using the properties of individual plies. The homogenized shell properties are then used in the analysis. After the analysis, the stresses and strains in the individual layers and between the layers can be calculated from the overall shell stresses and strains. These results may then be used to assess the failure indices of individual plies and of the bonding matrix.

Analysis of Composites Analysis of composite shells is very similar to the solution of standard shell elements. The primary difference is the use of the PCOMP or PCOMPG property card, instead of PSHELL, to specify shell element properties. From the ply information specified on the PCOMP entry, OptiStruct automatically calculates the effective properties of the shell element. After the analysis, the available results include shell-type stresses as well as stresses, strains, and failure indices for individual plies and their bonding. These results are controlled by the results flags on the PCOMP or PCOMPG entry and the usual I/O control cards. PCOMP and PCOMPG define the composite lay-up in two different ways. PCOMP defines the structure and properties of a composite lay-up which is then assigned to an element. The plies are only defined for that particular property and there is no relationship of plies that reach across several properties.

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PCOMPG defines the structure and properties of a composite lay-up allowing for global ply identification which is then assigned to an element. The plies of different PCOMPG definitions can have a relationship because of the use of global ply IDs. Some remarks are in place regarding the specifics of composite analysis: 1. The most typical material type used for composite plies is MAT8, which is planar orthotropic material. The use of isotropic MAT1 or general anisotropic MAT2 for ply properties is also supported. 2. While it is possible to specify ply angles relative to the element coordinate system, the results become strongly dependent upon the node numbering in individual elements. Thus, it is advisable to prescribe a material coordinate system for composite elements and specify ply angles relative to this system. 3. Depending on the specific lay-up structure, the composite may be offset from the reference plane of the shell element, i.e. have more material below than above the reference plane (or vice versa). 4. Stress results for composites include both shell-type stresses and individual ply stresses. Importantly, shell-type stresses are calculated using homogenized properties and thus only represent the overall stress-state in the shell. To assess the actual stress-state in the composite, individual ply results need to be examined. Interpretation of Results for Composites A number of composite-specific results are calculated for composite shell elements. Due to the specialized nature of these results, some explanation is required regarding their meaning. Ply Stresses and Strains Classical lamination theory assumes two-dimensional stress-state in individual plies (so-called membrane state). The values of stresses and strains are calculated at the mid-plane of each ply, i.e. halfway between its upper and lower surface. For sufficiently thin plies, these values can be interpreted as representing uniform stress in the ply. Ply stresses and strains are calculated in coordinate systems aligned with ply material angles as specified on the PCOMP card. In particular, correspond to the primary ply direction, is orthogonal to it, and represents in-plane shear stress. Inter-laminar Stress Inter-laminar bonding matrix usually has different material properties and stressstate than the individual plies. The primary stress that is of importance here is interlaminar shear with two components: Failure Indices To facilitate prediction of potential failure of the laminate, failure indices are calculated for plies and bonding material. While there are several theories available for such calculations, their common feature is that failure indices are scaled relative to allowable stresses or strains, so that: - the value of a failure index lower than 1.0 indicates that the stress/strain is within the allowable limits (as specified on the material data card), and - a failure index above 1.0 indicates that the allowable stress/strain has been

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exceeded. - according to the formula, some failure criteria (for example, Tsai-Wu and Hoffman) would produce the negative ply failure, depending on the problem. Here, a brief summary of failure theories available is provided. Hill's Theory of Ply Failure According to Hill's theory, the ply failure index is calculated as:

Where, X is the allowable stress in the ply material direction (1), Y is the allowable stress in the ply material direction (2), and S is the allowable in-plane shear stress. It should be noted that Hill’s theory does not differentiate between tensile and compressive stresses and it is strongly recommended to use the same values for both allowable stresses, i.e. Xt = Xc and Yt = Yc. If this suggestion is not adopted, warning messages will be output and the following rules will be applied: If otherwise, X = Xc, and similarly for Y and

> 0, X = Xt; . For the interaction term / X2, if

>

0, X = Xt; otherwise, X = Xc. Hoffman's Theory of Ply Failure In Hoffman's theory, the ply failure index is calculated as:

Tsai-Wu Theory of Ply Failure In Tsai-Wu theory, the ply failure index is calculated as:

Where, F12 is a factor to be determined experimentally. Maximum Strain Theory of Ply Failure In maximum strain theory, the ply failure index is calculated as the maximum ratio of ply strains to allowable strains:

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Where, is the allowable strain in the ply material direction (1), is the allowable strain in the ply material direction (2), and is the allowable in-plane engineering shear strain. If you provide different values of and for tension and compression, the appropriate values are used depending on the signs of , respectively. Note that if you prescribe allowable stresses rather than strains on the material data card, then the allowable strains are calculated via simple division by the relevant material module. Bonding Material Failure The primary failure mode of the bonding material is due to inter-laminar shear. The corresponding failure index is calculated as:

Where, SB is the allowable shear in the bonding material. Final Failure Index for Composite Element After calculation of failure indices for individual plies, the potential failure index for the composite shell element is obtained. This is based on the premise that failure of a single layer qualifies as failure of the composite. Thus, the failure index for composite element is calculated as the maximum of all computed ply and bonding failure indices (note that only plies with requested stress output are taken into account here).

Comparison of laminate modeled with PCOMP and PCOMPG

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Loads and Boundary Conditions The following boundary conditions are outlined here.

Static Loads and Boundary Conditions Static loads are applied at grid points in a variety of ways, including: Loads applied directly to grid points Pressure on surfaces Gravity loads Centrifugal forces due to steady rotation Equivalent loads resulting from: thermal expansion -

enforced deformations of structural elements

-

enforced displacements of grid points

Any number of load sets can be defined in the bulk data section of the input file. However, only those sets selected in the subcase information section (as described in the Linear Static Analysis, Inertia Relief, and Nonlinear Quasi-Static Analysis sections) will be used in the problem solution. The manner in which each type of load is selected is specified on the associated bulk data statement description. The FORCE statement is used to define a static load applied to a grid point in terms of components defined by a local coordinate system. The orientation of the load components depends on the type of local coordinate system used to define the load. The FORCE1 statement is used if the direction is determined by a vector connecting two grid points. The MOMENT and MOMENT1 statements are used to define the application of a concentrated moment at a grid point. Pressure loads on triangular and quadrilateral elements are defined with a PLOAD2 card. The positive direction of the loading is determined by the order of the grid points on the element connection card (using the right-hand rule). The magnitude and direction of the load is automatically computed from the value of the pressure and the coordinates of the connected grid points. The load is applied to the connected grid points. PLOAD pressure loads are used in a similar fashion to define the loading of any three or four grid points, regardless of whether or not they are connected with two-dimensional elements. Pressure loads on the HEXA, PENTA and TETRA solid elements are defined with the PLOAD4 card. The pressure is defined positively outward from the element. The magnitude and direction of the equivalent grid point forces are automatically computed using the isoparametric shape functions of the element to which the load has been applied. Pressure loads on the QUAD4 and TRIA3 elements can also be applied using the PLOAD4 card.

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The PLOAD1 card is used to describe concentrated, uniformly distributed or linearly distributed loads on the CBAR or CBEAM elements. The GRAV bulk data entry is used to specify a gravity load by providing the components of the gravity vector in any defined coordinate system. The gravity load is obtained from the gravity vector and the mass matrix of the structural model. Because the gravitational acceleration is not calculated at scalar points, you are required to introduce gravity loads at scalar points directly. The RFORCE statement is used to define a static loading condition due to a centrifugal force field. A centrifugal force load is specified by the designation of a grid point that lies on the axis of rotation and by the components of rotational velocity and acceleration in any defined coordinate system. In the calculation of the centrifugal force, the mass matrix pertains to a set of distinct rigid bodies connected to grid points. Deviations from this viewpoint, such as the use of scalar points or the use of mass coupling between grid points, can result in errors. Temperature loads can only be defined at grid points. The temperatures of the connected grid points are given on the TEMP and TEMPD bulk data entries. The thermal expansion coefficients are defined on the material definition cards. The mere presence of a thermal field does not imply the application of a thermal load. A thermal load will not be applied unless you make a specific request in the subcase information section. The LOAD card in the bulk data section defines a static loading condition that is a linear combination of load sets consisting of loads applied directly to grid points, pressure loads, gravity loads, and centrifugal forces. This card must be used if gravity loads are to be used in combination with loads applied directly to grid points, pressure loads, or centrifugal forces. The application of the combined loading condition is requested in the subcase information section by selecting the set number of the LOAD combination. It should be noted that the equivalent loads (thermal and enforced displacement) must have unique set identification numbers and be selected separately in the subcase information section. For any particular solution, the total load will be the sum of the applied loads (grid point loading, pressure loading, gravity loading, and centrifugal forces) and the equivalent loads. Zero enforced displacements may be specified on SPC or SPC1 cards. Zero displacements result in non-zero forces on the grid point constrained (SPC forces). The SPCADD statement allows the combination of different SPC sets. For inertia relief, the reaction degrees of freedom for the computation of the acceleration load are defined through SUPORT or SUPORT1 statements. Up to six degrees of freedom can be defined per subcase. Non-zero enforced displacements may be specified on SPC or SPCD cards. The SPC card specifies both the component to be constrained and the magnitude of the enforced displacement. The SPCD card only specifies the magnitude of the enforced displacement. When an SPCD card is used, the component to be constrained must be specified on either an SPC or an SPC1 card. The use of the SPCD card avoids the decomposition of the stiffness matrix when changes are only made in the magnitudes of the enforced displacements. The equivalent loads resulting from enforced displacements of grid points are calculated by the program and added to the other applied loads. If the magnitudes of the enforced displacements are specified on SPC cards, the application

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of the load is automatic when you select the associated SPC set in the subcase information section. If the magnitude of the displacement is defined on an SPCD card, the load is applied if you select the associated LOAD set in the subcase information section.

Prestressed Analysis Preloaded or prestressed analysis is any type of structural analysis performed on a structure under prior loading (also termed preloading or prestressing). The response of a structure is affected by its initial state and this is in turn affected by the various preloading/prestressing applied to the structure, prior to the analysis of interest. Examples of prestressed analysis include analysis of rotorcraft blades under centrifugal preloading, analysis of pillar-like structures under compressive preloading, etc. OptiStruct can be used to take into account such preloading or prestressing effects. The prestressing/ preloading loadcase is a linear or nonlinear static loadcase. Prestressed/preloaded analysis is currently only supported for linear statics, eigenvalue analysis and direct frequency response analysis. Specifying prestress in any other unsupported analysis will generate an appropriate user error. Prestressing is specified through the STATSUB(PRELOAD) Case Control card, which refers to the preloading static loadcase ID. Nested preloading is not supported and will generate an appropriate user error (that is: User error will be reported if Subcase C has preloading from Subcase B, which in turn has preloading from Subcase A). The preloading is captured or defined by a geometric stiffness matrix [KG] which is based on the stresses of the preloading static subcase. In prestressed analysis, this geometric stiffness matrix is subtracted from the original stiffness matrix [K] of the (unloaded) structure. Depending on preloading conditions, the resulting effect could be a weakened or stiffened structure. If the preloading is compressive, it typically has a weakening effect on the structure (example: column or pillar under compressive preloading). If the preloading is tensile, it typically has a stiffening effect (for example: rotorcraft blade under centrifugal preloading).

Prestressed Static Analysis Prestressed static analysis is governed by the following equation (where, F is the loading and U is the displacement).

K KG U

F

While linear static subcases can have prestressing, nonlinear static subcases under prestressing are not supported.

Prestressed Eigenvalue Analysis Prestressed eigenvalue analysis is governed by the following equation (where, M is the mass matrix, F is the eigenvector and l is the eigenvalue).

K KG lM

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0


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Prestressed eigenvalue analysis is currently supported by AMSES, AMLS and the Lanczos Method. However, if the specified preload is greater than the first critical buckling load, an appropriate error will be reported for AMSES/AMLS runs.

Prestressed Direct Frequency Response Analysis Prestressed direct FRF analysis is governed by the following equation.

K

KG ig K KG

iKGE iwC w2 M U

F

Where, M is the mass matrix, U is the complex displacement vector, KGE is the material damping matrix, C is the viscous damping matrix that includes the Area Matrix for fluidstructure coupling, w is the loading frequency and g is the structural damping coefficient.

Results All results that are supported for regular structural analyses are also available in the corresponding prestressed analyses. It is important to note that, while the prestressed analysis includes the effects of preloading as a weakening or a stiffening of the structure, the results from the prestressed analysis do not include the preloading results. For example, the displacements from prestressed static analysis do not include the preloading displacements. In order to get the overall deflection/stresses of the structure, the displacements/stresses from the prestressed analyses have to be carefully superposed with the preloading displacements/stresses while post-processing. Particularly, while postprocessing complex results from prestressed direct FRF, the correct approach would be to first obtain the complex results for a certain phase and then superpose the appropriate preloading result. Any other superposing approach would lead to incorrect results.

Pretensioned Bolt Analysis Overview Many engineering assemblies are put together using bolts, which are usually pretensioned before application of working loads. A typical sequence is described below.

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Figure 1: Step 1 of pretensioned assembly - application of pretensioning loads

In Step 1, upon preliminary assembly of the structure, the nuts on respective bolts are tightened, usually by applying prescribed torque (which translates into prescribed tension force according to the pitch of the thread). As the result, the working part of the bolt becomes shorter by a distance L . This distance depends upon the applied force, the compliance of the bolt and of the assembly being pretensioned. From the perspective of FEA analysis, it is important to recognize that: Pretensioning actually shortens the working part of the bolt by removing a certain length of the bolt from the active structure (in reality this segment slides through the nut, yet the net effect is the shortening of the working length of the bolt). At the same time the bolt stretches, since now the smaller effective length of the bolt material has to span the distance from the bolt mount to the nut. Calculation of each bolt’s shortening L , due to applied forces F, requires FEA solution of the entire model with the pretensioning forces applied. This is because the amount of nut movement due to given force depends on the compliance of the bolts, of the assembly being bolted and is also affected by cross-interaction between multiple bolts being pretensioned. At the end of Step 1, the amount of shortening L for each bolt is established and “locked”, simply by leaving the nuts at the position that they reached during the pretensioning step. In Step 2, with the shortening L of all the bolts “locked”, other loads are applied to the assembly (Figure 2). At this stage the stresses and strains in the bolts will usually change, while the length of material removed L remains constant for each bolt.

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Figure 2: Step 2 of pretensioned assembly – application of working loads with “locked” bolt shortening

Comments In practice, there may be variations of the application of pretensioning loads and more complex pretensioning sequences than that presented above. For example: In alternative assembly scenarios, instead of using a nut on top of the bolt, the bolt may be screwed into a base and thus compress the assembly, as illustrated in Figure 3.

Figure 3: An alternative approach to application of pretensioning loads

In this case, the shortening (removal of material) of the working part of the bolt happens at the thread within the base, rather than at the bolt-nut interface. Yet the final mechanical effect is the same. Sometimes the pretensioning by torque/force is augmented by “tightening” via prescribed number of turns. This means that on top of the L , due to pretensioning force, an

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additional

L ’ is added according to the number of turns and the pitch of the thread.

In an automated assembly process, usually all bolts are pretensioned simultaneously. Sometimes, however, the tensioning may happen in sequence or in groups. In such cases, while L is “locked” for bolts that have already been pretensioned, consecutive pretensioning force is applied to the next batch of bolts, which then become “locked” for the following step.

FEA Solution with Pretensioning In analysis of structures, the FEA model is usually defined in material reference frame and the amount of material is assumed to remain fixed, while the structure undergoes stretching and deformation. However, in the case of pretensioning, the actual working part of the model has some material removed by being driven through the nut (usually the protruding part of the bolt is not included in the working FEA model, since it does not participate in the balance of forces on the structure). The simulation of this phenomenon in Optistruct follows the approach described below: First, it is recognized that for straight bolts, from the viewpoint of balance of forces it does not matter at what location the removal of the material happens in the bolt. Therefore, instead of simulating the precise interaction between the nut and the bolt, the pretensioning is handled within the length of the bolt. Pretensioning in OptiStruct is implemented with the help of Multiple Point Constraints (MPC’s) via two different processes: 1D Bolt Pretensioning 3D Bolt Pretensioning Multiple point constraints (MPC’s) are used in both 1D and 3D pretensioning, the difference between the two implementations is the number of duplicate grid points created and controlled via MPC’s. 1D Bolt Pretensioning In this process, the bolt or its selected section is represented by single or multiple 1D element(s) (beam or rod). Note: If a bolt is meshed with 3D (solid) elements, 1D pretensioning can be applied by replacing a transverse section of the bolt with 1D beam/rod elements. In such cases; however, 3D pretensioning is recommended since it is easier to implement and accuracy is improved. If a bolt is constructed using 1D elements, 1D pretensioning can be used effectively. Step 1 The conceptual FEA handling of 1D bolt pretensioning is illustrated in Figure 4. A bolt is modeled using 3D elements and a beam or rod element represents the selected section of the bolt where pretensioning will be applied. 1. First, an imaginary cut is introduced into the beam (This is automatically done internally for a subcase that includes a PRETENSION command) and two duplicate grid points are created at the location of the cut.

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Figure 4: FEA implementation of Bolt Pretensioning applied to a 1D Bolt using a 1D element

Additionally, a scalar point (SPOINT) is automatically created to act as an independent grid point. A pair of self-balanced pretensioning forces is applied to both ends of the cut with the help of the newly created SPOINT. The specified pretensioning force is internally applied on the SPOINT and this is transferred to the duplicated grid points via an MPC. The MPC controls the movement of the newly created duplicate grid points and the scalar point based on the following equation:

uspo int

udg uig

Where,

uspoint

is the displacement of the independent scalar point (SPOINT)

udg

is the displacement of the dependent grid point

uig

is the displacement of the independent grid point

u

The reaction force on the scalar point due to an enforced displacement of spoint on it can be shown to be equal to the forces acting on the dependent or independent grid point.

Fspo int

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Fdg

Fig

or

Fspoint

Fdg

Fig

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Where,

Fspoint

is the total reaction force on the independent scalar point (SPOINT) due to an

enforced displacement of

uspoint

Fdg

is the force acting on the dependent grid point

Fig

is the force acting on the independent grid point

2. With these forces (plus other loads referenced in this subcase) applied, static analysis is performed to calculate deformation of the structure. Among the results of such analysis is the “overlap” L across the cut portion of the beam, which is equivalent to the distance that the bolt would move relative to the nut in Figure 1.

Figure 5: FEA implementation of Bolt Pretensioning applied to a 3D Bolt using a 1D element (Step 1)

Step 2 As shown in Figure 5, the amount of overlap L calculated in Step 1 is removed from the bolt length, and the bolt is reconnected at the cut location. This represents the shorter working length of a pretensioned bolt on which the nut has been tightened (Mechanically, this is similar to the effect of the DEFORM command). With bolt pretensioning “locked” in this way, additional working loads are applied and a FEA solution is performed.

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As mentioned already, it is possible to construct more complex sequences of pretensioning, wherein some bolts have already been pretensioned (Step 2) while the next batch of bolts is being pretensioned (Step 1). (Note that a specific tensioning sequence has an effect on the final result only in path-dependent problems, such as contact with friction or elastoplastic materials.) 3D Bolt Pretensioning In 3D Bolt Pretensioning, the bolt is represented (meshed) using 3D solid elements. A transverse surface in the beam is identified (cross-section) along which it is cut and the duplicate grids are then controlled by Multiple Point Constraints (MPC’s) and a SPOINT to simulate the pretensioning effect. Step 1 The first step in 3D pretensioning is to identify the transverse cross-sectional surface of the bolt. OptiStruct automatically cuts the bolt at the selected surface and duplicate grid points are created mirroring the existing grid points at the cut surface. This is automatically done internally for a subcase that includes a PRETENSION command.

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Figure 7: FEA implementation of Bolt Pretensioning applied to a 3D Bolt using a 3D element (Pretension Direction)

Additionally, a scalar point (SPOINT) is automatically created to act as an independent grid point. A pair of self-balanced pretensioning forces is applied to both ends of the cut with the help of the newly created SPOINT. The specified pretensioning force is internally applied on the SPOINT and this is transferred to the duplicated grid points via MPC’s. The MPC’s controls the movement of the newly created duplicate grid points and the scalar point based on the following equations: In the pretension direction:

Perpendicular to the pretension direction:

Where,

uspoint

is the displacement of the independent scalar point (SPOINT)

udgk

is the displacement of the k’th dependent grid point

uigk

is the displacement of the k’th independent grid point

N is the displacement of the k’th independent grid point

sdgk

is the displacement of the k’th dependent grid point perpendicular to the pretension direction

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sigk

is the displacement of the k’th independent grid point perpendicular to the pretension direction

Figure 8: FEA implementation of Bolt Pretensioning applied to a 3D Bolt using a 3D element (Perpendicular to the Pretension Direction)

u

The reaction force on the scalar point due to an enforced displacement of spoint on it can be shown to be equal to the sum of the magnitudes of the forces acting on either the dependent or independent grid points.

Where,

Fspoint

is the total reaction force on the independent scalar point (SPOINT) due to an

enforced displacement of

uspoint

Fdgk

is the force acting on the k’th dependent grid point

Figk

is the force acting on the k’th independent grid point

When these forces (including other loads referenced in this subcase) are applied, static analysis is performed to calculate deformation of the structure. Among the results of such analysis is the “overlap” L across the cut portion of the bolt, which is equivalent to the distance that the bolt would move relative to the nut.

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Step 2 As shown in Figure 5, the amount of overlap L calculated in Step 1 is removed from the bolt length, and then the bolt is reconnected at the initial cut surface. This represents the shorter working length of a pretensioned bolt on which the nut has been tightened. (Mechanically, this is similar to the effect of the DEFORM command.) With bolt pretensioning “locked” in this way, additional working loads are applied and FEA solution is performed.


As mentioned earlier (in 1D pretensioning), it is possible to construct complex pretensioning sequences wherein some bolts have already been pretensioned (Step 2) while the next batch of bolts is being pretensioned (Step 1). Note: A specific tensioning sequence has an effect on the final result only in path-dependent problems, such as contact with friction or elasto-plastic

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materials.

Analysis of Pretensioned Assemblies in OptiStruct In OptiStruct, the solution of problems involving pretensioning fits into the standard sequences of static subcases, linear or nonlinear. (Step 2, analysis of pretensioned structure, is also available in natural frequency, frequency response, buckling and transient subcases). Respective user’s input requires definition of pretensioning sections, loads and adjustments in the Bulk Data section, plus specification of tensioning sequences in the Subcase section. The available commands are outlined below – for more details, see individual card descriptions. Bulk Data Section PRETENS

Defines the pretension section. Presently this identifies the respective 1D element.

PTFORCE

Defines the pretensioning force F (actually a pair of forces) and assigns it to the respective pretension section.

PTFORC1

A simplified format that allows assigning force to multiple pretension sections.

PTADJST

Defines the tensioning adjustment L’ and assigns it to the respective pretension section.

PTADJS1

A simplified format that allows assigning one adjustment amount to multiple pretension sections.

PTADD

Combines multiple pretensioning forces or adjustments into a single load ID.

Subcase Section PRETENSION Identifies pretensioning forces / adjustments to be activated in this static subcase. (Corresponds to Step 1 described above.) STATSUB (PRETENS)

Identifies the static subcase that created pretensioned bolts, which are to be included in the present subcase. (Corresponds to Step 2 described above.)

It is allowed to have PRETENSION and STATSUB(PRETENS) in the same static subcase – this can be used to emulate more complex pretensioning sequences.

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A Simple Illustration A simple illustration of typical flow of pretensioned analysis is shown below. This is not a complete input deck, merely an illustration of a typical arrangement of respective commands. Refer to the tutorial OS-1390: 1D and 3D Pretensioned Bolt Analysis of an IC Engine Cylinder Head, Gasket and Engine Block System Connected Using Head Bolts for more information on setting up 1D and 3D pretensioned analysis.

Comments Subcases that Support Pretensioning Pretensioning Steps 1 and 2 require the solution of a static FEA problem. Therefore, PRETENSION and STATSUB(PRETENS) commands can appear only in linear or nonlinear static subcases of the default NLSTAT type.

Referencing Pretensioning in Other Types of Subcases Because pretensioning produces stresses in the FEA model, it can through nonlinear geometric stiffness effects, affect the static and vibrational response of the structure, such as increase of natural frequency of a pretensioned bolt. Such geometric stiffness effects are captured by a STATSUB(PRELOAD) command, which is available in static, natural frequencies and frequency response subcases. In problems with pretensioning, it is allowed for STATSUB(PRELOAD) to point to a pretensioned subcase or any of the follow-on static subcases that references pretension. The stresses created by pretension (and other loads in such subcase) will be used as the respective preload.

Sequencing of Pretensioning Subcases The subcases with PRETENSION and STATSUB(PRETENS) can be used to create various sequences of pretensioning, such as tightening bolts in sequence or in groups. A single pretension section (1D bolt) can receive consecutive cumulative pretensioning loads so as to model cases where bolt tightening with a force is followed up by an

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additional adjustment by a prescribed distance (number of turns of the nut). Such a stacked sequence is presented in the simple illustration above. The specific rules for sequencing pretensioning subcases on the same section are as follows: 1. Pretensioning force (PTFORCE) can only be activated in the new or “fresh” pretensioning subcase for a given section. In other words, subcase with PRETENSION pointing to PTFORCE cannot also include STATSUB(PRETENS) referencing a subcase that had already pretensioned this section. 2. Pretensioning adjustment (PTADJST) may be activated in any of the pretensioning subcases for a given section. The effect of adjustment is cumulative relative to the pretensioning status reached in the respective previous subcase, as referenced by STATSUB(PRETENS). In nonlinear path-dependent problems, this sequencing of pretensioning can be combined with continuation of nonlinear subcases, as controlled by subcase command CNTNLSUB, in quite arbitrary combination. STATSUB(PRETENS) controls the sequencing of pretensioning steps and CNTNLSUB controls the sequencing of nonlinear aspects (plasticity, contact with friction, and so on) for quite arbitrary loading scenarios.

Inertia Relief Inertia relief allows the simulation of unconstrained structures. Typical applications are an airplane in flight, suspension parts of a car, or a satellite in space. With inertia relief, the applied loads are balanced by a set of translational and rotational accelerations. These accelerations provide body forces, distributed over the structure in such a way that the sum total of the applied forces on the structure is zero. This provides the steady-state stress and deformed shape in the structure as if it were freely accelerating due to the applied loads. Boundary conditions are applied only to restrain rigid body motion. Because the external loads are balanced by the accelerations, the reaction forces corresponding to these boundary conditions are zero. This calculation is automated. Inertia relief boundary conditions may be defined in the bulk data section of the input deck or they may be determined automatically by the solver. The SUPORT and SUPORT1 bulk data entries are used to define up to six reaction degrees of freedom of the free body. SUPORT entries will be used in all relevant subcases and therefore do not need to be referenced in the Subcase Information section. SUPORT1 entries need to be referenced by a SUPORT1 data selector statement for use within a subcase. Inertia relief boundary conditions may be generated automatically by using PARAM, INREL, -2. In OptiStruct, inertia relief can be applied to linear static, nonlinear gap, modal frequency response (with residual vectors), and transient response (with residual vectors) analyses. A static case with inertia relief cannot be referenced in a linear buckling analysis. Inertia relief is meaningless in normal modes analysis.

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Geometric Nonlinear Analysis Loads and Boundary Conditions Any number of load sets can be defined in the bulk data section of the input file. However, only those sets selected in the subcase information section (as described in the Geometric Nonlinear Analysis section) will be used in the problem solution. The manner in which each type of load is selected is specified on the associated bulk data statement description. Geometric nonlinear analysis, whether implicit or explicit, (quasi-)static or dynamic, is performed as a time-dependent event with time-dependent loads. A termination time TTERM needs to be defined. Subcases can be combined to a successive load history using the CNYNLSUB subcase statement. Each continued subcases starts from the end time and final load of the previous (reference) subcase. TTERM is defined in terms of total time. The NLOAD1 bulk data statement defines a time-dependent load of the form:

f (t )

A*C * F

t B

The load history function F is defined using TABLED1. For both definitions, a DAREA or SPCD statement defines the force, displacement, velocity, or acceleration amplitude A, respectively. Aside from DAREA, FORCE, MOMENT, PLOAD, and PLOAD4 entries can be used to define the load amplitude. The quantities B and C are scale factors. The NLOAD card in the bulk data section defines a loading condition that is a linear combination of load sets consisting of loads applied directly to grid points. The application of the combined loading condition is requested in the subcase information section by selecting the set number of the NLOAD combination. Rigid walls may be defined using RWALL. Multiple rigid wall sets can be combined into a single set using RWALADD. The subcase selection is made by RWALL. Zero enforced displacements may be specified on SPC or SPC1 cards. The SPCADD statement allows the combination of different SPC sets. In dynamic analysis (explicit and implicit), initial velocities can be defined using TIC and TICA bulk data entries. TIC defines an initial velocity on a grid point, while TICA defines the initial velocities of a grid set along and/or about an axis. A subcase selection must be made with IC. In (quasi-)static analysis, static loads can also be defined by using FORCE, MOMENT, PLOAD, and PLOAD4 directly thru a LOAD reference in the subcase. These loads are then treated as linear ramp-up. In this case, TTERM is not mandatory, but if defined the load ramps up from the end time of the previous subcase to TTERM. If TTERM is absent it will be determined from the subcase identification number SID such that TTERM = SID.

Frequency Response Loads and Boundary Conditions Frequency dependent dynamic loads are applied at grid points. There are two different definitions available. Any number of load sets can be defined in the bulk data section of the input file. However,

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only those sets selected in the subcase information section (as described in the Frequency Response Analysis section) will be used in the problem solution. The manner in which each type of load is selected is specified on the associated bulk data statement description. The RLOAD1 bulk data statement defines a frequency dependent excitation of the form:

The RLOAD2 bulk data statement defines a frequency dependent excitation of the form:

For both definitions, a combination of DAREA, FORCE, FORCE1, FORCE2, MOMENT, MOMENT1, MOMENT2, PLOAD, PLOAD1, PLOAD2, PLOAD4, and RLOAD, or SPCD define the amplitude A of an excitation force or motion, respectively. A DPHASE reference defines the phase angle , and a DELAY reference defines the delay . The quantities B, C, D, and , are frequency dependent. They are defined using TABLED1, TABLED2, TABLED3, or TABLED4. It is recommended that SPCD be used for enforced motion. If the old inferior Large Mass Method is used for modal frequency response analysis with EIGRA, use PARAM,AMSESLM for better accuracy. The range for the loading frequency is defined using the FREQ, FREQ1, FREQ2, FREQ3, FREQ4, or FREQ5 bulk data statements. The DLOAD card in the bulk data section defines a static loading condition that is a linear combination of load sets consisting of loads applied directly to grid points. The application of the combined loading condition is requested in the subcase information section by selecting the set number of the DLOAD combination. Zero enforced displacements may be specified on SPC or SPC1 cards. The SPCADD statement allows the combination of different SPC sets. Combinations of dynamic loads (DAREA) with static loads (FORCE, FORCE1, FORCE2, MOMENT, MOMENT1, MOMENT2, PLOAD, PLOAD1, PLOAD2, PLOAD4, and RLOAD), is supported.

Transient Response Loads and Boundary Conditions Transient dynamic loads are applied at grid points. Two different definitions are available. Any number of load sets can be defined in the bulk data section of the input file. However, only those sets selected in the subcase information section (as described in the Transient Response Analysis section) will be used in the problem solution. The manner in which each type of load is selected is specified on the associated bulk data statement description. The TLOAD1 bulk data statement defines a time dependent load of the form:

f (t )

AF (t

)

where, F is defined using TABLED1, TABLED2, TABLED3, or TABLED4.

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The TLOAD2 bulk data statement defines a time dependent load of the form:

The quantities T1, T2 are time constants, a frequency and exponential coefficient, and B is a growth coefficient.

a phase angle, C is an

For both definitions, a DAREA or SPCD statement defines the force or displacement amplitude A, respectively. A DELAY statement defines the delay . The DLOAD card in the bulk data section defines a static loading condition that is a linear combination of load sets consisting of loads applied directly to grid points. The application of the combined loading condition is requested in the subcase information section by selecting the set number of the DLOAD combination. Transient initial conditions are defined using a TIC bulk data entry. Initial displacements and initial velocities can be defined. Zero enforced displacements may be specified on SPC or SPC1 cards. The SPCADD statement allows the combination of different SPC sets. Combinations of dynamic loads with static loads are not currently supported. It is recommended that SPCD be used for enforced motion. If the old inferior Large Mass Method is used for modal transient analysis with EIGRA, use PARAM,AMSESLM for better accuracy.

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Modeling Errors Warning #340 RBE3 using 123456 DOF coupling at the independent GRID points can produced unexpected results. OptiStruct will issue the following warning: *** WARNING #340 RBE3 element 6300346 has nonzero weight at rotational (456) dofs of independent grids. This practice may result in undesirable load distribution - use with caution.

Warning #1265 Use of very thin plate elements (.001 thickness or less) on surfaces of solid elements will produce near singularities, if MID2 and MID3 are specified on the PSHELL data. OptiStruct will issue the following warning. *** WARNING #1265 PSHELL 10003383 has thickness 0.001 or less and bending properties defined. This can lead to matrix singularities, causing message 153. If this element is intended to be only a membrane element, please leave MID2 and MID3 blank on the PSHELL data. For thin skin elements, leave MID2 and MID3 blank on the PSHELL data.

Warning #1942 Confirm that Field 9, CID is specified with either 0 (zero) or the appropriate local coordinate system on CELAS GRID points. Below is an example of the OptiStruct output for the two types of modeling errors: For CELAS grids having different local coordinate systems. *** WARNING # 1942 CELAS2 8820024 references GRID data with different CD. This may constrain rigid body motion

Elements Do not connect CBEAM or CBAR to skin elements that have MID2 and MID3 blank on the PSHELL data, as this will cause mechanisms. Use RBE2 elements instead with 123 DOFs specified.

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Rigid Body Modes Welding Methods Do not use welding methods that use MPC equations. This, in general, will give very poor rigid body modes that are not equal to zero. Use rigid elements instead.

Extra Rigid Body Modes Do not use an RBE3 using 123456 DOF on the dependent GRID to only 1 or 2 GRID points with 123456 DOF on the independent points. Use of these types of elements can produce extra rigid body modes. If you use just on independent GRID, OptiStruct will issue the following warning: *** WARNING #341 RBE3 element 10159308 has only two nodes. This practice may result in undesirable load distribution - use with caution. Use at least 3 non co-linear grid points with only 123 DOF on the independent GRID.

Poor Rigid Body Modes CELAS elements between two non-coincident points will produce poor rigid body modes. Use CBUSH elements instead. The use of GROUNDCHECK will catch misaligned CELAS elements. Below is a list of elements that have non-zero rigid body strain energies: These elements can cause GROUNDCHECK to FAIL CELAS elements in this list are probably misaligned. Elem no: 1 type: CELAS1

Six rigid mode energies and ratios:

energy - 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 ratio - 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 4.000E+00

Torsion Motion When representing torsion motion through a ball joint connection, like the intermediate shaft to steering column using MPC equations, be sure to remove the rigid body reaction from the MPC equations. Otherwise, the model will have poor rigid body modes.

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Results The following features are outlined here.

Results of a Finite Element Analysis The primary results in a finite element analysis are grid point displacements and rotations. Element results such as stresses, strains, and strain energy density are derived from those results. Other results include element forces, MPC forces, SPC forces, and grid point forces. Results of a finite element analysis are post-processed using a graphical tool. The definitions of the output options can be found in the I/O Options Section. An overview of the result files can be found in the Results Output by OptiStruct section. Information on stress, strain and force definitions regarding their coordinate system definition can be found in the section Element Results Representation in OptiStruct and on the respective element definitions.

Displacements Displacements and rotations are computed in linear static, and frequency response analyses. In addition, in frequency response velocities and acceleration are computed. Eigenvectors are the primary result in a normal modes and buckling analyses. In a normal modes analysis, they are normalized with respect to the mass matrix or with respect to the maximum vector component. In a buckling analysis, the latter always applies. Displacements, velocities, accelerations, and eigenvectors are grid point results. They are plotted as a deformed structure, or as a contour on the undeformed structure. Some postprocessors, such as Altair HyperMesh and Altair HyperView, also allow the animation of the displacements.

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Deformed displacement contour plot

Stresses The stresses are secondary results in a static analysis. Stresses near notches and other sharp corners, point loads and boundary conditions, and rigid elements are often unreliable due to the singularities in these points. This is not a trait unique to OptiStruct, but is inherent in the finite element method itself. A mesh refinement in such places can improve the stress prediction. A theoretically infinite stress cannot be predicted by finite elements. Stresses are primarily calculated at the Gauss integration points. These give the most accurate prediction. However, element stresses, corner stresses, and grid point stresses are provided. Element stresses are calculated at the centroid of the element. They should only be postprocessed using an assign plot. Contouring of element stresses vastly underestimates the extreme values due to the smearing across element boundaries. The stresses of interest are usually found on the surface of a structure. Mesh refinement will actually not just improve the stress prediction but also change the location of the point of stress evaluation. Therefore, it is common practice to use a skin of thin membrane elements in 3D modeling, or rod elements in 2D modeling, to evaluate the stresses on element surfaces or edges, respectively. This method is accurate since it considers the correct condition of a stress-free boundary if no load is applied to the boundary. The method of skinning a model also has the advantage of much faster post-processing of solid models because only the membrane skin needs to be displayed.

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Besides assign plots, elements stresses can be viewed in tensor plots that can help in the evaluation of the load path in a structure by evaluating the principal stress directions. Corner stresses are computed by extrapolating the stresses from the Gauss points to the element grid points. Corner stresses are plotted in a contour plot. Corner stresses for solid elements are not available for normal modes analysis. Grid point stresses are computed by averaging the corner stresses contributions of the elements meeting in a grid point. The averaging does not consider the condition of a stress-free boundary. Further, interfaces between different materials, where a stress jump normally can be observed, are not considered correctly because of the smearing of the stress. Grid point stresses are plotted in a contour plot. For first order elements, grid point stresses do not provide higher accuracy over element stresses. For second order elements, the stress prediction might improve by using grid point stress over element stresses, considering the weaknesses mentioned above.

Assign plot of maximum principal element stress

Strains Strains are secondary results. They are calculated as elements strains. Remarks made above on element stresses apply here too.

Strain Energy Densities Strain energy densities are secondary results in static and normal modes analysis.

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They are calculated as element strain energy densities. Remarks made above on element stresses apply here too.

Forces Element forces, MPC forces, SPC forces, and grid point forces are printed as tabulated output.

Grid Point Stresses The default method of calculating stresses in OptiStruct produces values of stress components at the centroids of elements. (Typically a post-processor, such as HyperView, will then average these values to produce smooth contour plots). This method, while useful for viewing stress distribution, may underestimate stress maxima, especially on the surface of the body. To provide higher accuracy stresses, OptiStruct supports grid point stress calculation. Grid point stresses are computed using the following steps: 1. Calculate stress components at integration points of elements (these are generally the most accurate stress locations). 2. Extrapolate stress values to element nodes (grid points). 3. Calculate average at each grid point using values from surrounding elements. 4. Calculate derived quantities, such as von Mises stress or principal stresses, at each grid point (this assures that these values are consistent and make physical sense). The above approach produces continuous stress field, typically in the entire domain. Since, however, stresses can be discontinuous between different materials, OptiStruct supports calculation of separate grid point stress field per each material sub domain. The presence of more than one material is detected automatically and then grid point stresses are calculated for the entire domain and as a separate field for each material region. Grid point stress calculation is activated through the I/O subcase command GPSTRESS. When activated, grid point stresses are produced in addition to default stress results – they can be found in a separate results subcase. The present support for grid point stress capability has the following scope: It is supported for nodes connected to three-dimensional solid elements It is available for static load cases and non-design elements Grid point stresses are always calculated in the basic coordinate system

Element Results Representation for Models in OptiStruct Elemental results (namely stresses, strains and element forces) may be provided with reference to either the material system or the elemental system. For the HM, PUNCH, and OPTI output formats, results are provided with reference to the material system; for first order shell element results, PARAM, OMID may be used to output with either representation. The H3D and OUTPUT2 output formats, to which these elemental results

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are written as tensors, always contain results with reference to the elemental systems (unaffected by PARAM, OMID). Since OptiStruct 10.0 optimization responses always match with the results written to the HM, PUNCH and OPTI formats and first order shell responses are consistent with the PARAM, OMID setting.

First Order Shell Elements For first order shell elements (CQUAD4, CTRIA3), the material system is defined as follows: 1. With blank MCID/THETA (default behavior), stresses, strains and plate forces are presented in the default element material system, which has x-axis aligned with line G1-G2, and other axis built accordingly to make an orthonormal triad. 2. With MCID > 0, results are presented in the material system CID projected onto the element plane (projected material system). 3. With MCID = 0, results are presented in the basic coordinate system projected onto the element plane (projected basic system). 4. With THETA specified (including zero), results are presented in a rotated element material system, which is rotated by angle THETA from the edge G1-G2. The elemental system is the bi-sector system for CQUAD4 elements (see figure) and the G1-G2 system for CTRIA3 elements.

Bi-sector coordinate system

For the H3D and OUTPUT2 formats this representation allows HyperView to perform coordinate system transformations on stress and strain tensors.

Second Order Shell Elements The results for second order shells (CQUAD8 and CTRIA6), including shell strains, stresses and forces, are always presented in the local material coordinate system, as described in the manual for CQUAD8 and CTRIA6 elements.

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Composite Shells Shell-type strains and stresses for composite shells use the same representation as homogeneous shells. By shell-type results, we mean strains and stresses calculated at Z1, Z2 using homogenized shell properties. Strains and stresses for individual plies are always presented in the respective ply coordinate system.

Solid Elements For solid elements (CHEXA, CTETRA, CPENTA and CPYRA), the results are always provided in the material coordinate system. 1. With blank CORDM on the PSOLID card (default behavior), strains and stresses are presented in the basic coordinate system. 2. With CORDM > 0, strains and stresses are presented in the material system CID. 3. With CORDM = -1, stresses are presented in the local element coordinate system (described in detail on respective solid element manual pages).

Gap Elements For gap elements, gap forces are represented in the gap coordinate system, as described on respective gap element manual pages (CGAP and CGAPG). Compression is positive.

Saving and Retrieving Normal Modes Analysis Results OptiStruct allows Normal Modes Analysis results to be retrieved for use in Frequency Response Analysis or Transient Response Analysis using the modal method. Thus, multiple dynamic loading analyses can be performed using the eigenvalue results of a single normal modes analysis. The following input I/O options and subcase information section entries may be used for this purpose: EIGVSAVE EIGVRETRIEVE EIGVNAME Saving Eigenvalues and Eigenvectors from a Normal Modes Analysis EIGVSAVE is a subcase information entry that, if used within a normal modes analysis subcase, causes the eigenvalues and eigenvectors of that subcase to be written to an external data file. The external data file will use the default output file prefix unless the EIGVNAME I/O option is present, followed by an underscore, then followed by the EIGVSAVE integer argument and the .eigv extension.

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For example, the input: EIGVNAME = test_file $ Subcase 10 spc = 1 method = 20 EIGVSAVE = 50 will save the eigenvector and eigenvalue results from a normal modes analysis to the file "test_file_50.eigv."

Retrieving Eigenvalues and Eigenvectors for a Modal Frequency Response Analysis or for a Modal Transient Analysis EIGVRETRIEVE is a subcase information entry that, if used within a modal frequency response analysis or a modal transient response analysis subcase, retrieves eigenvalues and eigenvectors from external data files. EIGVRETRIEVE may have multiple integer arguments, each referring to a different external data file. The external data files must have the default output file prefix unless EIGVNAME I/O option is present, followed by an underscore, followed then by the EIGVRETRIEVE integer argument and the extension .eigv. For example, the following input can be used in a frequency response analysis subcase using the modal method to retrieve the eigenvalues and eigenvectors that were saved in the example above: EIGVNAME = test_file $ Subcase 40 Spc = 1 Dload = 30 Method = 20 EIGVRETRIEVE = 50

Combining Eigenvalues and Eigenvectors from Two or More Normal Modes Analyses for a Single Modal Frequency Response or Modal Transient Response Analysis The results of two or more normal modes analyses can be retrieved in combination for a modal frequency response analysis. For example, a normal modes analysis is performed with the real eigenvalue extraction (EIGRL) data: (1)

(2)

EIGRL

20

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

50.0

The results are written to an external data file as follows: EIGVNAME = test_file $ Subcase 10 spc = 1

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method = 20 EIGVSAVE = 50 In this case, all of the eigenmodes up to 50 Hz have been calculated and written to the file "test_file_50.eigv." In order to perform a modal frequency response analysis with all of the modes up to 70 Hz, another normal modes analysis can be performed with the real eigenvalue extraction data: (1)

(2)

(3)

(4)

EIGRL

20

50.0

70.0

(5)

(6)

(7)

(8)

(9)

(10)

This time, the results are written to an external data file as follows: EIGVNAME = test_file $ subcase 10 spc = 1 method = 20 EIGVSAVE = 70 All eigenmodes between 50 Hz and 70 Hz are written to the file "test_file_70.eigv." You can now run a modal transient response analysis with: EIGVNAME = test_file $ subcase 40 spc = 1 dload = 30 method 20 tstep(time) = 100 EIGVRETRIEVE = 50, 70 The real eigenvalue extraction data referenced in the modal transient response analysis subcase must not request eigenvalue and eigenvector results outside of the range of retrieved values. If it does, OptiStruct will terminate with an error. In this example, the following EIGRL cards are valid: (1)

(2)

(3)

(4)

EIGRL

20

0.0

70.0

(1)

(2)

(3)

(4)

EIGRL

20

0.0

50.0

(1)

(2)

(3)

(4)

EIGRL

20

30.0

40.0

414

(5)

(6)

(7)

(8)

(9)

(10)

(5)

(6)

(7)

(8)

(9)

(10)

(5)

(6)

(7)

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The following EIGRL cards would cause error terminations for this example: (1)

(2)

(3)

(4)

EIGRL

20

0

100.0

(1)

(2)

(3)

(4)

EIGRL

20

50.0

70.01

(5)

(6)

(7)

(8)

(9)

(10)

(5)

(6)

(7)

(8)

(9)

(10)

It is recommended to use a frequency range without the maximum number of modes on the EIGRL bulk data entries referenced in normal modes analyses from which eigenvalue results are saved. If the maximum number of modes is specified and these eigenvalue results are retrieved by a modal frequency response analysis, and it cannot be determined whether all of the modes are obtained for the requested range, OptiStruct will terminate with an error. For example, assume there are exactly 300 modes in the frequency range 0.0 to 5.0.0 Hz. Now assume that a normal modes analysis is performed referencing the EIGRL bulk data entry. (1)

(2)

(3)

(4)

(5)

EIGRL

20

0.0

50.0

300

(6)

(7)

(8)

(9)

(10)

The eigenvectors and eigenvalues are saved as follows: EIGVNAME = $ Subcase 10 spc method EIGVSAVE

test_file = 1 = 20 = 50

All 300 modes in the range of 0 to 50.0 Hz are extracted and saved to the file "test_file_50.eigv." Now try to retrieve these results to use in a modal frequency response analysis, as follows: EIGVNAME = test_file $ subcase 40 spc = 1 dload = 30 method 20 EIGVRETRIEVE = 50 where the referenced EIGRL definition is: (1)

(2)

(3)

(4)

EIGRL

20

0.0

50.0

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(6)

(7)

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(9)

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This will cause an error termination because it is known (through the external data file) that there are 300 modes within the 0.0 to 50.0 Hz range, but do not know if this is all of the modes. If the EIGRL definition referenced in the normal modes analysis were specified as: (1)

(2)

(3)

(4)

(5)

EIGRL

20

0.0

50.0

301

(6)

(7)

(8)

(9)

(10)

and only 300 modes were found, you would know that these are all of the modes within the 0.0 to 50.0 Hz range, and would retrieve the saved eigenvalue results in this case. OptiStruct would not terminate with an error.

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Coupling OptiStruct with Third Party Software The following features can be found in this section:

Using the AMLS (Automatic Multi-Level Sub-structuring) Eigensolvers For the solution of large eigenvalue problems, the AMLS (Automatic Multi-Level Substructuring) eigensolver developed by the University of Texas can be used instead of the internal OptiStruct Lanczos eigensolver. The AMLS eigensolver is a separate program from OptiStruct and must be installed and licensed separately. OptiStruct interfaces with AMLS by writing AMLS input files, launching AMLS, and then reading the AMLS results back into OptiStruct once the AMLS execution is complete. The resulting eigenvalues and eigenvectors can then be used by OptiStruct for eigenvalue analysis, modal frequency response, and modal transient analysis. In addition, the AMLS solver can be used during topology and sizing optimizations. OptiStruct only supports AMLS version 3.2.0128 or later. To use AMLS version 5 or later, OptiStruct version 13.0 or later must be used. To use AMLS, the following should be defined: 1. The environment variable AMLS_EXE must be set by you to point to the AMLS executable. On UNIX and Linux platforms the script that is used to invoke OptiStruct (~altair/ scripts/invoke/optistruct) contains a "placeholder" where AMLS_EXE may be defined (search for AMLS_EXE). The definition contained in the invoke script will only be used if there is no pre-existing AMLS_EXE environment variable at invoke. Example: setenv AMLS_EXE /share/ams/cdhopt/2005/AIX-5.3/3.2.r159_exe/ amls.main_AIX.5 2. PARAM, AMLS must be set to YES in the OptiStruct input file. The run option –amls can also be used to activate AMLS. AMLS is a multithreaded application and can use 1, 2, or 4 processors. PARAM, AMLSNCPU may be defined in the OptiStruct input file to define the number of processors that are to be used by AMLS. If PARAM, AMLSNCPU is not set, then the AMLS eignersolver will use only 1 CPU. Note that when PARAM, AMLSNCPU is defined, it is possible for OptiStruct and AMLS to use different numbers of processors.

Parameters Affecting AMLS AMLS controls the accuracy and the cost of a solution primarily with three parameters. The “optimal” values of these parameters for typical NVH analysis have been established through extensive testing. The parameters and their values are: PARAM,SS2GCR,5.0 PARAM,GMAR,1.1 PARAM,GMAR1,1.7 In case of predominately solid models, such as engine blocks, SS2GCR should be set to 10.0 and GMAR1 should be set to 2.1. In case of typical shell models, such as car bodies,

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a slight improvement in FRF accuracy can be obtained without large increases in elapsed time by setting two of the parameters as follows: PARAM,SS2GCR,7.5 PARAM,GMAR1,2.1 However, you are discouraged from adjusting these values unless the accuracy improvement is known to be worth the increase in resource requirements. The default upper limit on eigenvalues that are taken to be associated with rigid body modes is 1.0 (equivalent to a natural frequency of about 0.16 Hz). This parameter can be adjusted by parameter RBMEIG, which can be set by the command: PARAM,RBMEIG,0.2 AMLS distinguishes between rigid body modes and flexible modes to improve the numerical conditioning, and hence accuracy, with which the flexible eigenvalues are computed. Control of the singularity processing is performed using PARAM, AMLSMAXR. If AMLSMAXR is exceeded in the process of factoring a stiffness matrix, this indicates a singularity in K. If the mass of this DOF is also zero, there is a "massless mechanism", and an SPC is applied and a message is written to the .out file. If there is mass, then this is a mechanism, which is treated as a rigid body mode, and a message is written to the .out file. By default AMLS does not handle disconnected structures. There are two solutions for handling disconnected structures: PARAM,AMLSUCON,YES PARAM,DISJOINT,n If PARAM,AMLSUCON is set to YES then OptiStruct will SPC out the disconnected components if there is a total of less than 4000 disconnected grids. This works with all versions of AMLS. When PARAM,DISJOINT is set to a value that is at least one larger than the number of disconnected parts then AMLS will be able to solve the eigenvalue calculation problem. This feature is only available in AMLS versions 4.2r22 or newer. For AMLS Versions 5 and later, the run option –amlsmem, the environment variable AMLS_MEM or the parameter PARAM, AMLSMEM can be used to set the amount of memory in Gigabytes used by AMLS. By default, AMLS will use the same amount of memory used by OptiStruct. The run option –amlsmem, the environment variable AMLS_MEM or the parameter PARAM, AMLSMEM can be used to override this default value. The run option overrides the value set by the environment variable and the parameter. If both AMLS_MEM and PARAM, AMLSMEM are set, then the value specified by the environment variable is used.

Residual Vector Calculations When the AMLS eigensolver is used, OptiStruct’s Residual Vector calculations are ignored. The AMLS eigensolver calculates its own residual vectors for each of the following: USET U6 data Frequency Response Dynamic Loads

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Transient Response Dynamic Loads Damping DOF from CBUSH, CDAMPi or CVISC data One Residual Vector is calculated for each USET U6 degree of freedom, each DAREA degree of freedom, and each damping degree of freedom associated with the CBUSH, CDAMPi and CVISC data. The Residual Vector calculations are controlled by the Solution Control data RESVEC. To control Residual Vector calculations with AMLS, use the following commands: Use RESVEC=NO to turn off Residual Vector calculations with AMLS Use RESVEC(NODAMP)=YES to turn off Residual Vectors associated with Damping DOF

Singularities If AMLS detects a large number of singularities in the model this is most likely due to thin CQUAD4/CTRIA3 elements used to “skin” solid models. These singularities cause numerical ill-conditioning and increase run time. The singularities are caused by the very low bending stiffness of these thin shell elements. To remove the singularities, convert the thin bending elements to membrane only elements by removing the MID2 and MID3 MID’s from the associated PSHELL data. The thin membrane elements will still calculate the correct surface stresses, but the singularities will not be present as the elements will have no bending stiffness. PARAM, AMLSMAXR is used to determine singularities in the stiffness matrix. If the value of AMLSMAXR is exceeded in the process of factoring a stiffness matrix, this indicates a singularity in K. If the mass of this degree-of-freedom is zero, there is a "massless mechanism"; an SPC is applied and a message is written to the .out file. If there is mass, then this is a mechanism which is treated as a rigid body mode and a message is written to the .out file. The list of GRID identification numbers of singular grids during an AMLS run is output to the .amls_singularity.cmf file.

Remote File Systems If the execution directory is on a remote file system, long run times will result as the AMLS scratch files will have to be accessed over the NFS mounted file system. Use the environment variable TMPDIR to redefine the scratch directory to be on the local machine. Note that the environment variable TMPDIR is different from the scratch file directory specified by the command line argument –tmpdir. The input and output files from AMLS (generic_real_file, generic_integer_file, and generic_amls_output) are stored in the directory specified by the environment variable AMLSDIR. In general, the environment variable AMLSDIR should be set to be the same directory as the environment variable TMPDIR.

Limitations 1. AMLS is designed for large problems. Problems less than a few hundred degrees of freedom cannot be solved by AMLS.

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2. The model must consist of only one structure. Models of unconnected parts cannot be solved by AMLS. When the CBN method of creating CSM Super Elements is used on the CMSMETH data, unconnected models can be generated if the center GRID of an RBE2 is an ASET GRID. If unconnected are found, a file named filename.unconnected.cmf is generated. This file can be used in HyperMesh to show the unconnected parts. If the parts are small, PARAM, AMLSUCON,1 can be used to SPC out the unconnected structure and AMLS will run correctly. If the unconnected part is large, you can: Remove one spider GRID of the RBE2 to make the structure connected Use a small CBAR, CBEAM, or CROD to connect the two structures

FastFRS Usage (Fast Frequency Response Solver) FastFRS is a solver developed by the University of Texas at Austin. It is very efficient for a certain class of large modal frequency response problems, such as NVH problems. OptiStruct has an interface to FastFRS. OptiStruct writes the file FastFRS_Gen.in as input for FastFRS, and reads results from FastFRS_gen.out. FastFRS will run in the directory specified by the environment variable AMLSDIR, or the current directory if AMSLDIR is not specified. The following parameters can be used within the OptiStruct to control the FastFRS solver. 1. Set the environment variable FASTFRS_EXE to point to the location of the FastFRS executable. 2. The run option –ffrs yes or the parameter PARAM,FFRs,YES can be used to activate FastFRS: 3. Add the following optional parameters to adjust the settings for FastFRS runs: PARAM,FFRSLFRQ PARAM,LOWRANK PARAM,K4CUTOFF PARAM,CSTOL PARAM,FFRSNCPU (or the run option –ffrsncpu) PARAM,FFRSMEM (or the run option –ffrsmem or the environment variable FFRS_MEM) Note: 1. OptiStruct version 13.0 or above is required to run FastFRS version 2 or above. 2. If FFRSNCPU is not set (using either the parameter or the run option), and AMLSNCPU is set, then FastFRS will use the number of CPU’s specified by AMLSNCPU. 3. For FastFRS versions 2 and later, the run option –ffrsmem, the environment variable FFRS_MEM, or the parameter PARAM,FFRSMEM can be used to set the amount of memory in Gigabytes used by FastFRS. By default, FastFRS will use the same amount of memory used by OptiStruct. The run option –ffrsmem, the environment variable, FFRS_MEM, or the parameter PARAM, FFRSMEM can be used to override this default value. The run option overrides the environment variable and the parameter. If both FFRS_MEM and PARAM, FFRSMEM are set, then the value specified by the environment

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variable is used.

Creating Output for Third Party Software The section describes how to create output from OptiStruct for third party programs for Fatigue Analysis (FEMFAT, Design Life, and FE-SAFE), Multi-Body Dynamic Analysis (AVL/ EXCITE, SIMPACK, ROMAX, ADAMS, RecurDyn and Virtual Lab), and Fluid Structure Interaction (AcuSolve).

Fatigue Analysis The .op2 file from OptiStruct can be used directly by Third Party Fatigue Analysis software programs FEMFAT, Design Life, FE-Fatigue, and FE-SAFE. Just request stress output to the .op2 file: STRESS(OP2) = SET or ALL

Multi-Body Dynamic Analysis AVL/EXCITE To create the condensed CMS Super Element information for AVL/EXCITE, use the CMSMETH CBN Method with ASET data for the connection DOF. The MODEL data can used specify interior grid data to be included in the Super Element for viewing in AVL/EXCITE. The MODEL data format is: MODEL=Element Set, Grid Set, RIGID/NORIGID. All grids associated with elements in the element set and rigid elements, if RIGID is specified, are combined with the grids in the Grid Set and output to AVL/EXCITE. In addition, the keyword PLOTEL can be used instead of an Element Set ID to specify all the grids associated with all of the PLOTEL data in the model. GPSTRESS is used to specify set of grids for which Grid Point Stresses are calculated for AVL/EXCITE. The PARAM,EXCEXB data controls the output of the AVL/EXCITE .exb file directly from OptiStruct. The PARAM,EXCOUT data is used to specify what data is written out for AVL/EXCITE. The EXCOUT values produce the information for AVL/EXCITE specified below: $ EXCOUT - -1: no output $ 0: all output (default) $ 1: DOF, geometry, and elements tables $ (GEOM1,GEOM2,EQEXIN,USET), $ reduced mass and stiffness matrices (MAA and KMAA) $ 3: 1 + GOA transpose $ 4: 3 + unreduced mass matrix (MFF) (CON6) $ 5: 1 + eigenvectors of condensed system (PHA) and $ grid point stress table, SORT 1 (OGS1) $ 6: 4 + eigenvectors of condensed system (PHA) and $ grid point stress table, SORT 1 (OGS1) $ 7: 3 + eigenvectors of condensed system (PHA) and

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$

grid point stress table, SORT 1 (OGS1)

After running AVL/EXCITE, a residual run can be made to recover displacement, velocity, acceleration, stress, and strain results for interior grids and elements in the CMS Super Element based on the modal participation results from AVL/EXCITE. Note that the results are only calculated for GRID and elements specified by the MODEL data in the CMS Super Element creation run. The residual run can be any combination of frequency response and transient analysis SUBCASE’s. After running AVL/EXCITE, a resulting filename.INP4 file is created that contains the modal participation factors for the modes of the CMS Super Element for each loading frequency or transient analysis time step. In the residual run, the CMS Super Element .h3d file and the AVL/EXCITE modal results file are specified using the ASSIGN data: ASSIGN,H3DDMIG,AX,'Crank_split2h_all.h3d' ASSIGN,EXCINP,10,'Crankshaft_SOL109_time.INP4' Where the 10 in the ASSIGN,EXCINP data corresponds to the SUBCASE for which the modal participation results will be used. In SUBCASE 10, instead of performing a frequency response or transient response analysis, OptiStruct will just use the modal participation results from AVL/EXCITE. Note that since the analysis is skipped, it does not matter if the residual run is modal or direct frequency response/transient analysis. For transient analysis, the number of time steps in the transient analysis residual run must match the number of time steps used in the AVL/EXCITE analysis. For frequency response analysis, the number of loading frequencies in the frequency response analysis residual run must match number of loading frequencies used in the AVL/EXCITE analysis. While the frequency response/transient analysis data is ignored, there must still be some dummy loading data (TLOAD/RLOAD, DAREA, and TABLED data). A sample of input data for a transient analysis residual run is shown below: ASSIGN,H3DDMIG,AX,'Crank_split2h_all.h3d' ASSIGN,EXCINP,10,'Crankshaft_SOL109_time.INP4' $ DISPLACEMENT = ALL STRESS = ALL $ SUBCASE 10 DLOAD = 10201 TSTEP = 10133 $ BEGIN BULK $ GRID,80001,,0.,-62.,0. $ TLOAD1,10201,10202,,,10301 DAREA,10202,80001,1,1.0 TABLED1,10301, ,0.0,1.0,0.1,1.0,0.2,1.0,0.3,1.0 ,ENDT TSTEP,10133,143,2.7778-4 ENDDATA SIMPACK To create the condensed CMS Super Element information for SIMPACK, you must use the CMSMETH CBN Method with ASET or CSET data for the connection DOF. If CSET connection DOF is used, then the AMSES solver must be specified on the CMSMETH data.

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The PARAM,SIMPACK data is used to specify what data is written out for SIMPACK. The SIMPACK values produce the information for SIMPACK.

Recover SIMPACK results in OptiStruct ASSIGN, SIMPIMP identifies an external .unv file generated after running a multibody dynamic analysis in SIMPACK. The resulting CMS flexbody modal participation factors in the .unv file can be used by OptiStruct to recover the dynamic displacements, velocities, accelerations, stresses and strains. The format is as follows: ASSIGN, SIMPINP, Subcase ID, Filename Subcase ID is used to specify which SUBCASE the modal participation factors should be used for.

RecurDyn To create the modal CMS Super Element information for RecurDyn, you must use the CMSMETH CBN Method with ASET data for the connection DOF. The PARAM,RFIOUT,YES data is used to turn on the generation of the .rfi file, which contains the modal super element that is used by RecurDyn. Note: This .rfi file can be created only by OptiStruct executables running on 64-bit Windows machines. This file cannot be created while using OptiStruct on Linux or Mac OS X machines.

ROMAX To create the condensed CMS Super Element information for ROMAX, use the CMSMETH CBN Method with ASET data for the connection DOF. Use PARAM,EXTOUT,DMIGPCH to create a PUNCH file containing the Super Element data. This data can be read by any version of ROMAX after release R12.6.2. ADAMS To create the condensed Flex Body information for ADAMS, use the CMSMETH CC or CB Method. The MODEL data used can specify interior grid and element data to be included in the Flex Body for viewing in ADAMS. The MODEL data format is: MODEL=Element Set, Grid Set, RIGID/NORIGID. All grids associated with elements in the element set and rigid elements if RIGID is specified are combined with the grids in the Grid Set and output to ADAMS. In addition, the keyword PLOTEL can be used instead of an Element Set ID to specify all the grids associated with all of the PLOTEL data in the model. GPSTRESS is used to specify set of grids for which Grid Point Stresses are calculated for ADAMS.

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The OUTPUT command is used to generate the .mnf file for ADAMS. The command is: OUTPUT=ADAMSMNF Virtual Lab To create the condensed Flex Body Modes and Full Diagonal Mass Matrix to the .op2 file for Virtual Lab, use the CMSMETH CB or CC Methods. The PARAM,LMSOUT data is to trigger the output of the condensed Flex Body Modes and full Diagonal Mass Matrix to the .op2 file. PARAM,POST is not required. OUTPUT=OP2 is not required.

Fluid Structure Interaction Analysis The .op2 file from OptiStruct can be used directly by the AcuSolve CFD code. Just request the eigenvector output to the .op2 file: DISP(OP2) = ALL Then run the Python script from ACUSIM called acuNASTRAN2pev.py: python acuNastran2Pev.py problem.op2 This will create the nodes.dat, elems.dat, and modeXX.dat files. Be sure to use the latest version of acuNASTRAN2pev.py from ACUSIM.

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Design Optimization The following features can be found in this section: Optimization Problem Responses Topology Optimization Free-size Optimization Topography Optimization Size Optimization Shape Optimization Free-shape Optimization Manufacturing Constraints Reliability-based Design Optimization (Beta) Optimization of Arbitrary Beam Sections Optimization of Composite Structures Equivalent Static Load Method (ESLM) Gradient-based Optimization Method Global Search Option Multi-Model Optimization

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Optimization Problem The following features can be found in this section:

Minimize Objective Function OptiStruct solves the following structural optimization problem:

min f ( x)

f ( x1 , x2 , K , xn )

Subject to:

j 1,K , m

g j ( x) 0 xiL

xi

xiU

i 1,K , n

The objective function f(x) and the functions g(x) in the constraint function are structural responses obtained from a finite element analysis. A constraint is considered active if it is satisfied exactly (g = 0); it is considered inactive if g < 0; it is considered violated if g > 0. The selection of the vector of design variables x depends on the type of optimization being performed. In topology optimization, the design variables are element densities (see Design Variables for Topology Optimization). In size optimization (including free-size), the design variables are properties of structural elements (see Design Variables for Size Optimization). In topography and shape (including free-shape) optimization, the design variables are the factors in a linear combination of shape perturbations (see Design Variables for Topography Optimization and Design Variables for Shape Optimization). The objective function is defined using a DESOBJ entry in the subcase information section. DESOBJ references a response defined by either the DRESP1, DRESP2, or DRESP3 bulk data entry. Depending on the type of response, DESOBJ is located inside or outside of a SUBCASE. The constraints are defined using a DESSUB or DESGLB entry in the subcase information section, depending on if the type of response is subcase related or global, respectively. DESSUB and DESGLB refer to DCONSTR or DCONADD bulk data entries. DCONSTR relates the constraint value or bound to a response defined by DRESP1, DRESP2, or DRESP3.

Minmax Objective Function The minmax optimization problem is given as:

min max f1 x / f1 , f 2 (x) / f 2 ,K , f K ( x) / f K Subject to:

j 1,K , m

g j ( x) 0 xiL

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fK

are the reference values

The reference values can take different values for positive or negative objective functions. These problems are solved using the Beta-method. In this method, the problem is transformed into a regular optimization problem through the introduction of an additional design variable

such that:

min Subject to:

fi (x) / fi

i 1,K , k

g j ( x) 0

j 1,K , m

The functions fi(x) and the functions g(x) in the constraint function are structural responses obtained from a finite element analysis. A constraint is considered active if it is satisfied exactly (g = 0); it is considered inactive if g < 0; it is considered violated if g > 0. The selection of the vector of design variables x depends on the type of optimization being performed. In topology optimization, the design variables are element densities (see Design Variables for Topology Optimization). In size optimization (including free-size), the design variables are properties of structural elements (see Design Variables for Size Optimization). In topography and shape (including free-shape) optimization, the design variables are the factors in a linear combination of shape perturbations (see Design Variables for Topography Optimization and Design Variables for Shape Optimization). The objective function of a minmax problem is defined using MINMAX or MAXMIN statements in the subcase information section. MINMAX or MAXMIN references a DOBJREF statement in the bulk data section, which again refers to a DRESP1, DRESP2, or DRESP3 response definition. The reference values are defined on the DOBJREF entry. The constraints are defined as stated above. The constraints are defined using a DESSUB or DESGLB entry in the subcase information section, depending on if the type of response is subcase related or global, respectively. DESSUB and DESGLB refer to DCONSTR or DCONADD bulk data entries. DCONSTR relates the constraint value or bound to a response defined by DRESP1, DRESP2, or DRESP3.

System Identification For system identification, OptiStruct solves the following two structural optimization problems:

n

2

f ( x) Ti Wi i Ti i 1 g j (x) 0 j 1,K , m

min

or

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min

with W1

g j (x)

0

f1 ( x) Ti Ti

j 1,K , m

The functions fi(x) and the functions g(x) in the constraint function are structural responses obtained from a finite element analysis. A constraint is considered active if it is satisfied exactly (g = 0); it is considered inactive if g < 0; it is considered violated if g > 0. The values Ti are the target value for the particular response, Wi is a weighting factor. The selection of the vector of design variables x depends on the type of optimization being performed. In topology optimization, the design variables are element densities (see Design Variables for Topology Optimization). In size optimization (including free-size), the design variables are properties of structural elements (see Design Variables for Size Optimization). In topography and shape (including free-shape) optimization, the design variables are the factors in a linear combination of shape perturbations (see Design Variables for Topography Optimization and Design Variables for Shape Optimization). The objective function is defined using a DESOBJ entry or a MINMAX, MAXMIN entry in the subcase information section. DESOBJ, MINMAX, or MAXMIN reference a DSYSID entry that defines target values for responses defined by either a DRESP1, DRESP2, or DRESP3 bulk data entry. The constraints are defined using a DESSUB or DESGLB entry in the subcase information section, depending on if the type of response is subcase related or global, respectively. DESSUB and DESGLB refer to DCONSTR or DCONADD bulk data entries. DCONSTR relates the constraint value or bound to a response defined by DRESP1, DRESP2, or DRESP3.

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Responses The following responses can be found in this section:

Internal Responses OptiStruct allows the use of numerous structural responses, calculated in a finite element analysis, or combinations of these responses to be used as objective and constraint functions in a structural optimization. Responses are defined using DRESP1 bulk data entries. Combinations of responses are defined using either DRESP2 entries, which reference an equation defined by a DEQATN bulk data entry, or DRESP3 entries, which make use of user-defined external routines identified by the LOADLIB I/O option. Responses are either global or subcase (loadstep, load case) related. The character of a response determines whether or not a constraint or objective referencing that particular response needs to be referenced within a subcase.

Subcase Independent Mass and Volume Both are global responses that can be defined for the whole structure, for individual properties (components) and materials, or for groups of properties (components) and materials. It is not recommended to use mass and volume as constraints or objectives in a topography optimization. Neither is very sensitive towards design modifications made in a topography optimization. In order to constrain the mass or volume for a region containing a number of properties (components), the SUM function can be used to sum the mass or volume of the selected properties (components), otherwise, the constraint is assumed to apply to each individual property (component) within the region. Alternatively, a DRESP2 equation needs to be defined to sum the mass or volume of these properties (components). This can be avoided by having all properties (components) use the same material and applying the mass or volume constraint to that material. Fraction of Mass and Fraction of Design Volume Both are global responses with values between 0.0 and 1.0. They describe a fraction of the initial design space in a topology optimization. They can be defined for the whole structure, for individual properties (components) and materials, or for groups of properties (components) and materials. The difference between the mass fraction and the volume fraction is that the mass fraction includes the non-design mass in the fraction calculation, whereas the volume fraction only considers the design volume. Formulation for volume fraction: Volume fraction = (total volume at current iteration – initial non-design volume)/initial design volume Formulation for mass fraction:

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Mass fraction = total mass at current iteration/initial total mass If, in addition to the topology optimization, a size and shape optimization is performed, the reference value for the volume fraction (the initial design volume) is not altered by size and shape changes. This can, on occasion, lead to negative values for this response. Therefore, if size and shape optimization is involved, it is recommended to use the Volume responses instead of the Volume Fraction response. These responses can only be applied to topology design domains. OptiStruct will terminate with an error if this is not the case. Center of Gravity This is a global response that may be defined for the whole structure, for individual properties (components) and materials, or for groups of properties (components) and materials. Moments of Inertia This is a global response that may be defined for the whole structure, for individual properties (components) and materials, or for groups of properties (components) and materials. Weighted Compliance The weighted compliance is a method used to consider multiple subcases (loadsteps, load cases) in a classical topology optimization. The response is the weighted sum of the compliance of each individual subcase (loadstep, load case).

CW

Wi Ci

1

2

Wi uiT fi

This is a global response that is defined for the whole structure. Weighted Reciprocal Eigenvalue (Frequency) The weighted reciprocal eigenvalue is a method to consider multiple frequencies in a classical topology optimization. The response is the weighted sum of the reciprocal eigenvalues of each individual mode considered in the optimization.

fW

Wi l i with

K

i M ui

0

This is done so that increasing the frequencies of the lower modes will have a larger effect on the objective function than increasing the frequencies of the higher modes. If the frequencies of all modes were simply added together, OptiStruct would put more effort into increasing the higher modes than the lower modes. This is a global response that is defined for the whole structure. Combined Compliance Index The combined compliance index is a method to consider multiple frequencies and static subcases (loadsteps, load cases) combined in a classical topology optimization. The index is defined as follows:

S

Wi Ci

NORM

Wj l j Wj

This is a global response that is defined for the whole structure.

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The normalization factor, NORM, is used for normalizing the contributions of compliances and eigenvalues. A typical structural compliance value is of the order of 1.0e4 to 1.0e6. However, a typical inverse eigenvalue is on the order of 1.0e-5. If NORM is not used, the linear static compliance requirements dominate the solution. The quantity NORM is typically computed using the formula:

NF

Cmax min

where, Cmax is the highest compliance value in all subcases (loadsteps, load cases) and min is the lowest eigenvalue included in the index.

In a new design problem, you may not have a close estimate for NORM. If this happens, OptiStruct automatically computes the NORM value based on compliances and eigenvalues computed in the first iteration step. von Mises Stress in a Topology or Free-Size Optimization The von Mises stress constraints may be defined for topology and free-size optimization through the STRESS optional continuation line on the DTPL or the DSIZE card. There are a number of restrictions with this constraint: The definition of stress constraints is limited to a single von Mises permissible stress. The phenomenon of singular topology is pronounced when different materials with different permissible stresses exist in a structure. Singular topology refers to the problem associated with the conditional nature of stress constraints, i.e. the stress constraint of an element disappears when the element vanishes. This creates another problem in that a huge number of reduced problems exist with solutions that cannot usually be found by a gradient-based optimizer in the full design space. Stress constraints for a partial domain of the structure are not allowed because they often create an ill-posed optimization problem since elimination of the partial domain would remove all stress constraints. Consequently, the stress constraint applies to the entire model when active, including both design and non-design regions, and stress constraint settings must be identical for all DSIZE and DTPL cards. The capability has built-in intelligence to filter out artificial stress concentrations around point loads and point boundary conditions. Stress concentrations due to boundary geometry are also filtered to some extent as they can be improved more effectively with local shape optimization. Due to the large number of elements with active stress constraints, no element stress report is given in the table of retained constraints in the .out file. The iterative history of the stress state of the model can be viewed in HyperView or HyperMesh. Stress constraints do not apply to 1-D elements. Stress constraints may not be used when enforced displacements are present in the model. Bead Discreteness Fraction This is a global response for topography design domains. This response indicates the amount of shape variation for one or more topography design domains. The response varies in the range 0.0 to 1.0 (0.0 < BEADFRAC < 1.0), where 0.0 indicates that no shape variation has occurred, and 1.0 indicates that the entire topography design domain has assumed the maximum allowed shape variation.

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Subcase Dependent Linear Static Analysis Static Compliance The compliance C is calculated using the following relationship:

C

1

C

1

2

uT f

2

uT Ku

with Ku

f

or

1

T 2

dv

The compliance is the strain energy of the structure and can be considered a reciprocal measure for the stiffness of the structure. It can be defined for the whole structure, for individual properties (components) and materials, or for groups of properties (components) and materials. The compliance must be assigned to a linear static subcase (loadstep, load case). In order to constrain the compliance for a region containing a number of properties (components), the SUM function can be used to sum the compliance of the selected properties (components), otherwise, the constraint is assumed to apply to each individual property (component) within the region. Alternatively, a DRESP2 equation needs to be defined to sum the compliance of these properties (components). This can be avoided by having all properties (components) use the same material and applying the compliance constraint to that material. Static Displacement Displacements are the result of a linear static analysis. Nodal displacements can be selected as a response. They can be selected as vector components or as absolute measures. They must be assigned to a linear static subcase. Static Stress of Homogeneous Material Different stress types can be defined as responses. They are defined for components, properties, or elements. Element stresses are used, and constraint screening is applied. It is also not possible to define static stress constraints in a topology design space (see above). This is a linear static subcase (loadstep, load case) related response. Static Strain of Homogeneous Material Different strain types can be defined as responses. They are defined for components, properties, or elements. Element strains are used, and constraint screening is applied. It is also not possible to define strain constraints in a topology design space. This is a linear static subcase (loadstep, load case) related response. Static Stress of Composite Lay-up Different composite stress types can be defined as responses. They are defined for PCOMP(G) components or elements, or PLY type properties. Ply level results are used, and constraint screening is applied. It is also not possible to define composite stress constraints in a topology design space. This is a linear static subcase (loadstep, load case) related response.

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Static Strain of Composite Lay-up Different composite strain types can be defined as responses. They are defined for PCOMP(G) components or elements, or PLY type properties. Ply level results are used, and constraint screening is applied. It is also not possible to define composite strain constraints in a topology design space. This is a linear static subcase (loadstep, load case) related response. Static Failure in a Composite Lay-up Different composite failure criterion can be defined as responses. They are defined for PCOMP(G) components or elements, or PLY type properties. Ply level results are used, and constraint screening is applied. It is also not possible to define composite failure criterion constraints in a topology design space. This is a linear static subcase (loadstep, load case) related response. Static Force Different force types can be defined as responses. They are defined for components, properties, or elements. Constraint screening is applied. It is also not possible to define force constraints in a topology design space. This is a linear static subcase (loadstep, load case) related response. Single Point Force at a constrained grid point This response can be defined using the DRESP1 bulk data entry (with RTYPE=SPCFORCE). This response is defined for constrained grid points. Constraint screening is applied to this response. This is a linear static subcase (loadstep, load case) related response. Grid Point Force This response can be defined using the DRESP1 bulk data entry (with RTYPE=GPFORCE). This response defines the contribution to a specific grid point force component from a nonrigid element (which is connected to that grid). Constraint screening is applied to this response. If ATTi specify multiple elements, then multiple responses will be generated, where, each response calculates a specified element’s contribution to the grid point force component at the specified grid. This is a linear static subcase (loadstep, load case) related response.

Linear Heat Transfer Analysis Temperature Temperatures are the result of a heat transfer analysis, and must be assigned to a heat transfer subcase (loadstep, load case). Temperature response cannot be used in composite topology or free-size optimization.

Normal Modes Analysis Frequency Natural frequencies are the result of a normal modes analysis, and must be assigned to the normal modes subcase (loadstep, load case). It is recommended to constrain the frequency for several of the lower modes, not just of the first mode.

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Mode Shape Mode shapes are the result of a normal modes analysis. Mode shapes can be selected as a response. They can be selected as vector components or as absolute measures. They must be assigned to a normal modes subcase.

Linear Buckling Analysis Buckling Factor The buckling factor is the result of a buckling analysis, and must be assigned to a buckling subcase (loadstep, load case). A typical buckling constraint is a lower bound of 1.0, indicating that the structure is not to buckle with the given static load. It is recommended to constrain the buckling factor for several of the lower modes, not just of the first mode.

Frequency Response Function (FRF Analysis) Frequency Response Displacement Displacements are the result of a frequency response analysis. Nodal displacements, i.e. translational, rotational and normal*, can be selected as a response. They can be selected as vector components in real/imaginary or magnitude/phase form. They must be assigned to a frequency response subcase (loadstep, load case). *The normal at a grid point is calculated based on the normals of the surrounding elements. The normal frequency response displacement at a grid point can be selected as a response and it is the displacement in the normal’s direction. The normals are also updated when shape changes occur during shape optimization. Frequency Response Velocity Velocities are the result of a frequency response analysis. Nodal velocities, i.e. translational, rotational and normal, can be selected as a response. They can be selected as vector components in real/imaginary or magnitude/phase form. They must be assigned to a frequency response subcase (loadstep, load case). *The normal at a grid point is calculated based on the normals of the surrounding elements. The normal frequency response velocity at a grid point can be selected as a response and it is the velocity in the normal’s direction. The normals are also updated when shape changes occur during shape optimization. Frequency Response Acceleration Accelerations are the result of a frequency response analysis. Nodal accelerations, i.e. translational, rotational and normal, can be selected as a response. They can be selected as vector components in real/imaginary or magnitude/phase form. They must be assigned to a frequency response subcase (loadstep, load case). *The normal at a grid point is calculated based on the normals of the surrounding elements. The normal frequency response acceleration at a grid point can be selected as a response and it is the acceleration in the normal’s direction. The normals are also updated when shape changes occur during shape optimization.

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Frequency Response Stress Different stress types can be defined as responses. They are defined for components, properties, or elements. Element stresses are not used in real/imaginary or magnitude/ phase form, and constraint screening is applied. The von Mises stress for solids and shells can also be defined as direct responses. It is not possible to define stress constraints in a topology design space. This is a frequency response subcase (loadstep, load case) related response. Frequency Response Strain Different strain types can be defined as responses. They are defined for components, properties, or elements. Element strains are used in real/imaginary or magnitude/phase form, and constraint screening is applied. The von Mises strain for solids and shells can also be defined as direct responses. It is not possible to define strain constraints in a topology design space. This is a frequency response subcase (loadstep, load case) related response. Frequency Response Force Different force types can be defined as responses. They are defined for components, properties, or elements in real/imaginary or magnitude/phase form. Constraint screening is applied. It is also not possible to define force constraints in a topology design space. This is a frequency response subcase (loadstep, load case) related response.

Random Response Analysis PSD and RMS Responses PSD displacement, PSD velocity, PSD acceleration, PSD acoustic pressure, PSD stress, PSD strain, RMS displacement, RMS velocity, RMS acceleration, RMS acoustic pressure, RMS stress and RMS strain responses are available.

Coupled FRF Analysis on a Fluid-structure Model (Acoustic Analysis) Acoustic Pressure Acoustic pressures are the result of a coupled frequency response analysis on a fluidstructure model. This response is available for fluid grids. It must be assigned to a coupled frequency response subcase (loadstep, load case) on a fluid-structure model.

Multi-body Dynamics Analysis Flexible Body Responses For Multi-body Dynamics problems, the Mass, Center of gravity, and Moment of Inertia of one or more flexible bodies are available as responses. This is in addition to other usual structural responses.

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MBD Displacement MBD displacements are the result of a multi-body dynamics analysis. They must be assigned to a multi-body dynamics subcase (loadstep, load case). MBD Velocity MBD velocities are the result of a multi-body dynamics analysis. They must be assigned to a multi-body dynamics subcase (loadstep, load case). MBD Acceleration MBD acceleration are the result of a multi-body dynamics analysis. They must be assigned to a multi-body dynamics subcase (loadstep, load case). MBD Force MBD forces are the result of a multi-body dynamics analysis. They must be assigned to a multi-body dynamics subcase (loadstep, load case). MBD Expression MBD expression responses are the result of a multi-body dynamics analysis. They are the result of the evaluation of an expression. They must be assigned to a multi-body dynamics subcase (loadstep, load case).

Fatigue Life/Damage Life and Damage are results of a fatigue analysis. They must be assigned to a Fatigue subcase.

Dynamic/Nonlinear Analysis Equivalent Plastic Strain Equivalent plastic strain can be used as an internal response when a nonlinear response optimization is run using the equivalent static load method. This is made possible through the use of an approximated correlation between linear strain and plastic strain, which are calculated in the inner and outer loops respectively, of the ESL method.

User Responses Function A function response is one that uses a mathematical expression to combine design variables, grid point locations, responses, and/or table entries. Whether the function is subcase (loadstep, load case) related or global, is dependent on the response types used in the equation.

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External An external response is one that uses an external user-defined routine to combine design variables, grid point locations, eigenvectors, responses, and/or table entries. Whether the function is subcase (loadstep, load case) related or global is dependent on the response types used in the routine. Refer to External Responses below for more information.

External Responses The DRESP3 bulk data entry, in combination with the LOADLIB I/O option entry, allows for the definition of responses through user-defined external functions. The external functions may be written in HyperMath Language (HML), FORTRAN, C or a Microsoft Excel workbook. The resulting libraries and files should be accessible by OptiStruct regardless of the coding language, providing that consistent function prototyping is respected, and adequate compiling and linking options are used.

Writing External Functions The OptiStruct installation provides "barebone" functions for FORTRAN (dresp3_barebone.F) and for C (dresp3_barebone.c) with proper function definition, arguments, and compilation directives. These files can be used as starting points to write your own functions. Refer to Referencing External Files for information on response definition through a user-defined Microsoft Excel workbook. HyperMath Language (HML) functions are defined as follows: integer function myfunct(iparam, rparam, iresp,

rresp, userdata)

If sensitivities need to be requested, then the following external function can be used. integer function myfunct(iparam, rparam, iresp, userdata)

rresp, dresp, isens

FORTRAN functions are defined as follows: integer function myfunct(iparam, rparam, nparam, iresp, rresp, nresp, userdata)

If sensitivities need to be requested, then the following external function can be used. integer function myfunct(iparam, rparam, nparam, iresp, rresp, dresp, nresp, isens userdata) character*32000 userdata integer nparam, nresp integer iparam(nparam), iresp(nresp) double precision param(nparam), rresp(nresp), dresp(nparam,nresp)

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C functions are defined as follows: int myfunct(int* iparam, double* rparam, int* nparam, int* iresp, double* rresp, int* nresp, char* userdata)

If sensitivities need to be requested, then the following external function can be used. int myfunct(int* iparam, double* rparam, int* nparam, int* iresp, double* rresp, double* dresp, int* nresp, int* isens, char* userdata) Note that the functions' arguments are identical in both languages so as to preserve compatibility. However, since FORTRAN always passes arguments by address, it is important to understand that external C functions receive pointers instead of variables. In order to ensure portability, the following must be adhered to: Function names should be written using either all lower-case or all upper-case characters. Only alphanumeric characters should be used. Underscore characters are prohibited. Names cannot be longer than eight characters. Regarding implementing of external, user-defined routines using HyperMath, refer to the online documentation for writing scripts in HyperMath. HyperMath is supported on the Windows and Linux operating systems only. Function Return Values External functions should return 0 or 1 for successful completion, where 1 indicates that a user-defined information message should be output by OptiStruct. External functions should return -1 in case of fatal error, in which case OptiStruct will terminate after outputting a user-defined error message. See below for more information about error and information messages. Function Arguments The following table briefly describes the arguments which are passed from OptiStruct to the external functions. Input / Output

Argument

Type

iparam

integer (table)

Input

Input parameters types (optional use)

rparam

double (table)

Input

Input parameters values

nparam

integer

Input

Number of parameters

iresp

integer (table)

Input

Output responses requests (optional use)

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Input / Output

Description

double (table)

Output

Output responses values

dresp

double (table)

Output

Output sensitivity values

nresp

integer

Input

Number of responses

isens

integer

Input

Sensitivity output flag

userdata

string

Input / Output

User data / Error or information message

Argument

Type

rresp

Parameters: nparam is the number of input parameters that were defined on the DRESP3 card. rparam(nparam) contains the values of the input parameters as evaluated by OptiStruct. iparam(nparam) indicates the types of the input parameters as described below. Parameter values are passed in the exact order in which they were defined on the DRESP3 card, regardless of their type. Using the parameter types table is optional, for instance to perform verifications or code-branching. The following types are currently supported: Parameter type iparam value DESVAR

1

DTABLE

2

DGRID/DGRIDB

3

DRESP1

4

DRESP2

5

DRESP1L

6

DRESP2L

7

DVPREL1

8

DVPREL2

9

DVMREL1

10

DVMREL2

11

DVCREL1

12

DVCREL2

13

DVMBRL1

14

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Parameter type iparam value DVMBRL2

15

DEIGV

16

DGRIDB

18

Responses: nresp is the maximum number of responses which the function is able to compute, as defined on the MAXRESP field of the DRESP3 card. rresp(nresp) returns the values of the responses as evaluated by the external function. iresp(nresp) contains the responses requests as described below. The responses requests table indicates which of the available responses are actually needed by OptiStruct. Entries in iresp(nresp) are flagged as 1 for requested responses and as 0 otherwise. Using that information is optional, and allows for saving computational effort by not evaluating responses which OptiStruct does not need. Userdata String Upon entering the function, the userdata string contains data as defined in the USRDATA field of the DRESP3 card. It provides a convenient mechanism to pass constants or any other relevant information to the function. There are no restrictions regarding the contents of the string, but its length must be 32,000 characters at most. Upon exiting the function, the string may contain a user-defined error or information message. The updated string is then returned to OptiStruct, where it is printed to the standard output (.out file and/or screen). Here again, the contents of the string are not restricted as long as its length does not exceed 32,000 characters. The error or information messages may be formatted by using the character "|" as a linebreak indicator. Standard C escape sequences are supported as well. It is advised, but not necessary, to format messages in such a way that each line does not exceed 80 characters, since the same convention is used in OptiStruct's output files. Sensitivity Flag isens indicates whether sensitivities are requested in the code. It is recommended to skip the calculation of sensitivities when isens is turned off. This will avoid unnecessary computations.

Building External Libraries Windows Systems with Microsoft Developer Studio Creating dynamic libraries under Windows with Microsoft Developer Studio is an extremely easy task. Simply start a new project and select either "FORTRAN Dynamic Link Library" for FORTRAN, or "Win32 Dynamic-Link Library" for C. For FORTRAN libraries, you need to change the argument passing conventions in the

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project settings. Under the "FORTRAN" tab, select the category "External Procedures" and then change the "Argument Passing Conventions" to "C, By Reference". On Windows systems, %PATH% must be set correctly to ensure that the right compiler DLLs are picked up at runtime. UNIX Systems Under UNIX, the general syntax to build a shared library starting from a FORTRAN or C file is: FC [options] -c myfile.F -o myfile.o

(for FORTRAN)

CC [options] -c myfile.c -o myfile.o

(for C)

LD [options] myfile.o -o mylib.so where, FC refers to the FORTRAN compiler (for instance f77), CC refers to the C compiler (for instance cc or gcc), and LD refers to the linker (for instance ld) installed on your computer. Refer to your system's manuals for more information. The compiler and linker options provide information about the platform you are building the library for. The linker options also specify that you are building a shared library. Other options, such as code optimization parameters, are left to your discretion and should not usually affect the compatibility with OptiStruct. The following table defines options for each of OptiStruct's release platforms, which have been verified to work correctly on various systems. Keep in mind that these options might change depending on the compilers and linker installed on your computer, so refer to your operating system manual for further information. In most cases GNU compilers can be used in place of Intel compilers. Use the appropriate compiler linker options to create a shared library with the compiler of your choice. The compilers and versions in the following table are the ones used to build OptiStruct. FORTRAN Compiler Version

FORTRAN Compiler Options

Win32

Intel FORTRAN 12.1

Win64

Macosx64

Platform

C Compiler Version

C Compiler Options

Linker Options

/iface:default / libs:dll /threads

Intel C++ 12.1

/MD

/LD

Intel FORTRAN 12.1

/iface:default / libs:dll /threads

Intel C++ 12.1

/MD

Intel FORTRAN 10.1.006

-fPIC

Intel C++ 10.1.006

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Platform

Linux64

FORTRAN Compiler Version

FORTRAN Compiler Options

Intel FORTRAN 12.1

–fPIC

C Compiler Version

C Compiler Options

Linker Options

Intel C++ 12.1

-fPIC

-shared

(If Compaq Visual Fortran is used to build the external response functions called by DRESP3, the following compiler directive is required to export the functions appropriately: For a function – “myfunc” integer function myfunc (iparam, rparam, nparam, iresp, rresp, dresp, nresp, isens, userdata) cDEC$ ATTRIBUTES DLLEXPORT, C, REFERENCE :: myfunc Compiler and linker options for Compaq Visual Fortran, similar to those given in the above table, will be required to build and use multithreaded dynamic runtime libraries). Once your library has been built, you can verify that the functions have been exported correctly by using nm mylib.so on UNIX systems and dumpbin /exports mylib.dll on Windows systems with Microsoft Developer Studio. This command will display the list of symbols found in the library, among which you should recognize the function(s) which you have written. Note that some FORTRAN compilers convert function names to lower-case or upper-case symbols, and some compilers also append an underscore to these names. However, in your input decks, you do not have to worry about the exact symbol name. Simply use the function name as it is defined in your code, and OptiStruct will automatically locate the appropriate symbol.

Using External Libraries All files referenced here are located in the HyperWorks installation directory under /demos/os/manual/. To locate the HyperWorks installation directory, , use the following approach: From the permanent menu, select the global panel and review the path next to template file: is the portion of the path preceding the templates/ directory on PC, and the hm/ directory on UNIX. HyperMath Example Refer to the OptiStruct tutorial, OS-4095 (Size Optimization using External Responses (DRESP3) through HyperMath), for information on using DRESP3 with HyperMath to implement external, user-defined routines. Simple Example The files dresp3_simple.F and dresp3_simple.c contain source code for simple examples

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of external functions written in FORTRAN and C, respectively. Both functions are named mysum and compute two responses – the sum of the parameters and the averaged sum of the parameters. The input deck dresp3_simple.fem contains an example problem calling both of these external functions. Two LOADLIB cards referring to the FORTRAN and C libraries are defined: LOADLIB DRESP3 LOADLIB DRESP3

FLIB CLIB

dresp3_simple_f.dll dresp3_simple_c.dll

You have created four DRESP3 cards, which are pointing to the FORTRAN and C functions and requesting the first and second responses in each of those functions. Two DRESP1 responses are used as parameters: DRESP3 + DRESP3 + DRESP3 + DRESP3 +

6 DRESP1 7 DRESP1 8 DRESP1 9 DRESP1

SUMF 2 AVGF 2 SUMC 2 AVGC 2

FLIB 3 FLIB 3 CLIB 3 CLIB 3

MYSUM

1

2

MYSUM

2

2

MYSUM

1

2

MYSUM

2

2

For verification purposes, you have also defined two DRESP2 cards that are pointing to two simple equations which evaluate the sum and the averaged sum of their parameters: DEQATN DEQATN

1 2

DRESP2 + DRESP2 +

4 DRESP1 5 DRESP1

F(x,y) = x+y F(x,y) = avg(x,y) SUME 2 AVGE 2

1 3 2 3

Running this input deck through OptiStruct shows that the FORTRAN external functions, the C external functions and the internal equations always return the same values, and are updated simultaneously throughout the optimization process. Advanced Example The file dresp3_advanced.F contains the FORTRAN source code of the second example, in which you are making use of advanced features of the DRESP3 functionality. The external function is able to compute the von Mises and maximum principal stresses (strains) of an element based on its stress (strains) components. Either 3 or 6 components can be passed as parameters – 3 components for a shell element and 6 components for a solid element. The following features are used: The USRDATA string is parsed to determine whether stresses or strains are requested, and an error message is returned otherwise. The number of parameters is used to determine whether a shell or solid element is treated, and an error message is returned if that number is not equal to 3 or 6. An error message is returned if the parameters are not of type DRESP1 or DRESP1L, since stress or strain components are expected. Even though the function is able to compute two different responses, only the response(s) actually requested by OptiStruct are computed when the function is

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called. An information message is returned indicating which responses were evaluated. The input deck dresp3_advanced.fem gives a simple example of problem making use of this external function, for analysis only. The DRESP1 responses 10-12 and 13-18 correspond to the stress components of a 2-D and a 3-D element, respectively. The DRESP1 responses 20-23 evaluate the von Mises stress and the maximum principal stress of the same two elements: DRESP1 DRESP1 DRESP1 DRESP1 DRESP1 DRESP1 DRESP1 DRESP1 DRESP1

10 11 12 13 14 15 16 17 18

SXX2D SYY2D SXY2D SXX3D SYY3D SZZ3D SXY3D SXZ3D SYZ3D

STRESS STRESS STRESS STRESS STRESS STRESS STRESS STRESS STRESS

ELEM ELEM ELEM ELEM ELEM ELEM ELEM ELEM ELEM

SX1 SY1 SXY1 SXX SYY SZZ SXY SXZ SYZ

100 100 100 50 50 50 50 50 50

DRESP1 DRESP1 DRESP1 DRESP1

20 21 22 23

SVM2D-1 SMP2D-1 SVM3D-1 SMP3D-1

STRESS STRESS STRESS STRESS

ELEM ELEM ELEM ELEM

SVM1 SMP1 SVM SMP

100 100 50 50

In addition, you have defined DRESP3 cards which compute the same stress results through our external library. You are also using the SLAVE feature to clone the parameters of similar cards: DRESP3 + + DRESP3 + DRESP3 + + DRESP3 +

30 DRESP1 USRDATA 31 SLAVE 32 DRESP1 USRDATA 33 SLAVE

SVM2D-3 10 STRESS SMP2D-3 30 SVM3D-3 13 STRESS SMP3D-3 32

STRLIB 11

GETSTR 12

1

2

STRLIB

GETSTR

2

2

STRLIB 14

GETSTR 15

1 17

2 18

STRLIB

GETSTR

2

2

16

Referencing External Files Microsoft Excel workbooks can be referenced via the LOADLIB entry to define user-defined responses. Both Implicit and Explicit options are available and are defined as follows: Implicit definition This is a simple implementation wherein two columns in an Excel worksheet are used to define the input and output parameters. Column 1 can be used to list input parameters and Column 2 can be used to list output parameters. LOADLIB DRESP3 DRESP3 +

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10 DRESP1

ELIB SUM 5

dresp3_excel.xlsx ELIB 6

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Explicit definition In this advanced implementation the cell input number is specified. Cells for input and output data are listed. LOADLIB DRESP3 DRESP3 + + + + +

20 DRESP1 DESVAR CELLIN CELLIN CELLOUT

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ELIB FUNC 5 1 B3 C10 E10

dresp3_excel.xlsx ELIB 6

MYFUNC 7

THRU

B6

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Topology Optimization Topology Optimization is a mathematical technique that produces an optimized shape and material distribution for a structure within a given package space. By discretizing the domain into a finite element mesh, OptiStruct calculates material properties for each element. The OptiStruct algorithm alters the material distribution to optimize the user-defined objective under given constraints. Convergence occurs in line with the description provided on the Iterative Solution page. The following responses (see Responses for a description) are currently available as the objective or as constraint functions: Mass

Volume

Volume or Mass Fraction

Center of Gravity

Moment of Inertia

Static Compliance

Static Displacement

Natural Frequency von Mises Stress on Entire Model (only as constraint)

Buckling Factor (special Frequency case) Response Displacement, Velocity, Acceleration

Temperature

Weighted Compliance

Combined Compliance Index

Weighted Frequency

Function The von Mises stress constraints may be defined for topology and free-size optimization through the STRESS optional continuation line on the DTPL or the DSIZE card. There are a number of restrictions with this constraint: The definition of stress constraints is limited to a single von Mises permissible stress. The phenomenon of singular topology is pronounced when different materials with different permissible stresses exist in a structure. Singular topology refers to the problem associated with the conditional nature of stress constraints, i.e. the stress constraint of an element disappears when the element vanishes. This creates another problem in that a huge number of reduced problems exist with solutions that cannot usually be found by a gradient-based optimizer in the full design space. Stress constraints for a partial domain of the structure are not allowed because they often create an ill-posed optimization problem since elimination of the partial domain would remove all stress constraints. Consequently, the stress constraint applies to the entire model when active, including both design and non-design regions, and stress constraint settings must be identical for all DSIZE and DTPL cards.

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The capability has built-in intelligence to filter out artificial stress concentrations around point loads and point boundary conditions. Stress concentrations due to boundary geometry are also filtered to some extent as they can be improved more effectively with local shape optimization. Due to the large number of elements with active stress constraints, no element stress report is given in the table of retained constraints in the .out file. The iterative history of the stress state of the model can be viewed in HyperView or HyperMesh. Stress constraints do not apply to 1-D elements. Stress constraints may not be used when enforced displacements are present in the model. The buckling factor can be constrained for shell topology optimization problems with a base thickness not equal to zero. Constraints on the buckling factor are not allowed in any other cases of topology optimization. The following responses are currently available as the objective or as constraint functions for elements that do not form part of the design space: Static Stress

Static Strain

Static Force

Composite Stress

Composite Strain

Composite Failure Criterion

Frequency Response Stress

Frequency Response Strain

Frequency Response Force

If an element is in the topology design region, its individual stress/strain or force criterion value cannot be constrained.

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Generating and Evaluating a Design using Topology Optimization The following example illustrates how OptiStruct is used to generate a design for a control arm and how engineering analysis is used to evaluate the design. 1. The package space for the control arm is filled with a finite element mesh. 2. Parts of the mesh are designated as nondesign, and the elements that make up these areas are placed in a nondesign component. The darker elements represent attachment points for the frame, shock, spring seat, stabilizer bar, and spindle. Nondesign elements are placed where loads and constraints are applied to the model.

Nondesign and design space of FEM model.

3. Loads and constraints are applied to the finite element model. Three load cases are applied at the spindle and stabilizer bar attachment points. Constraints are applied at the frame connections and shock point. The model and parameters are submitted to OptiStruct for topology optimization. 4. Elements with material densities below 60% are masked out.

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Density plot of control arm with elements below 60% material density removed from the display.

5. A finite element model of the control arm using the suggested layout as a guide is generated.

Finite element model of control arm design based on OptiStruct results.

6. Stress analysis is performed on the model using the loads and boundary conditions from the topology optimization run.

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Stress contour plot of control arm model during braking load.

7. The performance of the part is evaluated. Subsequent size and shape optimization is performed to minimize the mass while meeting stress and deflection criteria.

Final design with shape-optimized structural member.

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Design Variables for Topology Optimization OptiStruct solves topological optimization problems using the density method, also known as the SIMP method in the research community. Under topology optimization, the material density of each element should take a value of either 0 or 1, defining the element as being either void or solid, respectively. Unfortunately, optimization of a large number of discrete variables is computationally prohibitive. Therefore, representation of the material distribution problem in terms of continuous variables has to be used. With the density method, the material density of each element is directly used as the design variable, and varies continuously between 0 and 1; these represent the state of void and solid, respectively. Intermediate values of density represent fictitious material. The stiffness of the material is assumed to be linearly dependent on the density. This material formulation is consistent with our understanding of common materials. For example, steel, which is denser than aluminum, is stronger than aluminum. Following this logic, the representation of fictitious material at intermediate densities does reflect engineering intuitions. In general, the optimal solution of problems involves large gray areas of intermediate densities in the structural domain. Such solutions are not meaningful when you are looking for the topology of a given material, and not meaningful when considering the use of different materials within the design space. Therefore, techniques need to be introduced to penalize intermediate densities and to force the final design to be represented by densities of 0 or 1 for each element. The penalization technique used is the "power law representation of elasticity properties," which can be expressed for any solid 3-D or 2-D element as follows:

K

pK

Where, K and K represent the penalized and the real stiffness matrix of an element, respectively, is the density and p is the penalization factor which is always greater than 1. In OptiStruct, the DISCRETE parameter corresponds to (p - 1). DISCRETE can be defined on the DOPTPRM bulk data entry. p usually takes a value between 2.0 and 4.0. For example, compared to the non-penalized formulation (which is equivalent to p=1) at =0.3, p=2 reduces the stiffness of the element from 0.3 to 0.09 times the stiffness of the fully dense element. The default DISCRETE is 1.0 for shell dominant structures, and 2.0 for solids dominant structures (the dominance is defined by the proportion of number of elements). An additional parameter, DISCRT1D, can also be defined on the DOPTPRM bulk data entry. DISCRT1D allows 1-D elements to use a different penalization to 2-D or 3-D elements. When minimum member size control is used, the penalty starts at 2 and is increased to 3 for the second and third iterative phases. This is done in order to achieve a more discrete solution. For other manufacturing constraints such as draw direction, extrustion, pattern repetition, and pattern grouping, the penalty starts at 2 and increases to 3 and 4 for the second and third iterative phases, respectively. Obviously, due to the existence of semidense elements, the analysis results may change dramatically when the design process enters a new phase using a different penalization factor. Three types of finite elements can be defined as topology design elements in OptiStruct: Solid elements, shell elements, and 1-D elements (including ROD, BAR/BEAM, BUSH, and WELD elements).

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Design Elements Solid Elements The SIMP method (Solid Isotropic Material with Penalty) is used in OptiStruct. In the SIMP method, a pseudo material density is the design variable, and hence it is often called density method as well. The material density varies continuously between 0 and 1, with 0 representing void state and 1 solid state. The SIMP method applies a power-law penalization for stiffness-density relationship in order to push density toward 0/1 (void/ solid) distribution: pK

K Where,

K is the penalized stiffness matrix of an element. K is the real stiffness matrix of an element. is the density.

p is the penalization factor (Always greater than 1, with default penalty at 3.0 if no manufacturing constraints are applied).

Shell Elements The SIMP method (Solid Isotropic Material with Penalty) is used in OptiStruct. In the SIMP method, a pseudo material density is the design variable, and hence it is often called density method as well. The material density varies continuously between 0 and 1., with 0 representing void state and 1 solid state. The SIMP method applies a power-law penalization for stiffness-density relationship in order to push density toward 0/1 (void/ solid) distribution. pK

K Where,

K is the penalized stiffness matrix of an element. K is the real stiffness matrix of an element. is the density.

p is the penalization factor (Always greater than 1) For isotropic material a non-zero base plate thickness can be defined. For a composite plate or a plate with anisotropic material, the base plate thickness must be zero (the limitation of the current development). Topology optimization of composites has certain unique characteristics and is discussed in Composite Topology and Free-size Optimization.

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1-D Elements Only the density method is implemented for topology optimization of 1-D elements. Currently available elements include ROD, BAR/BEAM, BUSH, and WELD elements. Each element is controlled by a single design variable that is the material density of this element that varies between 0 (numerically a small value is used) and 1.0. In essence, 0 represents nonexistence and 1.0 represents full existence of the corresponding element. The following power law representation of elastic properties is used to penalize intermediate density:

K

pK

Where, K and K represent the penalized and the real stiffness matrix of an element, respectively, p is the penalization factor which is always bigger than 1. The penalty is controlled by the DISCRETE or DISCRT1D parameters, the value of these parameters correspond to (p - 1).

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Topology Optimization (Level Set Method) A new topology optimization algorithm based on the level set method was implemented in OptiStruct 12.0. In the level set method, the boundary of the design is implicitly represented as the isosurface (the zero level set) of a function ( x)defined on the finite element mesh, as shown in Figure 1. In the level set model, the domain is defined based on the value of the level set function:

( x) 0 : x / ( x) 0 : x ( x) 0 : x D / Where, D denotes the design domain; represents the material region, stands for the boundary, and D/ denotes the region with no material. The dynamic motion of the boundary is governed by the so-called, level set equation:

t

Vn

Where, Vn is the normal velocity and is the norm of the gradient of the level set function. The basic idea of the level set equation is to map the boundary evolution into an evolution of the level set function ( x).

Figure 1: A 2-D Design and its corresponding Level Set representation

Level-set based topology optimization can be considered as advanced shape optimization. It works in a way like conventional shape optimization, where the design is changed by moving the boundary, while at the same time topological changes such as boundary emerging and splitting can be handled naturally.

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Composite Topology and Free-size Optimization Input Definition For composite structures, topology and free-size optimization are defined through the DTPL and DSIZE bulk data entries, respectively. Both are supported in the HyperMesh optimization panel. Features available include: minimum member size control, symmetry, pattern grouping and pattern repetition. Stress or failure constraints are not supported at this stage. Prior to OptiStruct 8.0, composite topology optimization was based on the notion that the homogenized properties of an element remain unchanged. This construct does not allow the freedom for material redefinition. However, if this is indeed a preferred assumption, the HOMO option can be set on the MAT line of the DTPL card. Otherwise, an individual plybased formulation (discussed below) will be the default option. Topology and free-size methods target a system level composite design where laminate family definition is the objective. Therefore, the PCOMP model should not reflect a detailed stacking of plies of the same orientation. For example, even though 10 layers of 0 degree graphite cloth might be separated in the stacking of the final structure, the modeling for a concept study using topology and free-size should group them together in one ply in the PCOMP so that the optimal total thickness distribution of a 0 degree ply is optimized throughout the structure. Involving both topology and free-size in the same optimization problem is not recommended since the penalization on topology components creates a bias that could lead to sub-optimal solutions.

Problem Formulation For a composite shell element (shown in the figure below), the thickness ti of each ply is a variable between 0 and Ti defined on the PCOMP card.

Composite element

The only difference between topology and free-size here is that the former targets a discrete final solution of 0 (or Ti) for ti, while free-size allows ti to vary freely between 0 and Ti. The discrete solution is achieved by penalizing intermediate thickness. Most general characteristics of regular shell topology and free-size optimization also apply to composite. It is recommended that you become familiar with free-size before proceeding. The major differences between topology optimization and free-size can be illustrated through a simple example.

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Example: Cantilever Plate The cantilever plate is shown in the following figure. A symmetric lay-up of (0, +45, -45, and 90) degree plies are used. The optimization problem is stated as: Minimize Compliance Subject to Volume fraction < 0.3

Composite cantilever plate

For topology optimization, the thickness distribution of individual plies in the final design is shown in the following figure.

Topology result – thickness of individual plies

The total thickness of the laminate is shown next.

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Topology result – total thickness of the laminate

It can be seen that a rather discrete thickness for each ply is obtained. Note that while little overlapping of different orientations is shown in this result, it should be expected that overlapping of plies of different angles might be more pronounced when multiple load cases exist. The thickness distribution of free-size optimization for this example is shown below.

Free-size result – thickness of individual plies

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Free-size result – total thickness of the laminate

As expected, free-size created a design in which variable ply thickness appears in a large area of the structure. The compliance of both designs are compared in the figure below. It is not surprising to see that the free-size design outperforms the topology design in terms of compliance since a continuous variation of thickness offers more design freedom.

Compliance of topology and free-size results

While ply angles are not variables for topology and free-size optimization, thickness optimization of plies indirectly leads to a discrete optimization of angles. The available angles in the PCOMP can be interpreted as discrete angle variables. Also, while free-size often creates variable thickness distribution without extensive cavity, it does not prevent cavity if the optimizer demands it. For this example, you can see cavity in the free-size results in the 45 degree region, adjacent to the support, and in the upper and lower corners of the free end.

Comparing Design Characteristics of Topology and Free-size Most of the characteristics for general shell discussed in free-size section also apply to composite structures. One important exception is that the manufacturing cost is no longer a restrictive factor for composites. The reason for this is that a composite structure is manufactured by laying very thin plies of fiber cloth over each other and binding them with a matrix material, like epoxy resin. Therefore, an almost continuous change of laminate thickness can be achieved seamlessly by dropping/adding plies freely.

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Characteristics of Composite Topology vs. Free-size Composite Topology

Composite Free-size

Angle optimized indirectly.

Angle optimized indirectly.

Goal – 0/Ti discrete thickness of individual plies

Goal – variable thickness of individual plies -> "Free" under upper bound Ti

-> Restricted freedom Results – Truss-like design concepts.

Variable thickness panel likely for in-plane loading, 0/1 thickness likely when bending is dominant.

Not useful compared to Freesize?

Always better design?

Manufacture – not a factor for composite unless premanufactured laminate is used.

Manufacture – naturally achieved with no additional cost.

Concentrated full thick members are stronger against out of plane buckling.

Spread thin shell could be prone to buckling.

Functionality may need holes for Cavity is controlled by optimality, other non-structural components or and is usually not extensive under for passing lines/pipes. in-plane loading.

Interpreting Topology and Free-size Results Interpretation of topology results is rather straight-forward. For free-size, the change in thickness of individual plies provides insight for ply dropping/adding zones. The thickness of each ply in each individual zone can then be defined as a design variable in a detailed size optimization. At this stage, discrete variables can be used to reflect the discrete nature of ply thickness change. Overlapping all zones of individual plies can than help to generate PCOMP zones, where a ply traveling through different zones can be defined using PCOMPG.

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Free-size Optimization Input Definition Free-size optimization is defined through the DSIZE bulk data entry that is supported in the HyperMesh Optimization panel. Features available for free-size include: minimum member size control, symmetry, pattern grouping and pattern repetition, and stress constraints applied to von Mises stresses of the entire structure. Involving both topology and free-size in the same optimization problem is not recommended since penalization on topology components creates a bias that could lead to sub-optimal solutions.

Problem Formulation For a shell cross-section (shown below), free-size optimization allows thickness t to vary freely between T and T0 for each element; this is in contrast to topology optimization which targets a discrete thickness of either T or T0. The differences of topology optimization and free-size can be illustrated through a simple example.

Shell cross-section

Example: Cantilever Plate The cantilever plate is shown in the following figure. Base-plate thickness T0 is zero. The optimization problem is stated as: Minimize Compliance Subject to Volume fraction < 0.3

Cantilever plate

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The next figure shows the final results of topology and free-size optimization as performed on this plate, side by side. As expected, the topology result created a design with 70% cavity, while the free-size optimization arrived at a result with a zone of variable thickness panel.

Topology result

Free-size result

The compliance of both designs are compared in the following figure.

It is not surprising to see that the free-size design outperforms the topology design in terms of compliance since continuous variation of thickness offers more design freedom. It should be emphasized that free-size offers a concept design tool alternative to topology optimization for structures modeled with 2-D elements. It does not replace a detailed size optimization that would fine tune the size parameters of an FEA model of the final product. To illustrate the close relationship between free-size and topology formulation, consider a 3-D model of the same cantilever plate shown previously. The thickness of the plate is modeled in 10 layers of 3-D elements.

Cantilever plate – 3-D model

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3-D topology result

The topology design of the 3-D model shown above looks similar to the free-size results shown previously. This should not be surprising because when the plate is modeled in 3-D, a variable thickness distribution becomes possible under the topology formulation that seeks a discrete density value of either 0 or 1 for each element. If infinitely fine 3-D elements are used, a continuous variable thickness of the plate can be achieved via topology optimization. The motivation for the introduction of free-size is based on the conviction that limitations due to 2-D modeling should not become a barrier for optimization formulation. In regards to the 3-D modeling of shell, topology optimization is equivalent to the application of extrusion constraint(s) in the thickness direction of a 3-D modeled shell. It is important to point out that while free-size often creates variable thickness shells without extensive cavity, it does not prevent cavity if the optimizer demands it. For the example already shown, you can see cavity in the free-size result in the 45 degree region, adjacent to the support, and in the upper and lower corners of the free end. If a plate is predominantly under a bending load, free-size design can converge to a discrete 0/1 thickness distribution similar, or even identical to, the result of a topology optimization. The reason is that bending stiffness is a function of t3 and, therefore, maximum thickness is heavily favored. In other words, intermediate thickness is naturally penalized for bending performance. In the following figure, the free-size result of a plate under bending clearly demonstrates this behavior.

Free-size result of a plate under bending

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Stress Constraints for Free-size Optimization The von Mises stress constraints may be defined for topology and free-size optimization through the STRESS optional continuation line on the DTPL or the DSIZE card. There are a number of restrictions with this constraint: The definition of stress constraints is limited to a single von Mises permissible stress. The phenomenon of singular topology is pronounced when different materials with different permissible stresses exist in a structure. Singular topology refers to the problem associated with the conditional nature of stress constraints, i.e. the stress constraint of an element disappears when the element vanishes. This creates another problem in that a huge number of reduced problems exist with solutions that cannot usually be found by a gradient-based optimizer in the full design space. Stress constraints for a partial domain of the structure are not allowed because they often create an ill-posed optimization problem since elimination of the partial domain would remove all stress constraints. Consequently, the stress constraint applies to the entire model when active, including both design and non-design regions, and stress constraint settings must be identical for all DSIZE and DTPL cards. The capability has built-in intelligence to filter out artificial stress concentrations around point loads and point boundary conditions. Stress concentrations due to boundary geometry are also filtered to some extent as they can be improved more effectively with local shape optimization. Due to the large number of elements with active stress constraints, no element stress report is given in the table of retained constraints in the .out file. The iterative history of the stress state of the model can be viewed in HyperView or HyperMesh. Stress constraints do not apply to 1-D elements. Stress constraints may not be used when enforced displacements are present in the model.

Comparing Design Characteristics of Topology and Free-size The differences in the characteristics of topology and free-size are summarized in the following table. It is important to note that while the free-size design concept generally achieves better performance when buckling constraints are ignored, the topology concept could outperform free-size if buckling constraints become the driving criteria during the size and/or shape optimization stage. The reason for this is that topology optimization eliminates intermediate thicknesses, which leads to a more concentrated material distribution and a shell that is stronger against out-of-plane buckling. The performance of topology and freesize is compared in the next practical example. Since it is usually not possible to know what criteria are most critical for a given structure, it is recommended to follow both design concepts until detailed size and shape optimization is complete and can be evaluated. If it is not possible to derive two designs for every structural component, a benchmark of the relative performance of both concepts for every type of commonly evaluated structure should be established so that general guidelines can be used for reference. Manufacturing and functional considerations may favor topology optimization. Two cases in which free-size may not be the best choice from the start include those in which:

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(1) A variable thickness shell is typically far more expensive to manufacture and may not be a viable choice; as with most shell structures of an automobile that are manufactured using standard sheet metal, for example. (2) The functionality of the structure might require extensive cavity in the design; as with an airplane fuselage floor supporting beam which may need a significant amount of cavity to allow for the pass-through of wires, pipes or other equipment.

Characteristics of Shell Topology vs. Free-size Shell Topology Optimization

Free-size

GOAL: 0/1 thickness -> Restricted freedom

GOAL: variable thickness -> "Free" under upper bound T

Results: Truss-like design concepts.

Variable thickness panel likely for in-plane loading, 0/1 thickness likely when bending is dominant.

Equivalent to extrusion constraints when shell is modeled in infinitely fine 3-D elements.

Equivalent to model with infinitely fine 3-D elements.

Not useful compared to freesize?

Always better design?

Manufacturing constraint – punched sheet metal of constant thickness.

Manufacture – expensive and only used in industries less sensitive to cost.

Concentrated full thick members are stronger against out of plane buckling.

Spread thin shell could be prone to buckling.

Functionality may need holes for Cavity is controlled by optimality, other non-structural components or and is usually not extensive under for passing lines/pipes. in-plane loading.

Interpreting Free-size Results In most cases, variable thickness of a shell structure is achieved through step-wise change of thickness. Free-size results provide a different concept about how the zones of different thicknesses should be designed. Detailed size optimization can then be performed to fine tune the final design. This process is illustrated in the following example.

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Example: Supporting Beam of an Airplane Door Structure This example was discussed in a paper by Cervellera, Zhou and Schramm in 2005 (Proceedings of 6th World Congresses of Structural and Multidisciplinary Optimization, Rio de Janeiro, 30 May - 03 June 2005, Brazil). Free-size optimization is applied to improve the traditional beam design consisting of an "I" cross section with circular cut-outs. A model representing the design space of a beam component has been generated, in which a portion of outer skin and vertical frames is included (shown in following image).

Supporting beam of an airplane door structure

The design areas include the upper flange and the web, while the lower flange and the attachment ribs of vertical frames remain unchanged. Free-size optimization allows element thickness to vary between 0.05 mm and 10.0 mm. The design problem is to minimize the mass subject to a beam center deflection of 3 mm. The free-size result is shown on the left in the figure below. This result is interpreted into zones of different thicknesses as shown with the different colors on the right in the figure.

Left: Free-size result; Right: Interpreted zones of constant thickness

For comparison, topology optimization is applied to the same problem for shell thickness of 5 mm in the design area. The result and its interpretation is shown below.

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Left: Topology result; Right: Interpreted zones of constant thicknesses

Detailed size optimization is then carried out for both concepts, allowing all shell thickness to vary between 1.6 mm and 20 mm. The optimization problem is formulated as minimization of the beam mass subject to the following constraints: Maximum deflection of the beam < 3.0 mm. Maximum von Mises stress in the beam design area < 300 MPa. Buckling load factors > 1.0. In order to study the behavior of the design concepts under different design criteria, size optimization is carried out for different permissible deflection constraints (1.5 mm, 2.0 mm, 3 mm, 4.0 mm, and 5.0 mm). The results are summarized with the figure below, in which critical constraints are highlighted in red numbers. The figure also shows the optimum mass of the two concepts with respect to the maximum displacement. Note that the plate design is more efficient than the truss-like concept if high stiffness is required, while it is less efficient if stability and strength requirements dominate the final designs. More details of this example and additional discussions about free-size can be found in the paper by Cervellera, Zhou and Schramm in 2005.

Comparison of results for different deflection constraints

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Topography Optimization Topography optimization is an advanced form of shape optimization in which a design region for a given part is defined and a pattern of shape variable-based reinforcements within that region is generated using OptiStruct. The approach in topography optimization is similar to the approach used in topology optimization, except that shape variables are used rather than density variables. The design region is subdivided into a large number of separate variables whose influence on the structure is calculated and optimized over a series of iterations. The large number of shape variables allows you to create any reinforcement pattern within the design domain instead of being restricted to a few. The following responses (see Responses for a description) are currently available as the objective or as constraint functions: Mass*

Volume*

Center of Gravity

Moment of Inertia

Static Compliance

Static Displacement

Natural Frequency

Buckling Factor

Static Stress, Strain, Forces

Static Composite Stress, Strain, Failure Index

Frequency Response Frequency Displacement, Velocity, Response Stress, Acceleration Strain, Forces

Weighted Compliance

Weighted Frequency


Function

Bead discreteness fraction

Temperature

* Mass and Volume are not recommended for use as objectives or constraints since mass and volume are not very sensitive to design changes in topography optimization.

Design Variables for Topography Optimization OptiStruct solves topography optimization problems using shape optimization with internally generated shape variables. One or more design domains are defined using the DTPG card. These cards must, in turn, reference PSHELL, PCOMP or DESVAR definitions. If a DESVAR definition is referenced, it must be a shape design variable, meaning that it must, in turn, be referenced by one or more DVGRID cards. If a PSHELL or PCOMP definition is referenced, OptiStruct generates shape variables using the parameters defined on the DTPG card, creating internal DVGRID data for the nodes associated with the PSHELL or PCOMP definitions. In both cases, the end result is that each DTPG card references a single shape variable. This shape variable then gets converted into topography shape variables.

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Basic topography shape variables follow the user-defined parameters on the DTPG card (minimum bead width, and draw angle), they are circular in shape, and they are laid out across the design domain in a roughly hexagonal distribution. Each topography shape variable has a circular central region of diameter equal to the minimum bead width. Grids within this region are perturbed as a group, which prevents the formation of any reinforcement bead of less than the minimum bead width. Grids outside of the central circular region of the topographical variables are perturbed as the average of the variables to which they are nearest. This results in smooth transitions between neighboring variables. If two adjacent variables are fully perturbed, all of the nodes between them will be fully perturbed. If one variable is fully perturbed and its neighbor is unperturbed, the nodes in between will form a smooth slope connecting them at an angle equal to the draw angle. The spacing of the variables is determined by the minimum bead width and the draw angle in such a way that no part of the bead reinforcement pattern forms an angle greater than the draw angle. Pattern grouping options link topographical variables together in such a way that the desired reinforcement patterns are formed. Linear, planar, circular, radial, etc. shaped reinforcements are controlled by single variables, ensuring that the reinforcements follow the desired pattern. One-plane, two-plane, three-plane and cyclical symmetry pattern grouping options also use a similar approach to ensure that symmetry is created in the solution. Although topography optimization is primarily a tool for creating bead type reinforcements in shell elements, it can accommodate solid models as well. Many pattern grouping options (such as planar and cylindrical) are intended to be used with solid models since they effectively reduce 3-D problems into 2-D ones.

Variable Generation There are three methods of automatically generating shape variables for topography optimization using the DTPG card. The first two, element normal and draw vector, are performed entirely in OptiStruct. The third (user-defined) requires that the input data contain one or more shape design variables that are used as the design domain. Method

Description

Element normal

This method is the easiest one to use. When norm is entered for the draw direction, the normal vectors of the elements are used to define the draw vector for the shape variables. This method is especially effective for curved surfaces and enclosed volumes where the beads are intended to be drawn normal to the surface.

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Method

Description

Beads created using the element normal method of determining draw vector.

Draw vector

This method allows you to define the draw vector that is used for generating the shape variables. The X, Y, and Z components of the draw vector in the nodal coordinate system are entered. This method is useful when all beads must be drawn in the same direction. Note that the draw angle may not be maintained while using this method.

Beads created using the Draw vector method of determining draw vector.

User-defined

This method allows you to set up the vectors and heights for the topography optimization. A DESVAR card is referenced in place of a PSHELL or PCOMP card. All of the grids with DVGRID cards associated with that DESVAR card are considered part of the design domain. The DESVAR and DVGRID entries are redefined to reflect the minimum bead width and draw angle parameters that have been set by you. The vectors and magnitudes of the displacement vectors on each DVGRID card for each grid are

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Method

Description retained, so these entries must be left blank on the DTPG card. This allows you to create a design domain in which each node can have its own draw vector and draw height.

Multiple Topography Design Regions OptiStruct generates topography shape variables for each design domain defined by a DTPG card. It allows for overlapping of design domains. A grid that is in more than one design domain will be a part of shape variables for each design domain. For automatically generated bead variables, the draw height is divided by the number of bead variables acting on that grid. Thus, if a grid is a part of two DTPG cards that have draw heights of 3.0mm and 5.0mm, the draw heights become 1.5mm and 2.5mm. If this is not desired, simply make sure that no grid is in more than one design domain. In cases where two design components touch each other and the design domains are not user-defined (i.e. PSHELL or PCOMP definitions are referenced), a row of non-design elements needs to be inserted between them to prevent averaging. If the bead variables are user-defined (i.e. DESVAR definition is referenced), no averaging will be performed. It is assumed that you intend to have the shape variables overlap, which will result in the grid deflection being cumulative between multiple influencing bead cards. Bead Discreteness Fraction The bead discreteness fraction is a response that can be used to control the amount of shape variation for topography design domains. This response indicates the amount of shape variation for one or more topography design domains. The response varies in the range 0.0 to 1.0 (0.0 < BEADFRAC < 1.0), where 0.0 indicates that no shape variation has occurred, and 1.0 indicates that the entire topography design domain has assumed the maximum allowed shape variation.

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Size Optimization OptiStruct has the capability of performing size optimization. Size optimization can be performed simultaneously with the other types of optimization. In size optimization, the properties of structural elements such as shell thickness, beam cross-sectional properties, spring stiffness, and mass are modified to solve the optimization problem. Defining size variables in OptiStruct is done very similarly to other size optimization codes. Each size variable is defined using a DESVAR bulk data entry. If a discrete design variable is desired, a DDVAL bulk data entry needs to be referenced for the design variable values. The DESVAR cards are related to size properties in the model using a DVPREL1 or DVPREL2 bulk data entry. Each DVPREL bulk data entry must reference at least one DESVAR bulk data entry to be active during the optimization. HyperWorks includes a pre-processor called HyperMesh that can be used to set up any number of size variables for the properties. The following responses (see Responses for a description) are currently available as the objective or as constraint functions: Mass

Volume

Center of Gravity

Moment of Inertia

Static Compliance

Static Displacement

Natural Frequency

Buckling Factor



Frequency Response Displacement, Velocity, Acceleration

Frequency Response Stress, Strain, Forces

Weighted Compliance

Weighted Frequency


Function

Temperature

Design Variables for Size Optimization In finite elements, the behavior of structural elements (as opposed to continuum elements), such as shells, beams, rods, springs, and concentrated masses, are defined by input parameters, such as shell thickness, cross-sectional properties, and stiffness. Those parameters are modified in a size optimization. Some structural elements have several parameters depending on each other; like beams in which the area, moments of inertia, and torsional constants depend on the geometry of the cross-section. The property itself is not the design variable in size optimization, but the property is defined as a function of design variables. The simplest definition, as defined by the design-variable-to-property relationship DVPREL1, is a linear combination of design

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variables defined on a DESVAR statement such that

p C0

DVi Ci

Where, p is the property to be optimized, and Ci are linear factors associated to the design variable DVi. Using the equation utility DEQATN, more complicated functional dependencies using even trigonometric functions can be established. Such design-variable-to-property relations are then defined using the DVPREL2 statement. For a simple gage optimization of a shell structure, the design-variable-to-property relationship turns into:

T DVi Where, the gage thickness, T is identical to the design variable. If a discrete design variable is desired, a DDVAL bulk data entry needs to be referenced on the DESVAR bulk data entry for the design variable values.

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Shape Optimization OptiStruct has the capability of performing shape optimization. In shape optimization, the outer boundary of the structure is modified to solve the optimization problem. Using finite element models, the shape is defined by the grid point locations. Hence, shape modifications change those locations. Shape variables are defined in OptiStruct in a way very similar to that of other shape optimization codes. Each shape variable is defined by using a DESVAR bulk data entry. If a discrete design variable is desired, a DDVAL bulk data entry needs to be referenced for the design variable values. DVGRID bulk data entries define how much a particular grid point location is changed by the design variable. Any number of DVGRID bulk data entries can be added to the model. Each DVGRID bulk data entry must reference an existing DESVAR bulk data entry if it is to be a part of the optimization. The DVGRID data in OptiStruct contains grid location perturbations, not basis shapes. The generation of the design variables and of the DVGRID bulk data entries is facilitated by the HyperMorph utility, which is part of the HyperMesh software. The following responses (see Responses for a description) are currently available as the objective or as constraint functions: Mass

Volume

Center of Gravity

Moment of Inertia

Static Compliance

Static Displacement

Natural Frequency

Buckling Factor



Frequency Response Frequency Response Displacement, Stress, Strain, Forces Velocity, Acceleration

Weighted Compliance

Weighted Frequency Combined Compliance Index

Function

Temperature

Design Variables for Shape Optimization In finite elements, the shape of a structure is defined by the vector of nodal coordinates (x). In order to avoid mesh distortions due to shape changes, changes of the shape of the structural boundary must be translated into changes of the interior of the mesh. The two most commonly used approaches to account for mesh changes during a shape optimization are the basis vector approach and the perturbation vector approach. Both approaches refer to the definition of the structural shape as a linear combination of vectors.

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Using the basis vector approach, the structural shape is defined as a linear combination of basis vectors. The basis vectors define nodal locations.

x

DVi BVi

Where, x is the vector of nodal coordinates, BVi is the basis vector associated to the design variable DVi. Using the perturbation vector approach, the structural shape change is defined as a linear combination of perturbation vectors. The perturbation vectors define changes of nodal locations with respect to the original finite element mesh.

x

X0

DVi PVi

Where, x is the vector of nodal coordinates, X0 is the vector of nodal coordinates of the initial design, PVi is the perturbation vector associated to the design variable DVi. The initial nodal coordinates are those defined with the GRID entity. The perturbation vectors are defined on the DVGRID statement, which is referenced by the design variable entity DESVAR. If a discrete design variable is desired, a DDVAL bulk data entry needs to be referenced on the DESVAR bulk data entry for the design variable values. Note: In OptiStruct, only the perturbation vector approach is available. The DVGRID cards must contain perturbation vectors.

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Free-shape Optimization Free-shape optimization uses a proprietary optimization technique developed by Altair Engineering Inc., wherein the outer boundary of a structure is altered to meet with predefined objectives and constraints. The essential idea of free-shape optimization, and where it differs from other shape optimization techniques, is that the allowable movement of the outer boundary is automatically determined, thus relieving you of the burden of defining shape perturbations. Free-shape design regions are defined through the DSHAPE bulk data entry. Design regions are identified by the grids on the outer boundary of the structure (the edge of a shell structure or the surface of a solid structure). These grids are listed on the DSHAPE entry. Free-shape optimization allows these design grids to move in one of two ways: 1. For shell structures; grids move normal to the surface edge in the tangential plane. 2. For solid structures; grids move normal to the surface. During free-shape optimization, the normal directions change with the change in shape of the structure, thus, for each iteration, the design grids move along the updated normals.

Defining Free-shape Design Regions Ideally, free-shape design regions should be selected where it can be assumed that the shape of the structure is most sensitive to the concerned responses. For example, it would be appropriate to select grids in a high stress region when the objective is to reduce stress. Free-shape design regions should be defined at different locations on the structure where it is desired for the shape to change independently. For solid structures, feature lines often define natural boundaries for free-shape design regions. Containing any feature lines inside a free-shape design region should be avoided unless the intention is to smooth the feature lines during an optimization. Likewise for a shell structure, sharp corners should not be contained inside a free-shape design region unless the intention is to smooth out such corners. The DSHAPE card identifies the design region through the GRID continuation card, shown here: (1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

GRID

GID1

GID2

GID3

GID4

GID5

GID6

GID7

GID8

GID9

…

…

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A free-shape design region is defined on the curved edge of the plate by selecting the edge grids; the grids are free to move in the normal direction on the tangential plane.

A free-shape design region is defined on a surface of the solid structure by selecting the face surface grids; the grids are free to move normal to the surface.

Free-shape Parameters The five parameters that affect the way in which the free-shape design region deforms are the direction type, the move factor, the number of layers for mesh smoothing, the maximum shrinkage, and the maximum growth.

Direction Type This provides a general constraint on the direction of the movement of the free-shape design region. It is defined on the PERT continuation line of the DSHAPE entry in the DTYPE field, as shown: (1)

(2)

(3)

PERT

DTYP E

(4)

(5)

(6)

MVFACTOR NSMOOTH MXSHRK

(7)

(8)

(9)

MXGROW

SMETHO D

NTRANS

(10)

DTYPE has three distinct options: 1. GROW – grids cannot move inside of the initial part boundary. 2. SHRINK – grids cannot move outside of the initial part boundary. 3. BOTH – grids are unconstrained.

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GROW

SHRINK

BOTH

Undeformed Deformed

Move Factor The maximum allowable movement in one iteration of the grids defining a free-shape design region is specified as: MVFACTOR* mesh_size where "mesh_size" is the average mesh size of the design region defined in the same DSHAPE card. MVFACTOR is defined on the PERT continuation line of the DSHAPE entry. (1)

(2)

(3)

PERT

DTYPE

(4)

(5)

(6)

MVFACTO NSMOOT MXSHR R H K

(7) MXGROW

(8)

(9)

(10)

SMETHO NTRAN D S

The default value of MVFACTOR is 0.5. A smaller MVFACTOR will make free-shape optimization run slower but with more stability. Conversely, a larger MVFACTOR will make free-shape optimization run faster but with less stability.

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MVFACTOR affects the maximum movement in one iteration. Undeformed shape Shape at iteration 1 with MVFACTOR = 0.5 (default) Shape at iteration 1 with MVFACTOR = 1.0

Number of Layers for Mesh Smoothing With free-shape optimization, internal grids adjacent to those grids defining the design region are moved to avoid mesh distortion. The number of layers of grids to be included in the mesh smoothing buffer may be defined by the NSMOOTH field on the PERT continuation line of the DSHAPE entry. (1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

PERT

DTYPE

MVFACTO R

NSMOO TH

MXSHR K

MXGRO W

SMETHO D

NTRAN S

(10)

The default value of NSMOOTH is 10. A larger NSMOOTH will give a larger smoothing buffer, and consequently will work better in avoiding mesh distortion; however, it will result in a slower optimization.

NSMOOTH=5, 5 layers of grids move along with the design boundary.

NSMOOTH=1, only 1 layer of grids move along with the design boundary.

Maximum Shrinkage and Growth The maximum shrinkage and growth provide a simple way to limit the total amount of deformation of the free-shape design region. These parameters are defined on the PERT continuation line of the DSHAPE entry. (1)

(2) PERT

478

(3)

(4)

(5)

DTYP MVFACTO NSMOOT E R H

(6)

(7)

MXSH RK

MXGRO W

(8)

(9)

(10)

SMETHO NTRAN D S


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The design region is offset to form two barriers; MXSHRK is the offset in the shrinkage direction and MXGROW is the offset in the growth direction. The design region is then constrained to deform between these two barriers.

Deformation space defined by the maximum growing/shrinking distance

For more details and an example, refer to the section on the Mesh Barrier Constraint below.

Additional treatment to grids in the Transition Zone When the entire surface or edge of a system is not a design zone and both design and nondesign regions exist adjacent to one another, a transition zone can be defined using NTRANS which helps to smooth out the transition. Sharp changes can occur in the design region during optimization and the sections of the design region closest to the non-design region are designated as a transition zone where the corresponding location of the adjacent non-design region is taken into consideration allowing for a smoother transition from the design to non-design region. NTRANS defines the number of design grid layers in the transition zone to non-design area, where additional treatment will be applied to produce smooth transition. (1)

(2) PERT

(3)

(4)

(5)

(6)

DTYP MVFACTO NSMOOT MXSHR E R H K

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(8)

(9)

(10)

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Defining the Transition Zone grid points for a smooth transition between Design and Non-Design regions (NTRANS=3)

The resulting optimized design will incorporate the effect of non-design regions while moving the transition zone grid points to achieve a smoother final design. The three regions illustrated in the figure above consist of the following highlighted nodes: The non-design nodes (marked by yellow circles), which do not move during Freeshape optimization. The design nodes are separated into two groups: - Design nodes in transition zone (highlighted nodes enclosed by red circles, defined by NTRANS=3) - Design nodes that are NOT in the transition zone (highlighted nodes enclosed by a black circle) The design nodes in the transition zone will be adjusted during Free-shape optimization to build a smooth transition between “(1) non-design nodes” and “(3) Design nodes that are NOT in the transition zone”. Otherwise, you may get discontinuous or sharp sections, which can be explained in the illustration below.

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Defining the Transition Zone grid points for a smooth transition between Design and Non-Design regions (NTRANS=3)

Constraints on Grids in the Design Region It is possible to identify additional constraints on certain grids in free-shape design regions. Three types of constraints are available for specified grids as defined by CTYPE# on the GRIDCON continuation line of the DSHAPE entry: 1. FIXED – grid cannot move due to free-shape optimization. 2. VECTOR – grid is forced to move along the specified vector. 3. PLANAR – grid is forced to remain on a plane for which the specified vector defines the normal direction. Note: VECTOR is used to constrain a grid to move along a line, thus it makes no difference by rotating the vector by 180 degrees. Constraints are defined on the GRIDCON continuation line as follows: (1)

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Example Showing CTYPE = VECTOR This example demonstrates a simple case where it is necessary to use the "DIR" constraint type to force grids to move in a predefined direction. A free-shape optimization is performed on a quarter model of a rectangular plate with a hole, shown here:

The curved edge is the free-shape design region. Without any constraints on the freeshape design region, the grids at the ends of the curved edge do not move exactly along the line of the straight edge, but move slightly outward, as shown here:

In order to prevent this phenomenon, the grids at the ends of the curved edge (shown in yellow below) are both constrained to move along the vector indicated by the red arrows.

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Using these constraints - corner grids moving along the constrained direction - the grids at the ends of the curved edge now move as desired, along the line of the straight edge, as shown here:

Example showing CTYPE = PLANAR In this example, the total volume of a cantilever beam is to be minimized subject to a displacement constraint in the loading direction at the free-end of the beam. The model is shown here:

Two free-shape design regions are defined in this example. Both of the vertical sides of the beam are selected as design regions and a free-shape optimization is performed.

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Without any constraints on the free-shape design region, the top and bottom surfaces of the beam do not remain strictly on the X-Z plane.

To ensure that the top and bottom surfaces remain on the X-Z plane, the grids along the edges of the design regions DSHAPE1 and DSHAPE2 are constrained to move only on the X-Z plane.

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Using these constraints – constrained grids moving only on the X-Z plane – the top and bottom surfaces of the beam remain on the X-Z plane as desired.

1-plane Symmetry Constraint It is often desirable to produce a symmetric design. Even if the loads and boundary conditions are perfectly symmetric, there is no guarantee that the resulting design will be perfectly symmetric. In order to ensure a symmetric design, a symmetry constraint must be defined. An additional advantage of this constraint is that it will produce symmetric designs regardless of the initial mesh, loads or boundary conditions. The 1-plane symmetry constraint is defined on the PATRN continuation line: (1)

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Example Showing 1-plane Symmetry Constraints in 2-dimensions In this example, the objective is to minimize the total volume subject to a stress constraint using free-shape optimization. Results are shown with and without symmetry constraints.

2-D model showing free-shape design grids

Result without symmetry constraint

Result with symmetry constraint (XZ plane)

Example Showing 1-plane Symmetry Constraints in 3-dimensions In this example, the objective is to minimize the compliance subject to a volume constraint using free-shape optimization. Results are shown with and without symmetry constraints.

3-D Model showing free-shape design grids

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Result without symmetry constraint

Result with symmetry constraint (XZ plane)

Extrusion Constraint It is often desirable to produce a design with a constant cross-section along a given path, particularly in the case of parts manufactured by an extrusion process. By using extrusion manufacturing constraints with free-shape optimization, constant cross-section designs can be attained for solid models (regardless of the initial mesh, loads or boundary conditions). The extrusion constraint is defined on the EXTR continuation line: (1)

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Two types of extrusion path are available for free-shape optimization – straight line and circular.

Example Showing Extrusion Constraint Along a Straight Line The FE model, optimization problem and design variables definition are the same as in the previous example, so the result without the extrusion constraint is the same as shown above. The result with the extrusion constraint (straight line) is shown here.

Result with extrusion path (along x-axis)

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Example Showing Extrusion Constraint Along a Circular Path In this example, the objective is to minimize the von Mises stress subject to a volume constraint using free-shape optimization. A circular extrusion path is defined using a cylindrical coordinate system ( direction). Results are shown with and without extrusion constraints (circular).

Model showing free-shape design grids

Result without extrusion path

Result with extrusion path (circular)

Draw Direction Constraint In the casting process, cavities that are not open and lined up with the sliding direction of the die are not feasible. Draw direction constraints may be defined for the design region so that the optimized shape will allow the die to slide in a prescribed direction. Only a single die is considered for each design region (defined in each DSHAPE card), and nondesign regions will not be considered for this constraint. The draw direction constraint is defined on the DRAW continuation line: (1)

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Example Showing Draw Direction Constraint The FE model, optimization problem and design grids definition are the same as those in the example showing 1-plane symmetry constraints in 3 dimensions. Results with draw direction constraint are shown here.

Result with draw direction constraint (along Y-axis)

Example Showing Combination of 1-plane Symmetry and Draw Direction Constraints The FE model, optimization problem and design grids definition are the same as those in the example showing 1-plane symmetry constraints in 3 dimensions. Results with 1-plane symmetry and draw direction constraints are shown here.

Result with both draw direction constraint (Y-axis) and 1-plane symmetry constraint (XY-plane)

Side Constraints Similar to the maximum shrinkage and growth parameters as defined on the PERT continuation line, it is possible to limit the extent of the total deformation of the design region by way of side constraints. Side constraints allow the deformation space to be defined as a coordinate range; i.e. between (x1, y1, z1) and (x2, y2, z2). These ranges may be with reference to rectangular, cylindrical or spherical systems.

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Example Showing Side Constraints In this example, the objective is to minimize the von Mises stress subject to a volume constraint using free-shape optimization. Results are shown with and without side constraints.

Model showing side constraints defined by the radii R1 and R2 (1-direction of cylindrical system)

Result with side constraints

Result without side constraints

Mesh Barrier Constraints Aside from shrinkage and growth parameters and side constraints, a more general capability to limit the extent of the total deformation of the design region is available by way of defining a mesh barrier constraint. The mesh barrier is composed of special shell elements (BMFACE), and in order to keep computational effort to a minimum, as few elements as possible should be used in its definition. The mesh barrier is defined on the BMESH continuation line. (1)

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Example Showing Mesh Barrier Constraint The FE model, optimization problem and design grids definition are the same as those in the example showing 1-plane symmetry constraints in 3 dimensions. A mesh barrier is added (large red tria elements).

Model with mesh barrier

Result without mesh barrier constraint

Result with mesh barrier constraint

Result combining mesh barrier constraint and 1-plane symmetry constraint

From the results, you can see how the mesh barrier constrains the model deformation, but if the mesh barrier is not big enough, the design region deformation is unconstrained beyond its limits.

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Example Showing Maximum Shrinkage and Growth Parameters The FE model, optimization problem and design grids definition are the same as those in the example showing 1-plane symmetry constraints in 3 dimensions. In addition, maximum shrinkage and growth parameters (2.0) and a 1-plane symmetry constraint (XZplane), are defined.

Result with shrinkage and growth parameters. Max. growth distance = 2.0 Max. shrinkage distance = 2.0

Result with shrinkage and growth parameters and 1plane symmetry constraint (XZ-plane). Max. growth distance = 2.0 Max. shrinkage distance = 2.0

Additional Comments 1. In the case where multiple constraints are defined for the same design region, while the optimizer tries its best to satisfy all the different constraints, it is possible that it may not be able to coordinate all these constraints. 2. It should be pointed out that if constraints like mesh barrier, maximum growth and shrinkage, or side constraints are applied to avoid of interference between structural parts, you should define the constraints in such a way that clearance is guaranteed under manufacturing tolerance and structural deformation. In other words, the barrier surface should contain an offset from the potentially interfering parts.

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Manufacturing Constraints The following optimization features can be found in this section: Manufacturability for Topology Optimization Manufacturability for Topography Optimization Manufacturability for Free-size Optimization Multi-Model Optimization

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Manufacturability for Topology Optimization A concern in topology optimization is that the design concepts developed are very often not manufacturable. Another problem is that the solution of a topology optimization problem can be mesh dependent, if no appropriate measure is taken. OptiStruct offers a number of different methods to account for manufacturability when performing topology optimization: Member Size Control Draw Direction Constraints Extrusion Constraints Pattern Repetition Pattern Grouping

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Member Size Control for Topology Optimization Member size control allows you some control over the member size in the final topology and the resulting degree of simplicity of the final design. This feature may be added one of the two ways described below. 1. The DOPTPRM card: (1)

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Here, both the preferred minimum, MINDIM, and the maximum, MAXDIM, diameter of members may be defined on the MEMBSIZ continuation line. Member size dimensions can be defined differently for each DTPL in this way.

Minimum Member Size Control Although minimum member size control penalizes the formation of small members, results that contain members significantly under the prescribed minimum member size can still be obtained. This is because a small member in the structure can be very important to the load transmission and may not be removed by penalization. Minimum member size control functions more as a quality control than a quantity control. A discrete solution is achieved in two iterative steps. The first step converges to a solution with a large number of semi-dense elements. The second step tries to refine this solution to a solution with fully dense members. Each step consists of a number of iterations. The first step consists of two entire convergence phases - the first run with the initial discreteness values (defined by DISCRETE and DISCRT1D parameters on the DOPTPRM bulk data entry), followed by a run with the discreteness values increased by 1.0. This procedure is implemented in order to achieve a solution with clearly defined members. If this step could not create a solution with clearly defined members, the preferred minimum member size will not be preserved in the second step. In which case, you need to increase the discreteness parameters and/or reduce the convergence tolerance (defined by the OBJTOL parameter on the DOPTPRM bulk data entry) to improve the solution of the first phase. The default discreteness is set to 1.0 for 1-D elements, plates and shells, and 2.0 for 3-D solids.

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In general, once MINDIM is activated, checkerboarding is controlled by the methods applied for this feature, eliminating the need for the CHECKER parameter. In rare circumstances, checkerboards may still be introduced in the second phase described above for 3-D solids. If this happens, an additional checkerboard control algorithm can be activated with the MMCHECK parameter. (The CHECKER and MMCHECK parameters are defined using the DOPTPRM bulk data entry). The use of this card will assure a checkerboard-free solution, although with the undesired side effect of achieving a solution that involves a large number of semi-dense elements, similar to the result of setting CHECKER equal to 1. Therefore, use this card only when it is necessary. It is recommended that MINDIM be at least 3 times, and no greater than 12 times, the average element size for all elements referenced by that DTPL (or all designable elements when defined on DOPTPRM). The average element size for 2-D elements is calculated as the average of the square root of the area of the elements, and for 3-D elements, as the average of the cubic root of the volume of the elements. This recommendation is enforced when combined with other manufacturing constraints, and if the defined MINDIM is less than this value, it will be reset to a default value equal to 3 times the average element size. Similarly, if the defined MINDIM is larger than 12 times the average element size, it will be reset to a value equal to 12 times the average element size. This limit has been set to trim memory requirements that can become too large as a result of having to keep track of a much larger number of elements needed to satisfy the MINDIM constraint. In structures where the mesh is aligned with the draw or extrusion direction, setting MTYP as ALIGN on the MESH continuation line of the DTPL card may circumvent this constraint. The following examples demonstrate significant improvement in the manufacturability of results through the use of minimum member size control: Michell-truss Example MBB-Beam Example Arch Example 3-D Bridge Model Example

Maximum Member Size Control Maximum member size control penalizes the formation of large members. The control is not directional, meaning that if the thickness of a member is less than MAXDIM in any direction, this constraint is satisfied. This reflects the need to control the rib thickness of casting parts. MAXDIM must be at least twice MINDIM, and hence the minimum mesh requirement is that MAXDIM has to be at least 6 times the average element size for all elements referenced by that DTPL. This constraint is strongly enforced and an error termination will occur when this criteria is not met. In addition, MAXDIM should be less than half the width of the thinnest part of the design region. Based on the constraints mentioned above, a fine mesh is required to achieve good results with this manufacturing constraint.

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It is to be noted that use of the maximum member size control induces further restriction of the feasible design space and should therefore only be used when it is truly desirable. Also note that this feature is a new research development, and the techniques are still undergoing improvement. An undesired side effect that has been noticed for some examples is that it might result in more intermediate density in the final solution. Therefore, it is recommended that this feature be used sparingly until the technology becomes more robust. While MAXDIM also enforces a spacing of members of the same dimension, the maximum reachable volume fraction is 0.5. For problems involving constraints on structural responses, this could interfere with constraint satisfaction. It is strongly recommended that the behavior of the design problem be studied without MAXDIM first in order to determine if the use of MAXDIM would be advantageous, and if the target volume allows for it to be applied. The following examples demonstrate the impact of maximum member size control to the design outcome. Example 1: Engine Bracket

Engine Bracket Design with Draw Direction Constraint

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Engine Bracket Design with Draw Direction and Maximum Member Size Constraints

Example 2: Steering Wheel Bracket

Steering Wheel Bracket Design with Draw Direction Constraint

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Steering Wheel Bracket Design with Draw Direction and Maximum Member Size Constraints

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Draw Direction Constraints for Topology Optimization In the casting process, cavities that are not open and lined up with the sliding direction of the die are not feasible. Designs obtained by topology optimization often contain cavities that are not viable for casting. Transformation of such a design proposal to a manufacturable design could be extremely difficult, if not impossible. In some cases where this transformation is made, the likelihood of severely affecting the design optimality is high. OptiStruct allows you to impose draw direction constraints so that the topology determined will allow the die to slide in a given direction. These constraints are defined using the DTPL card. Different constraints can be applied to different structural parts, specified by PSOLID IDs. There are two DRAW options available. The option 'SINGLE' assumes that a single die will be used and it slides in the given drawing direction. The bottom surface of the considered casting part is the predefined contra part for the die. The option 'SPLIT' implies that two dies splitting apart in the given draw direction will be used to cast the part described in this DTPL card. The splitting surface of the two dies is optimized during the optimization process. It is often a requirement of certain designs that no through – holes exist. These holes can be prevented from forming in the direction of the draw through use of the ‘NO HOLE’ option. This parameter is also defined on the DTPL card. With ‘NO HOLE,’ the topology can only evolve gradually from the boundary one layer at a time, and in certain cases, it may take several iterations to remove one layer. Available with the ‘SINGLE’ draw option is a stamping or sheet metal manufacturing constraint. This option forces the evolution of 3-D shell interpretable structure from a 3-D design domain. This allows the design of 2-D shell or stamped parts from a 3-D design domain allowing greater design flexibility. The ‘STAMP’ option is also specified on the DTPL card along with a thickness value that represents the desired thickness of the resulting shell or stamped part. A casting may contain a non-designable region in addition to a designable region. These non-designable regions must be defined as obstacles for the casting process on the same DTPL card. This preserves the casting feasibility of the final structure. Also note that there is a default minimum member size for use with draw direction constraints. This is determined internally to be three times the average mesh size of the relevant components. Therefore, the mesh density of the model and the target volume fraction should be chosen so that enough material is available to fill members of the default minimum size. You can specify a desired minimum member size for each design part defined by a DTPL card. This value must be bigger than the default value or else it will be replaced by the default value. Example 1: Beam under Torsion The considered beam is clamped on one end and loaded with a pair of twisting forces on the free end. The finite element model is shown in Figure 1.1. The design problem is to minimize the compliance with a volume fraction constraint of 0.3. The final design without draw direction constraints is shown in Figure 1.2. The chosen draw direction is along the Z-axis. The designs with the options 'SINGLE’ and 'SPLIT' for draw direction constraint are shown in Figures 1.3 and 1.4, respectively.

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Figure 1.1: Finite element model of the beam under torsion

Figure 1.2: Design without draw direction manufacturing constraint

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Figure 1.3: Design with draw direction Z and die option 'SINGLE'

Figure 1.4: Design with draw direction Z and die option 'SPLIT'

As expected, the result without manufacturing constraints is a tube-like structure that is indeed the optimal topology for torsion load. However, this design does not permit the sliding of the die in the Z direction. The result that allows the sliding of the die for casting is not very intuitive, it forms a periodical X pattern to cross the pair of twist loads until they reach the supported end. Significantly more material at the crossing point reflects the doubled shear force at this point. Compared to an upward facing C channel solution, the cross pattern has the advantage that the stress is periodical in every X cell, thus eliminating higher order influence of the span of the beam for any solution that has system level bending action. Example 2: Engine Bracket The example shown below is an engine bracket model of a car. The finite element model of the design domain is shown in Figure 2.1, in which 9046 elements are used and the design domain is shown in blue color. Six load cases were considered, which reflect the following driving and service status: 1) start; 2) backwards; 3) into a pothole; 4) out of a

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pothole; 5) loads from an attaching part; and 6) loads during engine transport. The final topology that allows a single die sliding upwards is shown in Figure 2.2. The design that allows two dies to slide up and down, respectively, is shown in Figure 2.3.

Figure 2.1: Finite element model of a engine bracket

Figure 2.2: Design with draw direction Z and die option 'SINGLE'

Figure 2.3: Design with draw direction Z and die option 'SPLIT'

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Example 3: Compressor Mounting Bracket In this example, a topology optimization is run on a compressor mounting bracket, and the effects of the ‘NO HOLE’ manufacturing constraint will be showcased. The finite element model is shown in Figure 3.1. Regions in red indicate non-design space and the region in blue indicates design space. Four different loading conditions were considered representing operating conditions, and the model was built up with approximately 90,000 elements. The design problem was formulated to minimize the compliance as the objective function, with a constraint on the design volume fraction. Figure 3.2 shows the design proposal for the compressor mounting bracket without the use of the ‘NO HOLE’ manufacturing constraint. As seen in the image, the design contains through-holes. Figure 3.3 on the other hand, shows the design proposal of the bracket with the use of the ‘NO HOLE’ constraint. In this case now, the resulting design proposal does not contain any through-holes.

Figure 3.1: Finite element model representing the design and non-design spaces of the compressor mounting bracket

Figure 3.2: Design proposal without the ‘NO HOLE’ option

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Figure 3.3: Design proposal with the ‘NO HOLE’ option

Example 4: Automotive Bracket The bracket in this example was designed incorporating the ‘NO HOLE’ constraint along with the ‘STAMP’ constraint. The goal of the study was to generate a design proposal without any through-holes, while at the same time being interpretable as a sheet metal design. The ‘STAMP’ constraint here forces the formation of a 3-D shell interpretable structure representing a sheet metal design. Four subcases representing four different loading conditions were considered and the model was comprised of approximately 20,000 elements. The optimization problem was formulated to minimize the weighted compliance (one compliance value per subcase) for a constrained design volume fraction. This essentially results in evolving the stiffest design for a given amount of material. Figure 4.1 represents the finite element model of the design and non-design spaces of the bracket. The regions in red are the non-design regions and the region in blue is the design space. Figure 4.2 is the design proposal without the consideration of the ‘STAMP’ and ‘NO HOLE’ manufacturing requirements. While it is possible to cast such a design, stamping such a part is not feasible. Figures 4.3 and 4.4 showcase the design proposal after incorporating the ‘STAMP’ and ‘NO HOLE’ manufacturing constraints. The ‘NO HOLE’ constraint prevents the formation of any through-holes in the design space and helps keep a continuous shell layout, while the ‘STAMP’ constraint leads to the formation of a uniform thickness design proposal. The thickness is defined along with the ‘STAMP’ constraint and represents the desired thickness of the sheet metal part. From this proposal, it is now possible to interpret the design as a sheet metal or stamped part.

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Figure 4.1: Finite element model representing the design and non-design spaces of the bracket

Figure 4.2: Design proposal without ‘STAMP’ and ‘NO HOLE’

Figure 4.3: Design proposal with ‘STAMP’ and ‘NO HOLE’ (view 1)

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Figure 4.4: Design proposal with ‘STAMP’ and ‘NO HOLE’ (view 2)

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Extrusion Constraints for Topology Optimization In some cases, it is desirable to produce a design characterized by a constant cross-section along a given path, particularly in the case of parts manufactured through an extrusion process. By using extrusion manufacturing constraints in topology optimization, constant cross-section designs can be attained for solid models – regardless of the initial mesh, boundary conditions or loads. Extrusion constraints can also be used for the conceptual design study of structures that do not specifically need to be manufactured using an extrusion procedure. Those requirements can be regarded as specific geometric constraints and can be used for any design that desires such characteristics. For instance, it might be desirable to have ribs going through the entire depth of a solid domain. As with other manufacturing constraints, extrusion constraints can be applied on a component level, and can be defined in conjunction with minimum member size control using the DTPL card.

Setting up the Problem Extrusion constraints can be applied to domains characterized by non-twisted cross-sections (left figure) or twisted cross-sections (right figure) by using the NOTWIST or TWIST parameters respectively in the ETYP field. The structure is non-twisted when the local coordinates systems associated with each cross-section, projected onto a reference plane, remain parallel to each other.

Defining the Extrusion Path It is necessary to define a ‘discrete’ extrusion path by entering a series of grids in the EPATH1 field. The curve between these grids is then interpolated using parametric splines. The minimum amount of grids depends on the complexity of the extrusion path. Only two grids are required for a linear path, but it is recommended to use at least 5-10 grids for more complex curves.

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In the example above, four grids are used to define the extrusion path (left figure). As you can see, the path computed by OptiStruct is inaccurate. To obtain a more accurate approximation, more grids are included in the extrusion path (right figure). For twisted cross-sections, a secondary extrusion path needs to be defined in a similar manner through the EPATH2 field. Example 1 In this example, a curved beam is considered to be a rail over which a vehicle is moving. Both ends of the beam are simply supported. A point load applied over the length of the rail as five independent load cases simulates the movement of the vehicle. The objective is to minimize the sum of the compliances, and the material volume fraction is constrained at 0.3. The rail should be manufactured through extrusion. The 13 grids represented as black dots on the right figure define the extrusion path.

The optimized topologies without and with extrusion constraints are shown below. Reanalyzing the final designs without penalty for intermediate density, the compliances for these two designs are 29.9396 and 37.4377 respectively, which implies a 20% loss in performance due to extrusion constraints. The extruded design represents a clean proposal that requires little refinement. On the other hand, the design obtained without manufacturing constraints may require significant modifications that could cause efficiency loss in performance.

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Example 2 In this example, a "stairs" shaped structure is submitted to two lateral pressure loads defined in two separate subcases. The objective is to minimize the sum of the compliances under both load cases. The extrusion path is defined as a straight line parallel to the global Y-axis. The cross-section of the finite elements model along that path is not constant.

Clearly, this type of structure is not suitable to be manufactured through an extrusion process. However, extrusion constraints can be applied to obtain a manufacturable design characterized by ribs going through the entire depth of the structure. The optimized design gives a good idea of the layout of the resulting stiffening panels.

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Example 3 This example illustrates how extrusion constraints can be used to develop common components in different areas of a structure. The extrusion path can be defined through a solid mesh that is not continuous.

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Pattern Repetition for Topology Optimization Pattern repetition is a technique that allows different structural components to be linked together so as to produce similar topological layouts. To achieve this goal, a master DTPL card needs to be defined, followed by any number of slave DTPL cards which reference the master. The master and slave components are related to each other through local coordinate systems, which are required, and through scaling factors, which are optional. Other manufacturing constraints, such as minimum or maximum member size, draw direction constraints or extrusion constraints, can be applied to the master DTPL card. These constraints will then automatically be applied to the slave DTPL card(s) as described in the next sections. The following procedure should be followed to set up pattern repetition: 1. Create a master DTPL card. 2. Apply other manufacturing constraints as needed. 3. Define the local coordinate system associated to the master DTPL card. 4. Create a slave DTPL card. 5. Define the local coordinate systems associated to the slave DTPL card. 6. Apply scaling factors as needed. 7. Repeat steps 4-6 for any number of slave DTPL cards.

Local Coordinates Systems Local coordinates systems are generated by providing four points. These points can be defined either by entering explicit coordinates or by referencing existing grids, as follows: 1. CAID defines the anchor point for the local coordinates system. 2. CFID defines the direction of the X-axis. 3. CSID defines the XY plane and indicates the positive sense of the Y-axis. 4. CTID indicates the positive sense of the Z-axis. The definition of the fourth point allows for both right-handed and left-handed coordinate systems, which facilitates the creation of reflection patterns.

Right-handed coordinates system

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Alternatively, local coordinate systems can be defined by referencing an existing rectangular coordinate system in the CID field, and by defining an anchor point in the CAID field. Note that if the fields defining CFID, CSID, CTID, and CID are left blank, then the global coordinates system is used by default. The anchor point CAID, however, is always required.

Scaling Factors Scaling factors in the X, Y, and Z directions can be defined for each slave DTPL card. These factors are always related to the local coordinate system. By playing with the local coordinate systems and the scaling factors, a wide range of effects can be obtained as illustrated with the figure below.

Pattern Repetition with Draw Direction Constraints Draw direction constraints can be applied simultaneously with pattern repetition. To achieve this, simply define the draw direction for the master DTPL card, and the draw direction for the slave(s) will automatically be generated based on the local coordinate system. Even if some components are not naturally identical, the optimized design for each component will still satisfy the draw direction constraints. In particular, if different components contain different obstacles, the combination of all obstacles will always be considered.

Pattern Repetition with Extrusion Constraints Extrusion constraints can also be used in conjunction with pattern repetition. This allows for creating parts which have identical cross-sections. The components do not need to be identical in a three-dimensional sense; each part can have its own extrusion path. If the components have different extrusion paths, these paths have to be defined explicitly on each DTPL card. However, if the components have identical extrusion paths, the paths for the slave(s) will automatically be computed based on the master's extrusion path.

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Example 1 This example shows how pattern repetition may be used to generate the same topology in different parts. The first figure shows two similar blocks loaded in two different ways. The optimization problem is to minimize the compliance with 30 percent volume fraction.

If pattern repetition is not used, you can clearly see that the optimized topologies are different, as shown in the figures below:

Same view as in the first figure.

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Viewed from behind showing that the turquoise block is hollowed out.

Using pattern repetition, both of the loads on the master (the left hand block in first figure) and the loads on the slave are taken into account, and the optimized topology is repeated for both blocks, as shown below:

Example 2 This example shows a simplified wing model.

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The internal wing structure consists of 2 spars and 11 ribs. In this example, each rib is subdivided into three sections; the nose section, the center section and the tail section, and each of these sections is chosen as a topology design region.

The optimization problem is to minimize compliance for 30 percent of the design volume fraction. Here you see the optimized topology when each region is independent.

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Pattern repetition is used to group all of the noses together, all of the centers together and all of the tails together, resulting in 3 master pattern definitions, each with 10 slave definitions. Notice how different meshes are used for each rib; pattern repetition is mesh independent. Also the wing tapers, so the outboard ribs are shorter and thinner than the inboard ribs, scaling is defined for the slaves so that the pattern fits in the design space.

The optimized topology achieved using pattern repetition is shown below, and you can clearly see how the same topological layout is repeated for each rib.

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Pattern Grouping for Topology Optimization Pattern grouping is a feature that allows you to define a single part of the domain that should be designed in a certain pattern.

Planar Symmetry It is often desirable to produce a design that has symmetry. Unfortunately, even if the design space and boundary conditions are symmetric, conventional topology optimization methods do not guarantee a perfectly symmetric design. By using symmetry constraints in topology optimization, symmetric designs can be attained regardless of the initial mesh, boundary conditions, or loads. Symmetry can be enforced across one plane, two orthogonal planes, or three orthogonal planes. A symmetric mesh is not necessary, as OptiStruct will create variables that are very close to identical across the plane(s) of symmetry. To define symmetry across one plane, it is necessary to provide an anchor grid and a reference grid. The first vector runs from the anchor grid to the reference grid. The plane of symmetry is normal to that vector and passes through the anchor grid.

To define symmetry across two planes, a second reference grid needs to be provided. The second vector runs from the anchor grid to the projection of the second reference grid onto the first plane of symmetry. The second plane of symmetry is normal to that vector and passes through the anchor grid.

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To define symmetry across three planes, no additional information is required, other than to indicate that a third plane of symmetry is to be used. The third plane of symmetry is perpendicular to the first two planes of symmetry, and also passes through the anchor grid.

Uniform Element Density Pattern grouping also provides the possibility to request a uniform element density throughout selected components. This pattern group ensures that all elements of selected components maintain the same element density with respect to one another.

Cyclical Symmetry Cyclical symmetry can also be defined through the use of pattern grouping. With cyclical pattern grouping, the design is repeated about a central axis a number of times determined by you. Furthermore, the cyclical repetitions can be symmetric within

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themselves. If that option is selected, OptiStruct will force each wedge to be symmetric about its centerline. To define cyclical symmetry, it is necessary to provide an anchor grid and a reference grid. The axis of symmetry runs from the anchor grid to the reference grid. It is also necessary to specify the number of cycles; the repetition angle will be automatically computed.

To add planar symmetry within each wedge, a second reference grid needs to be provided. The plane of symmetry is determined by the anchor grid and the two reference grids.

Pattern Grouping with Draw Direction Constraints Draw direction constraints can be combined with pattern grouping. As illustrated below for one-plane symmetry, the sense of the vector defining the symmetry also determines the primary side of the design space to which the draw direction applies. The draw direction for the secondary side of the design space is automatically computed.

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Vector defining the symmetry Vector defining the draw direction Primary domain Secondary domain Vector defining the draw direction for the secondary domain (automatically created)

Plane of symmetry

The same reasoning applies for two-plane and three-plane symmetries, as well as for cyclical symmetry. Caution should be used in order to achieve manufacturable designs. With cyclical symmetry, for instance, the draw direction should be parallel to the axis of symmetry.

Pattern Grouping with Extrusion Constraints Currently extrusion constraints cannot be used simultaneously with pattern grouping. Example 1 In this example, a solid block of material is used. The grids located on the rear of the block (in the YZ plane) are fully constrained. Axial loading is applied to the block's upper edge in the direction of the negative Y-axis. The objective is to minimize the compliance with a constraint on the volume fraction. Single-die draw direction constraints are applied in the direction of the positive Z-axis.

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The figures below illustrate the results obtained for various symmetry combinations. As the loading is not symmetric with respect to the XY and YZ planes, the design is not symmetrical about these planes when symmetry constraints are not prescribed. Enforcing symmetry conditions about the XY or YZ planes yields significantly different results.

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Example 2 Here, solid elements are used to model a car wheel. The outer layers as well as the bolts are non-designable. Twenty load cases are considered. The objective is to minimize the weighted compliance with a constraint on the volume fraction. Split-die draw direction constraints are applied in the direction of the X-axis. Cyclical pattern grouping is defined with planar symmetry within each cycle.

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As the results show, a clean and reasonably manufacturable design is achieved. Cyclical symmetry is obtained even though the loading is not symmetric.

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Combining Pattern Repetition and Pattern Grouping with other Manufacturing Constraints for Pattern repetition is a feature that allows you to define multiple parts of the model that should be characterized by identical or similar designs. Pattern grouping is a feature that allows you to define a single part of the model that should be designed in a certain pattern. Pattern repetition and pattern grouping can be used with solid and shell elements. They can also be applied in conjunction with minimum and maximum member size constraints, with draw direction constraints, and (to some extent) with extrusion constraints. The following combinations are allowed: Solids Shells

Pattern repetition

Pattern grouping

Simple

Draw

Extrusion

Without scaling

Yes

Yes

Yes

Yes

With scaling

Yes

Yes

Yes

No

1-plane symmetry

Yes

Yes

Yes

No

2-planes symmetry

Yes

Yes

Yes

No

3-planes symmetry

Yes

Yes

Yes

No

Cyclical symmetry

Yes

Yes

Yes

No

Pattern grouping can be combined with draw direction constraints, but you should use caution in order to achieve manufacturable designs. For extrusion, pattern repetition generates identical cross-sections for different components. Therefore, scaling is not supported for this combination.

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Manufacturability for Topography Optimization Manufacturing methods can place constraints on the types of reinforcement patterns available for a given part. Some examples of this are: channels, which must have a continuous cross-section; discs, which must be turned on a lathe; and stampings, which cannot have the die lock conditions. These constraints can be accounted for in topography optimization by using Pattern Grouping Options, and a design with a manufacturable reinforcement pattern can be generated.

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Pattern Repetition for Topography Optimization Pattern repetition is a technique that allows different structural components to be linked together so as to produce similar topographical layouts. To achieve this goal, a master DTPG card needs to be defined, followed by any number of slave DTPG cards which reference the master. The master and slave components are related to each other through local coordinate systems, which are required, and through scaling factors, which are optional. Other manufacturing constraints, such as pattern grouping, can be applied to the master DTPG card. These constraints will then automatically be applied to the slave DTPG card(s). The following procedure should be followed to set up pattern repetition: 1. Create a master DTPG card. 2. Apply other manufacturing constraints as needed. 3. Define the local coordinate system associated to the master DTPG card. 4. Create a slave DTPG card. 5. Define the local coordinate systems associated to the slave DTPG card. 6. Apply scaling factors as needed. 7. Repeat steps 4-6 for any number of slave DTPG cards.

Local Coordinates Systems Local coordinates systems are generated by providing four points. These points can be defined either by entering explicit coordinates or by referencing existing grids, as follows: 1. CAID defines the anchor point for the local coordinates system. 2. CFID defines the direction of the X-axis. 3. CSID defines the XY plane and indicates the positive sense of the Y-axis. 4. CTID indicates the positive sense of the Z-axis. The definition of the fourth point allows for both right-handed and left-handed coordinate systems, which facilitates the creation of reflected patterns.

Alternatively, local coordinate systems can be defined by referencing an existing rectangular coordinate system in the CID field, and by defining an anchor point in the CAID field.

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Note that if the fields defining CFID, CSID, CTID, and CID are left blank, then the global coordinates system is used by default. The anchor point CAID, however, is always required.

Scaling Factors Scaling factors in the X, Y, and Z directions can be defined for each slave DTPG card. These factors are always related to the local coordinate system. By playing with the local coordinate systems and the scaling factors, a wide range of effects can be obtained as illustrated with the figure below.

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Pattern Grouping Options for Topography Optimization There are over 70 pattern grouping options and variations available for topography optimization. A summary of the major categories is shown below: Variable grouping pattern

Pattern Option

Type #

Required Vector Definitions

None

-

0

-

One plane symmetry

-

10

One

Reflection of variables across one plane normal to first vector.

Two plane symmetry

-

20

Two

Reflection of variables across two planes, one normal to the first vector and one normal to second vector.

Three plane symmetry

-

30

Two

Reflection of variables across three planes, one normal to first vector, one normal to the second vector and one perpendicular to both vectors.

Linear

-

1

One

Variables grouped as lines extending in direction of first vector.

+1 plane

21

Two

Reflection of variables across one plane normal to second vector.

+2 planes

31

Two

Reflection of variables across two planes, one normal to second vector and one perpendicular to both vectors.

2

One

Variables grouped as circles around anchor node lying in a plane normal to first vector.

12

One


3

One

Variables grouped as planes extending normal to first vector.

13

One

Reflection of planes across plane normal to first vector.

4

One

Variables grouped as rays extending radially and normal to first vector.

Circular

-

+1 plane

Planar

-

+1 plane

Radial 2-D -

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Description

Variables grouped as points.


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Variable grouping pattern

Pattern Option

Type #


+1 plane

14

One

Reflection of rays across plane normal to first vector.

+2 planes

24

Two

Reflection of rays across two planes, one normal to the first vector and one normal to second vector.

+3 planes

34

Two

Reflection of rays across three planes, one normal to first vector, one normal to the second vector and one perpendicular to both vectors.

Cylindrical -

5

One

Variables grouped as endless cylinders extending along and centered around first vector.

Radial 2-D & Linear

6

One

Variables grouped as a combination of radial and linear patterns.

+1 plane

26

Two

Reflection of radial planes across plane normal to second vector.

+2 planes

36

Two

Reflection of radial planes across plane normal to both first and second vectors.

7

-

+1 plane

17

One

Reflection of rays across plane normal to first vector.

+2 planes

27

Two

Reflection of rays across two planes, one normal to the first vector and one normal to second vector.

+3 planes

37

Two

Reflection of rays across three planes, one normal to first vector, one normal to the second vector and one perpendicular to both vectors.

8

-

Variables grouped along vectors defined by the draw vectors of the individual nodes.

Radial 3-D -

Vector defined

532

-

Description

Variables grouped as rays extending radially outward from anchor node.


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Variable grouping pattern

Cyclical*

Pattern Option

Type #


+1 plane

18

One

Reflection of variables across plane normal to first vector.

+2 planes

28

Two

Reflection of variables across two planes, one normal to the first vector and one normal to second vector.

+3 planes

38

Two

Reflection of variables across three planes, one normal to first vector, one normal to the second vector and one perpendicular to both vectors.

-

40,41

Two

Cyclical repetition of variables about axis defined by first vector.

+1 plane

50,51

Two


+ linear

60,61

Two

Cyclically repeated variables grouped as lines extending in direction of first vector.

+ radial

70,71

Two

Cyclically repeated variables grouped as rays extending radially and normal to first vector.

+ radial & linear

80,81

Two

Cyclically repeated variables grouped as a combination of radial and linear patterns.

Description

* For cyclical symmetry, the UCYC parameter (field 30) controls the number of repetitions (and thus the repetition angle) for the cycles. If the TYP option selected for cyclical symmetry is 40, 50, 60, 70, or 80, the cyclical repetition pattern will be non-reflective. If the TYP option selected for cyclical symmetry is 41, 51, 61, 71, or 81, the cyclical repetition pattern will be reflective. These options can be used with shell and solid models to create reinforcement patterns that obey manufacturing constraints and which conform to the shapes of the parts. Examples of pattern grouping options are given in the following sections: Cross-section Optimization of a Spot Welded Tube Optimization of the Modal Frequencies of a Disc Using Constrained Beading Patterns Multi-plane Symmetric Reinforcement Optimization for a Pressure Vessel Shape Optimization of a Stamped Hat Section Shape Optimization of a Solid Control Arm Using Topography Optimization to Forge a Design Concept Out of a Solid Block

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None If no variable grouping pattern is selected, OptiStruct will automatically generate circular bead variable definitions throughout the design variable domain as shown below:

TYP = 0: No symmetry

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Two Planes For two planes of symmetry (TYP = 20), the planes of symmetry are defined normal to both the first and second vectors as shown below. Note that the second grid does not have to be in the plane defined by the first vector, OptiStruct will calculate the second vector by projecting the second grid (or vector) onto the plane defined by the first vector.

TYP = 20: Two planes of symmetry

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Three Planes For three planes of symmetry (TYP = 30), the symmetry plane definitions are identical to those for two planes of symmetry with the third plane being placed perpendicular to the first two and located at the anchor node as shown below:

TYP = 30: Three planes of symmetry

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One Plane - Simple Symmetry Options For a single plane of symmetry (TYP = 10), the plane is defined normal to the first vector and is located at the anchor node as shown below:

TYP = 10: One plane of symmetry

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Linear Pattern Grouping Linear pattern grouping allows you to force OptiStruct to create beads in a given direction along the entire length of the part. This can be very useful for optimizing the shape of extruded parts which must maintain a constant cross section. It is also very useful when optimizing the side walls of stamped plates whose beads must run from the top to the bottom so that the part can be drawn from the die. In solid models, where the variable needs to control the movement of all grids through the thickness, linear pattern grouping is also very useful. For linear pattern grouping (TYP = 1, 21, or 31), OptiStruct generates shape variables that run along a line parallel to the first vector. These shape variables have a width equal to the minimum bead width parameter but have no limit on length. For simple linear pattern grouping, the anchor point and first vector can be located anywhere as shown below:

TYP = 1: Linear pattern grouping

For one and two plane linear symmetry, the anchor point locates the plane(s) of symmetry. For one plane linear symmetry (TYP = 21), the second vector defines the symmetry plane (since the first vector has been used to define the direction of the pattern).

TYP = 21: One plane linear symmetry

For two plane symmetry (TYP = 31), the symmetry planes are defined by the second vector and the cross product of the first and second vectors as shown below. There is no three plane linear pattern grouping since the pattern is automatically symmetric in the direction of the first vector.

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TYP = 31: Two plane linear symmetry

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Circular Pattern Grouping Circular pattern grouping allows you to force OptiStruct to create beads that form concentric circles around a user-defined axis. This can be very useful for optimizing the shape of circular parts that must have a circular reinforcement pattern such as a part turned on a lathe. For circular pattern grouping (TYP = 2 or 12), OptiStruct generates shape variables that form circles about an axis defined by the first vector. These circular beads have a width equal to the minimum bead width parameter. The anchor point can be located anywhere, but the first vector must be collinear with the desired central axis for the circular beads. The simple circular pattern grouping (TYP = 2) is shown below:

TYP = 2: Circular pattern grouping

For one plane circular pattern grouping (TYP = 12), the circular patterns are reflected about a plane located at the anchor node and defined by the first vector. One plane circular symmetry ensures that nodes equal distances above and below the plane of symmetry will be grouped into the same variables. See below:

TYP = 12: One plane circular symmetry

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Planar Pattern Grouping Planar pattern grouping allows you to force OptiStruct to create variables that consolidate the perturbations of active grids in a given plane. This can be very useful in forming beads that run in a fixed direction across an uneven part or in solid models to control the changes in the shape of a cross section. For a planar pattern grouping (TYP = 3 or 13), OptiStruct generates a series of parallel planar shape variables that are defined by the first vector. These shape variables have a width equal to the minimum bead width parameter, but have no limit on length. Beads formed using planar pattern grouping can turn vertical corners. For simple planar pattern grouping, the anchor point and first vector can be located anywhere as shown below:

TYP = 3: Simple planar pattern grouping

For one plane planar symmetry (TYP = 13), the planes are symmetric about a plane located at the anchor point as shown below. There is no need for two and three plane planar symmetry.

TYP = 13: One plane planar symmetry

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Radial (2-D) Pattern Grouping Radial (2-D) pattern grouping allows you to force OptiStruct to create beads in a radial direction extending outward from a central axis. This can be very useful for optimizing circular parts in which radial reinforcements are desired. For radial (2-D) pattern grouping (TYP = 4 , 14, 24, and 34), OptiStruct generates shape variables that run radially away from a central axis defined by the first vector. Radial beads, at their closest point to the central axis, have a width equal to the minimum bead width parameter. The width of the beads increases with distance from the center. There is no limit on the bead length. The anchor point can be located anywhere, but the first vector must be collinear with the desired central axis for the radial beads. The simple radial (2-D) pattern grouping (TYP = 4) is shown below:

TYP = 4: Simple radial (2-D) pattern grouping

For one plane radial (2-D) pattern grouping (TYP = 14), the radial patterns are reflected about a plane located at the anchor node and defined by the first vector. One plane radial symmetry ensures that nodes equal distances above and below the plane of symmetry will be grouped into the same variables. See below:

TYP = 14: One plane radial pattern grouping

For two and three plane radial (2-D) pattern grouping (TYP = 24 and 34), two symmetry planes are determined by the first and second vectors as shown below.

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TYP = 24: Two plane radial (2-D) pattern grouping

TYP = 34: Three plane radial (2-D) pattern grouping

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Cylindrical Pattern Grouping Cylindrical pattern grouping allows you to force OptiStruct to create variables that consolidate the perturbations of active grids along the surface of a cylinder. This can be useful in assigning a circular pattern grouping through the thickness of a solid model. For a cylindrical pattern grouping (TYP = 5), OptiStruct generates a series of concentric cylinders that run parallel to and are positioned about the first vector. The cylindrical pattern grouping is essentially the linear pattern grouping combined with the circular pattern grouping. The anchor point can be located anywhere, but the first vector must be collinear with the desired central axis for the cylinders.

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Radial 2-D and Linear Pattern Grouping Radial linear pattern grouping allows you to force OptiStruct to create variables that consolidate the perturbations of active grids along planes running radially from a central axis. This can be useful for assigning radial pattern groupings through the thickness of a solid model. For a radial linear pattern grouping (TYP = 6), OptiStruct generates a series of planes that run radially away from, and in the same plane as, the first vector. The radial linear pattern grouping is essentially the linear pattern grouping combined with the radial pattern grouping. The anchor point can be located anywhere, but the first vector must be collinear with the desired central axis for the radial planes. One and two planes of radial linear pattern grouping (TYP = 26 and 36) can be created by using the second vector to define the planes of symmetry. The symmetry planes are assigned in a manner similar to that for two and three plane radial symmetry.

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Vector Defined Pattern Grouping Vector defined pattern grouping allows you to force OptiStruct to create variables which are grouped according to the direction and magnitude of the individual draw vectors of the grids. This pattern grouping option is similar to the linear pattern grouping option except that the linear vector is not constant for the entire model. Instead, the direction of the draw vector for each grid is used to determine the variable groupings in place of a global linear vector. Additionally, unlike the linear pattern grouping option, the lengths of the beads are not infinite. The lengths of the beads are equal to the magnitude of the draw vectors for the grids. This pattern grouping option can be very effective for optimizing the shape of amorphous solid models. For vector defined pattern grouping (TYP = 8, 18, 28, and 38), OptiStruct generates shape variables by consolidating the perturbations of active grids that are within a cylindrical region about evenly spaced grids in the model. For a selected grid, a cylindrical zone of influence is created around it which has a radius defined by the minimum bead width and draw angle parameters, a length defined by twice the draw vector for the selected grid, and is oriented in the direction of the draw vector. See the figure below:

Note that for solid models, the internal grids will move along with the surface grids provided that the internal grids have draw vectors associated with them. This allows for large scale perturbations both inward and outward from the surface of a solid part while maintaining an acceptable mesh quality. You must create perturbations for all of the grids in the model to effectively use vector defined pattern grouping. If perturbations are defined for the surface grids only, those grids may end up passing through the second layer of grids if the variables are perturbed inward. The best way to use this pattern grouping option is to create a single shape variable by uniformly collapsing all of the grids in a solid model towards the center and then creating a DTPG card which points at the DESVAR for that shape variable.

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Radial (3-D) Pattern Grouping Radial (3-D) pattern grouping allows you to force OptiStruct to create variables that consolidate the perturbations of active grids in a radial direction away from a central point. This can be very useful for optimizing spherical models with solid elements. For radial (3-D) pattern grouping (TYP = 7, 17, 27, and 37), OptiStruct generates shape variables that run radially away from a central point defined by the anchor node. Radial beads, at their closest point to the central axis, have a width equal to the minimum bead width parameter. The width of the beads increases with distance from the center. There is no limit on the bead length. The anchor point can be located anywhere, but is ideally located at the center of a sphere. The planes for one, two, and three plane radial (3-D) symmetry are established in a manner identically to one, two and three plane symmetry without radial (3-D) pattern grouping (TYP = 10, 20, and 30).

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Cyclical Pattern Grouping The cyclical pattern grouping allows you to force OptiStruct to create a series of symmetric shape variables about a central axis that repeat a number of times determined by you (with the UCYC field). This can be useful in assigning a reinforcement pattern in a circular plate that matches an angularly repeated load in a symmetric fashion. For cyclical pattern groupings (TYP = 40 and 41), OptiStruct generates a series of symmetric shape variables about an axis defined by the cross product of the first and second vectors. The axis of rotation is positioned at the anchor point. The first vector defines a plane establishing one side of the cyclical wedge. The other side of the cyclical wedge is defined by the angle of repetition. The figure below shows cyclical pattern grouping for three "wedges".

TYP = 40: Cyclical pattern grouping for 3 repetitions

OptiStruct allows any number of repeated cyclical wedges. You enter the number of desired wedges into field 30 (UCYC). OptiStruct internally calculates the repetition angle according to the formula 360.0 / UCYC. For example, setting UCYC to three results in three wedges of 120.0 each, and setting UCYC to 6 results in six wedges of 60.0 each. You can also control whether the cyclical repetitions will be symmetric within themselves. This is done by choosing one of the cyclical TYP options ending in ‘1’ (41, 51, 61, 71, and 81). If the symmetric wedge option is selected, OptiStruct will force each wedge to be symmetric about its centerline. Selecting one of the cyclical options ending in ‘0’ (40, 50, 60, 70, and 80) will result in the wedges being non-symmetric. See the figures below:

TYP = 40 (non-symmetric) with UCYC = 5

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TYP = 41 (symmetric) with UCYC = 3


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Other Forms of Cyclical Pattern Grouping OptiStruct supports the combination of cyclical pattern grouping with one plane symmetry, linear pattern grouping, radial pattern grouping, and radial linear pattern grouping. Each option is specified with a different base TYP number to which the number denoting the repetition angle is added. For one plane cyclical pattern grouping (TYP = 50 and 51), the cyclical patterns are reflected about a plane located at the anchor node and defined by the cross product of the first and second vectors. One plane cyclical symmetry ensures that nodes equal distances above and below the plane of symmetry will be grouped into the same variables. See below:

TYP = 50: One plane cyclical symmetry with 3 wedges

For linear cyclical symmetry (TYP = 60 and 61), OptiStruct generates shape variables that run along a line parallel to the cross product of the first and second vectors. These shape variables have a width equal to the minimum bead width parameter but have no limit on length. The full lengths of the linearly drawn shape variables will be cyclically symmetric:

TYP = 60: Linear cyclical pattern grouping with 3 wedges

For radial cyclical pattern grouping (TYP = 70 and 71), OptiStruct generates shape variables that run radially away from a central axis defined by the cross product of the first and second vector. Radial beads have a width equal to the minimum bead width parameter but have no limit on length. The width of the beads does not change depending on the distance from the center. The full lengths of the radially drawn shape variables will be cyclically symmetric:

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TYP = 70: Radial cyclical pattern grouping with 3 wedges

For radial linear cyclical pattern grouping (TYP = 80 and 81), OptiStruct generates a series of planes that run radially away from and in the same plane as the first vector. The radial linear cyclical pattern grouping is essentially the linear cyclical pattern grouping combined with the radial pattern grouping. The full lengths of the radially drawn shape variables will be cyclically symmetric.

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Manufacturability for Free-size Optimization A concern in free-size optimization is that the design concepts developed are very often not manufacturable. Another problem is that the solution of a free-size optimization problem can be mesh dependent, if no appropriate measure is taken. OptiStruct offers a number of different methods to account for manufacturability when performing free-size optimization:

Member Size Control Member size control allows you some control over the member size in the final free-size design and the resulting degree of simplicity therein. This feature may be added one of the two ways described below. 1. The DOPTPRM card: (1)

(2)

DOPTPRM MINDIM

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

VALUE

Here, only the preferred minimum diameter (width in 2-D) of members may be defined as the VALUE field, following the MINDIM keyword. A global minimum member size is defined in this way. 2. The DSIZE card: (1)

(2)

(3)

MEMBSI Z

MINDIM

(4)

(5)

(6)

(7)

(8)

(9)

(10)

Here, the preferred minimum, MINDIM member may be defined on the MEMBSIZ continuation line. Member size dimensions can be defined differently for each DSIZE entry in this way.

Minimum Member Size Control Although minimum member size control penalizes the formation of small members, results that contain members significantly under the prescribed minimum member size can still be obtained. This is because a small member in the structure can be very important to the load transmission and may not be removed by penalization. Minimum member size control functions more as a quality control than a quantity control. A discrete solution is achieved in two iterative steps. The first step converges to a solution with a large number of semi-dense elements. The second step tries to refine this solution to a solution with fully dense members. Each step consists of a number of iterations. The first step consists of two entire convergence phases - the first run with the initial discreteness values (defined by DISCRETE and DISCRT1D parameters on the DOPTPRM bulk data entry), followed by a run with the discreteness values increased by 1.0. This

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procedure is implemented in order to achieve a solution with clearly defined members. If this step could not create a solution with clearly defined members, the preferred minimum member size will not be preserved in the second step. In which case, you need to increase the discreteness parameters and/or reduce the convergence tolerance (defined by the OBJTOL parameter on the DOPTPRM bulk data entry) to improve the solution of the first phase. The default discreteness is set to 1.0 for 1-D elements, plates and shells, and 2.0 for 3-D solids. In general, once MINDIM is activated, checkerboarding is controlled by the methods applied for this feature, eliminating the need for the CHECKER parameter. In rare circumstances, checkerboards may still be introduced in the second phase described above for 3-D solids. If this happens, an additional checkerboard control algorithm can be activated with the MMCHECK parameter. (The CHECKER and MMCHECK parameters are defined using the DOPTPRM bulk data entry). The use of this card will assure a checkerboard-free solution, although with the undesired side effect of achieving a solution that involves a large number of semi-dense elements, similar to the result of setting CHECKER equal to 1. Therefore, use this card only when it is necessary. It is recommended that MINDIM be at least 3 times the average element size for all elements referenced by that DSIZE (or all designable elements when defined on DOPTPRM). The average element size for 2-D elements is calculated as the average of the square root of the area of the elements, and for 3-D elements, as the average of the cubic root of the volume of the elements. This recommendation is enforced when combined with other manufacturing constraints, and if the defined MINDIM is less than this value, it will be reset to a default value equal to 3 times the average element size.

Pattern Repetition Pattern repetition is a technique that allows different structural components to be linked together so as to produce similar topological layouts. To achieve this goal, a master DSIZE card needs to be defined, followed by any number of slave DSIZE cards which reference the master. The master and slave components are related to each other through local coordinate systems, which are required, and through scaling factors, which are optional. Other manufacturing constraints, such as minimum or maximum member size, can be applied to the master DSIZE card. These constraints will then automatically be applied to the slave DSIZE card(s) as described in the next sections. The following procedure should be followed to set up pattern repetition: 1. Create a master DSIZE card. 2. Apply other manufacturing constraints as needed. 3. Define the local coordinate system associated to the master DSIZE card. 4. Create a slave DSIZE card. 5. Define the local coordinate systems associated to the slave DSIZE card.

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6. Apply scaling factors as needed. 7. Repeat steps 4-6 for any number of slave DSIZE cards.

Local Coordinates Systems Local coordinates systems are generated by providing four points. These points can be defined either by entering explicit coordinates or by referencing existing grids, as follows: 1. CAID defines the anchor point for the local coordinates system. 2. CFID defines the direction of the X-axis. 3. CSID defines the XY plane and indicates the positive sense of the Y-axis. 4. CTID indicates the positive sense of the Z-axis. The definition of the fourth point allows for both right-handed and left-handed coordinate systems, which facilitates the creation of reflected patterns.

Alternatively, local coordinate systems can be defined by referencing an existing rectangular coordinate system in the CID field, and by defining an anchor point in the CAID field. Note that if the fields defining CFID, CSID, CTID, and CID are left blank, then the global coordinates system is used by default. The anchor point CAID, however, is always required.

Scaling Factors Scaling factors in the X, Y, and Z directions can be defined for each slave DSIZE card. These factors are always related to the local coordinate system. By playing with the local coordinate systems and the scaling factors, a wide range of effects can be obtained as illustrated with the figure below.

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Pattern Grouping Pattern grouping is a feature that allows you to define a single part of the domain that should be designed in a certain pattern.

Planar Symmetry It is often desirable to produce a design that has symmetry. Unfortunately, even if the design space and boundary conditions are symmetric, conventional free-size optimization methods do not guarantee a perfectly symmetric design. By using symmetry constraints in free-size optimization, symmetric designs can be attained regardless of the initial mesh, boundary conditions, or loads. Symmetry can be enforced across one plane, two orthogonal planes, or three orthogonal planes. A symmetric mesh is not necessary, as OptiStruct will create variables that are very close to identical across the plane(s) of symmetry. To define symmetry across one plane, it is necessary to provide an anchor grid and a reference grid. The first vector runs from the anchor grid to the reference grid. The plane of symmetry is normal to that vector and passes through the anchor grid.

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To define symmetry across two planes, a second reference grid needs to be provided. The second vector runs from the anchor grid to the projection of the second reference grid onto the first plane of symmetry. The second plane of symmetry is normal to that vector and passes through the anchor grid.

To define symmetry across three planes, no additional information is required, other than to indicate that a third plane of symmetry is to be used. The third plane of symmetry is perpendicular to the first two planes of symmetry, and also passes through the anchor grid.

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Uniform Element Thickness Pattern grouping also provides the possibility to request a uniform element thickness throughout selected components. This pattern group ensures that all elements of selected components maintain the same element thickness with respect to one another.

Cyclical Symmetry Cyclical symmetry can also be defined through the use of pattern grouping. With cyclical pattern grouping, the design is repeated about a central axis a number of times determined by you. Furthermore, the cyclical repetitions can be symmetric within themselves. If that option is selected, OptiStruct will force each wedge to be symmetric about its centerline. To define cyclical symmetry, it is necessary to provide an anchor grid and a reference grid. The axis of symmetry runs from the anchor grid to the reference grid. It is also necessary to specify the number of cycles; the repetition angle will be automatically computed.

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To add planar symmetry within each wedge, a second reference grid needs to be provided. The plane of symmetry is determined by the anchor grid and the two reference grids.

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Reliability-based Design Optimization (Beta) Introduction Reliability-based Design Optimization (RBDO) is an optimization method that can be used to provide optimum designs in the presence of uncertainty. A gradient-based local approximation of responses is used to perform reliability analysis. The design variables, constraints, and objective are tested for reliability based on user-defined reliability requirements. Deterministic and Random design variables can be selected for a design optimization run, while a Random parameter is also available to check for Robustness of a particular design within specified bounds. Similarly, Design Constraints and Objectives can also be specified as Deterministic (mean) or Percentile values.

Implementation The Single Loop Approach (SLA) is used to optimize structures using Reliability-based Design Optimization in OptiStruct. Reliability-based optimization methods test the reliability of designs for each optimization iteration. The traditional, double-loop RBDO algorithm requires nested optimization loops, where the design optimization (outer) loop repeatedly calls a series of reliability (inner) loops. The computational time can be prohibitive for practical problems due to the nested optimization-reliability loops. The SLA converts the nested loops into a single loop using Karush-Kuhn-Tucker (KKT) conditions of the inner reliability loops in the outer optimization loop. The probabilistic optimization problem is converted into a deterministic optimization problem. SLA is highly efficient compared to the traditional double-loop RBDO process. The Single Loop Approach (SLA) is terminated if one of the following conditions is met: 1. The Sequential Quadratic Programming (SQP) convergence criterion is achieved. 2. Design variable convergence criterion is achieved. 3. Maximum number of allowable iterations is attained. Variables The following design variables and parameters can be used to define the structural design space in OptiStruct: Random Design Variables Random design variables are defined via the RAND continuation lines on the DESVAR bulk data entry. Various random distribution types can be selected and their parameters are defined accordingly. In an RBDO process, during reliability and/or robustness analysis, the design should satisfy optimality based on the specified distributions. Random parameters The definition of Random parameters is similar to that of Random Design Variables, using RANP definition. However, the important difference is that, while the mean values of random variables are changed to improve the design, the mean values of random parameters remain constant. For example, typically sheet metal thickness can be a random variable, due to

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fabrication variance, while the Young’s modulus of a material would typically be a random parameter, if its variance is accounted for. Deterministic Design Variables The deterministic design variables are the regular design variables used in an OptiStruct optimization run. Note: Due to the deviation of the random distribution, the design region should be defined carefully. For example, if a design variable value is intended to be positive, then its lower bound should not be defined lower than n*σ; where, σ is the standard deviation of the variable; n is a constant multiplier (a value of n=6 is recommended).

Objective The following design objective types are available in OptiStruct: Percentile value (RBDO) The minimum or maximum of the percentile based objective function can be defined on the DESOBJ subcase information entry. The MINP/MAXP options and the PROB argument can be used to define the required parameters.

Figure 1: Illustration depicting Percentile Value based Objective

The percentile value based objective is defined as follows: min

or max

Where, is the objective function, and r is the probability level (for example, 95%). The right and left percentile values are available. MINP minimizes the right percentile value and MAXP maximizes the left percentile value. Deterministic (mean) value The deterministic value based objective is the regular objective used in an OptiStruct optimization run. The mean value based objective is defined as follows: min Where,

or max is the objective function.

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Constraints The following design constraint types are available in OptiStruct: Percentile value (RBDO) The probability of one constraint satisfying its bounds should not be less than the predefined reliability value. The reliability value is defined via the PROB field on the DCONSTR bulk data entry. The reliability-based constraints are defined as follows:

Where,

c(x) is the constraint value ub is the upper bound of the constraint lb is the lower bound of the constraint r is the probability level (for example, 95%) For the “ ” constraint, the right percentile value of c(x) is forced to be less than or equal to the upper bound “ub”. For the “ ” constraint, the left percentile value of c(x) is forced to be greater than or equal to the lower bound “lb”. Deterministic (mean) value The deterministic value based constraint is the regular constraint definition used in an OptiStruct optimization run. The mean value based constraint is defined as follows:

c(x) < ub c(x) > lb Where, c(x) is the constraint value, lower bound of the constraint.

ub is the upper bound of the constraint, and lb is the

OptiStruct RBDO The RBDO process is illustrated in Figure 2.

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Figure 2: Flowchart depicting the OptiStruct RBDO implementation

Note that while all deterministic optimization capabilities in OptiStruct use the same approximation approach, design requirements are accurately evaluated after each FEA analysis. This premise does not hold for RBDO as accurate reliability analysis would need more FEA analyses for a given design. Therefore, it is important to assess the usability of the implemented algorithm. Seventeen examples are tested to verify the reliability of this approach. In these examples, distribution types of random design variables are normal distribution and constraints have a reliability requirement of 99%. A Monte Carlo simulation with 1000 sampling points based on accurate FEA analysis is used to check the reliability status (Table 1). Example

Start Design Status

Optimized Design Status

NME T

N

M

Objective

Reliability

Objective

Reliability

1

5

28

3

40.4

98.0%

27.6

98.5%

2

2

26

7

4.51

0.0%

3.24

100.0%

3

3

4

4

3.84e-5

70.4%

3.99e-5

98.8%

4

4

159

5

9.0e-4

100.0%

4.47e-4

98.9%

5

3

2

4

1.98

0.0%

0.472

98.1%

6

3

4

13

0.5

0.0%

2.98e-4

99.0%

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Example

Start Design Status

Optimized Design Status

NME T

N

M

Objective

Reliability

Objective

Reliability

7

2

20

8

4.83

100.0%

3.41

98.8%

8

8

170

13

661.4

0.0%

552.8

98.5%

9

3

316

8

5.785

0.0%

6.825

98.8%

10

2

1

6

26.07

100.0%

21.03

100.0%

11

8

2

7

3.9e-5

0.0%

1.229e-4

99.0%

12

6

16

8

0.01

0.0%

3.45e-3

96.4%

13

2

3

8

232.4

0.0%

240.1

100.0%

14

5

3

4

40.0

0.0%

39.3

98.1%

15

10

168

3

49.4

100.0%

71.9

97.3%

16

3

22

11

4.83

0.0%

2.70

98.8%

17

2

1060

6

232.5

0.0%

61.5

100.0%

T: test number of the example N: number of random design variables M: number of reliability constraints NME: number of model evaluations Objective: the objective value of the design; in these examples, objective value is to be minimized Reliability: the probability of the design that satisfies the constraint requirement The following conclusions can be drawn from the results in Table 1: 1. The OptiStruct RBDO approach is very efficient, since it consumes just a few model evaluations. 2. The reliability of the optimized design is close to the predefined requirement in most cases. Error does exist, however, and can be quite significant as observed in examples 12 and 15. The OptiStruct RBDO approach (based on local approximation) offers an efficient tool to consider uncertainty involved in design. For most problems this approximate reliability analysis yields reasonable estimates. Local approximation based reliability analysis does,

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however, contain some error of varying degree. Accurate reliability analysis should be carried out if accurate satisfaction of reliability requirements is critical. Also note that for the approximation-based OptiStruct RBDO approach, reliability analysis is performed only for retained constraints for which sensitivity is available. You can adjust the screening criteria using the DSCREEN bulk data entry, if required.

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Optimization of Arbitrary Beam Sections Optimization of Thin-walled Sections Optimization of thin-walled sections is facilitated through the use of the dimension fields on the PBARL and PBEAML bulk data entries and the DIM entry within the section definition. The DIM entry is used in the section definition to link either the thickness of a PSEC entry or the Y or Z coordinate of a GRIDS entry to the value of the corresponding DIM field of the referencing PBARL or PBEAML. Should the DIM field on the referencing PBARL or PBEAML be referenced by a DVPREL, the thickness or grid coordinate in the section definition will be affected by changes in the design variable. Solid sections are not designable at this time. The following is an example of a thin-walled section, where the wall thickness and the y coordinate of G2 are designable: $ DESVAR,1,THK,0.1,0.05,0.15 DESVAR,2,G2Y,1.0,0.5,1.5 $ DVPREL1,1,PBARL,1,DIM1,,,0.0 ,1,1.0 DVPREL1,2,PBARL,1,DIM2,,,0.0 ,2,1.0 $ PBARL,1,1000,HYPRBEAM,SQUARE ,0.1,1.0 $ MAT1,1000,2e5,,0.33,7.85e-9 $ BEGIN,HYPRBEAM,SQUARE $ GRIDS,1,0.0,0.0 GRIDS,2,1.0,0.0 GRIDS,3,1.0,1.0 GRIDS,4,0.0,1.0 $ CSEC2,10,100,1,2 CSEC2,20,100,2,3 CSEC2,30,100,3,4 CSEC2,40,100,4,1 $ PSEC,100,1000,0.1 $ DIM,1,T,100 DIM,2,G,2,Y $ END,HYPRBEAM $

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Optimization of Composite Structures OptiStruct offers a comprehensive optimization solution aimed at guiding and simplifying the design of laminate composite structures. The solution includes the following optimization phases and associated techniques: Phase I – Concept. Free-sizing optimization is used to generate design concepts, while considering global responses and optional manufacturing constraints. Phase II – System. Sizing optimization – with ply-based modeling – is performed to control the thickness of each ply bundle, while considering all design responses and optional manufacturing constraints. Phase III – Detail. Ply-stacking optimization is applied to determine the detailed stacking sequence, again while considering all behavioral responses, manufacturing constraints and various ply book rules. While these techniques can be applied independently, it is recommended to use them together as a three-phase integrated process guiding the design from concept to completion. This is particularly important when manufacturing constraints are involved. In order to satisfy such constraints at the finishing stage, they should be incorporated at the beginning so that the design concept can be carried forward. Automated tools are provided to facilitate the transition between the design optimization phases.

Composite Structures - Free-sizing Optimization The purpose of composite free-sizing optimization is to create design concepts that utilize all the potential of a composite structure where both structure and material can be designed simultaneously. By varying the thickness of each ply with a particular fiber orientation for every element, the total laminate thickness can change ‘continuously’ throughout the structure, and at the same time, the optimal composition of the composite laminate at every point (element) is achieved simultaneously. At this stage, a super-ply concept should be adopted, in which each available fiber orientation is assigned a superply whose thickness is free-sized. In other words, a super-ply is the total designable thickness of a particular fiber orientation. In addition, in order to neutralize the effect of ply stacking sequence, the SMEAR option is usually a good choice for this design phase unless it is intended to follow through with the stacking preference of the super-ply laminate model. In OptiStruct, additional manufacturing constraints can be defined for free-sizing optimization. As a composite laminate is typically manufactured through a stacking and curing process, certain manufacturing requirements are necessary in order to limit undesired side effects emerging during this curing process. For example, one such typical constraint for carbon fiber reinforced composites is that plies of a given orientation cannot be stacked successively for more than 3 or 4 plies. This implies that a design concept that contains areas of predominantly a single fiber orientation would never satisfy this requirement. Therefore, to achieve a manufacturable design concept, manufacturing requirements for the final product need to be reflected during the concept design stage. For the particular constraint mentioned above, for instance, the design concept would offer enough alternative ply orientations to break the succession of plies of the same orientation

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if the percentage of each fiber orientation is controlled (for example, no ply orientation should drop below 15%). In addition, balancing of a pair of ply orientations could be useful for practical reasons. For example, balancing 45° and -45° plies would eliminate twisting of a plate under bending along the 0 axis. In order to address these needs, the following manufacturing constraints are made available for composite free-sizing: Lower and upper bounds on the total thickness of the laminate. Lower and upper bounds on the thickness of individual orientations. Lower and upper bounds on the thickness percentage of individual orientations. Constant thickness of individual orientations. Thickness balancing between two given orientations. As for the constraints on laminate thickness, ply thickness and thickness percentage, these can be applied locally through the definition of element sets. Therefore, multiple instances of these constraints are supported. Advantages include being able to allow different constraints in different regions while preserving the continuity of plies. For example, different thickness requirements in critical regions, such as bolted areas, can be factored into the design process. Additionally, zone based free-sizing optimization can be performed. Zones are defined through groups of elements and there can be elements that do not belong to any zone. Zones are typically defined to simplify the design interpretation process and improve manufacturability. Instead of having to interpret manufacturable zones from the solution of a free-sizing optimization, the optimization is run based on pre-defined zones. While the interpretation process is simplified, there is a loss in design freedom as now the optimization is restricted to some extent due to the defined zones. Refer to the DSIZE card for detailed information regarding composite free-sizing optimization and its associated manufacturing constraints. Note also that other generic manufacturing constraints, such as pattern grouping or member size control, can be activated for composite free-sizing as well. The standard result from a free-sizing optimization is the thickness contribution of each orientation defined on the PCOMP(G) or STACK card referenced by the DSIZE card in the optimal laminate design. But, using free-sizing as part of the three-phase composite design and optimization process, and the mechanism to automatically generate an input file for phase two of this process, an additional level of detail/results can be drilled down to in terms of the thickness contributions per orientation. The automatically generated input file for phase two contains ply bundle data that can be reviewed in HyperMesh. A ply bundle is a continuous stack of plies of the same shape (or coverage area). Each super-ply results in the formation of 4 ply bundles. This is the default behavior and can be changed, that is a different number of ply bundles can be requested from the super-plies. However, in most cases, it is recommended to use the default approach. As described above, multiple ply shapes per orientation (through ply bundles) can be determined and generated from a free-sizing optimization. Note: Automatic offset control is available in composite free-size and sizing optimization wherein the specified offset values are automatically updated based on thickness changes. The offset values can be specified on the PCOMP(P/G) property entries or the CTRIA3/6, CQUAD4/8 element entries using the Z0 or ZOFFS fields.

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Composite Structures - Ply-based Sizing Optimization Ply-based Laminate Modeling Complimentary to the conventional property-based composite definition, a new ply-based modeling technique was introduced in OptiStruct 9.0. In this format, laminates are defined in terms of ply entities and stacking sequences, which reflect the native language of ‘plybook’ standards to composite laminate modeling and manufacturing. This is also analogous with how a laminate composite is manufactured. The PLY card specifies the thickness, orientation and material data for each ply, as well as its layout in the structure. The STACK card ‘glues’ the PLYs together to produce the laminate structure. Properties of every zone of unique laminate lay-ups are uniquely, albeit implicitly, defined. This allows you to simply focus on the physical buildup of the composite structure and eliminates the burden associated with identifying patches (PCOMPs) of unique lay-ups, which can be especially complicated for a free-sizing generated design.

Ply-based Optimization In property-based sizing optimization, the designable entities are the ply thicknesses associated with the PCOMP(G) properties. In ply-based sizing optimization, the PLY thicknesses are directly selected as designable entities. This approach greatly simplifies the design variables definition, since ply continuity across patches is automatically taken into account. As with free-sizing optimization, several composite manufacturing constraints are available to control the thickness of the laminate or the thicknesses of specific orientations. These constraints are defined on the DCOMP card and should generally be inherited from the concept phase. A mechanism exists whereby the composite manufacturing constraints defined in the free-sizing phase are automatically carried over into the ply sizing optimization phase. This is part of the same mechanism that also generates the input file for the ply based sizing optimization (phase 2), containing ply bundles as explained in the section on Phase 1: Free-Sizing Optimization. Through this, the ply bundles are automatically set up for optimization with the necessary DESVAR and DVPREL cards defined. The ply bundles are now ready to be sized to determine the optimum thickness per bundle per fiber orientation. In addition, discrete optimization is automatically activated when TMANUF, the thickness of the basic manufacturable ply, is specified for the PLY associated with a given design variable. This feature forces ply bundles to reach thicknesses reflecting a discrete number of physical plies. Therefore, from a ply bundle sizing optimization, the number of plies required per orientation can be established. Typically, additional behavioral constraints such as failure, strain, etc. are added to the problem formulation at this stage. To proceed to the final phase, an input file for phase 3 can be automatically generated from running phase 2, i.e. ply bundle sizing optimization.

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Note: Automatic offset control is available in composite free-size and sizing optimization wherein the specified offset values are automatically updated based on thickness changes. The offset values can be specified on the PCOMP(P/G) property entries or the CTRIA3/6, CQUAD4/8 element entries using the Z0 or ZOFFS fields.

Composite Structures - Stacking Sequence Optimization Composite plies are shuffled to determine the optimal stacking sequence for the given design optimization problem while also satisfying additional manufacturing constraints, defined on the DSHUFFLE card, such as: The stacking sequence should not contain any section with more than a given number of successive plies of same orientation. The 45° and -45° orientations should be paired together. The cover and/or core sections should follow a predefined stacking sequence. Using the three-phase process, composite plies are generated from running a discrete ply bundle sizing optimization (through TMANUF) in phase 2. Additionally, an input file for phase 3, i.e. stacking sequence optimization, is automatically generated from phase 2. An efficient proprietary technique is developed to allow the process to evaluate a huge number of stacking combinations from both performance and manufacturability perspectives.

Phase Transitions in the Optimization of Composite Structures In order to simplify the transition between the three design phases, OptiStruct is able to automatically generate input decks after the free-sizing optimization or the sizing optimization stages have converged. OUTPUT,FSTOSZ (free-sizing to sizing) is an output request that can be used during the free-sizing optimization phase to write a ply-based sizing optimization input deck. For each orientation, the composite interpreter identifies regions of similar thickness and creates PLYs for these regions. The resulting deck contains PCOMPP, STACK, PLY, and SET cards describing the ply-based composite model, as well as DCOMP, DESVAR, and DVPREL cards defining the optimization data. Manufacturing constraints are transferred from the DSIZE card to the DCOMP card. Typically, additional design responses such as stress/ failure constraints would be introduced at this optimization stage. OUTPUT,SZTOSH (sizing to shuffling) is an output request that can be used during the plybased sizing optimization stage to write a ply stacking optimization input deck. Each PLY bundle is divided into multiple PLYs whose thickness is equal to the manufacturable thickness TMANUF, and the STACK card is updated accordingly. The DESVAR and DVPREL cards from the previous stage are removed, and a bare DSHUFFLE card is introduced. As required by the design, additional ply-book rules or manufacturing constraints can be defined on the DSHUFFLE card.

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Example of Composite Structure Optimization - Rectangular Composite Panel In this example, a rectangular composite panel is clamped on one side and subjected to a tip load on the other side. This simple model can be used to demonstrate how the three phases of the composite optimization package interact with each other to ultimately generate a manufacturable design.

Free-sizing Optimization During the concept phase, the composite panel is modeled with four orientations (0°, 90°, 45° and -45°) of uniform thickness, and the SMEAR option is applied to eliminate stack biasing. At this stage, you are minimizing the compliance of the structure while maintaining its volume fraction below 30%. Manufacturing constraints are introduced to limit the total thickness of the panel and to ensure that each orientation accounts for at least 10% of the total thickness. In addition, the thicknesses of the 45° and -45° orientations should be balanced. The resulting DSIZE card is: DSIZE + + +

1 COMP COMP COMP

PCOMP 1 LAMTHK PLYPCT ALL BALANCE 45.0

3.2 0.10 -45.0

The optimization converges in seven iterations, after which the ply-based interpreter identifies thickness zones and generates a ply-based input deck. As illustrated by the following image (Figure 1), four ply bundles are created for each orientation. Figure 2 shows the thickness of each orientation after free-sizing optimization, while Figure 3 shows the equivalent thicknesses after going through the ply-based interpreter. As expected, the thickness zones are considerably more discrete in the interpreted design.

Figure 1: Ply bundles after ply-based interpretation

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Figure 2: Thicknesses after free-sizing optimization

Figure 3:Thicknesses after ply-based interpretation

Ply-based Sizing Optimization Once a concept design has been established, additional performance measures should be introduced. In this example, you are changing the formulation to minimize the volume while constraining the maximum principal stress in every ply. Also, the composite manufacturing constraints from the previous stage are preserved and transferred to the DCOMP card. The optimization converges in 19 iterations, at which point the objective function has been slightly reduced while satisfying the design constraints and the manufacturing constraints. Figure 4 shows the thickness of each orientation after convergence has been achieved. Note that, due to the implicit discrete variables formulation, each ply bundle’s thickness is

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equal to a multiple of TMANUF=0.05.

Figure 4: Thicknesses after ply-based sizing optimization

Ply Stacking Optimization At this stage, you are keeping the formulation that was introduced in the previous phase, while adding detailed stacking constraints. Specifically, you are requesting that no more than four successive plies of same orientation be present in the stack, and that -45° and 45° orientations be paired together in the reverse manner to minimize angle changes. The resulting DSHUFFLE card is: DSHUFFLE 1 + MAXSUCC + PAIR

STACK ALL 45.0

1 4 -45.0 REVERSE

The optimization converges in seven iterations, and the resulting stacking sequence strictly satisfies all constraints. The image below illustrates how the ply-based sizing optimization and the ply stacking optimization phases work together once free-sizing optimization has been performed. Figure (a) shows the initial stack for the sizing optimization phase; it consists of four ply bundles for each orientation as determined by the ply-based interpreter. Figure (b) shows the optimized stack; the thickness of each ply bundle has changed, and the total thickness has been slightly reduced to satisfy the manufacturing constraints. Figure (c) shows the initial stacking sequence for the shuffling optimization phase, where the ply bundles have been converted to actual plies. Figure (d) shows the optimized stacking sequence, which now satisfies the detailed manufacturing constraints as well as the design constraints.

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Figure 5: Stacking sequences during sizing and shuffling optimization

Note that, because most plies only cover part of the laminate structure, the stacking sequence for each zone of unique lay-ups is different from the one illustrated above. However, the manufacturing feasibility is evaluated for every individual zone.

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Equivalent Static Load Method (ESLM) The equivalent static load method, originally published by Dr. Park, Hanyang University, is a technique suitable for optimization of designs undergoing dynamic loads. The method has been implemented for the optimization of the following solutions: Multi-body dynamics problems including flexible bodies. Nonlinear responses from implicit static analysis, implicit dynamic analysis and explicit dynamic analysis. The equivalent static load method takes advantage of the well established static response optimization capabilities of OptiStruct.

Equivalent Static Load

Figure 1.1: Displacement – time history response

The equivalent static load is that load which creates the same response field as that of the dynamic/nonlinear analysis at a given time step. From Figure 1.1, an equivalent static load is calculated corresponding to each time step in the time history of the solution, thereby replicating the dynamic/nonlinear behavior of the system in a static environment. The calculated equivalent static loads from the analysis (as explained above) are considered as separate load cases, and these multiple load cases are used in the linear response optimization loop. An updated design from the optimization loop is then passed back to the analysis for validation and overall convergence. The design is validated against the original dynamic/nonlinear analysis. Based on the outcome of this validation, the solution converges or an updated set of equivalent static loads is calculated for the updated geometry, and the entire process is repeated till convergence. Figure 1.2 is a graphic description of the equivalent static load method for optimization.

Figure 1.2: Equivalent static load method

The equivalent static load method is completely automated in OptiStruct and is an efficient approach for the optimization of responses from transient, dynamic and nonlinear solutions. Apart from others, the equivalent static load method offers the following benefits:

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It can be applied at the concept design phase as well as design fine tuning phase, i.e. it can be used with topology, free-sizing, topography, size, shape and free-shape optimization. A design is optimized for updated loads due to an updated design during the optimization process.

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ESLM for MBD The equivalent static load method has been implemented for the solution of multi-body dynamics problems that include flexible bodies. Size, shape, free-shape, topology, topography, free-size, and material optimization can be applied to flexible bodies. Responses are mass, volume, center of gravity, moment of inertia, stress, strain, compliance (strain energy), and displacement of the flexible bodies. Displacement responses are taken into account with respect to local boundary conditions defined on the flexible body (see definition below). The optimization problem setup follows the setup typical for size, shape, free-shape, topology, topography, free-size, and material optimizations in OptiStruct. Responses are defined using DRESP1, DRESP2, and DRESP3 bulk data entries. Responses can only be defined on flexible bodies (PFBODY) using the properties, elements and grid points included in such bodies as reference. Design variables are defined using DESVAR, DVPREL1, DVPREL2, DVMREL1, DVMREL2, DSHAPE, DTPL, DTPG, DSIZE, and DVGRID bulk data entries. Free-size optimization is currently only available when PTYPE=PSHELL on the DSIZE card. Design variables can only be defined on flexible bodies (PFBODY) using the properties included in such bodies as reference. Constraints are defined using DCONSTR, DCONADD and DOBJREF bulk data entries. Constraints and objectives are referenced in the multi-body dynamics subcase or globally through DESOBJ, DESSUB, DESGLB, and MINMAX, respectively. ESLM specific parameters can be set through the DOPTPRM bulk data entry (see Parameters for the ESLM).

The Method An optimum solution can be found through a series of static response optimizations with the equivalent static load set, that is:

f eq

Ku

f

Ma Cv

Where, feq, u, a, v, and f are the equivalent static load, deformation, acceleration, velocity, and external load, respectively. The steps involved in the ESLM can be summarized as: Step 1

Initial design.

Step 2

Dynamic analysis.

Step 3

Static response optimization with the multiple subcases that consist of multiple equivalent static load sets.

Step 4

If design converged, Stop. Otherwise go to Step 2.

The iteration at the third step is referred to as an inner iteration; Steps 2 through 4 form the outer loop. The converged solution at Step 3 is the starting point of the next outer loop (if the design does not converge at the current outer loop). Note that if there are time steps in the multi-body dynamics analysis at Step 2, subcases in static response optimization are generated at Step 3 (provided that the time step screening option is deactivated). See Parameters for the ESLM.

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Boundary Conditions for Structural Analysis and Optimization To perform structural optimization with ESL, you must specify boundary conditions for each flexible body. In the solution of the dynamic analysis, the flexible and rigid bodies are connected by joints to form a multi-body system. When performing ESLM on the flexible bodies, these joints are not included in this static subcase based structural optimization. This means that each flexible body will have 6 rigid body modes. The 6 rigid body modes of each flexible body must be removed for structural analysis. Exactly 6 degrees of freedom (DOF) of each flexible body must be fixed to remove the 6 rigid body modes. If more than 6 DOF are fixed in a flexible body, the additional fixed DOF become the constraint of the flexible body, which may not result in an optimal solution and consequently increases the required ESLM outer loops. Due to the way ESL is calculated, each flexible body is in its equilibrium at the 0-th inner iteration of each outer loop. This is why you can fix an arbitrary 6DOF (usually single node) to get rid of rigid body modes in order to do static analysis with ESL. Reaction force at the fixed node is zero at the 0-th inner iteration. Thus, no additional load is applied to a flexible body although 6DOF of the flexible body are fixed. However, design changes occur as the inner iteration goes on. This means the original configuration that maintains equilibrium also changes. As a result, the equilibrium status does not hold true anymore from the 1st inner iteration in each outer loop. An undesirable effect caused by a broken equilibrium status (disequilibrium status) is the reaction force at the fixed point is not zero anymore, which means additional load is applied to that fixed point. This effect, due to shape/size changes, can be minimized by fixing a proper node, as explained below. A common way to remove the 6 rigid body modes is to fix all 6 DOF of a node. When looking for a node to fix, choose one that is not in a high stress region. If the fixed point is located in a high stress region, the optimization can be very slow. It is highly recommended that the 6 DOF of an independent node of a spider or rigid element (usually used to model joints) be selected to be fixed. On solid models, where the nodes do not have rotational stiffness; if a solid model does not have a rigid element to represent a joint, the next best way to remove the 6 rigid body modes is to fix 3 translational DOF (123) of one node, 2 translational DOF (23) of another node, and 1 translational DOF (3) of a third node. However, still make sure that the 3 nodes are not in a high stress region, otherwise this method of removing 6 rigid body modes does not work. One way or another, all of the 6 rigid body modes of each flexible body must be removed. SPC, SPC1, or SPCADD (referenced in the subcase information section) can define the fixed DOF. The example that follows shows a solid model with joints at the centers of two holes.

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Stress contours of this model at two time steps are shown in the following images.

Nodes A and B are the locations of joints. The best option here is to fix 6 DOF of either node (A or B) in order to remove 6 rigid body motions in this model. To fix alternative nodes other than node A or B, it would work to fix 3 DOF (123) of node E1, 2 DOF (23) of node E2, and 1 DOF (3) of node E3. These three nodes are located in a relatively low stress region. In this model, nodes C1, C2, and C3 or nodes D1, D2, and D3 would take a long time to converge if fixed. Again, the best and simplest way to remove 6 rigid body modes of each flexible body is to fix one of the joint locations in each flexible body. When the boundary conditions are properly defined, displacement constraints in an optimization can be applied to limit the deformation of the flexible body. An example is to optimize a rotating cantilever. You could fix either the left end or the middle point to retrieve deformation as long as the point has nothing or little to do with the shape perturbation vectors. If a relative displacement at the right end with respect to the left end is to be constrained, it would be most convenient to fix the 6 DOF of the left end to measure the relative displacement. Constraining a relative displacement of the left end with respect to the middle point of the cantilever, it would be best to fix the 6 DOF of the middle point. Regardless of whether the left end or the middle point (or even the right end) is fixed, the

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stress is the same. If the optimization problem is simply to constrain stress or minimize the maximum stress of a flexible body, all that is needed is to fix one proper node of the flexible body.

Rotating cantilever

Deformation when the left end is fixed

Deformation when the middle point is fixed

Output Files Generated by the Optimization Process Once ESL optimization converges, the following output files are available: .eslout

This text file contains brief and useful information about the optimization process. Open this file first to find out what occurred during the ESL optimization. This file is often enough to understand the overall optimization process. The following information is stored in this file: Involved time steps and corresponding subcases. The number of involved time steps. Design results. The number of inner iterations. CPU time.

_mbd_#.h3d

This binary output file contains the MBD analysis results of the #-th outer loop. Displacement, stress, and deformation are available in this file. This file is a modal h3d format (PARAM,MBDH3D,MODAL), which is only available format in ESL optimization. HyperView can display this file.

_des_#.h3d

This binary output file contains design change of the #-th outer loop. By selecting the contour button, you can see the design change. HyperView can display this file.

_mbd_#.abf

This binary output file contains information about rigid body dynamics and modal participation factors of the #-th outer loop. HyperGraph can display the data stored in this file.

Other than above output files, .desvar, .prop, .grid, .fsthick, and .oss will be found, if available.

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Convergence Enhancement for the ESLM It is important to catch the critical responses of a structure properly when optimizing the structure. Generally, it is a good practice to define many time steps in the time intervals where the critical responses of interest show up. By doing this, the optimizer can consider more precise responses at the critical time intervals, resulting in a decreased probability that the optimizer will go in the wrong direction in the current outer loop. Here is an example that shows how to refine specific time intervals. Suppose the analysis time interval is from 0 seconds to 1.0 seconds. In order to find out at which steps the critical responses of interest show up, you could use equi-spaced time steps first. MBSIM

4

TRANS

END

1.0

NSTEPS

100

The above card divides the time interval from 0 seconds to 1.0 seconds into 100 time steps, which is an equi-spaced time step. After analyzing with the above card, you can obtain the following time history for stress: At the time interval of around 0 seconds to 0.02 seconds and 0.60 seconds to 0.63 seconds, critical values of the stress show up. Generally speaking, these critical responses are dominant responses that control the optimization process. It is desirable to consider more critical responses at around these time intervals. In order to consider more critical responses at around these time intervals, increase the number of times steps at these time intervals. $more time steps from 0.0 sec to 0.02 sec. MBSIM 1 TRANS END 0.02 NSTEPS + VSTIFF MBSIM 2 TRANS END 0.61 NSTEPS + VSTIFF $more time steps from 0.61 sec to 0.63 sec. MBSIM 3 TRANS END 0.63 NSTEPS + VSTIFF MBSIM 4 TRANS END 1.0 NSTEPS $ MBSEQ 10 1 2 3 4

200 200

200 100

Time history of a response

In the above MBSIM cards, time steps in the time interval 0 to 0.02 seconds and time interval 0.61 and 0.63 seconds have been increased. Generally, the more information the optimizer knows, the better the solution it can provide. It is beneficial to define many time

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steps; enough to catch precisely the critical responses in the time intervals where critical response of interest show up. This will enhance convergence of the optimization process.

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Parameters for the ESLM Control the maximum number of outer loops executed in the ESLM by using the following parameter: DOPTPRM, ESLMAX In this case, the ESLMAX is an integer value, greater than or equal to 0. If the ESLMAX is equal to 0, then optimization process is not activated. The default value for the ESLMAX is 30. The ESLM has a feature that screens out the time steps that are not dominant during the optimization. In the most cases, this feature is very helpful to improve the efficiency of the ESL optimization process in terms of CPU time. This feature can be activated or deactivated by using: DOPTPRM, ESLSOPT The ESLSOPT can be either 0 or 1. If ESLSOPT is 0, then the screening is deactivated. If ESLSOPT is 1, then this screening is activated. The default value is 1. Unless deactivated by DOPTPRM, ESLSOPT, 0, this feature is always activated. The time step screening feature is different from the constraint screening of the static response optimization in Step 3, although the concepts of both screening capabilities are similar. The time step screening is performed at time Step 2, which is right before entering the static response optimization phase. By performing the time step screening before entering the static response optimization phase, the huge amounts of CPU time required for structural analysis and subsequent constraint screening strategy can be saved at the 0-th iteration in the static response optimization phase. Once ESLSOPT is activated, the number of time steps that should be screened out can be controlled by using: DOPTPRM, ESLSTOL ESLSTOL is a real number between 0.0 and 1.0. The smaller the value of ESLSTOL, the fewer the number of time steps involved in the optimization process. If ESLSTOL is 1.0, all of the time steps in multi-body dynamic analysis will be involved in the optimization process. Obviously, the smaller the ESLSTOL value, the less total CPU time will be used. Note, however, that assigning too small a value to the ESLSTOL could cause the optimization process to diverge. Therefore, if the number of time steps retained by ESLSTOL is less than 10, the 10 most dominant time steps will be involved in the optimization process. The default value for the ESLSTOL is 0.3.

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Output Requests and ESLM Users are not allowed to control analysis output in ESL optimization. All information from the MBD analysis is already available in _mbd_#.h3d, including displacement, stress, and deformation. All analysis output requests are invalidated during ESL optimization except that Output Request = option is allowed above the first subcase. For example, STRESS = ALL or DISPLACEMENT = SID is allowed above the first subcase. Optimization output is not controllable in ESL optimization. See Equivalent Static Load Method (ESLM) for more details. Output and format in output format controls are not controllable in ESL optimization. All requests on output and format are invalidated during ESL optimization, except for FSTHICK and OSS.

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MBD System Level Response Optimization The original ESLM implemented in OptiStruct is for the optimization of flexible bodies in MBD systems through the control of structural responses, such as stress or deformation. The original ESLM is not capable of handling system level responses, such as joint force of a joint or velocity of a node of a body. System-level responses referred to in this section can be displacement, velocities, acceleration, joint force, and functions of those 4 quantities defined by MBVAR. In addition to the original ESLM, MBD system level response optimization is also available in OptiStruct. ESLM and MBD system level optimization can be combined together to optimize MBD systems. One of typical optimization formulations that can be solved by OptiStruct could be: Minimize joint force of a joint subject to stress < allowable value deformation < allowable value velocities of a node < allowable value MBD system level responses can be controlled by the properties of PRBODY, CMBUSH(M), CMBEAM(M), and CMSPDP(M) as well as usual design variables in structural optimization such as thickness of bodies, shape of bodies, etc. DVMBRL1 and DVMBRL2 are available to define relationships between design variables and properties of PRBODY, CMBUSH(M), CMBEAM(M), and CMSPDP(M).

Large Number of Design Variables MBD system level responses are approximated by adaptive response surfaces. The basic solution strategy for MBD system level optimization is the same as that implemented in HyperStudy. In general, response surface based optimization cannot handle large number of design variables. This issue has been resolved in OptiStruct through the introduction of intermediate design variables for bodies. All the design variables defined on rigid/flexible bodies are converted to some predefined intermediate design variables. Adaptive response surfaces are built based on those intermediate design variables. As a result, even if there are hundreds of design variables defined on bodies, adaptive response surfaces for MBD system level responses can be built with only several intermediate design variables. This way, OptiStruct can handle MBD system level responses with large number of design variables.

Change of Length of Bodies as Shape Optimization When you try to achieve a specific velocity value less than an allowed value, changing the length of bodies is an efficient way to achieve it. You can set up shape optimization problem to change the length of bodies. The designable bodies can be either rigid bodies or flexible bodies. When defining shape perturbation vectors with DVGRID, you have to make sure that each perturbation vector does not break down the original joint configuration as the design changes. For example, a revolute joint must have two coincident nodes attached to two different bodies. Be aware while defining shape perturbation vectors, that the perturbation vector could cause a location change for only one of the two coincident nodes so that the revolute joint definition would not be valid after one design iteration.

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Limitations of MBD System Level Response Optimization Design variables associated with PRBODY, CMBUSH(M), CMBEAM(M), and CMSPDP(M) cannot control structural responses, such as deformation or stress. They can only control MBD system level responses such as MBDIS, MBVEL, MBACC, MBFRC, and MBEXPR. Design variables associated with structures such as length, shape, and thickness can control MBD system level responses. Thus, if interaction between structural responses and MBD system level responses are mutually strong, this feature is not applicable. Rigid bodies cannot have structural design variables (shape, properties, thickness, etc.) and the design variables associated with DVMBRL1/2 with TYPE=PRBODY at the same time. Currently, system level responses must be scalar values. OptiStruct provides an option to pick up the maximum, minimum, maximum absolute, or minimum absolute value of displacement/velocity/acceleration/ joint force when defining MBD system level responses using the DRESP1 card. If a system level response is an objective function, and the time when the objective function is picked up jumps around during the optimization process, convergence could be slow or even diverge in some cases.

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ESLM for Nonlinear Response Optimization The equivalent static load method has been extended to support nonlinear response optimization. The following geometric nonlinear analyses are supported for optimization in OptiStruct: Implicit (quasi-) static analysis (NLGEOM) Implicit dynamic analysis (IMPDYN) Explicit dynamic analysis (EXPDYN) Refer to the Geometric Nonlinear Analysis section for more details on the supported analysis types. This method is implemented to support various types of responses that are available in a usual static response optimization problem, i.e. displacement, strain, stress, etc., for example. Only responses defined by DRESP1 are currently supported. Concept level design techniques such as topology, free-sizing and topography optimization, and design fine tuning techniques such as size, shape and free-shape optimization are supported for nonlinear response optimization using ESLM. In the current implementation, for topology optimization, PSHELL and PSOLID are supported and PSHELL is supported for free-size optimization. The optimization setup using ESLM for nonlinear response optimization in terms of design variables, responses, constraints and objective function is the same as the setup for a typical static response optimization problem for topology, topography, free-size, size, shape and free-shape optimization.

Output Files Generated by the Optimization Process In addition to the standard optimization outputs, the following files are output: [model]_#.h3d: Analysis results for each #th outer loop. [model].eslout: Outer loop iteration history summary.

Parameters of ESLM Several parameters are available to control the optimization process: DOPTPRM, ESLMAX This parameter controls the maximum number of outer loops. DOPTPRM, DESMAX This parameter controls the maximum number of inner loop iterations. DOPTPRM, NESLNLGM This parameter controls the number of ESLs generated for each NLGEOM subcase. DOPTPRM, NESLIMPD This parameter controls the number of ESLs generated for each IMPDYN subcase.

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DOPTPRM, NESLEXPD This parameter controls the number of ESLs generated for each EXPDYN subcase. For more details, refer to DOPTPRM parameters.

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Gradient-based Optimization Method The following features can be found in this section:

Iterative Solution OptiStruct uses an iterative procedure known as the local approximation method to solve the optimization problem. This method determines the solution of the optimization problem using the following steps: 1. Analysis of the physical problem using finite elements. 2. Convergence test; whether or not the convergence is achieved. 3. Response screening to retain potentially active responses for the current iteration. 4. Design sensitivity analysis for retained responses. 5. Optimization of an explicit approximate problem formulated using the sensitivity information. Back to 1. To achieve a stable convergence, design variable changes during each iteration are limited to a narrow range within their bounds, called move limits. The biggest design variable changes occur within the first few iterations and, due to an advanced formulation and other stabilizing measures, convergence for practical applications is typically reached with only a small number of FE analyses. The design sensitivity analysis calculates derivatives of structural responses with respect to the design variables. This is one of the most important ingredients for taking FEA from a simple design validation tool to an automated design optimization framework. The design update is generated by solving the explicit approximate optimization problem, based on sensitivity information. OptiStruct has two classes of optimization methods implemented: dual method and primal method. The dual method solves the optimization problem in the dual space of Largrange multipliers associated with active constraints. It is highly efficient for design problems involving a very large number of design variables but much fewer constraints (common to topology and topography optimization). The primal method searches the optimum in the original design variable space. It is used for problems that involve equally as many design constraints as design variables, which is common for size and shape optimizations. The choice of optimizer is made automatically by OptiStruct, based on the characteristics of the optimization problem.

Regular or Soft Convergence Two convergence tests are used in OptiStruct and satisfaction of only one of these tests is required. Regular convergence (design is feasible) is achieved when the convergence criteria are satisfied for two consecutive iterations. This means that for two consecutive iterations, the change in the objective function is less than the objective tolerance and constraint violations are less than 1%. At least three analyses are required for regular convergence, as the convergence is based on the comparison of true objective values (values obtained from an analysis at the latest design point). An exception (design is infeasible) is when the constraints remain violated by more than 1%, and for three consecutive iterations the change in the objective function is less than the objective tolerance and the change in the

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constraint violations is less than 0.2%. In this case, the iterative process will be terminated with a conclusion ‘No feasible design can be obtained.’ Soft convergence is achieved when there is little or no change in the design variables for two consecutive iterations. It is not necessary to evaluate the objective (or constraints) for the final design point, as the model is unchanged from the previous iteration. Therefore, soft convergence requires one less iteration than regular convergence.

OptiStruct Optimization Algorithms OptiStruct utilizes gradient-based optimization algorithms to solve the optimization problem. The default optimization algorithm is known as the Method of Feasible Directions (MFD). You can select a different algorithm using the DOPTPRM, OPTMETH parameter. The following algorithms are available in OptiStruct: Method of feasible directions (MFD) Sequential quadratic programming (SQP) Dual Optimizer based on separate convex approximation (DUAL) Large scale optimization algorithm (BIGOPT) MFD, SQP, and DUAL are standard optimization algorithms. For further information, refer to relevant books and papers in the OptiStruct References section.

Large Scale Optimization Algorithm (BIGOPT) The Large scale optimization algorithm (BIGOPT) is a gradient-based method. It consumes less memory and is relatively more computationally efficient, compared to MFD and SQP. Implementation Consider an optimization problem that involves minimizing f(X) based on a set of constraints. It is described as follows:

Where,

f(X) is the objective function, gi(X) is the i’th constraint function, me is the number of equality constraints, m is the total number of constraints, X is the design variable vector, X L and XU are the lower and upper bound vectors of design variables, respectively.

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If BIGOPT is selected, OptiStruct converts this problem to an equivalent problem using the penalty method, as follows:

Where, r and qi are penalty multipliers. BIGOPT considers the bound constraints separately. So the original problem is converted to an unconstrained problem. Polak-Ribiere conjugate gradient method is used to generate search direction. After the search direction is calculated, a one-dimensional search can be accomplished by parabolic interpolation (Brent’s method). Terminating Conditions Optimization runs, based on the BIGOPT algorithm, will be terminated if one of the following conditions is met: 1.

and the design is feasible.

2.

and the design is feasible.

3. The number of iteration steps exceeds Nmax. Where, is the gradient of

k is the k’th iteration step is the convergence parameter

Nmax is the allowed maximum iterations

Sensitivity Analysis The response quantity, g, is calculated from the displacements as:

qT u

g

The sensitivity of this response with respect to the design variable x, or the gradient of the response, is:

g x

qT u u qT x x

Two approaches to sensitivity analysis, the direct and adjoint variable method, are possible. Given the equation of motion:

Ku

f

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and its derivative with respect to design variable x,

f x

K u u K x x

one can calculate the sensitivity of the displacement vector u as:

K

f x

u x

K u x

Using this equation, the largest cost in the calculation of the response gradient is the forward-backward substitution required for the calculation of the derivative of the displacement vector with respect to the design variable. This is called the direct method. One forward-backward substitution is required for each design variable. If constraints are active in more than one load case, and the load is a function of the design variable (say body force or pressure loads for shape optimization), then the set of forward-backward substitutions must be performed for each active load case. If the loads are not a function of the design variables, but there are active load cases with multiple boundary conditions, then the set of forward-backward substitutions must be performed for each active boundary condition. For the adjoint variable method of sensitivity analysis, the vector (adjoint variable) a is introduced, which is calculated as:

Ka q Then the derivative of the constraint can be calculated as:

g x

qT u aT x

f x

K u x

When the adjoint variable method for sensitivity analysis is used, a single forwardbackward substitution is needed for each retained constraint. This forward-backward substitution is needed to calculate the vector a. There are typically a small number of design variables in shape and size optimization (say 5 to 50) and a large number of constraints. The large number of constraints comes from stress constraints. If there are 20,000 elements, each with a single stress constraint, and 10 load cases, there are a total of 200,000 possible stress constraints. There are typically a large number of design variables in topology optimization (between 1 and 3 per element) and a small number of constraints. Because stress constraints are not usually considered in topology optimization, it makes sense that the adjoint variable method of sensitivity analysis be used for topology optimization (in order to reduce computational costs). For shape and sizing optimization, it is often beneficial to use the direct method for sensitivity analysis. However, in some cases, when there are a large number of design variables and a small number of constraints, the adjoint variable method should be used. For example, in a topography optimization, the number of constraints that gradients need to be calculated for can be reduced using constraint screening. With constraint screening, constraints that are not close to being violated are ignored. Only constraints that are violated, or nearly violated, are retained. Also, if there are many stress constraints that are retained in a small region of the structure, say at a stress concentration, only a few of the most critical need to be retained.

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The sensitivities of responses with respect to design variables can be exported to an Excel spreadsheet (see OUTPUT, MSSENS) or plotted in HyperGraph (see OUTPUT, HGSENS). For contouring in HyperView, the sensitivities of topology, free-sizing and gauge design variables can be exported to H3D format (see OUTPUT, H3DTOPOL and OUTPUT, H3DGAUGE, respectively). Sensitivity output in ASCII format for topology and free-sizing variables can be requested through OUTPUT, ASCSENS. The Excel spreadsheet allows the modification of design variables and then computes approximated responses. This can be used to make design studies without running OptiStruct again. See the image below.

Example spreadsheet output showing that modification of field C10 yields approximate results in the lower right of the spreadsheet, identified by a border surround here.

Move Limit Adjustments As the design moves away from its initial point in the approximate optimization problem, the approximate values become less accurate. This can lead to slow overall convergence, as the approximate optimum designs are not near the actual optimum design. Move limits on the design variables, and/or intermediate design variables, are used to protect the accuracy of the approximations. They appear as:

X

Xm

X

Xm

X

Small move limits lead to smoother convergence. Many iterations may be required due to the small design changes at each iteration. Large move limits may lead to oscillations between infeasible designs as critical constraints are calculated inaccurately. If the approximations themselves are accurate, large move limits can be used. Typical move limits in the approximate optimization problem are 20% of the current design variable value. If advanced approximation concepts are used, move limits up to 50% are possible. Even with advanced approximation concepts, it is possible to have poor approximations of the actual response behavior with respect to the design variables. It is best to use larger move limits for accurate approximations and smaller move limits for those that are not so

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accurate. Note that the same set of design variable move limits must be used for all of the response approximations. It is important to look at the approximations of the responses that are driving the design. These are the objective function and most critical constraints. If the objective function moves in the wrong direction, or critical constraints become even more violated, it is a sign that the approximations are not accurate. In this case, all of the design variable move limits are reduced. However, if the move limits become too small, convergence may be slowed, as design variables that are a long way from the optimum design are forced to change slowly. Therefore, the move limits on the individual design variables that keep hitting the same upper or lower move limit bound are increased. Move limits are automatically adjusted by OptiStruct.

Constraint Screening At each iteration of the optimization process, the objective function(s) and all constraints of the design problem are evaluated. Retaining all of these responses in the optimization problem has two potential disadvantages: 1. This can result in a big optimization problem with a large number of responses and design variables. Most optimization algorithms are designed to handle either a large number of responses or a large number of design variables, but not both. 2. For gradient-based optimization, the design sensitivities of these responses need to be calculated. The design sensitivity calculation can be very computationally expensive when there are a large number of responses and a large number of design variables. Constraint screening is the process by which the number of responses in the optimization problem is trimmed to a representative set. This set of retained responses captures the essence of the original design problem while keeping the size of the optimization problem at an acceptable level. Constraint screening utilizes the fact that constrained responses that are a long way from their bounding values (on the satisfactory side) or which are less critical (i.e. for an upper bound more negative and for a lower bound more positive) than a given number of constrained responses of the same type, within the same designated region and for the same subcase, will not affect the direction of the optimization problem and therefore can be removed from the problem for the current design iteration. Consider the optimization problem where the objective is to minimize the mass of a finite element model composed of 100,000 elements, while keeping the elemental stresses below their associated material's yield stress. In this problem, you have 100,000 constraints (the stress for every element must be below its associated material's yield stress) for each subcase. For every design variable, 100,000 sensitivity calculations must be performed for each subcase, at every iteration. Because design variable changes are restricted by move limits, stresses are not expected to change drastically from one iteration to the next. Therefore, it is wasteful to calculate the sensitivities for those elements whose stresses are considerably lower than their associated material's yield stress. Also the direction of the optimization will be driven primarily by the highest elemental stresses. Therefore, the number of required calculations can be further reduced by only considering an arbitrary number of the highest elemental stresses. Of course there is trade-off involved in using constraint screening. By not considering all of the constrained responses, it may take more iterations to reach a converged solution. If too many constrained responses are screened, it may take considerably longer to reach a converged solution or, in the worst case, it may not be able to converge on a solution if the

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number of retained responses is less than the number of active constraints for the given problem. Through extensive testing it has been found that, for the majority of problems, using constraint screening saves a lot of time and computational effort. Therefore, constraint screening is active in OptiStruct by default. The default settings consider only the 20 most critical (i.e. for an upper bound most positive and for a lower bound most negative) constraints that come within 50 percent of their bound value (on the satisfactory side) for each response type, for each region, for each subcase. The DSCREEN bulk data entry controls both the screening threshold and number of retained constraints. Different DSCREEN settings are allowed for all of the response types supported by the DRESP1 bulk data entry. Responses defined by the DRESP2 bulk data entry are controlled by a single DSCREEN entry with RTYPE = EQUA. Likewise, responses defined by the DRESP3 bulk data entry are controlled by a single DSCREEN entry with RTYPE = EXTERNAL. It is important to ensure that DRESP2 and DRESP3 definitions that use the same region identifier use similar equations. (In order for constraint screening to work effectively, responses within the same region should be of similar magnitudes and demonstrate similar sensitivities, the easiest way to ensure that is through the use of similar variable combinations). In order to reduce the burden on the user, it is possible to allow the screening criteria to be automatically and adaptively adjusted in an effort to retain the least number of responses necessary for stable convergence. Setting RTYPE=AUTO on the DSCREEN bulk data entry will enable this feature. Region definition is also automated with this setting. This setting is useful for less experienced users and can be particularly useful when there are many local constraints. However, there are some drawbacks; experienced users may be able to achieve better performance through manual definition of screening criteria, more memory may be required to run with RTYPE=AUTO, and manual under-retention of constraints will require less memory and may, therefore, be desirable for very large problems (even with compromised convergence stability and optimality).

Regions and Their Purpose In OptiStruct, a region is a group of responses of the same type. Regions are defined by the region identifier field on the DRESP1, DRESP2, and DRESP3 bulk data entries used to define the responses. If the region identifier field is left blank, then each property associated with the response forms its own region. The same region identifier may be used for responses of different types, but remember that because they are not of the same type they cannot form the same region. Example 1 (1)

(2)

(3)

(4)

(5)

DRESP1

1

label

STRESS

PSHELL

2

3

(6)

(7) SMP1

(8)

(9)

(10)

1

DRESP1 with ID 1 defines stress responses for all the elements that reference the PSHELL definitions with PID 1, 2, or 3. As no region identifier is defined, the stress

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responses for each PSHELL form their own regions. So, all of the stress responses for elements referencing PSHELL with PID1 are in a different region than all of the stress responses for elements referencing PSHELL with PID2, which in turn are in a different region than all of the stress responses for elements referencing PSHELL with PID3. If this response definition is constrained in an optimization problem, and the default settings for constraint screening are assumed, then 20 elemental stresses are considered for each of the three PSHELL definitions, i.e. 20 for each region, giving a total of 60 retained responses. Example 2 (1)

(2)

(3)

(4)

(5)

(6)

(7)

DRESP1

2

label

STRESS

PSHELL

1

SMP1

(8)

(9)

(10)

1

2

(1)

(2)

(3)

(4)

(5)

(6)

(7)

DRESP1

3

label

STRESS

PSHELL

1

SMP1

(8)

(9)

(10)

3

All of the stress responses defined in the DRESP1 entries above form a single region notice the entries (not blank) in field 6. Now, if these response definitions which are of the same type (STRESS) with the same entry (not blank) in field 6 are constrained in an optimization problem (assuming the default settings for constraint screening), then 20 elemental stresses are considered in total for the three PSHELL definitions because they form a single region.

Discrete Design Variables OptiStruct uses a gradient-based optimization approach for size and shape optimization. This method does not work well for truly discrete design variables, such as those that would be encountered when optimizing composite stacking sequences. However, the method has been adopted for discrete design variables where the discrete values have a continuous trend, such as when a sheet material is provided with a range of thicknesses. The adopted method works best when the discrete intervals are small. In other words, the more continuous-like the design problem behaves, the more reliable the discrete solution will be. For example, satisfactory performance should not be expected if a thickness variable is given two discrete values 0 and T. It is known that rigorous methods such as branch and bound are very time consuming computationally. Therefore, we developed a semi-intuitive method that is targeted at solving relatively large size problems efficiently. It is recommended to benchmark the discrete design against the baseline continuous solution. This helps to quantify the tradeoff due to discrete variables and to understand whether the discrete solution is reasonable. As local optima are always a barrier for none convex optimization problems, and discrete variables tend to increase the severity of this phenomenon, it could be helpful to run the same design problem from several starting points, especially when the optimality of a

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solution is in doubt. It is also possible to mix these discrete variables with continuous variables in an optimization problem. Discrete design variables are activated by referencing a DDVAL entry on a DESVAR card. The DDVOPT parameter on the DOPTPRM card allows you to choose between a fully discrete optimization or a two phased approach where a continuous optimization is performed first, and a discrete optimization is started from the continuous optimum.

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Global Search Option A common discussion that arises when an optimization problem is solved is whether or not the obtained optimum is a local or global optimum. Local approximation based methods (gradient-based optimizations) are more susceptible to finding a local optimum, while global approximation methods (response surface methods) and exploratory techniques (genetic algorithms) are less susceptible than local approximation based methods to finding a local optimum. In other words, these techniques improve the chances of finding a more global optimum. However, no algorithm can guarantee that the optimum found is in fact the global optimum. An optimum can be guaranteed to be the global optimum only if the optimization problem is convex. For a convex optimization problem, the objective function and feasible domain need to be convex. Unfortunately, in reality, most engineering problems being solved cannot be shown to be convex. Therefore, for practical problems, a global optimum remains elusive. Different algorithm types simply alter one’s chances of finding a more global optimum, not guarantee it. With that consideration, it is important to keep in mind that algorithms which improve the chances come at a computational cost. And most often this can be significant. The following image illustrates the concept of a convex problem as discussed above. A convex optimization problem has just one minimum (or maximum). This minimum (Point A in the image) is the global minimum.

Convex function, f(x)

In the case of non-convex problems solved using gradient-based techniques, the inherent behavior is that the optimized result obtained is dependent on the initial design starting point. This makes these types of algorithms all the more susceptible to finding local optimum. Recently implemented in OptiStruct version 11.0, is a new global search algorithm – an extension to the gradient-based optimization approach. The approach is called Multiple Starting Points Optimization. This global search algorithm performs an extensive search of the design space for multiple starting points to improve the chances of finding a more global optimum. Being dependant on the initial design starting point, n different design starting points could potentially result in n different optimum solutions. It is also highly likely that different design starting points could result in the same optimum solution. However, this

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does not mean that the optimum solution found is the global optimum. This concept is illustrated in the following image.

Non-convex function, f(x)

Consider the non-convex function, f(x), bounded by –a < x < b. Optimizing a design from design starting point A will result in the optimum solution, P. Similarly, optimizing a design from starting point B will result in the same optimum solution, P. On the other hand, optimizing a design from initial design starting point C will result in the optimum solution, Q. From this, it can be seen that through the multiple starting points approach, a global optimum cannot be guaranteed (as with any other algorithm), but at the same time, the chances of finding a more global optimum are improved. As of version 11.0, the Global Search Option (GSO) in OptiStruct supports those optimization disciplines with user-defined variables. Identifying a global search optimization study is done through the DGLOBAL entry in the I/O section of the input deck, and the parameters required to setup and run a GSO are defined on the DGLOBAL bulk data entry.

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Design Interpretation - OSSmooth OSSmooth is a semi-automated design interpretation software, facilitating the recovery of a modified geometry resulting from a structural optimization, for further use in the design process and FEA reanalysis. The tool has two incarnations: a standalone version that comes with the OptiStruct installation, and a dependent version that is embedded in HyperMesh. OSSmooth can be used in three different ways: OSSmooth for geometry, FEA topology reanalysis, and FEA topography reanalysis. Note: FEA topology reanalysis and FEA topography reanalysis are features which are available only in the version of OSSmooth that is embedded within HyperMesh, they are not supported by the standalone OSSmooth software. OSSmooth (for geometry) has several uses and can be used to: Interpret topology optimization results, creating an iso-density boundary surface (Isosurface). Interpret topography optimization results, creating beads or swages on the design surface. Recover and smooth geometry resulting from a shape optimization. Reduce the amount of surface data from a given set of triangular patches by combining smaller patches. Smooth surface data given as triangular patches. For FEA topology reanalysis and FEA topography reanalysis, OSSmooth can be used to: Preserve component boundaries for multiple design components. Recover geometry with or without an artificial layer of elements around a non-design space optionally. Tetramesh Iso-surfaces ‘by property’. Preserve boundary conditions upon geometry recovery to enable quick reanalysis. The following flowchart provides an overview of how OSSmooth works to interpret optimization results from OptiStruct:

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Each of the three applications of OSSmooth has a corresponding sub-panel in the OSSmooth panel in HyperMesh. OSSmooth (for geometry) is generally used to recover geometry by interpreting topology, topography, and shape optimization results, while FEA topology and FEA topography are used to generate recovered geometry with boundary conditions for FEA reanalysis. OSSmooth (for geometry) requires a parameter file (generally has the file extension .oss) to run. This parameter file may be generated from the OSSmooth panel in HyperMesh, or it may be generated manually through a text editor. At the completion of an optimization run, OptiStruct automatically exports an OSSmooth parameter file .oss with certain default settings depending on the type of optimization run. In addition to the parameter file, OSSmooth (for geometry) also requires the input file (.fem), the shape file (.sh), and/or the grid file (.grid) from an OptiStruct run. The grid file .grid contains the grid point locations after a topography or shape optimization and is output at the end of a topography or shape optimization run. The shape file, .sh, contains the element density information of a topology optimization and is output at the end of a topology optimization run. FEA topology requires the input model (.fem) to be loaded into HyperMesh before running, which is different from OSSmooth (for geometry). It also requires the shape file (.sh) generated by a topology optimization. For processing of the non-design elements, two options (Keep smooth narrow layer around and Split all quads) are provided to recover geometry.

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FEA topography requires a grid file (.grid) to run. Similar to FEA topology, it also requires that the input model (.fem) be loaded into HyperMesh first, with the option for iso-surface that performs the same functionality as FEA topology. Note: OSSmooth currently does not recognize OptiStruct longformat input data. A possible work-around for this problem is to import the long-format input file into HyperMesh and export it using the regular OptiStruct template before running OSSmooth. The interpreted design from OSSmooth can be exported as a finite element mesh in the bulk data format, as IGES surfaces, as a stereolithography file, or as a Hyper3D file.

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OSSmooth Parameter File The OSSmooth parameter file is composed of a number of parameter statements, each of which has the following format: parameter_name

arg1,arg2,...,argn

The parameter_name and arguments can be separated either by spaces or commas. The file is not case sensitive. Comment lines in the OSSmooth parameter file should start with either ‘#’ or ‘$’. The following is a list of allowable parameters and their respective arguments: Parameter

Description

input_file

Identifies the files to be interpreted by OSSmooth. arg1

output_file

Name of the file to be output by OSSmooth. arg1

output_code

The file name (without extension) of the OptiStruct .fem, .sh, and/or .grid files to be interpreted by OSSmooth.

Full name of the file output by OSSmooth.

Identifies the type of output. Argument Description arg1

Output format for iso-surface: 1 2 3 4

arg2

– – – –

Bulk data trias IGES patches STL trias [default] H3D trias

Output control for tet-meshing of volume enclosed by the iso-surface. [Default: no tet-meshing] 1 2 3 4

– – – –

Tetra4 + Tria3 elements Tetra10 + Tria6 elements Tetra4 elements Tetra10 elements

The tet-mesh is written in OptiStruct input format to a file _mesh.fem. In this file, the design space of the file .fem is replaced with a tet-mesh for immediate reanalysis. Loads, boundary conditions, and non-design elements are carried over from the input file.

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Parameter

Description

units

Defines output units for IGES format. This information gets written to the header of the IGES file and may be recognized by your CAD system. 1 2 4 6 10

autobead

– inch [default] – mm – foot –m – cm

Improves the recovered geometry from a topography optimization by applying automatic geometry creation. Argument Description arg1

Operation flag [integer]: 0 – autobead off 1 – autobead on [default]

arg2

Threshold value for creating autobead. [real, between 0.0 and 1.0, default = 0.3]

arg3

Bead layer [integer]: 1 – create 1 layer bead [default] 2 – create 2 layers bead

isosurface

Generate threshold surface from a topology optimization by applying automatic geometry creation. Argument Description arg1

Operation flag [integer]: 0 – isosurface off 1 – isosurface on [default]

arg2

Type of surface created [integer]: 0 1 2 3

– – – –

isosurface only isosurface with Optimization-based smoothing Element threshold surface isosurface with Laplacian smoothing [default]

For topology results, smoothing maintains the topology as suggested by OptiStruct, but it can deviate from the given density distribution. If option 1 or 3 is used, check the maximum and average smoothing error output by OSSmooth. arg3

Density threshold for creating isosurface. [real, between 0.0 and 1.0, default = 0.3]

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Parameter

Description

opti_smoothing

Optimization-based smoothing [Only used if isosurface C2=1]. Argument Description arg1

Unit-less surface distance coefficient [real, default = 0.0] Defines closeness of the smooth surface from the threshold surface. The effect of this coefficient varies for different input meshes. Higher magnitudes (both positive and negative) give smoother results, but the surface deviates more from the original density distribution. The recommended range is from -50 to 50. When the coefficient is set between 0 and 50, the surface usually tends to smooth and shrink. When the coefficient is set between 0 and -50, the surface usually tends to smooth and expand.

arg2

Smooth isosurface boundary flag [integer]: 0 – boundary not included in smoothing [default] 1 – boundary included in smoothing

laplacian_smooth Laplacian smoothing [Only used if isosurface C2=3]. ing Argument Description arg1

Number of iteration for Laplacian smoothing [integer > 0, default = 10]

arg2

Feature angle threshold in degrees [real, default = 30.0] The feature angle is defined as the angle of normal between two intersected element planes. All corners with a feature angle larger than the threshold will be preserved in the smoothing process.

arg3

Smooth isosurface boundary flag [integer]: 0 – boundary not included in smoothing 1 – boundary included in smoothing [default]

remesh

Remesh autobead surface and/or isosurface flag [integer]: 0 – remesh off [default] 1 – remesh on Remesh detects 2-layer elements around bead shape and/or boundary of isosurface. Use mixed type remesh if input mesh contains any QUAD elements, otherwise remesh with TRIA elements.

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Parameter

Description

surface_reductio Reduces the number of surfaces representing the geometry. Can reduce n the number of surfaces by up to 80%. Argument Description arg1

Surface reduction flag [integer]: 0 – no surface reduction [default] 1 – do surface reduction

arg2

Feature angle threshold in degrees [real, default = 10.0] The feature angle is defined as the angle formed by the surface normal of two adjacent elements. The surface reduction will be performed on any two adjacent elements in which the feature angle between the two elements is smaller than the threshold. The greater the threshold, the more surface reduction will be conducted. The valid range of the threshold is [1.0, 80.0].

pure_surf_smooth Surface smoothing only. ing Argument Description arg1

Pure surface smoothing flag [integer]: 0 – no surface smoothing [default] 1 – Optimization-based smoothing 2 – Laplacian smoothing

arg2

Number of iteration [Only used if G1=2] [integer > 0, default = 10]

arg3

Feature angle threshold in degrees [Only used if G1=2] [real, default = 30.0] The feature angle is defined as the angle of normal between two intersected element planes. All corners with a feature angle larger than the threshold will be preserved in the smoothing process.

pure_surf_reduct Surface reduction only. ion Argument Description arg1

Pure surface reduction flag [integer]: 0 – no surface reduction [default] 1 – do surface reduction

arg2

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Parameter

Description [real, default = 10.0] The feature angle is defined as the angle formed by the surface normal of two adjacent elements. The surface reduction will be performed on any two adjacent elements in which the feature angle between the two elements is smaller than the threshold. The greater the threshold, the more surface reduction will be conducted. The valid range of the threshold is [1.0, 80.0].

OSS Example Input File Parameter

Description

input_file example

Identifies the root of the input files as example, so OSSmooth will look for the files example.fem, example.grid, and example.sh.

output_file example.stl

The resulting output will be example.stl.

output_code 3

The output will be in stereolithography format.

Autobead 1

Isosurface 1

0.3

3

1

Topography results will be interpreted using the autobead feature with a threshold value of 30% creating single depth beads.

0.3

Topology results will be interpreted by creating an iso-density boundary surface with at a density value of 30% and smooth using laplacian smoothing.

laplacian_smoothing 10

Remesh 1

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30

1

The Laplacian smoothing will run for 10 iterations, consider a feature angle of 30-degrees and including the boundary in the smoothing. The two rows of elements around the recovered geometry will be remeshed in an attempt to smooth the mesh transition.


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Running OSSmooth To run OSSmooth from the command line, type: ossmooth .oss

To run OSSmooth from the HyperMesh solver panel: 1. Select Solver from the Tools menu. 2. Click the switch and select OSSmooth. 3. After input file=, enter .oss. 4. Click solve. Note: OSSmooth standalone, which can also be invoked from the solver panel in HyperMesh, checks out 50 HyperWorks Units.

To run OSSmooth from the HyperMesh ossmooth panel: 1. Select the ossmooth panel on the post page. 2. Choose either OSSmooth (for geometry), FEA topology or FEA topography. 3. Select the OptiStruct input file ( and/or and/or ) using the file= browser. 4. Edit the OSSmooth input data by making selections on the screen. 5. Click ossmooth. 6. OSSmooth (for geometry) will write a new file with the screen settings and load the geometry recovered into HyperMesh if the data format is IGES, STL, or Nastran. FEA topology and FEA topography will update the model in HyperMesh without outputting the result; if required, data can be exported from HyperMesh. Note: OSSmooth invoked from the ossmooth panel in HyperMesh checks out 42 HyperWorks Units (21 leveled and 21 stacked).

To run OSSmooth from the HyperWorks Run Manager: 1. Select the .oss file using the Input file(s) browser. 2. The Options field must be empty. 3. Click Run.

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Interpretation of Topology Optimization Results The purpose of this functionality is to provide an iso-density surface based on the volumetric density information of a topology optimization, which is conducted using OptiStruct. OSSmooth can handle both shell and solid elements with the same parameter setting. One example of post-processing of shell element topology optimization is shown below with the following parameter setting in the OSSmooth parameter file: #general parameters input_file output_file output_code

mattel mattel.stl 3

#specific parameters isosurface laplacian_smoothing surface_reduction

1 3 0.300 10 30.000 1 1 10.000

Surface reconstruction of shell element topology optimization.

The parameter laplacian_smoothing is used for additional smoothing. In most cases, the threshold surface (isosurface with second argument 0) already creates a smooth shape. Additional smoothing (isosurface with second argument 3) maintains the topology as suggested by OptiStruct, but it can deviate from the given density distribution. If this option is used, the maximum and average smoothing error output by OSSmooth should be checked. The surface_reduction parameter is used to reduce the number of elements.

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Laplacian Smoothing Laplacian smoothing can be used in the smoothing of the results of topology optimization. The laplacian_smoothing statement controls the iteration number of when the Laplacian smoothing will be performed and the feature angle threshold to preserve normal discontinuity at corners. One smoothing result is shown below with the following parameter setting in the OSSmooth parameter file. #general parameters input_file surf output_file surf.stl output_code 3 isosurface 1 3 0.300 #specific parameters laplacian_smoothing

10

30.000

1

Laplacian smoothing creates smooth boundary iso-surface by entering 1 as the 3rd argument of the laplacian_smoothing parameter statement. The comparison of the following two figures shows that the second figure is almost ready for casting.

Fix boundary of iso-surface.

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Smooth boundary of iso-surface.

The advantages of the laplacian_smoothing statement in OSSmooth include: The flexibility of controlling the number of smoothing iterations to obtain different degrees of smoothing (possibly a smoothing quality ready for casting). Normally, the iteration number ranges from 5 to 20. Smooth boundary of iso-surface with feature angle constrain are seamlessly incorporated into the smoothing process, which is more challenging in a pure CAD system.

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Interpretation of Topography Optimization Results The autobead feature of OSSmooth allows OptiStruct topography optimization results to be interpreted as one or two level beads. The following figure shows the level of detail captured in both cases; while the 2-level approach captures more details, it is more complicated to manufacture than the 1-level interpretation, often without significant performance gain.

Autobead interpretation of topography optimization result.

One example of post-processing of topography optimization is shown below with the following parameter setting in the OSSmooth parameter file: #general parameters input_file output_file output_code

decklid decklid.fem 1

#specific parameters autobead remesh

1 1

0.300

1

Autobead result from topography optimization.

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Some topography performances are relying on the half translation part. OSSmooth can interpolate topography optimization results to 2-layer autobead (autobead third argument 2). Here is one example of creating 2-layer autobead with the following parameter setting in the OSSmooth parameter file: #general parameters input_file output_file output_code

decklid decklid.nas 1

#specific parameters autobead

1

0.300

2

2-layer autobead result from topography optimization.

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Shape Optimization Results, Surface Reduction and Surface Smoothing OSSmooth may also be used to reduce and smooth surfaces or the surfaces of a domain. The parameter statements pure_surf_reduction and pure_surf_smoothing may be used for this purpose. The file defined by input_file must be in OptiStruct, and OSSmooth can smooth the surface or the surfaces of a domain of the model. The usage of in the OSSmooth parameter file is as follows: #general parameters input_file output_file output_code

surf surf.stl 3

#specific parameters pure_surf_smoothing pure_surf_reduction

2 1

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FEA Topology for Reanalysis Note: This feature is available only in the version of OSSmooth that is embedded within HyperMesh, it is not supported by the standalone OSSmooth software. The purpose of this functionality is to provide an iso-density surface based on the volumetric density information from a topology optimization. Through tetrameshing for 3-D models and inheriting boundary conditions, the results from FEA topology can be used for quick reanalysis. FEA topology support is available for first and second order shell and solid elements. For 3-D models, the recovered iso-surface can be tetrameshed-by-property automatically. FEA topology provides two options for the processing of non-design elements: Keep smooth narrow layer around and Split all quads. Keep smooth narrow layer around will retain an artificial layer of elements around the non-design space in the interpretation, and Split all quads will split quad elements in the non-design space, if present, to generate a tetra connection between design and non-design regions. Finally, FEA topology preserves boundary conditions by inheriting them from the original model (.fem). Those boundary conditions unattached to nodes/elements after geometry recovery are deleted to ensure reanalysis. An example of FEA topology for reanalysis is shown below with the following input data definition: file

block

density threshold

0.300

Keep smooth narrow layer around

off

Split all quads

on

Result of FEA topology for reanalysis

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The same model is run, this time with Keep smooth narrow layer around on, and Split all quads off. This approach creates a layer of elements around the non-design region and pyramids around the quad elements, if quads exist, to connect to the design space tetrahedral elements.

Result of FEA topology with a layer of elements around non-design space

Tetramesh will be applied on the iso-surface result if there is one close volume at least. The advantages of the tetramesh in FEA topology include: Tetramesh can be performed by property. The flexibility of controlling the number of tetramesh retries by perturbing the density threshold value, in cases where tetramesh sometimes fails.

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FEA Topography for Reanalysis Note: This feature is available only in the version of OSSmooth that is embedded within HyperMesh, it is not supported by the standalone OSSmooth software. The FEA topography option in OSSmooth allows the results from an OptiStruct topography optimization to be interpreted as one or two level beads and recover boundary conditions upon geometry extraction. An option for iso surface is also provided for combined use, which performs the same functionality as FEA topology, with FEA topography. The following figure shows the level of detail captured in a 1-level bead and 2-level bead case while preserving boundary conditions for quick reanalysis. FEA topography support is available for first and second order elements. The input data definition for a 1-level Autobead extraction is as follows: Grid file

brkt

Threshold

0.300

Layers

1

1-level autobead result from FEA Topography

A 2-level Autobead extraction is activated with the following input data: Grid file

brkt

Threshold

0.300

Layers

2

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2-level autobead result from FEA Topography

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OptiStruct References Finite Element Analysis – Books Bathe, K-J., Finite Element Procedures (Prentice Hall, 1996). Davies, G.A.O., Fenner, R.T., and Lewis, R.W., eds., NAFEMS - Background to Benchmarks (NAFEMS, 1992). Hitchings, D., ed., A Finite Element Dynamics Primer (NAFEMS, 1992). N.N., NAFEMS - A Finite Element Primer (NAFEMS, 1992). N.N., The Standard NAFEMS Benchmarks (NAFEMS, 1990). Zienkiewicz, O.C., and Taylor, R.L., The Finite Element Method 4th Edition (McGraw-Hill, 1989). Golub, G. H., and Van Loan, C. F., Matrix Computations (The Johns Hopkins University Press).

Finite Element Analysis – Papers MacNeal, R.H., and Harder, R.L., A Proposed Standard Set of Problems to Test Finite Element Accuracy, Finite Elements in Analysis and Design, 1 (1985) 3-20.

Multi-body Dynamics – Books Haug, E., Computer Aided Kinematics and Dynamics of Mechanical Systems: Basic Methods (Prentice Hall, 1989). Shabana, A.A., Dynamics of Multibody Systems (Cambridge University Press, 1998).

Powerflow – Papers Hambric, S.A., Power Flow and Mechanical Intensity Calculations in Structural Finite Element Analysis, Journal of Vibration and Acoustics, 112 (October 1990) 542-549. Hambric, S.A. and Szwerc, R.P., Predictions of Structural Intensity Fields Using Solid Finite Elements, Noise Control Engineering Journal, 47(6) (Nov-Dec 1999) 209-217. Kuhn, M.S.; Lee, H.P. and Lim, S.P., Structural Intensity in Plates with Multiple Discrete and Distributed Spring-Dashpot Systems, Journal of Sound and Vibration, 276 (2004) 627-648.

Design Optimization – Books Arora, J., Introduction to Optimum Design (McGraw-Hill, 1989). Bendsoe, M.P., and Sigmund, O., Topology Optimization - Theory, Methods and Applications (Springer, 2003). Haftka, R.T., and Guerdal, Z., Elements of Structural Optimization (Kluwer Academic Publishers, 1996). Rozvany, G.I.N., Topology Optimization in Structural Mechanics (Springer, 1997).

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Design Optimization – Papers Bendsoe, M., and Kikuchi, N., Generating Optimal Topologies in Optimal Design using a Homogenization Method. Computer Methods in Applied Mechanics and Engineering, 71 (1988), 197-224. Canfield, R.A., Design of Frames Against Buckling using a Rayleigh Quotient Approximation. AIAA Journal, 31 (1993) 1143-1149. Canfield, R.A., High-quality Approximation of Eigenvalues in Structural Optimization. AIAA Journal, 28 (1990) 1116-1122. Choi, W.S., and Park, G.J. Structural Optimization Using Equivalent Static Loads at All the Time Intervals. Computer Methods in Applied Mechanics and Engineering, 191 (2002) 21052122. Duysinx, P., and Bendsøe, M., Topology Optimization of Continuum Structures with Local Stress Constraints. DCAMM Report, Technical University of Denmark, 1996. Duysinx, P., and Sigmund, O., New Developments in Handling Stress Constraints in Optimal Material Distribution. Proceedings of the 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, St. Louis, Missouri, 1997, 1501-1509. Fleury, C., Structural Weight Optimization by Dual Methods of Convex Programming. International Journal for Numerical Methods in Engineering, 14 (1979) 1761-1783. Fleury, C., and Braibant, V., Structural Optimization: A New Dual Method using Mixed Variables. International Journal for Numerical Methods in Engineering, 23 (1986) 409-428. Kang, B.S., Park, G.J., and Arora, J.S. Optimization of Flexible Multibody Dynamic Systems Using the Equivalent Static Load Method. AIAA Journal, 43 (4) (2005) 846-852. Kidd, B.J.G., Kemp, M.D., and Jones, R.D., Application of Topology Optimization Technology to an Automotive Bracket Component. Engineering Design Optimization: Product and Process Improvements, 1st ASMO UK Conference, West Yorkshire, 1999. Kirsch, U., On Singular Topologies in Optimum Structural Design. Structural Optimization, 2 (1990) 133-142. Meyer-Prüßner, R., Prozeßkette Topologieoptimierung. Beiträge zum NAFEMS Seminar zur Topologieoptimie rung, Aalen, GY, 1997, 12.1 - 12.5. Park, G.J., and Kang, B.S. Validation of a Structural Optimization Algorithm Transforming Dynamic Loads into Equivalent Static Loads. Journal of Optimization Theory and Applications, 119 (1) (2003) 191-200. Rozvany, G.I.N., A Critical Investigation of Industrially used Structural Topology Optimization Methods, 7th World Congress on Structural and Multi-disciplinary Optimization, Seoul, Korea, 2007. Schmit, L.A., Structural Design by Systematic Synthesis, Proc. of the 2nd Conference on Electronic Computation, ASCE, (1960), New York, 105-132. Schmit, L.A., and Farshi, B., Some Approximation Concepts for Structural Synthesis, AIAA Journal, 12 (1974), 692-699. Schmit, L.A., and Fleury, C., Structural Synthesis by Combining Approximation Concepts and Dual Methods, AIAA Journal, 18 (1980) 1252-1260. Schramm, U., Structural Optimization – An Efficient Tool in Automotive Design. ATZ – Automobiltechnische Zeitschrift, 100 (1998) Part 1: 456-462, Part 2: 566-572. (In German, English in ATZ Worldwide). Schramm, U., Thomas, H.L., Zhou, M., and Voth, B., Topology Optimization with Altair OptiStruct, Proceedings of the Optimization in Industry II Conference, Banff, CAN, 1999.

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Sigmund, O., and Petersson, J., Numerical Instabilities in Topology Optimization: A Survey on Procedures Dealing with Checkerboards, Mesh-dependencies and Local Minima. Structural Optimization, 16 (1998), 68-75. Svanberg, K., Method of Moving Asymptotes – A New Method for Structural Optimization. International Journal for Numerical Methods in Engineering, 24 (1987) 359-373. Thomas, H. L., Vanderplaats, G. N., and Shyy, Y.-K., A study of move limit adjustment strategies in the approximation concepts approach to structural synthesis, Proceedings of the 4th AIAA/USAF/NASA/OAI Symposium on Multidisciplinary Design Optimization, Cleveland, OH, 1992, 507-512. Vanderplaats, G.N., and Salajehgeh, E., A New Approximation Method for Stress Constraints in Structural Synthesis. AIAA Journal, 27 (1989) 352-358. Vanderplaats, G.N., and Thomas, H.L., An Improved Approximation for Stress Constraints in Plate Structures. Structural Optimization, 6 (1993) 1-6. Zhou, M., Difficulties in Truss Topology Optimization with Stress and Local Buckling Constraints. Structural Optimization, 11 (1996) 134-136. Zhou, M., Geometrical Optimization of Trusses by a Two-Level Approximation Concept. Structural Optimization, 1 (1989) 47-72. Zhou, M., and Haftka, R.T., A Comparison Study of Optimality Criteria Methods for Stress and Displacement Constraints. Computer Methods in Applied Mechanics and Engineering, 124 (1995), 253-271. Zhou, M., and Rozvany, G.I.N., DCOC: An Optimality Criteria Method for Large Systems, Part I: Theory; Part II: Algorithm. Structural Optimization, 5 (1992) 12-25, 6 (1993) 250-262. Zhou, M., Shyy, Y.K., and Thomas, H.L., Checkerboard and Minimum Member Size Control in Topology Optimization. Proceedings of the 3rd WCSMO, Buffalo, NY, 1999. Zhou, M., and Thomas, H.L., Alternative Approximation for Stresses in Plate Structures. AIAA Journal, 31 (1993) 2169-2174. Zhou, M., and Xia, R.W., Two Level Approximation Concept in Structural Synthesis. International Journal for Numerical Methods in Engineering, 29 (1990) 1681-1699.

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