Structural Optimization

U. Kirsch

Structural Optimization Fundamentals and Applications

With 113 Figures

Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo Hong Kong Barcelona Budapest

Prof. Uri Kirsch Technion - Israel Institute ofTechnology Department of Civil Engineering Technion City Haifa 32000 Israel

ISBN-13:978-3-540-55919-1 e-ISBN-13:978-3-642-84845-2 DOl: 10.1007/978-3-642-84845-2 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1993

The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera ready by authors 6113020-543210 - Printed on acid-free paper.

To my wife, Ira

About the Author

Uri Kirsch holds the Sigmund Sommer Chair in Structural Engineering at the Technion - Israel Institute of Technology. He brings to this book a background of over twenty years' experience in research and development related to structural optimization. He has been involved as a consultant in various engineering design projects and has taught graduate courses on design optimization at several universities. Dr. Kirsch received his B.Sc., M.Sc. and D.Sc. degrees from the Technion. He was Fulbright Research Scholar (1970 - 1971) at the University of California in Los angeles and served as Visiting Professor at Case Western University, the University of Waterloo, Carnegie-Mellon University, Virginia Polytechnic Institute and State University, Heriot-watt University and the University of Essen. He was Carnegie Fellow (1989) in the United Kingdom. Dr. Kirsch has published over sixty papers on structural optimization. He is the author of the book "Optimum Structural Design", published by McGraw-Hill, New York (1981) and Maruzen, Tokio (1983). He is also a co-author of several books. Dr. Kirsch is a member of Editorial Boards of four international journals and has been a member of various international committees on structural optimization. He also acted as chairman of the Division of Structural Engineering and Vice Dean in the Department of Civil Engineering at the Technion.

Preface

This book was developed while teaching a graduate course at several universities in the United States. Europe and Israel. during the last two decades. The purpose of the book is to introduce the fundamentals and applications of optimum structural design. Much work has been done in this area recently and many studies have been published. The book is an attempt to collect together selected topics of this literature and to present them in a unified approach. It meets the need for an introductory text covering the basic concepts of modem structural optimization. A previous book by the author on this subject ("Optimum Structural Design". published by McGraw-Hill New York in 1981 and by Maruzen Tokyo in 1983). has been used extensively as a text in many universities throughout the world. The present book reflects the rapid progress and recent developments in this area. A major difficulty in studying structural optimization is that integration of concepts used in several areas. such as structural analysis. numerical optimization and engineering design. is necessary in order to solve a specific problem. To facilitate the study of these topics. the book discusses in detail alternative problem formulations. the fundamentals of different optimization methods and various considerations related to structural design. The advantages and the limitations of the presented approaches are illustrated by numerous examples. Most of the material in the book is general and covers a wide range of applications. However. the presentation is concentrated on well established and developed areas of structural optimization. The text is intended to both the student and the practising structural engineer. Previous knowledge of optimization methods is not required; the reader is expected to be familiar with basic concepts of matrix structural analysis and structural design. but the necessary material on structural analysis is included in the book. Chapter 1 deals with the problem statement. Some typical characteristics of structural optimization are discussed and the role of automated numerical optimization in the overall design process is outlined. The background material of structural analysis needed in the rest of the book is presented. terminology used throughout the text is defined. and the general problem is mathematically formulated. Various problem formulations are introduced and some typical formulations are discussed in detail and compared. Chapter 2 presents general optimization methods used in the solution of various structural optimization problems. Some optimization concepts are flfSt introduced. and unconstrained minimization and constrained minimization methods are then

X

Preface

discussed in detail. It is demonstrated that there is no one method that can be considered as the best one, and each method has its own typical characteristics, strengths and weaknesses. To choose the most effective method for any specific application, the user should be familiar with the different methods discussed in this chapter. Chapter 3 is devoted to approximation concepts which are essential in the solution of many practical design problems. Some general approximations are developed and methods for design sensitivity analysis are presented. Approximate behavior models, intended to improve the efficiency of the repeated analyses during the solution process, are introduced. Typical approximations, often used in structural optimization, are presented and recent developments in this area are discussed. Chapter 4 shows how various problem formulations, analysis models. optimization methods and approximation concepts are integrated to introduce effective design procedures. The special problems associated with such integration are discussed. emphasizing the physical aspects and the engineering considerations. Several design procedures which combine concepts introduced in previous chapters are developed. special purpose methods are introduced and structural layout optimization, which is perhaps the most challenging and economically the most rewarding task of structural optimization. is presented. Geometrical and topological optimization are discussed considering approximations. multilevel formulations and multistage procedures. Some sections in the book are necessary for continuity. while others are needed only for those interested in greater depth in a particular area. Many sections are independent and can be omitted, or their order can be changed. As a text, the book can be used in a one-semester or two-semester course in departments of civil, mechanical or aerospace engineering. The exercises, at the end of each chapter. cover the main topics discussed in the text. The author wishes to express his appreciation to many people who helped in preparing the manuscript. In particular, to my graduate students who pointed out errors and helped clarify the presentation and to Dr. R. Levy for his useful comments. The author is indebted to the Faculty of Civil Engineering at the Technion for the technical assistance and to McGraw-Hill Book Company for the permission to use the following illustrations from the book "U. Kirsch, Optimum Structural Design. McGraw-Hill Book Company. New York. 1981": Figs. 1.6. 1.8. 1.9. 1.12, 1.15.2.1.2.3.2.4.2.6.2.7.2.10 - 2.13. 2.16 - 2.23. 3.3 - 3.5. 3.10.4.4,4.5,4.13,4.14,4.29.4.31. Uri Kirsch

Contents

1

Problem Statement

1.1 1.1.1 1.1.2 1.1.3 1.1.4 1.2 1.2.1 1.2.2 1.3 1.3.1 1.3.2 1.3.3 1.3.4 1.4 1.4.1 1.4.2

Inttoduction Automated Structural Optimization Structural Optimization Methods Historical Perspective Scope of Text Analysis Models Elastic Analysis Plastic Analysis General Formulation Design Variables Constraints Objective Function Mathematical Formulation Typical Problem Formulations Displacement Method Formulations Force Method Formulations Exercises

1 1 3 5 8 9 9 20 25 25 27 30 31 44 44 45 52

2

Optimization Methods

57

2.1 2.1.1 2.1.2 2.2 2.2.1 2.2.2 2.3 2.3.1 2.3.2 2.3.3 2.3.4 2.4 2.4.1 2.4.2 2.4.3

Optimization Concepts Unconstrained Minimum . Constrained Minimum Unconstrained Minimization . Minimization Along a Line Minimization of Functions of Several Variables Constrained Minimization: Linear Programming Inttoduction Problem Formulation Method of Solution Further Considerations Constrained Minimization: Nonlinear Programming Sequential Unconstrained Minimization The Method of Feasible Directions Other Methods Exercises

1

58 58 59 66 66 72 80 80 80 85 94 97 98 110 116 120

XII

Contents

3

Approximation Concepts

125

3.1 3.1.1 3.1.2 3.1.3 3.2 3.2.1 3.2.2 3.2.3 3.2.4 3.2.5

General Approximations . Design Sensitivity Analysis . Intermediate Variables Sequential Approximations Approximate Behavior Models Basic Displacement Approximations Combined Displacement Approximations Homogeneous Functions . Displacement Approximations along a Line Approximate Force Models Exercises

126 126 133 139 145 146 150 160 164 169 175

4

Design Procedures

179

4.1 4.1.1 4.1.2 4.2 4.2.1 4.2.2 4.3 4.3.1 4.3.2 4.3.3 4.3.4 4.4 4.4.1 4.4.2 4.4.3 4.5 4.5.1 4.5.2 4.6 4.6.1 4.6.2 4.7 4.7.1 4.7.2 4.7.3 4.7.4 4.8 4.8.1 4.8.2 4.8.3

Linear Programming Formulations Plastic Design Elastic Design Feasible-Design Procedures General Considerations Optimization in Design Planes Optimality Criteria Procedures Stress Criteria Displacement Criteria Design Procedures The Relationship Between OC and MP . Multilevel Optimal Design General Formulation Two-Level Design of Prestressed Concrete Systems Multilevel Design of Indeterminate Systems Optimal Design and Structural Control . Optimal Control of Structures Improved Optimal Design by Structural Control Geometrical Optimization Simultaneous Optimization of Geometry and Cross Sections Approximations and Multilevel Optimization Topological Optimization Problem Statement Types of Optimal Topologies Properties of Optimal Topologies Approximations and Two-Stage Procedures Interactive Layout Optimization . Optimization Programs Graphical Interaction Programs Design Procedure Exercises References Subject Index

180 180 191 197 197 · 206

· 210 · 210 · 217

.220

· 223 .225 .225 · 233 · 241 .246 .246 · 250 .254 · 255 · 259 .262 .262 .264

.267 · 275 · 278 · 278 · 280 .280 .284

· 293

.299

1 Problem Statement

1.1 Introduction 1.1.1 Automated Structural Optimization The motivation of optimization is to exploit the available limited resources in a manner that maximizes utility. The object of optimal design is to achieve the best feasible design according to a preselected measure of effectiveness. A growing realization of scarcity of the raw materials resulted in a demand for light weight and low cost structures. This demand emphasizes the need for weight and cost optimization of structures. The Design Process. The structural design process may be divided into the following four stages [99]:

a. Formulation of functional requirements, which is the fIrst step in any design procedure. Functional requirements are often established already before the structural engineer enters the design process. Examples of such situations include the required number of lanes on a bridge or the required space in an industrial building. b. The conceptual design stage, characterized by ingenuity, creativity and engineering judgment of the designer, is a critical part of the design process. It deals with the overall planning of a system to serve its functional purposes. At this stage, the designer experiences the greatest challenges as well as chances of success or failure. Selection of the overall topology and type of structure are some of the decisions made by the designer at the conceptual design stage. c. Optimization. Within a selected concept there may be many possible designs that satisfy the functional requirements, and a triaI-and-error procedure may be employed to choose the optimal design. Selection of the best geometry of a truss or the cross sections of the members are examples of optimal design procedures. The computer is most suitable to carry out this part of the design, using methods of automated search for the optimal solutions. Thus, optimization in the present context is an automated design procedure giving the optimal values of certain design quantities. d. Detailing. After completing the optimization stage, the results obtained must be checked and modified if necessary. In the fmal detailing stage, engineering judgment and experience are required.

2

1 Problem Statement

Itemtive procedures for the four stages are often required before the final solution is achieved. The portion of the structural design process that can be optimized automatically has been considembly increased in recent years. The significant progress in this field is a result of developments in structural analysis, optimization methods and automated design procedures.

Computer-Aided Analysis, Design and Optimization. The central purpose of structural analysis is to predict the behavior of trial designs. A typical example is the calculation of stresses and displacements in a structure that result from applied loads. The results of structural analyses are used to assess the adequacy and relative merits of alternative trial designs with respect to established design criteria. Structural design is the process of defining the system itself. A typical example is the evaluation of the sizes and locations of members necessary to support a prescribed set of loads. Clearly, analysis is a subproblem in the design process. The design process is usually a trial-and-error procedure where the structure is analyzed repeatedly for successive modifications in the design. The availability of high-speed digital computers has played a central role in the development of analysis capabilities. It is now possible to develop similar tools for the design of complex structures. In formulating a structural optimization problem the design variables are those quantities defining a structural system that are varied by the design modification procedure. A failure mode is any structural behavior chamcteristic subject to limitation by the designer. Afeasible design satisfies all the requirements placed on it and the objective function is a function of the design variables which provides a basis for choice between alternative acceptable designs. The term load condition refers to one of several distinct sets of loads that approximately represent the effect of the environment on the structure. The structural optimization problem is to select optimal values of the design variables such that the specified objective function is minimized and a set of specifred constraints are satis/red. In general, it is not pmctical to introduce one automated progmm that solves the complete design problem without human intemction. Optimization methods are usually used to solve specific subproblems and the field of automated design is strongly connected with computer-aided design. Computer-aided design involves user-machine intemctions and it is characterized by the designer's decisions based on displayed information supplied by the computer. The use of gmphical input-output devices facilitates crossing the user-machine interface. Automated procedures for optimal design, on the other hand, seek the optimum in a defined sense and are chamcterized by preprogmmmed logical decisions based upon internally stored information. The two approaches of automated optimal design and interactive computer-aided design are not mutually exclusive, but rather they complement one another. Both approaches are suitable for the effective use of large amounts of information associated with matrix analysis methods. As the techniques of interactive computer-aided design develop further, the needs to employ standard routines for automated design of structural subsystems will become increasingly apparent.

1.1 Introduction

3

Structural Design Approaches. Characterization of a structural design philosophy involves many considerations. In examining a particular application of optimization methods to the design of structures it will be useful to classify the design philosophy as deterministic or probability based, identify the kinds of failure modes to be guarded against, and classify with respect to consideration of service load and/or overload conditions [106].

a. Deterministic or probability-based design philosophy. Structural systems are usually subjected to external loadings that are complex and continuously changing in time. In design practice, the environment is usually replaced by a finite number of distinct loading conditions which may be evaluated based on deterministic or probabilistic design philosophies. In addition, the design parameters are often affected by random variables. If any of the quantities involved in the structural design (loadings, material properties, etc.) are treated as random variables, the formulation is classified as probability based. If all the quantities are treated as deterministic, then the formulation is so classified. b. Kinds of failure modes. There are various ways of seeking to ensure that a structural system will perform its specified functional purposes and to avoid the occurrence of various kinds of failure modes. Adequate performance of structural systems may be sought by trying to avoid failure modes such as initial yielding, excessive deflection, and local damage under service load conditions, and by preventing failure modes such as rupture, collapse, and general instability under overload conditions. The definition of failure may vary from one design task to another. c. Service load and overload conditions. The kinds of failure modes considered under service load conditions will usually differ from those considered under overload conditions. While the former are defined as design load conditions representative of normal use, the latter are defined as load conditions representative of certain anticipated extraordinary or emergency situations. Based on this classification, various design approaches can be considered [62]. 1.1.2 Structural Optimization Methods The available methods of structural optimization may conveniently be subdivided into two distinctly different categories called analytical methods and numerical methods. While analytical methods emphasize the conceptual aspect, numerical methods are concerned mainly with the algorithmical aspect. Analytical methods are beyond the scope of this text, but are briefly described herein. Analytical Methods. Analytical methods are usually employing the mathematical theory of calculus, variational methods, etc., in studies of optimal layouts or geometrical forms of simple structural elements, such as beams, columns, and plates. These methods are most suitable for such fundamental studies of single structural components, but are usually not intended to handle larger structural systems. The structural design is represented by a number of unknown functions and the goal is to find the form of these functions. The optimal design is

4

1 Problem Statement

theoretically found exactly through the solution of a system of equations expressing the conditions for optimality. An example for this approach is the theory of layout, which seeks the arrangement of uniaxial structural members that produces a minimum-volume structure for specified loads and materials. The basic theorems of this approach were established by Maxwell [95) and Mitchell [97). Since they are applied without meaningful constraints on the geometric form of the structure, such theorems often yield impractical solutions. Work on analytical methods, although sometimes lacking the practicality of being applied to realistic structures, is nonetheless of fundamental importance. Analytical solutions, when they can be found, provide valuable insight and the theoretical lower bound optimum against which more practical designs may be judged. Problems solved by analytical methods, often formulated by functions describing continuous distribution of material over the structure, are called continuous problems, or distributed parameter optimization problems. It is instructive to note that some distributed parameter problems can be solved numerically. Numerical methods. Numerical methods usually employ a branch in the field of numerical mathematics called mathematical programming. The recent developments in this branch are closely related to the rapid growth in computing capacities. In the numerical methods, a near optimal design is automatically generated in an iterative manner. An initial guess is used as a starting point for a systematic search for better designs. The search is terminated when certain criteria are satisfied, indicating that the current design is sufficiently close to the optimum. Rapid developments in the programming methods as well as in the application of such methods in design facilitate the solution of realistically large practical design problems. Problems solved by numerical methods are called finite optimization problems or discrete parameter optimization problems. This is due to the fact that they can be formulated by a finite number of variables. Assignment of numerical values to these variables specifies a unique structure. Design optimization of practical structures is accomplished mainly by the use of finite formulations. Some of the mathematical programming methods, such as linear. quadratic. dynamic. and geometric programming algorithms, have been developed to deal with specific classes of optimization problems. A more general category of algorithms, referred to as nonlinear programming. has evolved for the solution of general optimization problems. Though the history of mathematical programming is relatively short, there has been a large number of algorithms developed for the solution of numerical optimization problems. However, there is no single best method for all optimization problems. There is an obvious need, therefore, for familiarity with the basic concepts of numerical optimization. Another approach for numerical optimization of structures is based on derivation of a set of necessary conditions that must be satisfied at the optimum design and development of an iterative redesign procedure. It has been shown that the latter special purpose optimality criteria methods and the general mathematical programming approach have coalesced to the same method under certain assumptions.

1.1 Introduction

5

Advantages and Limitations of Numerical Methods. Of the engineering disciplines, sttuctural design has probably seen the most widespread development and application of numerical optimization techniques. Using numerical optimization as a design tool has several advantages: - Reduction in design time and improving the quality of the design. Optimization is an effective tool to reach a high quality design much faster. Even in cases where optimization by itself does not save design time or cost, the final result is a product that is superior. - Dealing with large numbers and a wide variety of design variables and constmints relative to traditional methods. - Applying systematized logical design procedures may lead to improved, nontraditional and unexpected results, particularly in a new design environment. One of the most effective uses of numerical optimization is to make early design trade-offs using simplified models. The advantage is that we can compare optimal designs instead of just comparing nonoptimal solutions. On the other hand, numerical optimization has some limitations to be aware of : - The quality of the result is only as good as the assumed analysis model. That is, optimization techniques are limited to the range of applicability of the analysis method. - Incomplete problem formulation, such as ignoring an important constraint, may lead to meaningless if not dangerous design. Furthermore, improper formulation might reduce the real factors of safety that now exist. - The number of design variables is restricted due to the computational effort involved in solving large problems by many optimization methods. - Most optimization algorithms can solve problems with continuous functions. In addition, highly nonlinear problems may converge slowly or not at all. - In many problems it cannot be guaranteed that the global optimum design will be obtained. Therefore, it may be necessary to restart the optimization process from several different designs and compare the results. In summary, optimization techniques can gready reduce the design time and yield improved, efficient and economical designs. However, it is important to understand the limitations of these techniques. In addition, it should be recognized that the absolute best design will seldom be achieved. Thus, optimization methods can be viewed as a valuable and convenient tool to achieve improved designs rather than theoretical optima

1.1.3 Historical Perspective Several review papers on developments in sttuctural optimization have been published. Among others, Refs. [51, 90, 92, 93, 100, 103, 110, 114, 115, 118120, 126, 137, 140, 142] review the trail of developments in this area. A chronological bibliography covering the period 1940 - 1956 is contained in an Appendix of Ref. [41]. During the 1980's, Schmit [119] and Vanderplaats [137] reviewed these developments. The historical perspective presented here is based mainly on the latter reviews.

6

1 Problem Statement

Early Developments. The structural optimization literature prior to 1960 can be divided into three main categories as follows:

a. the classical literature dealing with the least weight layout of highly idealized frameworks; b. optimum design of structural components based on weight-strength analysis or structural index methods; and c. minimum weight optimum design of simple structural systems based on the plastic collapse or limit analysis design philosophy.

Perhaps the fIrst analytical work on structural optimization was by Maxwell in 1890 [95], followed by the better-known work of Mitchell in 1904 [97]. These works provided theoretical lower bounds on the weight of trusses, and, although highly idealized, offer considerable insight into the structural optimization problem and the design process. Minimum weight optimum design of basic aircraft structural components, such as columns and stiffened panels, subject to compressive loads was initially developed during World War II. Subsequently, during the late 1940's and the early 1950's a great deal of effort went into the development of minimum weight design methods for aircraft structural components subject to buckling constraints [41, 125]. During the 1940's and the early 1950's availability of the digital computer led to application of linear programming techniques to plastic design of frames [54]. This early numerical work is particularly significant in that it used mathematical programming techniques developed in the operations research community to solve structural design problems. During this period plastic design problems could be fonnulated as linear programming problems, and the application of mathematical programming techniques to structural optimization was limited to truss and frame structures. This type of structural optimization was focused primarily on steel frame structures and it did not consider stress, displacement or buckling constraints under service load conditions. During the late 1950's, the space programs created a strong demand for light weight structures and provided the resources necessary to develop new design approaches. In addition, digital computers were becoming commonly available and the fInite element method was offering the designer a powerful tool for analysis of complex structures. Thus, the setting of the late 1950's was in many ways ideal for major advances in structural optimization. Modern Structural Optimization. Schmit [117] in 1960 was the fIrst to offer a comprehensive statement of the use of mathematical programming techniques to solve the nonlinear-inequality-constrained problem of designing elastic structures under a multiplicity of loading conditions. This work is signifIcant, not only in that it ushered in the era of modem structural optimization, but also because it offered a new philosophy of engineering design which only in the 1980's began to be broadly applied. The important unique contribution of this work was that it introduced the idea and indicated the feasibility of coupling fInite element structural analysis and nonlinear mathematical programming to create automated optimum design capabilities for a rather broad class of structural

1.1 Introduction

7

systems. Working within the elastic design philosophy, it was shown that the minimum weight optimum design of elastic statically indeterminate structures could be stated as a nonlinear mathematical programming problem in a design variables space. Mathematical programming (MP) techniques were shown to be an effective tool for design of numerous civil, aeronautical, and space structures. This promising tool was new and much development required to establish the methodology. Indeed, these structural synthesis concepts were considered by many researchers to be a revolutionary change in the traditional approach to design. In the 1960's, enough computational experience had been documented to indicate that MP techniques applied to structural design were limited to only a few dozens design variables. Thus, despite the generality of MP, applications were limited to relatively small structures. In the late 1960's an alternative approach, called Optimality Criteria (OC), was presented in analytical form by Prager and coworkers [108, 109] and in numerical form by Veokayya and coworkers [141]. Although this approach was largely intuitive, it was shown to be most effective as a design tool. Its principal attraction was that the method was easily programmed for the computer, was relatively independent of problem size, and usually provided a near-optimum design with a few structural analyses. This last feature represented a remarkable improvement over the number of analyses required for MP methods to reach a solution. Consequently, much research effort was devoted to OC methods during the early and mid 1970's. MP was attractive due to its generality and rigorous theoretical basis. On the other hand, OC had no clear theoretical basis and would, on occasion, lead to nonoptimum designs. However, OC offered a solution for a variety of practical design problems. The main obstacles to the implementation of efficient MP methods prior to 1970 were associated with the large problem size (large numbers of design variables and constraints) and the need to repeat the structural analysis many times. Much effort has been devoted to solve these problems since the mid 1970's. The introduction of approximation concepts, via reduction of the problem size and the construction of high quality explicit approximations for the constraints, has led to the emergence of MP based structural optimization methods that are computationally efficient During the late 1970's and 1980's, development continued in both OC and MP approaches to structural optimization. The dual MP formulation was interpreted as a generalized OC method, and was presented as a basis for coalescing of the two approaches. Approximation concepts have been used for efficient solution of the optimization problem, and have been combined with the dual formulation to create new tools. In recent years, the range of applicability of structural optimization has been widened and much progress has been made in various topics associated with this area. Efficient techniques for derivative calculation have been developed, and problems with complex analysis model and various types of constraints and objective function have been investigated. Extensive research and development is continually being done on such topics as shape and layout optimization, decomposition of large scale problems, optimal control of structures and application of genetic algorithms. The significant progress in these advanced topics

8

1 Problem Statement

emphasizes the need for a deeper insight and understanding of the fundamentals of structural optimization. There are clear indications that optimum structural design methodology is coming of age. It has matured significantly and has grown in the last three decades from an abstract concept to a practical tool which is currently serving the quest for better structural designs. Although structural optimization has not yet achieved the near universal acceptance level enjoyed by fmite element analysis methods, a ftrlO knowledge and experience base exists for the further development of rather general and efficient capabilities.

1.1.4 Scope o( Text There is a growing demand for general-purpose optimization methods that are suitable for use with general-purpose software packages for structural optimization. In addition, the high computational cost associated with the analyses of many practical structures motivated the development of design procedures that do not involve multiple repeated analyses. Employing general-purpose mathematical programming methods and approximation concepts is the remedy for this obstacle in many structural optimization problems. Following these considerations, this text is mainly dealing with approximate problem formulations and general-purpose analysis and optimization methods. The broad scope of the structural optimization field is such that there are many different possible problem classifications, including the following problem areas : - The mathematical type of design variables : continuous, discrete or mixed continuous-discrete design variables. - The physical significance of design variables, describing the topology, geometry (configuration), material properties, or cross-sectional dimensions of elements. - The design philosophy: deterministic or probability based design philosophy. - The kinds of failure modes: initial yielding, excessive deflections and local damage under service load conditions; or rupture, collapse and general stability under overload conditions. The corresponding constraints are related to elastic (service load) and plastic collapse (overload) conditions. - The type of objective function: single criterion or multicriterion, representing cost, weight, performance, reliability, etc. - The structural response: static or dynamic (time dependent), linear or nonlinear response. - The type of optimization problem: continuous (distributed parameter) optimization problems or finite (discrete parameter) optimization problems. - The solution method: numerical or analytical optimization methods. Most of the material in this text is general and covers a wide range of applications. However, the presentation is concentrated on well established and developed areas of structural optimization. The design variables describe the form or the shape of the structure: the cross-sectional dimensions of elements, and the geometry and topology of the structure. The design philosophy is deterministic, and the assumed failure modes are related either to service load or to overload conditions. The

1.2 Analysis Models

9

constraints are related to the common stress and displacement limitations under service loads, and plastic collapse constraints under overload conditions, as well as technological constraints on the design variables. Static linear structural response and fmite (discrete parameter) optimization problem formulations are considered. A single criterion objective function is assumed, representing the cost or the weight of the structure, and numerical methods of structural optimization are used to solve the design problem. The remainder of this chapter deals with general analysis tools and various formulations of optimal design problems. General optimization methods that can be used to solve various optimal design problems are discussed in Chap. 2. Approximation concepts, which are essential in optimal design of practical structures, are presented in Chap. 3. Various problem formulations, optimization methods and approximate analysis models are combined to introduce design procedures in Chap. 4. The special problems associated with such integration are discussed, emphasizing the physical aspects and the engineering considerations. Finally, optimal design of the structural geometry and topology, that can greatly improve the design, is demonstrated.

1.2 Analysis Models Structural analysis is a main part of any optimal design formulation and solution. Furthermore, it has been noted that in many formulations the analysis must be repeated many times during optimization. In this section a general description of the most commonly used analysis methods is given. Only the background material needed in the rest of the book is covered. A detailed discussion is given in many texts related to structural analysis (e.g. [143]). Specifically, the following analysis methods will be reviewed:

a. Methods for elastic analysis of framed structures, such as beams, frames and trusses. The basic relations are first presented; the force method, the displacement method and the virtual load method, which have widely been used in structural optimization, are then briefly described. b. Plastic analysis methods for framed structures. Only the static approach, which is often used in optimal design formulations, is presented.

1.2.1 Elastic Analysis In linear elastic analysis we assume that displacements (translations or rotations) vary linearly with the applied forces, that is, any increment in displacement is proportional to the force causing it. All deformations are assumed to be small, SO that the resulting displacements do not significantly affect the geometry of the structure and hence do not alter the forces in the members. The majority of actual structures are designed to undergo only small and linear deformations.

10

I Problem Statement

Referring to behavior under working loads, the objective of the analysis of a given structure is to detennine the internal forces, stresses, and displacements under application of the given loadings. The forces must satisfy the conditions of equilibrium and produce defonnations compatible with the continuity of the structure and the support conditions. That is, any method of elastic analysis will ensure that both conditions of equilibrium and compatibility are satisfied. Basic Relations. The relations presented here fonn the basis for elastic analysis by various methods. The equilibrium equations are C A=R

(1.1)

in which the elements of matrix C depend on the undefonned geometry of the structure ; A is the vector of members' forces ; and the vector R represents the extemalloads. t The constitutive law is (1.2)

or

(1.3)

where F d and Kd are diagonal matrices of member flexibilities and member stiffnesses, respectively, and e is the vector of member displacements. The compatibility equations relate the member displacements e to the nodal displacements r by Qr=e

(1.4)

Q=CT

(1.5)

in which

Various analysis methods can be derived from these basic relations.

Displacement method. Substituting (1.3), (1.4) and (1.5) into (1.1) yields (1.6) Denoting the system stiffness matrix by K, where (1.7) tThe following symbols have been used throughout this text: bold letters represent matrices or column vectors; superscripts T represent transposed matrices or vectors; subscripts d denote diagonal matrices.

1.2 Analysis Models

11

the displacement method equilibrium equations (1.6) become

Kr=R

(1.8)

Force method. Equation (1.4) can be rewritten as (1.9) in which subscripts B and R are related to the basic statically determinate structure and the remaining redundants, respectively. From (1.9) (1.10) in which (1.11) Rewrite (1.1) and (1.2), respectively, as (1.12) (1.13)

Substituting (1.13) into (1.10) yields (1.14) Substituting AB from (1.12) into (1.14) gives

(1.15) Rearranging this equation and substituting

(1.16) (1.17) (1.18) gives the force method compatibility equations

12

1 Problem Statement

(1.19)

F N= ~

in which F is the system flexibility matrix, N is the vector of redundant forces and ~ are displacements corresponding to redundants.

Example 1.1. The purpose of this example is to demonstrate the basic relations of elastic analysis presented in this section. Considering the four-bar truss shown in Fig. 1.1, the equilibrium equations (1.1) aret

[..fia ..fia

01 -..fia -1] {~:} =..fi {10} ..fia 0 A3 10 ~

and the constitutive law [(1.2) or (1.3)] is

where E is the modulus of elasticity and ai is the the cross-sectional area of the ith member. The compatibility relations (1.4) are

and the displacement method equilibrium equations (1.8) are

Choosing members 3 and 4 as redundants, then from (1.9), (1.10), (1.12) and (1.13) we have

12 ..fi 12] r ={e3 } [-..fi -1 0 e4 t All

dimensions throughout this text are arbitrary, unless certain dimensions are specified.

1.2 Analysis Models 100

100

20

Fig. 1.1. Four-bar truss.

The inverse of CB is

C-1 =[...fi

0] 1

-1

B

and the force method compatibility equations (1.19) are

...fi2...fi

2 ...fi

-+-+-

100 lit E

llz

OJ

2 ...fi

-+al

a2

...fi

-+lit

llz

{~}=_ 20~100

2...fi

1

1

al

a2

a4

-+-+-

al

2 al

13

14

1 Problem Statement

Force Method. In theforce. or flexibility method. redundant forces are chosen as the analysis unknowns. Sufficient releases are provided by removing the redundant forces, their number equal to the degree of statical indeterminacy, to obtain a statically determinate structure, called the basic structure, or the primary structure. The primary structure undergoes inconsistent deformations, and the inconsistency in geometry is then corrected by the application of the redundant forces. The value of the latter is computed from the conditions of compatibility. With the redundant forces known, all internal forces. stresses, and displacements are determined by superposition of the effects of the external loads and the redundant

forces.

The compatibility equations to be solved by the force method are FN+Sp=So

(1.20)

in which F = flexibility matrix, whose elements, Fii. represent the displacement at i due to a unit redundant atj in the primary structure; both i andj are coordinates corresponding to the unknown redundants; N =the vector of unknown redundant forces; Sp =the vector of displacements corresponding to redundants due to loading in the primary structure; So =the vector of actual displacements corresponding to redundants (in general, So =0). Derming the vector

(1.21) the compatibility equations (1.20) become [see (1.19)] F N=S

(1.22)

The vector of unknown redundants. N, is computed by solving the set of simultaneous linear equations (1.22). It is important to note that the flexibility matrix, F, is dependent on the choice of redundants. With different redundants, the same structure would result in a different flexibility matrix. Final displacements D and forces A at any desired points in the structure are expressed explicitly in terms of N by the following superposition equations of the effect of external loads and the effect of redundants on the primary structure D=Dp+DNN

(1.23)

A =Ap+ANN

(1.24)

in which Dp , Ap = vectors of displacements and forces. respectively, due to loads in the primary structure, and D N, AN = matrices of displacements and forces. respectively, due to unit value of redundants in the primary structure. Equations (1.22) , (1.23) and (1.24) are related to the action of a single loading condition. In the case of several loading conditions all vectors will be transformed into matrices so that each of their columns will correspond to a certain loading condition.

1.2 Analysis Models

15

In the present formulation the elements of So are constants, the elements of F,

Sp DN and Dp are functions of both the geometry and members' cross section, and ~ elements of AN and Ap depend only on the geometry of the structure. Example 1.2. To illustrate solution by the force method, consider the simple continuous beam shown in Fig. 1.2a. The beam has a constant flexural rigidity EI, and the object is to find the forces and the rotations at the supports. The structure is statically indeterminate to the second degree, so that two redundant forces must be determined. The chosen redundants, Nb N 2 , and the corresponding primary structure (a cantilever beam) are shown in Fig. 1.2. The required coefficients, computed in the primary structure, are

(a)

(b)

AN11

AN11

C~

t

at

ANII

AHll

(c)

Fig. 1.2. Continuous beam example: a. Loads and redundants. b. External loading on the primary structure, c. Unit redundants on the primary structure.

16

1 Problem Statement

~ _{~PI}_~ {26} p -

Ap

~p2

-

48EI

~o ={~}

97

ApI} { 2P }

={ Ap2 = -Pi/2

Substiblting into (1.22) we fmd the unknown redundant forces

i 3 [2 5] {NI} Pi3 {26} 6EI 5 16 N2 + 48EI 97 =

{a}°

NI}=.!.. { 69} {N2 56-64

The desired displacements, D, and forces, A, are [(1.23) and (1.24)]

D

~} Ptl {10} i 2 ={ D2 =8EI 13 + 2EI

AI} {2P } A= {A2 = -Pi/2 +

[II 43] 56P {-6469} =112EI Pi {17} -5 2

[-I -I] {69} = {107} -i -21

P

56

-64

P

56

3U

Displacement Method. In the displacement. or stiffness method, restraints are added to prevent movement of the joints, and the forces required to produce the restraints are determined. Joint displacements, chosen as the analysis unknowns, are determined from the conditions of equilibrium . The internal forces, stresses, and displacements (other than joint displacements) are then determined by superposition of the effects of the external loads and the separate joint displacements. The equilibrium equations to be solved by the displacement method are (1.25)

K=

stiffness matrix, whose elements Kij represent the force in the ith in which coordinate due to unit displacement in the jth coordinate (Kij are computed in the restrained structure and i and j are coordinates corresponding to displacement degrees of freedom); r = the vector of unknown displacements; RL = the vector of forces corresponding to the unknown displacements in the restrained structure; Ro =the vector of extemalloads corresponding to the unknown displacements (Ro =0 if there are no loads acting in the direction of degrees of freedom). Defining the load vector R =Ro - RL • then (1.25) becomes [see (1.8)]

Kr=R

(1.26)

1.2 Analysis Models

17

The vector of unknown displacements r is computed by solving the set of simultaneous equations (1.26). The stiffness matrix K (in contrast to the flexibility matrix F), is determined uniquely for a given structure. Final displacements D (other than those included in r) and forces A at any desired points in the structure are given by the following superposition equations (1.27) (1.28) in which Dl,o AL= vectors of displacements and forces, respectively, due to loads in the restrained structure; Dr, Ar = matrices of displacements and forces, respectively, due to unit value of the components of r in the restrained structure. In many cases AL=O and the stresses (J can be determined from (1.28) by (J=Sr

(1.29)

in which S is the stress transformation matrix. In some structures, such as trusses, the elements of S are independent of the cross sections of the elements. All equations are related to the action of a single loading. In the case of several loading conditions all vectors will be transformed into matrices so that each of their columns will correspond to a certain loading condition. It can be observed that the elements of K, DL , Dr and Ar are functions of both the geometry and members' cross section. If the loads on the structure are predetermined, the elements of R and AL depend only on the geometry of the structure.

Example 1.3. To illustrate solution by the displacement method, consider the continuous beam shown in Fig. 1.3a. The beam has a constant flexural rigidity EI, and the object is to fmd the forces at the left-end support, Ai and A2• The structure has two degrees of freedom, the two support rotations ri and r2, which are the unknown displacements. The coefficients computed in the restrained structure are

Ro

AL

R=Pl {-3}

={_p~}

48

ALl} ={PIl} ={AL2 Pl/8

Ar =[~11 Ar2i

-43

[3 00]

Ar12] = 2EI 12 I

~22

Substituting into (1.26) we fmd the unknown displacements

.± 3

EI I

[5 I] {Ii} = 1 5

r2

-3}

PI { 48-43

Pl2 { 7} {r2Ii} = 384EI -53

18

1 Problem Statement

IP

Pi

~r·_"""",==t==::;;;;--i~~' ...I~Cf1= =U!Ol. . l.QI. !;P: !:/: :f: 1::1=p:;;;;;!:.I1i'~jOJOOf:Il~':""'oI.,l .I..,"Uy..I..i!lL.JI"i~~ ~ ::::::==' _

A2 (

Alt.~ 0-

__

;:::>

_

i_ _

rl

'2

~_____I_.5_i_ _ _ _~_____i____~ _I •

'1-

'1

(a)

RI.\

A 1.2 (

R,.2

~L_~lLP_...:....D..l~~%,j':J':J':J,:r':r,:rtj(j:~III'I' I'IIDII :t;~:C,1 II

I:I I:ICII~I~

1:'I::1:11:'

tAI.\

I:'

(h)

Am

C~

............

t Arll

--- i

1\21

1\11

1)

1.0

7¥

n

K~'!.

KI2

Am

C~

t Ar12

'"~

~

--------

1.0

- ;Yr1 ----- ~ /"

/

fe)

Fig. 1.3. a. Continuous beam example, b. Loads on the restrained structure, c.Unit displacements on the restrained structure.

The resulting forces A are computed by (1.28)

A=

2EI[3 0] Piz {7} P {117} {AAI}z = {P12} Pi/8 +7 i 0 384EI -53 = 192 3U

Virtual-Load Method. Application of the virtual-load method is convenient in many optimal design problems. Using this method, the displacements D are expressed in terms of the members forces by

1.2 Analysis Models

D= LTj(Aj)lgj(Xj )

19

(1.30)

j

where the elements of the vectors T j are given by T;j =

~

J

Aj

A;9dl j

(1.31)

j

Aj is the force in the jth member due to the actual loads;

A;9

is the force in the

jth member due to a virtual load Qi= 1.0 applied in the ith direction; lj is the member length; E is the modulus of elasticity; and glXj ) is a function of Xj' representing the cross-sectional properties of the jth member (for example, crosssectional area in truss elements, or moment of inertia in beam elements). The displacements expression (1.30) is used particularly in problems where only a small number of displacements are to be considered. It should be noted that (1.30) is based on the assumption that a single force (such as axial force or bending moment) is sufficient to describe the response behavior of each member. However, this fonnulation can be extended to the more general case of multiple force members. Assuming the common case where (1.32) then (1.30) becomes D= L

TjlXj =T l/X

(1.33)

j

in which matrix T consists of the vectors T j and the vector 1/X consists of the elements l/Xj. Writing (1.33) for the displacement degrees of freedom r=T l/X

(1.34)

the stresses become [see (1.29)] cr=S r=ST l/X=P l/X

(1.35)

where matrix P is defined by P=ST. It has been noted that in truss structures the elements of S are independent of the members' sizes. If the truss is statically detenninate, the forces Aj and

A;9 are also independent of the members' sizes, and

the elements of matrix T depend only on the truss geometry. For indetenninate structures, where the force distribution depends on the members' sizes, the elements of matrix T are usually implicit functions of the cross-sectional dimensions.

20

1 Problem Statement

Example 1.4. Consider the four-bar truss shown in Fig. 1.1, with the following forces obtained for the given cross-sectional areas Xj =1.0 AT = {8.28, 8.28,0, -8.28}

The forces A are implicit functions of the areas Xj and satisfy the conditions of equilibrium and compatibility. The forces AQ, due to unit loads in the directions of rl and r2' must satisfy only the conditions of equilibrium and may be calculated for a statically equivalent system where some redundant forces will be arbitrarily set to zero. Assuming, for example, A2 =0 for QI = 1.0 and A4 =0 for Q2 = 1.0, the resulting forces are given by Q T

(A ) =

[0.354 0 -0.354 -0.5] 0.354 0.5 0.354 0

The corresponding displacements expression is given by (1.34)

r

=

100[4.14 E

4.14

o

1/ Xl] [I/X

0 4.14] 1/ X2

4.1400

3

1/ X4

and the stresses are given by (1.35)

2.07 o 0.5 0.5] [ 4.14 E 0 1 4.14 4.14 o 0"=- [ r= 0 2.07 o -2.07 VX3 100 -0.5 0.5 0 -1 0 -4.14 -4.14 0 o VX 4 -8.28

2.~7 ]{~::} {::~}

1.2.2 Plastic Analysis Modem design of structures is based on both the elastic and plastic analyses. The plastic analysis cannot replace the elastic analysis but supplements it by giving useful information about the collapse load and the mode of collapse. An elastic analysis of a structure is important to study its performance, especially with regard to serviceability, under the loading for which the structure is designed. However, if the load is increased until yielding occurs at some locations, the structure undergoes elastic-plastic deformations. On further increase a sufficient number of plastic hinges are formed to transform the structure into a mechanism. The main object of plastic analysis is to determine the collapse load of a structure when resisting capacities of its members are known. The design of structures based on plastic approach, referred to as limit-design, is increasingly used and accepted by various codes of practice.

1.2 Analysis Models

21

... t:

'" E

Yield stress

o

M pl= fully plastic moment

~

Curvature

Strain (b)

(a)

Fig. 1.4. 8. Idealized stress-strain relation, b. Idealized moment-curvature relation.

While the exact calculation of the collapse load of a structure requires the solution of a costly nonlinear system of equations, it is possible to obtain a conservative estimate of that load by assuming an elastic-perfectly-plastic behavior. The material is assumed to deform in the idealized manner shown in Fig. 1.4. The strain and stress in Fig. 1.4a are proportional to one another up to the yield stress, at which the strain increases without any further increase in stress. In members subjected to bending the idealized relation shown in Fig. I.4b, between the bending moment and curvature at a section, is assumed. The curvature and bending moment are assumed to be proportional to one another up to the fully plastic moment Mpl. At the fully plastic moment a plastic hinge is formed, and the curvature (rotation at the hinge) is increased without any increase in the moment The rotations at the cross section before Mpl is reached are considered to be relatively small and the equilibrium equations are referred to the undeformed geometry of the structure. It is assumed that plastic hinges are concentrated at critical sections with ductility being unlimited. In addition, the loads applied to the structure are assumed to increase proportionally. Existing methods for plastic analysis are based on either the kinematic approach or the static approach [55]. The latter approach, which is often used in optimal design formulations, is briefly described herein. Static Approach. According to the static theorem of plastic analysis [55] , the bending moment distribution at collapse is such that the corresponding load factor is the largest statically admissible multiplier, i.e.,

A= max(AJ

(1.36)

The number of statically admissible bending-moment distributions (satisfying the equilibrium and yield conditions) is infinite. Consider such a distribution, Mj (j 1,... , J), for the given structure under the admissible loads, AR, so that nE independent equilibrium equations and the yield conditions for all critical sections are satisfied. The problem of collapse load analysis under proportional loading can be formulated as follows [18]: Find Aand Mj (j 1,... , J) such that

=

=

22

1 Problem Statement

A ~ max J

L C/cjMj=AR"

(1.37)

k

= t .... nE

j

=

(equilibrium equations)

(1.38)

j=l

1. ... .J

(yield conditions)

(1.39)

The number of independent equations of equilibrium is nE= J - nR, where nR is the degree of statical indeterminacy; Rk are the loads; and Ckj are constant coefficients. The equilibrium equations (1.38) could be obtained from (1.1), where A and R are replaced by M and AR, respectively. Equations (1.37) through (1.39) represent a linear programming problem with (J+ 1) variables and (2J+nE) constraints. If the directions of M j are known, the number of inequalities in (1.39) is J. and the number of constraints is reduced to (J+nE). Example 1.5. Consider the continuous beam shown in Fig. 1.5 with a uniform plastic moment Mpl. The number of critical sections is 3 and the number of equilibrium equations is nE = 3 - 1 = 2. The linear programming problem of plastic analysis is to fmd Aand M j (j = 1,2,3) such that A~

4Ml

max

+ 2M2

2M2 + 4M3 -Mpl~ M j ~ Mpl

AP

A?

I.

! 1

i

= APt = 2APt j= 1, 2. 3

2AP 2

2\:

.1.

! 3

i

Do

.1

(b)

Fig. 1.5. Continuous beam example: a. Collapse loads. b. distribution.

Bending-moment

1.2 Analysis Models

23

4

3 ::!

max A

-3

Fig. 1.6. Graphical presentation of beam example in the space of ). and M 2 .

The equilibrium equations can be obtained from the bending-moment distribution shown in Fig. 1.5b. Using the latter equations we may eliminate the variables M 1 and M 3. and obtain the following linear programming problem in terms of only two variables: fmd Aand M2 such that

A-+ max -Mpl

~( ~i A-~M2)

-Mpl

~

-Mpl

~(~i A-~M2)

M2

This formulation could be obtained also from (1.24). where all bending moments are expressed in terms of the chosen redundant force M 2 = N. A graphical presentation of this problem in the space of A and M2 is shown in Fig. 1.6. The solution is A=3Mpl

Pi

24

1 Problem Statement

p

Fig. 1.7. Plane truss example.

Example 1.6. Plastic analysis procedures can be employed also in statically indeterminate trusses subjected to axial forces. To illustrate this possibility, consider the plane truss shown in Fig. 1.7 subjected to a single load P. The given ultimate axial forces in members 1, 2 and 3 are ApI' 2Apl, and 1.5Apl, respectively. The corresponding members' forces are Ah A z, A3 . The number of equilibrium equations is nE = 3 - 1 = 2, and the plastic analysis problem is: find A,At,Az andA3 such that A~max

AI /{2 + Az + A3 /{2 = AP

At /{2

-A3/{2 = 0

At ~Apl Az ~2Apl A3 ~1.5Apl

Using the equilibrium equations to eliminate Al and A z , the problem is formulated in terms of only A and A z as follows: fmd A and A z such that A~max

(AP - Az )/ {2 ~ Apl

Az ~2Apl (AP-Az)/..fi ~1.5Apl

The solution of this problem is (see Fig. 1.8)

A = 3.414Apl l P

1.3 General Formulation

25

Fig. I.S. Solution of plane truss example.

1.3 General Formulation 1.3.1 Design Variables A structural system can be described by a set of quantities, some of which are viewed as variables during the optimization process. Those quantities defining a structural system that are fixed during the automated design are called preassigned parameters and they are not varied by the optimization algorithm. Those quantities that are not preassigned are called design variables. The preassigned parameters, together with the design variables, will completely describe a design. Quantities are designated as preassigned parameters for a variety of reasons. It may be that the designer is not free to choose certain parameters, or it may be known from experience that a particular value of the parameter produces good results. Often, by considering some quantities fixed, i.e., invariant during the optimization process, the problem is greatly simplified. From a physical point of view, the design variables that are varied by the optimization procedure may represent the following properties of the structure [123]:

a. the mechanical or physical properties of the material; b. the topology of the structure, i.e., the pattern of connection of members or the number of elements in a structure; c. the geometry or configuration of the structure; d. the cross-sectional dimensions or the member sizes.

From a mathematical point of view, it is important to distinguish between continuous and discrete design variables. In cases of discrete variables with a large

26

1 Problem Statement

number of values uniformly distributed over a given interval, use of a continuous variable representation is often satisfactory, followed by selection of the nearest available discrete value. When a strictly discrete design variable is handled in this way, it will be categorized as pseudodiscrete. However, it should be recognized that situations arise when it will be essential to employ discrete or integer variables; the latter represent the number of elements in the structure, for example. In general, the design variables are the independent ones in the optimization problem. Once their values are chosen in one way or another, the structure is completely determined, and its behavior can be evaluated from the analysis equations. By behavior we mean quantities that are the result of an analysis, such as forces, stresses, displacements, etc. Since the behavior of the structure is dependent on the value of the design variables, it can be represented by a set of dependent behavior variables. Other possibilities of selecting the independent variables will be discussed later in this section. Material Design Variables. Material selection presents a special problem with conventional materials, as they have discrete properties, i.e., a choice is to be made from a discrete set of variables. Such discrete variables might be considered in the optimization process at the expense of a considerable increase in complexity and time of computation. When there are only a small number of available materials, it would probably be more efficient to perform the optimization separately for each material and to compare the results at the end. Application of high-performance composite materials in structural components has encouraged further consideration of material properties as design variables [48]. For example, in fiber composites the volume fraction of fibers or the modulus of elasticity in the longitudinal direction of carbon fibers could be considered as continuous design variables. Optimization of composite laminates assuming integer design variables is reviewed elsewhere [44]. Topological Design Variables. The topology of the structure can be optimized automatically in certain cases when members are allowed to reach zero size. This permits elimination of some uneconomical members during the optimization process. However, it has been shown that topological optimization problems can have singular global optima that cannot be reached by assuming a continuous set of variables. This suggests that it may be necessary to represent some design variables as integer variables and to declare the existence or absence of a structural element. An example of an integer topological variable is a truss member joining two nodes which is limited to the values 1 (the member exists), or (the member is absent). Other examples of integer topological variables include the number of spans in a bridge, the number of columns supporting a roof system, or the number of elements in a grillage system. Optimization procedures, in general, do not permit a transition from one type of structure to another within a continuous design process. For example, the transition from a truss with axial forces to rigid frame with flexural behavior is usually not permitted.

o

Geometrical Design Variables. Geometrical or configurational variables may represent, for example, the coordinates of joints in a truss or in a frame.

1.3 General Formulation

27

Other examples for this class of variable include the location of supports in a bridge, the length of spans in a continuous beam, and the height of a shell structure. Although many practical structures have geometry which is selected before optimization, geometrical variables can be treated by most optimization methods. In general, the geometry of the structure is represented by continuous variables. Cross-Sectional Design Variables. Cross-sectional dimensions are the simplest design variables. The cross-sectional area of a truss member, the moment of inertia of a flexural member, or the thickness of a plate are some examples of this class of design variable. In certain cases a single design variable is adequate to describe the cross section, but a more detailed design with several design variables for each cross section may be necessary. For example, if the axial buckling of members is considered, the cross-sectional dimensions which define the area and the moment of inertia can be taken as design variables. It is often useful to choose quantities other than the obvious physical ones as design variables. In the above example, instead of the cross-sectional dimensions, we may use the area and the moment of inertia as variables. Such transformation of variables may simplify the problem formulation and can also yield considerable advantage in the solution. In practical design, cross-sectional variables may be restricted to some discrete values. Such are the rolled steel members, which are produced in distinct sizes with unevenly spaced cross-sectional properties. In such cases the design variable is permitted to take on only one of a discrete set of available values. However, as discrete variables increase the computational time, the cross-sectional design variables are usually assumed to be continuous. 1.3.2 Constraints Any set of values for the design variables represents a design of the structure. Clearly, some designs are useful solutions to the optimization problem, but others might be inadequate in terms of function, behavior, or other considerations. If a design meets all the requirements placed on it, it will be called afeasible design. The restrictions that must be satisfied in order to produce a feasible design are called constraints. From a physical point of view we may identify two kinds of constraints:

a. Constraints imposed on the design variables and which restrict their range for reasons other than behavior considerations will be called technological constraints or side constraints. These constraints, which are explicit in form, may derive from various considerations such as functionality, fabrication, or aesthetics. Thus, a technological constraint is a specified limitation (upper or lower bound) on a design variable, or a relationship which fixes the relative value of a group of design variables. Examples of such constraints include minimum slope of a roof structure, minimum thickness of a plate, or maximum height of a shell structure. b. Constraints that derive from behavior requirements will be called behavior constraints. Limitations on the maximum stresses, displacements, or buckling

28

1 Problem Statement

strength are typical examples of behavior constraints. Explicit and implicit behavior constraints are both encountered in practical design. Explicit behavior constraints are often given by formulas presented in design codes or specifications. However. behavior constraints are generally implicit. as will be illustrated later in Sect. 1.4. In any case the constraints must be a computable function of the design variables. From a mathematical point of view. both design and behavior constraints may usually be expressed as a set of inequalities j = 1•...• ng

(1.40)

where ng is the number of inequality constraints and X is the vector of design variables. Often. in a structural design problem. one has also to consider equality constraints of the general form j= 1•...• nil

(1.41)

where nil is the number of equalities. In many cases equality constraints can be used to eliminate variables from the optimization process. thereby reducing their number. The constraints (1.41) may represent the analysis equations or various design considerations such as a desired ratio between the width of a cross section and its depth. Such a simple and explicit constraint can easily be used to reduce the number of independent variables. However. in certain cases the elimination procedure may be complex and time consuming and some equality constraints must be considered. The constraints (1.40) and (1.41) may be linear or nonlinear functions of the design variables. These functions may be explicit or implicit in X and may be evaluated by analytical or numerical techniques. However. except for special classes of optimization problems. it is important that these functions be continuous and have continuous first derivatives in X . Design Space. We may view each design variable as one dimension in a design space and any particular set of variables as a point in this space. In cases with two

variables the design space reduces to a plane. In the general case of n variables. we have an n-dimensional hyperspace. Considering only the inequality constraints (1.40). the set of values of the design variables that satisfy the equation gj{X) = 0 forms a surface in the design space. It is a surface in the sense that it cuts the space into two regions: one where gj> 0 and the other where gj < O. The design space and the constraint surfaces for a typical truss (example 1.7) are shown in Fig. 1.9. A design which satisfies all the constraints is a feasible design. and the set of all feasible designs form the feasible

region. t

tIn all figures. the convention will be to hatch the feasible region, i.e., the acceptable side of the constraints.

1.3 General Formulation

29

00 II

II

"'_

Ov>

I

+

3.0

Intersection point

2.0

/design

Feasible 0",

1.0

region

- - 70 - "'0

1.0

0.5

Fig. 1.9.

1.5

Design space, three-bar truss.

=

Points within the feasible region [i.e., where #X) < 0, j 1,... , ng] are called unconstrained designs. Points on the surface [i.e., feasible designs for which at least one gJ{X) = 0] are called constrained designs. The subspace where two or more constraints gj 0 the constraint is violated and the corresponding design is infeasible . The equality constraints hj(X) = 0, j = 1, ... , nit , introduce couplings between the variables and may be thought of as surfaces in the n-dimensional design space. The feasible design points are required to be located in the intersection of these surfaces. The number of equality constraints nit must not exceed the total number of (dependent and independent) variables nT' In a case with nlt=nT the variables may, in principle, be determined as solutions to the equations hJ{X) = 0 and there is no optimization problem in the proper sense. When nit > nT the system of equations is overdetermined and either there are some redundant equality constraints or the formulation is inconsistent.

=

Typical Constraints. Typical inequality constraints considered in this text are DL~D ~Du

aL ~ a

~

aU

XL~X ~Xu

(displacement constraints) (stress constraints) (side constraints)

(1.42) (1.43) (1.44)

30

1 Problem Statement

in which L and U are superscripts denoting lower and upper bounds, respectively; and D and a are vectors of displacements and stresses, respectively. Both the displacements and the stresses are in general nonlinear and implicit functions of the design variables, given by the analysis equations. The lower and upper bounds are usually preassigned parameters. An exception is a lower bound on stresses which might depend on the design variables if buckling strength is considered.

1.3.3 Objective Function There usually exists an infmite number of feasible designs. In order to find the best one, it is necessary to form a function of the variables to use for comparison of feasible design alternatives. The objective function (also tenned the cost, criterion. or merit function) is the function whose least value is sought in an optimization procedure. It is usually a nonlinear function of the variables X, and it may represent the weight, the cost of the structure, or any other criterion by which some possible designs are preferred to others. We always assume that the objective function, Z = AX), is to be minimized, which entails no loss of generality since the minimum of -f(X) occurs where the maximum ofJ(X) takes place, i.e., maxJ(X) =-min [-f(X)]

(1.45)

The selection of an objective function can be one of the most important decisions in the whole optimal design process. The mathematical formulation of the objective function may be a very difficult task as, for instance, when important aesthetical values are influenced by the design variables. In general, the objective function represents the most important single property of a design, but it may represent also a weighted sum of a number of properties. Weight is the most commonly used objective function due to the fact that it is readily quantified, although most optimization methods are not limited to weight minimization. The weight of the structure is often of critical importance, but the minimum weight is not always the cheapest. Cost is of wider practical importance than weight, but it is often difficult to obtain sufficient data for the construction of a real cost function. A general cost function may include the cost of materials, fabrication, transportation, etc. In addition to the cost involved in the design and construction, other factors such as operating and maintenance costs, repair costs, insurance, etc., may be considered. In cases where a general objective function is considered, the result might be a "flat" function which is not sensitive to variations in the design variables and the optimization process, practically, will not improve the design. In most practical applications the objective function is indeed flat near the optimum (Fig. 1.10). Thus, a near optimal solution, rather then the theoretical optimum, is often sufficient Another approach is to consider both the initial cost of the structure and the failure costs which depend upon the probabilities of failure. The assumption is that the failure cost is given by the damage cost associated with a particular failure multiplied by its probability of occurrence. It is, however, recognized that answering the moral question of what constitutes an appropriate failure damage

1.3 General Formulation

31

cost is likely to be as difficult as selecting an acceptable probability of failure and estimating the probability of failure of an actual structure. In some optimal design problems a multicriterion objective function. representing several criteria. is considered [23]. However. dealing with multicriterion objective functions is complicated and is usually avoided. This can be done by generating a composite objective function. where each criterion is multiplied by a constant reflecting its relative importance. Alternatively. the most important criterion is selected as the only objective function and limits are imposed on the other objective functions. 1.3.4 Mathematical Formulation Formulation in the Design Variables Space. The structural optimization problem is to select optimal values of the design variables such that the specified objective function is minimized and a set of specified constraints are satisfied. The behavior of the structure can usually be computed for any given value of the design variables by the analysis equations. The latter equations can be excluded from the mathematical formulation and used as a computational rule to evaluate the constraints which are given in implicit form. Explicit formulation of the constraints is possible only in statically determinate or simple structures. Assuming that all equality constraints can be eliminated. the optimal design problem can be formulated mathematically as one of choosing the vector of design variables X such that (1.46) Z =j(X) ---+ min

j

= 1•...• n,

(1.47)

Equation (1.46) means thatj(X) goes to a minimum. This problem is said to be stated in the design space. since the design variables are the only independent ones. Formulation in the design variables space may be viewed as a two-level problem where. at each step. the structure is analyzed and the constraints are evaluated at the first-level by the analysis equations. The design is then modified at the secondlevel. In this nested approach the structural analysis is nested inside the optimization procedure. repeated again and again for a sequence of trial designs.

z

x

Fig. 1.10 Region £\ of a near optimal solution.

32

1 Problem Statement

Equations (1.46) and (1.47) represent a mathematical programming problem. In general, part of the terms in these equations are nonlinear functions of the variables, and the problem is called nonlinear programming (NLP). The following points should be considered in the problem formulation. - All inequality constraints are written as .. ~ 0". Any inequality constraint can be converted to this form by transferring the right hand side terms to the lefthand side, and multiplying by -1 if necessary. Also, there is no restriction on the number of inequality constraints. If the objective function value is scaled by multiplying it with a positive constant, the optimal design does not change. The optimal objective function value, however, changes. Also, any constant can be added to the objective function without affecting the optimal design. Similarly, the inequality constraints can be scaled by any positive constant. This will not affect the feasible region and hence the optimal solution. It is important to note that the feasible region usually shrinks when more constraints are added and expands when some constraints are deleted. When the feasible region shrinks, there are fewer feasible designs and the minimum value of the objective function is likely to increase. The effect is the opposite when some constraints are dropped. Example 1.7. Consider the three bar truss shown in Fig. 1.11. The structure is subjected to two distinct loadings, Pi and P 2' respectively, and the design variables are the cross-sectional areas. Due to symmetry of loading and geometry, the number of design variables is reduced to two (Xl and Xi) and only one loading condition may be considered. The constraints of the design problem are o~x

-15

~ (J ~

20

(side constraints)

(a)

(stress constraints)

(b)

The displacements r l , r2 are computed by the displacement analysis equations (1.26) (c)

in which E is the modulus of elasticity. The stresses (J are computed by (1.29)

{::}= 1~ [~.5 ~.5l {;:} (J3

(d)

-0.5 0.5

In this simple example it is possible to express explicitly the displacements in terms of the design variables. From (c) we fmd

1.3 General Formulation

I~

100

33

100

100

Fig. 1.11. Three-bar truss example.

(e)

Substituting (e) into (d), we obtain the stresses expressed in terms of the design variables

Only constraints which may affect the design must be considered. Since a1 and a2 will always be positive, and a 3 negative, some constraints can be deleted and we may consider only the stress constraints al-20~O a2-20~O -a3-15~O

Substituting (f) into (g) we fmd the following explicit stress constraints

(g)

34

1 Problem Statement o

11 ."

-.

-I-

3.0

2.0

1.0

Objective function

~~-~r---~4--contours

Fig. 1.12. Design space and objective function contours, three-bar truss.

(h)

Graphical representation of these constraints, in the space of Xl and X2 ' is shown in Fig. 1.9 . Assuming the volume of material as the objective function, we obtain the following linear expression Z = 282.8 Xl + l00x2

(i)

The locus of all points satisfyingj{X) =constant forms a surface. For each value of the constant there corresponds a different member of a family of surfaces. Figure 1.12 shows the family of constant volume (or weight) contours, called objective function contours. Every design on a particular contour has the same volume. It

1.3 General Formulation

35

can be observed that the minimum value of f(X) in the feasible region occurs at point A, which represents the optimal design min Z= 263.9

Xl = 0.788 It can be noted that only the constraint cr 1 -20 cr1 =20.

~

0 is active at the optimum, i.e.,

Simultaneous Analysis and Design. In some design problems it may be worthwhile to integrate the analysis and design procedures so that solution of the analysis and determination of the optimal design occur at the same time. This form of problem statement is called simultaneous analysis and design (SAND), or the integratedJormulation. In this approach both behavior variables, Y, and design variables, X, are assumed as independent variables, all treated in a similar way. In addition, the analysis equations are included in the problem formulation as equality constraints. The mathematical programming problem (1.46) and (1.47) is stated in this case as follows: fmd the design variables X and the behavior variables Y such that (1.48) Z=f(X) ~ min gj(X, Y) ~ 0

j = 1,... , ng

(1.49)

hiX, Y)= 0

j

= 1,... , nIl

(1.50)

This type of formulation is employed in cases where it is impossible or impractical to eliminate the equalities (1.50). For example, in problems with geometric nonlinearity, the nonlinear analysis equations must be solved for each value of the design variables if the problem is formulated in the design variables space. The repeated nonlinear analyses are not required in cases where the SAND formulation is used [35]. In general, the SAND approach eliminates the need for continually reanalyzing the structure at the expense of a larger optimization problem. This major shortcoming of additional variables and equality constraints makes the approach less attractive in many optimal design problems where elastic analysis models are considered. In such problems the nested design variables formulation is the rule, and the analysis equations are repeatedly solved for the modified designs. The main advantage of the nested formulation is that the number of variables and constraints is reduced. It should be noted that in the SAND approach only the implicit analysis equations might be considered in (1.50). The explicit equations can be eliminated from the problem formulation, and the total number of variables is reduced accordingly, as will be demonstrated in Sect. 1.4. Reduction or Problem Size. The size of an optimal design problem is mainly determined by the number of variables. The solution of large scale problems requires much more computational effort and there is a definite advantage in reducing the problem size.

36

1 Problem Statement

The number of independent design variables is often reduced by assuming several elements to have prescribed ratios between their sizes. In many optimal design problems, the number of elements needed in the analysis is much larger than the number of design variables required properly to describe the design problem. Frequently, it is neither necessary nor desirable for each element to have its own independent design variable. Design variable linking or basis reduction [l05, 121] fixes the relative size of some preselected group of elements so that some independent variables control the size of all elements. Variable linking can be accomplished by relating the vector of original design variables X to the vector of independent variables XI according to the expression

(1.51) where L is the matrix of linking constants giving the predetermined ratios between variables X and XI' In (1.51) the variables X are taken as a linear combination of XI' In many cases in which only simple design variable linking is used, the matrix L takes on a special form, in which each row contains only one nonzero element. The reduced-basis concept further reduces the number of independent design variables by expressing the vector XI as a linear combination of s basis vectors b j • giving

L yjbj =by s

XI =

(1.52)

j=l

Substituting (1.52) into (1.51) gives (1.53)

X=Lby=ty

where t is the matrix of prelinked basis vectors and y is the vector of a reduced set of design variables.

xG

X4 Xs

.d77T

x7

X 13

:::n:::

:±L

:::b..-

Xg

x~

~

Fig. 1.13

X 12 XII

X9 X3

~

l

Continuous beam.

~

XIO

1.3 General Formulation

37

Example 1.S. To illustrate variable linking and basis reduction, consider the continuous beam shown in Fig. 1.13, with a given span I, six geometrical variables (Xl> X 2 • X 3 • Xg• X 9 • XlO) representing the elements length. and seven cross-sectional variables (X4' X S,X6 , X 7 , Xu. X12, XI3 ). Assuming symmetry, the problem can be stated in terms of only seven independent variables Xlo •••• X7 . From the relation (1.51)

{jJ

1 0 0 0 0 0 0 1 0 0 0 0 0

0 1 0 0 0 0 0 0 1 0 0 0 0

0 0 1 0 0 0 0 0 0 1 0 0 0

0 0 0 1 0 0 0 0 0 0 1 0 0

0 0 0 0 1 0 0 0 0 0 0 1 0

0 0 0 0 0 1 0 0 0 0 0 0 1

0 0 0 0 0 0 1 0 0 0 0 0 0

{z} (a)

To further reduce the number of independent design variables, the following relations have been assumed

X2 = (Xl + X 3 )fl. Xs = (X4 + X6 )12

(b)

or, in the form of (1.52)

{JJ

1 0 0 0 0 0.5 0.5 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0.5 0.5 0 0 0 0 1 0 0 0 0 0 1

Finally, the matrix t is given by

Xl X3 X4 X6 X7

(c)

38

1 Problem Statement

1 0 0 0 0 0.5 0.5 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0.5 0.5 0 1 0 0 0 0 0 1 0 0 t=Lb= 0 1 0 0 0 0 0.5 0.5 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0.5 0.5 0 0 0 0 1 0

(d)

and the original variables X are expressed in tenus of the reduced set of independent variables y by (1.53). Scaling or Design Variables. It is often desirable to eliminate wide variations in the magnitudes of design variables and the value of constraints by normalization. Design variables may be nonualized to order 1 by scaling. This operation may enhance the efficiency and reliability of the numerical optimization process. Consider for example the variables XI and X2 , limited by the side constraints

The original variables can be replaced by new variables Y I and Y 2 defmed by

and the side constraints become

To illustrate the effect of scaling on the objective function, consider the function (Fig. 1.14a)

1.3 General Formulation

39

(b)

(a)

Fig. 1.14. Objective function contours:

8.

Original variables, b. Scaled variables.

Assuming

the scaled function becomes (Fig. 1.14b) !(y)=y? +Yi

which is much easier to minimize. Constraint Normalization. Since different constraints involve different orders of magnitude, it is often desirable to normalize all the constraint functions. Consider for example the typical constraints [see (1.42) through (1.44)] (1.54) that are normalized to obtain (D-DU) / DU

=(D/DU) - 1.0:s; 0

(cr-crU) / crU =(cr/cr U) : 1.0 :s; 0 (XL_X) / XL

= 1.0 - (X/XL) :s; 0

(1.55)

40

1 Problem Statement

This nonnalization does not affect the feasible region. The denominators of (1.55) represent nonnalization factors which place each constraint in an equal basis. For example, if the value of a stress constraint is -0.1 and the value of a displacement constraint is -0.1, this indicates that each constraint is within 10% of its allowable value. Without nonnalization, if a stress limit is 20,000, it would only be active (within 10%) if its value was 19,999.9. This accuracy is difficult to achieve on a digital computer. Also, it is not meaningful since loads, material properties and other physical parameters are not known to this accuracy. Using nonnalization, the constraint values are of the order of one, and do not depend on the units used. Constraint Deletion Techniques. The number of inequality constraints in optimal design problems may be very large, particularly in structures consisting of many elements and subjected to multiple loading conditions. Constraint deletion techniques [121] can be used to reduce the number of constraints. It is recognized that, during each stage of an iterative design process, it is only necessary to consider critical or potentially critical constraints. On the basis of analysis of the structure, all the inequality constraints may be evaluated. Constraint deletion techniques are then used to temporarily eliminate redundant and noncritical constraints that are not likely to influence significantly the design process during the subsequent stage. For each constraint type the most critical constraint value is identified using regionalization and truncation techniques. An example of the regionalization technique is that only the most critical stress constraint in each region under each load condition is retained. The regionalization idea works well provided the design changes made during a stage are small enough that they do not result in a shift of the critical constraint location within a region. A truncation technique, on the other hand, involves temporary deletion of constraints for which the ratio of the stress to its allowable value is so low that the constraint will clearly be inactive during the stage. Evidently, none of the constrains included in the original problem statement are pennanently deleted unless they are strictly redundant Consider for example the nonnalized constraints (1.55). We can delete all but the most critical (most positive) constraints. Alternatively, we may delete any constraint whose value is less than (more negative than) some cutoff value gc' say gc =-0.5 at the beginning of the optimization. This value can be reduced gradually at the later stages. Other techniques may be used. For example, we may delete first some constraints that are more expensive to evaluate and solve the problem. These constraints are then evaluated and if they are not violated the optimum is reached, having avoided much costly computation. If these constraints are violated, we add them to the constraint set and proceed from there. Unlike the reduction in the number of design variables (by linking or basis reduction) where the reduction is global in character, constraint deletion is a local strategy. Relative Minima. One difficulty in solving a nonlinear programming (NLP) problem is that there can be multiple relative minimum points. A point is said to be a relative (local) minimum if it has the least function value in its neighborhood, but not necessarily the least function value for all X . Relative minima may occur

1.3 General Formulation

41

in NLP problems due to the nature of either the objective function or the constraints, or of both. Consider for example a two-dimensional problem with inequality constraints. It is evident that the minimum may be a point where the constraints have no influence (Fig. LISa) and yet the problem has a relative minimum . Relative minima may exist also in problems where the constraints are active (Fig. USb). In both cases the multiple-optimum points are due to the form of the objective function. A relative minimum that occurs due to the form of the constraints is shown in Fig. USc. Example 1.9. The grillage shown in Fig. 1.16a is subjected to two concentrated loads P = 10.0. The width of the rectangular cross sections is b = 12.0, the members' lengths are lx = 1.0 and ly = 1.4, and the depths of the longitudinal and transverse beams (Xl and X 2 , respectively) are chosen as design variables. Neglecting torsional rigidity of the elements, the force method of analysis is considered, with N being the vertical contact force between beams at the intersection (Fig. U6b). The single compatibility equation is (a)

The bounds on stresses are a U=-01- = 1.0, and the allowable moments are given by

MF = aU (12X; /6) = 2X; Mf

=a L (12X; /6) =-2X;

i = 1,2

(b)

where the subscripts i denote the beam numbers. The optimal design problem is to

find Xl and X2 such that

Z

= 12(3Xl + S.6X2) ~ min

(c)

-2X'f ~ (10- N) ~ 2X'f

(d)

-2xi ~(0.7N)~2Xi

(e)

where N is given in terms of Xl and X2 by (a). The topology of the structure can be optimized indirectly by letting Xl =0 or X2 =0, resulting in elimination of the longitudinal or transverse beams, respectively. In either case a statically determinate structure is obtained (with N =0 or N = 10). The design space is shown in Fig. 1.17. It can be noted that three relative optimum points, representing three different topologies have been obtained. The optimal design values for the variables and objective function are given in Table 1.1. This result is typical to many grillages, where local optima fall into three categories:

42

1 Problem Statement

(a)

Global minimum

(b)

Global /minimum Relative minimum

(c)

Fig. 1.lS.

Relative minima.

1.3 General Formulation

~"/-<

l,

43

!"~', f tP

P

;00-1(

;00-1(

N

-

N

,/li;~_ _N...L-!- - & r

l,

I_ .

~+-~+~~

~

l,

;00

I

I(

l,

;00

I

(b)

(a)

Fig. 1.16. a. Grillage example. b. Redundant forces.

a. Heavy beams in the ix direction (X;) b. Heavy beams in the iy direction (X;) c. Designs with sizes intermediate to the ftrst two categories (X;). The global optimum is at X; , where the beams in the iy direction are eliminated. Table 1.1. Local optima, grillage example. Point

X*T

z*

1 2 3

(2.24. O) (1.63. 1.28) {O l.87}

80.6 144.7 125.7

Fig. 1.17. Design space, grillage example.

44

1 Problem Statement

1.4 Typical Problem Formulations In this section the general mathematical formulations discussed in Sect. 1.3.4 will be demonstrated for the typical constraints (1.42) through (1.44). The formulations presented here for the displacement method and for the force method of analysis will be considered throughout this text. Many of the examples in the text are related to truss structures. The truss is the most thoroughly investigated structure in relation to design optimization due to the following reasons:

a. Various practical structures are trusses or can be approximated as trusses,

including many bridge supports, transmission towers, ship masts and roof supports. b. A finite-element code for truss analysis is easily written, so it is not necessary to expend major effort on the analysis portion of the design program. c. Truss structures can be created which span the range of complexity from very simple to highly nonlinear. This type of structure provides excellent test cases for the study of optimization techniques.

1.4.1 Displacement Method Formulations Formulation in the Design Variables Space. For simplicity of presentation it is assumed that displacement constraints are related to all degrees of freedom. Thus, r will be considered instead of D in (1.42). Assuming the formulation (1.46) and (1.47), the optimal design problem is: fmd X such that Z=j(X)

~

min

rL ~ r(X) ~ r U

(1.56)

where r(X) and O'(X) are given in terms of X by the analysis equations (1.26) and (1.29)

r(X) =K-iR

O'(X) = S r = SK-IR

(1.57)

Assuming cross-sectional and geometrical variables, the elements of K and S are explicit functions of all variables. (The elements of S depend only on the geometrical variables in some structures such as trusses.) The matrix K-l is usually a nonlinear implicit function of all design variables. Thus, both r and 0' are also nonlinear implicit functions of these variables. In the above formulation the analysis equations are used as a computational rule for relating the value of the constraint functions to the design vector. Many structural-design problems possess this characteristic: the behavior to be limited cannot, for all practical purposes, be expressed explicitly in terms of the design variables.

1.4 Typical Problem Formulations

45

Simultaneous Analysis and Design. Considering the integrated formulation (1.48) through (1.50). the problem (1.56) and (1.57) can be stated as follows: find X • r and C1 such that

z =.f(X) -+ min

(1.58) Kr=R

C1=Sr In this formulation the analysis equality constraints are included in the problem formulation. These constraints are satisfied at the optimum but not necessarily at intermediate designs. before the optimum is reached. Since the stresses are given explicitly in terms of the displacements. it is not necessary to consider C1 as independent variables. Substituting the explicit stress-displacement relations into the stress constraints. the problem (1.58) can be stated in terms of only X and r as

z =.f(X) -+ min (1.59)

Kr=R

In this formulation the equilibrium conditions are the only equality constraints. 1.4.2 Force Method Formulations Formulation in the Design Variables Space. Considering the constraints (1.42) through (1.44). the problem (1.46) and (1.47) becomes: find X such that Z =.f(X) -+ min DLS D(X) SDu

(1.60)

46

I Problem Statement

where D(X) and O'(X) are given by the analysis equations. Assume the common force-stress relations (1.61) Aj W.{XJ O'j

=

where Wi is the ith modulus of section, which is a function of the ith design variable Xi [in the case of b'Uss structures, the cross sectional area Xj is used instead of W.{XJ in (1.61)]. The displacements and the stresses are then given in tenns of X by (1.22) through (1.24) D(X) = Dp + DNF-tl) O'(X) = Wit (Ap + ANF-tl)

(1.62)

where Wit is a diagonal matrix of the reciprocals wj- t . In the above fonnulation the elements of Dp, DN, F and l) are explicit functions of both the cross-sectional and the geometrical variables. The elements of Ap and AN are given explicitly in tenns of only the geometrical variables and the elements of W tl are explicit functions of only the cross-sectional variables. The matrix F-t is usually a nonlinear implicit function of all design variables. Simultaneous Analysis and Design. Considering the integrated formulation (1.48) through (1.50), the problem (1.60) and (1.62) can be stated as follows: find X, N, D and 0' such that Z=f(X)

~

min

DLSD SDu O'L SO'S aU XLSX SXU

FN

(1.63)

= l)

D = Dp + DNN 0' = Wit (Ap + ANN) Since the displacements and the stresses are explicit functions of X and N, these relations can be substituted into the constraints and the problem can be expressed in tenns of only X and N as Z=f(X) ~ min DL S Dp + DNN S D U a L SWi1(Ap+ANN)Sau XLSX SXU

(1.64)

1.4 Typical Problem Formulations

47

In this formulation the implicit compatibility conditions are the only equality constraints. Explicit Formulations. In some problems, for example in statically determinate structures, it is not necessary to consider the compatibility conditions (1.22). Neglecting the latter conditions in a general statically indeterminate structure, the problem (1.64) can be formulated explicitly in terms of X and N as Z=f{X)

~

min (1.65)

Since the forces in Ap and AN satisfy the equilibrium conditions, any selection of N also will result in a corresponding set of forces that will satisfy these conditions but not necessarily the compatibility conditions. Considering the equilibrium conditions (1.1) as equality constraints, the problem (1.65) can be formulated in terms of X and A. The virtual-load method may be used in this case to obtain the displacement expressions (1.33). The resulting explicit problem is to fmd X and A such that Z=f{X)

~

min

DL ~ T 1/X ~ DU

(1.66)

C A=R

The optimal solutions of problems (1.65) and (1.66) are identical. However, the following differences can be observed in the problem formulation: -

The number of variables in problem (1.66), where all members' forces are considered as variables, is larger. In problem (1.65) only the redundant forces are considered as variables. In problem (I.66) the equilibrium conditions are considered as additional equality constraints. As noted earlier, any solution of the problem (1.65) will satisfy these conditions.

Assume the common case of a structure with n elements and cross-sectional design variables such that

48

1 Problem Statement

L liXi =lTX II

Z=

i=l

(1.67)

These are typical relations in truss structures where Xi are the cross-sectional areas and li are the members' lengths. Substituting (1.67) into (1.65) and considering only stress and side constraints, the problem becomes: fmd X and N such that Z = (l'X -+ min (1.68)

XL:5:X:5:XU In this formulation (J~ and (J~ are diagonal matrices of bounds on stresses, and the elements of Ap and AN are constant, computed in the primary determinate structure. Since the objective function and all the constraints are linear functions of the variables, this is a linear programming (LP) problem. Alternatively, substituting (1.67) into (1.66) and considering only stress and side constraints, the following LP problem is obtained: find X and A such that Z =(l'x -+ min (1.69) C A=R

Since the elements of C are independent of the cross-sectional variables, they are constant. The two LP problems (1.68) and (1.69) are equivalent but, as noted earlier, the number of variables is larger and the equilibrium conditions are considered as additional equality constraints in problem (1.69). Both formulations have been used extensively in various optimal design applications and will be discussed throughout this text . In problems of optimal plastic design (Sect. 4.1.1), only equilibrium and yield conditions are considered, and both formulations can be viewed as simultaneous analysis and design (SAND). In the above presentation, explicit formulations of optimal design problems have been obtained by neglecting the implicit analysis equations. Explicit exact formulations can be achieved in simple systems and elements, or in statically determinate structures where the stresses and displacements are given explicitly in terms of the design variables. It will be shown now that such formulations are possible also for some statically indeterminate structures. It has been shown that, in general, the design variables are chosen as the independent ones and the behavior (dependent) variables are determined by the analysis equations. Alternatively, if

1.4 Typical Problem Formulations

49

compatibility conditions are neglected, both design and behavior variables are assumed as independent variables [formulations (1.65) through (1.69)]. Fuchs [37] proposed to choose cross-sectional design variables of the basic statically determinate structure and the remaining redundant forces as independent variables. To illustrate this possibility, consider again the basic relations (1.12) and (1.14), where AR =N [see (1.18)] (1.70) FdRN

=G FdB AB

(1.71)

These equations can be rearranged as

(1.72) (1.73) where F R is a vector of the diagonal elements of F dR, and N;i is a diagonal matrix of the inverse elements of N. The chosen independent variables are the elements of F dB and N, and the corresponding dependent variables are the elements of AB and FR, given explicitly by (1.72) and (1.73) [instead of using the implicit compatibility equations (1.22) to calculate N]. Thus, the implicit optimal design problem (1.60) can be formulated now explicitly as follows: find F dB and N such that (1.74)

in which the relations (1.72) and (1.73) are substituted as necessary. It should be noted that this formulation is not general and involves several limitations, including: - it is suitable only for certain cross-sectional design variables; - it is not suitable for problems with linking of variables and multiple loading conditions; - it might be effective for problems where the force method of analysis is assumed; in problems where the displacement method of analysis is used, the number of variables might be considerably larger. Example 1.10. In this example, various problem formulations are demonstrated for the four-bar truss shown in Fig. 1.1. Assume the four crosssectional areas as design variables XT = {Xl' X2• X3 • X4 } ,the volume of material as an objective function

50

1 Problem Statement (a)

and the stress constraints (b)

Displacement method formulations. Assuming the displacement method of analysis, r is given implicitly in terms of X by [see (1.26)]

and (J is given explicitly in terms of r by [see (1.29)]

E

(J

= 100

0.5

0.5]

[ 0

1

-0.5 0.5 -1 0

{rJ 'i

(d)

Substituting (d) into (b), the stress constraints become

0.5 0.5] L E O 1 Ii. u [ (J <-<(J - 100 -0.5 0.5 -1 0

{rJ-

(e)

The assumed variables and constraints for the formulations (1.56) through (1.59) are summarized in Table 1.2. Force methodformulations. For the force method of analysis, the redundant forces NT={A3 • A4 } are given implicitly in terms of X by (1.22)

v'2( 100 E

~+l.. )+~ Xl X3 Xl 2

v'2

Xl

Xl

-+-

2 v'2 -+Xl Xl 2v'2

20v'2

1

1

Xl

4

{~}=_I~

--+-+X Xl

The members' forces are given explicitly in terms ofN by (1.72)

Xl (f) 40

Xl

1.4 Typical Problem Formulations

51

or by the general fonn (1.24)

(h)

The stresses are given by

(i)

Substituting (h) and (i) into the stress constraints (b) yields

These constraints can be expressed as the following linear inequalities Table 1.1.

Various problem formulations, four-bar truss.

Analysis Method Displacement

Force

Force (explicit)

Formulation (1.56),(1.57) (1.58) (1.59) (1.60),(1.62) (1.63) (1.64) (1.68) (1.69) (1.74)

Variables ~

X, r, cr X, r X X, A, cr X,N X,N X,A X"Io X2,A 3,A 4

Constraints

Analysis rules

(b) (b),(c),(d)

(c),(d)

(c)'(~J

-

-

(b) (f),( h),( i) (b)'(f),(h),(i) (fl,(j)

(k) (l),(m) (p)

-

52

1 Problem Statement

(k)

The equilibrium equations (1.1) are

[..fia ..fia

01 -..fia -1] {~}= ..fi{10} ..fia 0 A3 10

(I)

~

and the stress consttaints, expressed in terms of A, are

(m)

Finally, from (1.73)

..fi

..fi

X3

Xl

1

=[All

X4

A41

]

..fi X2

2

1

Xl

X2

{~:}

(n)

Substituting (g), (i) and (n) into (b) gives the explicit constraints aLS;

a (Xl, X 2 , A 3 , A4) s; aU

(p)

The assumed variables and constraints for the various force method formulations are summarized in Table 1.2 .

Exercises A. In exercises 1.1, 1.2 and 1.3, assume the displacement method of analysis. Formulate the optimal design problem:

Exercises 100

53

100

1------1-·- - - I

Fig.

1.18.

a. in the space of displacements and design variables [SAND, see (1.59)];

b. in the design variables space [see (1.56), (1.57)].

Sketch the feasible region and the objective functions contours. Find graphically the optimal solution and the active constraints at the optimum. 1.1 The symmetric truss shown in Fig. 1.18 has two design variables: the crosssectional area XI and the angle X2• The bounds on stresses are aU = 20.0, 01- = 15.0. The upper bound on the vertical displacement at the free node is

rf = 0.02

xf

= 300, X¥ =6()0. The modulus of elasticity is and the bounds on X2 are: 30,000 and the objective function represents the volume of the truss. 1.2 The symmetric truss shown in Fig. 1.19 has two design variables: XI = cross-sectional area in members 1 and 4; X2 =cross-sectional area in members 2 and 3. The objective function represents the volume of the truss, the bounds on stresses are aU = 20.0, 01-= -15.0 and the modulus of elasticity is 30,000. 1.3 The two-span beam shown in Fig. 1.20 is subjected to two separate loading conditions, PI and P 2 .The design variables Xi (i = I, 2) represent the crosssectional areas in the two spans. The following relations have been assumed for the cross sections:

54

1 Problem Statement

100

Fig.

1.19.

Modulus of section Wi =Xi Moment of inertia Ii = 3Xi

i= 1,2

The bounds on stresses are aU = -01- = 20.0. The constraints are related to the stresses in sections A, B, C, D and the objective function represents the volume of the beam. The modulus of elasticity is 30,000. B. In exercises 1.4 - 1.8 assume the force method of analysis. Formulate the optimal design problem:

a. in the space of redundant forces and design variables [SAND, see (l.64)]; b. in the design variables space [see (l.60) - (l.62)]; c. in the LP form (l.68); d in the LP form (1.69). 1.4 Solve the truss of exercise l.2, assuming N I and N z =redundant force in member 4.

= redundant force in member 3,

1.5 Solve the beam of exercise l.3, assuming NI = bending moment over the interior support under PI' and N z bending moment over the interior support underPz·

=

X,

~

I: Fig.

lP' =8 I A

100 _\' 100 i) =200

1.20.

~P2 BC I zs; I

:I:

= 16

I D

100

i2

'I'

=200

X2 ~

100

:1

Exercises 10.0 5.0--

8)

@

CD

CD

55

®]

100

Fig.

I·

100

.1.

100

.1

1.21

1.6 The frame shown in Fig. 1.21 is subjected to a single-loading condition of two concenttated loads. Assume the following design variables and redundant forces: Xl cross-sectional area in the left hand column; X2 cross-sectional area in the beam; X 3 = cross-sectional area in the right hand column; N = bending moment in section 3. The modulus of section, the moment of inertia. the bounds on stresses and the modulus of elasticity are as given in exercise 1.3. The constraints are related to stresses in sections I, 2, 3, 4, 5, and the objective function represents the volume of the frame. Illusttate graphically the feasible region in formulation c for X2 =X3 =2XI ; find the optimal solution and verify by substituting into the constraints of formulation d.

=

=

1.7 Assume the symmetric grillage shown in Fig 1.16 with the following data: P

= 10.0,

Lx

= 100,

Ly

= 150.

Xi (i = I, 2) are the cross-sectional areas. The modulus of section, the moment of inertia, the bounds on stresses and the modulus of elasticity are as given in exercise 1.3. The constraints are related to stresses in all critical sections and the objective function represents the volume of the grillage. Choose the vertical interaction force in the intersection of the two beams as the redundant force and neglect torsional rigidity of the elements. Illusttate graphically the feasible region in formulation c for Xl X2 ; find the optimal solution and verify by substituting into the constraints of formulation d.

=

1.8 The symmetric grillage shown in Fig. 1.22 is subjected to a single-loading condition of two concenttated loads. The bounds on stresses are aU -oL 20.0, and the modulus of elasticity is 30,000. The constraints are related to stresses in all critical sections and the objective function represents the volume

= =

where Xh X2 are design variables representing the cross-sectional areas of the two beams. The following relationships have been assumed:

56

1 Problem Statement

/

Fig.

1.22.

Modulus of section Mcmnentofine~a

i= 1,2

Choose the ve~cal interaction force in the intersection of the two beams as the redundant force and neglect torsional rigidity of the elements.

2 Optimization Methods

Optimization problems discussed in this chapter can be formulated in the general form presented in Sect. 1.3.4, where the objective function and the constraints are nonlinear functions of the variables. The solution methods commonly used for obtaining the optimal design may be divided into several categories. One classification of solution methods considers specific versus general methods. Specific optimality criteria methods, used exclusively in structural optimization, will be presented in Sect. 4.3. In this chapter general-purpose mathematical programming (MP) methods, which are commonly applied to optimization problems in several fields, will be discussed. These methods have the advantage of wider applicability and base of resources. As a result, efficient and reliable algorithms are continually developed. Applying MP methods to structural design, a wide variety of problems can be considered, including:

a. Complex structural systems subject to different failure modes in each of several load conditions.

b. General design variables representing cross-sectional dimensions, the geometry or the topology of the structure.

c. Various constraints on the structural behavior and on the design variables.

d. A general objective function representing the cost or the weight of the structure.

Mathematical programming methods will find the optimum in problems where there exists a single global optimum. In some structural applications, however, multiple relative optima (or, local optima) may exist, in addition to the global optimum which is sought. Relative minima may occur in MP problems due to the nature of both the objective function and the constraints (Fig. 1.15). In such cases, the solution is liable to depend on the initial design from which the search procedure is started. This difficulty can be alleviated by repeating the computations from different starting points and comparing the solutions until reasonable confidence is built up that the global optimum has been achieved. Numerous MP methods have been developed. However, no single general purpose method can solve efficiently all optimization problems and there are no accurate methods for optimizing effectively large structures. In this chapter only those methods which are most commonly used in structural optimization are discussed. Methods applicable to problems of specialized form such as dynamic programming, geometric programming and optimal control techniques are not considered. However, these methods have been applied successfully in various structural design problems [14,62]. Efficient MP methods for practical problems

58

2 Optimization Methods

with large numbers of variables and constraints are often based on approximation concepts. Some of the latter concepts will be presented in Chap. 3. Nonlinear programming (NLP) problems can be divided into unconstrained and constrained problems. In unconstrained optimization we do not consider constraints and the problem is much easier to solve. Occasionally it is possible to eliminate some or all of the constrains from a problem. Some optimization concepts related to unconstrained minimum and constrained minimum are discussed in Sect 2.1. Unconstrained minimization along a line in the design space, which is common to many optimization algorithms, and methods for unconstrained minimization of functions of several variables are described in Sect. 2.2. Methods for constrained optimization are discussed in Sects. 2.3 (linear programming problems) and 2.4 (nonlinear programming problems).

2.1 Optimization Concepts 2.1.1 Unconstrained Minimum A point X* is a relative minimum of the function f(X) if there is a region containing X* in its interior, such that

f(X*) ~ft..X)

(2.1)

for all X in that region. Assume the Taylor series expansion of I about X* up to

quadratic tenns

(2.2)

r,

where Vf"T, and H* are computed at X*. In (2.2), vr is the vector of first derivatives, or the gradient vector (G) of I, and H is the matrix of second derivatives, or the Hessian matrix, given by

(2.3)

The vector of changes in the design variables, AX, is given by dX =X -X*

(2.4)

2.1 Optimization Concepts

59

If we assume a relative minimum at X* then from (2.2) (2.5) Concentrating only on the first-order term, we observe that Ilf~ 0 for all possible L1X when

vf* = 0

(2.6)

Equation (2.6) represents necessary conditions for the minimum of a function of n variables with continuous derivatives. A point satisfying these conditions is called a stationary point off{X). Considering the second term in (2.5) evaluated at a stationary point, the positivity of Ilf is assured if

q == AXT H* L1X > 0

(2.7)

for all L1X -:F- O. Equation (2.7) in this case is a sufficient condition for a local minimum of f{X) at X*. Both (2.6) and (2.7) ensure that a point is a relative minimum. These conditions involve derivatives but not the value of the function. It should be noted that if we add a constant to f{X), or if we multiply f{X) by any positive constant, the minimum point X* is unchanged although the value j(Xj of the function is altered . The Hessian H* is a positive definite matrix if the condition (2.7) is satisfied for every L1X. It is a positive semi-definite matrix if (2.8)

and it is indefinite if q is positive for some vectors L1X and negative for others. The matrix H* is positive definite if and only if all its eigenvalues are positive; it is positive semi-defmite if and only if all eigenvalues of H* are nonnegative; and it is indefinite if some eigenvalues are positive and some others are negative. Several methods can be used for checking the form of H*. One way is by using its principal minors. A principal minor Hi is a square sub-matrix of H of order i whose principal diagonal lies along the principal diagonal of matrix H. The matrix H is positive-definite if the determinants of all the principal minors located at the top left corner of the matrix are positive numbers. The matrix H is positive semi-defmite if the determinants of all principal minors are non-negative. If H is positive semi-definite but not positive-defmite, then the determinant of at least one of the principal minors is zero and higher order derivatives of f(X) are needed to establish sufficient conditions for a minimum. 2.1.2 Constrained Minimum Lagrange Multipliers. Consider the problem of minimizing a function subject to equality constraints. At the optimum, the differential change in the objective functionf(X), in terms of the differential change in X, must still vanish

60

2 Optimization Methods

(2.9)

where the differential changes dXb dX2> ... , dXn are related through the constraints. Considering only a single constraint

h{X) =0

(2.IO)

then the differential change in h is

oh oh oh dh=-a dX1 + - dX2 +···+a1X2 aIX,. dX,.=O 1X1

(2.ll)

From (2.9) and (2.ll) we obtain

in which A is an unknown, called Lagrange multiplier. From (2.12)

Of oh -+A-=O

oXi

i = 1, ... , n

oXi

(2.13)

Thus, we have a system of n+I equations [(2.1O) and (2.13)] and unknowns (X and A). In the case of multiple equality constraints

j

= 1,... , nit

(2.14)

we have to introduce a Lagrange multiplier for each constraint. Define the Lagrangian function

L Ajhj{X) ,.~

C\>{X, A) = f{X)+

(2.15)

j=l

In order to find a stationary point of f over all X and A, we have to satisfy

i = I, ... ,n

(2.16)

j = l, ... ,n"

(2.17)

2.1 Optimization Concepts

61

giving the necessary (n+nJ conditions for a minimum

....

Vf+

I, AjVh j =0

(2.18)

j=l

j

=1,..., nit

(2.19)

It should be noted that these expressions might represent a nonlinear system of equations that have several solutions. Not all solutions will be constrained minima, some might be constrained maxima or saddle points. Further tests are needed to ensure that a point is a minimum. The geometric interp-etatioo of (2.18) is that at the minimum V f must be expressible as a linear combination of the normals to the surfaces given by (2.19). We can apply the concept of Lagrange multipliers to the inequality constraints

j

#X)~O

= 1,... , n,

(2.20)

by adding slack variables, Sj.

hj(X. Sj)=gj(X)+SJ=O

=

(2.21)

=

If Sj 0 then gj(X) 0 ; if Sj :j:. 0, then gj(X) < O. Equation (2.21) cannot be satisfied if #X) > O. Applying the Lagrange multiplier method to inequalities, we define

,(X.

s.

",

A)=f+I,Aj(gj+SJ) j=l

(2.22)

The stationary conditions for, are

i= 1..... n

(2.23)

i= 1..... n,

(2.24)

=l, .... n,

(2.25)

i

Equations (2.25) ensure that the inequalities gj ~ 0 are satisfied. Equations (2.24) state that either Aj or Sj is zero, which implies that either the constraint is active (gj 0) and must be considered in testing (2.23), or it is inactive (Aj 0). Equations (2.23) require that Vf lie in the subspace spanned by those Vlj which correspond to the active constraints.

=

=

62

2 Optimization Methods

Kuhn-Tucker Conditions. The object is now to establish a test which can be applied to a given point rather than solving the set of equations (2.23), (2.24) and (2.25). Define a set of integers j = 1,... , J (J ~ n,) as the subscripts of those constraints 8i that are active at the point being tested. A point X may be a minimum if all the constraints 8i~ 0 are satisfied [see (2.25)] and if there exist Ai such that

L AiVgi =0 J

Vf+

(2.26)

i=l

Equations (2.26) are based on the conditions (2.23), considering only the active constraints. With this definition, the conditions (2.24) can now be excluded. To avoid situations in which the conditions (2.26) are satisfied and yet X is not a local minimum, we require j= 1,... , J

(2.27)

Equations (2.26) and (2.27) are the Kuhn-Tucker (KT) conditions for a relative minimum [88]. Define a cone as a set of points such that if Vg is in the set, AVg is also in the set for A~ o. The set of all nonnegative linear combinations (2.28)

forms a convex cone. The KT conditions require that -Vf be within the convex cone comprised by the active constraint normals Vgi (j = 1,... , I). These are necessary conditions for a point to be a relative minimum, but they are not sufficient to ensure a relative minimum. (This can be seen for example by the case of point A in Fig. USb). In convex programming problems discussed in the next subsection, the KT conditions are necessary and sufficient for a global minimum. A two-dimensional geometric interpretation of the conditions is shown in Fig. 2.1. In the case of Fig. 2.1a, - Vf is not within the cone formed by Vgi' The point is not optimal because/may be decreased without violating the constraints. KuhnTucker conditions are not satisfied, since we cannot find nonnegative Ai for which -Vf is expressed as a linear combination of Vgi' In the case of Fig. 2.1b the conditions are satisfied and the point is optimal. It can be seen that - Vf is within the cone of Vgi' and hence we cannot make any move that reduces/ in the feasible domain. We can find nonnegative Ai for which -Vf is expressed as a linear combination of Vgi. To illustrate the physical meaning of the Lagrange multipliers Ai' consider the problem Z=/(X, b)

~

min

(2.29) j

= 1,... , n,

(2.30)

2.1 Optimization Concepts

63

Vg j cone (b)

(a)

Fig. 2.1.

8.

KT conditions are not satisfied, b. KT conditions are satisfied.

where bj are some given parameters. The KT conditions for this problem are given by (2.26) and (2.27). It can be shown that at the optimum

dZ -=-1... db. J J

(2.31)

That is, Aj is the marginal price that we pay in terms of an increase in the objective function for making the constraints more difficult to satisfy. This explains why at the optimum all the Lagrange multipliers have to be non-negative. A negative Lagrange multiplier would indicate thatj(X) can be reduced by making a constraint more difficult to satisfy which is irrational. In summary, the KT conditions can be used to check whether or not a given point is a candidate minimum. Practical application of KT conditions as a test for a minimum usually requires the solution of simultaneous linear equations for the Aj. A procedure to compute Aj is discussed elsewhere [48]. The KT conditions provide also the basis for some of the constrained NLP methods.

Example 2.1. Consider the optimization problem (Fig. 2.2)

z =Xl + X2 ~

min.

gl ==xf+xi-8~O

g2 ==-XI-2.5~O g3 == -X2 - 2.5 ~ 0

64

2 Optimization Methods

-3.0

-2.0

-1.0

-1.0

-2.0

-3.0

Fig. 2.2. Space of example 2.1.

The conditions (2.26) are

Assuming all possibilities of two active constraints we fmd Case A B

C

Active constraints gl andg2 gl andg3 g2 and g3

X' {-2.5, -1.323} {-1.323,-2.5} {-2.5, -2.5}

Z

-3.823 -3.823 -5.0

In case C, the solution XTis infeasible because it does not satisfy the constraint gl SO. In case A Al = 0.37S, ~ = -O.S90, A3 = 0, and in case B, Al = 0.37S, A2 = 0, A3 = -O.S90, both being nonoptimal solutions. The optimum is XT = -{2.0, 2.0} , Z = -4, Al = 0.25, and A2 = A3 = 0, only the constraint gl S 0 being active. Modifying the constraint

xl+x~ SS such that Xf +X~ SS.1 we find the new optimum XT = -{2.0125, 2.0125}, Z = -4.025. At this point [see (2.31)] dZ == lJZ

dbt

llht

= -4.025-(-4) =-o.25=-AI S.1-S.0

2.1 Optimization Concepts

f

a!CX2 )+(l-a)!CXI

!

)

65

!

(X 2 )

x (b)

Fig. 2.3. a. Convex function, b. Nonconvex function.

Convex Functions and Convex Sets. The nature of the objective function and the feasible region can be determined using the definitions of convex function and convex set. A function f(X) is said to be convex if, on the line connecting every pair of points Xl and X 2 in its domain of definition, the value of the function is less than or equal to a linear interpolation offtX l ) andf(Xz), i.e.,

0< a < 1

(2.32)

The function is strictly convex if the strict inequality holds. A convex function is illustrated in Fig. 2.3a. If a functionf is convex, then (1) is concave. A linear function is both convex and concave, but neither strictly convex nor strictly concave. A function may be neither convex nor concave (see Fig. 2.3b).

~-------------------- XI

(a)

Fig. 2.4. a. Convex domain, b. Nonconvex domain.

(b)

66

2 Optimization Methods

A set of points is called convex if the line segment joining any two points Xl and X2 is contained entirely within the set. Mathematically, the set is convex if for all Xl and X 2 in the set, and 0 < a < I, the point Y;;;; aX I + (1 - a)X2 is also in the set (Fig. 2.4). The set may be bounded or unbounded. The functionftX) defined on a convex set is convex if and only if the Hessian matrix is positive semi-definite or positive-definite at all points in the set. A convex programming problem for minimization is one with a convex objective function ftX) and convex inequality constraint functions gj(X). In this case, the feasible domain formed by a single inequality constraint can be shown to be convex. Furthermore, the intersection of convex domains is convex. Thus, if the individual domains gj(X) ~ 0 are convex, the domain that is defined by all of them is also convex. A problem with equality constraints is convex if the hJ(X) are linear and if f(X) and gj(X) are convex. The intersection of linear equality constraints is convex, since a single linear equality constraint is a convex domain. Since all linear functions are convex, a linear programming problem is always a convex programming. A nonlinear equality constraint always defmes a nonconvex feasible region for the problem. For convex programming problems the Kuhn-Tucker necessary conditions are also sufficient. The significance of the above definitions is that in a convex programming problem any local minimum is a global one. However, it is often difficult to ascertain whether the functions in a given problem are convex. Problems which are not convex programs may still have only a global minimum or they may be solved for their relative minima, which provide useful information. Most optimal design problems cannot be shown to be convex. However, some of the approximate problems presented later in Chap. 3 are convex.

2.2 Unconstrained Minimization 2.2.1 Minimization Along A Line Problem Statement. Consider the case in which a point Xq+l is to be found by (2.33) where the point Xq and the direction vector Sq are given, and the scalar a is a single variable chosen as to minimize fl..X q + aS q ) with respect to a. From (2.33) (2.34) The object is to find the value of a, denoted a*, which minimizes f(a). This problem of finding a minimum of a function of a single variable a is one of the most important of unconstrained optimization problems, because this operation is basic to many techniques. (Note that a* does not produce the global minimum of J, unless the line X = Xq + aS q contains the global minimum point.) Different methods for this step are available. In general, the problem cannot be solved in a

2.2 Unconstrained Minimization

67

finite number of operations. and we often attempt to find only an estimate of the minimum. In this section. the golden section method and polynomial fitting techniques. which are commmly used. are described. In general. it is assumed thatj{a) is a unimodal lunction. That is. a minimum exists and it is unique in the interval of interest. For functions that are not unimodal. we locate only a local minimum point. Most solution procedures can be divided into two phases:

a. The location of the minimum point is bracketed and the initial interval of uncertainty is established. b. The interval of uncertainty is refmed by eliminating regions that cannot cmtain the minimum. This is done by computing and comparing function values in the interval of uncertainty. Golden Section Method. Using this method. it is assumed that the function I is unimodal. but it need not have continuous derivatives. Define f1l'St the Fibonacci sequence by

10

=1

I,. = 1,../ + 1,..2

11 = 2

n

= 2.3 •...

That is. any number is obtained by adding the previous two numbers. so the sequence of numbers is 1.2.3.5.8. 13.21 •.... The sequence has the property

lor n-+ oo This ratio between two successive numbers. as n becomes large. is called the

Golden Ratio.

Starting at a = O. we first evaluate j{a) for a = l). where l) > 0 is a small number. If j{l)
(2.35)

a 3 = 5.236l) + 1.6183 l) = 9.472l)

Assuming that (2.36) then the mlD1mUm lies between

uncertainty. I, is given by

aq

and

a q .2

•

and the initial interval 01

68

2 Optimization Methods /(0.)

~~

________

~

aq_2

(I-P)

ab

CXa

PI

~

________

>J-"E I

L-~a

I

1

at.

Initial bounds

__

(1-

P)

I

PI

J

I

I

First update

Fig. 2.S.

Golden section partition.

= Cl q = L q

Cl u

5(1. 618)q

ClL

= Cl q-2 =

j=O

L 5(1.618)q

q-2

j=O

q-l

(2.37)

1 =Clu - ClL = 2.618(1.618) 5

where Clu and of- are upper and lower limits on the interval of uncertainty. The object now is to reduce the interval of uncertainty. Assume two function values within the interval 1 symmetrically located at a distance of fY from either end, including the known value at Cl q_l' The new interval of uncertainty, 13/, is determined such that either the left or the right portion of 1 is eliminated. Considering the left portion of the interval, it can be seen from Fig. 2.5 (2.38) giving, 13 = 1/1.618 = 0.618 (the second root is not meaningful). Thus, the two points, Cl,. and Clb, are located at a distance of 0.6181 or 0.382/ from either side of the interval. The solution procedure for reducing the interval is as follows:

a. For the given Clq-2, Clq-lt Clq [see (2.35)] and 5 (a chosen small step size in the interval 1 is calculated by (2.37).

Cl),

2.2 Unconstrained Minimization

69

b. f(a,.) andf(ab) are computed at 0.,. =a L + 0.382/ and ab = a L + 0.6181. At the ftrst iteration 0.,. = a q_1 sof(a,.) needs no calculation. c. Iff(a,.) f(ab) then the minimum point 0.* lies between 0.,. and aU. Assume a L = 0.,." 0.,. = ab, compute JIaL+ 0.618(a U- a L )] and go to step f. e. Iff(a,.) =f(ab), assume a L =0.,.. aU =ab. I =aU - a L and go to step b. f. If the new interval of uncertainty I = 0. U - a L is small enough to satisfy a convergence criterion (i.e. I < e) let 0. = (aL + a U )/2 and stop. Otherwise, return to step c.

The method is most reliable and it is easily programmed for solution on digital computers. On the other hand, it requires a relatively large number of function evaluations.

The Quadratic Fitting. Polynomial-fitting techniques are most efficient and tend to give solutions with sufftcient accuracy in cases where 1 can be well approximated by low-order polynomials. For the quadratic fttting we assume that the functionf(a) can be approximated by the quadratic function q(a) = a + bOo + co.2

(2.39)

which has an easily determined minimum point At the minimum of q(a) we have to satisfy

dq

do.

or

= b + 2ca = 0

(2.40)

•

b (2.41) 2c The constant coefficients b and c (a is not needed) can be determined by computing 11./2./3' the value off(a) at three different 0. values, ah 0.2' 0.3, and solving the equations 0. = - -

It = a + ba1 + cat fz = a + ba2 + ca~ 13 = a + ba3 + ca~ If we use 0.1 =0, 0.2 =dO., and 0.3 equations (2.42) become

(2.42)

=2da, where dO. is a preselected trial step,

It =a fz = a + bda + c(.1a)2

13 = a + 2Ma + 4c(da)2

(2.43)

70

2 Optimization Methods

/(01.)

•I•

.6.01.

•I

Fig. 2.6. Quadratic approximation of /(a).

Solving for a, b, and c, we find

3ft - 13 c - ~/3~+_h~I_-....,,2,..=.f.~2 (2.44) 2(l\o.l 2l\o. Substituting (2.44) into (2.41), the approximated value of a. corresponding to the minimum value of q(o.) is a=ft

b = 4/2 -

a. *

(2.45)

For 0.* corresponding to a minimum and not a maximum of q(o.), we require (2.46) Based on (2.44) this condition can be expressed as

13+ft>/2 2

(2.47)

This means that the value of 12 must be below the line connecting 11 andl3 (see Fig. 2.6). 0.* is computed by (2.45) only ifl2 <11 and 12 <13. The Cubic Fitting. If the derivatives of I with respect to a. are readily computed, a two-point cubic interpolation can be used.j{o.) is approximated by C(o.) =a + bo. + co.2 + do. 3

(2.48)

2.2 Unconstrained Minimization

71

The parameters a, b, c, and d can be determined by solving the following equations for points a =A and a =B /(A) ==/A = a + bA + cA2 + dA 3 /(B) ==/B = a + bB + cB2 + dB 3

(2.49)

(df/da)A == fA = b + 2cA + 3dA2 (d/lda)B == fB = b + 2cB + 3dB 2

The minimum would be at one of the two points where

dC/da = b + 2ca + 3da2 =0

(2.50)

Solving (2.50) we find (2.51) Assuming A

= 0 and B = L\a, and solving (2.49) we obtain a=/A b =fA c- - -1- (3/A -3/B + 2/'A+ /,) B L\a

d=_I_ (L\a)2

f

L\a

(2/A -2/B + /' +/' ) L\a

(ex)

A =0

I•

ex* D.ex

B = D.ex

•I

Fig. 2.7. Cubic approximation of flu).

A

B

(2.52)

72

2 Optimization Methods

Substituting (2.52) into (2.51), a* can be found. To ensure that a* will be between A and B, we require fA < 0 and f B > 0 (see Fig. 2.7). The requirement for the minimum is d 2C - - 2 =2c+6da>O

da

(2.53)

Substituting (2.52) into (2.53), we may find that only the plus sign in (2.51) is required

2.2.2 Minimization of Functions of Several Variables Consider an unconstrained minimization in which the objective function Z =f(X) is to be minimized where no constraints are imposed on the choice of X. The significance of this class of problem stems from the following reasons:

a. Some design problems are either unconstrained or can be treated as

unconstrained during certain stages of the solution. b. Some of the most powerful and convenient methods of solving constrained problems are based on transformation of the problem to one of unconstrained minimization. Such methods are discussed in Sect. 2.4.1. c. Familiarity with unconstrained minimization methods provides a good conceptual base for studying constrained methods. Furthermore, a number of methods suitable for unconstrained optimization can be extended and applied to constrained problems. The following essential features should be considered before selecting a suitable solution method:

a. Differentiability and continuity of f(X) . It is well known that numerical

differentiation may result in errors due to poor approximation (too large interval) or excessive cancellation errors (too small interval). Therefore, it is preferable to compute the derivatives Vf analytically. b. Required accuracy of X* and accuracy of the objective function defmition. If the latter definition is not too accurate a rough estimate of X* might be sufficient. c. Special structure of f(X). Although no particular cases are assumed in this section, it is often profitable to use methods which take advantage of a special structure of the problem under consideration. Most optimization methods can seek only a relative minimum and convergence to the global minimum is not ensured, unless f(X) is a convex function. In practical problems where the functions f(X) are complex and their evaluations are time consuming, the number of function evaluations is usually an adequate measure of the algorithm's performance. Since no single method can be efficiently applied to all problems, many algorithms have been developed to handle different types of optimizations. The development of algorithms for unconstrained minimization is an area of active research and as such old algorithms are continuously being

2.2 Unconstrained Minimization

73

improved and new algorithms are emerging. The methods discussed in this section fall into the following categories:

a. Direct search methods. where the solution of the minimization problem is

found by solving a sequence of minimizations along direction vectors. but without the need of using the derivatives Direct search methods depend upon a direct comparison of the value of the objective function at several points. therefore the amount of effort required by the user is relatively low. h. Gradient methods. which are based on calculation of the gradient vector Using the derivatives of the objective function. more information is available and efficiency is increased. However. the evaluation of the derivatives is not always an easy task. c. Newton and Quasi-Newton methods. which are based on the matrix of second derivatives off. H [see (2.3)]. or its approximation.

vr.

vr.

Direct Search. Direct search methods are based on comparison of the value of the objective function without the need of using derivatives. These methods are usually reliable. easy to program. can deal effectively with nonconvex and discontinuous functions. and in many cases can work with discrete values of the design variables. The price paid for this generality is that these methods often require many function evaluations to achieve the optimum. Therefore. they are most useful for problems in which the function evaluation is not expensive. and will be competitive for relatively small optimization problems.

quadratically convergent methods. In the discussion that follows we consider the

quadratic function

q = XT a X + XT b+ c

(2.54)

where a is a given positive defmite symmetric matrix. b is a given vector. and c is a given scalar. The importance of this form is that many functions are reasonably approximated by a quadratic function near their minima. If a minimization method always locates the minimum of a quadratic function in no more than a predetermined number of operations. and if the limiting number of operations is directly related to the number of variables. then the process is said to be quadratically convergent. Most quadratically convergent methods are based. in one way or another. on the concept of conjugate directions. A set of n nonzero direction vectors S10 S2•... ,s" are said to be conjugate to each other with respect to a given (n>
for all i:1= j

(2.55)

A set of such directions possesses an extremely powerful property [107]. namely.

if a quadratic function q is minimized sequentially. once along each of a set of n

linearly independent conjugate directions. the global minimum of q will be located

74

2 Optimization Methods

at or before the nth step, regardless of the starting point. Note that the order in which the directions are used is immaterial to this property. Consider again the Taylor series expansion of a general function f about its minimum X* [see (2.2)]. It has been shown [see (2.6)] that = 0 at the minimum. As AX approaches zero. higher-order terms of the series can be neglected and f(X) approaches the quadratic form. Moreover. if U* is positive definite. the approximate quadratic function has its minimum at X*. Applying quadratically convergent methods to a general function for which the Taylor series is dominated by the quadratic terms near the minimum. then a rapid convergence in the neighborhood of X* is expected.

vr

Conjugate directions method. Among the various methods based on the concept of conjugate directions. powell's method [107] is one of the most efficient. reliable and successful. The method requires that the functionf(X) be unimodal. that is. it has only one minimum. The differentiability requirement onf is implicit in the exact line searches of the method. Consider the two-dimensional example shown in Fig. 2.8. The function is first minimized in each of the coordinate directions (directions S 1 and S 2). Minimization is then performed in the pattern direction. formed by a line through the initial and the last points of the previous step (direction S3). One of the coordinate directions is discarded now (direction Sl) and the pattern direction is included in the next minimization. A new pattern direction. S4. is generated and again one of the coordinate directions. S2. is replaced. The above steps are repeated until convergence. It can be shown that the pattern directions S3 and S4 are conjugate. A single iteration of this simple version of Powell's method is: a. Choose an initial point X and n initial independent directions (e.g .• the coordinate directions). Sq. q = 1.2•...• n . b. Set Y f- X . c. Find to minimize f(X + o.S q) and set X f- X + o.*Sq for q = 1. 2 •...• n. d. Set S,,+l f- X - Y. find 0.* to minimize f(X + o.S,,+I). and set X f- X + o.*S,,+1 . e. Replace Sq f- Sq+l for q = 1.2•...• n . f. Repeat from step b.

0:

Since the conjugate directions are not uniquely defined. different sets of n independent mutually conjugate directions can be found. The various ways for generating such directions form the basis for different methods all of which are quadratically convergent. Some difficulties may arise in the practical application of Powell's method. One problem is that each step is required to be a minimizing step in the given direction. Computing the exact minimum in each iteration step may require much computational effort. Usually. the functions to be minimized are not quadratic, and the number of iterations becomes large. For quadratic functions, we require three function evaluations per step and the number of steps is n2 , thus the total number of function evaluations is 3n2 • For nonquadratic functions this number may

2.2 Unconstrained Minimization

75

become 5n3 or more function evaluations, which makes the method prohibitive in problems with a large number of variables. Another problem is that the method can come to a halt before the minimum is reached. Both this failure and the previously mentioned inefficiency are because the Sj may become dependent or almost dependent. The original set of Sj is, of course, independent, and in theory each of the directions that are generated should be a linear combination of all the preceding Sj' unless some ex = O. One way to overcome this difficulty is to reset the directions to the original coordinate vectors whenever there is some indication that the directions are no longer productive. Powell [107] proposed a procedure for such modified directions. Gradient methods. Gradient methods are usually more efficient than direct search methods. The price paid for this efficiency is that gradient information must be supplied and that these methods often perform poorly for functions which have discontinuous fIrSt derivatives. While in direct-search methods only the values of the objective function are used to determine the direction of minimizations in the individual steps, gradient methods use information available by computing the gradient vector G ofj[see (2.3)]. Geometrically, G is normal to the tangent plane. That is, the direction of G coincides with that of greatest rate of change of j, the direction of steepest ascent. Therefore, -G is the direction of steepest descent.

----~~----------------------~~------------------------XI

Fig. 2.S. Progress of search along conjugate directions.

76

2 Optimization Methods

Given a point Xq , the best direction to move to reduce the function value would seem to be the one in which the function decreases most rapidly, namely, (2.56) where G q is the gradient vector at point X q • The solution is obtained by computing successively G q [by (2.3)], a.. (by minimization along a line), and Xq+l [by (2.56)] until the minimum is found. The method, called the method of steepest descent, can be useful in some problems, but better directions than that of -Gq can often be found. One difficulty which may arise even in problems of low dimensionality is that the process of solution can be very slow. For functions with significant eccentricity the method gradually settles into a steady n-dimensional zigzag. The method may work well for mildly distorted hyperspheres (Fig. 2.9). It can be shown that successive directions of steepest descent are orthogonal to one another. Another problem is concerned with computation of the gradient vector. Solving by gradient methods, one has to consider the following three situations:

a. Expressions for apaXj can be derived but are time consuming to compute . b. The gradient vector exists everywhere, but expressions for iJj(i)Xj are impractical or impossible to derive. c. The gradient vector is not defined everyWhere. In the frrst and second circumstances, consideration can be given to finite difference approximations of the partial derivatives. The third situation is more difficult, since the use of finite difference at a point along a discontinuity will be meaningless.

Conjugate gradient method. The convergence difficulties of the steepest descent method can be greatly reduced by a simple modification which converts it to the conjugate gradient method of Fletcher and Reeves [29]. The conjugate gradient directions generated by this method are not orthogonal to each other. Rather, these directions tend to cut diagonally through the orthogonal steepest descent directions. The general procedure for solving a problem is as follows:

a. Choose an initial point Xo- Compute Go = Vfo and determine So = -Go. b. For each iteration step q calculate

where

GTG

Sq =-G q + G T q G q q-l

q-l

S

q-l

2.2 Unconstrained Minimization

77

~------------------------------~Xl

Fig. 2.9. Two-dimensional example of steepest descents.

Since Sq is a linear combination of So. S1 ..... Sq.lt and G q • it is also a linear combination of Go. Glt....G q • It can be shown [29) that for the case of a quadratic function. XT a X + XT b + c. the conjugate gradient method generates a set of mutually conjugate directions. Sq. with respect to matrix a. Thus. theoretically the process should converge in n or fewer steps; however. for functions with highly eccentric contours. it can take considerably more than n cycles. It is proposed that the process be periodically restarted in order to clear out possible errors. A technique which can materially improve the rate of convergence of eccentric functions is the scaling of variables (Sect 1.3.4). Newton and Quasi-Newton Methods. We described in the preceding subsections direct-search methods in which no derivatives were used. and gradient methods which use rust derivatives. In this subsection methods based on the matrix of second derivatives H. or its approximation .will be discussed. Consider again the Taylor series expansion of f about the point X q • the qth approximation to the minimum point The expansion up to the quadratic terms is given by (2.2). At the optimum the condition (2.6) must be satisfied. Differentiating (2.2). we can find a vector X which satisfies the above conditions from

(2.57) which can also be written (2.58) Premultiplying by H;1 and designating X as the new value Xq+lt we obtain (2.59)

78

2 Optimization Methods

This is the traditional Newton-Raphson method, which is one of the preferred methods of solving simultaneous nonlinear equations. If! is a quadratic function, we obtain the minimum in a single step, since the Taylor series expansion is exaCL In many problems the process can converge rapidly but it may diverge in some cases. It will converge to saddle points and maxima, and sometimes the matrix Hq can be singular or near singular. The method of (2.59) can be improved by adding a scalar a* which minimizes!q+l (2.60)

This modification has a number of advantages. First, it will usually speed convergence, and can even secure convergence when the direct method diverges. Second, it will usually avoid convergence to a saddle point or a maximum. However, some disadvantages of the method still make it impractical for large or complicated problems. First, it may be difficult or impossible to compute the elements of H q • Even for simple functions, the second derivatives of! may be complicated or too time-consuming to compute. Second, computation of the inverse H;1 is impractical in large problems. Another principal difficulty is that H may be singular, or at least not positive-definite as is required to guarantee a solution for a minimum of f(X). In cases where! is linear in one or more of the design variables, the solution will be unbounded and the matrix H will be singular. If H can easily be calculated, Newton's method is often the preferred approach.

Variable metric method. The variable metric method of Davidon [21] Fletcher and Powell [28] is based on replacing the inverse of the Hessian H;1 by an approximate matrix Jq • The method eliminates the need for evaluating second derivatives and performing matrix inversions. Yet the method is quadratically convergent, and the matrix Jq which is improved at each iteration converges to

H;I. It is a conjugate direction method, and it can be viewed also as a quasiNewton or Newtonlike method. Quasi-Newton methods are used when only first derivatives are available, thus Newton's method cannot be implemented directly, but the spirit of this type of method is preserved. Using the variable metric method, we start with a given initial Xo and derme an initial positive definite matrix Joo The identity matrix I usually serves this purpose, namely, Jo=I. An initial direction So is determined by So = -Jo Go (where Go = Vfo ) and the qth iteration step is performed as follows:

b. Compute Jq+l

=Jq + Aq+ Bq• where

2.2 Unconstrained Minimization

79

Y q = Gq+1 - Gq

In applying the method, care must be taken to ensure that J q is not updated with data arising from poor approximations to Cl •q • In practice the algorithm is a powerful method and difficulties seldom arise, except on very badly eccentric functions. As with gradient methods, the computation of Vf by finite difference can be considered for the variable metric method. Stewart [133] developed an estimate of the incremental size in finite difference that will produce maximum accuracy. With Stewart's modifications, this method becomes competitive with powell's method for situations in which formulas for derivatives are not available or are impractical to compute.

Direct update methods. Another approach, known as the Broyden-Fletcher-

Goldfarb-Shanno (BFGS) [42] is to update the Hessian rather than its inverse. The solution procedure is as follows. For the initial design Xo. the Hessian of the objective function is estimated by a symmetric positive definite matrix Ho. (i.e. H o, =I), and the gradient vector, Go = Vfo, is calculated. The qth iteration step is performed as follows:

a. Calculate the norm of the gradient vector IIG qll. If IIG qll < £ then stop the iterative process. b. The search direction Sq is determined by solving the linear system of equations [see (2.57)]

c. The optimum step size Cl* is computed to minimize f(Xq + ClSq} •

=

d. The design is updated by Xq+l Xq + Cl*Sq. e. The Hessian approximation is updated by

where

80

2 Optimization Methods

YqY:

D = --'--"--q (Yq ·.1X q ) L\Xq

E

q

=_G--'-qG .......~_ (Gq·S q )

=U·S q

It can be shown that the BFGS update formula keeps the Hessian approximation positive defmite if exact line search is used. Numerical difficulties can arise since the Hessian can become singular or indefinite due to approximate line search or round-off errors. Some modifications can be used to overcome this problem.

2.3 Constrained Minimization: Linear Programming 2.3.1 Introduction Linear programming (LP) is a fundamental mathematical programming method. The special characteristic of the problem is that all the constraints and the objective function are expressed in linear terms of the variables. The constraints might be either equalities or inequalities and the objective function is minimized or maximized. Although only a relatively small part of structural design problems can be formulated as LP, the method is widely used. Some of the reasons for its popularity are:

a. The exact global optimum is reached in a finite number of steps and there are no local optima.

h. Computer programs of LP are most efficient. Problems with a relatively large number of variables and constraints can be solved in a reasonable computation time. c. Reliable computer programs which can be used as "black boxes" are available and data preparation is simple. d. Due to the high efficiency and reliability of LP programs they are often used as subroutines in solving nonlinear programming problems. LP is a powerful tool in methods such as feasible directions or sequential LP, described later. e. Some practical nonlinear problems are often approximated by a linear formulation and solved by LP algorithms. In this chapter formulation of the LP problem is presented and the simplex method [20] for solution is briefly discussed. The reader interested in more details, insight, and mathematical background is referred to the extensive literature in this area [20,

39,45].

2.3.2 Problem Formulation The general LP problem can be stated as one of choosing the variables XT = (Xl> X 2, ... , XII) such that

2.3 Constrained Minimization: Linear Programming

L

81

II

Z=

CjX j

~ min

(2.61)

j=l

subject to

L II

ajjXj {

~. =. ~ )bj

i=I. .... m

(2.62)

= 1,... , n

(2.63)

j=l

Xj ~

0

j

where ajj, bj and Cj are constant coefficients. The notation (~, =,~) means that the constraints might be either equalities or inequalities (~ or ~). This formulation is general and covers a wide range of problems, as can be seen from the following observations:

a. Although the objective function Z in (2.61) is minimized, problems of maximization are also included in the present formulation: instead of maximizing Z, the solution can be reached by minimizing (-Z). b. The constraints (2.62) might be either equalities or inequalities (~ or ~). From computational considerations the coefficients bj are required to be nonnegative. This requirement can always be satisfied since in cases of b j < 0 we may multiply the corresponding constraint by (-1) so that bj will become positive. c. For purposes of computation, (2.63) requires that all variables be limited to the nonnegative range. If in the original problem variables may possess negative values, as in many structural design problems, we can express them as the difference between two nonnegative variables

(2.64)

where X'l; ~ 0 and X\ ~ O. Thus the constraints (2.63) can always be satisfied; however, the number of variables is increased. Other transformations can be used to ensure nonnegativity of the variables (see for example Sect. 4.1.1). If any specific variable Xl; has a lower bound limitation (2.65)

t

where X < 0, then we may transform to a new variable, X'l;, given by (2.66)

The constraint (2.65) will become (2.67)

82

2 Optimization Methods

The advantage in using the transformation (2.66) is that the number of variables remains the same as in the original problem. In addition, we do not have to consider the constraint (2.67), since it is included in the nonnegativity constraints (2.63). For purposes of computation, a standard form of LP is utilized. It permits use of a standard algorithm and simplifies the discussion of its application. It will be shown that all linear programming problems can be written in the following standardform: fmd XT=(Xh ••• , X,,} that minimizes Z=

L" CjXj

(2.68)

j=l

subject to

" L aijXj =b j=l

j

i=I, ... ,m

(2.69)

j= 1,... , n

(2.70)

In order to convert the inequalities in (2.62) to equalities (2.69) we introduce new variables. If the hth constraint is an inequality of the form

Lj=l" altjXj :S::b

lt

(2.71)

a nonnegative slack variable, X,,+1t ~ 0, is added to the left-hand side, resulting in the equality

Lj=l" altjXj +X,,+1t =bit

(2.72)

If the kth constraint is an inequality of the form

Lj=l" akjXj ~bk

(2.73)

a nonnegative surplus variable, X,,+k, is subtracted so that we have

L" akjXj - X,,+k =bk

(2.74)

j=l

The slack and surplus variables satisfy the nonnegativity constraints (2.63). Thus, any vector X that satisfies the equalities (2.72) and (2.74) will satisfy also the

2.3 Constrained Minimization: Linear Programming

83

original inequalities (2.71) and (2.73) . In addition, since all the coefficients of the new variables in the objective function equal zero, their contribution is zero and the original function (2.61) is unchanged. Using the transformation (2.72) or (2.74) for all inequalities, the original problem can be formulated in the standard form of (2.68), (2.69) and (2.70). If m = n in the standard form problem and none of the equations (2.69) are redundant, there is only one solution to the system of equations. If m > n, then there are redundant equations that can be eliminated or the system has no solution. In all cases of interest m < n, the system possesses an infinity of solutions of which we seek the one that minimizes Z and satisfies the nonnegativity constraints Xj ~ O. A number of standard definitions can be made now. A vector that satisfies the equalities (2.69) represents a solution. If the nonnegativity constraints (2.70) are also satisfied, this is afeasible solution. The optimal feasible solution is the feasible solution which minimizes the objective function Z. A basic feasible solution is a feasible solution with no more than m nonzero Xj' In other words, it has at least (n-m) Xj that are zero. A nondegenerate basic feasible solution has exactly m positive Xj'

4

3 2

4

\\\ 3

Xl

2X l + X 2 = 4 max Z = Xl + X 2 (b)

(a)

3

3 (e)

Fig. 2.10.

Examples, bounded region.

Cd)

84

2 Optimization Methods

Geometrical Interpretation. Some observations can be made for twodimensional examples (Figs. 2.10 and 2.11). Considering only inequalities in the original problem, the constraints define a feasible region bounded by portions of some of the planes ailXI + aizX2 =bi . It can be seen that the optimum value of Z is obtained at one of the vertices (extreme points). Each vertex represents a basic feasible solution of the standard form problem. If the region is not bounded, then Z may still take on its minimum values at a vertex or it may approach 00 (or -00). In the latter case the solution is said to be unbounded. If the LP has a bounded solution, it will always take on its minimum at a vertex; but it may take on this same value all along an "edge" joining two vertices, or over the hyperplane containing three vertices, etc. These observations can be stated rigorously as theorems and are proven in most standard texts on LP [20,45]. In Fig. 2.10, LP problems with the constraints 2X I + X2~4 Xl + 3X 2 ~ 6 Xl, X2~ 0

and different cases of objective functions are demonstrated. The feasible region is bounded and different optimal solutions are obtained. In Fig. 2.11 the constraints

are

2X I + X 2 ~ 4 Xl + 3X2~ 6 XI,X2~0

The feasible region is unbounded and it can be seen that the optimal solution might be either at a vertex or it may approach 00 •

X2

Xl

4

4

3

3

2

2

(a)

Fig. 2.11.

Examples. unbounded region.

QI"

~";~ (b)

2.3 Constrained Minimization: Linear Programming

2.3.3

85

Method of Solution

It can now be stated that if the standard form linear programming problem has a bounded solution. the minimum of Z is attained at one of the basic feasible solutions of the program. This is an extremely powerful result. since there is a finite number of basic solutions; that is. there can be no more than the number of ways m variables can be selected from a group of n variables. or

n! (n-m)!m!

(2.75)

=(n) m

Usually there are much fewer possibilities. since many cQmbinations are infeasible. The simplex method. which will be described briefly here. is a powerful computational scheme for obtaining basic feasible solutions. If a solution is not optimal. the method provides a procedure for finding a neighboring basic feasible solution which has an improved value of Z. The process is repeated until. in a finite number of steps (usually between m and 2m). an optimum is found. Rewrite (2.69) in the expanded form auXI a21XI

+ alzX2 + ... + al..){" =bl + a2zX'2 + ... + a~" =~

(2.76)

It is possible to place this system into a form from which at least one solution can readily be deduced. We obtain this form by replacing the original system of equations with another one of the same size which is a linear combination of the equations of the original system. This procedure. called a pivot operation. produces the new system which is said to be equivalent to the original one. Both systems have exactly the same solutions. We may repeat operations on the system until there will be m columns. each containing zeros and a single 1.0 . These can always be arranged in the form +a'I,III+1 XIII+! + .. ·+a'I,,, X" = b'l +a'2, ...+1 X III+1+... +a'2,,, X" = XIII

b'2

(2.77)

+a'.......+l X...+l +.·.+a'...." X" = b'...

where the primes indicate that the a'ij and b'i are changed from the original system. Such a system is said to be canonical or in a canonical form. One solution to the system is always

j= 1.2..... m

j

= (m+ 1), (m+2), ... , n

(2.78) (2.79)

86

2 Optimization Methods

This solution is a basic solution and the ftrst m of the Xj are basic variables. If all the bj are nonnegative, then the solution given by (2.78) and (2.79) is a basic feasible solution. The general procedure for solving LP problems consists in going from one basis, or canonical form, to an improved one until the optimum is found. In discussing the method of solution the questions that should be answered are:

a. What are the criteria for choosing an improved basic feasible solution? b. How to go from one basic feasible solution to another one? c. How to ftnd an initial basic feasible solution in a simple way?

Criteria for Choosing An Improved Feasible Solution.

Given a system in canonical form (2.77) corresponding to a basic feasible solution, we can pivot to a neighboring basic feasible solution by bringing some speciftc variable X t into the basis in place of some Xs. The objective function Z can be expressed in terms of the nonbasic variables by eliminating the basic ones

Z = ZO +

L"

C'j Xj

(2.80)

j=III+1

where ZO is a ftxed value and C'j are constant coefftcients of Z in its new form. Note that ZO represents the value of Z for the current basic solution, since Xj = o for j = (m+l), ... , n [see (2.79)]. The question now is how to choose X t and Xs. If any speciftc coefftcient C't (corresponding to a nonbasic variable X,) in the objective function of the canonical form (2.80) is negative, then the value of Z can be reduced by increasing the variable X t from zero, while keeping all the other nonbasic variables at zero. Therefore, to improve the solution, we can bring any such X t into the basis. If more than one C'j is negative, we can choose among them, and one possibility is to choose the t for which C't

=m~n(c'j )

(2.81)

}

This choice selects the variable for which the objective function reduces at the greatest rate. It should be noted that some other choice may produce a greater improvement in Z. However, the above criterion is simple, and experience has shown it to be efftcient. If min (cj) is not negative, then there is no variable that can be increased to improve the objective function, and the optimality condition is satisfted. Bringing the chosen Xt into the basis, we have to determine now what variable Xs will leave the basis. Going from one basic solution to another, we increase X t from zero to some positive value during the process, while Xs becomes zero. If we are to keep the solution feasible, we cannot increase X, from zero by more than it takes to make the "frrst" Xs just go to zero, since to increase it further would make Xs negative. Suppose we take the canonical system (2.77); then keeping all the nonbasic variables zero except Xt, we can see the effect of increasing XI on the basic variables from

2.3 Constrained Minimization: Linear Programming

Xi

= b'i - a'it{,

i= 1,... , m

t> m

87

(2.82)

If any specific a'n is positive, the largest value of X, before Xs becomes negative is

Xl --~

(2.83)

a'n

If a'n is negative, no positive X, will make Xs negative. Therefore, X, is bounded by

X, = m~n(b'i la'il )

for

a'il>O

(2.84)

I

In other words, since we wish to make X, basic (nonzero) in place of some other variable Xs which will become nonbasic (zero), we must choose the row s for which X. = 0 by (2.82), and all others are nonnegative. Hence row s must produce the minimum value in (2.84). Note also that the chosen variable Xs has a coefficient of 1.0 in the sth row. In summary, once it has been detennined to bring the currently nonbasic variable X, into the basis [using the criterion (2.81)], we compute s by the condition (2.84) b's -,-= aSI

. (b'i I ail ' )

m~n

for

a'il>O

(2.85)

I

If no such s exists, i.e., there is no a'il > 0, X, can be increased indefinitely without the solution becoming infeasible and we have an unbounded solution. Otherwise we pivot to a new basic feasible solution that includes X, in place of XS'

Going From One Basic Feasible Solution to Another One. Once t and s have been detennined, transfonnation to the new basis is perfonned by the following pivot operations, where a'sl is the pivot tenn • Ysj

iSj

=-,a sl

(2.86)

(2.87)

•

Yij and iij are the coefficients of the new and current bases, respectively, in the ith row and the jth column. The transformations (2.86) and (2.87) are carried out also for the column of b'i and for the row of the objective function Z. Example 2.2. As an example consider the canonical fonn

88

2 Optimization Methods

3

1

Xl-4X2+~3-4X4

1 3 --X2 +3X3 --X4 +Xs

4

4 8X2 -24X3 + 5X4

=3 =5

= Z-28

The objective function is expressed here in tenos of the nonbasic variables. This can be done by eliminating the basic variables Xl and Xs. The basic solution is Xs = 5 From the last row it can be observed that Z = 28. Since C'3 < 0, we can improve the objective function by bringing X3 into the basis. To detenoine the variable s to leave the basis we compute

1)

b's = min(l =1 2 '3 2

a's3

Thus, s = 1 and the pivot is a'st = a'l3 (the underlined term). Using the transfonoations (2.86) and (2.87), the new canonical system is

131 -Xl --X2 +X3 --X4 2 8 8 3 7 3 --Xl +-X2 --X4 +Xs 2 8 8 12Xl -X2 +2X4

3 =2 1 =2 =Z+8

X3 and Xs are the new basic variables, and the new basic solution is

The improved value of the objective function is Z = -8. Since C'2 < 0, the objective function can be improved by bringing X2 into the basis. Thus we may find the new basis and proceed with the above procedure until the optimum is found. Special Cases. If we pivot on a row for which the right-hand side b'i is zero, the value of the objective function or of the basic variables will not change. In such a case, the value of the basic variable corresponding to the zero b'i is zero, and we have a degenerate basic feasible solution. In the degenerate case, the objective function is not improved even though a variable which has a negative coefficient in the objective function is brought into the basis. In the optimization procedure that will be described subsequently, we will still perfono such pivot operations, since doing so may allow us to proceed to other nondegenerate solutions.

23 Constrained Minimization: Linear Programming

89

If all the coefficients a'it in any column t corresponding to a negative c', (not necessarily that with the smallest cj) are negative. Xt can be increased indefinitely without causing any variable to become zero [see (2.82)]. The result is that the objective function can be reduced indefmitely and we have unboUllded solution. Initial Basic Feasible Solution. A basic feasible solution is required as a starting point. The object is to form an initial canonical system in which the coefficients aij of the basic variables form a unit mattix. so that a simple basic feasible solution can be found. If all the constraints in the original problem are inequalities (~) with nonnegative bi • then in order to get the standard form we have to add slack variables to all the constraints. The coefficients of the slack variables form in this case a unit matrix. thus they can be chosen as initial basic variables of the canonical system. This is not the case if some of the constraints are equalities or ~) inequalities. In order to start the solution with a unit matrix. we may add to each of the constraints a new variable called artificial variable. The original system of equations (2.76) in its standard form will become

allX1+···+alJ. X" + X,,+l 0:l1X1+"'+O:l"X" +X,,+2

(2.88)

Equations (2.88) are expressed in a canonical form where Xj.j =(n+l) •...• (n+m) are artificial variables. A basic feasible solution of this system is (bi ~ 0) j= 1.2•...• n

Xj=O X"+j =bj

j

= 1.2,... , m

(2.89)

The coefficients of the artificial variables in the original objective function are set equal to zero. The algorithm for solving the problem with artificial variables consists of two phases: phase I to find a basic feasible solution. if one exists. in which all artificial variables equal zero. and phase II to compute the optimal solution. In phase I we defme an "artificial" objective function

=L

"+111

Z'

Xj

(2.90)

j=,,+l

which is to be minimized. If the minimum of Z' is zero. then all the artificial variables have been eliminated from the basis and a new basic feasible solution is available which contains only the variables of the original system Xj' j = 1. 2 •..• n. Then the artificial variables and objective function (X) can be dropped and we proceed and solve phase II. namely. the problem with the original variables and objective function. If the minimum of X is greater than zero, then no basic feasible solution to the original problem exists.

90

2 Optimization Methods

General Iterative Procedure. Starting with the system in canonical fonn with a feasible basis, we can now state a general procedure for going from one basic feasible solution to an improved one. In addition, we may establish a criterion to identify whether a basic feasible solution is optimal or not. We have noted that if any C't corresponding to a nonbasic variable X t is negative, the objective function can be improved. A simple and efficient criterion for choosing Xt is given by (2.81). We have seen also that the criterion for choosing a variable X. to be eliminated from the basis is given by (2.85). So we have a general iterative procedure for proceeding from one basic feasible solution to an optimal solution, if one exists. In addition, we can determine if no feasible solution exists by checking the value of the objective function after completing the procedure for phase I. If min z: > 0, then the problem has no feasible solution. We can also detennine the unboundedness of the solution if that is the case. It has been shown that if all the coefficients a'it in any column t corresponding to a negative C't are also negative, then the solution is unbounded. Finally, we have a procedure for establishing an initial canonical system or initial basic feasible solution. We can now use all the above to fonnulate a general iterative procedure for solving LP problems as follows:

a. Formulate the LP problem in a standard fonn, namely:

-

b. c. d.

e.

nonnegative variables [use the transformations (2.64) or (2.66), if necessary] nonnegative bi (multiply the constraints by -I, if necessary) all constraints are made into equalities (add slack or surplus variables, if necessary) the objective function is minimized (change sign, if necessary) Fonn a starting canonical system with a basic feasible solution. Add artificial variables to the constraints, if necessary. Solve phase I, with the objective function (Z:) equal to the sum of artificial variables, by steps e through h. If min Z > 0, the problem has no feasible solution. Terminate procedure. If min z: = 0, eliminate all artificial variables and proceed with variables and objective function of the standard fonn problem. Find the variable X t to bring into the basis by computing C't

f.

= m~n(c'j )

(2.91)

J

If C't ~ 0, the optimum has been found. Terminate procedure. If variable X. to be eliminated from the basis by computing b'. m~n . (b'i I ait ' ) -,-=

a.t

I

for

a'it>O

C't < 0,

find the

(2.92)

g. If no such s exists, the solution is unbounded. Terminate procedure. If s has been found, a new basis and objective function are computed by pivoting on a'.t

2.3 Constrained Minimization: Linear Programming •

illj

II

a'.

y.=-

91

(2.93) i;#s

(2.94)

s is the row in which XII has a coefficient of 1.0, and Yij• and

iij

are the

coefficients of the new and the current canonical systems, respectively. h. Proceed with steps e.jand g until tennination. Application of the iterative procedure is demonstrated in the following examples. Example 2.3. Find the optimal solution of the LP problem (Fig. 2.10a)

2XI+ X2 S;4 Xl + 3X2 S; 6 Xl + X 2 =Z-+ max

°

In addition, the nonnegativity constraints XI, X2 ~ must be satisfied. Converting the objective function and adding slack variables, the standard fonn problem is

2XI + X 2 +X3 =4 X I +3X 2 +X4 =6

-Xl -

X2

X3

X4

1 0 0

0 1 0

=Z -+ min

or in a tableau fonn

XI

Z

1 -1

1t

Xa 1 3 -1

• ~

b 4 6

Z

•

Since all the constraints are (S;) inequalities with nonnegative bit artificial variables are not required and we start directly with phase II of the solution. The initial basic feasible solution is X3 4, X4 6, Xl X2 0, and Z =0, which is the vertex {O,O} in Fig. 2.10a. Since c't C'2 -I, either Xl or X2 can enter the basis. Choosing X, = Xl we compute the variable to be eliminated from the basis by

=

= = = = =

2 ' ~}=2 I

min{i

=

namely, X3 is eliminated from the basis. The next pivot is au 2, the underlined term in the tableau. The • denote the basic variables, 1t shows the new basic

92

2 Optimization Methods

variable, and .u the variable to leave the basis. Performing the pivot operations we fmd the new canonical system Xl 1 0 0

X2 1/2

X3 1/2 -1/2 1/2

m

-1/2

*

X4

b

2

0

1

4

0

Z+2

*

.u

1t

The new basic solution is Xl =2, X4 =4, X2 =X3 =0 and Z = -2, which is the vertex {2,O} in Fig. 2.10a. The new pivot is an = and after transformation we have the new canonical form

sa,

1

o

o *

o

o

b

3/5

-1/5 2/5

1

*

-1/5

2/5

1/5

6/5 8/5 Z+14/5

As all e'j ~ 0, this is the optimal solution: Xl =6/S = 1.2, X2 = 8/S = 1.6, X3 = X4 =0, and Z = -14/5 = -2.8. (Note that for the original problem, Z = 2.8). It represents the vertex {1.2, 1.6} in Fig. 2.10a. In this example we see how the solution proceeds from one vertex, or basic solution, to a neighboring and improved one until we fmd a vertex in which the objective function is better than in all its neighboring vertices. We do not have to check points inside the feasible region during the solution. Example 2.4. Find the optimal solution of the LP problem 5X l

-

4X2 + 13X3 - 2X4 + Xs

=20

In order to have an initial basic solution we add two artificial variables, X6 and X7.The new problem is

2.3 Constrained Minimization: Linear Programming

93

Eliminating X6 andX7 from Z, we have the starting system for phase I Real variables

X2 -4 -1 6 5

Xl 5 1 1 -6

Artificial

X4 -2 -1 1 3

X3

11

5 -7 -18

Xs 1 1 5 -2

X7 0 1 0 0

X6 1 0 0 0

*

1t

b

20 8 Z Z'-28

*

!!

Initial values of the objective functions are Z' = 28, Z = O. In phase I only Z is minimized, but transformations are made also for Z. Preforming the next two pivots, we have Artificial

Real variables

Xl 5/13 -12/13 48/13 12/13

X2 -4/13 7/13 50/13 -7/13

X2 -3/8

Xl 1/2 -3/2 12 0

~

-1 0

X3 1 0 0 0

*

Xs 1/13

Bill

72/13 -8/13

b

20/13 4/13 Z+140/13 Z'-4/13

*

1t X4 -1/8 -3/8 2 0

X3 1 0 0 0

I

X7 0 1 0 0

X6 1/13 -5/13 7/13 18/13

!!

Real variables

*

1t

X4 -2/13 -3/13 -1/13 3/13

Artificial

Xs 0 1 0 0

*

X6 1/8 -5/8 4 1

X7 -1/8 13/8 -9 1

b

3/2 1/2 Z+8 Z'+O

!!

Since Z' = 0 and the artificial variables are not in the basis, phase I is completed. Now we can drop the artificial variables and the row of Z'. Z is the objective function in phase II, and after performing the pivoting we have Xl -1/7 -12/7 72/7

X2 0 1 0

*

X3 1 0 0

*

X4 -2/7 -3/7 11/7

Xs 3/7 8/7 8/7

b

12/7 4/7 Z+60/7

Since all c] > 0 this is the optimal solution

X2 =4n

X3 = 12n

Xl =X4 =Xs = 0

Z = -60n

94

2 Optimization Methods

2.3.4 Further Considerations The Dual Problem. The original LP problem (2.61), (2.62) and (2.63) can always be formulated as one of choosing XT = {Xl' X2• .... X.. ) Such that

Z=

..

L

CjX j

(2.95)

-+ min

j=l

subject to

..

L

aijXj

i

=1, ....

m

(2.96)

j

= 1,... , n

(2.97)

~ bi

j=l

This can be done by expressing all inequalities in (~) form. Note that we do not require bi ~ 0, and that the equality constraints can also be expressed as inequalities. For example, the constraint

L II

akjXj

(2.98)

=bk

j=l

can be replaced by the following two inequalities

L

-L II

II

akjXj

~bk

j=l

~ -bk

akjXj

(2.99)

j=l

The problem (2.95), (2.96) and (2.97) is called the primal problem, for which a dual problem, with variables Ai' can be stated as follows

L biAi -+ max WI

cjI =

(2.100)

i=l

subject to

L WI

aijAi $, Cj

j

=1. ....

n

(2.101)

i=l

i= 1•...• m

(2.102)

2.3 Constrained Minimization: Linear Programming

95

in which the coefficients aij. bit and Cj are identical to those of the primal problem. If. for example. the primal problem is

Z=X1 +2X2~ min

then. the dual problem is

cII =6A1 + 10Az + A3 ~ max

The solutions of the two problems are identical. namely. min Z = max cII. If one of the two problems is solved. we can find the solution of the other. Since the computational effort in solving LP problems is a function of the number of constraints. it is desired to reduce this number. The number of constraints in the dual problem is equal to that of variables in the primal and vice versa, thus we can solve the problem with the smaller number of constraints. For example. if we have a primal problem with two variables (n=2) and 20 inequality constraints (m=20). then 20 surplus variables are needed to convert the problem into a standard form. At each stage we must deal with a 20 x 20-basis matrix. The dual problem. however. has two inequality constraints and 20 variables. This leads to a 2 x 2basis matrix and to a great computational advantage. If the primal problem is given in the standard form: find XT (Xl ..... X.. ) such that

=

Z=

..

L

..

L

CjX j

~min

(2.103)

j=l

aijXj

= hi

i=I ..... m

(2.104)

j=l

= 1..... n then the corresponding dual problem is: fmd AT=(A1 ..... A.".) such that j

(2.105)

96

2 Optimization Methods

(2.106) i=l

L III

aijAi

i=l

~ Cj

j=I, ... , n

(2.107)

(2.108)

~ unconstrained in sign

A primal problem in any of the forms (2.95) through (2.97) or (2.103) through (2.105) may be changed into any other by using the following devices:

a. replace an unconstrained variable by the difference of two nonnegative variables;

b. replace an inequality constraint with an equality by adding a slack or a surplus variable; c. replace an equality constraint by two opposing inequalities. We can make a one-to-one correspondence between the ith dual variable Ai and the ith primal constraint and between the jth primal variable and the jth dual constraint. Table 2.1 gives the primal-dual correspondence [89]. It can be shown that the dual variables Aj, called also shadow prices. give the variation of min Z per unit change of bi

A. = A(minZ) , Mi

(2.109)

Equation (2.109) is valid only in the range in which changes in bi do not change the basis of the optimal solution but only the value of the basic variables. Table 2.1 Primal-dual correspondence.

Primal Quantity

L"

Corresponding dual quantity

L biAi ~max L ~ L = III

CjXj

~min

i=l

j=l

III

aijAi

Cj

aijAi

Cj

i=l III

Xj

unconstrained in sign

L" L"

i=l

aijXj =bi

j=l

j=l

aijXj

~bi

Ai unconstrained in sign

2.4 Constrained Minimization: Nonlinear Programming

97

Sensitivity Analysis. The optimal solution is affected by the constant coefficients of the LP problem - aij. bit and Cj. It is sometimes important to know the sensitivity of the solution to changes in these coefficients, without solving again the LP problem. Changes in a limited range will not change the basis of the optimal solution. The sensitivity analysis might be a useful tool in computing the modified optimal solution for such cases. Define the matrix a and vectors b, c of the coefficients aij. bit and Cj. respectively. Changes L\a, L\b, or L\c can be made such that the modified values a"" bIll and c'" are given by a",=a +oAa bIll = b + pL\b c'" = c + yL\c

(2.110)

in which a, p and y are nonnegative scalars. Critical values of a, Pand y can be determined, for which the optimal basic variables should not be replaced and the new optimal solution can be found without solving the LP problem again. For cases of change in a it is often recommended to solve the complete problem again. Sensitivity analysis deals also with cases of adding variables and constraints to the original problem and computing the modified optimum. A detailed discussion is given in most LP texts.

2.4 Constrained Minimization: Nonlinear Programming In general, no single nonlinear programming method can solve efficiently all constrained optimization problems. The effectiveness of the optimization method depends on both the algorithm and the software. many algorithms have been developed and evaluated for practical optimization. The following factors should be considered in choosing a method: - Efficiency of the method is particularly important in large scale problems. Efficient methods are characterized by fast rate of convergence to the optimum point and a few calculations at each iteration cycle. Efficiency can greatly be improved by considering approximation concepts, discussed in Chap. 3. - Reliability (or robustness) of the method, that is, convergence to a minimum point is theoretically guaranteed, starting from any initial design. Reliable algorithms usually require more calculations during each iteration compared to algorithms that have no proof of convergence. Thus, robustness and efficiency are two conflicting factors that should be considered while selecting an algorithm. - Ease of use of the method is important in many practical applications. An algorithm requiring selection of input parameters might be difficult to use, because proper specification of the parameters usually requires previous experience. In considering the variety of algorithms available, there are no reliable rules to determine which method is best. It is most important to use an algorithm that

98

2 Optimization Methods

provides acceptable results for the problem of interest. Some of the more complicated algorithms, considered best by the theoreticians, are found to be less reliable for problems that are not carefully formulated. On the other hand, algorithms like the feasible directions method (Sect. 2.4.2) and the sequential LP (Sect. 3.1.3) are considered "poor" by the theoreticians, but usually perform reliably and efficiently in a practical design environment. Using constrained minimization methods, the design variables are modified successively during the design process by moving in the design space from one design point to another. Most methods consist of the following four basic steps:

a. Determination of the set of active constraints at the current design. b. Selection of a search direction in the design space, based on the objective function and the active constraint set. c. Calculation of how far to go in the direction found in step b. d. Convergence check which determines whether additional modifications in the design variables are required. Constrained optimization is a very active field of research, and many algorithms have been developed [42,91]. Only those methods which are most commonly used in structural optimization are presented here. Further considerations related to these methods are given elsewhere [2,48, 138]. Mathematical programming methods that are used to solve constrained optimization problems may be divided into indirect and direct methods. Indirect methods, described in Sect. 2.4.1, convert the problem first into an equivalent unconstrained optimization problem while direct methods. discussed in Sects. 2.4.2 and 2.4.3, deal with the constrained formulation as it is.

2.4.1 Sequential Unconstrained Minimization Unconstrained minimization methods discussed in Sect. 2.2 are quite general and useful for unconstrained minimization but are not suitable for constrained problems without modification. Sequential Unconstrained Minimization Techniques (SUMn are based on such modifications. The methods described in this section include:

a. Exterior penaltylunction methods, where all intermediate solutions lie in the infeasible region. One advantage of such methods is that the solution may be started from an infeasible point, eliminating the need for an initial feasible point. A major shortcoming is that we cannot stop the search with a feasible solution before the optimum is reached. b. Interior penalty (barrierJ1unction methods, in which all intermediate solutions lie in the feasible region and converge to the solution from the interior side of the acceptable domain. The advantage is that one may stop the search at any time and end up with a feasible and. hopefully, usable design. Moreover, the constraints become critical only near the end of the solution process; thus, instead of taking the optimal design we can choose a suboptimal but less critical design. Using the interior penalty-function approach, we keep the

2.4 Constrained Minimization: Nonlinear Programming

99

designs away from the constraint surfaces until final convergence. One drawback is that we have to start the solution always with a feasible design. c. The augmented Lagrange multiplier method, where Lagrange multipliers are incorporated into the optimization strategy to reduce the iII-conditioning often encountered in SUMT. The use of penalty function was fmt suggested by Courant [19]. Further work in this area is based on developments introduced mainly by Carroll [15] and Fiacco and McCormick [26]. The methods presented herein are widely used in structural design and have some practical advantages. The algorithms are general and suitable for various optimization problems. For problems of moderate complexity, the unconstrained formulations for constrained problems are usually simple and convenient to apply, provided an adequate minimization algorithm is available. In addition, the methods might work well with approximate behavior models discussed in Chap. 3. On the other hand, they may not be as efficient for some problems as the direct methods. An Exterior Penalty-Function. Consider a general nonlinear programming problem where the equalities are excluded: fmd X such that Z=j(X)

~

min j

= 1,..., n,

(2.111)

The idea behind penalty-function methods is simple. Rather than trying to solve the constrained problem, a penalty term that takes care of the constraints is added to the original objective function I and the problem is transformed to the minimization of a penalty function 'P(X, r)

".

'P(X,r)=I+rL, (gjr

(2.112)

j=l

The factor r performs the weighting between the objective function value and the penalty term, and it is often called the penalty parameter or response lac tor. The surfaces of 'P(X, r) are correspondingly termed response sUrfaces. The bracket operator means if gj > 0 (2.113) if gj SO

and the exponent y is a nonnegative constant. Usually, if the response factor r is positive, the minimum of 'II as a function of X will lie in the infeasible region. If r is chosen large enough, the minimum point of 'II will approach the constrained minimum ofI, subject to gj S O. The theoretical properties of this approach have been investigated by Zangwill [144]. Although several values for the selection of y are possible, y =2 , which is quite popular, will be used here.

100

2 Optimization Methods

The solution is obtained as follows. Initial values of X and r are chosen. X might be any point. not necessarily a feasible one. Some guidelines for the selection of r are discussed below. For the given r, a vector X· that minimizes 'I'(X, r) is calculated. If the point X· is in the feasible domain, the result is the optimum; otherwise r is increased r +- cr (c > 1), and "Starting from X· the function 'I'(X, r) is minimized again. The steps of increasing the values of r and minimizing the function 'I'(X, r) are repeated until the optimum point X· is in the feasible domain. There are a number of questions to consider in applying the method: a. How to choose the initial value for r and what is the desired rate of increase for r (c ?). To avoid an excessive number of minimizations of'll, it seems that a large initial r is desired. We might hope that this choice will force the minimum of'll toward the feasible region. However, for large values of r the function 'I' exhibits more distortion or eccentricity than for small ones. As r is increased, 'I' becomes more difficult and sometimes impossible to minimize. The conflict is thus clear: the initial value of r must be large in order to force the minimum to approach the feasible region, but still sufficiently small to enable the minimization of'll without excessive difficulty. This problem is the reason why r is sequentially increased from a moderate starting value. If the factor c is not excessively large, the use of X· as a starting point for the next minimization improves the likelihood that the minimum will be found. b. How to test x· for feasibility? Using the present method, it might be difficult. or even impossible, to satisfy strictly the inequality constraints. This is due to the fact that the optimum is approached from outside the feasible region. We may define the constraints gj = gj + Ej and minimize the new'll made up from the gJ. This may produce a strict satisfaction of gj < 0; however, choosing too large values of Ej may result in a solution which is far from the optimum. c. What are the special features of the method in minimizing'll? Applying minimization techniques to the penalty function, '1', the search must be kept out of certain zones, in which'll is not properly defined. We can guard against this by placing a test in the function evaluation step of the optimization procedure. In addition, since the Hessian matrix of second derivatives H 02 '1' /dX,-iJXj is discontinuous along the boundary of the feasible region, a quadratic approximation of 'I' is expected to be less effective. On the other hand, we may improve the minimization algorithms due to the nature of the method. Since the process is one of sequential minimization, the location of the minimum should change only incrementally from minimization to minimization. Thus we may preserve some information, such as the final set of directions in Powell's method, for the next r cycle. A possible source of trouble in the penalty-function methods lies in disparities between the various gj- The trouble arises when one gj changes much more rapidly than another and hence overpowers it over most of the unacceptable region. We may overcome this problem by using different r's for the gj. The penalty function might be defined, for example, by

=

=

2.4 Constrained Minimization: Nonlinear Programming

'I' = f +

",

L

rj

< gj

101

(2.114)

>2

j=l SO

that the constraints are weighted in accordance to their sensitivities.

Example 2.S. A simple two-dimensional example which demonstrates the exterior penalty-function formulation is to find Xl, X2 such that

f=XI2+xi

~min

gl ==4-XI -4X2 ~O

g2 == 4 - 4XI - X2

~

0

The penalty function to be minimized is:

=X2 (due to symmetry, the optimum of 'I' will always satisfy X; = X; = X·). The true optimum is X; = X; = 0.8,! =

Figure 2.12 shows contours of 'I' for Xl

1.28. The solution for r = 1 is X· =0.77, '1'. = 1.23,/ = 1.18, which is outside the feasible region. Variations of/, '1'., and X· with r are plotted in Fig. 2.13. It can be observed that, for r > 1, changes in the values of X· are small. For r = 10, for example, X· = 0.797,/ = 1.27 and '1'. = 1.275, which is very close to the true optimum.

>It >It(r = 10)

20 18 16 14 12 10 8 6 4 2 0.2

0.4

0.6

0.8

1.0

1.2

1.4

Fig. 2.12. Contours of'fl for X1= X2 = X, exterior penalty function.

102

2 Optimization Methods

X' f'

,It"

1.2

f'

0.8

X'

0.6

0.4

0.2

~--~----------------------------------------~r

10

Fig. 2.13.

Variation of 'P",! and X· with r, exterior penalty function.

An Interior Penalty (Barrier)-Function. Increasing the value of r in the exterior penalty-function method forces the minimum of 'I' toward the feasible region from the outside. In this subsection we discuss a method which always has its minimum inside the feasible region. It will be shown that for a decreasing sequence of values of the penalty parameter r, the minimum point X" is forced toward the constrained optimum from the interior. Like the method of exterior penalty function, we augment the objective function with a penalty term which is small at points away from the constraints in the feasible region, but which has very large values as the constraints are approached. One possibility for defining'll (X, r) is 'I'(X. r) =f

-

r

t

j=l

1

(2.115)

gj

but other forms of the penalty term can also be chosen. The idea is again to minimize (2.115) for a sequence of values of r, instead of solving the constrained problem (2.111). Since, at an interior point, all the terms in the sum are negative, a positive choice of r will result in a positive penalty term to be added to f. As a boundary of the feasible region is approached, some gj will approach zero and the penalty term will approach 00. Reducing successively the parameter r, the constrained minimum off is approached. It can be shown, however, that as with the exterior penalty-function method. the closer to the constrained optimum the

2.4 Constrained Minimization: Nonlinear Programming

103

minimum of 'II is forced to lie, the more eccentric the function becomes. Thus, it is necessary again to minimize'll sequentially. The solution process is as follows. Initial values of X and r are chosen. X must lie in the feasible region, namely, all the constraints gj(X) ~ 0 are satisfied. For the given r, the function 'I'(X, r) is minimized to obtain X·, and the convergence criterion of X· to the optimum is checked. If it is not satisfied, the parameter r is reduced r f- cr (c < 1), and starting from X· the function 'I'(X, r) is minimized again. The steps of reducing r and minimizing 'I'(X, r) are repeated until the convergence criterion is satisfied. The following points should be considered in practical application of the method:

a. Choosing a feasible starting point. In most structural-design problems it is

relatively easy to find a feasible point. For example, we may choose excessively large cross-sectional dimensions which will satisfy stress and displacement requirements. In other design situations, however, it might be more difficult to obtain an initial feasible design. Several methods intended to achieve such a design are presented in Sect. 4.2.1. b. Choosing an initial value for r. The matter of selecting an initial value for the penalty parameter r has been discussed in the literature [25]. If r is large, the function is easy to minimize, but the minimum may lie far from the desired solution to the original constrained problem. On the other hand, if r is small, the function will be hard to minimize. c. Convergence criteria. For decreasing values of r the minimum of'll should converge to the solution of the constrained problem. A simple criterion for checking convergence is to compute

£, == fmin('i-l)- fmin(rj)

(2.116)

fmin(rj)

and stop when £, is smaller than a predetermined value convergence test is, for example, to compute

o

£ ,.

Another

and stop when the absolute values of the components of £x are smaller than a desired value £ ~ .

d. Extrapolation techniques. Fiacco and McCormick [25] have shown that the

minimum points of 'II (r) obtained for decreasing values of r can be approximated by a continuous function of r, H(r), from data accumulated in two or more minimizations. Thus, H(O) can be used to obtain the approximate solution of the true optimum fmin (0) == fopt. Based on computational experience and some theoretical support, the following expression is proposed for H(r) [25] (2.118)

104

2 Optimization Methods H(r)

H(O) = a

~----~------------~----------~

r

Fig. 2.14. Typical H(r).

aj and bj for the ith approximation are determined by fitting through the two points of Hirj.]) and Hirj) 2 H·(r: " -I) = a·I. + b.r}'1 I. r.-

=!.rmn. (r..,- I)

Hj(r;) = aj + bj(c r;_d /2

(2.119)

=fmin(r;)

Solving for aj and bj, we obtain

(2.120)

g=E:

g= 0

1\

Feasible region

g=E:

I

IILinear extended

g=o

~

Feasible

Quadratic extended

\If

f Ci

(a)

(b)

Fig. 2.15. Extended penalty function: a. Linear, b. Quadratic.

2.4 Constrained Minimization: Nonlinear Programming

105

4 Optimum for 1

r=

3

4

True constrained optimum (a)

(b)

Fig. 2.16. Example, interior penalty function: a. Contours of 'P, r b. Contours of 'P, r =1.

= 10,

This solution must be checked for feasibility before it can be accepted. A typical approximation is shown in Fig. 2.14. The extrapolation scheme (2.118) can be used also to improve the starting points, X*(r), and to approximate the final answer to the problem. e. Extended penalty function. The penalty function 'i' defined by (2.115) can be minimized only in the interior of the feasible space, i.e., regions for which gj < O. The function is unbounded on the boundary of the feasible region, and as with an exterior penalty function in its regions of nondefinition, we must take special steps to keep the minimization in the proper portion of the space. One possibility to overcome this difficulty is to extend the penalty-function definition to the infeasible region. Kavlie [56], and Cassis and Schmit [16] proposed a linear extended function. A one dimensional example in terms of the single variable (l is shown in Fig. 2.15a, where £ is a small transition parameter, defining the transition between the two types of penalty terms.The transition point should be in the feasible region, to the right of the minimum point The linear extended penalty function has discontinuous second derivatives at the transition point. Thus, the function is not suitable for second-order optimization algorithms. This disadvantage can be overcome by introducing a quadratic extended function (Fig. 2.15b), as proposed by Haftka and Starnes [49]. /. Computational considerations. The penalty terms cause the function 'i' to have large curvature near the constraint boundary even if the curvatures of the objective function and constraints are mild. This effect permits an approximate calculation of the Hessian matrix which makes the use of Newton's method

106

2 Optimization Methods

more attractive. The number of iterations for Newton's method is independent of the number of design variables. Conjugate direction or quasi-Newton methods, on the other hand, require a number of iterations which is proportional to the number of design variables. Thus the use of Newton's method is most attractive when the number of design variables is large. Example 2.6. Example 2.5 will be demonstrated now by an interior penalty function. The function'll to be minimized [see 2.115)] is

Figures 2.16a and 2.16b show contours of'll for r = 10 and r = 1. It can be observed that the minimum for r = 1 is closer to the true constrained optimum of the original problem. In addition, the closer to the constrained optimum the minimum of 'I' is forced to lie, the more eccentric the function becomes. Variation of'll·,! and X; = X; = X· with r are shown in Fig. 2.17. The Augmented Lagrange Multiplier Method. Multiplier methods combine the use of penalty functions with that of Lagrange multipliers. When only Lagrange multipliers are employed, the optimum is a stationary point rather than a minimum of the Lagrangian function. When only penalty functions are used, ill-conditioning problems may be encountered during the solution process. The object in combining both is to obtain an unconstrained minimization problem that does not suffer from ill-conditioning. The conditions of optimality are included in the optimization algorithm in order to improve its efficiency and reliability. As a result, the dependency of the method on the choice of the penalty parameters is reduced. qt',

j', lOX'

50

40 30

20 10

100

200

Fig. 2.17. Variation of 'P.,! and X· with r, interior penalty function.

2.4 Constrained Minimization: Nonlinear Programming

107

The method was originally developed for the equality-contained problem

z =f(X) ~ min j

= 1,... , nit

(2.121)

Using an exterior penalty function, the following definition of 'II can be used [see (2.112)]

L r[h (X)]2 "l

'¥(X.r) = /(X)+

j

(2.122)

j=l

The Lagrangian function for the problem (2.121) is

L Aj "l

cj>(X.A) = /(X) +

h/X)

(2.123)

j=l

The stationary conditions acj>/aX j together with the equality constraints are the necessary Kuhn-Tucker conditions for optimality. It can be shown that the minimum of the Lagrangian function subject to the equality constraints provides the solution of the original problem (2.121). Thus, using the exterior penalty function approach we defme the augmented Lagrangian as A(X. A. r)=/(X)+

L {Ajhj(X)+r[hj(X)fl "l

(2.124)

j=l

If all Aj = 0 we get the usual exterior penalty functions (2.122). On the other hand,

for the optimum values A"j the minimum of A (X, A, r) provides the true minimum off{X) for any positive value of r. Then there is no need to use the large value of r required in the case of an exterior penalty function. In practice we do not know A" in advance, so we can modify A iteratively until the optimum is reached. Assuming 1..=0 and an arbitrary small value of r, the pseudo-objective function (2.124) is then minimized for the given A and r. To obtain an estimate for the Lagrange multipliers we compare the stationary conditions (2.125) with the exact conditions for the Lagrange multipliers

108

2 Optimization Methods

(2.126) Comparing (2.125) and (2.126), Hestenes [53] suggested the estimation

,

A.(~+l)

= A.(~) , + 2 r h.,(X(.t»)

(2.127)

where k is an iteration number. The value of r is then increased and the unconstrained minimization is solved for the modified values of r and A.. These steps are repeated until convergence. When the estimate of the Lagrange multipliers is good, it is possible to obtain the optimum without using larger r values. The value of r needs to be only large enough so that A has a minimum rather then a stationary point at the optimum. There are several ways the multiplier method may be extended to deal with inequality constraints. Assuming again the problem

z =f(X) ~ min j

= 1,..., n,

(2.128)

Fletcher [27] proposed to introduce the augmented Lagrangian function A(X. A.. r)=t(X)+r

", (A. )2 L -L+ gj j=l 2r

(2.129)

where < a > = max< a, 0 >. The condition of stationarity of A is

at

~ (A.. ago' =0 - ' +gj ) j=l 2r aX

-+2r £..J

aX

j

j

(2.130)

and the exact stationary conditions are

(2.131) where it is also required that

A. jgj = 0 (A.j = 0 for nonactive constraints).

Comparing (2.130) and (2.131), the following estimation is introduced (2.132) In summary, the method has several attractive features:

2.4 Constrained Minimization: Nonlinear Programming

109

a. It is relatively insensitive to the value of r and it is not necessary to increase r

to large values. h. Precise #X) 0 and hJ{X) 0 is possible. c. Acceleration is achieved by updating the Lagrange multipliers. d. The starting point may be either feasible or infeasible.

=

=

Multiplier methods have been studied exhaustively by numerous authors. A good survey is given in Ref. [13]. These methods have also been called primal-dual methods as they iterate on dual (A) as well as primal variables (X). Dual methods. where the original problem is transformed and solved for the dual variables, will be discussed in Sect. 2.4.3. Example 2.7. Consider again the problem of examples 2.5 and 2.6. The augmented Lagrangian function (2.129) is

A(X,

((~~ +4-XI - 4Xzr +(~+4-4XI-XZr)

A, r)=xr+xi+r

Assuming the initial values 1..(1) =0 and r(I) = 1, the solution of the unconstrained minimization is X(I)T = {0.769, 0.769}, gi =gz = 0.155. Using (2.132) the estimated Ais A(Z)T = {0.31 , 0.31}. Repeating the optimization for r(Z) = 10 and 1..(2) yields X(Z)T = {0.800 ,0.800], gi =gz =0, which is the true optimum. The

=A~ =A~ =0.32. Figure 4.18 shows the optimal X* =X; =X; obtained for various r and A =Al =Az values. It can be observed that for A· =0.32 the true optimum is obtained for any assumed r. optimal Lagrange multipliers are

1..*

x* r=1 0.82

r=1O 0.80

--====--n-=--r=100

0.78

A.*= 0.32

1~---l---'--'--J.......-L___"A. 0.2

0.4

0.6

Fig. 2.1S. Optimal solutions X· for various r and A..

110

2 Optimization Methods

2.4.2 The Method or Feasible Directions General Formulation. Methods of feasible directions are intended for problems with general inequality constraints [see (2.111)]. It is assumed that the derivatives of the objective and constraint functions are available. These methods consist of step-by-step solutions, where the direction vector, Sq' and the distance of travel, a, are chosen successively so that points Xq+l given by (2.133) are computed in a way that the objective function value is improved. In determining the direction vector, S, two conditions must be satisfied:

a. The direction must be feasible. i.e., we can take at least a small step from Xq along S that does not immediately leave the feasible domain. In problems where the constraints at a point curve inward, this requirement is satisfied if (2.134) If a constraint is linear or outward curving, we may require

(2.135) The interpretation of this condition is that the vector S must make an obtuse angle with all constraint normals except that, for the linear or outward-curving ones, the angle may go to 900. Any vector satisfying the strict inequality lies at least partly in the feasible region (see Fig. 2.19). b. The direction must be usable, i.e., the value off is improved. This requirement is satisfied if

~-----------------------------------------Xl

Fig. 2.19.

Feasible directions S.

2.4 Constrained Minimization: Nonlinear Programming

L -_ _ _ _ _ _-'---'---'---'--_ _ _ _ _ _

111

XI

Fig. 2.20. Usable directions S.

STVf < 0

(2.136)

A vector S satisfying the strict inequality (2.136) is guaranteed to produce, for some (l > 0, an Xq+! that reduces the value of f (see Fig. 2.20). Again, this condition means that the vector S must make an obtuse angle with Vf. A vector S satisfying both conditions a and b is said to be afeasible-usable direction from point X q • Methods of feasible directions produce improved feasible points by moving in a succession of feasible-usable directions. Selecting the Direction Vector. To find the direction S we note first that if there were no constraints active at a point X q , then we may choose S = - Vf, i.e., the steepest descent direction. When, however, constraints are active, the conditions (2.134) and (2.136) must be satisfied. Several possibilities exist for the selection of S in this case, of which the method of Zoutendijk [145] will be presented. The direction-finding problem can be formulated as the following LP problem of choosing the vector S and a scalar ~

X2

S(O-) 00)

Xl I - -_ _ _------------~-

Fig. 2.21. Effect of 9 on the search direction.

112

2 Optimization Methods

(2.137)

~~max

STVgj + OJ ~

S;

0

j= 1, ... , J

(2.138)

sTvr + ~ s; 0

(2.139)

-1 s; S s; 1

(2.140)

in which OJ are positive scalar constants, determined as a means of differentiating among the consttaints. The geometrical interpretation of OJ is shown in Fig. 2.21. The greater the value of OJ, the greater is the direction vector S pushed into the feasible region. The reason for introducing OJ is to prevent the iterations from repeatedly hitting the constraint boundary and slowing down the convergence. That is, the OJ prevent the vector S from lying exactly in the plane perpendicular to Vgj , so as to provide relief, when necessary, for the curvature of the constraints. Unless the problem has special characteristics, it is usually best to assume OJ = 1. Only the active consttaints (j = 1,... , J) are considered in (2.138). Clearly, if ~ > 0, the strict inequalities (2.134) and (2.136) are satisfied, and the vector S is a feasibleusable direction. The larger ~ can be made, the smaller (more negative) srvr is made [see (2.139)]; therefore, if S has a limited length, the vector S is more nearly aligned with - vr. Thus the purpose of maximizing ~ is to get the direction most nearly in line with the steepest descent direction. If Pm-ll =0, it can be shown that the Kuhn-Tucker conditions are satisfied. The constraints (2.140) are necessary to limit the length of S. Otherwise, ~ can be made large without bound for any vector 'Y S such that 'Y sTvr S; O. Although different constraints can ensure that S is bounded, (2.140) represents a set of simple linear constraints. Though this set of constraints has a tendency to direct S toward the comers of the hypercube defmed by (2.140), it generally produces good results. Selecting tbe Step Size. Assuming that a feasible-usable vector has been obtained, the problem is now to select the step size. The object is to find a so that Xq+l is feasible and/is minimized, without an excessive number of computations of the constraints (i.e., without a large number of analyses). There are two possibilities for the outcome of such a step (see Fig. 2.22). a. The point Xq+l is on the boundary of the feasible region. This first outcome occurs in many engineering problems with a nonlinear objective function (it is the only one possible in problems with a linear objective function which have no unconstrained minima). To find Xq+l on the boundary, we seek as large a move as possible without violating the constraints. In general, we take a trial step a and check the constraints; if they are in violation we reduce a and check again; if the check point is inside the feasible region, a is increased; and if the point is on the boundary, i.e., at least one of the constraints is active, we choose a new direction S. One problem is how to determine whether a constraint gj is active or not. many iterations may be necessary to reach sufficiently small values of gj; thus some sort of margin is required to make the search more efficient. We may state, for example, that a constraint gj is considered to be active if

2.4 Constrained Minimization: Nonlinear Programming

113

~-------/~~--------------------Xl

Optimum

Fig. 2.22. Possibilities for X q + lo minimizing

f along S. (2.141)

or

(2.142)

in which eg is defined as the constraint tolerance. Equation (2.141) requires that the point will be strictly feasible, whereas (2.142) allows a slight violation of the constraint. These criteria give the constraint some thickness and therefore it will be easier to find a constrained point It should be noted that even if the minimum of / along S is on the boundary, it might be a better strategy to find a point inside the feasible region and then to take a step in the - vr direction (Fig. 2.22). b. The point is inside the region. The second possible outcome in Fig. (2.22) is that/has a minimum along S inside the feasible region. In this case, the value of a* can be determined using one of the methods of minimizing a function along a line, discussed in Sect. 2.2.1. As in many engineering problems the step terminates at the constraints, a simple procedure is to find the maximum a for which Xq+1 is in the feasible region. and then compute (2.143)

If the result is negative. we proceed and find a ne,w direction at Xq+ I. otherwise the minimum of/is along the line segment Xq+l - Xq in the feasible region.

Example 2.S. To illustrate the procedure of solution by feasible directions, consider the problem (see Fig. 2.23): find XT = {Xl' X2 } such that

114

2 Optimization Methods

6 Objective function contours

4

3

2

~----~------L2------L~------4L------L------6L----Xl

Fig. 2.23. Solution of example by feasible directions.

(a)

gl ==

g2 ==

xl /20 - X2 + 1 ~ 0

(b)

xi 120 - Xl + 1 ~ 0

Assuming the initial unconstrained feasible point

xi = {6.

3}, we may choose (c)

Based on (2.133) we define (d)

2.4 Consuained Minimization: Nonlinear Programming

11 S

Minimizing f(a) along the line defined by (d), we find the boundary point X~ ={2.764, 1. 382}. To compute the direction S2' the following LP problem is formulated at X2

13-+ max

(e)

5.528} {Si' S2} { 2.764 +13 ~ 0

The solution is S~

={Si'

S2} ={-1.0. 1.0}, 13 = 1.276. To find X3 we defme

X3

2.764}

={ 1.382

+a

{-I.

O} 1.0

(j)

The value of a* is selected by minimizing the objective function along this line. The result is Xr new direction by

= {2.073.

2.073}. Since X3 is unconstrained, we choose the

S3

=-vr=-{ 4.146} 4.146

X4

={2.073}_a{4.146}

(g)

and ~ is computed by

2.073

4.146

(h)

Minimizing the objective function along S3, we fmd

X4 which is the optimal solution.

_{1.056} 1.056

-

(i)

116

2 Optimization Methods

2.4.3 Other Methods Dual Methods. Consider again the general nonlinear programming problem with inequality constraints [see (2.111)]. At the optimum, X* , we could use the Kuhn-Tucker conditions [(2.26) and (2.27)] to determine the optimum Lagrange multipliers A* corresponding to all critical constraints. It has been noted in Sect 2.4.1 that if A* is known in advance, the constrained optimization problem could be solved with only one unconstrained minimization. This was the basis for the development of the augmented Lagrange multiplier method, in which the values of the Lagrange multipliers are iteratively updated to improve efficiency and stability of the exterior penalty function method. Assuming that the problem (2.111) is the primal problem, then the corresponding dual problem is: find the dual variables A (Lagrange multipliers) such that (2.144)

c!>(A) ~ max j = 1,... , nil

(2.145)

in which c!>(A) is the dual function. The Lagrangian function is

",

c!>(X, A) = f(X)+

L Ajgj(X) j=l

(2.146)

and the dual function is given by c!>(A) = min c!>(X, A)

x

(2.147)

It may be advantageous in some cases to solve the dual problem and then retrieve the optimum primal variables X*. The motivation for such an approach is that in many design problems only a few constraints are critical at the optimum. Therefore only a few 'Aj. are nonzero. The point (X*, A*) defines a saddle point of the Lagrangian function given by (2.146). This will correspond to a maximum with respect to A and a minimum with respect to X. Thus, we can define the Lagrangian in terms of A alone as the dual function (2.147). Since the dual function c!>(A) is a maximum with respect to A at the optimum, the object is to fmd maxc!>(A) = max minc!>(X, A)

)..

)..

x

(2.148)

Alternatively, this problem could be stated as an equivalent min-max problem as

2.4 Constrained Minimization: Nonlinear Programming

117

Dual methods have been used extensively in linear programming to improve the optimization efficiency. In nonlinear programming the dual formulation is particularly attractive in cases where the primal problem is convex and mathematically separable. If the primal problem and its dual are both convex, their respective solutions satisfy the same optimality conditions. Both problems are equivalent and their optimal values are equal, that is cjI(A*) = JtX*)

(2.150)

Convex approximate problems will be discussed in Sect. 3.1.3. The objective and constraint functions are said to be separable if each can be expressed as the sum of functions of the individual design variables

f(X)

= L h(X "

j )

j=l

gj(X) =

(2.151)

L" gjj(Xj )

j

=1•...• n,

j=l

The primal formulation does not benefit much from the separability. However, the dual formulation does, because the Lagrangian function in this case

",,,

"

cjI(X. A)=

L h(X )+ L Aj L gjj(Xj ) j

j=l

j=l

(2.152)

j=l

is also a separable function and can be minimized by a series of one dimensional minimizations. Using (2.147) and the property that the minimum of a separable function is the sum of the minima of the individual parts, we can state the dual problem as: find A such that cjI(A) =

"

L j=l

Aj~

0

n:n[h(Xj )+

j=l

I

j= 1,...•

",

L Ajgjj(Xj)]--+ffiaX

n,

(2.153)

The Lagrangian cjl(A) is therefore easy to calculate. Furthermore, the single variable optimization problem has often a simple algebraic structure and it can be solved in closed form, yielding thus an explicit dual function. It should be noted, however, that because of the nonnegativity conditions that the dual variables must fulfil, a direct solution of the dual problem from the stationary conditions is usually difficult if not impossible.

118

2 Optimization Methods

An interesting property of the dual function is that its ftrst partial derivatives are given by the primal constraint values. that is acjl aA.

= g .[X(A)] J

J

(2.154)

The dual problem can therefore be solved by applying well-known first-order algorithms. Furthermore. if the Hessian matrix of the dual function is readily available. second-order methods can also be employed. Since dual methods operate in the space of A. they are particularly effective in cases where the number of constraints is small compared to the number of design variables. Example 2.9. Consider the primal problem: find Xlo X2 and X3 such that

z = xl' + xi + x; ~ min 1 1 gl =X2 + X2 -1 ~ 0 1 2 g2 =

1

1

xi + x; -1 ~ 0

The Lagrangian function (2.152) is

This is a separable function of Xl. X2 and X3

where

Each one of the functions iloh and h can be minimized separately to get the minimum of cjl(X. A) with respect to X

2.4 Constrained Minimization: Nonlinear Programming

X1 --

119

'1114

Jl.l

X3 --

'11/4

Jl.2

Substituting back into ~(X, A) yields the explicit dual function

~A) = -(AI + A2) + 2A1[2 + 2(Al + A2)1I2 + 2Ali2 Solving the dual problem ~(A) ~

max

A~O

we get the optimal solution

~(A·) = 5.828 Since the constraints A ~ 0 are not active, we have from (2.154)

and the corresponding primal solution is X;

= X; = 1.307

X; = 1.554

f(X·)

= 5.828

That is max ~(A)

=minJtX)

Gradient Projection and Reduced Gradient Methods. Many other methods for constrained optimization have been developed and evaluated in the literature [2, 42, 48, 91, 138]. Some of these methods that have been used in structural optimization are briefly described in this section. The gradient projection method [112] is based on a relatively simple procedure to obtain an approximate search direction in a closed fonn. However, it may not be as good as the one obtained by the feasible directions method by solving an LP subproblem. A direction that is tangent to the constraint surface is detennined by

120

2 Optimization Methods

projecting the steepest descent direction for the objective function to the tangent plane. Consequently, the new design will usually be infeasible and a series of correction steps need to be executed to reach the feasible region. The step size specification is arbitrary and the constraint correction process might be tedious. Despite these drawbacks the method has been applied to several structural design problems. The reduced gradient method is based on a simple variable elimination technique for equality constrained problems. Dependent and independent variables are identified in the linearized subproblem and the dependent variables are eliminated from it. The generalized reduced gradient (GRG) method is an extension of the reduced gradient method to accommodate nonlinear inequality constraints. A search direction is selected such that for any small move the current active constraints remain active. If some active constraints are not satisfied due to nonlinearity of the constraint functions, the Newton-Raphson procedure is used to return to the constraint boundary. Thus, the GRG method is similar, in this respect, to the gradient projection method. Although the method appears complicated, relative to SUMT or other methods, the efficiency is often improved. However, the method does have some drawbacks:

a. The main computational burden arises from the Newton-Raphson iterations

during the one-dimensional line search. If the problem is highly nonlinear, the use of these iterations may become ill-conditioned and may not converge. b. If there are many inequality constraints and inequalities are converted to equalities by adding slack variables, the problem size might become large. This problem can be overcome if only potential active constraints are considered. c. The method tends to move from one constraint vertex to the next. If the number of independent variables is large the convergence might be slow. d. A feasible starting point must be selected. In addition, the design process produces a sequence of infeasible designs. This feature is undesirable in many applications.

Vanderplaats [138] proposed a robust feasible directions method, which is intended to incorporate the best features of the method of feasible directions and the generalized reduced gradient method. The proposed method does share some limitations with the GRG method. However, it has been found that it offers a powerful technique for solution of many design problems.

Exercises 2.1 Find the minimum of the function

using the necessary and sufficient conditions for a relative minimum.

Exercises

121

2.2 Given the function

vr

xi

xf

and H at the points = {O. OJ. X~ = {2. 2}. = {t. I}. check if H is positive definite at each of these points. Are the necessary and sufficient conditions for a relative minimum satisfied at any of the points? b. Check numerically if the function is convex in the range determined by Xl and X 2 for a = 0.25, 0.50, 0.75 [see (2.32)]. c. Find the Taylor series expansion of the function about the points Xl and X3 up to quadratic terms. Express / in the quadratic form (2.54). Check if a is

a. Calculate

positive definite and if the coordinate directions

sf = {Xl'

OJ. s~ = {O. X2 }

are conjugate with respect to a at the points Xl and X3 (Note: a = 1/2 H). 2.3 Given the constrained optimization problem

z= 3XI + 2X2 ~ max 2XI+X2~IO XI+X2~8

Xl

~

4

XloX2~0

Check the Kuhn-Tucker conditions at the points

xf = {4.

2}.

xI = (2.

6).

Xr = {4. OJ. Show graphically the gradient vectors of the constraints and the objective function at the three points. 2.4 Given the general quadratic function

q = XT a X + XT b + c Substitute X = X* + a S* and derive a formula for a* (corresponding to the minimum value of q) in terms of a, b, X* and S*.

2.5 Given the function

/ = X't -2XIX2 +2xi +2 point xf = {O. I}, carry out

Assuming the initial mmlmlzations in two successive directions of the steepest descent Find the minimum at each iteration by:

a. The formula derived in Exercise 2.4; plot/versus a for the two directions; b. the golden section method; c. the quadratic fitting; and d. the cubic fitting.

122

2 Optimization Methods

2.6 Show that successive directions of steepest descent are orthogonal to one another in the algorithm of (2.56). 2.7 Given the function

a. Sketch the contours/= 4.5, /= 5, /= 6 in the space of Xl andX2.

xi

h. Minimize the function, starting at the initial point = {O, -I}, by the conjugate directions method. Use quadratic interpolation for the onedimensional minimizations and show graphically the directions of move. c. Solve part h by the conjugate gradient method. 2.8 Complete two iterations of the methods of conjugate directions and conjugate gradient, starting from the given initial point Xo. for the following functions:

a.

/ = X~ + 2xi - 4Xl - 2X1X2;

h.

/ = 25X~

+ 20xi - 2Xl

-

X~ = {I. I}. X~ = {3, I}.

X2 ;

2.9 Solve problems a, h, c by the procedure described in Sect. 2.3.3. Show graphically the constraints and the objective function contours in the space of Xl, X 2 • Find in what range we may vary Cl (the coefficient of Xl in the objective function) without changing the basis of the optimal solution. Solve the dual problem and verify that solutions of the primal and the dual problem are identical.

a.

Z= 3X l + 2X2 ~ max 2Xl+X2~IO Xl+X2~8 Xl~

4

XltX2~0

h.

Z=X l +X2

~

min

2Xl+X2~4 Xl + 3X 2 ~ 6 XltX2~0

C.

Z=2Xl+X2~max

5X l + X 2 ~ 30 3Xl+X2~20

3X l +

2X2~

28

XltX2~0

2.10 Solve problems a, h, c by the procedure described in Sect. 2.3.3.

Exercises

a.

123

Z=Xl + 2X2 ~ max -Xl + 3X2~ 10 Xl+X2~6 XI-X2~2 Xl+3X2~6

X h X2

b.

~

0

Z = 2Xl - 3X2 ~ min Xl +X2~ 1 -2Xl +X2 ~ 2 XhX2~0

c.

Z=2Xl +X2

~

max

-2Xl+X2~4 Xl +2X2~2 Xl>X2~0

2.11 Given the problem Z=Xl +X2~ min g == (Xl - 5)2+ (X2 - 5)2 - 9 ~ 0 Solve by the exterior penalty function for '1 = 0.1, '2 = 1.0. Show graphically 'II (X, ,) as a function of X = Xl = X2 for the two values of,. b. Solve by the interior penalty function for'l = 0.1,'2 = 0.01. Show graphically 'II (X, r) as a function of X = X I = X 2 for the two values of ,. Use the extrapolation for, = 0 to estimate the true optimum. c. Solve by the augmented Lagrange multiplier method for'l = 0.1, '2= 1.0.

Q.

2.12 Given the problem Z= (Xl - 1)2+ (X2 - 1)2~ min g==Xl+X2-1~0 Xl ~ 0

Solve by the exterior penalty function for '1 = 1.0, '2 = 10.0. Show graphically contours of'll = 1, 2 and 4 for each case. b. Solve by the interior penalty function for ·'1 = 1.0,'2 = 0.1. Show graphically contours of 'II = 5 ,10,15 for each case.

Q.

2.13 Given the problem Z = (Xl - 1)2 + (X2 - 1)2~ min

X I -X2 - 2

=0

124

2 Optimization Methods

Solve by the augmented Lagrange multiplier method as follows.

a. Write the expression for the augmented Lagrangian using r = 1. b. Beginning with Al = Az = 0 perform three iterations. c. Repeat part b, beginning with Al = Az = 1. d. Repeat part b, beginning with Al = Az =-1.

2.14 Solve the problem of Exercise 2.11 by the method of feasible directions. Perform two iterations, starting at the initial point the directions of move in the space of Xl and X2 •

xf = {7.

4}. Show graphically

2.1S Given the problem

z = xl + 2X~ --+ min -Xl -X2 + 2 ~ 0

Slate the dual problem in terms of A alone and solve for the optimum values of A, Xl, and X 2• What are the optimum values of the primal and the dual objective functions? 2.16 Given the problem

5 2 . Z =-+---+mm Xl

X2

=

Xl + X2 - 3 ~ 0 g2=4X l +X2-6~0 g3 Xl + 3X2 - 20 ~ 0

gl

=

Solve this problem using dual methods. Find the optimum values of the primal and dual variables and the primal and dual objective functions. Draw the design space for the primal problem and verify the solution.

3. Approximation Concepts

One of the main obstacles in the solution of optimal design problems is the high computational cost required for solving large scale problems. Applications of approximation concepts in structural optimization have been motivated by the following characteristics of the design problem: -

-

The problem size (number of variables and constraints) is usually large. Each element involves at least one variable, and various failure modes under each of several load conditions must be considered. The constraints are usually implicit functions of the design variables. That is, evaluation of the constraints for any given design involves solution of a set of simultaneous equations. In addition, it is often required to calculate constraint derivatives with respect to design variables. In general, the solution of optimal design problems is iterative and consists of repeated analyses followed by redesign steps. The number of redesigns (or repeated analyses) is usually a function of the problem dimensionality.

For practical design problems each redesign involves extensive calculations and the number of redesigns is large. Consequently, the total computational effort might become prohibitive. Introduction of approximate models of the structural behavior in terms of the design variables is intended to reduce the computational cost and allows the solution of practical design problems. It is recognized that only methods which do not involve many implicit analyses are suitable for practical design applications. In structural optimization, the analysis task will require most of the computational effort. As a result, approximation techniques used to solve a structural optimization problem might affect the overall computational cost more than the choice of the optimization method. In this chapter, approximation concepts for structural optimization are discussed. In Sect. 3.1 methods for calculating derivatives of displacements with respect to design variables are first introduced. These derivatives are needed for effective approximations of the constraints and efficient solution of the optimization problem. Intermediate variables, often used in structural optimization, and methods based on sequential approximations are then discussed. In Sect. 3.2 various approximate behavior models for evaluation of displacements, stresses and forces in terms of the design variables are presented.

126

3 Approximation Concepts

3.1 General Approximations 3.1.1 Design Sensitivity Analysis Calculation of derivatives of the constraint functions with respect to design variables. often called design sensitivity analysis. is required in using most of the optimization methods. This operation is necessary also in applying explicit approximations of the constraint functions. Since calculation of derivatives often involves a major computational cost of the optimization process. efficient computational techniques are essential in most applications. Considering the displacement analysis equations (1.26) Kr=R

(3.1)

the object is to fmd derivatives of the displacements r. Derivatives of the stresses (J can then be obtained by differentiation of the stress displacement equations (1.29) (J=Sr

(3.2)

Before taking up the topic of sensitivity analysis. it is instructive to note that efficient solution of (3.1) for the displacements requires often decomposition of the stiffness matrix K into a product of the upper triangular matrix U and the lower triangular matrix UT (3.3)

This can be done by a simple recursion algorithm. The displacements are then computed by forward and backward substitutions. For a problem with n design variables X j (i = 1..... n). finite difference derivative calculations of the displacements with respect to design variables require to repeat the analysis for (n+I) different stiffness matrices. However. the derivatives can be calculated analytically in more efficient ways. and the large number of analyses associated with fmite difference calculations can be avoided. In this section three alternative methods for such analytical calculations of derivatives are discussed. Further developments in this area are reviewed by Haftka and Adelman [47]. Direct Method. In this approach the displacements r are expressed in terms of the independent design variables X by (3.1). Implicit differentiation of (3.1) with respect to Xj yields (3.4)

in which

3.1 General Approximations

oR! IOXj} . { ax· = : , oR",loXj

or! IOXj} . ax. ={ : , or", loXj

oR

or

127

(3.5)

oK!", oXj

oK

oK"""

(3.6)

oXj

and m is the number of displacements r. The direct approach involves solution of (3.4) for or/oXj and then taking the desired component or/oXj. For multiple design variables, (3.4) must be solved repeatedly for each design variable. Defming the matrices

ar ={ar ax aXl

'

ar} ax"

..••

(3.7)

(3.8)

then (3.4) becomes (3.9) It should be noted that (3.9) and (3.1) have the same coefficient matrix K. Therefore, if the decomposed form (3.3) is available, then only forward and backward substitutions are needed to solve for To obtain derivatives of a single displacement rj we compute

arrax.

arJ _

T _

T

J

J

ar ax

-=Vr·-I·-

ax

(3.10)

where Ij is a vector having unit value at the jth location and zeros elsewhere. In many problems where the load vector R is independent of the design variables j =0, and (3.4) is reduced to

aR/aX

ar aX

aK aX

K-=--r j

j

(3.11)

128

3 Approximation Concepts

Adjoint-Variable Method. It has been shown that differentiation of (3.1) with respect to X; gives (3.4). Premultiplying the latter equation by substituting (3.10) yields

Ir

K - 1 and

(3.12)

The adjoint-variable vector

~j is defmed as

the solution of the the set of equations (3.13)

K~j=Ij

Substituting (3.13) into (3.12) gives (3.14) where use has been made of the symmetry of K. The adjoint-variable method involves solution of (3.13) for ~j and then calculation of or/oX; by (3.14). Assuming the vector X of all design variables, then (3.14) becomes

or!

- ' =Vr!=~~ V

oX

'

,

(3.15)

where matrix V is defined by (3.8). Once the system (3.13) is solved, the adjointvariable vector ~j is repeatedly used in (3.14) for all variables. Since (3.13) and (3.1) have the same coefficient matrix K, again only forward and backward substitutions are needed to solve for ~j • Virtual-Load Method. This method is also based on the assumption that the dependent displacements r are expressed in terms of the independent design variables X. To calculate derivatives of displacements with respect to design variables, any desired displacement rj is expressed as (3.16) where Qj is a virtual-load vector, having unit value at the jth location and zeros elsewhere. Differentiation of (3.16) with respect to X gives

or!

- ' =Vr?' =Q~

oX

'

Or

'ax

(3.17)

The virtual-displacement vector rf corresponding to the virtual-load vector Qj is given by

3.1 General Approximations

K r?

=Qj

129 (3.18)

Substituting (3.18) into (3.17), the latter becomes

Vr! J

=(r(2)TK 2!. J al(

(3.19)

Substimting (3.9) into (3.19), the following expression for the derivatives of rj is

obtained VrJ

=(r?)T V

(3.20)

where V is given by (3.8). In the calculations of r? by (3.18), again the previously calculated matrices U and uT [see (3.3)] can be usM. To fmd the derivative vector Vrj' the vector r? is flI'St computed by solving the set (3.18). The desired derivatives are then computed by (3.20). In cases where the loads are independent of the design variables aR/axi =0, and (3.20) is reduced to

i=I, ... ,n

(3.21)

In the calculations of aK/aXi only the stiffness matrix of the ith member, K i , can be considered and (3.21) becomes

ar. =_(r(2)T aK. 'r ax aX

_J

i

J

i

i= 1, ... ,n

(3.22)

The vectors r? and r can accordingly be reduced to contain only those degrees of freedom associated with the ith element. In many strucbJral design problems Ki is a linear function of Xi. Thus

i =l, ... ,n

(3.23)

i=I, ... ,n

(3.24)

and (3.22) can be expressed as

Comparison or Methods. Arora and Haug [3] analyzed the different methods to design sensitivity analysis. It has been shown that the virtual-load method can be derived from either the direct method or the adjoint-variable method.

130

3 Approximation Concepts

Using the adjoint-variable method. the unknown vector solving the set (3.13). From the defmitions of Ijand Qj

;j is calculated by

I j =Qj

(3.25)

and the adjoint equations (3.13) can be written as

K ;j= Qj

(3.26)

Comparing (3.26) and (3.18). it can be seen that the adjoint-variable vector identical to the virtual-displacement vector

rf

;j =rf

;j is

(3.27)

Substituting (3.27) into (3.15) gives (3.28) which is identical to (3.20). obtained by the virtual-load method. is computed from (3.9) Using the direct method.

ar/ax

~=K-IV

ax

(3.29)

Substituting (3.25) and (3.29) into (3.10) gives

Vr! =Q~K-IV J

(3.30)

J

Based on (3.18) and on the symmetry of K. this equation can be written in the fonn of (3.20). obtained by the virtual-load method. A summary of the various methods for calculating the derivatives of the displacements is given in Table 3.1. It has been shown that the three design sensitivity analysis methods give the same results. However. as noted by Arora and Haug [3]. there are some differences in generality and efficiency of the individual methods. The adjoint-variable and direct methods are more general than the virtualload method and can be extended to include other behavior functions. As to efficiency considerations. both the adjoint-variable and the virtual-load methods are superior to the direct method in cases where derivatives of a limited number of displacements must be calculated. Let J be the number of displacements to be considered. The adjoint-variable method then requires calculation of J adjoint vectors;j by (3.13). and the virtual-load method requires calculation of J virtualdisplacement vectors

rf by (3.18). Thus. the numbers of operations for the two

methods are identical. In the direct method. the number of vectors

ar/ax;

that must

3.1 General Approximations

131

be determined by (3.9) is n nL. where nL is the number of loading conditions. Depending on the values of J and n nLo one method is to be preferred over the other. In cases where J < n nL the direct method is less efficient than the other two methods. However, it should be emphasized that in most design cases stress constraints are also considered and the number of displacements to be calculated may become large. In many cases only derivatives of critical constraints must be calculated. The number of critical constraints does not change significantly with the number of load cases, and is usually of the same order as the number of design variables. Therefore, in a multiple-load case situation the adjoint-variable method is often prefernble. All three methods require calculation of 13K laX j [see (3.8)]. In some cases analytical expressions for these derivatives are cumbersome and expensive. For these reasons 13K laX j can be computed by fmite differences. This combination of analytical derivatives of r coupled with finite-difference evaluation of the stiffness matrix is known as the semi-analytical method. Unfortunately, this method is prone to large errors for some problems. Table 3.1. Summary of design sensitivity analysis. Method

Unknowns

Direct

ar/ax

Adjoint-variable

Virtual-load

Number of unknown vectors

Equations

n nL

K~=V ax T ar Vr·JT =1·J ax K ~j= I j VrJ =~]V K rf =Q j

J

~j

J

r~ J

VrJ =(rflV J

n

(3.9) (3.10) (3.13) (3.15) (3.18) (3.20)

=Number of displacements to be considered. =Number of design variables.

nL = Number of loading conditions.

Example 3.1. Consider the three bar truss shown in Fig. l.ll. The equilibrium equations are

°

° ]=

E [0.707X1 0.707X1 + Xz r 100

{14.14} 14.14

(a)

132

3 Approximation Concepts

where the modulus of elasticity is E =30,000. Assuming the initial design

X -T

={1.0,

r -T

1.0}

={0.0666,

0.0276}

the object is to find the derivatives at X·

art

art

axl aX2

orax

(b)

-=

or;

or;

axl ax2 Direct method. The matrix V is first calculated by (3.8)

Substituting (a) and (c) into (3.9) gives

300

[

°

0.707

0] 1. 707

orax =-

The solution of this equation is

or- [-0.0666 ax = -0.0114

[14.14 0] 5.86 8.28

0]

-0.0162

(d)

(e)

Adjoint-Variable Method. The adjoint vectors ~j are calculated by (3.13) 300 [0.

g}

~07 1. ~07]~l =

300 [0.

~07 1. ~07]~2 ={~}

and the solution is ~r

= {0.00471,

O}

Substituting (c) and (g) into (3.15) gives

~~

={O,

0.00195}

(g)

3.1 General Approximations

133

(h)

and

ar/ax is given by ar· [-0.0666 ax =(Vrl •Vr2) = ° T

••

-0.0114] -0.0162

(i)

Virtual-Load Method. The virtual-load vectors are given by

Q[ = (1.

Q~

o)

= (0.

(j)

I)

The virtual displacements are calculated by (3.18)

300

°

[ 0.707

0] Q 1. 707 r l

{I}°

300 [

=

0.707 0

0] 1. 707 r¥

{O}1

=

(k)

and the solution is (I)

Substimting (c) and (I) into (3.20) gives Vr?

=(rIQ)TV ={-o.0666.

O} (m)

Vr;T

=(r¥lV ={-o.0114.

-0.0162}

3.1.2 Intermediate Variables Direct and Reciprocal Approximations. Various approximations can be improved by using intermediate variables, Y;. defined by

Y;

=Y,{X;)

(3.31)

A typical example is the general fonn

Y;=Xt'

(3.32)

134

3 Approximation Concepts

where m is a predetennined parameter. One of the more popular intermediate variables is the reciprocal of Xj [121] 1 Xj

y=I

(3.33)

The reason for this is that displacement and stress constraints for detenninate structures are often linear functions of the reciprocal variables. For statically indetenninate structures, the use of these variables still proves to be a useful device to obtain better approximations. Intennediate variables are usually most effective for some homogeneous functions as will be shown later in Sect. 3.2.3. One disadvantage of the reciprocal approximation is that it becomes infinite if any Xj is zero. This difficulty can be overcome by the simple transfonnation [48]

1 Xj +OXj

y=--I

(3.34)

where the values of ~Xj are typically small compared to representative values of the corresponding X/so Consider the first order Taylor series expansion of a constraint function g in tenns of the design variables Xj , denoted as a direct approximation gD gD == g • + L,"

-og· (X. -

j=! oXj

I

x.•) I

(3.35)

To improve the quality of the results, g can be expressed in terms of the intennediate variables Y j [see (3.33)]. The resulting expression, denoted as a reciprocal approximation gR, is given by

•

gR == g +

Og. )]

[(Xj "o

Lj=! -( .) (r; - r; ) = g + L or; j=! oXj

. . "

(3.36)

Conservative and Convex Approximations. In some applications it is desirable to introduce conservative approximations. This is the case for example in feasible design procedures (Sect. 4.2) where all intennediate solutions lie in the feasible region. Such procedures have an advantage from a practical point of view. A hybrid form of the direct and reciprocal approximations which is more conservative than either can be introduced [132]. The fonn of this conservative approximation is derived by subtracting the reciprocal from the direct approximation

3.1 General Approximations

135

The sign of each term in the sum is determined by the sign of the ratio (ag"' axi ) , Xi which is also the sign of the product Xi (ag"' aXi ). Since the constraint is expressed as g ~ 0, a more conservative approximation is the one which is more positive. It is possible, therefore, to create a conservative approximation gc which includes the more positive term for each design variable

L II

gc =g"

+

(Xi

i=1

agO • (X·-X·) ' , aXi

(3.38)

where

a~ll , X:

-' Xi

~o

if

Xi

agO aXi

if

Xi

agO <0 aXi

(3.39)

It should be noted that this approach does not ensure that the approximate constraint is conservative with respect to the true constraint. The conservative approximation is only more conservative than both the linear and the reciprocal approximations. The conservative approximation has the advantage of being convex approximation [30]. However, it has been found that this approximation tends to be less accurate than either the direct or the reciprocal approximation. The method of moving asymptotes (MMA), proposed by Svanberg [134], is intended to introduce more accurate approximations. Using this method the intermediate variables are defined such that the degree of convexity, and hence conservativeness, of the approximation can be adjusted. Instead of using direct and reciprocal variables, the method employs the intermediate variables 1 y=-,

or

Xi-L i

(3.40)

1

y=-,

Ui-Xi

where Li and U i are specified parameters that may be changed. Based on this transformation, the moving asymptotes approximation gM is formulated as

-. + " -L a" (

gM =g

~ ,=1 ax.'

A.) '+-P-'X·-L., ,

(X.

U·, -X·,

(3.41)

136

3 Approximation Concepts

where

={ (Ui -

(x. &

0

Pi

={O •

_.

•

Xi• )2

-(Xi -Li)

8 =8 -

a8· laxi >0

if

a8· laxi ~O

if

a8·laxi ~O

if

a8· laxi <0

2

~ a8· ~ -

. ax·

&=1

if

&

(Xi

U·-X~ & &

Pi)

+---i.........X·-£. & &

(3.42)

(3.43)

Since all the coefficients in (3.41) are non-negative, the approximations for 8 are convex functions. The approximated functions are driven by the selected values for the parameters Li and Ui which act as asymptotes. The moving asymptotes approximation is general; the direct approximation and the conservative approximation can be viewed as the following special cases: - For Li -+_00, Ui -+00, no intermediate variables are considered, and the direct formulation (3.35) is obtained. - For Li 0, Ui -+00 , the conservative formulation (3.38) is obtained.

=

Other values of Li and Ui are acceptable, and these values may even be modified during the solution process. However, it is not at all straight forward to find suitable values for the asymptotes. To avoid the possibility of any unexpected division by zero, move limits

xl'

xf and XiU

can be chosen such that

4 < xf and

< Ui . The closer Li and Ui are chosen to X; , the more curvature is given to the approximate function, and the more conservative becomes the approximation of the original function. Choosing Li and Ui far away from Xi· , then the approximate function becomes close to linear. Both formulations (3.38) and (3.41) are based on first order convex approximations that attempt to simulate curvature of the functions. The latter method (MMA) offers more flexibility through the moving asymptotes Li and Ui. Example 3.2. To illustrate the various approximations, consider the constraints

xl 120 - X2 + 1 ~ 0 82 =.xi 120-X1 +1~0 81 =.

Assuming the given point X·T

obtained.

= (6, 3), the following expressions have been

3.1 General Approximations

137

Direct-linear approximations [see (3.35)] gl == -0.8 + O.6X I - X2 :5'; 0 g2 == 0.55 - Xl + 0.3X 2 :5'; 0

(a)

Conservative-convex approximations [see (3.38)] == O·6X I + 91X2 - 6.8:5'; 0 g2 == 36IXI + 0.3X 2 - 11.45:5'; 0

gl

(b)

Moving asymptotes approximations [see (3.41)]

(c)

The various approximations are demonstrated in Fig. 3.1. It can be observed that

xt,

the closer Li and U i are chosen to the more curvature is given to the approximate functions, and the more conservative become the approximations of the original constraints.

A

10

8 6

A - Equation (a), Linear

4

B - Equation (b), Convex C - Equation (c), L=O U=1O D - Equation (c), L={) U=20

2

6

8

10

Fig. 3.1. Various approximations of constraints.

138

3 Approximation Concepts

Linearized Segments of a Separable Function. In some problems it might be necessary to linearize a separable function by several linearized segments. Consider the sepamble function (3.44) which can be linearized by two sets of m linear segments as (Fig. 3.2) Z=

L 1)J:fik + L YZdZk 1ft

1ft

k=O

k=O

(3.45)

In this equation. fu andflk are values offl andlz at the preselected points Xu and Xlk (k =O•...• m). respectively. where

L 1)kXU

(3.46)

L YZkX

(3.47)

1ft

Xl =

k=O 1ft

Xz =

2k

k=O

and the unknown interpolation functions Yu • Ylk satisfy

L 1)k = L Y 1ft

1ft

k=O

k=O

2k

i

= 1.2

fll Xl

X

10

Fig. 3.2.

X II

X 12

X13

Linearized segments.

X 14

=1

(3.48)

k

= 0.1.2•...• m

3.1 General Approximations

139

We require that for every i (i = 1,2) at most two adjacent YiA: be positive. If, for example, Yll and YIZare nonzero with all other Yu; zero then the value of Xl is given by (3.49)

with Yll + Yl2 = 1

(3.50)

The two adjacent nonzero Y's identify the segment where the final solution lies. The number (m) of segments determines the degree of approximation; the larger the m the closer will be the approximation to the original function.

3.1.3 Sequential Approximations Sequential Linear Programming. Consider again the general nonlinear programming problem with inequality constraints [see (2.111)]. Sequential Linear Programming (SLP) methods are based on successive linearizations of the constraints and the objective function. Using the Taylor series expansion off and gj about a point X· up to linear terms, the original nonlinear programming problem is replaced by the following LP problem

l

+

" dj* L -(Xj-X;)~min j=l aXj

" agj* * g*.+ L -(X·-X-)
j=l

ax.

I

1-

(3.51)

j = 1•...• ng

I

The LP problem can be solved repeatedly, redefming X* each time as the optimal solution of the preceding problem. This procedure, however, will not always converge to the optimum. Problems which may arise during the solution process include:

a. If the true optimal solution does not correspond

to a vertex of the feasible region, the LP results will oscillate indefinitely between the vertices of that region. This might happen in underconstrained problems where the number of active constraints at the optimum is smaller than the number of design variables. b. Even though the true optimum may be at a vertex, the starting design point may be so far from the true optimum that the solution still does not converge. c. In nonconvex problems some of the linearizations of the constraints may cut off feasible portions of the space that include the optimal solution.

Various procedures can be used to overcome such difficulties.

140

3 Approximation Concepts

The cutting-plane method, developed by Cheney and Goldstein [17] and by Kelley [57], is based on the useful property that linearized constraints in convex problems always lie entirely outside the feasible region (Fig. 3.3a). We therefore can approximate the region by an envelope of the linearized constraints, and solve an LP problem. The constraints are first linearized in the neighborhood of a starting point X*. Assuming a linear (or linearized) objective function, we solve the resulting LP problem. The solution is substituted in the original nonlinear constraints, and the most violated constraint is linearized about the optimum point of the previous LP problem. This linearized constraint is added to the problem which is solved again. The steps of adding linearized constraints and solving the modified LP problem are being repeated until all nonlinear constraints are satisfied to a desired degree of accuracy. We note that the LP problem continuously expands. An obvious disadvantage of the method is that its applicability is restricted to convex problems (see Fig. 3.3b). Very few structural optimization problems can be guaranteed to be convex and hence the cutting plane method is of limited use for solving such problems. Another undesirable feature is that the method can be plagued by roundoff errors if the optimum solution does not coincide with a vertex of the feasible region. In this case, the computational process deteriorates due to ill-conditioning. Finally, the fact that the intermediate solutions are infeasible may be undesirable in design applications. A different approach is the method of approximate programming due to Griffith and Stewart [43]. The objective function and the constraints are, again, linearized by taking the first terms of the Taylor series expansion about the current point X*. However, we relinearize aU the nonlinear relations at each iteration and no part of the preceding LP problem is retained. The original NLP problem is approximated by linear terms, permitting the solution of nonconvex problems. To ensure that the approximation is adequate, we limit the variation of X by the constraints

(a)

(h)

Fig. 3.3. Linear approximations: a. Convex problem, b. Nonconvex problem.

3.1 General Approximations

141

First LP solution with move limits I

;1

/

/

True / OPtimu";......-//

First LP solution / ,// without move limits I / / /

//

)././ /,

Fig. 3.4.

Move limits.

(3.52) where AXL and AXu, called the move limits, are suitably chosen vectors of positive constants (see Fig. 3.4). To solve a problem we first choose a starting point X· and linearize the objective function and the constraints in the neighborhood of X·. We solve the linearized problem (3.51) and (3.52) and redefine X· as the optimum solution to the preceding problem. The nonlinear functions are relinearized about the new X· and the process is repeated until either no significant improvement occurs in the solution or successive solutions start to oscillate between the vertices of the feasible region. In the latter event we may reduce the values of the bounds AXL and ax U and continue. For computational efficiency it is desirable to choose large values for the move limits, so that the imposed limits will not slow convergence. However, these bounds should be gradually shrunk as the design approaches the optimum. One reason for the need to shrink the move limits is that the accuracy of the approximations is required to be higher when we get close to the optimum. The move limits are typically reduced by ten to fifty percent of their previous values until convergence. Convergence is assumed if the change in the optimal Z for two successive LP subproblems is smaller than a desired value and the nonlinear constraints are satisfied within a desired tolerance. The method of approximate programming is applicable to nonconvex problems and produces feasible or nearly feasible intermediate solutions with good accuracy. The method differs from the cutting-plane method in that there is no link between subsequent LP problems.

142

3 Approximation Concepts Objective function contours

~

3

2

1 iO'!!/~/~/;;:/~=,---_ X· .,.'

3

X;

~ optimum

2

3

4

5

6

-\

Fig. 3.5. Solution of example by SLP.

In summary, the main problem in using the method of approximate programming is that the selection of move limits is a trial-and-error process. Thus, the method cannot be used as a black box and the rate of convergence depends to a large extent on the selection of move limits. Although the method is considered unattractive by theoreticians, it has proved to be quite powerful and efficient for many structural design problems. Example 3.3. Consider the two-dimensional example (Fig. 3.5) Z = Xl2 +

xi ~ min

gl == Xf /20 - X2 + 1 ~ 0

g2 ==

xi 120 - Xl + 1~ 0

Using the approximation (3.51), we obtain the following LP problem: find Xb X2 such that · 2 )+2(XlXl • Z=-(Xl· 2 +X2

• . +X2X2)~mm

(X; 1l0)Xl - X2 + (1- X? /20) ~ 0

The original problem is a convex program with the optimal solution X; = X; = 1.056, and Z· = 2.229. Assume the starting point X? = {6. 3}, disregarding the move limits (3.52) and linearizations from previous steps. Results

3.1 General Approximations

143

for the fllSt three iterations are shown in Table 3.2 and in Fig. 3.5. Convergence is fast although the starting point is far from the optimum and no move limits have

been assumed.

Table 3.2. Results. solution by sequential LP.

q 1

2

3 4

(6.3) {O.378 • -0.573} {O.924 • 1.028} {1.055 • 1.055}

Quadratic Programming Subproblem. Some of the drawbacks encountered in SLP can be overcome if a quadratic programming (QP) subproblem is solved iteratively to find a search direction~. A step size is then determined along ~X that guarantees reduction of the objective function. Using this approach. the linear move limits (3.52) are replaced by a step size constraint IIdXII S!;

(3.53)

where II~II is the length of the search direction and !; is a specified small positive number. Substituting 1I2

II~XII= (~ Mf J

(3.54)

into (3.53) yields (3.55)

It can be noted (Fig. 3.6) that the new design is required to be in a hypersphere of radius!; with origin at the current poinL Thus. the approximate subproblem to be solved at each iteration is: find ~X such that

j

=l •...• n,

(3.56)

144

3 Approximation Concepts

Feasible region

Fig. 3.6. Quadratic step size constraint.

vr

where and Vg; are computed at X·, and the last equation is a quadratic step size constraint. A solution of this problem may not exist if I; is too small and the current design X· is infeasible. The problem (3.56) can be transformed into the following QP [2]: find aX such that Z =f + VrTaX + aXT aX ~ min j =

t .... n,

(3.57)

Solution of this QP problem is identical to that of the problem defmed by (3.56). This can be verified by the KT necessary conditions for an optimum point. The QP problem is convex and its solution, if one exists, is unique. The problem can be solved by several efficient methods [42,91]. Sequential Convex Approximations. Assuming the first-order Taylor series approximations (3.51), the objective function and the constraints are separable and linear functions of the design variables. Thus, the problem has a unique global optimum. It has been noted in Sect. 3.1.2 that better approximations can be obtained by using reciprocal variables. Employing a hybrid form of the direct and reciprocal variables, the resulting approximate problem is separable and convex. That is, each function is linearized with respect to a properly selected mix of variables so that a convex and separable function is generated. Assuming the convex approximations (3.38), the approximate problem is formulated as

~ (X. OJ. (X. -X~)~ min f * + 4.J 'ax. ' , i=l

'

(3.58) j = 1..... n,

where (Xi and (Xij are defined by (3.39). Fleury and Braibant [30] employed these approximations in the convex linearization method (CONLIN). It has been found that in some cases the convex approximations scheme used in CONLIN might not

3.2 Approximate Behavior Models

145

be appropriate, leading to inaccurate approximations, which are either too conservative (causing slow convergence) or not sufficiently conservative (causing oscillations). The method of moving asymptotes (MMA), proposed by Svanberg [134], is intended to overcome these difficulties. The intermediate variables (3.40) are defined such that the degree of convexity, and hence conservativeness, of the approximations can be adjusted depending upon the problem being solved. The resulting approximate problem is

!=!-* +

Li=1"

R.)

al'* ((X.-.:

_'J_

aXi

" ag~ _ -* ~) gj =gj + £.J -

i=1 aXi

Ui

Xi

+-P-~- ~min

((X"

I)

Ui -Xi

Xi

Li

R..) - 0 +--- < PI)

Xi -Li

(3.59) j

= l. .... n

,l

g;

where (Xi' J3i' (Xij' J3ij' j*. are defined by (3.42) and (3.43). It has been noted in Sect. 3.1.2 that the MMA formulation is general, and the formulations (3.51) and (3.58) can be viewed as two special cases. The approximate problem can be solved repeatedly, each solution followed by structural reanalysis and revised approximations, until convergence is achieved. Convex-separable approximations are most useful in cases where dual methods (see Sect 2.4.3) are applied. Since these formulations require only one-dimensional minimizations, they can be used for discrete design variables [122].

3.2 Approximate Behavior Models It has been noted that in most structural optimization problems the implicit behavior constraints must be evaluated for successive modifications in the design. For each trial design the analysis equations must be solved and the multiple repeated analyses usually involve extensive computational effort This difficulty motivated extensive studies on explicit approximations of the structural behavior in terms of the design variables [1, 5]. The various studies can be divided into the following classes:

a. Global approximations (called also multipoint approximations), such as a polynomial fitting, obtained by analyzing the structure at a number of design points. The approximation is valid for the whole design space (or, at least, large regions of it). b. Local approximations (called also single-point approximations), based on information calculated at a single design point. The approximation is only valid in the vicinity of a point in the design space. c. Combined approximations, which attempt to give global qualities to local approximations.

146

3 Approximation Concepts

In general, two conflicting factors should be considered in choosing an approximate behavior model for a specific optimal design problem:

a. the accuracy of the calculations, or the quality of the approximation; and b. the computational effort involved, or the efficiency of the method.

Global approximations may require much computational effort in problems with a large number of design variables. Local approximations, such as the Taylor series expansion about a given design, are most efficient but these methods are effective only for small changes in the design variables. For large changes, the accuracy of the approximations often deteriorates and they may become meaningless. Sections 3.2.1 through 3.2.4 of this chapter deal with approximations of displacements. Some local and global approximations, commonly used in structural optimization, are first presented in Sect. 3.2.l. Combined approximations, intended to improve the quality of local approximations, are introduced in Sect. 3.2.2; approximations of homogeneous functions, which are typical in many applications, are demonstrated in Sect. 3.2.3; and approximations along a line in the design space are discussed in Sect. 3.2.4. Finally, approximations of forces are presented in Sect. 3.2.5.

3.2.1 Basic Displacement Approximations Problem Statement. Considering the displacement method analysis formulation [see (l.56)]. the equilibrium conditions r(X) = K-l R are the only implicit equations. These implicit expressions can be eliminated by assuming explicit approximations of the displacements in terms of the design variables ra(X) == r(X)

(3.60)

Approximate displacement models, defmed by (3.60), usually will result in errors in satisfying the equilibrium equations. To evaluate these errors we may define a fictitious load vector Ra by

(3.61) where K is the stiffness matrix for any assumed design, and r a is a vector of approximate displacements computed for this design. If r a are the exact displacements, then Ra =R (the actual given loading). That is, the approximate displacements r a can be viewed as the exact displacements for Ra. The difference between the fictitious loading and the real loading (3.62) indicates the discrepancy in satisfying the original equilibrium conditions due to the approximate displacements. The problem considered in this section can be stated as follows:

3.2 Approximate Behavior Models Q.

147

Given an initial design variables vector X*, the corresponding stiffness matrix K*, and the displacements r*, computed by the equilibrium equations (1.26) K* r* = R

(3.63)

The elements of the load vector R are often assumed to be independent of the design variables and the stiffness matrix K* is usually given from the initial analysis in the decomposed form (3.64)

where U* is an upper triangular matrix. b. Assume a change .1.X in the design variables so that the modified design is

x =X* +.1.X

(3.65)

and the corresponding stiffness matrix is

K = K* +.1.K

(3.66)

where.1K is the change in the stiffness matrix due to the change .1X. c. The object is to find efficient and high quality approximations of the modified displacements r due to various changes in the design variables .1.X, without solving the modified analysis equations K r = (K* + .1.K)r = R

(3.67)

In this formulation, the elements of the stiffness matrix are not restricted to certain forms and can be general functions of the design variables. That is, the design variables X may represent coordinates of joints, the structural shape, geometry, members' cross sections, etc. Once the displacements are evaluated, the stresses can readily be determined explicitly by (1.29). Thus the presented approximations of r are intended only to replace the set of implicit analysis equations (3.67). Local Approximations: Series Expansion. A common approach is to consider the first terms of a series expansion, to obtain the approximate displacements r a (3.68) The Taylor series expansion is one of the most commonly used approximations in structural optimization. The first three terms, obtained by expanding r about X*, are given by rl =r*

r2 = Vr;.1.X r3j

T

(3.69) *

= lJ2.1.X H j .1.X

148

3 Approximation Concepts

where the displacements r*, the matrix of flfSt derivatives Vr; ,and the matrix of second derivatives H; ,are computed at X*. The scalar r3j is the jth component of vector r3. To reduce the computational effort, linear approximations are often used. These require evaluation of the flfSt derivatives which can readily be calculated by the methods discussed in Sect. 3.1.1. It should be noted, however, that the fll'storder approximations may be insufficient in many cases and second-order models might be needed. The latter methods can be divided into two groups:

a. Methods based on calculation of the complete Hj matrices. An advantage of this approach is that all available second-order information is used. However, the computational effort involved in this calculation might be prohibitive. b. Methods based on consideration of only the diagonal elements of matrices Hj [98]. Neglecting the off-diagonal elements of matrices Hj will considerably reduce the computational effort for the second order approximations. In addition, the use of diagonal second-order derivatives will provide separable approximations which are desired in some applications. In particular, this approach benefits from the possibility of using efficient dual methods discussed in Sect. 2.4.3. Alternative series approximations are obtained by rearranging (3.67) to read

K* r = R - AK r

(3.70)

Writing this equation as the recurrence relation

K* r(k+l) = R - AK r(k)

(3.71)

where r(k+l) is the value of r after the kth cycle, and assuming the initial value r(l)= r*, the following binomial series expansion is obtained

ra = (1- B + B2 - ... ) r*

(3.72)

where matrix B is defmed by

B == K*·l AK

(3.73)

That is, the flfSt three terms of the series are given by

rl =r* r2=-B r* r3= B2r*

(3.74)

Calculation of r a by (3.72) involves only forward and backward substitutions if K* is given in the decomposed form (3.64). The calculation of r2, for example, is carried out by means of this equation

3.2 Approximate Behavior Models

K*r2

=-L\K r*

149

(3.75)

We ftrst solve for t by the forward substitution (3.76) then r2 is calculated by the backward substitution (3.77) Similarly, r3 is calculated from (3.78) Problems of slow convergence or divergence may be encountered in applying the series (3.72) . The series converges if and only if (3.79) A sufftcient criterion for the convergence of the series is that IIBII

~

1

(3.80)

where IIBII is the norm of B. It will be shown later is Sect. 3.2.3 that the terms of the Binomial series (3.74) are equivalent to those of the Taylor series (3.69) for homogeneous displacement functions. Series expansions are local approximations, based on information of a single design. As a result, the quality of the approximations might be sufftcient only for a limited region. Several methods have been proposed to improve the series convergence. These include the Jacobi iteration, block Gauss-Seidel iteration, dynamic acceleration methods and scaling of the initial design [62, 67]. These means may considerably improve the results with a moderate computational effort. Improved displacement approximations, based on combining local and global approximations, will be discussed in Sect. 3.2.2. Global Approximations: The Reduced Basis Method. In this approach [34] it is assumed that the displacement vector r of a new design can be approximated by a linear combination of s linearly independent basis vectors rh r2, ... ,r.. of previously analyzed basis designs (where s is assumed to be much smaller than the number of degrees of freedom m), that is (3.81) or in matrix form (3.82)

150

3 Approximation Concepts

where

(3.83)

y is a vector of coefficients to be detennined. Substituting (3.82) into the modified analysis equations (3.67) and premultiplying by r~ yields r~

K

rB

Y = rj R

axm mxm mXa ax!

axm mx!

(3.84)

Introducing the notation (3.85) and substituting into (3.84) we obtain

KR Y = RR axa ax!

ax,

(3.86)

For cases of s« m, the approximate displacement vector can be obtained by solving the smaller (sxs) system in (3.86) for Y instead of computing the exact solution by solving the large (mxm) system in (3.67). The approximate displacements r II are then computed for the given y by (3.82). The reduced basis method usually involves analysis of the structure at a number of design points, therefore it may be classified as global approximations. A basic question in using the method lies in the choice of an appropriate set of the linearly independent vectors r It r 2, ••• , r a that span the design variables space. Displacement vectors of previously analyzed designs can be used, but it should be emphasized that an ad hoc or intuitive choice may not lead to satisfactory approximations. In addition, calculation of the basis vectors requires several exact analyses of the structure for the basis design points, which involve extensive computational effort. It is instructive to note that although a small number of basis vectors may be adequate to model the displacements, it may not be adequate to model the displacement derivatives with respect to design variables. The latter derivatives are often required by the optimization algorithm. 3.2.2 Combined Displacement Approximations Combined Series Expansion and Scaling. Various studies have shown that the quality of the results obtained by series expansion is often insufficient. Poor approximations might be obtained for large changes in the design and the

3.2 Approximate Behavior Models

151

series might even diverge. Scaling of the initial design can greatly improve the quality of the approximations. Scaling of the initial stiffness matrix K· is defined by [67] K =)JK.

(3.87)

where J.1 is a positive scalar multiplier. From (3.63), (3.67) and (3.87) it is clear that the exact displacements after scaling can be calculated directly by (3.88) Note that (3.87) does not require linear dependence of K on X. Furthermore, in many cases, where the elements of K are nonlinear functions of X, the matrix )JK. does not correspond to an actual design. That is, the matrix K computed by (3.87) does not have the usual physical meaning. Scaling of the initial stiffness matrix will improve the quality of the approximations, if the known displacements J.1- 1 r· [see (3.88)] provide better initial data than the original displacements r·. To obtain the displacement approximations, the modified stiffness matrix K [see (3.66)] is expressed in terms of J.1 by (Fig. 3.7) (3.89) That is, if an initial design )JK* is assumed instead of K*, the modified stiffness matrix K is expressed in terms of the corresponding changes in the stiffness matrix AK" instead of AK . From (3.89), AK" is given by (3.90) Consider the recurrence relation (3.71), with J.l.K·, AK~ and WI r· assumed instead of K*, .11K and r·, respectively. The resulting series is [see (3.72)] (3.91) where B~ is given by I-J.1

1

J.1

J.1

B =-- 1+- B

"

(3.92)

and B is defined by (3.73). For J.1 = 1 we find B~ =B and (3.91) is reduced to (3.72). Several criteria for selecting the value of J.1 have been proposed [50, 67, 85]. One possibility is to minimize the Euclidean norm of B", that is

(3.93)

152

3 Approximation Concepts

Fig. 3.7. Scaling of the initial stiffness matrix.

in which m denotes the order of matrix B. A major drawback of using the criterion (3.93) is that the elements of matrix B must be calculated. Since this opemtion involves much computational effort, we may use an alternative criterion that minimizes the Euclidean norm of the second term in (3.91) , that is [85], UB" r*U ~ min

(3.94)

Substituting B" from (3.92) into (3.94), differentiating and setting the resUlt equal to zero, yields a

J.l=-

b

(3.95)

where

L ('i;-ru)Z III

a=

;=1

L (rJ-rurz;)

(3.96)

III

b=

;=1

ru are the given elements of rl = r*, and ruare the computed elements of rz [see (3.74)]. The effect of J.l on the quality of the approximations will be demonstmted later in this section. Combined First-Order Approximations and Scaling. Because of efficiency considemtions, frrst-order series approximations are often used. If only two terms of the binomial series (3.72) are considered, the following first-order approximations are obtained r .. ={I-B)r*

(3.97)

3.2 Approximate Behavior Models

153

It is instructive to note that (3.97) can also be introduced by substituting r = r*+Ar in the right-hand side of (3.70) to obtain K*r = R - 11K r* - 11K Ar

(3.98)

Neglecting the second order tenn 11K I1r, premultiplying by K*·1 and substituting (3.63) and (3.73) gives (3.97). It will be shown now that by combining two types of scaling: -

scaling of the initial stiffness matrix K*; and scaling of the approximate displacements r a ;

the fIrst-order binomial series approximations (3.97) can be transfonned into a reduced basis fonn, with improved quality of the approximations. A similar procedure can be used for the Taylor series approximations. It has been noted that it is possible to assume J.1K*, I1KJI. and W1r* as initial values instead of K*, 11K and r*, respectively [see (3.87), (3.88) and (3.90)]. Substitution into (3.97) yields (3.99) where r1 and r2 are defIned by (3.74). Note that for J.1 = I, (3.97) is obtained. It has been shown that J.1 can be selected by the criterion (3.95). Alternatively, scaling of the approximate displacements, presented herein, can be introduced. DefIne the scaled displacements r.r by r.r=Qra

(3.100)

where Q is a scalar. By evaluating ra for any given J.1 by means of (3.99), the latter displacements can then be scaled by (3.100) such that the fInal displacements are improved. Substituting (3.99) into (3.100) yields (3.101) Thus, each evaluation of the displacements involves the following two steps:

a. Selecting J.1-scaling of the initial stiffness matrix K* and evaluation of the

approximate displacements r a by (3.99). b. Selecting .a-scaling of the approximate displacements r a by (3.100). Assuming the transfonnations Y1 =.a (J.1-2(2J.1-1)]

Y2 =Q 1.1"2 and substituting (3.102) into (3.101) yields

(3.102)

154

3 Approximation Concepts

(3.103)

where (3.104) (3.105) That is, (3.101) which is based on combining the two types of scaling, is equivalent to the two-terms expression of the reduced basis method (3.103). It should be noted that ~ and Q can be determined uniquely for any y by (3.102). The above combination of a first-order series expansion and the reduced basis method can be generalized to any number of terms in the series, as will be shown subsequently. Combined Series Expansion and Reduced Basis. The drawbacks of series expansion and the reduced basis method motivated combination of the two approaches to achieve an improved solution procedure [80]. In this procedure, the computed terms of a series expansion are used as high quality basis vectors in a reduced basis expression. The advantage is that the efficiency of local (series expansion) approximations and the improved quality of global (reduced basis) approximations are combined to obtain an effective solution procedure. The solution process involves the following steps:

a. The modified stiffness matrix K is introduced.

b. The basis vectors rj of a series expansion [i.e. (3.69) or (3.74)] are calculated,

and matrix rB [see (3.83)] is introduced. To maintain efficiency of the calculations, only two or three basis vectors might be considered. c. The elements of KR and RR [see (3.85)] are determined. d. The coefficients y are calculated by solving the set of (2x2 or 3x3) equations (3.86). e. The final displacements are evaluated by (3.82). To evaluate the quality of the results, we substitute the approximate displacements (3.82) into the expression of the errors in the modified analysis equations (3.62) to obtain [see (3.61)] AR(y) = Kr a - R = KrBY - R

(3.106)

It has been noted earlier that if r a is the vector of the exact displacements, then AR =O. Thus, AR can be used to evaluate the quality of the approximations. Let us derme the common measure of smallness of AR(y) by the quadratic form (3.107) Substituting (3.106) into (3.107) , differentiating with respect to y and setting the result equal to zero, we obtain the following linear equations in the form of (3.86)

3.2 Approximate Behavior Models

a y =b

155

(3.108)

where a and b are given by

a =(KrBl(KrB)

(3.109)

This alternative method for determining y can be used instead of (3.85) in step c of the solution process. The two criteria have been compared using several numerical examples [80]. It has been found that although the method (3.109) provides smaller q(y) values, better results might be obtained by the method (3.85). Another method, that combines the reduced basis approach and the first-order Taylor series approximations of the displacements, has been proposed by Noor and Lowder [101]. The assumed basis vectors are rl

= r*

and rj+l

=or" fiJX j

(j =I •... ,n) . These vectors are normalized to overcome numerical roundoff errors. An advantage of this choice is that it contains the sensitivity analysis vectors. However, many of the computed derivatives may correspond to nonactive constraints and are not needed for the optimization process. In addition, this method is not efficient in problems with a large number of design variables, where n derivative vectors are to be computed and an (n+l)x(n+l) system of equations must be solved. Recently, Noor and Whitworth [102] proposed to express the modified analysis equations in terms of a single parameter, and to choose the basis vectors as the various-order derivatives of the displacements with respect to the parameter, evaluated at the original design. It has been found that a small number of basis vectors is often sufficient to obtain good results. Computational Considerations. The quality of the results and the efficiency of the calculations are two conflicting factors that should be considered in selecting an approximate reanalysis model. That is, better approximations are often achieved at the expense of more computational effort. In this subsection, some computational considerations associated with the presented approximations are discussed. Consider first the two methods of calculating the basis vectors, namely the Taylor series and the binomial series. Assuming the common fmt-order Taylor series expansion, once the matrix Vr; is available each redesign involves only calculation of the product Vr;AX . This is probably the most efficient reanalysis method. However, it has been noted that the quality of the results might be insufficient for large changes in the design variables. In addition, the second-order Taylor series expansion is usually not practicable owing to the large computational effort involved in calculation of the second-order derivative matrices H j • An exception is the common case of homogeneous displacement functions, discussed in Sect. 3.2.3, where the Taylor series and the binomial series become equivalent. The advantage of using the binomial series is that, unlike the Taylor series, calculation of derivatives is not required. This makes the method more attractive in general applications where derivatives are not available. Calculation of each term of the binomial series involves only forward and backward substitutions, if K* is

156

3 Approximation Concepts

given in the decomposed form of (3.64). Thus. the second order terms can readily be calcuJated. In the case of fll'St-order approximations. determination of only Br· must be repeated for each trial design. This requires calculation of a single vector by forward and backward substitutions. As to the selection of the number of basis vectors to be considered in the combined approximations. it has been noted that. in general. second-order approximations (three basis vectors) provide better results than first order approximations (two basis vectors). To evaluate the computational effort involved in the combined approximations. compared with conventional local methods. assume the fll'St-order approximations. The series approximations require only calculation of the second basis vector [(3.69) or (3.74)]. The combined approximations require. in addition. introduction of the modified stiffness matrix K. calcu1ation of the products given by (3.85) or (3.109). determination of y by solving the set of (2 x 2) equations (3.86) or (3.108) and multiplication of the two basis vectors by y. Certainly. these operations increase the computational cost. However. the result is often considerably better approximations. particularly in cases of large changes in the design variables. Tttat is. high quality approximations can be obtained in cases where the local series approximations provide meaningless results. Consequently. exact analyses which involve more computational effort are not required in cases where the local approximations provide insufficient results. Finally. the local approximations may be viewed as a special case of the combined approximations where y 1.0 is selected. An additional advantage of the combined approximations is that the errors involved in the approximations can be evaluated by AR and q.

=

Example 3.4. Consider the truss shown in Fig. 3.8 with ten cross-sectional area variables Xj (i = 1•...• 10). subjected to two loading conditions. Assuming E=L =1.0 and the initial design X·=1.0. the corresponding displacements are

Loading A:

r*T=( 195.4.465.1.235.5. 1054.2. -264.5. 1094.3. -204.6. 500.6)

LoadingB:

r*T=(190.7. 447.3.221.0.1034.1. -279.0.1114.4. -209.3. 518.3)

I•

I

• ·'1

360

(I)

(2) '4

t 50

(b)

(a) 8.

t 50

36{ h50

FIg.3.S. Ten-bar truss:

(2)

Loading A. b. Loading B.

L50

3.2 Approximate Behavior Models

157

Table 3.3. First-order approximations, ten-bar truss, changes of cross sections. Load A

Method Exact (3.109) (3.85)

DisElacements 19.52 53.17 23.49 115.5 -26.51 120.52 -20.48 57.54 19.51 53.17 23.48 115.5 -26.52 120.51 -20.48 57.54 19.51 53.17 23.48 115.5 -26.52 120.52 -20.48 57.54

B

Exact (3.109)

19.04 50.99 21.98 113.1 -28.02 123.00 -20.96 59.72 19.03 50.98 21.95 113.1 -28.04 122.99 -20.96 59.72 19.03 50.99 21.96 113.1 -28.04 122.99 -20.97 59.72

~3.85~

To illustrate the physical meaning of the combined first-order approximations and the effectiveness of the solution procedure, consider the modifIed design

xT=

{lO, 10, 10, 10,8,8,8,8,8, 8}

The resulting displacements obtained by the methods (3.85) and (3.109) are summarized in Table 3.3. It can be observed that excellent results have been obtained for these large changes (up to 900%) in the design variables. The high quality of the results can be explained by the scaling procedure. That is, the modified design is relatively close to the scaling line J.1K*. Considering the optimal designs [62] Loading A: LoadingB:

xT =

{7.94, 0.1, 8.06, 3.94,0.1,0.1,5.75,5.57,5.57, 0.1}

XT = {5.95, 0.1, 10.05,3.95,0.1,2.05,8.56,2.75,5.58, 0.1}

results obtained for these very large changes in the cross-sections (up to +905% and -90% simultaneously) by various methods are shown in Table 3.4. It can be observed that solution by the Taylor series (3.69) is meaningless. SpecifIcally, since the series diverges the results obtained by the second-order Taylor series approximations (three terms of the series, or three basis vectors) are worse than those obtained by the first-order approximations (two terms of the series). Relatively good results have been obtained by both combined methods (3.85) and (3.109). Considering three terms of the series (second-order approximations), the quality of the results is further improved. It has been noted [80] that the results obtained by the method (3.85) are better than those obtained by (3.109) even in cases of larger q values. Example 3.5. Consider the thirteen-bar truss with the initial geometry and loading shown in Fig. 3.9a. The modulus of elasticity is 10,000, the initial cross sections are X* = 1.0, and the unknowns are the horizontal (to the right) and the vertical (upward) displacements in joints B, C, D, F, G and H, respectively. Assume first the following modifIed cross-sectional areas

XT = {10, 10, 10, 10, 8, 8, 8, 8, 6, 6, 6, 6, 6}

158

3 Approximation Concepts

Table 3.4. First- and second-order approximations, ten-bar truss, optimal solutions.

Load Method Tenns A Exact - 25.0 (3.69) 2 -1180 8468 3 (3.109) 2 21.1 3 22.8 (3.85) 2 23.0 3 24.1 B

-

Exact (3.69)

2 3 2 3 2 3

(3.109) (3.85)

25.0 -1078 7095 24.5 21.8 25.7 22.6

75.0 -2201 13620 68.8 67.4 75.0 70.1

Displacements 40.5 184.4 -50.0 -1223 -4831 1218 8615 31588 -8649 31.6 160.7 -40.1 35.6 172.5 -47.6 34.4 175.3 -43.8 37.7 181.4 -50.6

75.0 -2189 13729 66.0 63.5 69.6 66.1

38.1 -1111 7189 31.9 31.0 33.6 32.3

175.0 -5031 33615 153.3 156.8 161.7 164.0

200.0 -4873 31736 171.1 185.4 186.7 195.0

-50.0 200.0 1406 -5125 -1106133887 -40.1 172.6 -47.2 180.5 -42.3 182.4 -50.0 189.0

-25.0 1219 -8739 -22.7 -24.6 -24.7 -26.0

75.0 -2272 13814 77.0 75.9 84.0 79.1

-25.0 75.0 1360 -2469 -10949 15209 -22.4 78.2 -24.5 75.5 -23.4 82.5 -26.0 78.6

Results obtained for these large changes in cross sections (up to 900%) by various approximate methods are shown in Table 3.5. It can be observed that: - Results obtained by the local flrst- and second-order series expansion [(3.69) or (3.74), which are equivalent in this case] are meaningless. Once again, the series diverges due to the large changes in the design. 5.0

15.0

13

'''''

60

80

80

'''''

60

(a)

~,

Fig. 3.9. Thirteen-bar truss.

y (b)

""

3.2 Approximate Behavior Models

159

Table 3.S. First- and second-order approximations, thirteen-bar truss, modified cross sections.

Displacement Nwnber

1 2 3 4 5 6

7 8 9

10 11 12

(3.69) or (3.74) 2 Terms 3 Terms -3.90 35.7 -2.12 19.3 -11.85 103.8 26.3 -3.10 -20.70 177.2 -3.30 27.4 -4.00 36.5 2.10 -19.6 -11.70 102.5 3.10 -26.4 -21.00 178.4 3.30 -27.5

(3.109) 2 Terms 3 Terms 0.051 0.048 0.028 0.026 0.178 0.167 0.050 0.047 0.338 0.319 0.058 0.055 0.054 0.051 -.029 -.027 0.173 0.162 -.050 -.046 0.348 0.329 -.059 -.056

(3.85) 2 Terms 3 Terms 0.048 0.048 0.026 0.026 0.168 0.167 0.047 0.047 0.320 0.319 0.055 0.055 0.050 0.051 -.027 -.027 0.162 0.162 -.047 -.047 0.329 0.329 -.056 -.056

Exact Method 0.048 0.026 0.167 0.047 0.319 0.055 0.051 -.027 0.162 -.047 0.329 -.056

- Very good results have been obtained by the first- and the second-order approximations for both methods (3.85) and (3.109). Consider the modified cross-sectional areas X as in the previous case and a single geometric variable Y representing the span (Fig. 3.9b), with the optimal value Y = 153 [85]. Results obtained for these large changes in both the geometry (155% in Y) and the cross-sections (up to 900%) by various approximations are shown in Table 3.6. Improved results have been obtained for the second-order approximations (three terms) by both methods (3.85) and (3.109). Specifically, assuming three terms for the method (3.85), the errors in most displacements do not exceed 5%. Table 3.6. First- and second-order approximations, thirteen-bar truss, modified geometry and cross sections.

Displacement Exact Nwnber Method 1 0.0098 2 0.0101 3 0.0400 4 0.0162 5 0.0811 6 0.0134 7 0.0114 8 -.0101 0.0360 9 10 -.0166 11 0.0902 12 -.0133

(3.109) Error 2 Terms 3 Terms ~%l 0.0043 0.0105 7 0.0059 0.0091 10 0.0230 0.0364 9 0.0105 0.0142 12 10 0.0518 0.0726 0.0095 0.0114 15 0.0061 0.0108 5 -.0063 -.0089 12 0.0185 0.0339 6 -.0104 -.0141 15 0.0603 0.0817 9 -.0102 -.0120 10

(3.85) Error 2 Terms 3 Terms ~%l 0.0089 14 0.0112 0.0084 0.0099 2 0.0378 0.0396 1 0.0146 4 0.0155 0.0788 0.0795 2 0.0140 0.0127 5 0.0108 0.0117 3 -.0088 -.0096 5 0.0330 0.0367 2 -.0154 7 -.0145 0.0895 1 0.0879 0 -.0133 -.0147

160

3 Approximation Concepts

3.2.3 Homogeneous Functions First-Order Taylor series Expansion. In many structural design problems the displacements, stresses and forces are homogeneous functions of the design variables. In this subsection, some simplified approximations for this type of function are presented. Assume that the displacements r are homogeneous functions of degree n in the design variables X, for which we have by definition rijJ.Xj =J1" r(Xj

(3.110)

where 1.1 is a scalar. Euler's theorem on homogeneous functions states that (3.111) where Vr; is the matrix of first derivatives computed at X*. It is instructive to note that the derivatives of homogeneous functions of degree n are homogeneous functions of degree n-l, that is (3.112) These properties of homogeneous functions can be used to obtain simplified approximations [36]. Assuming the rust-order Taylor series expansion of the displacements r r" = r* + Vr;(X-X*)

(3.113)

and substituting (3.111) into (3.113), the latter approximations for homogeneous displacement functions become (3.114) Intermediate Variables. Assume intermediate variables of the form Y; = X;'" [see (3.32)]. The resulting displacements are homogeneous functions of degree nlm in V. Therefore, the rust-order Taylor series expansion (3.114) is r"

= (1- nlm)r * + Vr;V

(3.115)

For any given n, we can choose the value of m such that the approximations are improved. Considering the common reciprocal cross-sectional variables Y; = l/X; where n=m=-l, then (3.115) becomes (3.116) Since the displacements in this case are homogeneous functions of degree 1 in V, we have from (3.112)

Approximate Behavior Models

161

(3.117) That is. the approximations (3.116) are exact along the scaling line Y = J.1Y*. This illustrates the advantage of using the reciprocal variables for approximations near the scaling line in structures with cross-sectional variables. Combined Taylor Series and Scaling. For a point X along the scaling line (3.118)

X = J.1X* the displacements and displacement derivatives are [see (3.110) and (3.112)] r = J.1" r*

Vrx = J.1II-IVrx•

(3.119)

Expanding r about J.1X* and substituting (3.119) into (3.114) yields (3.120) The value of J.1 can be chosen such that the approximations are improved. The relationship between Various Approximations. Using the virtualload method. assume the common case of cross-sectional design variables where the displacements and the stresses are are given by (1.34) and (1.35). respectively. 1 r=T-=TY X

(3.121)

1 1 a=ST-=P-=PY

X

X

(3.122)

For a general statically indeterminate structure. the elements of T are implicit functions of Y. Assuming constant forces then the elements of T are also constant, T = T*. If the elements of S are independent of the design variables. then for constant forces P = p* and the resulting displacements and stresses become r=T*Y

(3.123)

a=P· Y

(3.124)

Differentiation of (3.121) with respect to Y gives (3.125)

162

3 Approximation Concepts

This equation is based on the relations (aT"' alj)Y" = 0 (i = 1, ... , n), which are true for a general statically indeterminate structure. Based on (3.125), it can be seen that the approximations (3.116) and (3.123) are equivalent In addition, it can be shown that (3.124) is equivalent to the first-order Taylor series expansion in the reciprocal variables. That is, the first-order Taylor series approximations in the

reciprocal variables are equivalent to the assumption of constant internal forces.

The latter assumption is justified in many statically indeterminate structures having normal action, where the internal forces are not appreciably affected by the design variables. Assuming constant forces in such cases, the displacements become linear functions of Y, and the linearized expressions in terms of Y usually represent high quality approximations of the constraints. If the elements of the stiffness matrix are linear functions of X and the elements of R are constant, then the displacements are homogeneous functions of degree 1 in X. Assuming a single variable X, the Taylor series expansion of r about X" becomes (3.126) From (3.11) we have

nvrx"_ - - K"-lnK" v x r"

(3.127)

Matrix K can be expressed as (3.128) where the elements of matrix KO are constant. Matrix VKx is therefore given by (3.129) Substituting (3.127) and (3.129) into (3.126) yields (3.130) Based on (3.128) matrix M( [see (3.66)] can be expressed as M(=KoM

(3.131)

Substituting (3.131) and (3.73) into (3.130) yields (3.132) This expression is identical to (3.97). That is, for the common case of homogeneous displacement functions of degree -1 in the design variables, the first-

order Taylor series and the first-order binomial series are equivalent.

Approximate Behavior Models

163

Example 3.6. Consider the three-bar truss shown in Fig. 1.11 with the following single stress constraint (Fig. 3.10) (a)

The stress 0"1 is an homogeneous function of degree -1 in the design variables X. At the point X*T = (1.0, 1.0), the value of the stress and its derivatives with respect to design variables are

=14.14

0";

*T VO"x =(-11.714. -2.426)

(b)

Assuming the fIrst-order Taylor series expansion about X·, the approximate stress constraint is (c)

The intersection of the scaling line X = IlX* through X· with the constraint surface 0"1 = 20 is at the point X**T = (0.707, 0.707). At this point

b ').\)

.,)

0\

Xz

\

\

\

2.0

~)

\

\ \

()"\

d-

01

v--""" \

\~ \ \

b ').\) \

1.6

\

I \

\

\

\

I I \

\

\ \

1.2

< 20

\

\ \

0.8

\

\

°1

r-

\

°1

(Y") <20

\

\

\ \

\

\ \ \ \ \

\

(Y') =

,

\

X·

\

0.4

o

0.2

0.4

Fig. 3.10. Linear approximations of the constraint

1.2 0"1 ~

20 .

164

3 Approximation Concepts

a~* = 20.0

Vax**T = {-23.428. - 4. 852}

(d)

and the approximate stress constraint is al(X**)=40.0-23.428X1 -4.852X2 :5;20

(e)

It can be observed from Fig. 3.10 that both approximate constraints (c) and (e) represent parallel lines in the design space. Also, the line (e) is tangent to the constraint surface al =20 at the intersection point X**. Repeating the above procedure for the reciprocal design variables Y, we obtain for both y*T = {1.0, 1.0} and y**T = {1.414. 1.414} al (y*) = a 1(y**) = 11.714l} + 2.426Y2:5; 20

if)

It can be noted that the stress a 1 is a homogeneous function of degree 1 in the reciprocal variables y. Figure 3.10 shows that, geometrically, the approximation if) represents a surface in the space of X, which is tangent to the exact constraint surface at X**. Also, it can be seen that this approximation in the reciprocal variables is superior to both approximations in the design variables (c) and (e). 3.2.4 Displacement Approximations Along A Line Approximations along given lines in the design space are often required in optimal design procedures. This problem is common to many mathematical programming methods presented in Chap. 2. In general, the lines (or direction vectors) are selected successively by the optimization method used. For each of the determined directions it is usually necessary to evaluate the constraint functions (or to repeat the analysis) several times. A line in the design space is defined in terms of a single variable ex by X = X*+ exdX*

(3.133)

in which X* is a given initial design, dX* is a given direction vector in the design space, and the variable ex determines the step size. Since only a single variable is involved, approximations along a line require much less computations. In this section, local (series expansion) and global (polynomial fitting) approximations along a line are presented. Other methods are discussed elsewhere [65]. Dependence of K on ex. The elements of the stiffness matrix are some functions of ex. A common form of the modified stiffness matrix is [see (3.66)] K = K* + f(ex)dK*

in which the elements of K* and dK* are given.

(3.134)

Approximate Behavior Models

165

In truss structures where X represents the cross-sectional areas, or in beam elements where the moments of inertia are chosen as design variables, the elements of the stiffness matrix are linear functions of X and (3.66) becomes ~ Kj!!Xj 0 K = K "' + £.J

(3.135)

j

where K? are matrices of constant coefficients, representing the contributions of the individual elements. For the line defined by (3.133) this equation is reduced to K=K"'+aAK"'

(3.136)

ax!

If the elements of K are functions of (where Xi is the naturally chosen design variables, and a and b are given constants) we may use the transformation

Y; = aX!

(3.137)

and obtain the linear relationship (3.135) in terms of L\Yj • In cases where such transformations are not possible, linear approximations of the nonlinear terms of K are often sufficiently accurate. Local Approximations: Series Expansion. Assuming approximations along the line defined by (3.133), the Taylor series expansion about X"' (a =0) is given by (3.138) The displacement derivatives can readily be calculated by the methods discussed in Sect. 3.1.1. Assuming that R is independent of a and considering the direct method, differentiation of (3.1) with respect to a gives

(3.139)

Obviously, since only a single variable, a, is involved, calculation of the derivatives or"'lOa and iJ2r"'/iJa2 requires a relatively small computational effort. It has been noted that the solution of (3.139) involves only forward and backward substitutions if K* is given from the initial analysis in the decomposed form (3.3). If the elements of K are linear functions of a [see (3.136)] then

166

3 Approximation Concepts

aK aa =M( •

(constant)

(3.140)

Assuming that the relationship (3.134) holds, then the binomial series (3.72) along the line (3.133) is reduced to the explicit expression r = [1- !(a) B* + f(a) B·2 - ... ]r·

(3.141)

in which the elements of B* are constant, given by (3.142) If the linear relationship (3.136) holds, then the explicit expression (3.141) becomes r = (1- aBo + a 2B"2 - ...)r"

(3.143)

The constant coefficient vectors (3.144) can readily be calculated by forward and backward substitutions and then used for multiple reanalyses along a line. The displacement derivatives in this case are given by [see (3.139), (3.140) and (3.142)]

(3.145)

Comparing (3.144) and (3.145) it can be seen, once again, that the Taylor series and the binomial series approximations are equivalent if the linear relationship (3.136) holds. Global Approximations: Polynomial Fitting. Polynomial fitting techniques, discussed in Sect. 2.2.1, can be used to obtain explicit approximations of the displacements along a line. While series expansions are based on a single exact analysis, polynomial fitting techniques usually require analyses or calculation of the displacement derivatives for several designs. Since more information is used in the latter techniques, the quality of the approximations is higher at the expense of more computational effort. Assuming, for example, the quadratic fitting

Approximate Behavior Models

r(a) = a + ba + ca2

167 (3.146)

the constants a, b, c can be determined from results of analyses of two or three designs. Assuming the following conditions for the interval 1 ~a ~ 1 for a=a' =0 (3.147)

r" = a + b + c

for a = a" = 1

and substituting into (3.146), we obtain

.)

'(

r(a)=r ,Or +-a+ r '" -r -Or- a 2 aa aa

(3.148)

This equation is based on two exact analyses and calculation of the displacement derivatives at a single point. Another possibility, that does not involve evaluation of derivatives, is to use results of three exact analyses. Substituting the computed values of r' (for a' = 0), r" (for a" =0.5), r'" (for a·" = 1) into (3.146) and solving for a, b, c, we find r = r' + (-3r· + 4r" - r··*) a + (2r' - 4r" + 2r"')a2

(3.149)

Similarly, the constants for the cubic fitting r(a)

=a + ba + ca2 + da3

(3.150)

can be determined from the values of r·, ar'/aa (for a' = 0) and r", ar"/aa (for a" = 1). The result, based on exact analyses and derivative calculations for two designs, is

r = r· + ar' a + (3r" _ 3r' _ 2 ar' _ ar ,. ) a 2 aa aa aa

+ ( 2r , -2r "ar· + - +ar") -- a3 aa

(3.151)

aa

Example 3.7. Consider the three-bar truss shown in Fig. 3.11. The modulus of elasticity is 30, 000 and the cross-sectional areas are 1.0. The geometric variables flo f2 represent the location of the free node. Two cases of changes in the geometry along a line in the design space have been considered

168

3 Approximation Concepts

20

Fig. 3.11.

Three-bar truss.

Case a

{ll} ={tOO} + {tOO} 100

Y2

100

(l

Caseb

Results obtained for the horizontal displacement by the following methods [85] are shown in Fig. 3.12:

r

r

(a)

0.25

(b)

0.25

l1....f

0.20

0.20

0.15

0.15

0.10

0.10

T

I

0.50

I

a ~

T

1.00

Fig. 3.12. a. Results, case a, b. Results, case b.

I

0.50

I

1.00

,..a

Approximate Behavior Models

169

- A =exact solution; - B =the flrst-order Taylor series (3.138) ; - C the second-order Taylor series (3.138) ; -D =the quadratic fitting (3.148) ; -E =the quadratic fitting (3.149); - F = the cubic fitting (3.151) ; - G =the fU'St-order binomial series (3.72).

=

It can be seen that in both cases methods B and G provide poor results. Better results are obtained by method C, and the best results are achieved by the polynomial fitting techniques (methods D. E. F). 3.2.5 Approximate Force Models Forces as Intermediate Response Quantities. In statically determinate structures the element forces are independent of the cross-sectional variables while the stresses are not. This implies that in statically indeterminate structures the stresses are more nonlinear functions of the variables than element forces. Consequently, improved approximations of stresses could be developed by using element forces as intermediate response quantities. That is, instead of using flrstorder approximations of the stresses

·I

II

~_.

0'.=0'.+

"

-ClUj

j=l

ax· I

• (X--X-) I

(3.152)

I

it is possible to use the forces as intermediate response quantities to obtain [139]

•+ I" -aA; aX

A.

0'.

,

A. J = _, = __ Xj

:=j-~l

j

(x. I

x·•) I

_ _ _ _ __

Xj

(3.153)

Equation (3.153) produces a more accurate approximation than (3.152) because it captures the cross coupling between the effect of Xj and Xj on the stress. This cross coupling is present because the stress is a nonlinear function of the element force Aj • which is a function of the X/s, and the element variable Xj' This can be seen by examining the second partial derivatives of (3.152) and (3.153). The second partial derivative of (3.152) is zero while the second derivative of (3.153) is

(3.154) It should be noted that the approximation (3.153) is exact for statically determinate structures.

170

3 Approximation Concepts

Force Method Formulation. In some optimal design applications it is advantageous to consider the force method of analysis. Using this method in statically determinate structures, the constraints are explicit functions of the design variables and therefore approximations are not needed. In statically indeterminate structures, the number of unknown redundant forces is often much smaller then the number of unknowns in the displacement method of analysis. In addition, forces are less sensitive to design changes than displacements are. Thus, application of the force method may contribute to the efficiency and to the quality of the approximations. Considering the force method analysis formulation [see (1.60)], the compatibility conditions N(X) =F-IB are the only implicit equations. The latter equations can be eliminated by assuming explicit approximations of the redundant forces in terms of the design variables. Such Approximate force models defined by (3.155)

NiX) == N(X)

can be used for simple evaluation of the constraint functions. The displacements and the forces corresponding to any approximate redundant forces N a can be calculated directly by (1.23) and (1.24). It should be noted that member forces corresponding to approximate redundant forces will satisfy equilibrium but not necessarily compatibility conditions. For any given design represented by a flexibility matrix F with approximate redundant forces N a, we may define a vector of fictitious displacements Ba by (3.156)

If Na are the exact redundant forces then Ba = B. That is, the approximate redundant forces can be viewed as the exact forces for a structure with displacements Ba in the direction of the redundants. The difference (3.157)

indicates the discrepancy in satisfying the original compatibility equations due to the approximate redundant forces. The problem discussed in this section can be stated as follows: given a design X* and the corresponding redundant forces N*, the object is to determine the forces N corresponding to various changes AX in the design without repeating the solution of the implicit compatibility equations (1.22). In general, methods similar to those described in previous sections for displacement approximations can be used also for forces. Some local approximations and combined series approximations are presented subsequently. Local Approximations. Assume that for a given initial design X *, the corresponding flexibility matrix F* and redundant forces N* [see (1.22)] are known. For a change AX in the design variables, the modified analysis equations become (F* + AF) (N* + AN)

=B* + AB

(3.158)

Approximate Behavior Models

171

where aF. aN and a~ are the corresponding changes in F. N and ~. Rearranging (3.158). neglecting the second order term aF· aN and derming. for brevity C = F·· l aF

(3.159)

the following First-Order Approximations (FOA) are obtained N

=N· + aN =NA - CN·

(3.160)

Another approximate model can be obtained by the binomial series expansion. Rewriting (3.158) in the form (F· + aF) N

=~. + a~

(3.161)

premultiplying (3.161) by F*·l and substituting (3.159) yields (I + C)N

=NA

(3.162)

Premultiplying (3.162) by (I + C).l and expanding the latter expression under certain conditions as (I + C).l = 1 - C + CZ ....

(3.163)

then (3.163) becomes equivalent to the Binomial Series Approximations (BSA) N = (I - C + Cz - ... )NA

(3.164)

To calculate the terms of the series (3.164) define

(3.165)

etc. The series then becomes (3.166) For the given initial triangularization F = U ~ U F (where UF is an upper triangular matrix) the determination of the vectors N z• N 3• etc. involves only forward and backward substitutions. The solution is based on the recurrence relation

172

3 Approximation Concepts

(3.167) in which k denotes the term in the series and (3.168) It can be noted that if only the rrrst terms of the BSA (3.164) are considered, then these approximations become equivalent to the FOA (3.160). The First-Order Taylor series expansion of N about X* is given by (3.169) It has been shown [63] that for cross-sectional variables the frrst-order Taylor series expansion in the reciprocal variables (3.170) is equivalent to the FOA (3.160). Similar to displacement approximations, first-order approximations of forces may provide poor results for large changes in the design variables. More accurate results can be obtained by the BSA when the series converges and several tenns are considered. However, the series might diverge for large changes in the design. Combined Series Expansion and Scaling. A procedure that greatly improves the quality of the BSA (3.164) is presented in this subsection. Similar to scaling of the stiffness matrix [see (3.87)] the method [83] is based on scaling of the initial flexibility matrix F* F

=JlF*

(3.171)

and employing nonn minimization to detennine the optimal scaling multiplier Jl in (3.171). The matrix F of the modified design can be expresses as [see (3.89)] F

= F* + AF =JlF* + AFIL

(3.172)

Substituting for AF from (3.172) into (3.161) and premultiplying by F*·l we obtain (3.173) Substituting AFIL from (3.172) into (3.173) and noting also (3.159) gives (3.174)

Define, for brevity, C IL as

Approximate Behavior Models

1-1l 1 C =--I+-C ... Il Il

173

(3.175)

Using this last expression in (3.174) and expanding (I + C...>-t, the following Improved Binomial Series Approximations (IBSA) of N in terms of Il are obtained

1 2 N=-(I-C ... +C ... - ... )N A Il

(3.176)

For Il = 1, C ... becomes equivalent to C and the IBSA is reduced to the BSA (3.164). The object now is to select Il such that the convergence properties and approximation qualities of the series (3.176) are improved over those of the series (3.164). It has been shown [67, 85] that various selections of Il may lead to improved approximations of displacements. One possibility is to assume a criterion similar to (3.94), that minimizes the Euclidean norm of the second term in the series (3.176) (3.177) This does not involve a prior determination of elements of matrix C and the calculation requires only forward and backward substitutions. By this criterion, the scaling factor Il is determined from

(3.178) in which NJj and N2i are the element of the vectors Nt and N 2, respectively, as defmed by (3.165). The advantage of this criterion is that all terms can readily be computed. The effect of Il on the quality of the approximations is demonstrated by the following example. Example 3.8. Consider the twenty-five-bar truss shown in Fig. 3.13 with the initial and modified designs

X·T= 10 L\XT = {95, 100,82,83,80,99, 88,95, 96, 88, 97, 82, 81,94,88, 88, 96, 93, 91,86,95, 87,92, 84, 85} where X is the vector of cross-sectional areas. The applied loads are Node

1

2

3

4

x 100 0

50 50

v

z

1000 -500 1000 -500

o o

0 0

174

3 Approximation Concepts

Fig. 3.13.

Twenty-five-bar truss.

Table 3.7. Redundant forces, twenty-five-bar truss. Member 19 20 21 22 23 24 25

(3.160) 77.5 168.8 -136.6 999.5 -1242.1 1400.1 875.8

(3.164) Two # 10.5 25.7 -39.2 186.4 -236.5 -270.8 157.8

Five # 27.1 59.1 -80.7 403.5 -510.8 -580.8 340.0

(3.176) ## Two # Five # 126.7 125.9 193.1 193.3 -147.2 -146.1 1007.9 1008.4 -1261.6 -1262.0 -1381.8 -1380.5 829.3 830.6

Exact 126.7 193.2 -146.0 1008.4 -1261.9 -1382.0 830.7

# Tenns ## Il = 0.10

The chosen redundants are the forces in members 19-25, and the changes in the design variables are about 900%. Redundant forces obtained by the FOA (3.160), the BSA (3.164) and the IBSA (3.176) are given in Table 3.7 [83]. It can be observed that poor results have been obtained by the FOA (3.160). The series (3.164) and (3.176) converge to the exact solution. However, it has been noted that only two terms are sufficient to obtain good results by the IBSA (3.176)

Exercises

175

while more than 50 terms are required by the BSA (3.164) to reach a similar accuracy level. This example illustrates the effectiveness of scaling on the quality of the results.

Exercises 3.1 Consider the three-bar truss shown in Fig. 1.11. with the given analysis equations (c), (d) in example 1.7 a. Assuming the initial design

xf

= {l.0. l.0}. calculate the derivatives of the stresses with respect to design variables by the direct method and the adjointvariable method. b. Introduce the direct-linear approximations (3.35). the reciprocal approximations (3.36) and the conservative-convex approximations (3.38) for 0'3.

3.2 The symmetric truss shown in Fig. 3.14 is subjected to a single load P = 10. The members' cross-sectional areas are Xl> X z and 1/2 Xl as shown and the modulus of elasticity is E =30.000. a. Show that the fIrst-order Taylor series expansions of the displacements rl • rz in

terms of the reciprocal variables are exact b. Calculate the elements of the matrix of derivatives Vry [orjc)Yj (j=I. 2; i=l. 2)] using (3.125). c. Calculate the elements of the matrix Vay [oajloYj (j=1 •...•5; i=l. 2)]. d. Assume that the joints C. D are supported in both the vertical and the horizontal directions and consider only rl. rz as displacement degrees of freedom. Calculate the derivatives Vrh Vrz by the direct method and the adjoint-variable method at the point X*T = {1.0. 1.414}. Introduce the fIrst-order Taylor series expansion forrT = {rlt r z} and aT = {al' az} about X*. 3.3 Given the problem

Perform three iterations of the sequential linear programming method. Assume the

xf

initial point ={7.0, 4.0}. and the move limits L1Xu = L1XL [see (3.52)] of 2.0. 1.0. and 0.5 for the fIrst. the second and the third iteration. respectively. Show graphically the linearized feasible region for each of the iterations.

3 Approximation Concepts

176

C.L.

I

p= 10

4Xl

C

D

(5)

rl Xl

iXI

(4)

100

(I)

(3)

A

B I r2

4Xl

I. Fig.

100

t

.1

·1

100

c

3.14.

3.4 Assume the continuous beam shown in Fig. 3.15. The five design variables are Xi = (EII!)i (i = 1•...• 5). Assuming the initial design X* = 1.0 • the following results are given from the initial analysis

28

(K*)-l =_1 [ -7.5 209 2 -0.5

-7.5

2

30

-8

-8 2

30

-7.5

-7.5

28

~5l

r.

0.10526}

={ 0.07895 0.07895 0.10526

Assume a change in the initial design variables such that the modified design is given by (X)T = {5.0. 5.2.4.8.5.2. 5.0} Evaluate the modified displacements by the following methods. assuming only two terms (rl and r~ for each of the series: the Taylor series (3.69) ; b. the binomial series (3.74) ; c. the scaled binomial series [(3.91). (3.92). (3.95). and (3.96)] ; d. the combined series expansion (3.74) and reduced basis [(3.82) through (3.86)] ;

Q.

Exercises

177

Evaluate the errors (3.106) in the modified analysis equations obtained by each of the four methods. 3.5 Assume that X·· is a point on the constraint surface rj = rX. Show that for any point X· = J,1X·· on the scaling line through X·*, where r* = (lIJ,1)r**, the linearized consttaint is parallel to the consttaint tangent hyperplane at X** and can be expressed as " ':lo ** "'" ar· ~_J_X;

;=1

ax;

=J,1(J,1-2)rjU

3.6 Evaluate the modified displacements of exercise 3.4 along the line X =X*+a AX*

for a = 0.2S, O.SO, 0.7S, 1.0. Draw the displacements versus a as obtained by each of the four approximate methods. Compare the results with the exact solution. 3.7 Consider the problem formulated in exercise I.S. Assuming the initial design (X*)T = {SO, SO}, introduce the stress approximations in sections B. C under the loadP1 by:

a. the frrst-order approximations of stresses (3.1S2); b. the fJrst-order approximations of forces (3.1S3). Compare the results with those obtained by exact analyses for X[ X~

= {40, 4S} and

={30,40}.

a Fig.

I (>... 2i

3.15.

3 ,...,...

2

2

"

2i

3

2i

4 ,...,... 4

2i

5

~

4 Design Procedures

In establishing an optimal design procedure, the following steps should be taken:

-

-

-

-

The design problem is formulated. The design variables are chosen, the constraints and the objective function are defined and an analysis model is introduced. This step is of crucial importance for the solution process. A poor problem formulation might lead to incorrect results and/or prohibitive computational cost. Various formulations have been discussed in Chap. 1. The optimization method is selected. One of the methods presented in Chap. 2 might be suitable for the solution process. In general, the reliability and ease of use of the method are more important than its computational efficiency. Since most of the cost of optimization is associated with the exact analysis and derivative calculations, efficiency of the method used to solve the problem is not a major consideration in choosing the method. Approximations are introduced. It has been noted that approximations are essential in most practical design problems. Using linking and basis reduction methods, it is possible to reduce the number of independent design variables. Scaling of variables, constraint normalization and constraint deletion techniques (Sect. 1.3.4) are all intended to improve the solution efficiency. Approximate behavior models, discussed in Chap. 3, are often necessary in order to reduce the number of exact analyses during the solution process. A design procedure is established. The problem formulation, the chosen optimization method and the approximation concepts are integrated to introduce an effective solution strategy. In this chapter, various design procedures demonstrate the solution methodology.

Section 4.1 deals with linear programming formulations of optimal design problems. Both plastic design and elastic design formulations are presented. Feasible design considerations and methods intended to find improved feasible designs are discussed in Sect. 4.2. Optimality criteria procedures are developed in Sect.4.3, multilevel optimal design is presented in Sect. 4.4, and optimal design of controlled structures is demonstrated in Sect. 4.5. The presented approaches combine various concepts of structural optimization and might involve multistage design procedures. Optimization of the structural layout is discussed in Sects. 4.6 through 4.8; geometrical optimization (Sect. 4.6), topological optimization (Sect 4.7) and interactive layout optimization (Sect. 4.8) are presented, and the benefits as well as the difficulties involved in this class of optimization are demonstrated.

180

4 Design Procedures

4.1 Linear Programming Formulations 4.1.1 Plastic Design Assuming the plastic analysis formulation, discussed in Sect 1.2.2, the optimal design problem can be cast in a linear programming (LP) form under the following assumptions [94]: 1. Equilibrium conditions are referred to the undeformed geometry. 2. The loads applied to the structure are assumed to increase proportionally. 3. Constraints are related only to yield conditions and to design considerations. In trusses, it is required that the yield stress will not be exceeded in any member under any load condition. In frames, the magnitude of the bending moment in each cross section can at most be equal to the plastic moment. Linear relations between plastic moments as well as limitations on the plastic moments may be considered in the problem formulation 4. The objective function represents the weight and can be expressed in a linear combination of the cross-sectional variables. Cross-sectional areas of truss members and plastic moments of frame members are chosen as design variables. (It has been found that the error involved in the latter assumption is of the order of 1%.) Designs that satisfy the conditions of equilibrium and yield are safe in the sense that the load factor at plastic collapse must be greater than or equal to the required load factor. The present formulation is based on the static (lower bound) theorem of limit analysis, which states that the equilibrium conditions and yield conditions represent a necessary and sufficient condition for the design to be capable of carrying the given loads. Compatibility requirements are not considered in this formulation of plastic design. Truss Structures. The plastic design problem of trusses can be stated as the following LP: find the cross-sectional areas X and the members' forces A such that

Z=11X

~

min

(J~X ~ A ~ (J~X

(yield conditions)

C A=Ru

(equilibrium)

(4.1)

This formulation is similar to that of (1.69). The elements of C depend on the undeformed geometry of the truss (members' direction cosines), thus they are constant during optimization of cross sections; Ru is a vector describing the ultimate load (representing constant extemalloads or self-weight expressed in linear terms of X), considering a given load factor; (J~ is a diagonal matrix of lower bounds on stresses (compressive limiting stresses, negative values); and (J~ is a diagonal matrix of upper bounds on stresses (tensile yield stresses). The number of

4.1 Linear Programming Fonnulations

18 1

independent equations of equilibrium nE relating the unknown forces A is equal to n - nR. where n is the number of members and nR is the number of redundants (the

degree of redundancy). In the general case of nL loading conditions, A and Ru will become matrices of nL columns. The number of variables in this problem is (nL+l)n, the number of equalities is nL n E and the number of inequality constraints is 2nL n. It has been noted in Sect. 1.4.2 that the formulations (1.68) and (1.69) are equivalent. Thus, the problem (4.1) can be stated [similar to (1.68)] as: find X and N such that

z = t1){ ~ min

(4.2)

(J~X ~ Ap +ANN ~ (J~X

The number of variables in this LP formulation [n+(nL nR)] is smaller than the number of variables in formulation (4.1). In addition, the equality constraints have been eliminated. A major complicating factor in the plastic design of trusses by LP is the variability of the compressive limiting stresses. Several approaches have been proposed to treat buckling in compression members. The ultimate stress in member i which buckles is (4.3) in which (JEi = Euler buckling stress, E = modulus of elasticity, li = unbraced length of member, and rGi =critical radius of gyration. In general, the stress in each member must satisfy (4.4)

However, if the Euler buckling stresses (JEi were entered into the yield conditions, a nonlinear programming problem would result. Russell and Reinschmidt [116] found that when evaluating the results of the LP problem, it is useful to ignore the computed member areas and design for the computed member forces, which constitute a force system in equilibrium. The compressive limiting stresses are then modified and the LP problem is solved repeatedly until convergence. Transformation of Variables. It has been noted in Sect. 2.3.2 that all variables in a standard LP formulation are assumed to be nonnegative. Since the member forces are not restricted to nonnegative values, transformation of variables may be used to account for the unrestricted variables. This can be done in several ways, briefly described herein. Considering the formulation (4.1), the member forces A can be represented by the difference of two vectors of nonnegative variables A' and A by II

A=A'_A"

(4.5)

182

4 Design Procedures

Substituting (4.5) into (4.1) the number of variables in the LP problem becomes

(2nL+1)n. However, the advantage of this approach is that only nL n yield stress conditions are required instead of 2nL n in the original problem. To see this

possibility, we may rewrite the yield conditions in terms of A' and An and obtain the following n inequalities for each loading condition (4.6)

A procedure can be established to guarantee that for any member i either the constraint Xj ~ A'j Icrf or Xj ~ _Anj IcrT will be considered, depending on the sign of the force. An alternative approach is to use an axis transfer of the form A = A' - Ao 1.0

(4.7)

where A' is a vector of nonnegative variables, Ao is a nonnegative scalar variable which is invariant for all forces and loading conditions, and all the elements of the vector 1.0 are equal to 1.0. That is, the number of variables is increased by one. Another possibility is to consider the transformation of variables A' = A - AL

(4.8)

where AL are lower bounds on A, chosen as constants or linear functions of X. Assuming (4.9) substituting (4.8) and (4.9) into the yield stress conditions in (4.1) and rearranging yields (4.10)

The variables A' and the inequalities (4.10) can be used instead of A and the original yield stress conditions. Since all variables in a standard LP formulation are assumed to be nonnegative, only the right-hand side of the inequalities (4.10) must be considered. Finally, we may introduce a vector AU of positive constants or linear functions of X which have numerical values larger than the expected values of A. Defming the new variables A' and replacing the original variables A by A'=A+AU

(4.11)

then all variables in the LP problem will be nonnegative. Choosing the upper bounds (4.12)

4.1 Linear Programming Formulations

183

the original yield stress constraints become

(4.13) It can be noted that in cases where cr~ = -cr~, the constraints (4.10) and (4.13) become identical. Considering the formulation (4.2), similar transformations can be applied. The transformation (4.7) becomes

(4.14) and the number of variables is increased by one. Alternatively, the transformations (4.8) and (4.11), respectively, become N' =N _NL

(4.15)

N'=N +Nu

(4.16)

The elements of NL and NU can be determined in a manner similar to that of (4.9) and (4.12), respectively. Example 4.1. Consider the three-bar truss shown in Fig. 4.1 and subjected to three alternative loadings. The limiting stresses and the three loadings are

PI =20.0

P2 = 30.0

P3 = 40.0

Assuming the formulation (4.1), the problem is to find the cross-sectional areas XT = {Xl> X2 • X3 } and the members force matrix A=[Al> A2 , A 3] (corresponding to the three loading conditions) such that Z = {141.4. 100.0. 141.4} X -+ min

-20] 20

(b)

cr~ [X. X. X] ~ A ~ cr~ [X. X. X]

(c)

[1 0 1 .J2

-1] [40 0 1 A = 40 30.J2

(a)

Alternatively, choosing the forces in member 2 as redundants NT = {NI' N 2 , N 3 } (corresponding to the three loading conditions), the LP plastic design problem (4.2) is to find the cross-sectional areas XT = {Xl. X2• X3 } and the redundants N such that

184

4 Design Procedures

100

~ I•

Xl

PI

100

= 20

P2 = 30

P3 = 40

Fig. 4.1. Three-bar truss, three-loading conditions.

Z = {141.4, 100.0, 141.4} X ~ min 0

(d)

2l.21

(J~ [X, X, X] ~ [ 0

0

20 21.21

(e)

The optimal LP solution is X

*T

z* = 1206

= {5.SS ,0.75, 2.12}

representing a statically indetenninate structure.

I'

360

360

(I)

(2)

~I

(6)

100

Fig. 4.2. Ten-bar truss.

360

100

(j)

4.1 Linear Programming Formulations

185

Example 4.2. The ten-bar truss shown in Fig. 4.2 is designed to resist a single ultimate loading. Limiting stresses of a L = -25, aU = 25 are assumed for all members. Choosing the forces in members 7 and 10 as redundants N1 and N 2• respectively, the LP plastic design problem can be stated in the form of (4.2) as follows: fmd the cross-sectional areas XT={X 1• X2 • •••• XlO } and the redundants NT={N1• N 2 } such that

L

Xj

+ 1.414 x 360

j=1

-~{~: }~ XlO

L 10

6

Z = 360

Xj

~ min

(a)

j=1

-0.707 300 0 0 0 -0.707 -100 -0.707 0 -100 0 -0.707 100 -0.707 -0.707 + 0 0 -0.707 0 1.0 0 -282.8 1.0 0 141.4 0 1.0 0 0 1.0

{::}s~f: }

(b)

XlO

The optimal solution is X*T = {8.0, 0, 8.0,4.0,0, 0, 5.66, 5.66, 5.66, O} N*T = {141.4, O}

Z' = 15,840

(c)

That is, the unnecessary members 2,5,6 and 10 have been eliminated by the LP. In order to compare the elastic and plastic optimal designs, limiting stresses and loadings are assumed to be the same in both cases. Since the optimal solution represents a statically determinate structure, the plastic and elastic optimal designs are identical. If lower bound constraints on X are considered so that 0.1 ~ X, the resulting optimal plastic design is X*T

= {8.0, 0.1,8.0,3.9,0.1,0.1,5.66,5.66,5.51, 0.14}

Z* = 15,910

(d)

The optimal elastic design in this case is X*T

Z*

= {7.94, 0.1,8.06,3.94,0.1,0.1,5.75,5.57,5.57, 0.1}

=15,934

(e)

Frame Structures. The frame optimal design problem can be stated as the following LP, similar to (4.1) : find Mpl and M such that

186

4 Design Procedures

Z=lMpl~min (4.17) C M=R,. where M is a vector of moments Mj,j = 1,... , I, in statically admissible moment field corresponding to collapse, and I is the number of critical sections where plastic hinges may form. L is a linking matrix of 0, 1 elements. If Lji = 0, the ith plastic moment does not govern section j. The inequality constraints require that the admissible moments nowhere exceed the plastic moment capacities of the members. Strictly, these constraints must apply at any point in the structure, but in practice it is necessary to confine their application to I possible hinge positions. For prismatic members this can be achieved by considering moments only at the ends of members and at the position of maximum moment in loaded elements. The number of variables in the LP problem (4.17) is I+I (I being the number of plastic moments). The number of independent equations of equilibrium is nE = I - nR • The frame design problem (4.17) can be stated as the following equivalent LP, similar to (4.2) : find Mpl and N such that Z = iT Mpl ~ min -L

Mpl ~ Mp

+ MNN

~

L

Mpl

(4.18)

in which Mp is the vector of moments due to the applied loading, and MN is the matrix of moments due to unit value of redundants, both computed in the primary structure. The number of variables in this LP formulation is l+nR and the number of inequality constraints is 21. Though most practical designs for regular frames will tend not to have reverse taper in the column members, we may consider constraints of the form (4.19) in which M pl. is plastic moment of columns at the ith story. We may consider also linking constraints to ensure a desired ratio between Mpl. and Mpl'+1 (4.20) where ~ is a given constant. The above linear constraints can be included in the LP formulation. Example 4.3. To illustrate the LP formulation (4.17) consider the frame shown in Fig. 4.3 with 14 critical cross sections and 4 groups of plastic moments. Bending moments with tension at inner fibers of columns or at lower fibers of

4.1 Linear Programming Formulations

187

beams are assumed to be positive. Results for this example have been presented by Cohn et al. [18]. The optimal design problem is to find M~l = (M pll' M p l 2' M pl 3' M pl.) and T

M = (MI' M 2•

•.••

M 14 ) such that

Z=

lTMpl

~ min.

(a)

6 0 0 0 0 0 0 0 0 0 0 0 -1 2 -1 12 0 0 0 0 -1 2 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 1 -1 0 0 0 3 -1 1 1 -1 0 0 0 0 0 0 0 0 0 0 9 M= 0 0 0 0 0 0 0 0 -1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 1 0 0 -1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 1 0 0 1 0 0 0 0 C

Pul

(b)

Ru

-L

Mpl

:5:M:5:L Mpl

(c)

in which IT

= (6l. 4l. 6l. 4l)

1 1 1 1 0 0 0 L= 0 0 0 0 0 0 0

0 0 0 0 1 1 1

0 0 0 0 0 0 0

0 0 0 0 0 0 0 1 1 1 1 0 0 0

0 0 0 0 0 0 0 0 0 0 0 1 1 1

(d)

188

4 Design Procedures

3Pu

Pu -

t

@

CD

@

M,'J

® M,/4

@ M,ll

6Pu

~

CD CD CD

2Pu -----+-

MplJ

MplJ

CD m 7?

I I

I' Fig. 4.3.

CD

2l

I,

•

I 13l ,I

0) @ M,a 8)

0

-t

2l

'77 ~ I

.. !

~

-!

I 13l I

-'

Frame example.

Using (4.8) with A, A' and AL replaced by M, M' and -L M pl' respectively, the resulting number of variables is 18, the number of equality constraints is 8 and the number of inequalities is only 14. The optimal solution is

Fig. 4.4. Bending-moment distribution for optimal frame.

4.1 Linear Programming Formulations

M~,

=P"t

189

(3, 4.5, 1.5, 1.5)

M'T = P"t (0, 3, 6, 0, 6, 9, 0, 0, 0, 3, 0, 0, 3, O)

z= 51P. t2

"

and the corresponding bending moments M are (Fig. 4.4)

MT =P ul (-3,0,3, -3, 1.5,4.5, -4.5, -1.5, -1.5, 1.5, -1.5, -1.5, 1.5, -1.5)

To illustrate the LP formulation (4.18), assume the six redundants as shown in Fig. 4.5a. The optimal design problem is: find M~ = (M pll' M plz' M pl,' M pl 4) T

and N = (N •• N20 N 3• N4• Ns.N61 such that Z= tTMpl ~ min 1 0 0 0 0 0 0 -1.5 0.5 -0.5 1 0.5 0.5 0 0 0 1 0 0 0 0 -10.5 -0.5 0.5 1 0.5 0.5 0 -1.5 0.5 -0.5 1 -0.5 0.5 0 0 0 0 1 0 0 0 -10.5 -0.5 0.5 1 0.5 -0.5 0 -LMpl '5.P"t N'5.LMpt + 1 0 0 0 0 0 0 -1.5 0 0 0.5 -0.5 1 0 0 0 0 0 0 1.0 0 -4.5 0 0 -0.5 0.5 1 0 -1.5 0 0 0.5 -0.5 1 0 1 0 0 0 0 0 0 -4.5 0 0 0 -0.5 0.5 1 Mp

(e)

(j)

MN

The physical interpretation of the elements of Mp and MN is shown in Figs. 4.5b and 4.5c. The number of variables in this formulation is 10 and the number of inequalities is 28. Using the transformation (4.16), with NU defined as (NU)T=put (10,10, 10, 10, 10, 1O)

the resulting nonnegative variables are Mpl and N'. The optimal solution is

190

4 Design Procedures

3p',

N6 ~r""

N4

~

r

l.5Pu l

p.,--

t

4.5Pu i

N)

"""''\r''''

Nl~

)NS

~

)N2 (b)

(a)

05 0.5 . h--::::::OO<~~-T""1

0.5

rT--~O-::::::-""""

1.0

0.5

1.0 r--r---<>--~-'11.0

1.0

Ns

= 1.0 (c)

Fig. 4.5. a. Redundants, b. Coefficients of M p , moments due to loading, c. Coefficients of MN , moments due to unit value of redundants.

4.1 Linear Programming Formulations

=P1.l (3, 4.5, 1.5, 1.5) N,T = Put (7, 13, 14.5, 8.5, 11.5,

191

M~l

11.5)

Z=51Put 2

and the corresponding redundants N are computed from (4.16) NT =Put (-3, 3,4.5, -l.5, l.5, l.5)

4.1.2 Elastic Design It has been noted that neglecting the compatibility conditions (l.22), assuming the relations (l.67) and considering only stress and side constraints, the optimal design problem can be stated in the LP form (l.68): fmd X and N such that

Z =(l'X -+ min (4.21)

This formulation is similar to that of (4.2), but the loadings and the limiting stresses are different; while service loads and allowable stresses are considered in (4.21), ultimate loads and yield stresses are assumed in (4.2). The LP formulation can be preserved, under certain circumstances, even if displacement constraints are considered. This is the case, for example, in continuous beams when the displacements are given in the form of (l.33). We note first that the internal forces A are linear functions of N. Also, a statically equivalent internal force system corresponding to the virtual loads may be selected so that the forces

A;9

[see (l.31)] are constant. Thus, the elements

expressed as

Iij = Iijo +

L

Iijk Nk

Tij

can be (4.22)

k

where T ijO and T ijk are constant coefficients. From (l.33) and (4.22), any displacementDi can be expressed as

(4.23)

Assuming a continuous beam and choosing the bending moments over the supports of the member under consideration as redundants Nil N 2 , then the displacementDi in the hth member becomes

192

4 Design Procedures D. = TiIIO I

+ Till! N! + TiII2 N2

(4.24)

Xit

Substituting this equation into the displacement constraint D; ~ rearranging gives the linear expression

Df

and

(4.25) which can be added to the LP problem (4.21). Effect or Compatibility. The LP formulation (4.21) is most suitable for optimal design of trusses. If no lower bounds on cross-sectional areas X are considered, the LP method has the ability to make unnecessary members to vanish from the structure. This topic of optimizing the topology will be discussed later in Sect. 4.7. If the resulting optimal solution represents a statically determinate structure, the compatibility conditions are always satisfied and the LP solution is the final optimum. In cases where the optimal solution represents a statically indeterminate structure the compatibility conditions may not be satisfied, and the LP solution is not the final optimum. To illustrate the effect of compatibility conditions on the optimum, solutions of two formulations have been compared [70]: the LP, where compatibility conditions are neglected and the NLP, where the latter conditions are considered. In general, the optimal LP solution, Z;;, will be at least as good as the optimal NLP solution Z~, that is (4.26) Therefore, Z;; can be viewed as a lower bound on ZNLP' It has been shown that under certain circumstances the optimal solutions of the two problems are identical [70]. In general, three different situations might be encountered : Q.

The usual case where

Z;'
(4.27)

b. The two optimal objective functions are identical, but the LP problem possesses multiple optimal force distributions, including the one corresponding to the optimal NLP force distribution, that is

(4.28)

4.1 Linear Programming Formulations

193

This situation might occur in cases where the LP objective function contours are parallel to the boundary of the feasible region. c. The two optimal solutions are identical

z:.r, =z~

(4.29)

N~=N~LP

That is, the compatibility conditions do not affect the optimal solution. This situation might occur for certain geometries and loading conditions. Several procedures may be used in case a to modify the LP solution such that compatibility requirements are satisfied. Denoting the strain in the redundant member i under the load condition q as Eiq' the corresponding redundant forces as Niq' and the implied areas as Xiq' then

N· X. =--.!!L "I

EE.

q= 1. 2•...• nL

(4.30)

"I

where nL is the number of loading conditions. Obviously the value of Xiq would be unique for all q if the design had been fully compatible. It has been shown [24] that the conditions of compatibility are equivalent to the conditions that lead toward the uniqueness of X iq for all q. That is, in a compatible design the following relations must be satisfied

q

=2, 3,..., nL

(4.31)

Equation (4.31) produces (nL - 1) equality constraints for the uniqueness of the area of each redundant member i. Based on (4.30), these constraints can be expressed as

q = 2, 3,... , nL

(4.32)

Assume that the variables of (4.32) are N iq and N i .q.l while Eiq and Ei,q.l are taken as known constants whose approximate magnitudes are supplied by elastic analysis of the optimal LP design X~. Thus (4.32) are linear and suitable for use together with the original LP. After the initial solution of the LP, subsequent iterations involve both the LP and (4.32) simultaneously, resulting in a series of modified designs X~ whose compatibility violations successively diminish until the final optimal design is reached. This procedure is quite efficient [24] but it may not converge to the true optimum in some cases [61]. An alternative iterative procedure to consider compatibility, buckling, and discrete rolled steel sections is as follows [111]: 1. Specify all given parameters and the initial values for the allowable compression stresses, to start the iteration.

194

4 Design Procedures

2. Solve the LP problem (4.21) for the member areas and forces, satisfying the equilibrium conditions, the stress constraints, and the area bounds. 3. For each member, using the computed member forces from step 2, select the minimum section (from the table of available rolled steel sections) which satisfies the stress constraints. The member areas computed by the LP are ignored; the use of the computed member forces gives improved convergence because the correct allowable compression stress can be computed for each candidate section, using code formulas and the tabulated radius of gyration. The LP is used to derive a distribution of forces in the optimal structure; the crosssectional areas determined by the LP are only an intermediate step. 4. If the process has converged, go to step 5. If it has not converged, replace the lower stress bound for each member in the LP by the allowable compression stress computed for the section selected in step 3, and return to step 2. In the case of discrete sections and code-allowable compression stresses, convergence is obtained when for each member the same section is selected in two successive cycles. 5. If the resulting structure is statically determinate, the design is complete. If the design is statically indeterminate, add the compatibility conditions, and obtain an exact elastic analysis. 6. Select new sections as necessary based on the code requirements and the member forces computed in step 5. If the convergence criterion is met, the design is complete; if not, return to step 5. As there is no explicit objective function in steps 5 and 6, the resulting design will be fully stressed but not necessarily minimum weight. Example 4.4. Consider again the three-bar truss shown in Fig. 4.1 and subjected to three loading conditions. In order to compare elastic and plastic optimal designs, limiting stresses and loadings are assumed to be the same in both cases. Choosing the forces in member 2 as redundants NT = {N I' N z, N 3 } (corresponding to the three loading conditions), the resulting LP problem is given by (d), (e) in example 4.1. The plastic and the elastic design formulations are similar, however, in the latter formulation compatibility conditions must be considered. The optimal LP solution is X

"T

= {5.88 ,0.75, 2.12}

z" = 1206

(a)

Analysis of this structure shows that the elastic forces NE do not match the optimal forces N" determined by LP. The strains £2q of the redundant member 2, are computed for all loading conditions (q = 1,2,3) and the LP is now solved with the added constraints (4.32). These requirements are modified after each cycle, and the global optimum, reached after six cycles, is [24] "T

X = {8.0, 1.5, O}

z· = 1281

(b)

It can be seen that the method has the ability to eliminate unnecessary members of the structure. Structural analyses are performed only to update the terms in (4.32). Solving the NLP problem by a numerical search technique, the solution is [127]

4.1 Linear Programming Formulations

X

-1

-=1597

= {7.02. 2.14. 2.75}

Z

195 (c)

which is not the optimum. That is. the true optimal solution could not be reached by the general NLP formulation. This phenomenon of singular solutions in optimization of structural topologies will be discussed later in Sect. 4.7.3.

or Prestressing. Prestressing by 'lack of fit' [74] may be applied to maintain compatibility at the optimum of the LP problem (4.21). To illustrate the effect of prestressing on the optimal solution assume the general case where the Effect

optimal LP solution X~, N~ represents a statically indeterminate structure (SIS). Elastic analysis of the optimal design will provide the corresponding elastic force distribution N~. If the compatibility conditions are not satisfied at the LP optimum, then N~ *- N~ and a set of prestressing forces. N~. given by

N~ =N~-N~

(4.33)

may be applied to maintain compatibility. In cases of a single loading condition the forces N~ can easily be determined by (4.33). For multiple loading conditions. different values of N~ might be obtained for the various loadings. i.e. at least for some i.j (4.34) i,j= 1•...• nL Here. i, j = loading conditions; and nL = the number of loadings. It should be noted. however. that the assumption of applying different sets of prestressing forces for various loading conditions is not practical. In cases where the LP optimal design represents a statically determinate structure (SDS). the redundant forces equal to zero and no prestressing forces are required to maintain compatibility. i.e. j

= 1•...• nL

(4.35)

This is the case also in SIS if the conditions (4.28) or (4.29) hold for all loading conditions. In summary. the various possible cases of LP solutions are characterized as follows:

a. The LP solution is an SDS. no prestressing forces are required [see (4.35)]. h. The LP solution is an SIS and compatibility conditions are satisfied at the optimum. The LP and NLP solutions are identical. no prestressing forces are required [see (4.28) or (4.29)]. c. The LP solution is an SIS and a single set of prestressing forces is required to maintain compatibility at the optimum [see (4.33)]. d. The LP solution is an SIS and different sets of prestressing forces are required for the various loading conditions [see (4.34)].

196

4 Design Procedures

100

(a)

(b)

Fig. 4.6. Eleven-bar truss

a. Initial design, b. Optimal design.

Example 4.5. The eleven-bar truss shown in Fig. 4.6a is subjected to two loading conditions (P l and P z, respectively). The allowable stresses for all members are aU -oL 20.0 and the assumed loads are P 1 P z 10.0. Conditions of symmetry are considered to reduce the number of variables and constraints. No lower bounds are imposed on cross sections, that is XL = O. Two selections of redundants have been assumed: the forces in elements 1 and i. and the force in element 4. The results are summarized in Tables 4.1 and 4.2, and the optimal design is shown in Fig. 4.6b [74]. The optimal objective function value is Z' = 125. It can be observed that the optimal structure is statically indeterminate, and a single set of prestressing forces can be chosen to maintain compatibility at the optimum for the two loading conditions [see (4.33)].

= =

= =

Table 4.1. Optimal LP solution, eleven-bar truss. Element 1 2 3 4 5 6

Load 1 3.54 -3.54 0 5.00 0 0

A~I

X~I

Load 2 -3.54 3.54 0 5.00 0 0

0.177 0.177 0 0.250 0 0

Table 4.1. Prestressing forces, eleven-bar truss. Redundant elements 1 and I 4

N*LP 3.54 5.00

Loading 1

N*E

N*P

N*LP

4.25 4.00

-0.71 1.00

-3.54 5.00

Loading 2

N*E

N;

-2.83 4.00

-0.71 1.00

4.2 Feasible-Design Procedures

197

4.2 Feasible-Design Procedures 4.2.1 General Considerations In many practical design problems the object is to find an improved feasible design rather than the theoretical optimum. Such an approach is particularly useful in cases where the objective function is not sensitive to changes in the design variables near the optimum. Two classes of optimization methods may be considered from this point of view [75]:

a. Exterior (or infeasible) methods. based on search for the optimum outside the feasible region. where all intermediate solutions lie in the infeasible region and converge to the optimum from the outside. A major shortcoming of these methods is that the search cannot be stopped with a feasible solution before the optimum is reached. Examples for this class of optimization methods include exterior penalty-function techniques and sequential linear programming. Such methods are usually not suitable for evaluating improved feasible designs. Intermediate solutions obtained by exterior methods may often be viewed as a lower bound on the optimum. b. Interior (or feasible) methods. where all intermediate solutions lie in the feasible region and converge to the optimum from the interior side of the acceptable domain. The advantage of these methods is that we may stop the search at any time and end up with an improved feasible design. Moreover. for some methods the constraints become critical only near the end of the solution process; thus instead of taking the optimal design we can choose a suboptimal but less critical design. Another advantage is that a near optimal solution can be achieved with a reduced computational effort. Examples for this class of optimization methods include interior penalty-function techniques and the method of feasible directions. Intermediate solutions obtained by feasible methods may be viewed as an upper bound on the optimum. Introducing Feasible Designs. One problem in using interior optimization methods is that it might be difficult to find an initial feasible design. particularly in problems with a narrow feasible region. Several methods can be used for this purpose. some of which are discussed in this section. Such methods are useful also in identifying situations where no feasible solutions exist. Assume the inequality constraints (2.111) j = 1•.•••

n,

(4.36)

If a given infeasible design X· violates p of the constraints (4.36). those may be arranged as the fU'St p constraints such that (4.37) The largest 8p is selected as the objective function for the following problem [33]

198

4 Design Procedures

j

=1..... p-l

(4.38)

j = p+1. ... ,ng This problem can be solved by an interior method (Le. the interior penalty-function method). The search is terminated as soon as gp(X) ~ O. A new test for feasibility is performed and the process is repeated until all the constraints are satisfied. Another method to achieve feasible designs is based on the KS function [87]

KS =

.!.In[t p

exp(pgj)]

(4.39)

j=!

It is a continuous and differentiable function that has the property of approximating the maximum constraint by a conservative envelope function (4.40) The KS function is positive when at least for one j. gj> O. The factor p is controlled by the user. For large values of p the function becomes closer to the constraints envelope. However. it may lose numerical differentiability by forming sharp "knees" at the constraint intersections. If the constraint tolerance is e (i.e.• for Igi ~ e the constraint is assumed to be active). it is recommended to choose p by In(ng ) (4.41) p=--

e

The problem of satisfying the constraints (4.36) is thus replaced by the unconstrained problem of minimizing K S . The latter function serves as a convenient single measure of the degree of constraint violation. The direction finding problem. used in the feasible directions method (Sect. 2.4.2) can be applied to find a search direction pointing toward the feasible region by a simple modification [138]. Deleting the usability requirement (2.136) and limiting the 9j for violated constraints [see (2.138)]. a direction is found which will be away from the violated constraints and toward the feasible region. It is instructive to note that in linear programming problems. the simplex method discussed in Sect. 2.3.3 always will find a feasible solution. if one exists. in a simple manner. Scaling of A Design. In various problems with cross-sectional design variables. a given infeasible design X· can be converted into a feasible one by the scaling procedure. Due to its simplicity and efficiency. this procedure is most

4.2 Feasible-Design Procedures

199

useful in problems where it is necessary to modify different infeasible designs into improved feasible ones during the solution process. Consider again the displacement, stress and side constraints (1.56)

(4.42) XL SXSX U

A design line through the given design point X* is defined by X= jlX*

(4.43)

in which J1 is a variable scaling multiplier (J1 > 0). Equation (4.43) defines the scaling of X*. Under certain circumstances, analysis of a scaled design is trivial, provided the initial design X* has already been analyzed. In the presentation that follows, the displacement method of analysis is considered. A similar approach can be applied for the force method, or other analysis methods. The scaling procedure will be used in several design procedures throughout this chapter. Assume that the elements of R in (1.26) are independent of X and the elements of the stiffness matrix are linear functions of the design variables, that is,

L XiK ? II

K=

i=1

(4.44)

in which K? are matrices of constant elements. The displacements for any design on the line defmed by (4.43) are then given by

1 *) r(jlX * ) = -r(X

(4.45)

J1

The relationship (4.44) is typical for various structures such as trusses, where Xi are the cross-sectional areas, or beams, where Xi are the moments of inertia. To find the resulting stresses, the scaled displacements are substituted into the stress-displacement relations. Assuming that the elements of Sin (1.29) (4.46)

cr=Sr

are constant, then cr(jlX *) = -1 cr(X *)

(4.47)

J1

In cases where the above conditions for stresses do not hold, simplified expressions can be obtained. For frame elements, cr can be expressed as an explicit function of J1 under certain assumptions [64].

200

4 Design Procedures

The significance of the relations (4.45) and (4.47) is that the exact displacements and stresses at any point along the design line (4.43) can readily be determined without solving the set of implicit equations (1.26). The scaling procedure can be used to achieve the best feasible design along a given design line, as will be demonsttate
rL~!r~rU ~

OL

~ !oo ~ OU

(4.48)

~

XL

~~X·~

XU

or, alternatively (4.49)

where

U•

rj0 /rj

II

r-

L}

rj° /rj

LO U ° L = max{ 0·/0· .0,/0, . . I.J

"

"

x!- / X~ I

II U = min(X~ ,.... . , / X~) , I

(4.50)

I

(4.51)

The elements of rL and oLin (4.50) are assumed to be negative. That is, all expressions in this equation are positive. From (4.49), the best feasible design along the design line (4.43) can readily be determined by

Fig. 4.7. Scaling of a design.

4.2 Feasible-Design Procedures

201

It can be noted that if JlL> JlU, no feasible designs exist along the design line. That is, a necessary and sufficient condition for a feasible design along this line is (4.53) For some standard beam cross sections the stiffness matrix elements are functions ofaXt , in which Xi are the cross-sectional areas and a and b are given constants. Choosing (4.54) as design variables, then the displacements can be scaled by (4.45). A typical relationship for a general frame structure is given by

L [XiKFi+d(Xi)KNd II

K=

(4.55)

i=l

in which KFi and KNi are matrices of constant elements, representing the contribution of flexural deformations and axial deformations, respectively. Analysis of the scaled design by (4.45) is possible if (4.56) where c is a constant. In cases where the relationship (4.56) does not hold, linear approximations may be assumed. Another approach, which might prove useful in cases where the conditions (4.45) cannot be used and the displacement derivatives are available, is to assume the fIrst-order approximations of rj in terms of Jl to obtain (4.57) Considering the constraint (4.58) then from (4.57) and (4.58) we have

)· L II

r· +

i=l

~.

urj • • )
ax•.......· · - }

(4.59)

202

4 Design Procedures

or, after rearranging,

J.I.>I+

-

r·U -r·" 1

1

ar; "

~ L.J - X ·

aXj

j=1

(4.60)

,

Similarly, for the reciprocal approximation (3.36) (4.61) we get, instead of (4.60),

L aXar~ X." II

_l

II

>

j

j=1

t""-

,

a" + L -X· lax.' II

r·"

1

-r·U

rj

j=1

(4.62)

"

'

Equations (4.60) or (4.62) may be used instead of (4.49) and (4.50) to find the best feasible design. ' Conditions of Feasibility. For some problems it is difficult to determine a priori whether feasible solutions exist or not. However, it might be possible to show for a specific problem that there is no feasible region. This is the case, for example, if the constraints (4.36) can be expressed in the form (4.63) where C is a vector of preassigned parameters. Based on the transitivity property, necessary conditions for a feasible solution are (4.64) It should be noted that the conditions (4.64) are independent of the design variables value and can be checked before solving the complete problem. Furthermore, in cases where the conditions (4.64) are not met it might be possible to evaluate modified values of C such that these conditions will be satisfied, as illustrated in the next subsection. In cases where the problem can be stated in a linear programming form it is possible to identify situations where no feasible solutions exist. It can be shown that in such cases the dual optimal solution is unbounded.

4.2 Feasible-Design Procedures

203

Modification of Preassigned Parameters. A given infeasible design X· can be converted into a feasible one by considering additional variables. For example, prestressing by 'lack of fit' or passive control devices may be used for this purpose. The latter possibility will be demonstrated in Sect. 4.5.1. Alternatively, a feasible design can be achieved by mod~fying some of the following preassigned parameters C [75]: a. oL, (Ju (functions of material properties). b. rL, r U, XL, XU (constraints on r and X). c. R (loads).

Assuming the displacement method formulation [see (1.26) and (1.29)], the constraints (4.42) for the given design X· can be expressed as

(4.65)

where X·, K· and S· are constant. It can be observed that all the constraints are linear functions of the preassigned parameters. Define a linear objective function (4.66) where b is a vector of constants and C is the vector of preassigned parameters to be modified. It is then possible to find optimal values of the preassigned parameters by solving the linear programming problem (4.65) and (4.66). A direct solution can often be achieved, as will be illustrated in the following example. Example 4.6. Consider the typical prestressed concrete member of a uniform cross section continuous beam shown in Fig. 4.8 (all dimensions are in tons and meters). A parabolic tendon is assumed with Yl' Y2 being the tendon's coordinate variables, and P is the variable prestressing force. Two loading conditions have been considered:

J Fig. 4.8. Prestressed concrete beam.

204

4 Design Procedures

- Dead Load DL = 2.0 and P; - Total Load TL = 3.0 and 0.8P;

crf = -IS00. crf = 100 . crf = -1200. crlf = 0 .

The modulus of elasticity is 3 x 106 , the allowable displacements are DU = -J)L = 0.04, and the minimum concrete coverage is O.OS. Assuming the given concrete dimensions, Bland B2 = 0.1, and only potentially critical constraints, the problem of minimizing the prestressing force can be stated as the following linear programming [see (a) in example 4.14]: findXo = P and Xl =P(Yl - Y:z} such that Z=Xo ~min

(a)

(stress)

112.S-416.7Bl :::;Xl :::;36+333.3Bl

(displacement)

(concrete coverage)

(b)

(c) (d)

The design space for three different values of B 1 is shown in Fig. 4.9 and the results are summarized in Table 4.3. In case A (Fig. 4.9a), a wide feasible region is obtained. However, if the constraint (d) is modified such that O.SX 1 :::; Xo no feasible solutions can be achieved. In case B (Fig. 4.9b), the feasible region is reduced to a line segment, where a transitivity condition becomes equality. Specifically, the first equation of the stress constraints (b) becomes -3214:S; -10.9Xo- 47.6Xl :s; -3214

(e)

In case C (Fig. 4.9c), the stress constraints are -37S0 :s; -12.5Xo - 62.5Xl :s; -4219 2719 :s; -12.5Xo + 62.5X l :s; 2350

if) (g)

Table 4.3. Results, prestressed concrete beam. Case A B C

1.000 0.916 0.800

Feasible Space Wide Line Segment

P

58.7 64.5

0.900 0.816

4.2 Feasible-Design Procedures

205

70

60

40

/5'0

,

,

70

90

Xo

4' '50

•

I

70

I

90

Xo

•

/~O

Fig. 4.9. Design space. a. Bl

= 1.000,

b. Bl

~ ~o

(e)

(b)

(a)

70

= 0.916,

C.

Bl

= 0.800.

That is, the transitivity conditions are not satisfied for both if) and (g), and no feasible solutions can be achieved for these concrete dimension~ Assuming the infeasible design Bl = 0.800, P =64.5, Yl - Y2=0.70, the stress constraints can be expressed in terms of the preassigned parameters as follows crf -1125DL:S; -3628.125:S; 1.25cr~ -1406. 25TL

(h)

1.2501 + 1406.25TL:S; 2015.625:s; crf + 1125DL

(i)

Two cases of direct solutions are demonstrated: For the given loadings, it is possible to find the modified allowable stresses needed to convert the design into a feasible one. The result is

Q.

crf:s; -1378

crf ~-234

cri:s; -1762.5

cr~ ~472.5

(j)

That is, the allowable stresses for the second loading condition must be modified. b. For the given allowable stresses, the modified loadings needed to convert the design into a feasible one are DL

~

1.89

TL:S; 2.50

That is, the live load must be reduced to 0.50 to achieve a feasible design.

(k)

206

4 Design Procedures

4.2.2 Optimization in Design Planes A typical optimal design procedure involves the following steps:

a. An initial design is assumed.

b. A direction of move in the design space is selected. c. An optimal step size is detennined for the selected direction. d. Steps b and c are repeated until the fmal optimum is achieved. The number of directions required to reach the optimum might be large, depending on the initial design, the nature of the problem under consideration, and the method used to introduce the direction vectors. In addition, several analyses are usually required for each selected direction. Consequently, the number of repeated analyses and the resulting computational effort involved in the solution process might become very large. The design procedure presented in this section is based on optimization in a design plane. instead of the common optimization along a selected direction (step c), at each iteration cycle. Applying the scaling procedure, optimization in a design plane can be reduced to a single-variable optimization. To further improve the efficiency, approximate behavior expressions in tenns of a single variable are used for each selected design plane. The result is an efficient feasible-design procedure. Selecting the Design Planes. Consider a modified design X, given by (4.67) where X* is the current design, AX* is a selected direction vector, and a is a variable step size. Equation (4.67) represents a line in the design space. A design plane is defined as a two-dimensional space detennined by the vectors X* and AX* (Fig. 4.10). An arbitrary two-dimensional coordinate system Wlo W2 , can be selected in a given design plane. Assuming an orthogonal system with the same origin as that of X, we obtain the relationships

W=TX (4.68) in which T is a (2 x n) rotation matrix and n is the order of X. The elements of T are the direction cosines

T = [COS(Wt.xl)... COS(Wt.x,,)] cos(W2•XI ).•. COS(W2 •X,,)

(4.69)

In Fig. 4.10 the direction of WI is selected along X*, and the direction of W 2 is normal to X* in the given design plane.

4.2 Feasible-Design Procedures

207

.. Fig. 4.10. A design plane determined by X· and l1X·.

Although various methods can be used to select .1X*, only the SLP method (Sect. 3.1.3) will be considered in this section. That is, the direction vector .1X· =X-X·

(4.70)

is determined by solving the LP problem (3.51) and (3.52). Solution method. Once a direction vector has been selected, it is advantageous to use the available information associated with that direction. To achieve this goal, the following solution procedure is employed [84]:

a. An initial design is assumed and scaled to the best feasible design. b. The vector .1X* is selected by solving the LP problem (3.51) and (3.52), and a design plane is introduced. c. Optimization in the given design plane is carried out, as described subsequently. d. Steps b and c are repeated until the optimum, or a satisfactory design, is achieved. Once the direction .1X· has been selected, the object is to find the optimum in the corresponding design plane (step c). Any point on the design line through X is given by (Fig. 4.10)

or, alternatively

X = J.1(X· + a.1X*)

(4.71)

w =J.1(W· + a.1W*)

(4.72)

in which J.1 is a scaling multiplier. It is instructive to note that scaling of X will always provide a design X in the given design plane. For any assumed a, the optimal J.1 value can be determined simply by the scaling procedure. As a result,

208

4 Design Procedures

optimization in a design plane is reduced to a one-dimensional search problem, with a being the independent variable and ~ the dependent one. One possible drawback of this procedure is that the direction L1X· might be close to that of a certain design line. In such cases very large values of a are required to arrive at the vicinity of the optimum. To overcome this difficulty, we choose a search direction for the independent variable normal to the design lines. Denoting the distance of move in this direction by t (Fig. 4.11), W can readily be determined for any assumed t. The design W is then scaled and Z is evaluated. These steps are repeated until the optimum in the design plane,Wopt , is reached. To further improve the computational efficiency, an approximate displacement model can be introduced for each selected plane in terms of the single step-size variable (Sect. 3.2.4). To evaluate the error in the scaled design, consider the displacement constraint (4.73) Derming the ratio between the approximate and the exact displacement at W by

P== ra(W) I r(W)

(4.74)

then it can be shown [84] that the same ratio is obtained also for the approximate scaled design Wa

P= ra(Wa) I r(Wa)

(4.75)

That is, the errors in the displacements at Wa depend only on the errors in W, and not on the distance between W and Wa.

w,

~

________

~-L

__

~_~

Fig. 4.11. Determination of W opt .

4.2 Feasible-Design Procedures

209

Example 4.7. Consider the ten-bar truss shown in Fig. 4.2 with a lower bound on cross sections XL =0.1. Two cases of stress constraints will be considered:

case A:

CJU =

-crL = 25 for all members.

case B: CJ~ = -CJ~ = 40

CJU

= -~ = 25 for other members.

The assumed initial design is XO = 10, the objective function represents the volume and the assumed convergence criterion is (Z
where k denotes the iteration cycle. The iteration history for the presented procedure is shown in Fig. 4.12 [84]. The optimal solution in case A is X°T = (7.94, 0.10, 8.06, 3.94, 0.10, 0.10, 5.74, 5.57, 5.57,0.1) Z* = 15,932 This solution is achieved after four iteration cycles (Fig. 4.12a). The results for

caseB are

X°T = (7.90,0.10,8.10,3.90,0.10,0.10,5.80,5.51,3.68,0.14) Z* = 14,976 It instructive to note that the true optimum, achieved after four iteration cycles, is not a fully-stressed design. Z/IOOO 35

35

25

25

15

15

-IL--1--11--+--1-1-.. k 2

4

TL--t--tl-t---l-I__ .. k 2 4

(a)

Fig. 4.12. Iteration history ten-bar truss.

(b)

a. Case A b. Case B.

21 0

4 Design Procedures

4.3 Optimality Criteria Procedures Optimality criteria (OC) methods and mathematical programming (MP) methods have the same objectives but they differ in the redesign step. While in MP methods the objective function is minimized directly by various numerical algorithms, in OC methods an a priori criterion is defined and the premise is that when this criterion is satisfied the optimum is found. Application of OC methods involves derivation of the appropriate criteria for the specialized design conditions and establishing an iterative procedure for achieving the final design. Based on the choice of the criteria, OC methods can be classified as follows: a. Physical (or intuitive) OC methods, in which explicit recurrence relations for redesign are derived, based on approximate expressions of the constraints in terms of the design variables (the latter expressions are exact for determinate structures). In some cases these methods show poor convergence and tend to be unstable. However, the efficiency (i.e. the number of analyses required) is usually good and independent of the problem size. The fully stressed design technique is probably the most successful OC method, and has motivated much of the interest in this area. Similar displacement criteria have been developed, as will be shown in this section. b. Mathematical (or rigorous) OC algorithms, based on the Kuhn-Tucker necessary conditions of optimality. These are nonlinear equations which can be solved iteratively. A major problem in using this approach is to identify the set of active constraints at the optimum a priori. Methods based on OC are usually not as general as those of MP. The latter methods have the generality to consider different constraints and objective functions. On the other hand, OC methods are computationally most efficient but they may depend on the specialized behavior of the structure and the convergence to the optimum is not always guaranteed. In this section physical OC methods for stress and displacement criteria are first presented. General design procedures are then introduced, and the relationship between OC and MP is discussed.

4.3.1 Stress Criteria Fully Stressed Design. Fully stressed design (FSD) procedures are based upon the intuitive assumption that in an optimal structure each member is subjected to its allowable stress under at least one of the loading conditions. Similar to most OC methods, this approach generally consists of the iterative application of analysis and a redesign rule. If analysis shows that a certain member is overstressed in a critical load condition, the redesign rule increases the size of that member to reduce the stress. The opposite is done if the member is understressed. Since modifications of the member sizes in statically indeterminate structures will change the force distribution to the members. a few cycles of iterative analysis and redesign are necessary to obtain a fully stressed design. In many structures having normal action. the forces in the members are little affected

4.3 Optimality Criteria Procedures

211

by relative variations in the size of the other members and the iterative process converges rapidly. However, for structures with the so-called hybrid action, the convergence might be slow. Reasons for the significance of FSD include:

a. Engineering experience indicates that a good design is often one in which each

member is subjected to its allowable stress. b. FSD can be proved to be optimal under certain circumstances. c. FSD procedures are relatively efficient in comparison with many MP methods. d. An FSD is often a good starting point for design procedures based on MP.

There is no explicit reference to an objective function in FSD methods. It was shown long ago that, for a statically determinate structure under a single load condition, the FSD is the minimum weight design. This consideration does not extend, however, to indeterminate structures with multiple load conditions. One reason for this is that an FSD is not unique. An indeterminate structure might have more than one FSD and there is no assurance that an algorithm for calculation of FSD will converge to the minimum weight FSD. However, experience with FSD methods does indicate in many problems, not selected for their exceptional behavior, that the resultant design is indeed either the optimum or close to iL Using stress-ratio procedures, the influence of force redistribution in the structure on the stress in a given member is not considered. The design variable in the (k+ l)th cycle,

x[ 1+1) , is calculated by the redesign rule (1)

X(l+l) I

=X(l) ~

(4.76)

esC!

I

I

The stress-ratio approach consists of a cyclic analysis procedure in which the results from a given kth cycle are used to resize the members to the fully stressed state by (4.76). The member sizes are increased or decreased by the ratio of the computed stress to the allowable stress for the member. From all load conditions only the largest (critical) ratio of stresses for any specific member i must be considered. The computed sizes are then used in the next analysis cycle. The process is continued to convergence, if one exists. It can be noted from (4.76) that

esf

if X(l+l) is along the scaling line through X(l), then is the exact stress at X(l+l). This result indicates that stresses computed by (4.76) at any point along the scaling line through is given by

X(l)

are exact. Assuming that the force in the member,

Af l ), (4.77)

and substituting (4.77) into (4.76), then

X[l+l)

A(l) ~_._ X (l+l) __ j

-

esC! I

can be computed by (4.78)

212

4 Design Procedures

That is, the new design variable is the computed critical member force divided by the allowable member stress. Geometrically, (4.76) means that the exact constraint surface is approximated by a plane normal to the ith axis in the design space. Such an approximation is called of zero order. Using a simple stress-ratio redesign method, convergence will occur in one step for a statically determinate structure. The number of iterations required for convergence in statically indeterminate structures is not, in general, linked closely to the number of variables in the problem. It is this fact which makes the use of FSD so attractive. Increasing the stiffness of a member will result in more force in that member. Thus, it might be reasonable to increase (or reduce) Xj by a factor which is larger (or smaller) than the stress ratio. To improve the convergence, an "over-relaxation" factor, v ,greater than unity can be used [40] in (4.76) v>1

(4.79)

Another possibility is to use the first-order Taylor series approximations. To find a new design X(k+l) in which the stresses afk+l} will be equal to aU, we may require (4.80) X(k+l) could be calculated by solving (4.80) for the set of all active constraints. However, in general, the stress constraints which are active at the optimum are not

known a priori. Alternatively, it is possible to evaluate X[k+l) for a change in a single design variable, Xj, and a specified stress, all, by (4.80) (4.81)

A critical stress can be determined, for which the required X[k+l) is the largest. The zero-order stress-ratio approximations (4.76) may not be sufficient in some cases and convergence might be slow. Using the first-order approximations (4.81) may lead to a divergent calculation when the initial design represents a poor

(oa load

k ) may require estimate of the final design. In addition, calculation of ll much computational effort. Several approaches have been proposed to overcome these difficulties. In the mixed mode approach [38], the stress-ratio rule is applied for a few cycles, after which the first-order rule of (4.81) is used. Another possibility is to use hybrid methods [32] based on zero and first-order approximations.

4.3 Optimality Criteria Procedures Minimum

Minimum

size

size

213

/

/

/

"

100

100

100 (b)

(a)

Fig. 4.13.

100

Load path of:

8.

Optimal solutions, b. FSD, cases B and C.

It should be noted that in certain cases convergence of the iterative fully stressed design procedure cannot be achieved since there is no FSD. To illustrate this situation, consider a truss with n members (or design variables), nR redundants, and nL load conditions. There are (n-nR) independent element equilibrium equations for each load condition. The requirement that the number of independent equations exceeds the n unknowns can be written as (4.82) If this requirement is violated, the member sizes cannot be chosen independently and a fully stressed design might not exist

Example 4.8. To illustrate nonoptimal fully stressed designs obtained by the stress-ratio rule, consider the ten-bar truss shown in Fig. 4.2. The truss is subjected to a single-loading condition, the objective function represents the volume, and the member-size constraints are 0.1 ~ Xi (i = 1, ... , 10). The following cases of stress constraints have been solved: Case A:

-25

~ (Ji ~

CaseB:

-25 -50

~ (J9 ~

-25 -70

~ (J9 ~

Case C:

~ (Ji ~

~ (Ji ~

25

for i = 1, ... , 10

25 50

for i = 1, ... , 8, 10

25 70

for i = 1,... , 8, 10

Results reported by Berke and Khot [11] are given in Table 4.4. For cases B and C the solution is not improved, it rather converged to a fully stressed but nonoptimal design. Figure 4.13 shows that different load paths have been obtained for case A and for cases B and C. In the latter cases application of the stress-ratio rule may

214

4 Design Procedures

lead to elimination of member 9 by dividing with its greater allowable stress in the stress-ratio algorithm. At the true optimum for cases B and C member 9 is not fully stressed (cr9 = 37.5). An identical optimum would be obtained for all similar problems with Table 4.4. Member 1 2 3 4 5 6 7 8 9 10 Volume

crlf ~ 37.5.

Results, ten-bar truss, stress and member-size constraints. Cross-sectional areas Case A Cases B and C FSD and optimum FSD True optimum 7.94 4.11 7.90 0.10 3.89 0.10 8.06 11.89 8.10 3.94 0.11 3.90 0.10 0.10 0.10 0.10 3.89 0.10 5.74 11.16 5.80 5.57 0.15 5.51 5.57 0.10 3.68 0.10 5.51 0.14 15,932 17,252 14,976

Combined Stress Criteria and Scaling. Some of the difficulties involved in applying stress criteria can be overcome by combining the stress-ratio rule and the scaling procedure. Considering stress and member-size constraints, an effective design procedure where all intermediate designs are feasible can be introduced. The solution process involves the following steps: 1. An initial design X Il} is assumed (k = 1). 2. The current design is analyzed to obtain the stresses (j( k) • 3. The design is scaled by [see (4.50) and (4.52)]

(4.83)

(4.84) The objective function value is then calculated for X(k), which is the best feasible design along the scaling line through X(k). If some convergence criteria are satisfied (that is, no further improvements of the objective function value can be achieved), then the solution process is terminated and the current scaled design is the final solution. 4. Each design variable is modified independently by

4.3 Optimality Criteria Procedures

- ( l+1) _

-max(

Xi

A(l) A(l) ) ·_ .:.:i.-.-a; L U '-L-,Xi ,-l, ... ,n (Ji

215

(4.85)

(Ji

considering the stress-ratio rule (4.78) and the member-size constraints 5. Steps 2, 3 and 4 are repeated until convergence. By iteration, using the redesign rule (4.85) in conjunction with the scaling procedure [(4.83) and (4.84)], an efficient path to the optimum can be constructed. However, it should be noted that depending on the initial design, the solution process may not converge to the true optimum. Several different initial designs may be assumed to guard against such cases, and the best solution is then chosen. Another point to be recognized is that iteration using the redesign rule (4.85) proceeds in fmite steps and it is possible to miss the optimum between two steps. In such cases we can change the step size in the iteration or, alternatively, a linear search in the vicinity of the optimum can be applied. Example 4.9. To illustrate the combined stress-ratio and scaling procedure, consider the three-bar truss shown in Fig. 1.11. The stress constraints and the objective function are as given in example 1.7. It can be seen (Table 4.5) that solution by the stress-ratio rule (4.78) converges slowly to the point (1.0, 0) which is a fully stressed but nonoptimal design (Z = 282.8).

1.0

0.8

0.6

0.4 -

/

/

/

/ /

/

/

/'

-(2)

X

/' /'

/'

/

/

/'./ /

/'./

_

-

0.2

..--

/'

"--X(3) /~(3)

./ /

X

0.4

X(4) 4

( 4 ) . _0-_ X_ _ _ _ 5

- - - - -

-.:::. -

,:::-::;"'.::;-::::=---

/'

:~~::~\\~~\ ./

I 0

/' / '

/

Objective function contours

2

/ / /' /' /' ././ ./ ./

0.2 -

o

/

/

/

-

0.6

-

-Q..( i X

-(5)

0.8

Fig. 4.14. Design space, three-bar truss.

x(S)

1.0

1.2

216

4 Design Procedures

Solving by the combined stress-ratio and scaling procedure, the assumed initial design variables are given by X(l)T 4.14). Computing J,1(1)

=(l.O,

l.O), forming the design line 0-1 (Fig.

= max(a(l) tau) and X(l) =J,1(l)X(l) we frod

J,1(1) =0.7CY7

X(l)T = (0.707,0.707)

i 1) =270.7

From (4.85), the new relative design variables X(2)T = (0.707, 0.414) are computed, forming the line 0-2. Scaling of the design yields J,1(2) = 1.094

X(2)T = (0.773,0.453)

Repeating the above steps we fmd X(3)T J,1(3) = 1.054

= (0.773,

X(3)T = (0.815,0.337)

Z(2)

=264.0

0.320) and z(3)

=264.3

Thus, X(2) represents the best design, which is close to the true optimum X·T = (0.788,0.410)

-r =263.9

It has been noted that, depending on the initial design, the solution may not converge to the true optimum. Choosing, for example, X(4)T initial design, we obtain J,1(4) =0.890 From (4.85),

X(S)T J,1(S)

X(4)T = (0.890,0.178)

=(0.890,

= 1.012

= (1,0.2)

z<4)

=269.5

Z
= 270.3

as the

0.156) and

X(S)T

= (0.900,0.158)

In this case the solution process does not converge to the true optimum. The initial design X(4) is the best, but nonoptimal, solution. Table 4.5. Three-bar truss example, solution by stress-ratio (4.78).

Cyelek 0 1 2 3

X(l)

X(l)

1 1.0 0.707 0.173 0.815

2 1.0 0.414 0.320 0.261

a{l)

~l)

a~l)

14.14 21.87 21.09 20.71

8.28 15.47 16.32 16.89

-5.86 -6.40 -4.17 -3.82

1.0

0

20.0

0

4.3 Optimality Criteria Procedures

217

4.3.2 Displacement Criteria Fully stressed design procedures can be employed only in problems with stress and member-size consttaints. To consider displacement consttaints, assume that one stress is sufficient to describe the response behavior of each element in the structure. For more complex elements involving multiple-stress components, a similar procedure with some modifications will apply. Single-stress elements are, for example, axial force members or bending members in which the shear effect is neglected. OC methods are easy to apply when there is only a single-displacement constraint, as will be shown subsequently. A Single-Displacement Constraint. The basic assumption used in the stress ratio technique for arriving at a fully stressed design is the insensitivity of internal forces to member sizes. The same assumption is used in deriving resizing rules for structures subject to displacement constraints. Consider a structure subjected to a single-loading condition with a single-displacement equality constraint r = rU. Assuming the virtual-load method of analysis (1.34), the problem is to fmd X T = {Xl ..... XIIl such that II

Z=

L ljXj ~min

(4.86)

j=l

~ T:

LJ -' =r

u

(4.87)

j=l Xj

From (2.15), we defme the Lagrangian function

(4.88) At the optimum the following conditions must be satisfied [see (2.16)]

h= t

.... n

(4.89)

In an indeterminate structure the quantities Tj are functions of the design variables because the forces Aj depend on the member sizes [see (1.31)]. However, it has been shown [10] that, based on the principle of virtual work, the last term in (4.89) is identically zero, therefore

'\ T" l"-"'2=0

X"

h=l, ...• n

(4.90)

218

4 Design Procedures

Equation (4.90) states that at the optimum. the quantity l"X~ I T" is the same for every design variable. This form of optimality criteria is useful only if every member is subject to change. In general. only part of the design variables are determined by the constraint (4.87). To consider this possibility. the variables in the structure are divided into two groups - active and passive variables. A variable becomes passive if its value is determined by considerations other than those associated with the displacement constraint (4.87). This topic of active and passive groups of variables will be discussed later in this section. At present it is assumed that there are I (~n) active variables. Denoting the contribution of the passive members to the displacement expression as ro and to the objective functions as Zo. then the problem (4.86) and (4.87) becomes I

Z=Zo+ I

~

L (Xj ~min

(4.91)

j=l

u

T:

£..J -' = r - ro == r *

(4.92)

j=l Xj

and (4.90) can be rewritten in the form

h = 1. ... .1 Substituting (4.93) into (4.92) for all variables (h gives 1 I

.,ff. = . .

(4.93)

= i = 1•...• /) and solving for A

L ~T;lj

(4.94)

r j=l

Finally. substituting (4.94) into (4.93). then X" is given by

h= 1. ... .1

(4.95)

Equation (4.95) represents the criteria which must be satisfied at the optimum of the problem (4.91) and (4.92). Multiple-Displacement Constraints. Multiple-displacement constraints may arise in the case of multiple-loading conditions. or multiple active constraints under a single loading. or both. A major difficulty in deriving the optimality criteria for such problems is associated with prediction of the set of active constraints. Consider the following optimal design problem with J equality displacement constraints and I active members: find XT = (Xl •...• XI) such that

4.3 Optimality Criteria Procedures

Z = Zo +

L

219

I

tjXj

-+ min

j=1

I

L j=1

(4.96) T·· • -L- r · =0 X. J

j

= 1, ... .1

'

(4.97)

and the necessary conditions to be satisfied at the optimum are h = 1. ... ,/

(4.98)

These conditions can be rewritten in the form

xf=

t A.j (Tjh) j=1

(4.99)

h=I, ... ,!

th

Equations (4.99) represent the optimality criteria which must be satisfied at the optimum. Recurrence Relations for Redesign. Different numerical procedures for redesign can be established for similar optimality criteria. In this subsection redesign rules to satisfy the displacement optimality criteria, based on physical considerations, will be presented. Consider frrst the optimality criteria for a single-displacement constraint as given by (4.95). The latter equation can be used as a recurrence relation for an iterative process of determining Xh. Similar to the stress-ratio rule for achieving a fully stressed design, (4.95) can be written in the form [9]

h= 1, ... ,/

(4.100)

where the superscript k denotes the iteration cycle. The iterative process consists of applying successively analysis of the structure and the rule of (4.100). This rule is based on the assumption that the elements of matrix T are constant at a given iteration, i.e., the force redistribution is not sensitive to changes in member sizes. This is an intuitive recurrence relation for modification of the design variables to

220

4 Design Procedures

satisfy the optimality criteria (4.95) with no guarantees on convergence. The latter depends on the force redistribution during the solution process. The optimality criteria for multiple-displacement constraints, given by (4.99). can be expressed as the following recurrence relations

h=I •... , /

(4.101)

These relations are similar to those of (4.100) and could be used to obtain X it . However, the multipliers Aj cannot be eliminated in a simple manner and explicit expressions for Xit cannot be obtained. The optimal solution has to satisfy the / conditions (4.99) and the J constraints (4.96). These are (/+J) nonlinear equations with / unknown design variables Xit and J unknown Lagrange multipliers Aj. Several methods have been proposed to solve this problem. One major difficulty common to all approaches is that the constraints which are active at the optimum are not known in advance. Thus any procedure has to be capable of deleting constraints which are inactive at the optimum. This can be done by considering the Kuhn-Tucker necessary conditions for optimality. The above considerations indicate that OC methods are most suitable for problems with a singledisplacement constraint. Passive Variables. Different considerations may affect the selection of the passive variables and several approaches of dividing the design variables into the two groups of active and passive variables are possible. If the variables are separated into active and passive ones then the optimality criteria are necessary conditions. On the other hand. if all variables are assumed to be active and can be modified, the final design will not be required to satisfy the optimality criterion. The latter is neither a necessary nor a sufficient condition. but only an expedient way to modify the design variables giving a practical solution. If for a member the size required to satisfy another constraint is greater than the size required to satisfy a certain displacement constraint. that member should be considered passive. Another criterion for the selection of passive members can be observed from an examination of (4.100). If Ti for a member is negative. its inclusion as an active member would require calculation of ~1il; • introducing unacceptable imaginary numbers. The corresponding criterion for a passive member is T; < O. indicating that reduction in displacement occurs when the member size is decreased rather than increased. This means that the member size will be determined by other constraints (such as minimum size. stress. or other displacement). Such a member may be treated as passive in the redesign to satisfy the appropriate displacement constraint. 4.3.3 Design Procedures Combined Optimality Criteria and Scaling. It has been noted that the OC method is most suitable for problems with a single displacement constraint. In the case of multiple-displacement constraints, (4.100) can be applied to each

4.3 Optimality Criteria Procedures

221

constraint separately and the largest value of each design variable is selected for the final design. This extension of the recurrence relation to multiple loading conditions and displacement constraints is accomplished in a manner similar to that used in the stress-ratio procedure. That is, the largest design variable computed by (4.100) is taken, considering all loading conditions and displacement constraints. If we have nL loading conditions and J displacement constraints, X" is computed by (4.100) for the J nL combinations. This procedure, proposed by Gellatly and Berke [40], is called the envelope method. Since the method is based on the optimality criteria for a single constraint (4.95), the solution usually does not satisfy the correct optimality criteria for multiple constraints (4.99). However, solutions reached by this approximate method are often very close to those obtained by the correct criteria for multiple constraints. The design procedure described in Sect. 4.3.1 for stress and member-size constraints can be extended to include also displacement constraints. The procedure presented here is based on combining the stress-ratio rule, the displacement criteria and the scaling procedure. The solution process involves the following steps:

1. An initial design

x(/c)

is assumed (k=I).

r(k) and 0°:). 3. The scaling multiplier is determined by (4.50), the design is scaled by (4.52) to obtain X(k), and the objective function value is calculated at X(k). If some convergence criteria are satisfied, the solution process is terminated and the current scaled design is the final solution. 4. Each design variable is modified independently considering the redesign rule for stress and member-size [see (4.85)].

2. The current design is analyzed to obtain

I·

100

•I •

100

-I

"

~

(2)

(I)

Xl

T (3)

X2

Xl

Fig. 4.15. Three-bar truss, two loadings.

100

222

4 Design Procedures

x,

Feasible region

Fig. 4.16. Design space, three-bar truss, two loadings.

5. The design variables are modified independently according to the displacement criteria (4.100). Initially all variables may be considered to be active; then, variables determined in previous cycle by step 4 or by other displacement constraints are assumed to be passive. 6. For each design variable the maximum value from steps 4 and 5 is chosen. 7. Steps 5 and 6 are repeated until there is no change in active and passive variables. 8. Steps 2 to 7 are repeated until convergence. Example 4.10. The symmetric truss shown in Fig. 4.15 is subjected to two distinct loading conditions P1 and P2. The cross-sectional area design variables are Xl, X 2 , and the objective function irepresents the volume of the truss. The allowable stresses are a U=20.0, a L =-15.0, for all members, and the allowable displacement in the vertical direction is r U=O.04. The modulus of elasticity of the material is 30,000 and the assumed initial design is X(1)=(3.0, 0.25JT. Starting with scaling of the initial design and solving the problem by the design procedure described in this section, the iteration history is shown in Table 4.6 and in Fig. 4.16. The final optimum is X*=(0.943, 1.0}T, '£=366.7. Table 4.6 . Iteration history, three-bar truss, two loadings. Stress Criteria k

1 2 3 4 5

Xl 0.943 0.943 0.943 0.943 0.943

X2

0.105 0.191 0.320 0.485 0.648

Dis:el. Cri teria

Xl

1.997 1.726 1.345 0.919 0.390

X2

0.333 0.575 0.897 1.225 1.014

Max. Values

Xl 1.997 1.726 1.345 0.943 0.943

X2

0.333 0.575 0.897 1.225 1.014

After Scaling

Xl 1.907 1.602 1.213 0.943 0.943

X2

0.318 0.534 0.809 1.225 1.014

Z 571.3 506.5 424.0 389.2 368.0

4.3 Optimality Criteria Procedures

223

Mathematical Optimality Criteria. Several iterative procedures based on application of the Kuhn-Tucker conditions have been proposed. An important characteristic of these methods is that they converge to a local optimum design whereas physical optimality criteria methods may converge to nonoptimal solutions. The algorithms involve two types of approximations. associated with:

o. the recurrence relations for redesign;

b. identifying the constraints that are active at the optimum.

Various relations can be used to modify iteratively the design variables X and the Lagrange multipliers A. in the redesign step. The optimality criteria are used to derive recurrence relations to modify the design variables. and the constraint equations are used to establish relations to update the Lagrange multipliers. In general. the constraints that are active at the optimum are not known in advance. At each iteration the set of active constraints can be checked and updated as necessary. Since the Lagrange multipliers must be nonnegative at the optimum. the constraints corresponding to negative multipliers at a given iteration may be deleted from the set of active constraints. Convergence difficulties may arise in cases when the initial design is far from the optimum or if constraint switching occurs from iteration to iteration. However. convergence near the optimum is usually very rapid. Various recurrence relations for redesign are discussed and compared elsewhere [12.58.59]. 4.3.4 The Relationship Between OC and MP The MP methods. presented in Chap. 2. are general and most suitable for problems with multiple types of constraints. These methods exhibit usually good convergence properties. The computational process is stable but the convergence near the optimum might be slow. In addition. the number of iterations increases rapidly with the number of design variables . Physical OC methods are not as general as MP methods. but their efficiency is usually good. Mathematical OC methods are more general. but convergence difficulties can arise in cases where the initial design is far from the optimum. The efficiency of MP methods has been greatly increased by using approximation concepts (Chap. 3). In addition. OC methods have been extended to include more complex design conditions and rigorous criteria. The different approaches have many common ideas and the positive features can be used to establish better solution methods. The relationship between approximate formulations and solution methods used in both approaches is briefly discussed in this section. Approximate Problem Formulations. The original design problem is usually transformed into a sequence of approximate problems. solved successively. The definition of the solution method used depends on the formulation and on the algorithm for solving the approximate problems. Classification of different approximate problem formulations and possible methods of solution have been proposed elsewhere [32]. Assuming homogeneous displacement and stress functions. it has been noted in Sect. 3.2.3 [(3.121) through (3.135)] that the

224

4 Design Procedures

assumption 0/ constant internal/orces (used in DC) is equivalent to afirst-order Taylor series expansion in the reciprocal variables (used in MP). From this statement, the following observations can be made: a. In statically determinate sttuctures, a fJl'St-order Taylor series expansion of the displacements and stresses in the reciprocal variables is exact

b. If the internal forces in statically indeterminate sttuctures are not sensitive to changes in the design variables, the fU'st-order Taylor series expansion in the reciprocal variables represents a good approximation of the behavior constraints. Although the approximations used in OC and MP may lead to equivalent problem formulations, the two approaches use different solution algorithms.

Solution Methods. The solution approach of the approximate problem can be interpreted as an MP, a generalized optimality criteria (GOC), or a mixed method, depending on the number of steps before analyzing the sttucture. Defining K. as the number of steps before updating the approximate constraints by analysis of the structure, the following methods of solution may be considered [32] :

a. K=I, corresponds to solution by MP. The sttucture is analyzed and the approximate constraints are updated after each change in the design variables.

Therefore, the solution process requires a large number of analyses and much computational effort However, the convergence is usually guaranteed. b. K ~ 00 , the approximate problem is solved exactly after each sttuctural analysis. This can be done by generalization of the optimality criteria, as proposed by Fleury and Sander [32]. The explicit GOC are derived from the Kuhn-Tucker conditions of the approximate problem. Since the approximate problem formulation is equivalent to using a first-order Taylor series expansion in the reciprocal variables, the problem can be solved exactly also by MP methods applied to the linearized constraints. This means that solution by GOC is equivalent to solving completely the approximate problem with the same linearized constraints by MP. The properties of solution by this approach are similar to those of OC methods, i.e. the solution is more efficient but the convergence is uncertain. c. K. limited number, the approximate problem is solved partially. Updating the approximate constraints after a limited number of steps, the approach becomes mixed with properties lying between those ot OC approaches and those of MP. Actually, both MP and OC methods often do not solve the approximate problem exactly.

=

Thus, K can be used as a convergence control parameter [31]. A large value of K. increases the efficiency but there is a possibility of divergence. A smaller value of K.. may improve the convergence but the number of analyses will be larger. Therefore, it is desired that the value of K. will be as large as possible for efficiency but small enough to avoid divergence. The above considerations explain the high efficiency and the possible convergence difficulties which may be encountered in solution by OC methods.

4.4 Multilevel Optimal Design

225

4.4 Multilevel Optimal Design 4.4.1 General Formulation Decomposition and Coordination. optimization are:

The main objectives of multilevel

a. Solution of problems with a naturally multilevel fonnulation. b. Efficient solution of large scale problems. c. Integrated optimization of complex structural systems. The two basic processes of multilevel optimization are decomposition and coordination.

Decomposition. Decomposition is the generation of subproblems by breaking down the system into subsystems with interactions. The coupling between subsystems prevents the direct solution of the overall problem. A multilevel structure generated through the process of decomposition is shown in Fig. 4. 17a. It is typically characterized by several subproblems in the lower levels. However, it is possible to generate a multilevel structure such that each level is fonnulated by a single problem [Fig. (4.17b)]. In general, decomposition models can be divided into the following two classes: a. Process oriented decomposition, where the analysis, design and optimization process is decomposed into different subsequent steps. Structural optimization is often a multilevel design problem involving more detailed design procedures at the component level than at the system level. At the system level gross proportioning, usually based on finite element analysis, is carried out. On the other hand, detailed design of structural elements is often carried out one component at a time, using special purpose analyses. A typical example for process decomposition is to assume global fmite element model for calculation of element forces and local model with constant element forces. b. System oriented decomposition, where the system to be optimized is decomposed into proper subsystems which are treated separately. Large-scale structures are often divided into several smaller substructures, each of which optimized separately. A typical example for system decomposition is a bridge consisting of the deck subsystem, the supporting subsystem, and the foundations.

~

!&Yd.3. Level 2

l.&Yill

(b)

(a)

Fig. 4.17. Multilevel structures:

8.

using decomposition, b. without decomposition.

226

4 Design Procedures

It is instructive to note that the multilevel (hierarchical) decomposition presented here is not the only decomposition scheme. Nonhierarchical decomposition, where there is no restriction on the couplings which might exist between the subproblems [130], is not considered in this section. Decomposition methods can be divided into formal methods, that can be shown to converge to a true optimum under certain assumptions, and intuitive methods, often based on physical understanding of the model. The latter methods are reasonable but are not guaranteed to lead to a true optimum. Despite this drawback, most applications of decomposition are carried out in an informal way by engineering judgement Coordination. Coordination is a scheme of revising the subproblem optimization so that the final solution is that of the original problem. The interconnection of the subsystems may take on many forms, but one of the most common is the hierarchical form [124] in which a second-level unit coordinates the units on the level below, called the first level. Central to the coordination process is the identification of coordinating variables (called also the global variables or the interaction variables). These are held fixed at the lower level, giving decoupling of the lower level subproblems which are separated and solved independently. The goal of the second level is to coordinate the action of the first-level units so that the solution of the original problem is obtained. The extension of this approach to the general case of multilevel formulation is straight forward. The main steps of most coordination methodologies are as follows: - The coordinating variables are chosen. These may include some of the design variables, behavior variables, Lagrange multipliers or penalty parameters. - The independent lower level subproblems are solved for fixed values of the coordinating variables. - The solutions of the lower level subproblems are used at the upper level to update the values of the coordinating variables such that the overall objective function value is improved. - The lower level subproblems are solved for the revised values of the coordinating variables and the procedure is repeated. Problem Structure. Consider again the general NLP design problem Z=j{X) g(X)

~

~

0

min (4.102)

h(X) = 0

It has been shown [4] that the relationship between the problem variables, objective function and constraint functions can be described by the problem matrix. An entry V in position i, j of this matrix indicates that function i depends on the jth subvector of variables. Depending on the variables selected to formulate it, a problem may have many different structures. In this section only some of the many possible problem formulations will be considered and classified according to the structure of the problem matrix. From the stand point of decomposition, a problem having an additively separable objective function and a block-diagonal problem matrix (Fig. 4.18) is ideal, since

4.4 Multilevel Optimal Design

227

it yields totally uncoupled subproblems which can be solved independently of each other. That is, the original problem can be formulated, assuming suitable reordering of the variables and constraints, as follows: find the design variable subvectors Xlo X 2, ••• , Xs such that

L h(X;) s

Z=

i=l

(4.103)

i = 1. ...• s resulting in s independent subproblems each having a certain!.{Xi) as an objective function. It can be observed that any design satisfying the optimality conditions for each of the subproblems will satisfy those conditions for the original problem. A nested structure may be often obtained by partitioning the variable vector into two subvectors Y and X. The resulting problem is to find Y and X such that Z =f(Y, X)

~

min

g(Y. X) $; 0 h(Y. X)

(4.104)

=0

This problem can be stated in a two-level form where the lower level variables X are optimized for fixed values of the upper level variables Y = yo. Satisfying the optimality conditions for the first- and second-level subproblems in this case guarantees that the conditions for the original problem are also satisfied. Complex design problems usually cannot be formulated with a block-diagonal structure. A more typical structure is block-angular with a number of coupling variables and/or constraints (Fig. 4.19) . This problem can often be formulated as a two-level problem. At the upper level. the coordinating second-level variables Y affect directly the upper level constraints. At the lower level the first-level local subsystem variables Xi affect directly the lower level constraints. Assuming additively separable objective function. a problem with only coupling variables (Fig. 4.19a) can be formulated as: find Y and Xi (i = 1•...• s) such that

L h(Y. X;) ~ min s

Z = fo(Y) +

i=l

(global constraints)

gi(Y' X)$;O} hi(Y. X)=O

i = 1•...• s (local constraints)

(4.105)

228

4 Design Procedures

Variables

Xl

X2

X3

X4

Objective function

V

V

V

V

V

Constraints

V V

V

Fig. 4.18.

Block-diagonal problem matrix.

It can be shown that a design satisfying the optimality conditions for the first- and second-level problems will satisfy those conditions for the original problem.

Example 4.11. To illustrate various problem formulations, assume a general truss structure with members' cross-sectional areas X, members' forces A, and members' lengths t. Considering the force method of analysis and only stress constraints, the optimal design problem is [see (1.22) and (1.68)]: find X and A such that z = (l'X ~ min. (a) (b)

F(X) N = o(X)

(e)

If all variables are optimized simultaneously, the general formulation (4.102) is obtained. This integrated formulation can readily be recognized as the simultaneous analysis and design (SAND), discussed in Sect 1.4.2. If the truss is a statically determinate structure then N = 0 and the members' forces are constant A =A L • Thus, the analysis equations (e) are eliminated and the problem can be stated in the block-diagonal form (4.103) as: find Xj (i = 1,... , n) such that Z=

L" ljXj -+ min

(d)

j=1

i= I, ... ,n

(e)

It should be noted that the optimal solution in this case can be determined directly. Alternatively, denoting the redundant forces N = Y, the problem can be formulated in the form of (4.104) as follows: find X and Y such that if) (g)

F(X)Y = o(Y)

(h)

4.4 Multilevel Optimal Design

229

Variables Objective function

V

V V V V

V V V V

Global constraints

V

V V V V

V V V V

V V V V V V V

V

V

Local constraints

V

V V

(a)

V V V (b)

V V V V V V (c)

Fig. 4.19. Block-angular problem matrix with: a. Coupling variables, b. Coupling constraints, c. Coupling variables and constraints.

This fonnulation is similar to the integrated problem fonnulation (a), (b) and (c), except that a two-level optimization is considered here, where the design variables X are selected at the frrst level for the assumed redundant forces Y =yo. The latter forces are then updated at the second level and the frrst-Ievel problem is solved for the modified Y. Example 4.12. Consider an optimal plastic design of the three bar truss shown in Fig. 4.15, subjected to two distinct loadings PI and P2 • Choosing the force in member 2 as a redundant (denoted Ni for the ith loading condition), the integrated problem is to find XT = {Xl' X 2 } and NT = {NI • N 2 } such that Z = 282.8 Xl + 100 X2 ~ min

(a)

(b)

Assuming N = Y the fonnulation (4.105) is obtained. where the cross sections XI and X2 are optimized independently at the first level for the assumed forces Y=Yo. The latter are then modified at the second level. Model Coordination. Although there may be many different ways of transfonning a given constrained optimization problem into a two-level problem, they are all essentially combinations of two different approaches which may be tenned [96] the model coordination method and the goal coordination method. The model coordination method is known as the feasible method. all intennediate designs being feasible. It has been noted in Sect. 4.2 that this is advantageous since the solution process can be tenninated always with an improved feasible

230

4 Design Procedures

design. The goal coordination method, on the other hand, will provide infeasible intermediate designs. That is, full convergence must be achieved in order to obtain a feasible design. Another drawback of the goal coordination method is that it is guaranteed to work, loosely speaking, only for convex programs. Due to these limitations only the model coordination method will be considered in the remainder of this section. Consider the typical formulation (4.105). It has been noted that a natural approach to solve the problem is to use a two-level optimization procedure where the optimization of the local (subsystem) variables Xi is nested inside an upperlevel optimization of the global variables Y. The resulting first- and second-level problems are formulated as follows:

First-level problem. For given values Y =yo the problem is decomposed into the following s independent subproblems: find Xi such that

gi(YO' Xi) ~ 0 hi (yo, Xi)

(4.106)

=0

Second-level problem. Denoting the solution of the fIrst-level subproblems Hi (Y)

=min Zi

(4.107)

the task in the second-level problem is to find Y such that s

H(Y) = fo(Y) +

L Hi(Y)~ min i=1

(4.108)

ho(Y) = 0 An additional constraint on Y is that the first-level problem has a feasible solution, i.e., that H(Y) exists. The two-level problem is solved iteratively as follows: 1. 2. 3. 4.

Choose an initial value for the coordinating variables yo. For a given yo solve the s independent fIrSt-level problems. Modify the value ofY so thatH(Y) is reduced. Repeat steps (2) and (3) until min H(Y) is achieved.

Multilevel optimization is particularly effective for problems with a small number of global variables and a large number of component variables which are only weakly linked. The choice of the coordinating variables depends on two considerations:

4.4 Multilevel Optimal Design

231

1. The second-level problem must be easy to solve. That is, either the space of Y must be small enough to search, or else gradients must exist so as to permit efficient optimization techniques. Calculation of derivatives of the subsystems optima with respect to Y is demonstrated elsewhere [128]. 2. The first-level problem must be formulated in such a way that it can be decomposed into simple independent subproblems. One problem associated with the two-level approach is that for some values of Y there may be no feasible solution to the first-level problem. To overcome this difficulty we may add constraints to the upper-level problem that help prevent infeasible solution of the lower level problem, as will be illustrated in Sects. 4.4.2 and 4.4.3. Other means such as an extended penalty function [46] and an envelope function which replaces a vector of constraints [6, 129] have been proposed. Example 4.13. The object of this example is to illustrate system oriented decomposition of a coupled statically indeterminate structure. Using this procedure, decomposition is associated with dividing the structure into a number of substructures along boundary lines. The coordinating variables represent the behavior of the structure (displacements and forces) along these lines, and each substructure corresponds to a proper first-level subproblem. Consider the continuous beam shown in Fig. 4.20a (all dimensions are in kilograms and meters) with the following functions of the four cross-sectional variables Xi cross - sectional areas

=6 x 10-2 Xi

Moments of inertia

= 1. 8 x

Allowable moments

A;u

.}

10-5 Xi

i

= 1. ... ,4

= _A;L =9 X 103 Xi Boundary line

(a) _

.....

1

(b)

£ = 10.0

~c------,~

1

.....

- _.

I

£-100 £=10.0 -~- -·----~-I

l=10.0 _I ...------

First substructure

Second substructure

Fig. 4.20. Continuous beam: a. Integrated system, b. Decomposed system.

232

4 Design Procedures

Since bending moments are independent of the modulus of elasticity E. we assume 1O-6Ell = 1.0. Considering the displacement method of analysis. only constraints on bending stresses at members' ends (cross sections 1•...• 6). and volume as the objective function. the optimal design problem is [81]: fmd XT = {Xl .X2 • X3.X4 } and the corresponding displacements rT = (rB. re. rD) such that Z=0.6(X I +X2 +X3 +X4} -+ min

-9 X 103

54 Xl 12X2

+300 -400

Xl X2 X2

~

X3

36X2 +400 + -400 0 +400 -300

X3 X4

[54Xl + 12X, 36X2

o

0 0

0

(a)

0

36X2 0 12X2 0 12X2 36X3 36X3 12X3 0 54X4

m~9XlO'

36X, 12X2 + 12X3 36X o3 36X3 12X3 + 54X4

Xl X2 X2

(b)

X3 X3 X4

1{"} rOO} re = 0 rD -100

(c)

The boundary line through C separates the structure into two substructures and the chosen coordinating variables are

re= Yl A3 = -A4 = Y2

(rotation at C) (bending moment at C)

(d) (e)

Equation (e) is based on the relation A3 +A4= O. The second equation of (c). which includes variables of both substructures. can now be decomposed by separating the contributions of the individual substructures ~

+ ~ = 400 + 36X2'B + 12X2re - 400 + 12X3re + 36X3rD = 0

,

~

"

-i;

,

(()

The first and the third equations of (c) include only variables of a single substructure. For assumed fixed values

11 = Yjo.

Y2

= y20

•

the integrated problem

(a) through (c) can be transformed into the following two independent first-level

subproblems. each with its own variables. constraints. and objective function:

Subproblem 1. Find Xl. X2 and rA such that

°

0 ] {rB} ~9xlO 3{Xl} -9xlO 3{Xl} ~ { 3oo} + [54Xl X2 -400 12X2 36X2 Yj X2

4.4 Multilevel Optimal Design

233

Subproblem 2. Find X3. X4 and rc such that

The second-level problem is to find Y1 and Y2 such that the first-level solution exists and the overall objective function (a) is minimized. A similar approach has been demonstrated also for the force method of analysis [81]. 4.4.2 Two-level Design of Prestressed Concrete Systems Problem Formulation. The object in this section is to illustrate a two-level optimal design of prestressed concrete systems. Considering a typical indeterminate structure (Fig. 4.21) , the problem under consideration is to find the vector of concrete dimensions B, the initial prestressing force P, and the vector of tendon's coordinates Y such that

z = Cc V(B) + CpP -+ min

(objective function)

aL ~ O'(B, P, Y) ~ O'u

(stress)

DL~

D(B, P, Y)

~

DU

(displacement)

BL~B ~Bu

(concrete dimensions)

pL~p ~pu

(prestressing force)

yL

(tendon's coordinates)

~

Y

~

yu

(4.109)

in which Cc = unit cost of concrete; Cp = unit cost of prestressing force; V = volume of concrete; 0' = vector of normal stresses under service load conditions;

234

4 Design Procedures

and D =vector of displacements. The fact that the three types of design variables in this formulation are of fundamentally different nature may produce numerical problems in the solution process. In addition, the number of variables and constraints might be large even in simple structures. In the presented two-level procedure the variables P and Y are optimized at the first level and the variables B are treated at the second level. That is, the two-level formulation is based on process oriented decomposition. Considering the formulation (4.104), it will be shown that the first-level problem can be stated in an LP form, thereby allowing an efficient solution. For purpose of illustration, uniform concrete dimensions have been assumed. That is, the relative stiffnesses of the members are constant, and the bending moments, stresses and displacements (due to both external loadings and prestressing) are explicit functions of the design variables. The number of stress and displacement constraints can be reduced by considering only potentially critical constraints. The latter can be identified before solving the problem, as will be shown subsequently. The normal stress constraints for a given cross section are L

YqP

a <---q -

.4.: (B)

Mq +YqM(P.Y) W,(B)


u q

(top fiber) (4.110)

in which q = subscript denoting the loading condition; Yq = given coefficient for prestressing losses; Ac = concrete cross-sectional area; W"~ Wb = top and bottom section moduli ; M q' M (P, Y) = moments due to the qth external loading and prestressing, respectively. (Moments causing tension in bottom fibers are assumed to be positive). For uniform concrete dimensions Mqare assumed to be constant and M (P, Y) are independent of B. Defining the following quantities, which are functions of only B

L

apr =

max q

(L a q +Mq)f - - Yq W,(B)

Mq)f aL aq - Yq pb = max (L q

Wb(B)

u= mm. (u +W,(B) Mq)f - - Yq

apr

q

aq

u . (u

(4.111)

Mq)f a pb = mm a - - Y q q Wb(B) q

and substituting into (4.110) , the following reduced set of stress constraints is obtained aL < __P__ M(P. Y) < aU Pr - Ac(B) W,(B) - Pr

a

L

P M(P.Y) u <---+
Pb -

Ac(B)

Wb(8) -

Pb

(4.112)

4.4 Multilevel Optimal Design

235

That is, for any assumed B there are only two stress constraints for each" point, instead of the original 2 nL constraints (nL =number of loading conditions). Similarly, the displacement constraints are (4.113)

=

=

in which Dq(B) displacement due to the qth external loading, and D(B, P, Y) displacement due to the initial prestressing force, both being explicit functions of the variables. Defming

D;

=max[D; - Dq (B)] I 'Yq q (4.114)

and substituting into (4.113), the following constraints is obtained

reduced set of displacement

D; ~ D(B,P, Y) ~ D}t

(4.115)

Based on (4.112) and (4.115), the problem (4.109) can be slated as the following explicit design problem for uniform concrete dimensions [68]: find B,P, Y, such that

Z=

CcV(B)+CpP~

min

CJL < __P__ M(P,Y)
CJ

Ac(B)

W,(B) -

PI

L < u - - P- + M(P,Y)< CJ Ph - Ac (B) Wh(8) Ph (4.116)

D; ~D(B,P,Y)~D}t

yL~y ~Yu

In this formulation the bounds CJ~, ' CJ~, ' CJ~h' CJ~h ' D;, D}t are functions of only B. The stress and displacement constraints are written for all potentially critical points in the structure. The elimination process (4.111) and (4.114) may considerably reduce the number of constraints in structures subjected to several loading conditions.

236

4 Design Procedures

Fig. 4.21. Typical prestressed concrete indeterminate structure.

First-Level Problem. The fIrst-level LP formulation presented here is general and covers a wide range of problems, including: -

Indeterminate systems with nonuniform cross sections. Structures subjected to multiple-load conditions and prestressing forces. Structures with any given tendon length and prestressing loss diagrams. Structures with any tendon shape.

Assuming the equivalent load method, the problem (4.109) can be stated for any assumed 8 8 0 as follows [60] : fmd P and Y such that:

=

Z=P

--+

min (stress)

(displacement)

=

(4.117)

pL<5:P <5:pu

(prestressing force)

yL<5: Y <5:Yu

(tendon's coordinates)

in which a. b constant coeffIcients; and n = the number of tendon coordinates. Similar to (4.111) and (4.114), defIne

4.4 Multilevel Optimal Design

237

(4.118)

To convert the NLP problem (4.117) into an LP. assume the transfonnation of variables

j= 1•...• n

(4.119)

Substituting (4.118) and (4.119) into (4.117). the problem can be stated as the following LP: find Xj (j = 0.1 •...• n) such that z=XO ~ min

L ajXj

$;

O"~

L bjXj

$;

D~

II

O"~

$;

j=O II

Dft $;

j=O

(4.120)

pL $;Xo $; pU

Similar to the problem (4.116) • only two behavior constraints are considered in this fonnulation at each point instead of 2 nL constraints in the original fonnulation (4.117). Second-Level Problem. The object at the second level is to optimize the concrete dimension B. We first introduce constraints on B that help prevent infeasible solution of the first-level problem. Considering the stress constraints (4.112). necessary conditions for a feasible solution (on the basis of the transitivity

property) are

L

U

O"PI $; 0" PI

(4.121)

Substituting (4.111) into (4.121). denoting q = 1.2.3.4 for the four tenns in the latter equation and rearranging yields

238

4 Design Procedures

(4.122)

w; (8) > "(4 M 3 b

-

U

"(3 M 4

= W;L

L-b

"(30'4 -"(40'3

The right hand side tenns in these equations ( w,L and WbL ) are constant lower bounds on the section moduli. Similarly, from the displacement constraints (4.115), necessary conditions for a feasible solution are U D pL
(4.123)

Substituting (4.114) into (4.123) , denoting q = 5,6 for the two tenns in the latter equation and rearranging yields (4.124) in which / (8) = moment of inertia of the concrete cross section; d s, d 6 = the displacements Ds(8), D 6(8), respectively, assuming / (8) = 1.0 ; and /L = constant lower bound on / (8). Equations (4.122) and (4.124) are necessary conditions for a feasible solution of the first-level problem. Thus, the second-level constraints are

W,(8) ~ W,L} Wb(8) ~ WbL / (8) ~/L

( stress)

(displacement)

(4.125)

(side constraints) Assuming the objective function

z= V (8) ~ min

(4.126)

it is possible to find the minimum concrete volume by solving the problem (4.125) and (4.126). The concrete dimensions 8 are then modified. For any assumed 8 the LP problem (4.120) is solved until the optimum is achieved. Example 4.14. Based on the two-level fonnulation presented in this section, a general design procedure has been introduced [68]. To illustrate the problem fonnulation and numerical results, consider the typical member of a unifonn crosssection continuous beam shown in Fig. 4.8 (all dimensions are in tons and meters). A parabolic tendon is assumed with fl. f2 being the tendon's coordinate variables. Two loading conditions have been considered:

4.4 Multilevel Optimal Design

239

crL =-1500, aU = 100. crL = -1200, aU = O.

a. Dead load = 2.0, and P; b. Total load 3.0, and 0.8P;

=

The modulus of elasticity is 3 x 106 ; the allowable displacements are DU =-JJL = 0.04; the bounds on concrete dimensions are Bf

= 0.5,

BIU

= to,

Bf

= 0.1,

= 0.2; the bounds on tendon's coordinates are fL=O.05, yU =Bl - 0.05; and the bounds on the prestressing force are pL = 10, pU = 100. Bf

C'l

8

Bl

0

....

II)

ci II

ci II

~

~

~~

8C'l ci II

~~

1.00

0.90

llmin

0.80

0.70

~~'--~1------~1----~172~--~~~ 0.10

0.15

O. 0

B2

Fig. 4.22. Design space, minimum concrete dimensions problem.

240

4 Design Procedures

The optimal design problem (4.116) is to fmd BIoB1' P, Y Io Y1, such that

-1500-144 I (B1!if) S -P 1(~B1)-4P(1l- Y1)/(B1B?) S -270 I (B1B?)

112.5-4167B1Bf SP(1l-Y1 )S36+3333B1Bf 0.50 S Bl S 1.00

(a)

0.10 S B1 S 0.20 lOSP S 100 0.05 S YI S BI - 0.05 0.05 S Y1 S Bl - 0.05

The minimum concrete dimensions can be determined by solving the second-level problem (4.125) and (4.126): fmd Bl and B1 such that

B1!if ~ 0.079 (b)

0.50 S Bl S 1.00 0.10 S B1 S 0.20

Assuming, for illustrative purposes, Bl = 1.00 and B1 = 0.16, the first-level problem (4.117) is to find P, Y1 and Y1 such that

4.4 Multilevel Optimal Design

z=p

~

241

min

-2400 ~ -6.25P - 25P(Y1 - Y:z} ~ -1687.5 187.5

~

-6.25P + 25P(Y1 - Y:z}

-554.2

~

10 ~P

~

0.05

~

Y1 ~ 0.95

0.05

~

Yz ~ 0.95

P(Y1 - Y:z}

~

~

1000

569.3

100

(c)

The design space of the problem (b) is shown in Fig. 4.22. The optimal solution is

B!"un

= {O.92,

O.IO} and the corresponding frrst level LP solution is P = 64.5,

Y1 = 0.87, Yz = 0.05. Solving the frrst-Ievel problem for B[max = {l. 00, 0.1} (maximum depth and minimum width) yields P =58.7. It has been shown [68] that, depending on the ratio CclCp , the optimal solution is, either at B min (for high CclCp ratios) or at B1max (for low CclCp ratios). 4.4.3 Multilevel Design or Indeterminate Systems In most structural optimization formulations, design variables are treated as the independent variables, while behavior variables are the dependent ones. This appears to be a rational choice, since for any assumed design the behavior can be determined uniquely by the analysis equations. However, it has been noted in Sect 1.3.4 that this approach requires multiple repeated analyses for successive modifications in the design variables. An alternative two-level formulation, presented in this section, is to choose some behavior quantities as second-level independent variables. The resulting frrst-Ievel problem is simple and can readily be solved. The presented approach is intended to reduce the number of repeated analyses in structures subjected to multiple loading conditions. Problem Formulation. For simplicity of presentation, continuous beams will be assumed in the presentation that follows. The problem under consideration is to find the independent behavior variables Y (bending moments) and the dependent design variables X such that

Z =f(Y, X) -+ min M (Y)

~

MpiX)

ML~M (Y)~MU

M~ ~ Mpt(X) ~ M~ XL ~X~XU

(4.127)

242

4 Design Procedures

in which M(Y) is the vector of absolute values of moments due to external loadings and Mp,(X) is the vector of moment capacities of cross sections. The bounds on M(Y) are intended to limit the deviation from conventional elastic theory (moment redistribution). Such deviation (partial redistribution) is allowed by various standards under some specified conditions. The bounds on Mpl(X) and X are technological constraints on the cross-sectional dimensions. Additional constraints related to displacements and shearing forces might be considered in a similar manner [69]. It can be noted that the probleltl (4.127) is formulated in the form of (4.104). In addition, the present multilevel formulation is based on system oriented decomposition, as will be shown subsequently. To introduce the fast-level problem, determine a fixed value for the variables Y through the constraints Y =yo and denote the vector of design variables of the ith element as Xi. The assumption of fixed values for Y is equivalent to the common approach of assuming constant internal forces in the structure. If Z is additively separable, then the problem of optimizing X can be decomposed into the following s independent subproblems [see (4.106)]: find Xi such that (4.128)

max[M~i' Mi(yO)]SMp'i(Xi)SM~'i

xf SXi sxf in which subscript i denotes quantities associated with the ith element. Since the number of variables in each subproblem is small and the constraints are expressed explicitly in terms of the variables, solution of the flfSt-level problem is simple. To introduce second-level constraints, we have from (4.127) (4.129) These constraints are necessary conditions for a feasible solution of the flfSt-level problem. Thus, the task in the second-level problem is to find Y such that

H

s

s

i=l

i=l

=I, minZi =I, IIi (Y) ~ min

(4.130)

and the constraints (4.129) are satisfied. The step of elastic analysis and modifying the bounds on M(Y), if necessary, may be viewed as the third-level problem. Only at this level the structural analysis is repeated. Three-Level Steel Design. As a typical three-level design problem, consider the steel continuous beam shown in Fig. 4.23a. A similar approach has been demonstrated for reinforced concrete structures elsewhere [69]. The structure consists of five elements and is subjected to four ultimate load conditions.

4.4 Multilevel Optimal Design

243

Assuming an initial uniform cross section, the elastic envelope moments M are shown in Fig. 4.23b. Choosing the moments over the interior supports as the independent variables, Y1 and Y2 , the moments M(Y) are shown in Fig. 4.23c. For any assumed Y, the moments M I (Y) , M 3(Y), and M s(Y) can readily be determined by the equilibrium conditions. Two cross-sectional variables, Xii and X i2 , are assumed for each of the five elements (Fig. 4.23d). (i

The integrated problem formulation is: fmd yT = {Yh Y 2 } and = 1,... , 5) such that

0.02 10.01

(a)

0.01

I

0.02

0.02

I O.QI

0.01

1 ~

tl

i 1

I

Q""'"

1000

Q

1

'<'«'

l3

t4

1000

0.02

q~

"""

t t

1000

(b)

M2=Y I

(c)

(d)

M4 =Y2

-
Xn 1.0

Fig. 4.23. Steel continuous beam.

xf ={Xii

X i2 }

244

4 Design Procedures 5

Z=

L A; (XJli(Y) --+ min i=!

(4.131)

in which Ai(X i ) is the cross-sectional area. and li(Y) is the length of the ith element. selected as the distance between zero moment sections (see Figs. 4.23a and 4.23c). Formulating the problem in the design space. the number of independent design variables is 10. Assuming the three-level formulation. the number of independent variables is two. The fIrst-level problem can be decomposed into fIve subproblems. each of which having only two design variables. The threelevel formulation is as follows [69].

First-level subproblems. For Y = yo. fInd Xi (i

= 1•...• 5) such that

(4.132)

Second-level problem. Find yT = {flo f 2 } such that

L minZi L Hi (Y) --+ min 5

H=

5

=

i=!

i=!

(4.133)

i = 1•...• 5

Third-level problem. Elastic analysis is carried out and the bounds MiL and MY are modifIed. if necessary .

4.4 Multilevel Optimal Design

245

Example 4.15. To illustrate numerical results, the following values have been assumed (all dimensions are in tons and centimeters; due to symmetry, Y1 =Y2 =Y)

Aj(XJ =2(Xil + Xji) 11(Y) = Is(Y) = 1000 - Y/IO,

13(Y) = (1
12 (y) = 14 (Y) = 500 + Y /10 - .!.(100 - 400y)1I2 2

Mpl.(X j ) = 2.4Xil Xj2 +2.4Xj2 +0.6X!t

M~l. = 2200

M;l. = 800. Xj~ =

xf2 = 10.

xg = xg = 40

Ml (Y) = Ms(Y) = (100- Y /100)2/4 M2 (Y) = M4 (Y) = Y. M3 (Y)=2500-Y Mf =Mf =0.6xI81O.

Mf =Mf =1810

Mf=M~=0.6x2170.

M¥ =MY =2170

Mf = 0.6 x 1000.

Mf = 1000

The expressions for Aj(Xj) and M pl. (Xj) are based on the cross section shown in Fig. 4.23d. The lower and upper bounds on Mj are Mf = 0.6Mj (40% maximum moment redistribution) and MY = Mj , where Mj are the envelope moments. From the second-level constraints, we obtain the feasible region for Yas 1500

~

Y

~

1900

At the ftrst level we ftnd the optimal cross-sectional dimensions for any assumed Y. Evidently, the feasible region of the ftrst-Ievel subproblems depends on the assumed values of Y. Solving the first- and the second-level problems, the resulting optimal moments are

M; =M; =0.95M =1715

M; =0.79M =785 Elastic analysis of the optimal structure and modiftcation of the bounds ML and MU (third-level) do not affect the optimal solution. That is, two elastic analyses are sufftcient to obtain the ftnal optimum [69].

246

4 Design Procedures

4.5 Optimal Design and Structural Control Control forces, or control gains for displacements, might improve the behavior of a structure under given external loads. In general, two classes of structural control systems may be considered:

1. Passive control systems, where there is no external energy supply that may generate control forces. Passive viscoelastic dampers and tuned mass dampers, used in some tall structures to reduce their dynamic response, are examples for this class of control systems. Similarly, base isolation devices can be used to reduce buildings seismic response. 2. Active control systems that rely on the availability of an external energy supply and generally consist of a feedback control system that is designed to generate corrective forces. Several such systems have been studied analytically and experimentally. These include active tuned mass dampers, active tendons and active pulse systems. Passive control is most appealing from the reliability and maintainability points of view. Active control systems, on the other hand, are usually expensive and difficult to maintain, but they can be designed to control the dynamic response more precisely. Two design problems will be discussed in this section: 1. The problem of optimizing a passive structural control system where the structure itself is given (Sect. 4.5.1). Some considerations for selecting optimal control systems, intended to improve the structural behavior under static loads, are discussed. It will be shown that the problem can be stated in a linear programming form where the control forces or control gains for displacements are the independent variables. 2. The problem of improving the optimal design by control variables (Sect. 4.5.2). Potential savings in weight achieved by introducing passive control devices are demonstrated, and a solution procedure for the optimal design of controlled structures subjected to static loads is presented. 4.5.1 Optimal Control of Structures Problem Statement. Consider a given structure subjected to external static loads, with given displacements ro and stresses (Jo, determined by analysis of the structure. It is assumed that some constraints are violated due to excessive loads or other reasons, and it is therefore necessary to introduce control forces or gains for displacements to improve the behavior of the structure. Ideally, a system of control forces could be applied in directions opposite to those of the external loads. However, this solution is usually not practical due to the large number of forces. A reasonable criterion in selecting a system of control variables is to consider both the number and the magnitude of control forces. Mathematically, either control forces, Pc, or control gains for displacements, Y e, may be chosen as independent variables. Assuming control gains for displacements

4.5 Optimal Design and Structural Control

247

as variables and considering the displacement method of analysis, the displacements r(yc) and the stresses a(Y c) due to Yc are computed by [see (2.26) and (2.29)] K r(Y c) =R(Yc)

(4.134)

where the elements of K and S are constant. Considering truss structures for illusttative purposes, the elements of the load vector R(Yc) are determined by (Fig. 4.24a) (4.135) where E, aj and ti are given constants. The resulting load vector can be expressed as a linear function of Yc as (4.136) where the elements of matrix c are independent of Y c. Substituting (4.136) into (4.134), the following linear expressions are obtained (4.137) where the elements of matrices ryand ayare independent of Yc. In the absence of specific requirements, the objective function Z is assumed as a linear function of Yc' that is (4.138) where the elements of vector Zyare constant. To account for negative values of Yc' it is possible to express Z in terms of new variables X as

Z=Z~X

(4.139)

where X is the vector of absolute values of the elements of Yc' that is (4.140) Based on (4.137) through (4.140), the problem under consideration can be stated as the following LP [79]: find Yc and X such that

248

(a) (b)

(c)

4 Design Procedures

~

0

FW£i'/Yffff4l

rU ____ 0

Ycj ~

rU ----

Pci

IE

y--

Yei

Y

R / Ycj) ----

R/Ycj1

l,

~i

R/Ycj)

o _ _ rj2

o _ _ rj2

~I

~i

Fig. 4.14. Truss element: a. Fixed ends, effect of control gains for displacements, b. Effect of joint displacements, c. Final position.

Z=Z~X~min

(4.141)

Since there is a linear relationship between control gains for displacements Yc and the corresponding control forces Pc (Fig. 4.24c), the problem (4.141) can be stated as the following equivalent LP in terms of Pc: fmd Pc and X such that

Z=Z~ X~min

(4.142)

where the elements of Zp, rp and CJp are constant. The two LP formulations can readily be extended to include structures subjected to multiple loading conditions. Design Considerations. For statically determinate structures, Y c will not cause any stresses and forces, and the resulting displacements can be determined directly by kinematic relations. For statically indeterminate structures, any gains for displacements Y c will result in self-equilibrating forces and corresponding stresses in the structure. These forces and stresses can be introduced by a reduced set of gains for displacements corresponding to a selected set of redtmdant forces N. Assuming Pc = N, the displacements and the stresses due to Pc are given by

4.5 Optimal Design and Structural Control

249

(4.143) where rN and aN are matrices of stresses and displacements, respectively, due to unit value of N in the primary structure. It can be observed that equivalent stress distributions can be obtained by various selections of the redundant forces N. The number of possible selections of N might be very large in structures with high degree of statical indetenninacy. Unlike the stresses, the displacements depend on both the magnitude and the location of the control variables. That is, various selections of N may result in equivalent stress distributions but different displacement fields in the structure. Thus, one consideration for selecting the set of control variables is their effect on the displacement constraints. To find the optimal location and magnitude of the control variables, it is possible to assume fIrst control forces or control gains for displacements in all potential locations (members). This initial set of variables is optimized by solving the LP problem (4.141) or (4.142). It is expected that due to the nature of the problem, only a small number of the variables will be non-zero at the optimum. Indeed, experience has shown [79] that usually there are only a small number of non-zero control variables at the LP optimum. The LP procedure can also identify situations where the problem has no feasible solution. For cases where the LP solution does not provide a sufficiently small number of non-zero control variables, an alternative solution procedure has been proposed [79]. Example 4.16. The seven-bar truss shown in Fig. 4.25 is subjected to a single loading condition of two loads. Assume E = 30,000, aU = -aL = 25 and an upper bound on the vertical displacement at joint 0, r U = 2.0. The cross-sectional areas and stresses due to the given loads are shown in Table 4.7. The given vertical displacement at joint 0 is ro = 2.21. That is, the constraints aL $; a6 and ro $; r U are violated. Assuming fIve cases of a single control force (at members 1, 2, 4, 5 and 6), the resulting constraints on the control force are summarized in Table 4.8. Identical force distributions can be obtained by assuming each of the five control forces if only stress constraints are considered. The optimal solutions in this case are Pel = -5.80, P e2 = -5.80, or P,;04= -5.80. For both stress and displacement constraints the only optimal solution is P e2 = -7.55. Table 4.7. Cross sectional areas Member i 1 2 3 4 5 6 7

a; 10.0 10.0 4.0 4.0 8.0 4.0 6.0

ai

and stresses due 10 loads

~

17.66 -22.23 25

-5.86 21.82 -27.05 23.57

(J0i.

seven-bar truss.

250

4 Design Procedures

o 100 360

Fig. 4.25.

100 360

Seven-bar truss.

Table 4.8. Constraints on control forces. Control force

Stress constraints -18.00 S Pel S -5.80 -18.00 S Pe2 S -5.80 -18.00 S P c4 S -5.80 8.20 S PeS S 25.44 8.20 S Pc6 S 25.44

Displacement constraints Pel S 7.90 P e2 S -7.55 Pc4 S -21.68 PeS S 30.66 P c6 S -54.56

Stress and displacement constraints 7.90SPel S -5.80 -18.00 S Pe2 S -7.55 -18.00 S Pc4 S-21.68 30.66 SPeS S 25.44 8.20 S P c6 S -54.56

4.5.2 Improved Optimal Design by Structural Control Problem Formulation. Although the displacement method is the prevalent tool in current computerized structural analysis practice, the force method formulation has been adopted in this section due to the following reasons: 1. The structural behavior can often be expressed in terms of a small number of independent redundant forces N. 2. The presented two-level solution approach is most suitable for this formulation. 3. The redundant forces are often not sensitive to changes in the design variables, thereby allowing an efficient solution. Based on (1.22) and (1.68) the problem under consideration is formulated as the following NLP: find the cross-sectional design variables X and the control variables Y such that Z= tTX -+ min (J~X:S; Ap + ANN:S; (J~X XL:s;X :S;XU yL:s;y:s;yU

F(X, Y)N

=~(X, Y)

(4.144)

4.5 Optimal Design and Structural Control

251

For simplicity of presentation, linear dependence of Z and the allowable forces on X has been assumed. The basic difference between this formulation and conventional optimal design problems is that control variables Y are included in the compatibility equations, allowing improvements of the optimal solution. The elements of Ap and AN, computed in the primary determinate structure, are independent of the variables X and Y. Two-Level Design Procedure. Assuming Y = 0, the problem is reduced to conventional optimal design without control. The optimal solution in this case can be viewed as an upper bound (VB) on the optimum .Neglecting the compatibility conditions, the problem becomes independent of Y, and the resulting LP problem (1.68) provides a lower bound (LB) on the optimum. Solving the VB and the LB problems, we can estimate potential savings that can be achieved by introducing control devices. In addition, reasonable convergence criteria for the solution process can be established. Two approaches might be considered for solving the NLP problem (4.144): 1. Simultaneous optimization of all variables. One drawback of this approach is that the problem size is usually large and the computational effort might become prohibitive. In addition, since the variables are of different nature, numerical problems may arise during the solution process. 2. Successive selection of the design variables and the control variables. Starting with the LB solution, the control variables are then selected for the given design and the LB constraints are modified, if necessary. The rationale of this approach is that the goal is to achieve an optimal solution that is as close as possible to the LB solution.

Considering the second approach, it has been shown [76] that the final optimal solution is often identical or very close to the LB solution. Therefore, a small number of iteration cycles is needed to arrive at the optimum and the solution efficiency is greatly improved. Starting with the LB solution X~. N~. Z~, we first check the compatibility conditions. In cases where the latter conditions are violated it is still possible to find Y and N such that the LP solution is the final optimum, as will be shown subsequently. Assuming the formulation (4.144) and the given LP solution X~, the problem of optimal control (OPC) is formulated as follows: find Y such that

(4.145) In this formulation the implicit expression for N, obtained from the compatibility conditions, is substituted into the stress constraints. One approach to obtain a feasible solution of the OPC problem is to use the KS function (4.39), where all the constraints (4.145) are expressed in the form #Y) :5: O. If the optimal solution,

252

4 Design Procedures

, Y~pc, does not satisfy all the constraints, no feasible control forces can be achieved and it is necessary to modify the LB problem constraints. Several methods can be used for this purpose [76]. Based on these considerations the following two-level design procedure can be

used: 1. The LB problem (1.68) is solved to obtain X~ (fIrst-level solution). 2. The ope problem (4.145) is solved to obtain Y~pc satisfying all the constraints (second-level solution). 3. If all the constraints are satisfIed, the fInal optimum is reached. Otherwise the LB constraints are modifIed. Steps 1 through 3 are repeated as necessary until the fInal solution is reached. It has been observed that in many cases the optimum is reached in one iteration cycle [76]. Example 4.17. The grillage shown in Fig. 4.26a is subjected to a single load P, acting at point C. The structure is simply supported at points A. E. F and J. A uniform cross section with EI Cl=-=2.5 GJ

has been assumed, where EI is the flexural rigidity and GJ is the torsional rigidity. A single design variable, X, representing the bending moment capacity is assumed, and the objective function is Z = 36 X . Two types of control devices are demonstrated in this example:

(a)

: l

t

3.75

t

0 3.75

rcJS:J Y=F

t

3.75

C

C

(b)

:}

I

C

B

3.75

1

;

I

A

J

H

G

(c)

[ltD 7 --.JfY=5 0

Fig. 4.26.

Grillage example.

4.5 Optimal Design and Structural Control

253

::J:::::~

(a)

lB75P

0

J.B75P

1111111111111111111111111111"'111111111111111111111111111

O.7P

(b)

I.B75P 3.05P

Fig. 4.27. Optimal moment diagrams, grillage example: a. LB and NLP solution b. UB solution.

=

1. A Linear Spring Device (LSD) at point C, whose flexibility Y F is designed to achieve the desired displacements under the applied loads (Fig. 4.26b). 2. A Limited Displacement Device (LDD) at point C, where the maximum displacement Y =50 is limited by the control device (4.26c). One basic difference between the two types of control devices is that for the LSD a unique statical scheme is considered, whereas for the LDD this scheme is changed after the maximum displacement occurs. A possible problem with the LSD is that flexibility and strength are two conflicting factors that should be considered in the spring design. Results for the LB (1.68), the complete NLP with control (4.144), and the UB (Y 0, optimal design without control) problems are shown in Fig. 4.27 and in Table 4.9. It can be seen that the control variable Y greatly improves the optimal moment diagram. That is, the conventional optimal structure (UB solution) is 62% heavier than the NLP optimal structure, which is identical to the LB optimum.

=

Table 4.9. Summary of results, grillage example. Problem 1B NI.P UB

Xi 1.875P 1.875P 3.050P

Z*

67.5P 67.5P l09.8P

254

4 Design Procedures

4.6 Geometrical Optimization Most of the work that has been done on optimum structural design is related to optimization of cross sections. Much less effort has been devoted to optimization of the geometry and relatively little work has been done on optimal topologies [73, 136]. It is recognized, however, that optimization of the structural layout (geometry and topology) can greatly improve the design. That is, potential savings affected by layout optimization are generally more significant than those resulting from fixed-layout optimization. Layout optimization of discrete problems by general optimization methods is presented in Sects. 4.6 through 4.8. Recent work on related subjects, such as optimization of continuous problems and solution by optimality criteria methods, are discussed elsewhere [7,8,113]. Because of the complexity in simultaneous optimization of the geometry, the topology and the cross sections, two classes of problems are often considered in optimization of the structural layout:

a. Geometrical optimization, where the coordinates of joints and the cross-

sectional sizes are treated as design variables and optimized simultaneously. In general, the design variables are assumed to be continuous and numerical search algorithms are used to find the optimum. The topology is usually assumed to be fixed, unless some of the joints coalesce during the solution process. The geometrical optimization problem is presented in this section. b. Topological optimization, where members are removed from a highly connected structure, called the ground structure, to derive an optimum topology with the corresponding cross sections. That is, in a topological optimization problem both the topological and the sizing variables are optimized simultaneously. The topological optimization problem is discussed in Sect. 4.7. In problems where the object is to optimize the geometry, the topology and the cross sections, multistage design procedures might prove useful. Interactive layout optimization, where structural optimization methods and graphical interaction are combined to achieve an effective design procedure, is demonstrated in Sect 4.8. Consider the following NLP problem of geometrical optimization: find the geometrical variables Y and the cross-sectional variables X such that Z = f(Y, X)

~

min (displacement constraints) (stress constraints)

(4.146)

4.6 Geometrical Optimization

255

In general, upper and lower bounds on design and behavior variables are assumed to be constant. If stability of members is considered, the elements of oL are functions of both Y and X. This situation will be discussed later in this section. The difficulties associated with the common problem of cross-sections optimization (Le., the implicit nature of the behavior constraints and the large numbers of variables and constraints) are magnified in problems of geometrical optimization. That is, the problem size is increased and the need for multiple repeated analyses is a major obstacle in applying optimization methods to large scale structures. In addition, changes in the geometrical variables may be of different magnitude than changes in cross sections. Experience has shown that combining these two types of variables, which are of fundamentally different nature, may produce a different rate of convergence and ill-conditioning problems. Thus, two-level optimization is most suitable. In this section two approaches for solving the geometrical optimization problem will be demonstrated: - Simultaneous optimization of all design variables (Sect. 4.6.1). _ Multilevel optimization, where the two types of variables are optimized in different levels (Sect. 4.6.2).

4.6.1 Simultaneous Optimization of Geometry and Cross Sections Considering the displacement method of analysis, the elements of the stiffness matrix K are functions of both Y and X, while the elements of the load vector R and the stress transformation matrix S are usually functions of only Y. Assuming the first-order Taylor series expansion about the design {yo, X*}, the problem (4.146) can beapproximated as the following LP [see (3.51)]: find X and Y such that r L :5: r * + Vrx* (X - X *) + Vry*(Y - Y *):5: r U * *)+VO"y(Y-Y • *):5:0" U 0"L :5:0" * +VO"x(X-X

(4.147)

X·L:5: X :5:X·u in which all derivatives are computed at X*, Y*, the elements of the bounds on Y and X are given by

Y

X

.L

{y

L

= max. L Y - 6Y

.L =max. {XL X

- 6X

L

Y

X

.U

.

{yu

= mm. Y

.U = mm. . {XU

+ 6Y

X + 6X

U

(4.148) U

256

4 Design Procedures

and _!!lyL. !!lYu • _IlXL • !!lXu. are vectors of given move limits [see (3.5.2)]. Using the SLP procedure (Sect. 3.1.3). the LP problem is solved iteratively. X· and Y· redefined each time as the optimum solution to the preceding problem. In addition. it might be necessary to modify the move limits in each iteration. As noted in Sect. 3.1.3. we may start with relatively large move limits and end up with limits corresponding to the required accuracy. One possible procedure to deal with the geometrical variables is to increase or decrease the corresponding move limit for each step of iteration. If a coordinate change goes in the same direction twice we increase the move limit. If. instead. the change is opposite to the one before. then we decrease the move limit. Example 4.18. Consider the symmetrical bridge truss shown in Fig. 4.28 (all dimensions are in newtons and meters). The bridge is subjected to five alternative loading conditions Pq =3 X 106 (q = 1•..• 5). The geometrical variables are the coordinates of the five arch joints. which lie initially on a parabolic arch. To consider stability constraints. assume a factor of safety against buckling. 'YE. in the elastic range. In the plastic range. a parabolic design formula is suggested that guarantees a continuous transition from the elastic range [104]. In order to know whether we are in the elastic or the plastic range. defme the limiting forces

AP

= - (~r

E~E

(a)

where aP is the stress at the limit of proportionality between stress and strain and a is a cross-sectional constant given by (b) rG

being the mdius of gyration. Defming

Fig. 4.28. Initial geometry of bridge truss.

4.6 Geometrical Optimization

257

we find for the allowable compressive forces in the elastic range for

(d)

For the suggested design formula in the plastic range, the necessary area given by

Xmin

is (e)

where crA is the allowable compressive stress corresponding to slenderness ratio

equal to zero. From (e) it follows that for the allowable compressive forces in the

plastic range

for Determining AL by (d) or by if) we may find crL and derivatives of the allowable compressive stresses can now be computed. The corresponding terms of cr*L + Vcr;L(X - X*) + Vcr;L(y - y*) are substituted in (4.147) in place of crL.

Four cases of constraints have been solved, as summarized in Table 4.10 [104]. The assumed fixed bounds on stresses, for the cases in which stability considerations are neglected, are

The parameters for calculation of bounds on compressive stresses when stability is

considered are

'YE

= 2.5

a.

= 0.8

and the modulus of elasticity and density are

E

= 2.1xlO11

p

= 7850 kg/m 3

Table 4.10. Constraints on stability and displacement, bridge truss. Case a b c d

Stability constraints Not considered Not considered Considered Considered

Maximum displacement Not considered 0.01 Not considered 0.01

258

4 Design Procedures

._

~

~

:

. . 6.29

1

5

·1~13.62

~-

(c)

1. 5.78

Cd) . cases a , b, c, d [104]. Fig. 4.29. Optimal geo metrles,

4.6 Geometrical Optimization

259

Table 4.11. Comparison of results, bridge truss [104]. Case Minimum weight, fixed geometry (kg) Minimum weight, variable geometry (kg) Percentage of change in weight

a 1868 1656 11.3

b 3526 2911 17.4

c 3141 2905 7.5

d 3668 3315 9.6

The optimal geometries are shown in Fig. 4.29. It can be seen that the horizontal changes of the positions of the joints are toward the supports for the four cases. To illustrate the effect of changes in the geometry, the above four cases were optimized also for the fixed initial geometry shown in Fig. 4.28. Comparison of results, given in Table 4.11, shows that savings in weight due to changes in geometry are 7.5 to 17.4 percent. Some of the optimal solutions are not fully stressed designs. 4.6.2 Approximations and Multilevel Optimization It has been noted in Chap. 3 that approximation concepts can greatly reduce the computational effort involved in cross-sections optimal design. Similar approximations may be applied also in geometrical optimization. Multilevel optimization, discussed in Sect. 4.4 is most suitable for the problem under consideration, where different types of variables are involved. It combines efficient suboptimization for member sizes, reduction in the number of design variables optimized simultaneously and improved convergence. In this section, approximation concepts and multilevel optimization are combined to achieve effective solution procedures. Consider again the geometrical optimization problem (4.146). Assuming the force method of analysis, the problem can be formulated as follows [see (1.64)] : find Y and X such that Z=j(Y,

X)~min

(4.149)

The redundant forces N are given in terms of Y and X by the compatibility equations (1.22). If the latter conditions are neglected [see (1.65)], the forces N are included in the set of independent variables, and the problem (4.149) is reduced to an approximate explicit problem (AEP). Similar to cross-sections optimization, solution of the AEP can be viewed as a lower bound on the optimum. The

260

4 Design Procedures

compatibility conditions can then be considered to modify the lower bound optimum. The advantage of this approach is that the AEP solution does not involve multiple explicit analyses and can readily be obtained. Multilevel Formulations. The AEP can be stated in a two-level form [66], where the fIrSt-level variables X and N are optimized for fixed values, Y yO, of the geometric variables. Since solution of the fIrSt-level problem consumes most of computational resources, it is essential to employ efficient optimization methods at this level. It has been shown that if the linear relations (1.67) are assumed and the displacement constraints are not considered, the problem is reduced to the LP form (l.68). To consider displacement consttaints in the LP formulation, the first-order Taylor series expansion of the displacements about X*, N* may be used. Adding the linearized displacement constraints, we may solve an augmented LP problem, where the steps of linearization and solution of the modified LP problem are repeated until convergence. The second-level problem is to fmd Y such that the consttaints are satisfied and the objective function is minimized. The solution efficiency is highly dependent on the number of candidate geometries at the second-level problem. The concept of design variable linking, often used in cross-sections optimal design, can be applied also in geometrical optimization. The number of design variables is reduced by expressing all the geometric variables in terms of a small number of independent ones. Moreover, design variable linking is often necessary due to such considerations as functional requirements, fabrication limitations, etc. Another possibility to reduce the number of candidate geometries at the second level is to use a coarse grid in the space of geometric variables, so that only a small number of Y values is considered. This is justified in many cases where the objective function is not sensitive to changes in the geometric variables near the optimum. To optimize the Y variables at the second level, one of the unconstrained minimization methods presented in Sect. 2.2.2 can be used, with provisions taken to ensure satisfaction of the side consttaints on Y. Since compatibility conditions are not considered, the two-level solution of the AEP will provide a lower bound on the optimum. Then, the precise fIrst-level problem (cross-sections optimization) can be solved for the final geometry. It has been found [66] that the optimal geometry is usually not appreciably affected by the compatibility conditions. Therefore, the compatibility conditions may be considered only for the fmal geometry obtained by solving the AEP, that can can be viewed as a near optimal solution. If the degree of statical indeterminacy is low, it might be advantageous to employ the following three-level solution procedure:

=

1. Assume initial geometry and force distribution. 2. Optimize the cross-sectional variables. 3. Modify the geometric variables (third level). Each trial geometry involves optimization of the force distribution (Step 4). 4. Modify the force distribution (second level). Each trial force distribution involves cross-sections optimization (Step 5). 5. Optimize the cross-sectional variables (fIrSt level).

4.6 Geometrical Optimization

261

If displacement constraints are not considered, Steps 2 and 5 (the fIrst level) are reduced to a direct determination of the cross-sectional variables. The choice of Y in Step 3 (the third level) is subject to the side constraints on Y. It should be noted that all intermediate values of the variables Y (two-level formulation) or the variables Y and N (three-level formulation) are feasible. That is, the iteration can be terminated always with a feasible - even though nonoptimal - solution, whatever the number of cycles. While the three-level formulation may prove useful for some specifIc applications, the two-level procedure is usually most effective [66].

Example 4.19. The symmetric four-bay frame shown in Fig. 4.30 is subjected to a single loading of six loads (all dimensions are in tons and meters). The allowable stresses are aU = -aL = 15,000. For simplicity of presentation, the following relationships have been assumed for the three cross-sectional design variables i = 1,2,3 in which ai = the i th cross-sectional area ; Wi = the modulus of section ; and Ii = the moment of inertia. Two geometric variables (Y1 and Y:0 have been considered with the following side constraints

The objective function is

Assuming a grid of points with intervals of 2.0 in the geometric variables space, the lower bound solution of the AEP is [66] Y1 = 12.0

Y2 = 4.0

Z = 602

The exact solution of the cross-sections optimization problem for this geometry is Z =682, and the true optimum for the assumed grid is

I 5.0 I 10.0 I lIE

lIE

~

)I

lIE

10.0

2.0--o~_-I---r---l_..,----L_.-----,-_..-_2.0

x,

x,

XI

X,

I 20.0-Y I ....

;Jr .'1(

I';

~

I

lIE

I';

I 20.0-Y I

)1'.

lIE

Fig. 4.30. Four-bay frame example.

..

J

262

4 Design Procedures Y1 = 14.0

Y2 = 4.0

Z = 668

That is, the true optimum is only 2% lighter than the final result obtained from the AEP solution (Z = 682).

4.7 Topological Optimization 4.7.1 Problem Statement The fact that topological optimization did not enjoy the same degree of progress as fixed-layout optimization may be attributed to some basic difficulties involved in the solution process. These difficulties make the problem perhaps the most challenging of the structural optimization tasks [73, 136]. One basic problem is that the structural model is itself allowed to vary during the design process. Discrete structures are generally characterized by the fact that the finite element model of the structure is not modified during the optimization process. In topological design, however, since members are added or deleted during the design process both the finite element model and the set of design variables change. These phenomena greatly complicate the design and analysis interactions. Another difficulty is that the number of possible element-joint connectivities grows dramatically as the number of possible joint locations is increased. This number might be very large particularly in structures of practical size. One solution to this problem is to specify a reduced set that does not include all possible element-joint connectivities. However, a fundamental disadvantage of this approach is that the optimal solution may not be included in the specified set. An additional difficulty that might be encountered during elimination of members is that the problem can have singular global optima that cannot be reached by assuming a continuous set of variables. This suggests that it may be necessary to represent some design variables as integer variables and to declare the existence or absence of a structural element An example of an integer topological variable is a truss member joining two nodes which is limited to the values 1 (the member exists), or 0 (the member is absent). Other examples of integer topological variables include the number of spans in a continuous beam or the number of elements in a grillage system. While methods for integer variables have been developed, these methods are still computationally costly for practical engineering applications. The above mentioned considerations have lead to various approximations and simplifications in the formulation of topological optimization problems. These include: -

approximate analysis models (e.g. rigid plastic); consideration of only certain constraints (e.g. stress constraints); simplified objective function (e.g. weight); consideration of simple structural systems (e.g. trusses); simplified sizing variables (e.g. cross-sectional areas); consideration of a limited number of loadings.

4.7 Topological Optimization

263

Most studies on optimal topologies deal with truss structures. This may be attributed to the fact that the truss by its very nature is most suitable for optimization of the topology. It possesses usually many nodes and elements that can be deleted or retained without affecting the functional requirements. In addition. the truss is a relatively simple. yet nontrivial. structure. It is therefore an ideal system for the demonstration of some properties and characteristics associated with optimal topologies. Although most of the discussion in this section is related to trusses. the extension to other classes of structure is often straightforward. While the displacement method formulation is the prevalent structural analysis tool in current computational practice. the force method formulation is adopted in many topological optimization problems due to several reasons:

a. The effect of some approximations can be evaluated directly. h. The analysis model is convenient to demonstrate properties of optimal topologies. c. A linear programming formulation is obtained under certain assumptions. The Ground Structure Approach. Most topological optimization studies are based on the assumption of an initial ground structure that contains many joints and members connecting them. Member areas are allowed to reach zero and hence can be deleted automatically from the structure. This permits elimination of uneconomical members during the optimization process [22. 52. 111]. In general. the problem of optimizing the topological and sizing variables subject to general constraints and objective function can be stated in a nonlinear programming form. For illustrative purposes. the following simplifications are assumed in the present formulation:

a. The objective function represents the weight and can be expressed in linear

terms of the sizing variables. h. Only stress constraints and side constraints on the sizing variables are considmld. c. Elastic compatibility conditions are temporarily neglected.

Assume a grid of points that may be connected by many potential truss members to form a ground structure with a finite number of members. Considering the above mentioned simplifications. the problem can be stated in the LP form (1.69): fmd the cross-sectional areas X and the members' forces A such that Z=lTX~min

(objective function) (stress constraints) (4.150) (side constraints)

CA

=R

(equilibrium conditions)

264

4 Design Procedures

where the stress constraints and the equilibrium conditions are formulated for all loading conditions. It has been noted that the LP problem (4.150) can be expressed in terms of the redundant forces N as [see (1.68)]: find X and N such that Z=(fX~min

(4.151)

In this formulation the equality constraints have been eliminated and the number of variables is reduced (only the redundant forces are considered as independent variables instead of all member forces). The stress constraints are formulated for all loading conditions. One advantage of the LP formulation is that the global optimum is reached in a finite number of steps. Thus, large structures with many members and joints can efficiently be solved. In addition, problems of singular optimal topologies are eliminated and some effects of the optimal solutions can be evaluated directly. In the case of zero lower bounds on cross sections (4.152) the LP method has the ability to make unnecessary members vanish from the structure to obtain the minimum weight design. The optimal structure in such cases satisfy equilibrium and stress constraints, but it might not satisfy compatibility conditions or may represent unstable configuration under a general loading. That is, the LP solution is not the final optimum and some modifications might be required. However, this solution may be viewed as a lower bound on the optimum. The various types of the optimal topologies are discussed subsequently in Sect. 4.7.2. Some properties of these topologies are introduced in Sect. 4.7.3 and two-stage design procedures are presented in Sect. 4.7.4.

4.7.2 Types or Optimal Topologies Assuming an arbitrary initial ground structure and solving the LP problem for zero lower bounds on cross sections, the resulting optimal topology might represent one of the following types of structure: - statically determinate structure (SDS); - mobile structure, or mechanism (MS); - statically indeterminate structure (SIS). Elimination of elements from the structure will change the numbers of variables and active stress constraints at the optimum. If buckling constraints are considered the constraints become nonlinear, and the areas of compression elements might not converge to zero. In this case, as the area of a compression element decreases the buckling stress also decreases, until it becomes critical; then the area of the element will increase.

4.7 Topological Optimization

265

Statically Determinate Structures. Solving the LP problem, it is possible that the resulting optimal design will represent an SOS. That is, for a certain selection of the redundant forces we obtain at the optimum N =O. In this case the element forces A are constant, compatibility conditions can always be satisfied and the LP solution is the final optimum. It has been shown [22] that for structures subjected to a single loading condition with XL 0, the optimal solution indeed will represent an SOS. This result, while not guaranteed, might be obtained also for structures subjected to multiple loading conditions.

=

Mobile Structures. The number of members eliminated during the LP solution might be larger than the degree of statical indeterminacy and the optimal topology might represent an MS. The optimal structure in this case will satisfy equilibrium and stress constraints but there are unstable members. This situation may occur for certain geometries or loading conditions, where the forces in some elements change from tension to compression or vice versa. Such elements are not required to maintain equilibrium for that particular geometry or loading condition, and will be eliminated during the LP solution process. To overcome this difficulty it is possible to add members to the optimal configuration. They are not needed to satisfy equilibrium, but they are required to satisfy the necessary relationship which exists between the joints and the members in a stable structure. Example 4.20. The seven-bar truss shown in Fig. 4.31 is subjected to three loads acting simultaneously: P lo P z and P 1 - Pz. Optimal topologies for various PdP z ratios are shown in Fig. 4.32. The main results are [72]: For P z < P 1/2 and P1/2 < P z < P 1- the optimal topologies are SOS (Figs. 4.32a and 4.32c). For P z =Pd2 (Fig. 4.32b) the optimal topology is an MS that can be viewed as a particular case of the former two topologies. A diagonal element (Fig. 4.32a) or the vertical element (Fig. 4.32c) become zero for this loads ratio. For P 1=P z (Fig. 4.32d) the optimal topology is reduced to a single mobile vertical element

-

These results illustrate how mobile topologies might be obtained for particular loading cases.

~

100

Fig. 4.31.

Seven-bar truss.

266

4 Design Procedures

; ffi P2 (= PI/Z)

/

/

I I

t

~

(a) Fig. 4.32.

(b)

t

~-P2

(c)

(d)

Optimal topologies, seven-bar truss.

Statically Indeterminate Structures. In cases where the optimal LP solution X LP , NLP , represents an SIS, compatibility conditions might not be satisfied. The vector M LP , defmed by [see (3.156) and (3.157)] (4.153) indicates the discrepancy in satisfying the compatibility conditions by the optimal LP solution. The subscripts LP in (4.153) denote optimal values of the LP problem. For certain geometries or loading conditions it is possible that (4.154) That is, the compatibility conditions are satisfied for the optimal SIS and the LP solution is the final optimum. A case of particular interest, discussed in the next section, is one where the LP possesses an infinite number of optimal solutions representing multiple optimal topologies. If the conditions (4.154) are not satisfied for the optimal SIS, it is still possible to maintain compatibility by applying a set of prestressing forces (Sect. 4.1.2) or control forces (Sect. 4.5.1). In cases where such forces are not considered, the optimal LP topology is a lower bound on the optimum and may be used as a reference for comparison with other solutions. Analysis of the optimal LP structure will give the resulting elastic forces N EL • A certain deviation from elastic force distribution is often allowed on account of the inelastic behavior. The allowable deviation can be defmed by the set of linear constraints (4.155) where d, ~, and d U are matrices of given parameters. In cases where the constraints (4.155) are not satisfied for any loading condition, modifications of the LP optimum are required. Several procedures have been proposed for this purpose [73, Ill, 127], some of which have been discussed in Sect. 4.1.2.

4.7 Topological Optimization

267

4.7.3 Properties of Optimal Topologies

Singular and Local Optima. One reason for excluding compatibility conditions from optimization of the ground structure layout is that the computational effort in the solution process is considerably reduced. Another difficulty in solving the complete problem is that the optimal topology might correspond to a singular point in the design space. This occurs since a change in the structural topology (elimination of one or more elements) will result in a corresponding modification of the compatibility equations and elimination of some constraints previously included in the problem statement. That is, a change in the structural topology might change the design space. If the optimal solution is a singular point in the design space, it might be difficult or even impossible to arrive at the true optimum by numerical search algorithms. The singularity of the optimal topology in truss structures was first shown by Sved and Ginos [135]. Singular optima of grillages have been studied later [71, 82]. To illustrate the effect of neglecting the compatibility conditions on singular optimal topologies, consider two cases of stress and displacement constraints: a. The accurate constraints, obtained by substituting (1.22) into (1.23) and (1.24) (4. 156a) (4. 156b)

b. The approximate constraints, where the compatibility conditions are neglected and both X and N are independent variables. From (1.23) and (1.24) we have (4.1 57a)

(4.157b) It will be shown in example 4.21 that elimination of members might change the design space, if the accurate stress constraints (4.156a) are considered. Problems of discontinuity due to deletion of members cannot occur if the linear approximate constraints (4. I 57a) are assumed. Assuming the accurate displacement constraints and decreasing the cross section of a specific member, its relative stiffness and internal forces are reduced. Consequently, its contribution to the accurate displacement expression (4.156b) might approach zero. That is, deletion of a member will not result in discontinuity or singular solutions similar to those obtained in the case of stress constraints. Considering the approximate displacement constraints (4.157b), the result might be meaningless in cases where members are eliminated from the structure [77]. Another difficulty that might be encountered in topological optimization is that several local optima representing different topologies might exist. Local optima representing three different topologies are shown in Fig. 1.17 for the grillage of Fig. 1.16a. Evidently, local optima do not exist in the LP problem.

268

4 Design Procedures

Example 4.21. Consider the three-bar truss shown in Fig. 4.33 and subjected to a single vertical load. The members length is [=l.0, and the allowable stresses are crU =-crL= 20.0. The design variables are X I =cross-sectional area of member 1, X2 =cross-sectional area of members 2 and 3, and the accurate stress constraints (4.1500) are

- 20 XI

~

30 Xl 3XI + X2

- 20 X2

~

14.14 -

- 20 X2

~

-10 +

~

20XI

42.42 XI 3XI + X2 30 XI 3XI + X2

~

~

20X2

(a)

20X2

Choosing the force in member 1 as a redundant N, the approximate stress constraints (4.157a) are N ~ 20 XI

- 20 XI

~

- 20 X2

~

14.14 - 1.414 N

- 20 X2

~

-10 + N

~

~

20 X2

(b)

20 X2

The design spaces for the accurate stress constraints and for the approximate stress constraints are shown in Fig. 4.34. To illustrate the singularity phenomenon, consider a parametric objective function Z = aXl + X 2

(c)

where a. is a non-negative parameter. Assuming fIrst the accurate constraints (a) and a numerical search procedure, the solution process will converge to the following points (Fig. 4.34a)

10.0

Fig. 4.33. Three-bar truss, vertical load.

4.7 Topological Optimization

269

X, (a)

®

~ ?

'"-:"

(I)

~

0.5

1.0

1.5

X,

2(q. ?: 3)

X,

!i

0.5

X,

Fig. 4.34. Design space: constraints.

B.

Accurate stress constraints, b. Approximate stress

Point A:

Xl

=0.5,

X2 =0

(for a S 3)

Point B:

Xl

=0,

X2 = 1.5

(for a

(d) ~

3)

It should be noted that at point B member 1 and the corresponding active constraint are eliminated, and the line segment BC is included in the feasible region. Therefore, the solution (d) is incorrect for certain a values and the true optimum points are given by (for a S 1.414)

Point A: Point C:

Xl

=0,

X2 =0.707 (for a

(e) ~

1.414)

That is, for 1.414 < as 3, the solution process will converge to point A and for a ~ 3 the solution process will converge to point B while the true optimum in both cases is at point C. For the approximate constraints (b) a two-dimensional feasible region can be introduced for any assumed N. Eliminating the latter variable, the resulting design space is shown in Fig. 4.34b and the solution process converges to the true

270

4 Design Procedures

optimum points (e). Since both points A and C represent statically determinate structures, compatibility conditions can always be satisfied and the approximate solution is the final optimum. Multiple Optimal Topologies. Solving the LP problem it is possible that, for certain geometries, identical optimal objective function values are obtained for multiple force distributions in the structure. The various optimal force distributions usually correspond to several different optimal topologies. Some properties of such particular geometries, where the optimal objective function value becomes independent of the force distribution, are demonsttated subsequently. For purpose of illustration, assume a structure subjected to a single loading condition such that each element is fully stressed. From (4.157a) we obtain for the force in the ith member Ap., + A I,. N =a~" X. nl

(4.158)

af

in which at is the allowable stress (at = or at = aT) and A ~i is the it h row of matrix AN' Substituting Xi from (4.158) into Z yields

-+ (Api

~ t·

Z = £.J

+

T ANi

N)

i=l ai

(4.159)

Defining the constant Zo by ,. lZo= L-+Api i=l a i

(4.160)

and substituting (4.160) into (4.159) yields

~ t·

Z = Zo + £.J

-T ANiT N

i=l a i

(4.161)

The objective function will become independent of the force distribution in the structure if

(4.162) The conditions (4.162) form a system of nR equations (nR = the number of redundant forces) expressed in terms of the geometric parameters of the structure. In some cases it is possible to find a geometry satisfying these conditions, where multiple optimal topologies (MOT) are obtained. MOT occur in cases where the

4.7 Topological Optimization

271

objective function contours are parallel to the boundary of the feasible region, and the LP problem possesses an infinite number of optimal solutions and several corresponding optimal topologies. Situations where MOT occur can be identified from the LP solution. Consider the LP problem where all inequalities are transformed into equalities by adding surplus or slack variables S. Assume a basic feasible solution, given in the canonical form

(4.163) where the elements of a, b, c and Zo are constant, and subscripts B and N denote basic and nonbasic variables, respectively. The LP solution is optimal if Cj ~ 0 (j=I, 2, 3). If the latter conditions hold and at least one C; equals zero, then the problem might have multiple basic optimal solutions representing MOT. If C; < 0 at least for one nonbasic variable, the solution is nonoptimal and an improved basic feasible solution with a corresponding topology can be introduced. Assume an MOT geometry where all basic optimal solutions with a corresponding set of T optimal topologies have been determined. The basic optimal topologies can be used now to introduce new optimal topologies, not corresponding to the basic solutions. This can be done by assuming linear combinations of the basic feasible solutions X t , Nt [78]

L atX t T

X =

tal

=L

(4.164)

T

N

atNt

t=l

in which at are some coefficients satisfying

o~ T

at

~

L at = 1

1

t= 1, ... ,T

(4.165)

t=l

Equations (4.164) and (4.165) provide multiple optimal cross-sectional areas and force distributions corresponding to all optimal topologies. New combined optimal topologies are obtained for some at values other than 0 and 1. Based on these definitions, any specific basic optimal topology is given by a certain at equals

272

4 Design Procedures

unity and all the remaining at equal to zero. The basic optimal topologies are usually statically detenninate structures, therefore the LP solution is the final elastic optimum.The combined topologies, on the other hand, represent SIS but compatibility conditions might be satisfied. An important property of the MOT geometry is that new optimal topologies are introduced from existing basic optimal topologies by combination rather than the common ground structure approach of elimination.

Example 4.22 [78]. The fifteen-bar symmetric truss shown in Fig. 4.35 is subjected to three loads P acting simultaneously. Assume nonuniform depth with a single geometric variable Y, allowable stresses (Ju=-aL=20.0 and P=10.0 (all dimensions are in kips and inches). Two SDS basic optimal topologies are obtained for Y = 100 (Figs. 4.300, 4.36b), and a combined SIS optimal topology (Fig. 4.36c) is introduced by

Variation of Z" with Y and the corresponding optimal topologies are shown in Fig. 4.37. A mobile topology (Fig. 4.36d) is obtained for Y 75, where active constraints in the internal members change from tension to compression. Assuming unifonn depth Yand two fixed supports (Fig. 4.38), nineteen optimal topologies can be introduced (Fig. 4.39). The five basic SDS optimal topologies are shown in Fig. 4.39a, four of which being mobile and two are asymmetric structures. However, these two types of structure are needed to introduce the complete set of fourteen combined SIS optimal topologies (Fig. 4.39b), five of which also being mobile structures. Variation of Z' with Y is shown in Fig. 4.40. It can be seen that the optimum for fixed supports (Z' = 632.4), is significantly better then the optimum for nonfixed horizontal support (Z' = 836.7). In both cases Z is not sensitive to changes in Y near the optimum. It has been found that compatibility conditions are satisfied for the combined SIS optimal topologies.

=

4x 100

Fig. 4.35. Fifteen-bar truss, nonuniform depth.

4.7 Topological Optimization

(a)

(b)

(e)

(d)

Fig. 4.36. Optimal topologies, fifteen-bar truss, nonuniform depth.

z· 1400

1300

1200

1100

1000

900

~~~~~~~~~~---y Fig. 4.37. Variation of Z· with Y, fifteen-bar truss, nonuniform depth.

273

274

4 Design Procedures

l

p

p

p

1

G

1

YI~ 4 x 100

.It

Fig. 4.38. Fifteen-bar truss, uniform depth.

(a)

2

3

I' ~ _.:.11'-

~ -

, I "

4

5

/IVTVr'\.

~

Lj,~~~

(b)

, I "

L_~:l_~

6

7

8

II

9

10

~ /--~~-~

~

12

13

14

15

16

17

,

18

I "

19

Fig. 4.39. Optimal topologies, fifteen-bar truss, uniform depth.

4.7 Topological Optimization

275

z· Non fixed horizonlal support

900

800

1'"

0

167.3

700 Z' ·632.4 600

L,

1'" .126.5 I

50

,

75

I

,

!

I

!

,

100 125 150 175 200 225

Y

Fig. 4.40. Variation of Z· with Y, fifteen-bar truss, uniform depth.

4.7.4 Approximations and Two-stage Procedures Two-Stage Design Procedures. The difficulties involved in the solution of topological optimization problems have motivated two-stage design procedures. A typical such procedure is to evaluate an approximate solution at the frrst stage and modify it at the second stage to achieve a near optimal design. Various formulations may be assumed at the first stage for the following two cases of lower bounds on X :

a, XL =£ ( > 0). No members can be eliminated from the ground structure and

the accurate NLP formulation (4.156) will give the optimum of this problem. It has been noted that the approximate formulation (4.157) usually will give a better solution, since compatibility conditions are not considered. Assuming mixed formulation, with approximate stress constraints (4.157a) and accurate displacement constraints (4.156b), the resulting frrst-stage solution will be in between the former two cases. b. XL = O. It has been shown that the accurate formulation (4.156) might lead to incorrect results in the case of a singular optimum. To overcome this difficulty, it is possible to assume the mixed formulation, or the approximate LP formulation (4.151), where compatibility conditions and displacement constraints are neglected. The approximate displacement constraints cannot be used if members are eliminated from the structure. The various possibilities for the first-stage formulations are given in Table 4.12 [77]. In summary, approximations assumed at the first stage may include: -

Approximations of stress constraints. Elimination of displacement constraints. Introduction of side constraints XL =£ •

276

4 Design Procedures

Table 4.12. Alternative first-stage formulations. Formulation Accurate Mixed Approximate Accurate Mixed Approximate

Size constraints XL =£ XL=O

Stress constraints Accurate Approximate Approximate Accurate Approximate Approximate

Displacement constraints Accurate Accurate Approximate Accurate Accurate Neglected

The criteria for selecting the type of approximation could reflect both the solution efficiency and the chance to arrive at the true optimum. Several procedures might be assumed at the second stage for modifying the flfStstage approximate solution. In case a (XL> 0), it is possible to eliminate members with X =XL and modify the design accordingly. Accurate formulation of both stress and displacement constraints is needed at this stage. In case b ( XL =0), modification of the first-stage solution is required to account for displacement constraints and compatibility conditions. The accurate NLP problem can be solved by available methods (such as those presented in Chap. 2) for the given topology obtained by the LP solution. Example 4.23. The ten-bar truss ground structure shown in Fig. 3.8a is subjected to two loads acting simultaneously. The allowable stresses are aU =-oL = 25, the modulus of elasticity is 30,000 and the upper bound on the vertical displacement at joint A is DU =2.0. To illustrate the effect of various constraints on the optimum, the following cases have been solved [77]: A. Only stress constraints.

B. Stress and side constraints XL = 0.1.

C. Stress and displacement constraints. D. Stress, displacement and side constraints XL =0.1. Results are given in Table 4.13 and the optimal topology for cases A and C is shown in Fig. 4.41a. It can be noted that in this example:

(a)

(b)

Fig. 4.41. Various topologies, ten-bar truss ground structure.

(e)

4.7 Topological Optimization

277

Table 4.13. Optimal cross-sections, effect of constraints, ten-bar truss. Case

Member 1 2 3 4 5 6 7 8 9 10 Z

-

B 7.99 0.1 8.06 3.94 0.1 0.1 5.74 5.57 5.57 0.1 15,932

A 8.0 0 8.0 4.0 0 0 5.66 5.66 5.66 0 15,840

C 9.9 0 8.0 4.94 0 0 5.66 7.0 7.0 0 18,210

D 9.9 0.1 8.0 4.94 0.1 0.1 5.6 7.0 7.0 0.1 18,343

The displacement constraint significantly affects the optimum but does not change the optimal topology; the lower bound constraints do not appreciably affect the optimum.

To illustrate the effect of the topology on the optimum, optimal solutions for various topologies (Fig. 4.41) are compared in Table 4.14. It can be seen that topology (4.41a) is much better than topologies (4.41b) and (4.41c). Optimal Topologies or Controlled Structures. An alternative approach for the second-stage solution is to apply a set of control forces such that the accurate constraints will be satisfied at the fl1'St-stage optimum [86]. The rationale of this approach is that the approximate optimum is usually better than the accurate one, since some constraints (compatibility conditions and displacement constraints) are neglected. The required control forces can readily be determined by solving an LP problem, as shown in Sect. 4.5.1. The optimal solution will give the minimum magnitude and number of control forces needed to convert the infeasible frrst-stage solution into a feasible one. Table 4.14. Optimal cross-sections, effect of topology, ten-bar truss. Member 1 2 3 4 5 6 7 8 9 10 Z

Only stress constraints Fig. Fig. Fig. 4.41c 4.41b 4.41a 4.0 8.0 12.0

Stress and displacement Fig. Fig. Fig. 4.41a 4.41b 4.41c 9.9 13.54 6.16

8.0 4.0 0

4.0 4.0 4.0

12.0 4.0 4.0

8.0 4.94

5.53 5.53 4.0

15.1 6.16 6.16

5.66 5.66 5.66

0 11.31 5.66

11.31 0 5.66

5.66 7.0 7.0

0 11.31 7.81

12.33 0 8.72

15,840

17,276

17,276

18,210

20,028

22,803

278

4 Design Procedures

4.8 Interactive Layout Optimization Some of the difficulties involved in layout optimization can be overcome by combining structural optimization methods and graphical interaction. In this section such a capability is described, where the automated optimization methods produce systematic modifications of the design and the graphical interaction programs provide the designer with the flexibility and control which are necessary in the practical design process. This control is useful to execute those decisions which either cannot be automated, or which are based on the designer's experience and judgement The Computer Aided Design (CAD) system described here [131] is intended to optimize the cross sections, the geometry and the topology of the structure. The object is to obtain a practical optimum design and not the theoretical optimum. The optimization programs used in the interactive optimization procedure are described in Sect 4.8.1, the graphical interaction programs are presented in Sect 4.8.2, and the general design procedure is discussed in Sect 4.8.3.

4.8.1 Optimization Programs Three optimization methods, discussed in previous sections, will be considered (Fig. 4.42): a. the Optimality Criteria (OC); b. the Linear Programming (LP); and c. the Unconstrained Minimization (UM).

Graphics

display scrcen

Fig. 4.42. Programs used in the design process.

4.8 Interactive Layout Optimization

279

Optimality Criteria. The OC method is used to optimize the member sizes for a structure whose topology and geometry are given. The method has been chosen because of its efficiency and relative simplicity as compared to mathematical programming. The method produces designs which are sufficiently good for most practical design purposes. The input to the program includes the structure built interactively on the graphical screen, including the loads and constraints on the stresses, displacements and minimum size. The structure is first analyzed using the displacement method. Scaling is performed to bring the design to the boundary of the feasible region. The members' areas are then resized, considering the constraints on stresses, displacements and members' sizes. The allowable compressive stresses were taken as variables considering Euler buckling. In addition, the user has the option of linking member sizes. At the end of every cycle all elements linked together are checked and the highest size is assigned to all of them. Linear Programming. Assuming the LP formulation (4.151), the optimal topology and member sizes are chosen for a given geometry, considering only equilibrium, stress and member size constraints. As a frrst step in the solution process a grid set is built to cover the design space. Load and support points are included in this grid set. An initial ground structure is then formed by interactively connecting members between the nodes. The input to the LP includes the structure built interactively on the screen together with the loads and constraints. If the allowable compressive stresses are taken as variables, the OC program is first applied. The allowable stresses obtained at the last resizing are used as input to the LP run. It has been noted in Sect. 4.7.1 that the LP method has the ability to eliminate unnecessary members from the structure layout. Unconstrained Minimization. The UM program optimizes the cross sections and the geometry. Powell's direct search method, described in Sect. 2.2.2, is used to optimize the geometry of the structure with a given topology. For problems with many variables, the method is not as efficient as minimization methods based on first- or second-order derivatives. However, in the design procedure described here only a small number of geometrical variables are assumed and the direct search method, which is based on comparison of the objective function values without the need of using derivatives, has been found to be adequate and convenient. For any candidate geometry in the solution process the OC method is used as the objective function calculator. That is, the member sizes are optimized by the OC program and the weight is determined accordingly. Each time the objective function has to be calculated, the following steps are carried out -

The algorithm is stopped. The output includes the current values of the geometrical variables and also a file containing all the information needed to restart the algorithm. The optimality criteria method is used to find the optimum weight of the structure obtained when the algorithm is stopped, taking into consideration the minimum size, stress, and displacement constraints. The unconstrained minimization algorithm is restarted with its input being the calculated optimum weight and the restart file saved when it was stopped.

280

4 Design Procedures

When the process converges the value of the geometrical variables and the optimum weight are output. and the optimum geometry is displayed on the screen. 4.8.2 Graphical Interaction Programs Two display programs are used for the graphical interaction (Fig. 4.42): Q.

The structure display program.

h. The convergence display program.

Structure Display Program. This program can be used either to interactively build the starting structural model or to interactively display and modify the model at any later stage in the process. Loads and constraints can be applied interactively and any information about the model can be extracted and displayed on the screen. The interactive changes, based on the designer's decisions, may include: - Elimination of members that either reached, or are almost at, minimum size after the optimality criteria method run. Although this decision could be automatically taken by the program, it has been found that it is not always beneficial to delete members that reached minimum size. Mter deletion of these members it might be necessary to delete or add other members to obtain a reasonable topology from a practical viewpoint. - Addition of members that were deleted by the linear programming method, but which are needed to produce a stable structure. - Deletion and/or addition of members, taking into consideration the construction limitations. - Rounding-off the values of the geometrical variables resulting from the UC minimization. Mter every interactive change in the topology or in the geometry of the structure, the member sizes are optimized by the OC program. The new optimum weight is compared with the weight before the change to decide whether the change is beneficial. Convergence Display Program. This program is used to observe the convergence of the optimization methods with the purpose of eliminating unnecessary cycles near the optimum. Convergence graphs that can be displayed on the screen include:

a. The weight against the number of cycles for the optimality criteria method. h. The weight against the value of a geometrical variable and the weight against the number of uni-dimensional minimizations for the UM method. 4.8.3 Design Procedure Using interaction, the display programs, and the three optimization methods, several alternative procedures can be used to optimize a structure whose topology,

4.8 Interactive Layout Optimization

281

geometry and member sizes are treated as variables. These alternative procedures could differ by the order in which they use the optimization methods and by the amount of interaction involved. The more the engineering experience of the user, the more he could depend on his own judgement to guide the design. The four steps of the procedure described here are: establishing starting structure, establishing preliminary topology and geometry, establishing final geometry and establishing fmal topology and member sizes. Establishing Starting Structure. A grid set covering the design space is first introduced. Obviously, this grid set must include the load and support points. A starting topology is established by connecting the grid points with as many members as desired. The designer may connect only those members which seem to be reasonable, based on engineering and practical considerations. Equal initial starting sizes are assigned and the applied loads are introduced. All these actions are done interactively on the screen which provides an instant graphical check of the data. The consuaints applied to the structure are also defined. Establishing Preliminary Topology and Geometry. The purpose of this step is to establish a preliminary topology by deleting members from the starting structure built in the first step. Grid points are also deleted if all the members connected to them are deleted. This indirectly establishes a preliminary geometry. Taking into account all the constraints, the OC program is first applied to evaluate the allowable compressive stress. If buckling and displacement consuaints are not considered, this substep is not needed. The LP program is then applied. The structure is displayed, after deleting members that reach zero size and grid points for which all connected members have been deleted. If the resulting topology is mobile, members that are needed to produce stability are interactively added. Finally, the OC program is applied to consider the compatibility conditions and the displacement constraints. Establishing Final Geometry. The geometrical variables are chosen, based on practical considerations. We may impose symmetry on the structure and decide which grid coordinates or geometrical dimensions will be treated as independent variables. The remaining coordinates or dimensions could either by linked to the ones taken as variables or could be fIXed. The unconstrained minimization method is applied, with the optimality criteria method as its function calculator. The input is the topology obtained in the previous step and equal initial starting sizes. The resulting geometry can be rounded off to practical dimensions; the optimality criteria method is used to obtain the weight of the rounded-off structure. Establishing Final Topology and Member Sizes. The purpose of this step is to apply fmal interactive changes to the structure. Starting with the structure obtained in the previous step, the designer can interactively delete or add members to produce a better structure based on his experience and taking into consideration the construction limitations. For each of the alternative structures resulting from the deletion or addition of members, the optimality criteria method is applied. The optimum weight of each of the alternative structures is recorded.

282

4 Design Procedures

The final optimal structure chosen is not necessarily the one with the least weight; for there could be a structure, with slightly higher weight than the least weight one, but still considered better because of other engineering or construction considerations. In this final step of the procedure the designer uses his experience to the utmost advantage in moulding the structure to its fmal design. Example 4.24. Consider the transmission tower with the initial grid set, including the load points and the supports, shown in Fig. 4.43a. The following design data have been assumed: -

Modulus of elasticity E = 10,000, density p = 0.001. Allowable tensile stress, aU = 21.6. Assuming hollow tube section with dlt =20 (where d is the mean diameter and t is the wall thickness) the Euler buckling stress in member i is:

a;E

7tEX· (d=~ 8l; t

t) = 7.87 x 10

+ d

4

X· -t l;

Allowable compressive stresses for the kth cycle, c:#/c), are computed by

-

This average is taken to prevent oscillations in the values of oL from one cycle to the other. Displacement limits: DU = 1.0 in the direction of P 2 • Minimum size XL = 0.1. Loading conditions (Fig.4.43a): loading a: PI + P 2 + P 3 + P4 ; loading b: PI + P 2•

The design procedure for this example involved the following steps [131]: -

Starting with the 8 point grid set, 19 members were interactively built (Fig 4.43b). Three cycles of the optimality criteria method were applied to obtain the starting allowable compressive stresses for the LP run. The LP was applied and members reaching zero size were deleted (Fig 4.43c). The optimality criteria method was applied to introduce a compatible structure, taking into account the minimum size, stress and displacement constraints. The final weight obtained is 4.487. Considering the structure as symmetrical around a vertical axis, the width at the base is taken as a geometrical variable. The width at the top is fixed and the width at intermediate stories is taken as varying linearly between them. Using as input the topology shown in Fig. 4.43c and equal starting sizes, the UM program is applied. The final weight obtained is 1.349 and the width at the base is 152.60. Rounding off the width at the base to a practical dimension of 153, the resulting structure is shown in Fig. 4.43d. Applying the optimality criteria

4.8 Interactive Layout Optimization

283

method to the rounded-off structure, the weight practically did not change. This step represents a 70 percent saving in weight, as compared to the starting geometry whose optimal weight was 4.487.

=40 t

P3 P =5

4~

t.<;=40

P=15 ~

80 0

0

0

0

.2

.2

80

BC

(a)

+

-.§!L+

(b)

(c)

(e)

(j)

Fig. 4.43. Transmission tower truss: a. Initial grid-set, b. Initial topology, c. Topology after LP run, d. Geometry after UM run, e. Structure after interactive deletion of members, f. 3D transmission tower truss.

284

4 Design Procedures

Deleting interactively the members reaching mlDlmum size and the corresponding grid points, the structure shown in Fig. 4.43e is obtained. Since the two lengthened members are in compression it is essential to provide them with a support at the middle so that only half their length is used for calculating the allowable compressive stresses. This support can be provided by re-adding two horizontal members. In cases where the truss is actually only one face of a four-faced 3D transmission tower (Fig 4.43.1), the arrangement of the trusses at the four faces gives support for the lengthened members in a plane perpendicular to the plane of the truss. Such a support affects also the buckling length of the member in the plane of the truss. Applying the OC program, an optimal weight of 1.348 is obtained. Thus, deleting interactively several members resulted in simpler topology without increasing the weight. The intermediate results for this example are summarized in Table 4.15. Table 4.15. Results, transmission tower truss.

Structure Figure 4.43c Figure 4.43d Figure 4.43e

Number of members

13 13 8

Weight 4.487 1.349 1.348

Exercises 4.1 Consider the three-bar truss shown in Fig. 1.11. Choose the force in member 2 as a redundant, N, and consider only the load Pl. Assume the bounds on stresses aU =- oL =40.

a. Ftlmulate the LP plastic design problem (4.1). b. Formulate the LP problem (4.2). c. Assuming uniform cross sections Xl =X2 =X, illustrate graphically the feasible region and find the optimal solution. 4.2 The three-span continuous beam shown in Fig. 4.44 is subjected to a singleloading condition of three concentrated loads. The plastic moments for the three spans are Mpll' Mpo. and Mpl3' respectively.

a. Formulate the LP plastic design problem (4.17).

b. Assuming the moments over the interior supports B and C as redundants N 1 and N 2, respectively, formulate the LP problem (4.18). c. Assuming N 1 =N 2 =N, and M pn =Mpo. =M pl3' illustrate graphically the feasible region and find the optimal solution.

Exercises 5.0

10.0

5.0

00

0)

0@

>L

100

285

100

200

600

Fig.

4.44.

4.3 The frame of example 4.3, shown in Fig. 4.3, is subjected to two loading conditions: loading 1 - the vertical loads, and loading 2 - the horizontal loads. Choose redundant forces as shown in Fig. 4.5.

a. Formulate the LP problem (4.18).

h. Solve the problem by an LP computer program and find the optimal solution.

c. lllustrate graphically the bending-moment distributions for the optimal frame under each of the loadings. 4.4 Consider the three-bar truss shown in Fig. 1.11, with a uniform crosssectional area X for all members. The objective function represents the volume of the truss and the modulus of elasticity is E =30,000. Choose the force in member 2 as a redundant, N, and consider only the loading Pl. The bounds on stresses are aU =20, aL = - 15, and the lower bound on X is XL = 0.4. Formulate and solve the LP problem (4.21). Calculate the prestressing force required in member 2 to obtain an identical optimal elastic design. 4.5 Consider the problem of example 4.10 (Fig. 4.15). Write a computer program and solve the problem by the design procedure that combines the stress-ratio rule, displacement criteria and scaling. Verify the results given in Table 4.6.

1.0

t

I yo.s

P

:1: 20.0

Fig.

4.45.

.. Y l

10.0

.1· 20.0

Typical cross section

10.0

286

4 Design Procedures

Boundary line

l20

llO

~

M"lJ

A

200

ZS

Mpl/

B

·1

1

M,,12

C

200

200

ZS D

I

! 40 M,,12

6. E

200

I

Second substructure

First substructure

Fig.

po

4.46.

4.6 The continuous prestressed concrete beam shown in Fig. 4.45 (all dimensions are in tons and meters) is subjected to the following loading conditions: l. P + (dead load = 2.0 tIm); bounds on stresses aU = ISO, 2. 0.8P + (total load 3.0 tIm); bounds on stresses aU 0,

=

=

crL =- 1500. crL = - 1200.

The design variables are P, YI> Y2 and the tendon is assumed to be parabolic. The constraints are related to the stresses in the cross sections of Yl and Y2 and to the design requirements j = 1,2

a. Formulate the linear programming problem of minimizing P. Consider only potentially critical constraints.

b. Show graphically the feasible region for Y2 = 0.9 and find the optimal P and Yl.

4.7 Consider the grillage of exercise l.8 shown in Fig. l.22.

a. Formulate the LP problem (4.17). b. Choosing the vertical interaction force in the intersection of the two beams as the redundant force, formulate the LP problem (4.18). c. Assuming Mpl =Mpll =2Mpa, find graphically the optimal solution. 4.8 Formulate and solve the optimal plastic design problem of the beam shown in Fig. 4.46 by model coordination. The objective function is 2

Z=400L M pli i=l

where M pli (i= 1, 2) are the plastic moments for the two substructures. Choose the moment Me as the coordinating variable. Illustrate graphically the variation of Zuun with the coordinating variable.

Exercises

287

A

"I

T

~I

1

+

l",,,,l,,,,,~,~:,J;:lI

I I

~~ 1 V".

I'. -

:

T

~1

I

I. Fig.

i.

I

8.0

_I.

A...J

• I.

8.0

Y '1' I 15.0 -2Y . 1 . . .Y. . 15.0 •i Se..:tion A-A

8.0

4.47.

4.9 Given the reinforced concrete system of a slab, two beams, and eight identical columns shown in Fig. 4.47 (all dimensions are in tons and meters). The slab is subjected to a uniformly distributed load of 2.0 tlm Z and the design variables are Xl' Xz, X3 and Y. Consider the following constraints: Design constraints,

3.5 ~ Y ~ 6.5

Slab constraints,

Ms(Y) ~ 190X1Z

Beam constraints,

Mb(Y) ~ 38Xi

Column constraints,

~(Y) ~650xi

in which Ms(Y), Mb(y), Ac(Y), = maximum bending moment at the slab support, maximum bending moment at the beam support, and maximum load at the columns, respectively. The objective function represents the concrete cost Z = 400X1 + 20Xz + 130xi ~ min Formulate the ftrst- and second-level problems to be solved by model coordination. Compute Zmin for Y = 3.5, 5.0, 6.5, and ftnd the optimal solution by quadratic interpolation. Use the following data:

y 3.5

5.0 6.5

9.0 5.0 7.5

90.0 70.0 62.0

123.7 96.8

84.7

288

4 Design Procedures

4.10 The truss shown in Fig. 4.48 is subjected to two loads, PI acting simultaneously. The objective function is

=20 and Pl =10,

Z =282.8XI + l00Xl where Xl and Xl are design variables representing the cross-sectional areas. The modulus of elasticity is 30,000, and the allowable stresses are aU=20.0, aL= -15.0.

a. Formulate the optimal design problem, using the displacement method of analyses. lllustrate graphically the feasible region.

b. Assume the initial design (X;)T = {I, I} and the initial move limits given by - 0.8 < IiX < 0.8. Apply one iteration of the sequential LP and scale the resulting design to obtain a feasible solution. 4.11 Consider the grillage of exercise 1.8, shown in Fig. 1.22. The bounds on stresses are aU aL 20 in beam 1, and aU aL= 10 in beam 2. The lower bound on X is XL 10 and the modulus of elasticity is E 30,000.

=- = =

=-

=

a. lllustrate graphically the feasible region and the objective function contours in the space of Xl and Xl . b. Solve the problem by the solution process for optimization in design planes. Assume the initial design (XjT = {20, ISO} and choose (l1X*)T = {I, -I} [see (4.71)]. Show the intermediate solutions in the design plane. c. Solve part b for XL= O. Show that the optimal solution is a singular point 4.12 Solve the problem of exercise 1.3 by the procedure of (4.83) through (4.85). Assume the lower bounds on the design variables design (X;)T

={10, 10}.

100

Fig.

4.48

xf = xf =16.0, and the initial

Exercises

I•

100

100

• I •

289

~I

10.0

Fig.

4.49.

4.13 The truss shown in Fig. 4.48 is subjected to two distinct loading conditions, PI = 20.0 and P2 = 20.0. The objective function is Z =282.8X1 + lOOX2 ' the allowable stresses are aU= 20.0, oL=-15.0, and the modulus of elasticity is 30,000.

a. Formulate the optimal design problem using the displacement method of analysis. Show that this is an LP problem and find the optimal solution.

h. Neglecting the compatibility conditions, formulate the LP lower bound problem (4.21). Choose the forces in member 2 as redundants NI and Nz (for

the two loading conditions). Find the optimal solution and compare the result with that of part a. c. Find the control forces in member 2, Nel and NeZ' required to maintain compatibility at the optimum of part h. 5.0

5.0

~

b· Fig.

4.50.

~

200

JOI

! JE 600

5.0

~

200

l

:::h....

-1. ~ 100

loE

290

4 Design Procedures

4.14 Consider the truss shown in Fig. 4.49 with two design variables: the crosssectional area X and the angle Y. The upper bound on the stresses is aU =20 and the bounds on Yare Yu = 6()0, yL = 300. The objective function is Z =X. Solve the nonlinear programming problem by sequential LP [see (4.147)] without move limits. Choose the initial design X(l) = 0.4. y(1) = 30 0 • 4.15 Assuming a two-level formulation, solve the problem of exercise 4.14, where X is the frrst-level variable and Y is the second-level variable. Choose the initial design yel) = 300. 4.16 The symmetric continuous beam shown in Fig. 4.50 is subjected to a single loading condition of three concentrated loads. The two design variables are the cross-sectional area X and the distance between the supports Y. Assume a uniform cross section with the modulus of section W = )(2/6. The bounds on stresses are aU = -aL = 20.0 and the bounds on Y are yL = 120, Yu = 280. The objective function represents the volume of material. Formulate the two-level problem, where X is the first-level variable and Y is the second-level variable, and find the optimal solution. 4.17 The symmetric truss shown in Fig. 4.51 is subjected to two concentrated loads, PI = 10.0 and P 2 = 10.0. The bounds on stresses are aU = -aL = 20.0, and the lower bound on cross-sectional areas is XL = O. The objective function represents the volume of material. Formulate the LP problem (4.151) and find the optimal topologies for Y = 50, Y = 100 and Y = 150 and for each of the following loading cases:

a. A single loading condition PI; h. two loading conditions, PI and P 2, respectively.

Show all optimal topologies for Y = 100.

l", . J~I20a li

Fig.

4.51.

Exercises

p

p

4 x 100

Fig.

291

P

f

4.52.

4.18 The symmetric twenty-one-bar truss shown in Fig. 4.52 is subjected to three concentrated loads P =10, acting simultaneously. The bounds on stresses are aU = _aL = 20.0, and the objective function represents the volume of the truss. Formulate and solve the LP problem (4.151) for Y = 100, Y = 150 and Y =200. Show all optimal topologies and find the optimal value of Y by quadratic interpolation. 4.19 The five-bar truss shown in Fig. 4.53 is subjected to two loads, PI and P 2 , acting simultaneously. The bounds on stresses are aU = _aL = 20.0, and the objective function represents the volume of material. Formulate and solve the LP problem (4.151) for:

a. PI = 10.0, P 2 = 5.0; b. PI = 5.0, P2 = 10.0; c. PI =P2 = 10.0. Show the optimal topologies and check whether the resulting structures are mobile (mechanisms).

Fig.

4.53.

References

1. Abu Kassim, A. M., and Topping, B. H. V.: Static reanalysis: a review. J. Struct. Engrg., ASCE 113, 1029-1045 (1987) 2. Arora, J.S.: Introduction to optimum design. Singapore: McGraw-Hill Book Company 1989 3. Arora, J.S., and Haug, E.J.: Methods of design sensitivity analysis in structural optimization. J. AIAA 17, 970-974 (1979) 4. Barthelemy, J-F.M.: Engineering design applications of multilevel optimization methods. In: Brebbia C.A., and Hernandez. S. (eds.): Computer aided optimum design of structures. Berlin: Springer-Verlag 1989 5. Barthelemy, J-P.M., and Haftka, R. T.: Recent advances in approximation concepts for optimum structural design. In: Proceedings of NATO/DFG ASI on Optimization of large structural systems. Berchtesgaden, Germany, September 1991 6. Barthelemy, J-F.M., and Riley, M.F.: An improved multilevel optimization approach for the design of complex engineering systems. J. AIAA 26, 353-360 (1988) 7. Bendsoe, M. P., and Mota Soares, C. A. (eds.): Proceedings of NATO ARW on Topology design of structures. Sesimbra, Portugal June 1992 8. Bendsoe, M. P., and Kikuchi, N.: Generating optimal topologies in structural design using the homogenization method. Computer Methods in Appl. Mech. and Engrg. 71, 197-224 (1988) 9. Berke, L.: An efficient approach to the minimum weight design of deflection limited structures. USAF-AFFDL-TR-70-4-FDTR, May 1970 10. Berke, L.: Convergence behavior of optimality criteria based iterative procedures. USAF AFFDL-TM-72-1-FBR, January 1972 11. Berke, L., and Khot, N.S.: Use of optimality criteria methods for large scale systems, AGARD-SL-70, 1974 12. Berke, L., and Khot, N.S.: Performance characteristics of optimality criteria methods. In: Rozvany, G.I.N. and Karihaloo, B.L. (eds.): Structural optimization. Dordrecht: Kluwer, 1988 13. Bertsekas, D.P.: Multiplier methods: a survey. Automatica 12, 133-145 (1976) 14. Carmichael, D.G.: Structural modelling and optimization. Chichester, UK: Ellis Horwood 1981 15. Carroll, C.W.: The created response surface technique for optimizing nonlinear restrained systems. Operations Res. 9, 169-184 (1961) 16. Cassis, I.H., and Schmit, L.A.: On implementation of the extended interior penalty function. Int. J. Num. Meth. Engrg. 10, 3-23 (1976) 17. Cheney, E.W., and Goldstein, A.A.: Newton's method for convex programming on Tchebycheff approximation. Num. Math I, 253-268 (1959) 18. Cohn. M.Z., Ghosh, S.K., and Parimi, S.R.: Unified approach to the theory of plastic structures. J. Eng. Mech. Div., ASCE 98, 1133-1158 (1972)

294

References

19. Courant, R.: Variational methods for the solution of problems of equilibrium and vibrations. Bull. Am. Math. Soc. 49, 1-23 (1943) 20. Dantzig, G.: Linear programming and extensions. Princeton, N.J.: Princeton University Press 1963 21. Davidon, W.C.: Variable metric method for minimization. Argonne Nat. Lab. ANL5990, Rev., 1959 22. Dorn, W.S., Gomory, R.E., and Greenberg, H.J.: Automatic design of optimal structures. Journal de Mecanique 3, 25-52 (1964) 23. Eschenauer, H., Koski, J., and Osyczka, A.: Multicriteria design optimization procedures and applications. Berlin: Springer-Verlag 1990 24. Farshi, B. and Schmit, L.A.: Minimum weight design of stress limited trusses. J. Struct. Div., ASCE 100, 97-107 (1974) 25. Fiacco, A.V., and McCormick, G.P.: Programming under nonlinear constraints by unconstrained minimization: a primal-dual method. Research Analysis Corporation, Bethesda, Md., Tech. Paper RAC-TP-96, 1963 26. Fiacco, A.V., and McCormick, G.P.: Nonlinear programming, sequential unconstrained minimization techniques. New York: John Wiley & Sons 1968 27. Fletcher, R.: An ideal penalty function for constrained optimization. J. Inst. Maths. Appl. 15, 319-342 (1975) 28. Fletcher, R., and Powell, M.J.D.: A rapidly convergent descent method for minimization. Comput. J. 6, 163 (1963) 29. Fletcher, R., and Reeves, C.M.: Function minimization by conjugate gradients. Comput. J. 7, 149 (1964) 30. Fleury, C. and Braibant, V.: Structural optimization: a new dual method using mixed variables. Int. J. Num. Meth. Engrg. 23, 409-428 (1986) 31. Fleury, C. and Geradin, M.: Optimality criteria and mathematical programming in structural weight design. Computers and Structures 8, 7-18 (1978) 32. Fleury, C. and Sander, G.: Structural optimization by finite element. University of Liege, Belgium, LTAS Report NBR-SA-58, January 1978 33. Fox, R.L.: Optimization methods for engineering design. Reading, Mass.: Addison-Wesley 1971 34. Fox, R.L. and Miura, H.: An approximate analysis technique for design calculations. J. AIAA 9, 177-179 (1971) 35. Fox R.L. and Schmit L.A.: Advances in the integrated approach to structural synthesis. 1. Spacecraft and Rockets 3, (1966) 36. Fuchs, M.B: Linearized homogeneous constraints in structural design. Int. J. Mech. Sci. 22, 33-40 (1980) 37. Fuchs, M.B.: Explicit optimum design. Int. 1. Solids Structures 18, 13-22 (1982) 38. Gallagher, R.H.: Fully stressed design. In: R.H. Gallagher and O.C. Zienkiewicz (eds.): Optimum structural design. New York: John Wiley & Sons 1973 39. Gass, S.I.: Linear programming. New York: McGraw-Hill Book Company 1958 40. Gellatly, R.A. and Berke, L.: Optimal structural design. USAF AFFDL-TR-70-165, April 1971 41. Gerard, G.: Minimum weight analysis of compressive structures. New York: New York Univ. Press 1956 42. Gill, P.E., Murray, W., and Wright, M.H.: Practical optimization. New York: Academic Press 1981 43. Griffith, R.E., and Stewart, R.A.: A nonlinear programming technique for the optimization of continuous processing systems. Manage. Sci. 7, 379-392 (1961) 44. Gurdal, Z., and Haftka, R. T.: Optimization of composite laminates. In: Proceedings of NATO/DFG ASI on Optimization of large structural systems. Berchtesgaden, Germany, September 1991 45. Hadley, G.: Linear programming. Reading, Mass.: Addison-Wesley 1962

References

295

46. Haftka, R.T.: An improved computational approach for multilevel optimum design. 1. Struct. Mech. 12, 245-261 (1984) 47. HaftIca, R.T., and Adelman, H.M.: Recent developments in structural sensitivity analysis. Structural Optimization I, 137-151 (1989) 48. HaftIca, R.T., Gurdal, Z., and Kamat, M.P.: Elements of structural optimization. Dordrecht: Second Edition, Kluwer Academic Publishers 1990 49. HaftIca, R.T., and Starnes, 1.H.: Application of a quadratic extended interior penalty function for structural optimization. 1. AIM 14, 718-724 (1976) 50. Hajali, R.M., and Fuchs, M.B.: Generalized approximations of homogeneous constraints in optimal structural design. In: Brebbia C.A., and Hernandez, S. (eds.): Computer aided optimum design of structures. Berlin: Springer-Verlag 1989 51. Haug, E.1.: A review of distributed parameter structural optimization literature. In: Haug, E.1., and Cea, 1. (eds.): Optimization of distributed parameter structures. Proceedings of NATO ASI. Iowa City: Sijthoff and Noordhoff, Alphen ann der Rijn 1981 52. Hemp, W.S.: Optimum structures. Oxford, UK.: Clarendon Press 1973 53. Hestenes, M.R.: Multiplier and gradient methods. 1. Optimization Theory and Applications 4, 303-320 (1969) 54. Heyman, 1.: Plastic design of beams and frames for minimum material consumption. Quarterly of Appl. Math. 8, 373-381 (1951) 55. Hodge, P.G. Ir.: Plastic analysis of structures. New York: McGraw-Hill Book Company 1959 56. Kavlie, D.: Optimum design of statically indeterminate structures. University of California, Berkeley, Ph.D. Thesis, 1971 57. Kelley, H.I.: The cutting plane method for solving complex programs. SIAM 1. 8, 703-712 (1960) 58. Khot, N.S.: Algorithms based on optimality criteria to design minimum weight structures. Engineering Optimization 5, 73-90 (1981) 59. Khot, N.S., Berke, L., and Venkayya, V.B.: Comparison of optimality criteria algorithms for minimum weight design of structures. 1. AIM 17, 182-190 (1979) 60. Kirsch, U.: Optimized prestressing by linear programming. Int. 1. Num. Meth. Engrg. 7, 125-136 (1973) 61. Kirsch, U.: Optimal design of trusses by approximate compatibility. Computers and Structures 12, 93-98 (1980) 62. Kirsch, U.: Optimum structural design. New York: McGraw-Hill Book Company 1981 63. Kirsch, U.: Approximate structural reanalysis based on series expansion. Compo Meth. Appl. Mech. Eng. 26, 205-223 (1981) 64. Kirsch, U.: Optimal design based on approximate scaling. 1. Struct. Div., ASCE 108, 888-910 (1982) 65. Kirsch, U.: Approximate structural reanalysis for optimization along a line. Int. 1. Num. Meth. Engrg. 18, 635-651 (1982) 66. Kirsch, U.: Synthesis of structural geometry using approximation concepts. Computers and Structures 15, 305-314 (1982) 67. Kirsch, U.: Approximate behavior models for optimum structural design. In: Atrek, E., et al (eds.): New directions in optimum structural design. New York: Iohn Wiley & Sons 1984 68. Kirsch, U.: A bounding procedure for synthesis of prestressed systems. Computers and Structures 20, 885-895 (1985) 69. Kirsch, U.: An improved multilevel structural synthesis. lour. of Structural Mechanics 13, 123-144 (1985) 70. Kirsch, U.: The effect of compatibility conditions on optimal design of flexural structures. Int. 1. Num. Meth. Engrg. 24, 1173-1185 (1987)

296

References

71. Kirsch, U.: Optimal topologies of flexural systems. Engineering Optimization 11, 141-149 (1987) 72. Kirsch, U.: Optimal topologies of truss structures. Computer Methods in Appl. Mech. and Engrg. 72, 15-28 (1989) 73. Kirsch U.: Optimal topologies of structures. Appl. Mech. Rev. 42,223-239 (1989) 74. Kirsch, U.: Effect of compatibility and prestressing on optimized structures. J. Struct. Eng., ASCE 115, 724-737 (1989) 75. Kirsch, U.: Feasibility and optimality in structural design. Computers and Structures 41, 1349-1356 (1991) 76. Kirsch, U.: Improved optimum structural design by passive control. Engineering with Computers 5, 13-22 (1989) 77. Kirsch, U.: On singUlar topologies in optimum structural design. Structural Optimization 2, 133-142 (1990) 78. Kirsch, U.: On the relationship between optimum structural geometries and topologies. Structural Optimization 2, 39-45 (1990) 79. Kirsch, U.: Optimal design of structural control systems. Engineering Optimization 17, 141-155 (1991) 80. Kirsch, U.: Reduced basis approximations of structural displacements for optimal design. J. AIAA 29, 1751-1758 (1991) 81. Kirsch, U., and Moses, F.: Decomposition in optimum structural design. J. Struct. Div., ASCE 105, 85-100 (1979) 82. Kirsch, U., and Taye, S.: On optimal topology of grillage structures. Engineering with Computers 1, 229-243 (1986) 83. Kirsch, U., and Taye, S.: High quality approximations of forces for optimum structural design. Computers and Structures 30, 519-527 (1988) 84. Kirsch, U., and Taye, S.: Structural optimization in design planes. Computers and Structures 31, 913-920 (1989) 85. Kirsch, U., and Toledano, G.: Approximate reanalysis for modifications of structural geometry. Computers and Structures 16, 269-279 (1983) 86. Kirsch, U., and Topping, B.H.V.: Minimum weight design of structural topologies. J. of Computing in Civil Engineering, ASCE 118, 1770-1785 (1992) 87. Kresselmeir, G., and Steinhauser, R.: Systematic control design by optimizing a vector performance index. In: Proceedings of IFAC Symposium on Computer Aided Design of Control Systems. Zurich, Switzerland: 1979, pp. 113-117 88. Kuhn, H.W., and Tucker, A.W.: Nonlinear programming. In: Proc. 2d Berkeley Symp. Math. Stat. Probab., Berkeley, Calif., 1950, pp. 481-492 89. Lasdon, L.S.: Optimization theory for large systems. London: The Macmillan Company 1970 90. Lellep, J., and Lepik, U.: Analytical methods in plastic structural design. Engineering Optimization 7, 209-239 (1984) 91. Leunberger, D.G.: Introduction to linear and nonlinear programming. Reading, Mass: Addison-Wesley 1984 92. Lev, O.E. (ed.): Structural optimization: recent developments and applications. New-York: ASCE 1981 93. Levy, R., and Lev, 0.: Recent developments in structural optimization. J. Struct. Eng. ASCE 113, 1939-1962 (1987) 94. Massonnet, C.E., and Save, M.A.: Plastic analysis and design, vol. 1. Blaisdell Publishing Company 1965 95. Maxwell, C.: Scientific papers ll. Cambridge University Press, 175 (1890) 96. Mesarovic, M.D., Macko, D., and Takahara, Y.: Two coordination principles and their application in large scale systems control. In: Proc. IFAC Congress, Warsaw, Poland, 1969

References

297

97. Mitchell, A.G.M.: The limits of economy of material in framed structures. Phil. Mag., series 6, 8, 589-597 (1904) 98. Miura, H., and Schmit, L.A.: Second order approximations of natural frequency constraints in structural synthesis. Int. J. Num. Meth. Engrg. 13, 337-351 (1978) 99. Moe J., and Gisvold K.M. (eds.): Optimization and automated design of structures. Div. of Ship Structures, Norwegian Institute of Technology, Trondheim, Rep. SK/M21, December 1971 100. Niordson, FJ., and Pedersen, P.: A review of optimal structural design. In: Proceedings, 13th Int. Congress on Theoretical and Applied Mechanics. Berlin: Springer-Verlag 1973 101. Noor, A. K., and Lowder, H. E.: Approximate techniques for structural reanalysis. Computers and Structures 4, 801-812 (1974) 102. Noor, A. K., and Whitworth, S.A.: Reanalysis procedure for large structural systems. Int. J. Num. Meth. Engrg. 26, 1729-1748 (1988) 103. Olhoff, N., and Taylor, J.E.: On structural optimization. J. Appl. Mech. 50, 1139-1151 (1983) 104. Pedersen P.: On the optimal layout of multi-purpose trusses. Computers and Structures 2, 695-712 (1972) 105. Pickett, R.M., Rubinstein, M.F., and Nelson, R.B.: Automated structural synthesis using a reduced number of design coordinates. J. AIAA II, 489-494 (1973) 106. Pope, G.G., and Schmit L.A. (eds.): Structural design applications of mathematical programming. AGARDograph No. 149, AGARD-NATO Publication, February 1972 107. Powell, M.J.D.: An efficient method of finding the minimum of a function of several variables without calculating derivatives. Compo J. 7, 155 (1964) 108. Prager, W., and Shield, R.T.: Optimal design of multi-purpose structures. Int. J. Solids Structures 4, 469-475 (1968) 109. Prager, W., and Taylor, J.E.: Problems of optimal structural design. Journal of Applied Mechanics 35, 102-106 (1968) 110. Reinschmidt, K.F.: Review of structural optimization techniques. In: Proceedings of The ASCE Annual Convention. Philadelphia: 1975 111. Reinschmidt, K.F., and Russell, A.D.: Applications of linear programming in structural layout and optimization. Computers and Structures 4, 855-869 (1974) 112. Rosen, J.B.: The gradient projection method for nonlinear programming. J. Soc. Industrial and Appl. Math. 9, 514-532 (1961) 113. Rozvany, GJ.N.: Structural design via optimality criteria. Dordrecht: Kluwer 1989 114. Rozvany, GJ.N., and Mroz, Z.: Analytical methods in structural optimization. Appl. Mech. Rev. 30, 1461-1470 (1977) 115. Rozvany, GJ.N., and Ong, T.C.: Update to analytical methods in structural optimization. App. Mech. Rev. Update. 289-302 (1986) 116. Russell, A.D., and Reinschmidt, K.F.: Discussion of optimum design of trusses for ultimate loads. J. Struct. Div., ASCE, 97, 2437-2442 (1971) 117. Schmit, L.A.: Structural design by systematic synthesis. In: Proceedings of the 2nd Conference on Electronic Computation. New York: American Society of Civil Engineering 1960, pp. 105-122 118. Schmit, L.A.: Structural synthesis 1959 - 69: A Decade of progress. In: Japan U.S. seminar on matrix methods of structural analysis and design. Tokyo: 1969 119. Schmit, L.A.: Structural synthesis - its genesis and development. J. AIAA 19, 1249-1263 (1981)

298

References

120. Schmit, L.A.: Structural optimization - some key ideas and insights. In: Atrek, E., et al (eds.): New directions in optimum structural design. New York: Iohn Wiley & Sons 1984 121. Schmit, L.A., and Farshi, B.: Some approximation concepts for structural synthesis. I. AIAA 12, 692-699 (1974) 122. Schmit, L.A., and Fleury, C.: Discrete-continuous variable structural synthesis using dual methods. I. AIAA 18, 1515-1524 (1980) 123. Schmit L.A., and Mallett R.H.: Structural synthesis and design parameter hierarchy. I.StrucL Div., ASCE, 89, August 1963 124. Schoeffler, I.D.: Static multilevel systems. In: D.A. Wismer (ed.): Optimization methods for large scale systems. New York: McGraw-Hill Book Company 1971 125. Shanley, F.R.: Weight strength analysis of aircraft structures. New York: McGraw-Hill Book Company 1952 126. Sheu, C.Y., and Prager, W.: Recent developments in optimal structural design. Appl. Mech. Rev. 21, 985-922 (1968) 127. Sheu, C.Y., and Schmit, L.A.: Minimum weight design of elastic redundant trusses under multiple static loading conditions. 1. AIAA 10, 155-162 (1972) 128. Sobiszczanski-Sobieski I., lames, B.B., and Dovi, A.R.: Structural optimization by multilevel decomposition. I. AIAA 23, 1775-1780 (1985) 129. Sobioszczanski-Sobieski, I., lames, B.B., and Riley, M.F.: Structural sizing by generalized multilevel optimization. 1. AIAA 25, 139-145 (1987) 130. Sobiszczanski-Sobieski: Optimization decomposition - a step from hierarchic to non-hierarchic systems. 2nd NASA Air Force symposium on recent advances in multidisciplinary analysis and optimization. NASA CP-3031, part 1 (1988) 131. Somekh, E., and Kirsch, U.: Interactive optimal design of truss structures. CAD lour. 13, 253-259 (1981) 132. Starnes, I.H. Ir., and Haftka, R.T.: Preliminary design of composite wings for buckling stress and displacement constraints. 1. Aircraft 16, 564-570 (1979) 133. Stewart m, G.W.: A modification of Davidon's minimization method to accept difference approximations to derivatives. I. ACM 14, 72 (1967) 134. Svanberg, K.: The method of moving asymptotes - a new method for structural optimization. Int. I. Num. Meth. Engrg. 24, 359-373 (1987) 135. Sved, G., and Ginos, L.: Structural optimization under mUltiple loading. Int. I. Mech. Sci. 10, 803-805 (1968) 136. Topping, B.H.V.: Shape optimization of skeletal structures: A review. I. Struct. Eng., ASCE 109, 1933-1951 (1983) 137. Vanderplaats, G.N.: Structural optimization - past, present and future. I. AIAA 20, 992-1000 (1982) 138. Vanderplaats, G.N.: Numerical optimization techniques for engineering design. New York: McGraw-Hill Book Company 1984 139. Vanderplaats, G. N., and Salagegheh, E.: A new approximation method for stress constraints in structural synthesis. I. AIAA 27, 352-358 (1989) 140. Venkayya, V.B.: Structural optimization: A review and some recommendations. Int. I. Num. Meth. Engrg. 13, 205-228 (1978) 141. Venkayya, V.B., Khot, N.S., and Reddy, V.S.: Optimization of structures based on the study of strain energy distribution, AFFDL-TR-68-150, 1968 142. Wasiutynsky, A., and Brandt, A.: The present state of knowledge in the field of optimum design of structures. Appl. Mech. Rev. 16, 341-350 (1963) 143. Weaver, W. Ir., and Gere I.M.: Analysis of framed structures, second edition. New York: Van Nostrand Reinhold 1980 144. Zangwill, W.I.: Nonlinear programming via penalty functions. Manage. Sci. 13, 344-358 (1967) 145. Zoutendijk, G.: Methods of feasible directions. Amsterdam: Elsevier 1960

Subject Index

Additively separable function, 17,138, 139 Adjoint-variable vector, 128 Analysis, 9, 27 displacement method, 10, 16-18, 44, 146 elastic, 9-29 fInite-element method, 6, 262 flexibility method (see force method) force method, 11, 14-16, 45, 170 kinematic approach, 21 nonlinear, 35 plastic, 20-24 static approach, 21-24 stiffness method (see displacement method) virtual-load method, 18-20, 47, 128129, 161, 191 Analytical optimization, 3 Approximate programming, 140-141 Approximations, 125-175, 259-262, 275-276 along a line, 164-167 combined, 145, 150-155, 172-173 conservative and convex, 134-136, 137 direct and reciprocal, 133-134, 137 displacements, 146-169 forces, 169-173 global, 145, 149-150, 166-167 local, 145, 147-149, 165-166 moving asymptotes, 135-137 reduced basis, 149-150, 154-155 sequential-convex, 144-145 series, 147-149 Augmented Lagrange multiplier, 99, 106109 Augmented Lagrangian, 107 Automated structural optimization, 1

Barrier function (see Penalty function, interior) Basic feasible solution, 83, 87, 89, 271 Basic optimal structure, 271 Basic variables, 86, 271 Basis reduction, 36 Basis vectors, 149 Beam examples, 15, 17, 22, 37, 203, 231, 238, 245 Binomial series, 148-149, 162, 171 Block-angular matrix, 227 Block-diagonal matrix, 226 Buckling constraints, 181, 256-257 Canonical form of equations, 85 Compatibility equations, 10, 192-194, 266 Composite materials, 26 Computer-aided design, 2, 278 Concave function, 65 Conceptual design stage, 1 Conjugate directions, 73 Conjugate directions method, 74-75 Conjugate gradient method, 76-77 Constitutive law, 10 Constraint deletion, 40 Constraint derivatives (see Sensitivity analysis) Constraints, 27-30 active, 29 behavior, 27 critical (see active) equality, 28 explicit, 28, 47, 48 implicit, 28 inequality, 28 passive, 29 side (see technological) technological, 27

300

Index

Continuous problems, 4, 8 Control of structures, 246-253, 211 active, 246 forces, 246 gains for displacements, 246 optimal, 246-249 passive, 246 Convergence criteria, 103 Convex cone, 62 Convex function, 65-66 Convex linearization, 144-145 Convex programming, 66, 111 Convex set, 65-66 Coordination, 225-226 Cost function (see Objective function) Cubic fitting, 10-12, 161 Cubic function, 10 Cutting-plane method, 140 Decomposed matrix, 141, 111 Decomposition, 225 process oriented, 225, 234 system oriented, 225, 231 Derivatives of constraints, (see Sensitivity analysis) Design: constrained, 29 deterministic, 3 probabilistic, 3 unconstrained, 29 Design approaches, 3 Design line, 199, 201 Design plane, 206-208 Design procedures, 119-289 Design process, 1 Design space, 28, 31 Design variables, 25-21 configurational (see geometrical) continuous, 25 cross-sectional, 25, 21 discrete, 25 geometrical, 25, 26 integer, 26 material, 25, 26 pseudodiscrete, 26 topological, 25, 26 Detailing stage, 1 Dimensions, 12n Direct search methods, 13 Direct update methods, 19-80 Direction vector, 110-111 Discrete parameter optimization, 4, 8, 9 Distributed parameter optimization, 4, 8

Dual methods, 116-118 Dual objective function, 94, 116 Dual problem. 94-96, 116 Dynamic programming, 51 Equilibrium equations, 10, 180 Equivalent load method, 236 Equivalent system of equations, 85 Euler's buckling, 181 Euler's theorem, 16 Extrapolation techniques, 103-104 Failure: cost of, 30 probability of, 30 Failure modes, 3, 1 Feasible design, 21, 191-198, 202-203 Feasible directions methods, 98, 110113 Feasible domain (see Feasible region) Feasible region, 28 Feasible solution, 83 Fibonacci sequence, 61 Finite difference, 19 Finite-element method, 6, 262 First-level problem, 230, 236 First-order approximations, 134, 152, 160, 169, 112, 212, 224, 255 (see also Taylor series expansion) Flexibility matrix, 12 Frame examples, 186, 261 Fully stressed design, 194, 210-213 Functional requirements, 1, Gauss-Seidel iteration, 149 Geometric programming, 4, 58 Geometrical optimization, 254-262 Global optimum, 51 Goal coordination method, 229 Golden ratio, 61 Golden section method, 61 Gradient methods, 13, 15-11 Gradient projection method, 119-120 Gradient vector, 58, 15 Graphical interaction, 280 Grillage examples, 41, 252 Ground structure, 263-264 Hessian matrix, 58, 18-80 Homogeneous functions, 160-162 Hyperspace, 28 Indirect methods (see Optimality criteria methods)

Index Integrated problem formulation (see Simultaneous analysis and design) Interactive optimization, 278 Interior methods, 197 Intermediate response, 169 Intermediate variables, 133-139, 160161 Interpolation functions, 138-139 Intersection, 29 Interval of uncertainty, 67 Jacobi iteration, 149 Kresselmier-Steinhouser function, 198 Kuhn-Tucker conditions, 62-63, 210 Lagrange multipliers, 59-60, 107, 116, 217-218, 223 Lagrangian function, 60, 217 Limit analysis, 180 Limit design, 20 Linear programming, 4, 22, 48, 80-97, 278 directions fmding by, 111-112 elastic design by, 191-198 frame design by, 185-186 geometrical interpretation of, 184 optimal topology by, 263-267 optimality condition of, 86 phase I of, 89 phase II of, 89 plastic design by, 180-191 prestressed concrete optimization by, 236-237 sensitivity analysis of, 97 sequence of, 98, 139-142, 255 standard form of, 82 truss design by, 180-181 Linking of design variables, 36 Load: overload, 3, 8, 9 service load, 3, 8, 9 factor, 21 Local minimum (see Relative minimum) Lower bound of the optimum, 197,251 Lower bound theorem of limit analysis, 180 Marginal price, 63 Mathematical programming, 4, 7, 57, 223 Mechanism of collapse, 20 Minimization along a line, 66-72

301

Mobile structures, 264-265 Model coordination method, 229-231 Move limit constraints, 141, 256 Moving asymptotes, 135-136, 145 Multilevel optimization, 225-245, 259262 Multiple optimal topologies, 270-272 Necessary conditions for a minimum, 62 Newton methods, 73, 77-80 Newton-Raphson method, 78, 120 Nondegenerate basic feasible solution, 83 Nonlinear programming, 4, 32, 58, 97120 Normalization of constraints, 39 Numerical optimization, 4 Objective function, 8, 9, 30, 31 Objective function contours, 34 Optimality criteria, 4, 7, 278 a single displacement, 217-218 generalized, 224 multiple displacements, 218-219 Optimality criteria methods, 210-224 envelope method, 221 hybrid, 212 mathematical, 210, 223 physical, 210 Optimization methods, 3 analytical, 3, 8 numerical, 4, 8, 9 Optimization stage, 3 Oscillation, 139 Pattern direction, 74 Penalty function, 98-109 extended, 105 exterior, 98-101 interior, 98, 102-106 Penalty-function methods, 98-109 Penalty parameter, 99-100, 103 Pivot, 87 Pivot operation, 85 Plastic collapse, 20 Plastic design, 180-191 Plastic hinges, 20 Plastic moment, 21, 186 Polynomial fitting, 69-72, 166-167 Positive defmite matrix, 59 Preassigned parameters, 25, 203 Prestressed concrete elements, 203-206 Prestressed concrete systems, 203-206, 233-241

302

Index

Prestressing by lack of fit, 195 Primal problem, 94 Principal minor, 59 Quadratic fitting, 69-70, 166-167 Quadratic form, 58, 69 Quadratic function, 69, 73 Quadratic programming, 143 Quadratically convergent methods, 73 Quasi-Newton methods, 73, 77-80 Recurrence relation for redesign, 219220,223 Reduced basis method, 149-150, 154-156 Reduced gradient method, 119-120 Reinforced concrete design, 242 Relative minimum, 40, 57, 267 Response factor (see Penalty parameter) Rotation matrix, 206 Saddle point, 61, 116 Scaling of a design, 151-153, 161, 172, 198-202, 207-208, 214, 220-221 Scaling multiplier, 151 Scaling of variables, 38 Second-level problem, 230, 237 Sensitivity analysis, 126-133 adjoint-variable method of, 128 direct method of, 126-127 semi-analytical method of, 131 virtual-load method of, 128-129 Sequential approximations, 139-145 Sequential unconstrained minimization, 98-109 Shadow prices, 96 Shape optimization, 7 Simplex method, 85-93 Simultaneous analysis and design, 35, 45, 46, 48, 228 Singular optimum, 267 Stability (see Buckling constraints) Static theorem, 180 Stationary point, 59 Steel design, 242-245 Steepest descent method, 75-76 Stiffness matrix, 10 Stress criteria, 210-216 Stress ratio, 211, 215

Taylor series expansion, 58, 77, 134, 147-148, 160 Topological optimization, 254, 262-277 Transformation of variables, 181-183, Trial and error, 1 Truss structures, 180-181 Truss examples: bridge, 256 eleven-bar, 196 fifteen-bar, 272 four-bar, 12, 20, 49 seven-bar, 249, 265 ten-bar, 156,185, 209, 213, 276 three-bar, 24, 32, 131, 163, 167, 183, 194, 215, 222, 229, 268 transmission tower, 157, 182 twenty-five bar truss, 173 Two-level optimization, 233-241, 251252 Ultimate load, 180 Unbounded solution, 84, 89 Unconstrained minimization methods, 66-80, 278 Unimodal function, 67 Upper bound of the optimum, 197,251 Variable metric method, 78-79 Variables: active, 218-220 artificial, 89 behavior, 26 coordinating, 227 dual,94 integer, 262 inverse (see reciprocal), local, 227 passive, 218, 220 reciprocal, 133-134, 162, 224 slack, 61, 82, 271 surplus, 82 Vertex, 84 Yield conditions, 180 Zero-order approximations, 212 Zigzag, 76

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