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Distributing Coupling Elements and Kinematic Coupling Constraints Courtesy, Hibbitt, Karlsson & Sorensen (Michigan), Inc.
Distributing Distributing coupling elements (DCOUP2D, DCOUP3D) and kinematic coupling constraints, introduced with ABAQUS/Standard Version 5.8, offer general capabilities for transmitting loads and associating motions between one node and a collection of “coupling” nodes. Both options associate the coupling nodes with a single node in a “rigid body” sense; translations and rotations of the node (the distributing distributing coupling element node or kinematic coupling reference node) are associated with the coupling node group as a whole. The distinction between the two options is in how the rigid body association is enforced. The following examples illustrate this distinction.
Kinematic Coupling Constraints The ∗KINEMATIC COUPLING option is a nonlinear generalization of the NASTRAN RBE2 element. For this constraint the rigid body association between the coupling nodes and the independent reference node is exact, similar to a BEAM B EAM multi-point constraint. Unlike the latter, however, however, with kinematic coupling constraints the user is free to constrain degrees of freedom selectively (in a corotational coordinate system when finite rotations occur) at the coupling nodes. The kinematic coupling generally results in a stiff constraint that can be tailored to specific needs. For example, consider the assembly shown below.
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Contents Distributing Coupling Elements and Kinematic Coupling Constraints
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Penalty Contact in ABAQUS/Explicit
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assembly would involve modeling contact between the relatively relatively stiff box section and the relatively compliant shaft. However, However, a simplified modeling approach can be pursued by using a kinematic coupling constraint to approximate the effect of the rigid box section on the shaft. The shaft is seen as transmitting only circumferential motion, which is achieved by constraining only the circumferential degree of freedom on the slots to the rigid body motion of a reference node on the cylinder axis. θ
z
z Reference node 500
r The rigid body motion of the reference node is transmitted to selected degrees degrees of freedom freedom of the coupling nodes. For positive rotation about the z-direction, only the red node regions would be included in the constraint.
The constraint on the circumferential displacement can be defined as follows (assuming a positive positive rotation about the z-direction): *orientation,system=cylindrical,name=kc 0.0, 0.0, 0.0, 0.0, 0.0, 1.0 *kinematic coupling,ref node=500,orientation=kc red_nodes, 2
y
z
x
The axial and radial displacements on the coupling nodes are not affected by this constraint. ABAQUS imposes the constraint on the circumferential displacement by eliminating that degree of freedom at the coupling nodes. (As with MPCs and equations, once any combination of translational or rotational degrees of freedom at a coupling node is constrained, additional translational or rotational constraints cannot be applied to that node.)
Distributing Coupling Elements Torque 3
Possible regions for kinematic coupling constraints
A notched shaft slides onto a stiffened box section with protrusions. A torque is applied to the stiff section, and we wish to understand the finite rotation response of the shaft, which is fixed at its far end. One approach to analyzing this
Distributing Distributing coupling elements, DCOUP2D and DCOUP3D, are nonlinear generalizations of the NASTRAN RBE3 elements. DCOUP elements influence the response of a collection of coupling nodes via a single node, which forms the DCOUP element. The connection is created by using the ∗DISTRIBUTING COUPLING option. NASTRAN is a registered trademark of NASA.
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The DCOUP element, unlike the kinematic coupling constraint, enforces a rigid body association between the coupling nodes and the element node in an average sense; and the user is free, through the use of nodal weight factors, to control the averaging. The user-defined averaging can be used to distribute applied forces and moments to the coupling nodes, to prescribe an average displacement and rotation to the coupling nodes, to distribute mass to the coupling nodes, and to create a flexible connection between the structural and solid elements. This average association between the nodes’ motions has some desirable properties: • The structure to which the coupling nodes are attached will not be stiffened by the DCOUP element. • Forces transmitted through the element, either through the application of a ∗CLOAD at the element node or through the force of the rigid body constraint, will be proportional to “weight” factors assigned to the coupling nodes. This proportionality is generally complex, but physically motivated, and similar to force distributions in a classic bolt-pattern analysis. • The element node is constrained to follow the average motion of the coupling nodes, but no strict constraint is implied in the opposite sense. Individual coupling nodes are not constrained to follow the rigid body motion of the element node, only the coupling node group as a whole. • Boundary conditions can be prescribed to nodes referenced by the ∗DISTRIBUTING COUPLING option. These distributing coupling elements are appealing when a load transfer path is known but it is feasible or desirable to suppress geometric details for the analysis at hand.
For example, consider the case of a global-local analysis of an offshore oil structure. A global beam or frame element model can provide resultant axial, bending, torsion, and shear forces near a complex connection. We can then use distributing coupling elements to apply those loads to a local, detailed shell model of the connection region. The use of the DCOUP element in an analysis such as this means that the cut sections are still free to ovalize and warp as the joint responds to the resultant forces and moment; thus, cuts can be made closer to the joint without adversely affecting the solution due to unnatural stiffening as would occur if a beam MPC or the kinematic coupling option was used. Using the ∗CLOAD option, the relevant section forces from the global model can be applied to nodes located at the centroid of each cut section of the connection region. These forces and moments should be applied in local coordinate systems (defined with ∗TRANSFORM) that correspond to the beam axis and normal directions in the global model. Each of the centroidal nodes is used to define a distributing coupling element (DCOUP3D) that references the circumferential nodes of the relevant cut section via the ∗DISTRIBUTING COUPLING option. To ensure that the shell model is in global equilibrium, we define minimal boundary conditions to constrain rigid body motion. The reaction forces in each of the six degrees of freedom chosen for this purpose should be minimal if the applied loads and moments are in equilibrium.
Penalty Contact in ABAQUS/Explicit A penalty contact algorithm was introduced in ABAQUS/Explicit Version 5.8 as an optional alternative to the default kinematic contact algorithm. The addition of penalty contact expands the range of contact problems that can now be addressed with ABAQUS/Explicit. There are fundamental differences in the way in which the kinematic and penalty contact algorithms enforce contact constraints. These differences can be illustrated with the simple diagrams shown in Figure 1. This figure illustrates a single slave node that is about to come into contact with a fixed master surface. i
pred
f i md i f i = -------------------2 ( ∆ t )
n predicted configuration
i+1
(a)
d pred
i
n
i+1 cu r
f i + 1 = kd
f i+1
i+1
cur
A detail of a tubular joint with resultant forces and moments applied through distributing coupling elements.
(b)
d
Figure 1: (a) Kinematic contact (b) Penalty contact
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(a)
(b)
Figure 2: Intermediate configuration of a frame rail impacting a rigid wall: (a) Kinematic contact (b) Penalty contact
The kinematic algorithm is a predictor/corrector method. When kinematic contact is active in an analysis, ABAQUS/Explicit carries out a predictor phase and a corrector phase in each time increment. In the predictor phase the kinematic state of the model is advanced by ignoring any contact conditions. This can result in overclosure or penetration, as shown by the predicted configuration of the slave node in Figure 1a. In the corrector phase of the time increment, an acceleration correction is applied to the slave and master nodes to correct for this predicted penetration, while conserving momentum. This correction results in a final configuration for increment i in which the slave node is exactly in compliance with the master surface. It can be seen that the kinematic algorithm is essentially implicit —it seeks to eliminate the contact penetration at the end of each time increment. The penalty algorithm uses an explicit approach to enforcing contact constraints. Figure 1b illustrates the same slave node penetrating the fixed master surface at the end of increment i (beginning of increment i+1). However, in contrast to the kinematic algorithm, a corrector phase is not processed for increment i. Rather, an interface “spring” is inserted automatically between the slave node and the master face in increment i+1 to minimize the contact penetration. The force associated with the interface spring is equal to the spring stiffness multiplied by the penetration distance. A small residual penetration will, therefore, exist, since contact forces are not generated unless there is some amount of penetration at the beginning of the increment. Thus, the explicit nature of the penalty algorithm is apparent, since it seeks to resolve contact penetrations that exist at the beginning of each time increment. Because the kinematic algorithm is implicit, it has no effect on the ongoing calculation of the stable time increment during the analysis. The penalty algorithm may have the effect of reducing the stable time increment, since the penalty springs increase the overall stiffness acting on the
interface nodes. This added stiffness can reduce the stable time increment in the same manner that increasing the stiffness of a material can. ABAQUS/Explicit automatically computes a default spring (penalty) stiffness using the mass and stiffness of the contacting bodies. The default penalty stiffness is calculated to minimize residual penetration, while reducing the stable time increment by no more than 4%. It is possible to override the default penalty stiffness by scaling it upward or downward using the ∗CONTACT CONTROLS, SCALE PENALTY=value option. If the penalty stiffness is scaled up significantly, the automatic time incrementation algorithm in ABAQUS/Explicit will account for this and automatically reduce the stable time increment accordingly. Though there are significant differences in the kinematic and penalty contact algorithms, for most problems they will yield similar results. Figure 2 shows the deformed configurations of a beam discretized with shell elements as it is crushed axially and collapses onto itself. Contact occurs between the shells and flat rigid bodies attached to either end of the beam (not shown), as well as between the different regions of the shell itself (self-contact). The two configurations shown are quite similar, even though different contact algorithms are used in each analysis. In addition, the choice of contact algorithm has little effect on the computational cost, since the overall CPU times are nearly identical in these two cases. Penalty contact is invoked by adding the PENALTY parameter to the ∗CONTACT PAIR option.
(a)
(b) Figure 3: Forging example (a) Kinematic contact (Pure master-slave) (b) Penalty contact (Balanced master-slave)
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It is possible to use a combination of kinematic contact pairs and penalty contact pairs in the same analysis. One advantage of penalty contact is that it allows a balanced master-slave approach for contact between faceted rigid bodies and deformable bodies. Figure 3 shows a twodimensional forging model, in which the billet must flow around two sharp corners in the rigid forging dies. Using the required pure master-slave approach for kinematic contact between a deformable and a rigid body, noticeable penetration of one of these corners into the billet occurs, even with adaptive meshing invoked for the billet. However, using a balanced master-slave approach with penalty contact, this penetration is greatly reduced, since the nodes at each corner of the rigid dies can now be considered as slave nodes in the contact algorithm. In Version 5.8 ABAQUS/Explicit uses a balanced master-slave approach by default for all penalty contact pairs between discretized surfaces. Penalty contact also provides the capability to simulate contact between two rigid bodies. At least one of the bodies must be discretized with elements (contact between two analytical rigid surfaces is not possible). Figure 4 shows a simple mechanism modeled in ABAQUS/Explicit. The primary goal of the analysis is to evaluate the kinematic motion and forces associated with actuating the mechanism. The stresses in each component are not critical, and the corresponding elastic deformations are considered negligible; hence, it is possible to consider each component as rigid. Since a rigid body in ABAQUS/Explicit can be any collection of nodes and elements assigned by the user, components of a model can be discretized initially with deformable elements, then designated as rigid using the ∗RIGID BODY option. The rigid vs. rigid contact approach allows the simulation to run much faster than if each component were considered deformable, since expensive
element calculations are not required and larger time increments can generally be used. Figure 5 shows time histories of displacement and reaction force obtained when both the cam and cam follower are considered as rigid. The main disadvantage of penalty contact is that the contact constraints are not enforced exactly—there is a small residual penetration that exists while two bodies are in contact. In most applications this residual penetration has a negligible impact on the solution of interest. However, in some small-deformation, displacement-driven Hertz contact problems, this small residual penetration may have a more significant effect on the predicted stresses in the contacting bodies. In such cases the default kinematic contact algorithm should be used. The following guidelines can, thus, be established regarding the choice of kinematic or penalty contact for a particular problem: • In most cases the two algorithms will yield very similar results at similar computational expense. • Penalty contact allows a balanced master-slave approach for all contact pairs between two discretized surfaces. This feature can be used to avoid contact penetrations that sometimes develop due to a pure master-slave approach. • If contact between two rigid bodies is desired, penalty contact must be used. • Kinematic contact should usually be considered in smalldeformation, displacement-driven Hertz contact problems.
Spring compressed
Cam node 3530
2 Follower pinned 1
(a) Cam pinned and rotated
3
Figure 4: A simple timing mechanism.
ABAQUS
(b)
Figure 5: (a) Normalized displacement time history of node 3530 on the cam. (b) Time history of normalized reaction force at the rigid body reference node of the cam.
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