Equivalent Static Load Method (ESLM)

The iteration at the third step is referred to as an inner iteration; Steps 2 through 4 form the outer loop. The converged solution at Step 3 is the starting point of the next outer loop (if the design does not converge at the current outer loop). If there are time steps in the multibody dynamics analysis at Step 2, subcases in static response optimization are generated at Step 3 (provided that the time step screening option is deactivated). See Parameters.

When performing ESLM on the flexible bodies, these joints are not included in this static subcase-based structural optimization. This means that each flexible body will have 6 rigid body modes. The 6 rigid body modes of each flexible body must be removed for structural analysis. Exactly 6 degrees of freedom (DOF) of each flexible body must be fixed to remove the 6 rigid body modes. If more than 6 DOF are fixed in a flexible body, the additional fixed DOF become the constraint of the flexible body, which may not result in an optimal solution and consequently increases the required ESLM outer loops.

Due to the way ESL is calculated, each flexible body is in its equilibrium at the 0-th inner iteration of each outer loop. This is why you can fix an arbitrary 6DOF (usually single node) to get rid of rigid body modes in order to do static analysis with ESL. Reaction force at the fixed node is zero at the 0-th inner iteration. Thus, no additional load is applied to a flexible body although 6 DOF of the flexible body are fixed. However, design changes occur as the inner iteration goes on. This means the original configuration that maintains equilibrium also changes. As a result, the equilibrium status does not hold true anymore from the 1st inner iteration in each outer loop. An undesirable effect caused by a broken equilibrium status (disequilibrium status) is the reaction force at the fixed point is not zero anymore, which means additional load is applied to that fixed point. This effect, due to shape/size changes, can be minimized by fixing a proper node, as explained below.

A common way to remove the 6 rigid body modes is to fix all 6 DOF of a node. When looking for a node to fix, choose one that is not in a high stress region. If the fixed point is located in a high stress region, the optimization can be very slow. It is highly recommended that the 6 DOF of an independent node of a spider or rigid element (usually used to model joints) be selected to be fixed. On solid models, where the nodes do not have rotational stiffness; if a solid model does not have a rigid element to represent a joint, the next best way to remove the 6 rigid body modes is to fix 3 translational DOF (123) of one node, 2 translational DOF (23) of another node, and 1 translational DOF (3) of a third node. However, still make sure that the 3 nodes are not in a high stress region, otherwise this method of removing 6 rigid body modes does not work. One way or another, all of the 6 rigid body modes of each flexible body must be removed. SPC, SPC1, or SPCADD (referenced in the Subcase Information Section) can define the fixed DOF.

Figure 3. Solid Model with Joints at the Centers of Two Holes

Figure 4. Stress Contours of Model at Two Time Steps

Figure 5.

Nodes A and B are the locations of joints. The best option here is to fix 6 DOF of either node (A or B) in order to remove 6 rigid body motions in this model. To fix alternative nodes other than node A or B, it would work to fix 3 DOF (123) of node E1, 2 DOF (23) of node E2, and 1 DOF (3) of node E3. These three nodes are located in a relatively low stress region. In this model, nodes C1, C2, and C3 or nodes D1, D2, and D3 would take a long time to converge if fixed. Again, the best and simplest way to remove 6 rigid body modes of each flexible body is to fix one of the joint locations in each flexible body.

When the boundary conditions are properly defined, displacement constraints in an optimization can be applied to limit the deformation of the flexible body. An example is to optimize a rotating cantilever, by fixing either the left end or the middle point to retrieve deformation as long as the point has nothing or little to do with the shape perturbation vectors. If a relative displacement at the right end with respect to the left end is to be constrained, it would be most convenient to fix the 6 DOF of the left end to measure the relative displacement. Constraining a relative displacement of the left end with respect to the middle point of the cantilever, it would be best to fix the 6 DOF of the middle point. Regardless of whether the left end or the middle point (or even the right end) is fixed, the stress is the same. If the optimization problem is simply to constrain stress or minimize the maximum stress of a flexible body, all that is needed is to fix one proper node of the flexible body.

Figure 6. Rotating Cantilever

Figure 7. Deformation When the Left End is Fixed

Figure 8. Deformation When the Middle Point is Fixed

In addition to the above output files, .desvar, .prop, .grid, .fsthick, and .oss will be found, if available.

By doing this, the optimizer can consider more precise responses at the critical time intervals, resulting in a decreased probability that the optimizer will go in the wrong direction in the current outer loop.

MBSIM 4 TRANS END 1.0 NSTEPS 100

$more time steps from 0.0 sec to 0.02 sec.
MBSIM 1 TRANS END 0.02 NSTEPS 200
+ VSTIFF 
MBSIM 2 TRANS END 0.61 NSTEPS 200
+ VSTIFF 
$more time steps from 0.61 sec to 0.63 sec.
MBSIM 3 TRANS END 0.63 NSTEPS 200
+ VSTIFF 
MBSIM 4 TRANS END 1.0 NSTEPS 100
$
MBSEQ 10 1 2 3 4

Figure 9. Response Time History

In the above MBSIM cards, time steps in the time interval 0 to 0.02 seconds and the time interval 0.61 and 0.63 seconds have been increased. Generally, the more information the optimizer is provided, the better the solution it can provide. It is beneficial to define many time steps; enough to catch precisely the critical responses in the time intervals where critical response of interest show up. This will enhance convergence of the optimization process.

DOPTPRM, ESLMAX

In this case, the ESLMAX is an integer value, greater than or equal to 0. If the ESLMAX is equal to 0, then optimization process is not activated. The default value for the ESLMAX is 30.

DOPTPRM, ESLSOPT

The ESLSOPT can be either 0 or 1. If ESLSOPT is 0, the screening is deactivated. If ESLSOPT is 1, this screening is activated. The default value is 1. Unless deactivated by DOPTPRM,ESLSOPT,0, this feature is always activated.

The time step screening feature is different from the constraint screening of the static response optimization in Step 3, although the concepts of both screening capabilities are similar. The time step screening is performed at time Step 2, which is right before entering the static response optimization phase. By performing the time step screening before entering the static response optimization phase, the huge amounts of CPU time required for structural analysis and subsequent constraint screening strategy can be saved at the 0-th iteration in the static response optimization phase.

DOPTPRM, ESLSTOL

ESLSTOL is a real number between 0.0 and 1.0. The smaller the value of ESLSTOL, the fewer the number of time steps involved in the optimization process. If ESLSTOL is 1.0, all of the time steps in multibody dynamic analysis will be involved in the optimization process. Obviously, the smaller the ESLSTOL value, the less total CPU time will be used. Note, however, that assigning too small a value to the ESLSTOL could cause the optimization process to diverge. Therefore, if the number of time steps retained by ESLSTOL is less than 10, the 10 most dominant time steps will be involved in the optimization process. The default value for the ESLSTOL is 0.3.

All analysis output requests are invalidated during ESL optimization except that Output Request = option is allowed above the first subcase. For example, STRESS=ALL or DISPLACEMENT=SID is allowed above the first subcase.

Optimization output is not controllable in ESL optimization. See Equivalent Static Load Method (ESLM) for more details.

Output and format in output format controls are not controllable in ESL optimization. All requests on output and format are invalidated during ESL optimization, except for FSTHICK and OSS.

Minimize joint force of a joint
subject to stress < allowable value
deformation < allowable value
velocities of a node < allowable value

MBD system level responses can be controlled by the properties of PRBODY, CMBUSH(M), CMBEAM(M), and CMSPDP(M) as well as usual design variables in structural optimization such as thickness of bodies, shape of bodies, etc. DVMBRL1 and DVMBRL2 are available to define relationships between design variables and properties of PRBODY, CMBUSH(M), CMBEAM(M), and CMSPDP(M).