Constraint screening is an automatic process for both design sensitivity and optimization and consists of two parts: regionalization and deletion. You can override some of the defaults associated with this process by
• Providing explicit region IDs on the DRESP1 and DRESP2 entries.
• Using DSCREEN entries to override key screening parameters.
There is often little need to override the constraint screening defaults, since these perform well in the vast majority of cases. However, computational efficiency can sometimes be gained by modifying these defaults in special cases. This section outlines constraint screening so that you can make effective use of it should the occasion arise.
Basic Concepts
The basic assumption behind constraint screening is that most structural optimization problems contain more constraints than are necessary to adequately guide the design. For example, suppose bending stresses for every beam element are required to be less than some allowable upper limit. Not every element in the model may be critically loaded, while those that are may undergo more or less similar changes as the design is modified. This is a case of extraneous and redundant design data. If the constraints can be “filtered” in a consistent manner, a considerable increase in efficiency may be realized.
A response that is to be constrained must first be defined on either a DRESP1 or DRESP2 entry. Response lower and upper bounds are set with a DCONSTR entry. For the j-th response this definition yields
Equation 3-5.
NX Nastran uses these bounds to create a pair of normalized constraints as
Equation 3-6.
where the absolute values of the bounds are used as normalizing factors.
This normalization provides a convenient method of screening the constraints. For example, any normalized constraint with a value of +0.5 has violated its bound by 50%, while a constraint with a value of –0.5 is within 50% of its bound. We might argue that it may not be necessary to retain every constraint, just those that are greater than some normalized value.
Constraint Deletion
Figure 3-4represents a group of constraints in bar chart form. In constraint deletion, any constraint exceeding the truncation threshold value (shown as TRS in the figure) is retained for the current design cycle. Those constraints whose values are less than this threshold are temporarily deleted for the current cycle. These deleted constraints may again become active during subsequent design cycles and thus retained.
Figure 3-4. Constraint Deletion
The constraints retained for the current design cycle are denoted by an ‘X’ in the figure. You can change the value of TRS from its default of –0.5 using a DSCREEN entry. TRS is defined on a per response-type basis.
Constraint Regionalization
Each set of structural responses is automatically assigned a unique “region” identifier. All constraints associated with this region for a given subcase comprise a unique group. The assumption is that constraints from this group are likely to contain redundant information from this group, only a maximum number of constraints is retained (as determined by NSTR on the DSCREEN entry).
Note
Responses are usually grouped into regions based on their DRESP1 entry identification or by their property group association. See the DSCREEN entry for default region specifications.
For example, imagine a highly stressed, constant thickness panel comprised of a large number of shell elements. Suppose we are seeking to redesign the panel thickness, subject to element stress constraints. Due to the large loads, a number of elements in this panel may yield stress constraints that are well above the truncation threshold TRS. They will thus pass the first screening test. However, this still gives us more design information than we really need since, as the panel thickness is changed, the stresses will probably vary more or less in unison. It is probably reasonable to consider only the few largest stresses in this region and temporarily ignore the rest even though they are greater than TRS. Grouping similar constraints together is the idea behind constraint regionalization.
Note
The default for NSTR is 20. In a region of high stress, modeled using a fine mesh of plate elements, reduction to an NSTR number of constraints could easily lead to a hundredfold reduction in the number of design constraints.
Suppose the first 30 constraints inFigure 3-4actually belong to three separate regions as shown inFigure 3-5. In Region 3, eight constraints are greater than or equal to the truncation threshold. It may be advantageous to consider only a few of the largest retained constraints from this region and disregard the others. Suppose that NSTR = 2 for Region 3. Then only two out of the eight possible constraints in this region are retained and six are discarded even though their values exceed the truncation threshold. In Regions 1 and 2, less than two constraints are retained; therefore, no further deletion takes place. In the end, only three constraints out of the original 30 are retained in the approximate problem.
Figure 3-5. Constraint Regionalization Constraint Screening and Sensitivity Analysis
Constraint screening precedes sensitivity analysis as can be seen from the flowchart inFigure 3-1 and in the detailed Solution 200 flowchart inDesign Modeling for Sensitivity and Optimization.
The default values of TRS and NSTR are usually fine for most design optimization problems;
however, for design sensitivity analysis, these default values may not be satisfactory. It may be that some of the responses for which you had requested that sensitivities be computed are screened out. To ensure that all responses are retained, you may need to set TRS to a large negative number, perhaps –10. or –100. Also, depending on the number of responses per region, it may be necessary to increase NSTR. This combination ensures that all constraints pass both levels of screening and all responses are retained for sensitivity analysis. This procedure is the only way to turn off the screening.
Formal Approximations
NX Nastran uses formal approximation to avoid the high cost associated with repeated finite element analyses during design optimization. Two basic types of approximation are used
• Direct variable approximations.
• Reciprocal variable approximations.
These approximations are used in various ways according to three different methods available in NX Nastran:
• Direct Approximations (APRCOD = 1).
• Mixed Approximations (APRCOD = 2 default).
• Convex Linearization (APRCOD = 3).
Note
A DOPTPRM entry is used to select APRCOD.
Choosing to override the default approximation is usually problem-dependent, and often without strict rules or guidelines. This subsection presents theoretical details that might help you decide when such an override might be appropriate.
The formal approximations used in NX Nastran are all based on simple first-order Taylor series expansions. The general form of this expansion is
Equation 3-7.
Approximation Errors
The required derivative information is available from the design sensitivity analysis as outlined inDesign Sensitivity Analysis. Note that the approximate response, , inEq. 3-7is linear in the design variable change Δx. Thus, we expect some error when we try to approximate responses that are actually nonlinear in Δx. Figure 1-16inStructural Optimizationshows how the error is a function of Δx.
Move Limits and Approximations
During approximate optimization, the code places limits on the maximum allowable moves in the design space in order to minimize the errors associated with these approximations. In NX Nastran, move limits are placed both on property as well as design variable changes. Move limits are discussed inOptimization with Respect to Approximate Models.
Direct Approximations
A direct variable approximation linearizes the function directly in terms of the design variables.
For the j-th constraint function, the direct approximation is written as
Equation 3-8.
where the subscript D indicates a direct approximation.
The quantity represents a total vector move in the design space from the initial design . The partial derivatives ∂gj/∂xiare available from the design sensitivity analysis. An equation similar toEq.3-8can be written for the objective function, where F replaces g in the expression.
Reciprocal Approximations
The first-order Taylor series expansions ofEq. 3-8can alternately be expressed in terms of reciprocal variables. This choice turns out to be quite useful for those responses that are inversely proportional to the design variables. This simple idea can be easily shown in the case of the axially loaded rod element ofFigure 3-6where the cross-sectional area A is taken as the design variable.
Figure 3-6. Axially Loaded Bar Element
The axial stress in the bar is equal to P/A, and the displacement is equal to PL/AE. If these responses are used in the design model, linearizing with respect to the quantity 1/A will produce approximations that are exact in both cases. In the general case, of course, these approximations are not exact due to the static indeterminacy of the structure. However, for all element types, the proportionality of the stiffness matrices to the inverse of the sizing quantities forms a reasonable basis for arguing the use of reciprocal approximations.
Reciprocal approximations can be derived from linear approximations if we first substitute the intermediate variables yi. For the approximate constraint ofEq. 3-8, we get
Equation 3-9.
Setting the intermediate variables to reciprocals of the design variables
Equation 3-10.
and noting that ∂y = –(1/x2)∂x, and that ▵y = –▵x/(x°x), we obtain:
Equation 3-11.
where the subscript R indicates a reciprocal approximation.
Approximation Methods
Three different approximation methods are available in NX Nastran. You can select any one of them using the APRCOD field on the DOPTPRM entry. These methods are as follows:
• APRCOD = 1: This specifies that direct approximationsEq. 3-8 are to be used for the objective function as well as all constraints. The resulting approximate optimization is then performed with respect to an entirely linear approximate design space. You should use this option if you know that the structural responses are well-approximated by linear functions in the design variables. Otherwise, without carefully chosen move limits, a greater number of approximate optimization cycles may be required before convergence is achieved.
• APRCOD = 2: (Default) This choice specifies that mixed approximations (or more precisely, a mixture of approximations) are to be used. Direct approximations are used for volume, weight, element force, and buckling load responses, while reciprocal approximations are used for all other response types. This method works well in a wide variety of problem types.
Because of its reliability, it has been selected as the default method.
• APRCOD = 3: This selects the convex linearization method. Essentially, this method chooses either a direct or reciprocal constraint approximation depending on which one provides the larger estimation of the constraint function. In other words, this method chooses the more conservative of the two approximations.
For example, the direct approximation for the j-th constraint is
Equation 3-12.
while the reciprocal approximation is
Equation 3-13.
Since both approximations are readily available (both use the same gradient information), the choice of which to use can be made just by looking at the sign of the difference between the two:
Equation 3-14.
Since the squared term in the expression is always positive, the choice depends on the sign of the product:
• if Use the direct approximation for xi since it yields a larger estimate of constraint value.
Equation 3-15.
• if Use the reciprocal approximation for xisince it yields a larger estimate of constraint value.
Equation 3-16.
This criterion is applied on an individual design variable basis. Thus, a combination of direct and reciprocal approximations can be made for a single constraint.
For convex linearization, the objective function is always linearized, regardless of its response type.
Modeling Guidelines
There are several situations where the default mixed method (APRCOD=2) is not applicable and therefore not recommended. The first is when basis vectors for either property or shape optimization are used. In this case, the physical justification for the use of reciprocal approximations does not apply (seeReciprocal Approximationsearlier in this chapter), and either direct approximations (or perhaps even convex linearization) should be used instead.
The second situation occurs if design variables can pass through zero during the course of optimization. This can occur if the lower bound on the DESVAR entry is a negative quantity and the upper bound is positive. In this case, APRCOD is automatically set to 1 to avoid ill-conditioning about the zero value of the design variable.
Finally, experience has also shown that the mixed method is not always the best performer for dynamic response optimization tasks. For these highly nonlinear responses, direct approximations, coupled with more stringent move limits are often more reliable.