A design sensitivity coefficient is defined as the rate of change of a particular response quantity r with respect to a change in a design variable x or, in other words, as ∂r/∂x. These coefficients are evaluated at a particular design characterized by the vector of design variables , giving
Equation 3-32.
where subscripts are used to indicate the i-th design variable and the j-th response. Eq. 3-32is just the slope of the response with respect to xias is shown inEq. 3-7.
Figure 3-7. Design Sensitivity Coefficient—Graphical Interpretation For Which Responses Are Sensitivities Computed?
Sensitivities are computed for all of the responses (both first- and second-level) that are used to define the objective function and retained constraints.
Recall fromApproximation Concepts in Design Optimizationthat sensitivity analysis is always performed after the constraint screening phases are completed (Figure 3-1). Thus, fewer response sensitivities may be computed than there are responses defined in the design model.
Of course, this is the goal since the computational effort associated with sensitivity analysis is reduced. (If you wish to force the computation of all sensitivities for all responses defined in the design model, you can effectively disable the screening process by using a large negative TRS on the DSCREEN Bulk Data entries along with a large NSTR if necessary. See the DSCREEN entry description inInput Datafor further details.)
Requesting Sensitivity Analysis Output
Design sensitivities are output in the form ofEq. 3-1. In Solution 200, you request design sensitivity analysis and output with either the DSAPRT Case control command or the following Bulk Data parameter definition:
PARAM, OPTEXIT, ±4
Note
You can also request design sensitivity coefficient output with an OPTEXIT value of 7. See Solution 200 Program Flow), Solution 200 Program Flow, for details.
An OPTEXIT of ±4 prints the sensitivity coefficients to the standard output file. A value of –4 outputs the coefficients in binary form using OUTPUT4 formats. DSAPRT has priority over the PARAM, OPTEXIT, ±4 (or 7) option. It does have somewhat more flexibility and options, such as formatting.
Identifying the Sensitivity Coefficients
If requested by a PARAM,OPTEXIT,±4 or 7, or by the “unformatted” DSAPRT Case Control command, sensitivity coefficients are output in the matrix DSCM2. If you request sensitivity coefficients with the formatted DSAPRT Case Control command, then the software prints the coefficients in a table with headers. You can use both options with an EXPORT option to write the sensitivities to an external binary file.
Each column in the matrix DSCM2 corresponds to a particular response quantity, while each row corresponds to a design variable. The output is formatted in one column after another, in sequence, such as response by response. The component of the i-th row and j-th column is
Equation 3-33.
A correlation matrix is also output to help identify the column order of the individual design responses. The row order in DSCM2 corresponds to the design variables defined on the set of DESVAR entries sorted by increasing design variable ID. SeeOutput Features and Interpretation, Design Sensitivity Output, for examples illustrating the interpretation of this output.
Internal Representation of the Sensitivity Coefficients in NX Nastran
For reasons of efficiency as well as accuracy, NX Nastran uses a slightly different internal representation of these coefficients than described above, depending on how the design variables are related to the analysis model properties.
For DVPREL1 type relations, the set of independent design variables is taken as the basis for the design sensitivity coefficients (see the DLINK entry for a definition of independent and dependent design variables). This is an efficient choice since a large number of properties may be a function of a much smaller set of design variables.
For DVPREL2 relations, the basis for sensitivity analysis is taken to be the set of analysis model properties referenced on all of these second-level relations. Inside the code, the sensitivities are then
Equation 3-34.
Choosing properties as the basis greatly improves the accuracy of the approximate responses used in design optimization. Since an explicit relation between the properties and the associated design variables is given by the DVPREL2–DEQATN entry pair, this information can be used to evaluate the properties for a given change in the set of design variables. Computing the sensitivities with respect to the design variables would have linearized this relation, resulting in a less accurate approximation.
To summarize, the sensitivity coefficients used internally in NX Nastran are
Equation 3-35.
where NIDV is the number of independent design variables and NIPR is the set of properties defined on all DVPREL2 entries.
To illustrate the advantage in using a mixed design variable-property basis, consider the normalized lower-bound constraint inEq. 2-29,Defining the Constraints.
Equation 3-36.
This is the form of the lower-bound constraint that is automatically generated in NX Nastran from the DCONSTR entry data. For a finite move in the design space Δx, the approximated constraint is
Equation 3-37.
Since the response is only approximate, the constraint is approximate as well. For responses that depend on properties that are linear functions of the design variables (DVPREL1), the approximation is
Equation 3-38.
while, for responses that depend on the properties that are nonlinear functions of the design variables (DVPREL2),
Equation 3-39.
where the properties at the perturbed design inEq. 3-39can be evaluated precisely from the input equations DEQATN.Eq. 3-38andEq. 3-39show the advantage of generalizing the basis to include not only design variables but also design properties.