OptiStruct can calculate and output design sensitivities for structural optimization problems.

Prerequisite: Create a shape or sizing optimization .fem deck for OptiStruct.

Sensitivity analysis is a technique to understand how certain responses in a design optimization problem are influenced by changing the values of the design variables. The design sensitivities are the derivatives of the response with respect to the individual design variables that are calculated to update the design variables in an optimization problem.

For a response *c*, the sensitivity with respect to design variable *x* can be calculated as the derivative for the response with respect to the design variable, *dc/dx.*

In OptiStruct, the sensitivity information can be output to Excel from an optimization run with the output commands OUTPUT or SENSITIVITY. It is often useful to understand the sensitivities from the initial iteration in the optimization. You can use these commands to run the optimization for the initial iteration and output only the first sensitivities to the Excel file:

DOPTPRM, DESMAX, 0

OUTPUT, MSSENS, FIRST

This outputs the sensitivities to the _.0.slk file. In the Excel file, a column for each response is shown with the title in blue. In each column, the sensitivities of the responses with respect to the different design variables are shown, with a design variable for each row. The below example shows the Excel SYLK file that has design variables for a Body in White analysis as well as two responses: total mass and the first mode frequency from Subcase 2. In this case the design variables are the gauge thicknesses of the sheet metal parts in the body.

The sensitivities that are calculated are taken as the derivative of the response with respect to the design variable. Thus, they can have a positive or negative value. A positive sensitivity indicates that an increase in the design variable would increase the response value, while a negative sensitivity would indicate that an increase in the design variable would decrease the response value. The highest magnitude sensitivities indicate the design variables that have the highest effect on the response. In the example above, the design variable with the highest influence on both responses is design variable with ID 10, so the mass and first mode frequency of the structure are most sensitive to the gauge of the part linked with that design variable.

The Excel file also gives a way to estimate how changing a design variable affects the response. You can do this by modifying the design variable values in the “New” column and seeing the approximated responses change at the bottom of the response columns. The values in blue show the approximated response value based on the change in the design variables.

A more in-depth document for the Excel .#.slk file and the different response approximations can be found here:

https://2022.help.altair.com/2022/hwsolvers/os/topics/solvers/os/slk_file.htm#slk_file

Based on the sensitivity results, one can improve the design by understanding which design variables have the most influence on the structural responses.

For further reference, review the Gradient-based Optimization section in the OptiStruct User Guide.

**Addendum: Example Displacement Sensitivity Calculation**

Here is an example sensitivity calculation for displacement *u* with respect to a design variable *x* using the direct method.

Linear static equation of motion:

Derivative with respect to design variable *x*:

Rearrangement:

Displacement sensitivity: