Semi-Analytical Sensitivity Calculation

In context of sensitivity-based topology optimization to obtain the sensitivity of design responses for some special situations semi-analytical approach is used.

Semi-Analytical Sensitivity Calculation is based on a finite difference of the elemental stiffness and mass matrices. That means that in addition to the original matrices perturbed matrices must be requested from the solver. This is done by a "matrix-step" as shown in the following figure:
Important:

This matrix step's only purpose is to perturb all pseudo densities to get the perturbed matrices - the original matrix is obtained in a usual solver step where also the results of the FE-problem are wanted. For situations where matrix step is needed the solver is called twice in each iteration. It would demand a lengthy discussion to explain in detail exactly when and why a matrix step is needed so it is avoided here.