Stability of finite element models for distributed-parameter optimization and topology design (Q2564556)

We study the numerical instability that is often encountered in finite element solutions of distributed-parameter optimization and variable-topology shape design problems. We show that the cause of this problem is numerical rather than physical in nature. We consider a two-field, distributed-parameter optimization problem involving a design field and a response field, and show that the optimization problem corresponds to a mixed variational problem. An improper selection of the discrete function spaces for these two fields leads to grid-scale anomalies in the numerical solutions to optimization problems, similar to those that are sometimes encountered in mixed formulations of the Stokes problem. We present a theoretical framework to explain the cause of these anomalies and present stability conditions for discrete models.