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

We address a problem of 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. The general theoretical framework is specialized to analyze the stability of specific optimization problems, and stability results for various mixed finite element models are presented. We propose patch tests that are useful in identifying unstable elements.