STABILITY OF FINITE-ELEMENT MODELS FOR DISTRIBUTED-PARAMETER OPTIMIZATION AND TOPOLOGY DESIGN

We address a problem of numerical instability that is often encountere d in finite element solutions of distributed-parameter optimization an d variable-topology shape design problems. We show that the cause of t his problem is numerical rather than physical in nature. We consider a two-field, distributed-parameter optimization problem involving a des ign 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 caus e of these anomalies and present stability conditions for discrete mod els. The general theoretical framework is specialized to analyze the s tability of specific optimization problems, and stability results for various mixed finite element models are presented. We propose patch te sts that are useful in identifying unstable elements.