Numerical analysis of a topology optimization problem for Stokes flow

TL;DR

This paper advances the mathematical understanding of topology optimization for Stokes flow by proving regularity results, extending numerical analysis, and demonstrating convergence of finite element solutions to local minimizers.

Contribution

It provides new regularity results, extends convergence analysis of finite element methods, and investigates convergence rates for topology optimization in Stokes flow.

Findings

Existence of strongly convergent finite element solutions to local minimizers.

First numerical investigation into convergence rates.

Extension of numerical analysis for topology optimization in fluid flow.

Abstract

T. Borrvall and J. Petersson [Topology optimization of fluids in Stokes flow, International Journal for Numerical Methods in Fluids 41 (1) (2003) 77--107] developed the first model for topology optimization of fluids in Stokes flow. They proved the existence of minimizers in the infinite-dimensional setting and showed that a suitably chosen finite element method will converge in a weak(-*) sense to an unspecified solution. In this work, we prove novel regularity results and extend their numerical analysis. In particular, given an isolated local minimizer to the infinite-dimensional problem, we show that there exists a sequence of finite element solutions, satisfying necessary first-order optimality conditions, that strongly converges to it. We also provide the first numerical investigation into convergence rates.