Shape Optimization via Control of a Shape Function on a Fixed Domain: Theory and Numerical Results

Abstract

We present a fixed-domain approach for the solution of shape optimization problems governed by linear or nonlinear elliptic partial differential state equations with Dirichlet boundary conditions, where shape optimization is facilitated via optimal control of a shape function. The method involves extending the state equation to a larger domain using regularization. Results regarding the convergence to the original problem are provided as well as differentiability properties of the control-to-state mappings. An algorithm for the numerical implementation of the method is stated and, in a series of numerical shape optimization experiments, the algorithm’s behavior is studied with regard to varying the regularization parameter and initial conditions.KeywordsShape Optimization ProblemFixed Domain MethodMaximal Monotone ExtensionCost Functionals17 Required Line SearchesThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.