Relating phase field and sharp interface approaches to structural topology optimization

¹ Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany
luise.blank@mathematik.uni-regensburg.de; harald.garcke@mathematik.uni-regensburg.de
² Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117 Berlin, Germany
Hassan.Farshbaf-Shaker@wias-berlin.de
³ Department of Mathematics, University of Sussex, Brighton, BN1 9QH, UK
v.styles@sussex.ac.uk

Received: 8 March 2013
Revised: 10 January 2014

Abstract

A phase field approach for structural topology optimization which allows for topology changes and multiple materials is analyzed. First order optimality conditions are rigorously derived and it is shown via formally matched asymptotic expansions that these conditions converge to classical first order conditions obtained in the context of shape calculus. We also discuss how to deal with triple junctions where e.g. two materials and the void meet. Finally, we present several numerical results for mean compliance problems and a cost involving the least square error to a target displacement.