A convex analysis approach to multi-material topology optimization

¹ Faculty of Mathematics, University Duisburg-Essen, 45117 Essen, Germany.
christian.clason@uni-due.de
² Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, 8010 Graz, Austria.
³ Radon Institute, Austrian Academy of Sciences, Linz, Austria.
karl.kunisch@uni-graz.at

Received: 5 June 2015
Revised: 14 January 2016
Accepted: 2 February 2016

Abstract

This work is concerned with optimal control of partial differential equations where the control enters the state equation as a coefficient and should take on values only from a given discrete set of values corresponding to available materials. A “multi-bang” framework based on convex analysis is proposed where the desired piecewise constant structure is incorporated using a convex penalty term. Together with a suitable tracking term, this allows formulating the problem of optimizing the topology of the distribution of material parameters as minimizing a convex functional subject to a (nonlinear) equality constraint. The applicability of this approach is validated for two model problems where the control enters as a potential and a diffusion coefficient, respectively. This is illustrated in both cases by numerical results based on a semi-smooth Newton method.