A convex analysis approach to multi-material topology optimization
Clason, Christian1 ; Kunisch, Karl2, 3
1 Faculty of Mathematics, University Duisburg-Essen, 45117 Essen, Germany.
2 Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, 8010 Graz, Austria.
3 Radon Institute, Austrian Academy of Sciences, Linz, Austria.
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 6, pp. 1917-1936.

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.

Reçu le :
Accepté le :
DOI : 10.1051/m2an/2016012
Classification : 49Q10, 49K20, 49M15
Mots-clés : Topology optimization, convex analysis, convexification, semi-smooth Newton method
Clason, Christian 1 ; Kunisch, Karl 2, 3

1 Faculty of Mathematics, University Duisburg-Essen, 45117 Essen, Germany.
2 Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, 8010 Graz, Austria.
3 Radon Institute, Austrian Academy of Sciences, Linz, Austria. 
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Clason, Christian; Kunisch, Karl. A convex analysis approach to multi-material topology optimization. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 6, pp. 1917-1936. doi : 10.1051/m2an/2016012. http://www.numdam.org/articles/10.1051/m2an/2016012/

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