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.
Accepté le :
DOI : 10.1051/m2an/2016012
Mots clés : Topology optimization, convex analysis, convexification, semi-smooth Newton method
@article{M2AN_2016__50_6_1917_0, author = {Clason, Christian and Kunisch, Karl}, title = {A convex analysis approach to multi-material topology optimization}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {1917--1936}, publisher = {EDP-Sciences}, volume = {50}, number = {6}, year = {2016}, doi = {10.1051/m2an/2016012}, zbl = {1354.49092}, mrnumber = {3580127}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2016012/} }
TY - JOUR AU - Clason, Christian AU - Kunisch, Karl TI - A convex analysis approach to multi-material topology optimization JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2016 SP - 1917 EP - 1936 VL - 50 IS - 6 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2016012/ DO - 10.1051/m2an/2016012 LA - en ID - M2AN_2016__50_6_1917_0 ER -
%0 Journal Article %A Clason, Christian %A Kunisch, Karl %T A convex analysis approach to multi-material topology optimization %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2016 %P 1917-1936 %V 50 %N 6 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2016012/ %R 10.1051/m2an/2016012 %G en %F M2AN_2016__50_6_1917_0
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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