This work is concerned with the determination of the diffusion coefficient from distributed data of the state. This problem is related to homogenization theory on the one hand and to regularization theory on the other hand. An approach is proposed which involves total variation regularization combined with a suitably chosen cost functional that promotes the diffusion coefficient assuming prespecified values at each point of the domain. The main difficulty lies in the delicate functional-analytic structure of the resulting nondifferentiable optimization problem with pointwise constraints for functions of bounded variation, which makes the derivation of useful pointwise optimality conditions challenging. To cope with this difficulty, a novel reparametrization technique is introduced. Numerical examples using a regularized semismooth Newton method illustrate the structure of the obtained diffusion coefficient.
Accepté le :
DOI : 10.1051/m2an/2017061
Mots clés : Topology optimization, total variation, convex analysis, non-smooth optimization, semi-smooth Newton method
@article{M2AN_2018__52_1_275_0,
author = {Clason, Christian and Kruse, Florian and Kunisch, Karl},
title = {Total variation regularization of multi-material topology optimization},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {275--303},
publisher = {EDP-Sciences},
volume = {52},
number = {1},
year = {2018},
doi = {10.1051/m2an/2017061},
zbl = {1397.49056},
mrnumber = {3808161},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2017061/}
}
TY - JOUR AU - Clason, Christian AU - Kruse, Florian AU - Kunisch, Karl TI - Total variation regularization of multi-material topology optimization JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 275 EP - 303 VL - 52 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2017061/ DO - 10.1051/m2an/2017061 LA - en ID - M2AN_2018__52_1_275_0 ER -
%0 Journal Article %A Clason, Christian %A Kruse, Florian %A Kunisch, Karl %T Total variation regularization of multi-material topology optimization %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 275-303 %V 52 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2017061/ %R 10.1051/m2an/2017061 %G en %F M2AN_2018__52_1_275_0
Clason, Christian; Kruse, Florian; Kunisch, Karl. Total variation regularization of multi-material topology optimization. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 1, pp. 275-303. doi : 10.1051/m2an/2017061. http://www.numdam.org/articles/10.1051/m2an/2017061/
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