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We consider the optimal distribution of several elastic materials in a fixed working domain. In order to optimize both the geometry and topology of the mixture we rely on the level set method for the description of the interfaces between the different phases. We discuss various approaches, based on Hadamard method of boundary variations, for computing shape derivatives which are the key ingredients for a steepest descent algorithm. The shape gradient obtained for a sharp interface involves jump of discontinuous quantities at the interface which are difficult to numerically evaluate. Therefore we suggest an alternative smoothed interface approach which yields more convenient shape derivatives. We rely on the signed distance function and we enforce a fixed width of the transition layer around the interface (a crucial property in order to avoid increasing “grey” regions of fictitious materials). It turns out that the optimization of a diffuse interface has its own interest in material science, for example to optimize functionally graded materials. Several 2-d examples of compliance minimization are numerically tested which allow us to compare the shape derivatives obtained in the sharp or smoothed interface cases.
@article{COCV_2014__20_2_576_0,
author = {Allaire, G. and Dapogny, C. and Delgado, G. and Michailidis, G.},
title = {Multi-phase structural optimization \protect\emph{via }a level set method},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {576--611},
publisher = {EDP-Sciences},
volume = {20},
number = {2},
year = {2014},
doi = {10.1051/cocv/2013076},
zbl = {1287.49045},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2013076/}
}
TY - JOUR AU - Allaire, G. AU - Dapogny, C. AU - Delgado, G. AU - Michailidis, G. TI - Multi-phase structural optimization via a level set method JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2014 SP - 576 EP - 611 VL - 20 IS - 2 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/cocv/2013076/ DO - 10.1051/cocv/2013076 LA - en ID - COCV_2014__20_2_576_0 ER -
%0 Journal Article %A Allaire, G. %A Dapogny, C. %A Delgado, G. %A Michailidis, G. %T Multi-phase structural optimization via a level set method %J ESAIM: Control, Optimisation and Calculus of Variations %D 2014 %P 576-611 %V 20 %N 2 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/cocv/2013076/ %R 10.1051/cocv/2013076 %G en %F COCV_2014__20_2_576_0
Allaire, G.; Dapogny, C.; Delgado, G.; Michailidis, G. Multi-phase structural optimization via a level set method. ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 2, pp. 576-611. doi: 10.1051/cocv/2013076
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