The goal of this paper is to study the so-called worst-case or robust optimal design problem for minimal compliance. In the context of linear elasticity we seek an optimal shape which minimizes the largest, or worst, compliance when the loads are subject to some unknown perturbations. We first prove that, for a fixed shape, there exists indeed a worst perturbation (possibly non unique) that we characterize as the maximizer of a nonlinear energy. We also propose a stable algorithm to compute it. Then, in the framework of Hadamard method, we compute the directional shape derivative of this criterion which is used in a numerical algorithm, based on the level set method, to find optimal shapes that minimize the worst-case compliance. Since this criterion is usually merely directionally differentiable, we introduce a semidefinite programming approach to select the best descent direction at each step of a gradient method. Numerical examples are given in 2-d and 3-d.
Classification : 49Q10, 74P10, 74P15, 74P20
Mots clés : robust design, worst-case design, shape optimization, topology optimization, level set method, semidefinite programming
@article{COCV_2008__14_1_43_0,
author = {Jouve, Fran\c{c}ois and Allaire, Gr\'egoire and Gournay, Fr\'ed\'eric de},
title = {Shape and topology optimization of the robust compliance via the level set method},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {43--70},
publisher = {EDP-Sciences},
volume = {14},
number = {1},
year = {2008},
doi = {10.1051/cocv:2007048},
mrnumber = {2375751},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv:2007048/}
}
TY - JOUR AU - Jouve, François AU - Allaire, Grégoire AU - Gournay, Frédéric de TI - Shape and topology optimization of the robust compliance via the level set method JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2008 DA - 2008/// SP - 43 EP - 70 VL - 14 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2007048/ UR - https://www.ams.org/mathscinet-getitem?mr=2375751 UR - https://doi.org/10.1051/cocv:2007048 DO - 10.1051/cocv:2007048 LA - en ID - COCV_2008__14_1_43_0 ER -
Jouve, François; Allaire, Grégoire; Gournay, Frédéric de. Shape and topology optimization of the robust compliance via the level set method. ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 1, pp. 43-70. doi : 10.1051/cocv:2007048. http://www.numdam.org/articles/10.1051/cocv:2007048/
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