Shape and topology optimization of the robust compliance via the level set method

Frédéric de Gournay; Grégoire Allaire; François Jouve

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 14, Issue: 1, page 43-70
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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de Gournay, Frédéric, Allaire, Grégoire, and Jouve, François. "Shape and topology optimization of the robust compliance via the level set method." ESAIM: Control, Optimisation and Calculus of Variations 14.1 (2010): 43-70. <http://eudml.org/doc/90865>.

@article{deGournay2010,
abstract = { 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. },
author = {de Gournay, Frédéric, Allaire, Grégoire, Jouve, François},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Robust design; worst-case design; shape optimization; topology optimization; level set method; semidefinite programming; robust design; worst-case design; shape optimization; topology optimization; level set method; semidefinite programming},
language = {eng},
month = {3},
number = {1},
pages = {43-70},
publisher = {EDP Sciences},
title = {Shape and topology optimization of the robust compliance via the level set method},
url = {http://eudml.org/doc/90865},
volume = {14},
year = {2010},
}

TY - JOUR
AU - de Gournay, Frédéric
AU - Allaire, Grégoire
AU - Jouve, François
TI - Shape and topology optimization of the robust compliance via the level set method
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 14
IS - 1
SP - 43
EP - 70
AB - 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.
LA - eng
KW - Robust design; worst-case design; shape optimization; topology optimization; level set method; semidefinite programming; robust design; worst-case design; shape optimization; topology optimization; level set method; semidefinite programming
UR - http://eudml.org/doc/90865
ER -

References

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