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.
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 SP - 43 EP - 70 VL - 14 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2007048/ DO - 10.1051/cocv:2007048 LA - en ID - COCV_2008__14_1_43_0 ER -
%0 Journal Article %A Jouve, François %A Allaire, Grégoire %A Gournay, Frédéric de %T Shape and topology optimization of the robust compliance via the level set method %J ESAIM: Control, Optimisation and Calculus of Variations %D 2008 %P 43-70 %V 14 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2007048/ %R 10.1051/cocv:2007048 %G en %F COCV_2008__14_1_43_0
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/
[1] Shape optimization by the homogenization method. Springer Verlag, New York (2001). | MR | Zbl
,[2] Structural optimization using topological and shape sensitivity via a level set method. Control Cyb. 34 (2005) 59-80. | MR
, , and ,[3] A level-set method for vibrations and multiple loads in structural optimization. Comp. Meth. Appl. Mech. Engrg. 194 (2005) 3269-3290. | MR | Zbl
and ,[4] A level set method for shape optimization. C. R. Acad. Sci. Paris 334 (2002) 1125-1130. | MR | Zbl
, and ,[5] Structural optimization using sensitivity analysis and a level-set method. J. Comp. Phys. 194 (2004) 363-393. | MR | Zbl
, and ,[6] Unconstrained variational principles for eigenvalues of real symmetric matrices. SIAM J. Math. Anal. 20 (1989) 1186-1207. | MR | Zbl
,[7] Methods for optimization of structural topology, shape and material. Springer Verlag, New York, 1995. | Zbl
,[8] Variational Methods for Structural Optimization. Springer Verlag, New York, (2000). | MR | Zbl
,[9] Optimal design for uncertain loading condition, in Homogenization, Series on Advances in Mathematics for Applied Sciences 50, V. Berdichevsky et al. Eds., World Scientific, Singapore (1999) 193-213. | MR | Zbl
and ,[10] Principal compliance and robust optimal design. J. Elasticity 72 (2003) 71-98. | MR | Zbl
and ,[11] Optimization and Nonsmooth Analysis. SIAM, classic in Appl. Math. edition (1990). | MR | Zbl
,[12] Bubble method for topology and shape optimization of structures. Struct. Optim. 8 (1994) 42-51.
, and ,[13] The topological asymptotic for pde systems: the elasticity case. SIAM J. Control Optim. 39 (2001) 1756-1778. | MR | Zbl
, and ,[14] Optimisation de formes par la méthode des lignes de niveaux. Ph.D. thesis, École Polytechnique, France (2005).
,[15] Velocity extension for the level-set method and multiple eigenvalues in shape optimization. SIAM J. Control Optim. 45 (2006) 343-367. | MR | Zbl
,[16] Études de problèmes d'optimal design. Lect. Notes Comput. Sci. 41 (1976) 54-62. | Zbl
and ,[17] The topological derivative of the dirichlet integral under the formation of a thin bridge. Siberian. Math. J. 45 (2004) 341-355. | MR | Zbl
and ,[18] Level-set methods for optimization problems involving geometry and constraints: frequencies of a two-density inhomogeneous drum. J. Comput. Phys. 171 (2001) 272-288. | MR | Zbl
and ,[19] Vector variational problems and applications to optimal design. ESAIM: COCV 11 (2005) 357-381. | Numdam | MR | Zbl
,[20] Optimal shape design for elliptic systems. Springer-Verlag, New York (1984). | MR | Zbl
, , and fast marching methods: evolving interfaces in computational geometry, fluid mechanics, computer vision and materials science. Cambridge University Press (1999). |[22] Structural boundary design via level-set and immersed interface methods. J. Comput. Phys. 163 (2000) 489-528. | MR | Zbl
and ,[23] Introduction to shape optimization: shape sensitivity analysis, Springer Series in Computational Mathematics 16. Springer-Verlag, Berlin (1992). | MR | Zbl
and ,[24] On the topological derivative in shape optimization. SIAM J. Control Optim. 37 (1999) 1251-1272. | MR | Zbl
and ,[25] An introduction to the homogenization method in optimal design, in Optimal shape design, A. Cellina and A. Ornelas Eds., Lecture Notes in Mathematics 1740, Springer, Berlin (1998) 47-156. | MR | Zbl
,[26] Semidefinite programming. SIAM Rev. 38 (1996) 49-95. | MR | Zbl
and ,[27] A level-set method for structural topology optimization. Comput. Methods Appl. Mech. Engrg. 192 (2003) 227-246. | MR | Zbl
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