Shape and topology optimization of the robust compliance via the level set method
ESAIM: Control, Optimisation and Calculus of Variations, Volume 14 (2008) no. 1, pp. 43-70.

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.

DOI: 10.1051/cocv:2007048
Classification: 49Q10,  74P10,  74P15,  74P20
Keywords: robust design, worst-case design, shape optimization, topology optimization, level set method, semidefinite programming
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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, Volume 14 (2008) no. 1, pp. 43-70. doi : 10.1051/cocv:2007048. http://www.numdam.org/articles/10.1051/cocv:2007048/

[1] G. Allaire, Shape optimization by the homogenization method. Springer Verlag, New York (2001). | MR | Zbl

[2] G. Allaire, F. De Gournay, F. Jouve and A.-M. Toader, Structural optimization using topological and shape sensitivity via a level set method. Control Cyb. 34 (2005) 59-80. | MR

[3] G. Allaire and F. Jouve, A level-set method for vibrations and multiple loads in structural optimization. Comp. Meth. Appl. Mech. Engrg. 194 (2005) 3269-3290. | MR | Zbl

[4] G. Allaire, F. Jouve and A.-M. Toader, A level set method for shape optimization. C. R. Acad. Sci. Paris 334 (2002) 1125-1130. | MR | Zbl

[5] G. Allaire, F. Jouve and A.-M. Toader, Structural optimization using sensitivity analysis and a level-set method. J. Comp. Phys. 194 (2004) 363-393. | MR | Zbl

[6] G. Auchmuty, Unconstrained variational principles for eigenvalues of real symmetric matrices. SIAM J. Math. Anal. 20 (1989) 1186-1207. | MR | Zbl

[7] M. Bendsoe, Methods for optimization of structural topology, shape and material. Springer Verlag, New York, 1995. | Zbl

[8] A. Cherkaev, Variational Methods for Structural Optimization. Springer Verlag, New York, (2000). | MR | Zbl

[9] A. Cherkaev and E. Cherkaeva, 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

[10] A. Cherkaev and E. Cherkaeva, Principal compliance and robust optimal design. J. Elasticity 72 (2003) 71-98. | MR | Zbl

[11] F.H. Clarke, Optimization and Nonsmooth Analysis. SIAM, classic in Appl. Math. edition (1990). | MR | Zbl

[12] H. Eschenauer, V. Kobelev and A. Schumacher, Bubble method for topology and shape optimization of structures. Struct. Optim. 8 (1994) 42-51.

[13] S. Garreau, P. Guillaume and M. Masmoudi, The topological asymptotic for pde systems: the elasticity case. SIAM J. Control Optim. 39 (2001) 1756-1778. | MR | Zbl

[14] F. De Gournay, Optimisation de formes par la méthode des lignes de niveaux. Ph.D. thesis, École Polytechnique, France (2005).

[15] F. De Gournay, Velocity extension for the level-set method and multiple eigenvalues in shape optimization. SIAM J. Control Optim. 45 (2006) 343-367. | MR | Zbl

[16] F. Murat and S. Simon, Études de problèmes d'optimal design. Lect. Notes Comput. Sci. 41 (1976) 54-62. | Zbl

[17] S.A. Nazarov and Y. Sokolovski, The topological derivative of the dirichlet integral under the formation of a thin bridge. Siberian. Math. J. 45 (2004) 341-355. | MR | Zbl

[18] S. Osher and F. Santosa, 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

[19] P. Pedregal, Vector variational problems and applications to optimal design. ESAIM: COCV 11 (2005) 357-381. | Numdam | MR | Zbl

[20] O. Pironneau, Optimal shape design for elliptic systems. Springer-Verlag, New York (1984). | MR | Zbl

[21] J.-A. Sethian, Level-Set Methods and fast marching methods: evolving interfaces in computational geometry, fluid mechanics, computer vision and materials science. Cambridge University Press (1999). | MR | Zbl

[22] J.-A. Sethian and A. Wiegmann, Structural boundary design via level-set and immersed interface methods. J. Comput. Phys. 163 (2000) 489-528. | MR | Zbl

[23] J. Sokolowski and J-P. Zolesio, Introduction to shape optimization: shape sensitivity analysis, Springer Series in Computational Mathematics 16. Springer-Verlag, Berlin (1992). | MR | Zbl

[24] J. Sokolowski and A. Zochowski, On the topological derivative in shape optimization. SIAM J. Control Optim. 37 (1999) 1251-1272. | MR | Zbl

[25] L. Tartar, 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] L. Vandenberghe and S. Boyd, Semidefinite programming. SIAM Rev. 38 (1996) 49-95. | MR | Zbl

[27] M-Y. Wang, X. Wang and D. Guo, A level-set method for structural topology optimization. Comput. Methods Appl. Mech. Engrg. 192 (2003) 227-246. | MR | Zbl

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