Hostname: page-component-8448b6f56d-mp689 Total loading time: 0 Render date: 2024-04-18T07:02:27.567Z Has data issue: false hasContentIssue false
  • English
  • Français

Shape and topology optimization of the robust compliance via the level setmethod

Published online by Cambridge University Press:  21 September 2007

Frédéric de Gournay ,
Grégoire Allaire  and
François Jouve
Frédéric de Gournay
Affiliation:
Centre de Mathématiques Appliquées (UMR 7641), École Polytechnique, 91128 Palaiseau, France; degourna@cmapx.polytechnique.fr; allaire@cmapx.polytechnique.fr; jouve@cmapx.polytechnique.fr
Grégoire Allaire
Affiliation:
Centre de Mathématiques Appliquées (UMR 7641), École Polytechnique, 91128 Palaiseau, France; degourna@cmapx.polytechnique.fr; allaire@cmapx.polytechnique.fr; jouve@cmapx.polytechnique.fr
François Jouve
Affiliation:
Centre de Mathématiques Appliquées (UMR 7641), École Polytechnique, 91128 Palaiseau, France; degourna@cmapx.polytechnique.fr; allaire@cmapx.polytechnique.fr; jouve@cmapx.polytechnique.fr
Get access

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.

Keywords

Robust designworst-case designshape optimizationtopology optimizationlevel set methodsemidefinite programming
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 14 , Issue 1 , January 2008 , pp. 43 - 70
Copyright
© EDP Sciences, SMAI, 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

G. Allaire, Shape optimization by the homogenization method. Springer Verlag, New York (2001).
Allaire, G., Gournay, F.de, Jouve, F. and Toader, A.-M., Structural optimization using topological and shape sensitivity via a level set method. Control Cyb. 34 (2005) 59–80.
Allaire, G. and Jouve, F., A level-set method for vibrations and multiple loads in structural optimization. Comp. Meth. Appl. Mech. Engrg. 194 (2005) 3269–3290. CrossRef
Allaire, G., Jouve, F. and Toader, A.-M., A level set method for shape optimization. C. R. Acad. Sci. Paris 334 (2002) 1125–1130. CrossRef
Allaire, G., Jouve, F. and Toader, A.-M., Structural optimization using sensitivity analysis and a level-set method. J. Comp. Phys. 194 (2004) 363–393. CrossRef
Auchmuty, G., Unconstrained variational principles for eigenvalues of real symmetric matrices. SIAM J. Math. Anal. 20 (1989) 1186–1207. CrossRef
M. Bendsoe, Methods for optimization of structural topology, shape and material. Springer Verlag, New York, 1995.
A. Cherkaev, Variational Methods for Structural Optimization. Springer Verlag, New York, (2000).
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.
Cherkaev, A. and Cherkaeva, E., Principal compliance and robust optimal design. J. Elasticity 72 (2003) 71–98. CrossRef
F.H. Clarke, Optimization and Nonsmooth Analysis. SIAM, classic in Appl. Math. edition (1990).
Eschenauer, H., Kobelev, V. and Schumacher, A., Bubble method for topology and shape optimization of structures. Struct. Optim. 8 (1994) 42–51. CrossRef
Garreau, S., Guillaume, P. and Masmoudi, M., The topological asymptotic for pde systems: the elasticity case. SIAM J. Control Optim. 39 (2001) 1756–1778. CrossRef
F.de Gournay, Optimisation de formes par la méthode des lignes de niveaux. Ph.D. thesis, École Polytechnique, France (2005).
Gournay, F.de, Velocity extension for the level-set method and multiple eigenvalues in shape optimization. SIAM J. Control Optim. 45 (2006) 343–367. CrossRef
Murat, F. and Simon, S., Études de problèmes d'optimal design. Lect. Notes Comput. Sci. 41 (1976) 54–62. CrossRef
Nazarov, S.A. and Sokolovski, Y., The topological derivative of the dirichlet integral under the formation of a thin bridge. Siberian. Math. J. 45 (2004) 341–355. CrossRef
Osher, S. and Santosa, F., Level-set methods for optimization problems involving geometry and constraints: frequencies of a two-density inhomogeneous drum. J. Comput. Phys. 171 (2001) 272–288. CrossRef
Pedregal, P., Vector variational problems and applications to optimal design. ESAIM: COCV 11 (2005) 357–381. CrossRef
O. Pironneau, Optimal shape design for elliptic systems. Springer-Verlag, New York (1984).
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).
Sethian, J.-A. and Wiegmann, A., Structural boundary design via level-set and immersed interface methods. J. Comput. Phys. 163 (2000) 489–528. CrossRef
J. Sokolowski and J-P.Zolesio, Introduction to shape optimization: shape sensitivity analysis, Springer Series in Computational Mathematics 16. Springer-Verlag, Berlin (1992).
Sokolowski, J. and Zochowski, A., On the topological derivative in shape optimization. SIAM J. Control Optim. 37 (1999) 1251–1272. CrossRef
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.
Vandenberghe, L. and Boyd, S., Semidefinite programming. SIAM Rev. 38 (1996) 49–95. CrossRef
M-Y. Wang, X. Wang, D. Guo, A level-set method for structural topology optimization. Comput. Methods Appl. Mech. Engrg. 192 (2003) 227–246. CrossRef