The goal of this paper is twofold. On one hand, our work revisits the minimization of the robust compliance in shape optimization, with a more natural and more general approach than what has been done before. On the other hand, following a more recent viewpoint on robust optimization, we study the maximization of the so-called stability radius for a fixed maximal compliance. We provide theorical as well as numerical results.
DOI : 10.1051/cocv/2014066
Mots clés : Robustness, stability radius, compliance, topological derivative, topology optimization
@article{COCV_2016__22_1_64_0,
author = {Amstutz, Samuel and Ciligot-Travain, Marc},
title = {A notion of compliance robustness in topology optimization},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {64--87},
publisher = {EDP-Sciences},
volume = {22},
number = {1},
year = {2016},
doi = {10.1051/cocv/2014066},
zbl = {1335.49069},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/cocv/2014066/}
}
TY - JOUR AU - Amstutz, Samuel AU - Ciligot-Travain, Marc TI - A notion of compliance robustness in topology optimization JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2016 SP - 64 EP - 87 VL - 22 IS - 1 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2014066/ DO - 10.1051/cocv/2014066 LA - en ID - COCV_2016__22_1_64_0 ER -
%0 Journal Article %A Amstutz, Samuel %A Ciligot-Travain, Marc %T A notion of compliance robustness in topology optimization %J ESAIM: Control, Optimisation and Calculus of Variations %D 2016 %P 64-87 %V 22 %N 1 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2014066/ %R 10.1051/cocv/2014066 %G en %F COCV_2016__22_1_64_0
Amstutz, Samuel; Ciligot-Travain, Marc. A notion of compliance robustness in topology optimization. ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 1, pp. 64-87. doi : 10.1051/cocv/2014066. http://archive.numdam.org/articles/10.1051/cocv/2014066/
G. Allaire, Conception optimale de structures, Vol. 58 of Mathématiques & Applications (Berlin)[Mathematics & Applications]. With the collaboration of Marc Schoenauer (INRIA) in the writing of Chapter 8. Springer-Verlag, Berlin (2007). | Zbl
and , A linearized approach to worst-case design in parametric and geometric shape optimization. Math. Models Methods Appl. Sci. 24 (2014) 2199–2257. | DOI | Zbl
, and , Structural optimization using sensitivity analysis and a level-set method. J. Comput. Phys. 194 (2004) 363–393. | DOI | Zbl
, Sensitivity analysis with respect to a local perturbation of the material property. Asymptot. Anal. 49 (2006) 87–108. | Zbl
, Analysis of a level set method for topology optimization. Optim. Methods Softw. 26 (2011) 555–573. | DOI | Zbl
and , A new algorithm for topology optimization using a level-set method. J. Comput. Phys. 216 (2006) 573–588. | DOI | Zbl
and , Optimality conditions for shape and topology optimization subject to a cone constraint. SIAM J. Control Optim. 48 (2010) 4056–4077. | DOI | Zbl
, and , Worst case scenario analysis for elliptic problems with uncertainty. Numer. Math. 101 (2005) 185–219. | DOI | Zbl
Y. Ben-Haim, Information-gap Decision Theory. Series on Decision and Risk. Academic Press Inc., San Diego, CA (2001). Decisions under severe uncertainty. | Zbl
A. Ben-Tal, L. El Ghaoui and A. Nemirovski, Robust optimization. Princeton Series Appl. Math. Princeton University Press, Princeton, NJ (2009). | Zbl
M.P. Bendsøe and O. Sigmund, Topology optimization. Theory, Methods and Appl. Springer-Verlag, Berlin (2003). | Zbl
J.F. Bonnans and A. Shapiro, Perturbation analysis of optimization problems. Springer Ser. Oper. Res. Springer-Verlag, New York (2000). | Zbl
A. Cherkaev and E. Cherkaev, Optimal design for uncertain loading condition. In Homogenization. Vol. 50 of Ser. Adv. Math. Appl. Sci.. World Sci. Publ., River Edge, NJ (1999) 193–213. | Zbl
and , Principal compliance and robust optimal design. Essays and papers dedicated to the memory of Clifford Ambrose Truesdell III. Vol. III. J. Elasticity 72 (2003) 71–98 | Zbl
, and , Shape and topology optimization of the robust compliance via the level set method. ESAIM Control Optim. Calc. Var. 14 (2008) 43–70. | DOI | Zbl
, , and , Duality and solutions for quadratic programming over single non-homogeneous quadratic constraint. J. Global Optim. 54 (2012) 275–293. | DOI | Zbl
and , Duality and sensitivity in nonconvex quadratic optimization over an ellipsoid. Eur. J. Oper. Res. 94 (1996) 167–178. | DOI | Zbl
and , The trust region subproblem and semidefinite programming. Optim. Methods Softw. 19 (2004) 41–67. | DOI | Zbl
and , The -procedure and the duality relation in convex quadratic programming problems. Vestnik Leningrad. Univ. (1 Mat. Meh. Astronom. Vyp. 1) 155 (1973) 81–87. | Zbl
and , The -procedure and the duality relation in convex quadratic programming problems. Vestnik Leningrad. Univ. 6 (1979) 101–109. | Zbl
, and , The topological asymptotic for PDE systems: the elasticity case. SIAM J. Control Optim. 39 (2001) 1756–1778. | DOI | Zbl
, Computing optimal locally constrained steps. SIAM J. Sci. Statis. Comput. 2 (1981) 186–197. | DOI | Zbl
, and , Robust structural topology optimization considering boundary uncertainties. Comput. Methods Appl. Mech. Engrg. 253 (2013) 356–368. | DOI | Zbl
D. Hinrichsen and A.J. Pritchard, Real and complex stability radii: a survey. In Control of uncertain systems (Bremen, 1989). Vol. 6 of Progr. Systems Control Theory. Birkhäuser Boston, Boston, MA (1990) 119–162. | Zbl
, and , Riccati equation approach to maximizing the complex stability radius by state feedback. Int. J. Control 52 (1990) 769–794. | DOI | Zbl
J.-B. Hiriart-Urruty and C. Lemaréchal, Convex analysis and minimization algorithms. II. Advanced theory and bundle methods. Vol. 306 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin (1993). | Zbl
A.D. Ioffe and V.M. Tihomirov, Theory of extremal problems. Vol. 6 of Stud. Math. Appl. Translated from the Russian by Karol Makowski. North-Holland Publishing Co., Amsterdam-New York (1979). | Zbl
, The omnipresence of Lagrange. Ann. Oper. Res. 153 (2007) 9–27. | DOI | Zbl
, Generalizations of the trust region problem. Optimiz. Methods Softw. 2 (1993) 189–209. | DOI
A.A. Novotny and J. Sokołowski, Topological derivatives in shape optimization. Interaction of Mechanics and Mathematics. Springer, Heidelberg (2013). | Zbl
and , Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79 (1988) 12–49. | DOI | Zbl
and , A semidefinite framework for trust region subproblems with applications to large scale minimization. Semidefinite programming. Math. Program. Ser. B 77 (1997) 273–299. | DOI | Zbl
R.T. Rockafellar, Conjugate duality and optimization. Lectures given at the Johns Hopkins University, Baltimore, Md., June (1973), Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 16. Society for Industrial and Applied Mathematics, Philadelphia, Pa. (1974). | Zbl
and , On the topological derivative in shape optimization. SIAM J. Control Optim. 37 (1999) 1251–1272. | DOI | Zbl
, Newton’s method with a model trust region modification. SIAM J. Numer. Anal. 19 (1982) 409–426. | DOI | Zbl
and , Trust region problems and nonsymmetric eigenvalue perturbations. SIAM J. Matrix Anal. Appl. 15 (1994) 755–778. | DOI | Zbl
and , Indefinite trust region subproblems and nonsymmetric eigenvalue perturbations. SIAM J. Optim. 5 (1995) 286–313. | DOI | Zbl
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