The topological sensitivity analysis consists in studying the behavior of a given shape functional when the topology of the domain is perturbed, typically by the nucleation of a small hole. This notion forms the basic ingredient of different topology optimization/reconstruction algorithms. From the theoretical viewpoint, the expression of the topological sensitivity is well-established in many situations where the governing p.d.e. system is of elliptic type. This paper focuses on the derivation of such formulas for parabolic and hyperbolic problems. Different kinds of cost functionals are considered.
Mots-clés : topological sensitivity, topology optimization, parabolic equations, hyperbolic equations
@article{COCV_2008__14_3_427_0,
author = {Vexler, Boris and Takahashi, Tak\'eo and Amstutz, Samuel},
title = {Topological sensitivity analysis for time-dependent problems},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {427--455},
publisher = {EDP-Sciences},
volume = {14},
number = {3},
year = {2008},
doi = {10.1051/cocv:2007059},
mrnumber = {2434060},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/cocv:2007059/}
}
TY - JOUR AU - Vexler, Boris AU - Takahashi, Takéo AU - Amstutz, Samuel TI - Topological sensitivity analysis for time-dependent problems JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2008 SP - 427 EP - 455 VL - 14 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv:2007059/ DO - 10.1051/cocv:2007059 LA - en ID - COCV_2008__14_3_427_0 ER -
%0 Journal Article %A Vexler, Boris %A Takahashi, Takéo %A Amstutz, Samuel %T Topological sensitivity analysis for time-dependent problems %J ESAIM: Control, Optimisation and Calculus of Variations %D 2008 %P 427-455 %V 14 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv:2007059/ %R 10.1051/cocv:2007059 %G en %F COCV_2008__14_3_427_0
Vexler, Boris; Takahashi, Takéo; Amstutz, Samuel. Topological sensitivity analysis for time-dependent problems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 3, pp. 427-455. doi : 10.1051/cocv:2007059. http://archive.numdam.org/articles/10.1051/cocv:2007059/
[1] , , and , Structural optimization using topological and shape sensitivity via a level set method. Control Cybern. 34 (2005) 59-80. | MR
[2] and , Reconstruction of small inhomogeneities from boundary measurements, Lecture Notes in Mathematics 1846. Springer-Verlag, Berlin (2004). | MR | Zbl
[3] and , Reconstruction of elastic inclusions of small volume via dynamic measurements. Appl. Math. Optim. 54 (2006) 223-235. | MR | Zbl
[4] and , Generalized polarization tensors, inverse conductivity problems, and dilute composite materials: a review, in Inverse problems, multi-scale analysis and effective medium theory, Contemp. Math. 408, Amer. Math. Soc., Providence, RI (2006) 1-67. | MR | Zbl
[5] , Sensitivity analysis with respect to a local perturbation of the material property. Asymptotic Anal. 49 (2006) 87-108. | MR
[6] and , A new algorithm for topology optimization using a level-set method. J. Comput. Phys. 216 (2006) 573-588. | MR | Zbl
[7] and , Topological sensitivity analysis in the context of ultrasonic nondestructive testing. RICAM report 2005-21 (2005).
[8] , and , Crack detection by the topological gradient method. Control Cybern. 34 (2005) 81-101. | MR
[9] , Topological sensitivity for 3d elastodynamic and acoustic inverse scattering in the time domain. Comput. Meth. Appl. Mech. Engrg. 195 (2006) 5239-5254. | MR | Zbl
[10] and , Sounding of finite solid bodies by way of topological derivative. Int. J. Numer. Methods Eng. 61 (2004) 2344-2373. | MR | Zbl
[11] , and , Incorporating topological derivatives into level set methods. J. Comput. Phys. 194 (2004) 344-362. | MR | Zbl
[12] and , Introduction aux problèmes d'évolution semi-linéaires, Mathématiques & Applications 1. Ellipses, Paris (1990). | MR | Zbl
[13] and , Analyse mathématique et calcul numérique pour les sciences et les techniques, Vol. 6. INSTN: Collection Enseignement, Masson, Paris (1988). | MR
[14] and , Topology and shape optimization procedures using hole positioning criteria - theory and applications, in Topology optimization in structural mechanics, CISM Courses and Lectures 374, Springer, Vienna (1997) 135-196. | MR | Zbl
[15] , and , Bubble method for topology and shape optimization of structures. Struct. Optimization 8 (1994) 42-51.
[16] , and , The topological asymptotic for PDE systems: the elasticity case. SIAM J. Control Optim. 39 (2001) 1756-1778 (electronic). | MR | Zbl
[17] and , The topological asymptotic expansion for the Dirichlet problem. SIAM J. Control Optim. 41 (2002) 1042-1072 (electronic). | MR | Zbl
[18] and , Topological derivative for the inverse scattering of elastic waves. Quart. J. Mech. Appl. Math. 57 (2004) 161-179. | MR | Zbl
[19] and , Problèmes aux limites non homogènes et applications. Vol. 2. Travaux et Recherches Mathématiques 18. Dunod, Paris (1968). | MR | Zbl
[20] and , The topological derivative of the Dirichlet integral under the formation of a thin bridge. Siberian Math. J. 45 (2004) 341-355. | MR | Zbl
[21] and , Isoperimetric Inequalities in Mathematical Physics. Annals of Mathematics Studies 27. Princeton University Press, Princeton, N. J. (1951). | MR | Zbl
[22] and , The topological asymptotic for the Helmholtz equation with Dirichlet condition on the boundary of an arbitrarily shaped hole. SIAM J. Control Optim. 43 (2004) 899-921 (electronic). | MR | Zbl
[23] and , Virtual mass and polarization. Trans. Amer. Math. Soc. 67 (1949) 130-205. | MR | Zbl
[24] , Topologieoptimierung von bauteilstrukturen unter verwendung von lopschpositionierungskriterien. Ph.D. thesis, Univ. Siegen (1995).
[25] and , On the topological derivative in shape optimization. SIAM J. Control Optim. 37 (1999) 1251-1272 (electronic). | MR | Zbl
Cité par Sources :
