Numerical Analysis
The topological asymptotic with respect to a singular boundary perturbation
Comptes Rendus. Mathématique, Volume 336 (2003) no. 12, pp. 1033-1038.

The aim of the topological sensitivity analysis is to obtain an asymptotic expansion of a design functional with respect to the insertion of a small hole in the domain. The question that we address here is what happens if the hole is located at the boundary of the domain and what happens if the boundary is not regular. The adjoint method and the domain truncation technique are proposed to solve this problem. As a model example, we consider the Laplace equation in a domain with a corner.

Le but de la sensibilité topologique est d'obtenir une expression asymptotique d'une fonctionnelle de forme par rapport à l'insertion d'un petit trou dans le domaine. Dans cette Note, nous considérons le cas d'un petit trou situé sur un coin du domaine. La méthode de l'état adjoint et la technique de troncature de domaine sont proposées pour résoudre ce probléme. Nous considérons comme exemple modèle, l'équation de Laplace posée dans un domaine avec un coin.

Received:
Accepted:
Published online:
DOI: 10.1016/S1631-073X(03)00258-9
Samet, Bessem 1

1 MIP, UMR 5640, CNRS-Université Paul Sabatier-INSA, 118, route de Narbonne, 31062 Toulouse cedex, France
@article{CRMATH_2003__336_12_1033_0,
     author = {Samet, Bessem},
     title = {The topological asymptotic with respect to a singular boundary perturbation},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1033--1038},
     publisher = {Elsevier},
     volume = {336},
     number = {12},
     year = {2003},
     doi = {10.1016/S1631-073X(03)00258-9},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/S1631-073X(03)00258-9/}
}
TY  - JOUR
AU  - Samet, Bessem
TI  - The topological asymptotic with respect to a singular boundary perturbation
JO  - Comptes Rendus. Mathématique
PY  - 2003
SP  - 1033
EP  - 1038
VL  - 336
IS  - 12
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/S1631-073X(03)00258-9/
DO  - 10.1016/S1631-073X(03)00258-9
LA  - en
ID  - CRMATH_2003__336_12_1033_0
ER  - 
%0 Journal Article
%A Samet, Bessem
%T The topological asymptotic with respect to a singular boundary perturbation
%J Comptes Rendus. Mathématique
%D 2003
%P 1033-1038
%V 336
%N 12
%I Elsevier
%U http://www.numdam.org/articles/10.1016/S1631-073X(03)00258-9/
%R 10.1016/S1631-073X(03)00258-9
%G en
%F CRMATH_2003__336_12_1033_0
Samet, Bessem. The topological asymptotic with respect to a singular boundary perturbation. Comptes Rendus. Mathématique, Volume 336 (2003) no. 12, pp. 1033-1038. doi : 10.1016/S1631-073X(03)00258-9. http://www.numdam.org/articles/10.1016/S1631-073X(03)00258-9/

[1] Eschenauer, H.; Schumacher, A. Bubble method for topology and shape optimization of structures, Structural Optim., Volume 8 (1994), pp. 42-51

[2] Garreau, S.; Guillaume, P.; Masmoudi, M. The topological sensitivity for linear isotropic elasticity, European Conference on Computationnal Mechanics (ECCM99), 1999 (rapport MIP 99.45)

[3] T. Lewinski, J. Sokolowski, Topological derivative for nucleation of non-circular voids, Rapport de recherche No. 3798, INRIA-Lorraine, 1999

[4] Masmoudi, M. The toplogical asymptotic (Kawarada, H.; Periaux, J., eds.), Computational Methods for Control Applications, Internat. Ser. GAKUTO, 2002

[5] B. Samet, S. Amstutz, M. Masmoudi, The topological asymptotic for the Helmholtz equation, SIAM J. Control Optim., accepted

[6] B. Samet, J. Pommier, The topological asymptotic for the Helmholtz equation with Dirichlet condition on the boundary of an arbitrary shaped hole, SIAM J. Control Optim., submitted

[7] A. Schumacher, Topologieoptimierung von Bauteilstrukturen unter Verwendung von Lochpositionierungkriterien, Ph.D. Thesis, Universität-Gesamthochschule-Siegen, Siegen, 1995

[8] Sokolowski, J.; Zolesio, J.-P. Introduction to shape Optimization. Shape Sensitivity Analysis, Springer-Verlag, 1992

[9] Sokolowski, J.; Zochowski, A. On topological derivative in shape optimization, SIAM J. Control Optim., Volume 37 (1999), pp. 1251-1272

[10] Sokolowski, J.; Żochowski, A. Topological derivatives for elliptic problems, Inverse Problems, Volume 15 (1999), pp. 123-134

Cited by Sources: