Partial differential equations/Calculus of variations
Higher-order topological sensitivity analysis for the Laplace operator
[Analyse de sensibilité topologique d'ordre supérieur pour l'opérateur de Laplace]
Comptes Rendus. Mathématique, Tome 354 (2016) no. 10, pp. 993-999.

Dans ce papier, on donne une analyse de sensibilité pour l'opérateur de Laplace par rapport à des perturbations géométriques de type Dirichlet. On pésente deux résultats. Le premier concerne l'influence de la perturbation géométrique sur la solution du problème de Laplace. On dérive une formule de représentation asymptotique d'ordre supérieur décrivant le comportement de la solution perturbée en fonction de la taille de la perturbation. Le deuxième concerne les dérivées d'une fonction de forme par rapport à la modification de la topologie du domaine. On donne un développement asymtotique topologique d'ordre supérieur valable pour une grande classe de fonctions de forme.

This paper deals with higher-order topological sensitivity analysis for the Laplace operator with respect to the presence of a Dirichlet geometry perturbation. Two main results are presented in this work. In the first one, we discuss the influence of the considered geometry perturbation on the Laplace solution. The second one is devoted to the higher-order topological derivatives. We derive a higher-order topological sensitivity analysis for a large class of shape functions.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2016.09.002
Hassine, Maatoug 1 ; Khelifi, Khalifa 1

1 Monastir University, FSM, 5019 Monastir, Tunisia
@article{CRMATH_2016__354_10_993_0,
     author = {Hassine, Maatoug and Khelifi, Khalifa},
     title = {Higher-order topological sensitivity analysis for the {Laplace} operator},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {993--999},
     publisher = {Elsevier},
     volume = {354},
     number = {10},
     year = {2016},
     doi = {10.1016/j.crma.2016.09.002},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2016.09.002/}
}
TY  - JOUR
AU  - Hassine, Maatoug
AU  - Khelifi, Khalifa
TI  - Higher-order topological sensitivity analysis for the Laplace operator
JO  - Comptes Rendus. Mathématique
PY  - 2016
SP  - 993
EP  - 999
VL  - 354
IS  - 10
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2016.09.002/
DO  - 10.1016/j.crma.2016.09.002
LA  - en
ID  - CRMATH_2016__354_10_993_0
ER  - 
%0 Journal Article
%A Hassine, Maatoug
%A Khelifi, Khalifa
%T Higher-order topological sensitivity analysis for the Laplace operator
%J Comptes Rendus. Mathématique
%D 2016
%P 993-999
%V 354
%N 10
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2016.09.002/
%R 10.1016/j.crma.2016.09.002
%G en
%F CRMATH_2016__354_10_993_0
Hassine, Maatoug; Khelifi, Khalifa. Higher-order topological sensitivity analysis for the Laplace operator. Comptes Rendus. Mathématique, Tome 354 (2016) no. 10, pp. 993-999. doi : 10.1016/j.crma.2016.09.002. http://www.numdam.org/articles/10.1016/j.crma.2016.09.002/

[1] Abdelwahed, M.; Hassine, M. Topological optimization method for a geometric control problem in Stokes flow, Appl. Numer. Math., Volume 59 (2009) no. 8, pp. 1823-1838

[2] Abdelwahed, M.; Hassine, M.; Masmoudi, M. Optimal shape design for fluid flow using topological perturbation technique, J. Math. Anal. Appl., Volume 356 (2009), pp. 548-563

[3] Ammari, H.; Kang, H. High-order terms in the asymptotic expansions of the steady-state voltage potentials in the presence of conductivity inhomogeneities of small diameter, SIAM J. Math. Anal., Volume 34 (2003) no. 5, pp. 1152-1166

[4] Ben Abda, A.; Hassine, M.; Jaoua, M.; Masmoudi, M. Topological sensitivity analysis for the location of small cavities in Stokes flow, SIAM J. Control Optim., Volume 48 (2009) no. 5, pp. 2871-2900

[5] Bonnet, M. Topological sensitivity for 3D elastodynamic and acoustic inverse scattering in the time domain, Comput. Methods Appl. Mech. Eng., Volume 195 (2006), pp. 5239-5254

[6] Bonnet, M. Inverse acoustic scattering by small-obstacle expansion of a misfit function, Inverse Probl., Volume 24 (2008) no. 3

[7] Dautray, R.; Lions, J. Analyse mathémathique et calcul numérique pour les sciences et les techniques, Collection CEA, Masson, Paris, 1987

[8] Garreau, S.; Guillaume, Ph.; Masmoudi, M. The topological asymptotic for PDE systems: the elastics case, SIAM J. Control Optim., Volume 39 (2001) no. 6, pp. 1756-1778

[9] Guillaume, P.; Sid Idris, K. The topological asymptotic expansion for the Dirichlet problem, SIAM J. Control Optim., Volume 41 (2002), pp. 1052-1072

[10] Guzina, B.; Bonnet, M. Small-inclusion asymptotic of misfit functionals for inverse problems in acoustics, Inverse Probl., Volume 22 (2006), pp. 1761-1785

[11] Hassine, M.; Jan, S.; Masmoudi, M. From differential calculus to 01 topological optimization, SIAM J. Control Optim., Volume 45 (2007) no. 6, pp. 1965-1987

[12] Hassine, M.; Masmoudi, M. The topological asymptotic expansion for the quasi-Stokes problem, ESAIM Control Optim. Calc. Var., Volume 10 (2004) no. 4, pp. 478-504

[13] Rocha de Faria, J.; Novotny, A.A.; Feijoo, R.A.; Taroco, E. First- and second-order topological sensitivity analysis for inclusions, Inverse Probl. Sci. Eng., Volume 17 (2009) no. 5, pp. 665-679

[14] Rocha de Faria, J.; Novotny, A.A.; Feijóo, R.A.; Taroco, E.; Padra, C. Second order topological sensitivity analysis, Int. J. Solids Struct., Volume 44 (2007) no. 14, pp. 4958-4977

[15] Samet, B.; Amstutz, S.; Masmoudi, M. The topological asymptotic for the Helmholtz equation, SIAM J. Control Optim., Volume 42 (2003) no. 5, pp. 1523-1544

Cité par Sources :