Topological sensitivity analysis for time-dependent problems
ESAIM: Control, Optimisation and Calculus of Variations, Volume 14 (2008) no. 3, p. 427-455

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.

DOI : https://doi.org/10.1051/cocv:2007059
Classification:  49Q10,  49Q12,  35K05,  35L05
Keywords: 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},
publisher = {EDP-Sciences},
volume = {14},
number = {3},
year = {2008},
pages = {427-455},
doi = {10.1051/cocv:2007059},
zbl = {pre05309725},
mrnumber = {2434060},
language = {en},
url = {http://www.numdam.org/item/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, Volume 14 (2008) no. 3, pp. 427-455. doi : 10.1051/cocv:2007059. http://www.numdam.org/item/COCV_2008__14_3_427_0/

[1] G. Allaire, F. De Gournay, F. Jouve and A.-M. Toader, Structural optimization using topological and shape sensitivity via a level set method. Control Cybern. 34 (2005) 59-80. | MR 2211063

[2] H. Ammari and H. Kang, Reconstruction of small inhomogeneities from boundary measurements, Lecture Notes in Mathematics 1846. Springer-Verlag, Berlin (2004). | MR 2168949 | Zbl 1113.35148

[3] H. Ammari and H. Kang, Reconstruction of elastic inclusions of small volume via dynamic measurements. Appl. Math. Optim. 54 (2006) 223-235. | MR 2239535 | Zbl 1102.35306

[4] H. Ammari and H. Kang, 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 2262349 | Zbl 1111.35101

[5] S. Amstutz, Sensitivity analysis with respect to a local perturbation of the material property. Asymptotic Anal. 49 (2006) 87-108. | MR 2260558 | Zbl pre05076682

[6] S. Amstutz and H. Andrä, A new algorithm for topology optimization using a level-set method. J. Comput. Phys. 216 (2006) 573-588. | MR 2235384 | Zbl 1097.65070

[7] S. Amstutz and N. Dominguez, Topological sensitivity analysis in the context of ultrasonic nondestructive testing. RICAM report 2005-21 (2005).

[8] S. Amstutz, I. Horchani and M. Masmoudi, Crack detection by the topological gradient method. Control Cybern. 34 (2005) 81-101. | MR 2211064

[9] M. Bonnet, Topological sensitivity for 3d elastodynamic and acoustic inverse scattering in the time domain. Comput. Meth. Appl. Mech. Engrg. 195 (2006) 5239-5254. | MR 2241411 | Zbl 1119.74026

[10] M. Bonnet and B.B. Guzina, Sounding of finite solid bodies by way of topological derivative. Int. J. Numer. Methods Eng. 61 (2004) 2344-2373. | MR 2103152 | Zbl 1075.74564

[11] M. Burger, B. Hackl and W. Ring, Incorporating topological derivatives into level set methods. J. Comput. Phys. 194 (2004) 344-362. | MR 2033389 | Zbl 1044.65053

[12] T. Cazenave and A. Haraux, Introduction aux problèmes d'évolution semi-linéaires, Mathématiques & Applications 1. Ellipses, Paris (1990). | MR 1299976 | Zbl 0786.35070

[13] R. Dautray and J.-L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques, Vol. 6. INSTN: Collection Enseignement, Masson, Paris (1988). | MR 944304

[14] H.A. Eschenauer and A. Schumacher, 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 1488756 | Zbl 0885.73050

[15] H.A. Eschenauer, V.V. Kobolev and A. Schumacher, Bubble method for topology and shape optimization of structures. Struct. Optimization 8 (1994) 42-51.

[16] S. Garreau, P. Guillaume and M. Masmoudi, The topological asymptotic for PDE systems: the elasticity case. SIAM J. Control Optim. 39 (2001) 1756-1778 (electronic). | MR 1825864 | Zbl 0990.49028

[17] P. Guillaume and K. Sid Idris, The topological asymptotic expansion for the Dirichlet problem. SIAM J. Control Optim. 41 (2002) 1042-1072 (electronic). | MR 1972502 | Zbl 1053.49031

[18] B.B. Guzina and M. Bonnet, Topological derivative for the inverse scattering of elastic waves. Quart. J. Mech. Appl. Math. 57 (2004) 161-179. | MR 2068401 | Zbl 1112.74035

[19] J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 2. Travaux et Recherches Mathématiques 18. Dunod, Paris (1968). | MR 247244 | Zbl 0165.10801

[20] S.A. Nazarov and J. Sokolowski, The topological derivative of the Dirichlet integral under the formation of a thin bridge. Siberian Math. J. 45 (2004) 341-355. | MR 2061421 | Zbl 1071.35037

[21] G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics. Annals of Mathematics Studies 27. Princeton University Press, Princeton, N. J. (1951). | MR 43486 | Zbl 0044.38301

[22] J. Pommier and B. Samet, 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 2114381 | Zbl 1080.49028

[23] M. Schiffer and G. Szegö, Virtual mass and polarization. Trans. Amer. Math. Soc. 67 (1949) 130-205. | MR 33922 | Zbl 0035.11803

[24] A. Schumacher, Topologieoptimierung von bauteilstrukturen unter verwendung von lopschpositionierungskriterien. Ph.D. thesis, Univ. Siegen (1995).

[25] J. Sokołowski and A. Żochowski, On the topological derivative in shape optimization. SIAM J. Control Optim. 37 (1999) 1251-1272 (electronic). | MR 1691940 | Zbl 0940.49026