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Topological sensitivity analysis for time-dependent problems

Published online by Cambridge University Press:  21 November 2007

Samuel Amstutz ,
Takéo Takahashi  and
Boris Vexler
Samuel Amstutz
Affiliation:
Laboratoire d'analyse non-linéaire et géométrie, Faculté des sciences, 33 rue Louis Pasteur, 84000 Avignon, France; [email protected]
Takéo Takahashi
Affiliation:
Institut Élie Cartan de Nancy, Nancy-Université, CNRS, INRIA, BP 239, 54506 Vandœuvre-lès-Nancy cedex, France; [email protected]
Boris Vexler
Affiliation:
Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstrasse 69, 4040 Linz, Austria; [email protected]
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Abstract

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.

Keywords

Topological sensitivitytopology optimizationparabolic equationshyperbolic equations
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 14 , Issue 3 , July 2008 , pp. 427 - 455
Copyright
© EDP Sciences, SMAI, 2007

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References

Allaire, G., de Gournay, F., Jouve, F. and Toader, A.-M., Structural optimization using topological and shape sensitivity via a level set method. Control Cybern. 34 (2005) 59–80.
H. Ammari and H. Kang, Reconstruction of small inhomogeneities from boundary measurements, Lecture Notes in Mathematics 1846. Springer-Verlag, Berlin (2004).
Ammari, H. and Kang, H., Reconstruction of elastic inclusions of small volume via dynamic measurements. Appl. Math. Optim. 54 (2006) 223–235. CrossRef
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.
Amstutz, S., Sensitivity analysis with respect to a local perturbation of the material property. Asymptotic Anal. 49 (2006) 87–108.
Amstutz, S. and Andrä, H., A new algorithm for topology optimization using a level-set method. J. Comput. Phys. 216 (2006) 573–588. CrossRef
S. Amstutz and N. Dominguez, Topological sensitivity analysis in the context of ultrasonic nondestructive testing. RICAM report 2005-21 (2005).
Amstutz, S., Horchani, I. and Masmoudi, M., Crack detection by the topological gradient method. Control Cybern. 34 (2005) 81–101.
Bonnet, M., Topological sensitivity for 3d elastodynamic and acoustic inverse scattering in the time domain. Comput. Meth. Appl. Mech. Engrg. 195 (2006) 5239–5254. CrossRef
Bonnet, M. and Guzina, B.B., Sounding of finite solid bodies by way of topological derivative. Int. J. Numer. Methods Eng. 61 (2004) 2344–2373. CrossRef
Burger, M., Hackl, B. and Ring, W., Incorporating topological derivatives into level set methods. J. Comput. Phys. 194 (2004) 344–362. CrossRef
T. Cazenave and A. Haraux, Introduction aux problèmes d'évolution semi-linéaires, Mathématiques & Applications 1. Ellipses, Paris (1990).
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).
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.
Eschenauer, H.A., Kobolev, V.V. and Schumacher, A., Bubble method for topology and shape optimization of structures. Struct. Optimization 8 (1994) 42–51. CrossRef
Garreau, S., Guillaume, P. and Masmoudi, M., The topological asymptotic for PDE systems: the elasticity case. SIAM J. Control Optim. 39 (2001) 1756–1778 (electronic). CrossRef
Guillaume, P. and Sid Idris, K., The topological asymptotic expansion for the Dirichlet problem. SIAM J. Control Optim. 41 (2002) 1042–1072 (electronic). CrossRef
Guzina, B.B. and Bonnet, M., Topological derivative for the inverse scattering of elastic waves. Quart. J. Mech. Appl. Math. 57 (2004) 161–179. CrossRef
J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 2. Travaux et Recherches Mathématiques 18. Dunod, Paris (1968).
Nazarov, S.A. and Sokolowski, J., The topological derivative of the Dirichlet integral under the formation of a thin bridge. Siberian Math. J. 45 (2004) 341–355. CrossRef
G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics. Annals of Mathematics Studies 27. Princeton University Press, Princeton, N. J. (1951).
Pommier, J. and Samet, B., 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). CrossRef
Schiffer, M. and Szegö, G., Virtual mass and polarization. Trans. Amer. Math. Soc. 67 (1949) 130–205. CrossRef
A. Schumacher, Topologieoptimierung von bauteilstrukturen unter verwendung von lopschpositionierungskriterien. Ph.D. thesis, Univ. Siegen (1995).
Sokołowski, J. and A. Żochowski, On the topological derivative in shape optimization. SIAM J. Control Optim. 37 (1999) 1251–1272 (electronic). CrossRef