Comptes Rendus
no. 7-8
An adaptive procedure for defect identification problems in elasticity
[Une procédure adaptative pour l'identification de défauts en élasticité]
Sergio Gutiérrez1  ; J. Mura1
1 Department of Structural and Geotechnical Engineering, Pontificia Universidad Católica de Chile, casilla 306 Correo 22, Santiago, Chile
Comptes Rendus. Mécanique, Volume 338 (2010) no. 7-8, pp. 402-411.

Dans le contexte de problèmes inverses en mécanique, il est bien connu que la situation typique est celle où ni l'intérieur du solide ni la totalité de sa frontière ne sont disponibles pour recueillir des données permettant l'identification d'inclusions ou de défauts. Nous proposons ici une approche adaptative qui utilise la connaissance du chargement et de déplacements mesurés sur une portion de la frontière pour détecter des défauts intérieurs à un solide élastique. Cette méthode repose sur l'homogénéisation à faible amplitude, c'est-à-dire l'hypothèse que le contraste des coefficients de Lamé entre les défauts et la matrice est modéré. A l'aide des données correspondant à un chargement et une configuration de capteurs, une méthode d'optimisation permet d'estimer l'emplacement de l'inclusion, cette estimation étant ensuite utilisée pour adapter le choix du chargement et de l'emplacement des capteurs et raffiner la solution. Des exemples numériques, reposant sur au maximum trois configurations successives de chargement et de capteurs, montrent l'efficacité de la méthode dans certains cas.

In the context of inverse problems in mechanics, it is well known that the most typical situation is that neither the interior nor all the boundary is available to obtain data to detect the presence of inclusions or defects. We propose here an adaptive method that uses loads and measures of displacements only on part of the surface of the body, to detect defects in the interior of an elastic body. The method is based on Small Amplitude Homogenization, that is, we work under the assumption that the contrast on the values of the Lamé elastic coefficients between the defect and the matrix is not very large. The idea is that given the data for one loading state and one location of the displacement sensors, we use an optimization method to obtain a guess for the location of the inclusion and then, using this guess, we adapt the position of the sensors and the loading zone, hoping to refine the current guess.

Numerical results show that the method is quite efficient in some cases, using in those cases no more than three loading positions and three different positions of the sensors.

Publié le :
DOI : 10.1016/j.crme.2010.07.004
Keywords: Continuum mechanics, Inverse problems, Defects in solids, Homogenization
Mot clés : Milieux continus, Problèmes inverses, Défauts dans les solides, Homogénéisation

Sergio Gutiérrez 1 ; J. Mura 1

1 Department of Structural and Geotechnical Engineering, Pontificia Universidad Católica de Chile, casilla 306 Correo 22, Santiago, Chile 
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Sergio Gutiérrez; J. Mura. An adaptive procedure for defect identification problems in elasticity. Comptes Rendus. Mécanique, Volume 338 (2010) no. 7-8, pp. 402-411. doi : 10.1016/j.crme.2010.07.004. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2010.07.004/

[1] M. Burger; S. Osher A survey on level set methods for inverse problems and optimal design, European J. Appl. Math., Volume 16 (2005) no. 2, pp. 263-301

[2] R. Gallego; G. Rus Identification of cracks and cavities using the topological sensitivity boundary integral equation, Comput. Mech., Volume 33 (2004) no. 2, pp. 154-163

[3] B. Guzina; M. Bonnet Topological derivative for the inverse scattering of elastic waves, Quart. J. Mech. Appl. Math., Volume 57 (2004) no. 2, pp. 161-179

[4] S. Gutiérrez; J. Mura Small amplitude homogenization applied to inverse problems, Comput. Mech., Volume 41 (2008) no. 5, pp. 699-706

[5] L. Tartar H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, Volume 115 (1990), pp. 193-230

[6] G. Allaire; S. Gutiérrez Optimal design in small amplitude homogenization, M2AN Math. Model. Numer. Anal., Volume 41 (2007) no. 3, pp. 543-574

[7] J. Mura, S. Gutiérrez, Detection of weak defects in elastic bodies through small amplitude homogenization, submitted for publication.

[8] G. Allaire Shape Optimization by the Homogenization Method, Springer-Verlag, 2002

[9] L. Tartar Remarks on optimal design problems, Marseille, 1993 (Ser. Adv. Math. Appl. Sci.), Volume vol. 18, World Sci. Publishing, River Edge, NJ (1994), pp. 279-296

[10] J. Qian; Y.-T. Zhang; H.-K. Zhao Fast sweeping methods for eikonal equations on triangular meshes, SIAM J. Numer. Anal., Volume 45 (2007) no. 1, pp. 83-107

[11] F. Hecht; O. Pironneau; A. Le Hyaric; K. Ohtsuka http://www.freefem.org (FreeFem++, code and user manual freely available at)

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