# Regularization in 3D for Anisotropic Elastodynamic Crack and Obstacle Problems

• Jean-Claude Nedelec
• Eliane Becache
• Naoshi Nishimura
Conference paper

## Abstract

The problems of wave scattering by obstacles or cracks appear very often in geophysics and in mechanics. In particular the linearized theory of elastodynamics for 3 dimensional elastic material is used frequently, because this theory keeps the analysis relatively simple. Even with this theory, however, a practical analysis is possible only with the use of some numerical methods. This has been the raison d’être of many numerical experiments carried out in the engineering community. Among those numerical methods tested so far, the boundary integral equation (BIE) method has been accepted favourably by engineers, presumably because it can deal with scattered waves effectively in external problems. In particular the double layer potential representation is considered to be an efficient tool of numerical analysis for wave problems including cracks. The only inconvenience of the double layer potential approach, however, is the hypersingularity of the kernel, which does not permit the use of conventional numerical integration techniques. Hence we can take advantage of this approach only after weakening the hypersingularity of the kernel, or only after ‘regularizing’ it. As a matter of fact, some of such attemps can be found in the articles by Sladek & Sladek [11], Bui [5], Bonnet [4], Polch et.al [10], Nishimura & Kobayashi [8], [9] who used the collocation method and in Nedelec [7], Bamberger [1] where the variational method has been used.

### Keywords

Convolution Geophysics Bonnet

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### References

1. [1]
Bamberger, A. (1983) Approximation de la diffraction d’ondes élastiques: une nouvelle approche (I), (II), (III), Internal report n°91, 96, 98 of Centre de Mathématiques appliquées, Ecole Polytechnique, France.Google Scholar
2. [2]
Becache, E & Ha Duong, T (1989) Formulation Variationnelle Espace — Temps Associée au Potentiel de Double Couche des Ondes Elastiques, Internal report n°199 of Centre de Mathématiques appliquées, Ecole Polytechnique, France.Google Scholar
3. [3]
Becache, E, Nedelec, J-C & Nishimura, N (1989) Regularisation in 3D for anisotropic elastodynamic crack and obstacle problems, Internal report n°205 of Centre de Mathématiques appliquées, Ecole Polytechnique, France.Google Scholar
4. [4]
Bonnet, M. (1987) Méthode des équations intégrales régularisées en élastodynamique, Bulletin de la direction des études et recherches, EDF, France.Google Scholar
5. [5]
Bui, H.D.(1977) An integral equations method for solving the problems of a plane crack of arbitrary shape, J. Mech. Phys. Solids, 25, 29–39.Google Scholar
6. [6]
Martin, P.A. & Rizzo, F.J. (1989) On boundary integral equations for crack problems, Proc. Roy. Soc. London (A) 421, 341–355.
7. [7]
Nedelec, J.-C. (1983) Le potentiel de double couche pour les ondes élastiques, Internal report n°99 of Centre de Mathématiques appliquées, Ecole Polytechnique, France.Google Scholar
8. [8]
Nishimura, N. & Kobayashi, S. (1989) A regularized boundary integral integral equation method for elastodynamic crack problems, Comp. Mech., 4, 319–328.
9. [9]
Nishimura, N., Guo, C & Kobayashi, S. (1987) Boundary Integral Equation Methods in Elastodynamic Crack Problems, Proc. 9th Int Conf BEM, vol2, ed Brebbia, Wendland, Kuhn, Springer Verlag, 279-291.Google Scholar
10. [10]
Polch, E.Z., Cruse, T.A & Huang, C.-J (1987) Traction BIE solutions for flat cracks, Comp. Mech., 2, 253–267.
11. [11]
Sladek, V. & Sladek, J. (1984) Transient elastodynamic three-dimensional problems in cracked bodies, Appl. Math. Model., 8, 2–10.