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Topological derivative for the inverse scattering of elastic waves. (English) Zbl 1112.74035
The focus of this study is an extension of the concept of topological derivative, rooted in elastostatic and shape optimization, to three-dimensional elastodynamical involving semi-infinite and infinite solids. The main result of the proposed boundary integral approach is a formula for topological derivative, explicit in terms of the elastodynamic fundamental solution obtained by an asymptotic expansion of the misfit-type cost functional with respect to the creation of an infinitesimal hole in an otherwise intact (semi-infinite or infinite) elastic medium. Valid for an arbitrary shape of the infinitesimal cavity, the formula involves the solution of six canonical exterior elastostatic problems, and becomes fully explicit when the vanishing cavity is spherical. A set of numerical results is included to illustrate the potential of topolocial derivative as a computationally efficient tool for exposing an approximate cavity topology, location, and shape via grid-type exploration of the host solid. For a comprehensive solution to three-dimensional inverse scattering problems involving elastic waves, the proposed approach can be used most effectively as a pre-conditioning tool for more refined, albeit computationally intensive minimization-based imaging algorithms. To the authors’ knowledge, an application of topological derivative to inverse scattering problems has not been attempted before, the methodology proposed in this paper could also be extended to acoustic problems.

74J20 Wave scattering in solid mechanics
74J25 Inverse problems for waves in solid mechanics
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