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The direct method of fundamental solutions and the inverse Kirsch-Kress method for the reconstruction of elastic inclusions or cavities. (English) Zbl 1166.74008
Summary: We consider the inverse problem of detecting inclusions or cavities in an elastic body, using a single boundary measurement on an external boundary. We discuss the identifiability questions on shape reconstruction, presenting counterexamples for Robin boundary conditions or with additional unknown Lamé parameters. Using the method of fundamental solutions (MFS) we adapt a method introduced twenty years ago by A. Kirsch and R. Kress [Inverse Problems, Proc. Conf., Oberwolfach/Ger. 1986, ISNM 77, 93–102 (1986; Zbl 0607.65089)] (in the context of an exterior problem in acoustic scattering) to this inverse problem in a bounded domain. We prove density results that justify the reconstruction of the solution from the Cauchy data using the MFS. We also establish some connections between this linear part of the Kirsch-Kress method and the direct MFS, through matrices of boundary layer integrals. Several numerical examples are presented, showing that with noisy data we were able to retrieve a fairly good reconstruction of the shape (or of its convex hull) with this MFS version of the Kirsch-Kress method.

MSC:
74G75 Inverse problems in equilibrium solid mechanics
35R30 Inverse problems for PDEs
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