THE DIRECT METHOD OF FUNDAMENTALSOLUTIONS AND THE INVERSE KIRSCH-KRESSMETHOD FOR THE RECONSTRUCTIONOF ELASTIC INCLUSIONS OR CAVITIES

摘要

In this work we consider the inverse prob-lem of detecting inclusions or cavities in an elastic body, us-ing a single boundary measurement on an external boundary.We discuss the identifiability questions on shape reconstruc-tion, presenting counterexamples for Robin boundary condi-tions or with additional unknown Lame parameters. Using themethod of fundamental solutions (MFS) we adapt a methodintroduced twenty years ago by Andreas Kirsch and RainerKress [20] (in the context of an exterior problem in acousticscattering) to this inverse problem in a bounded domain. Weprove density results that justify the reconstruction of the so-lution from the Cauchy data using the MFS. We also establishsome connections between this linear part of the Kirsch-Kressmethod and the direct MFS, through matrices of boundarylayer integrals. Several numerical examples are presented,showing that with noisy data we were able to retrieve a fairlygood reconstruction of the shape (or of its convex hull) withthis MFS version of the Kirsch-Kress method.