Interior Elastic Scattering by a Non-Penetrable Partially Coated Obstacle and Its Shape Recovering

Angeliki Kaiafa and Vassilios Sevroglou
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Angeliki Kaiafa: Department of Statistics and Insurance Science, School of Finance and Statistics, University of Piraeus, 80 Karaoli and Dimitriou Street, 18534 Piraeus, Greece
Vassilios Sevroglou: Department of Statistics and Insurance Science, School of Finance and Statistics, University of Piraeus, 80 Karaoli and Dimitriou Street, 18534 Piraeus, Greece

Mathematics, 2021, vol. 9, issue 19, 1-24

Abstract: In this paper, the interior elastic direct and inverse scattering problem of time-harmonic waves for a non-penetrable partially coated obstacle placed in a homogeneous and isotropic medium is studied. The scattering problem is formulated via the Navier equation, considering incident circular waves due to point-source fields, where the corresponding scattered data are measured on a closed curve inside the obstacle. Our model, from the mathematical point of view, is described by a mixed boundary value problem in which the scattered field satisfies mixed Dirichlet-Robin boundary conditions on the Lipschitz boundary of the obstacle. Using a variational equation method in an appropriate Sobolev space setting, uniqueness and existence results as well as stability ones are established. The corresponding inverse problem is also studied, and using some specific auxiliary integral operators an appropriate modified factorisation method is given. In addition, an inversion algorithm for shape recovering of the partially coated obstacle is presented and proved. Last but not least, useful remarks and conclusions concerning the direct scattering problem and its linchpin with the corresponding inverse one are given.

Keywords: direct and inverse scattering problem; partially coated obstacle; interior mixed boundary value problem; variational formulation; modified factorization method; inversion algorithm (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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