Ευθέα και αντίστροφα προβλήματα σκέδασης επιπέδων και σφαιρικών ελαστικών κυμάτων
Direct and inverse scattering problems of plane and spherical elastic waves
KeywordsΜεικτό πρόβλημα ; Σκέδαση ; Ελαστικά κύματα ; Μερικώς επικαλυμμένο αντικείμενο ; Θαμμένο αντικείμενο ; Αντίστροφο πρόβλημα ; Αλγόριθμος αντιστροφής
In this dissertation, we study direct and inverse scattering problems of elastic waves by a non-penetrable partially coated object located in a homogeneous and isotropic elastic medium. We formulate the scattering problem via the Navier equation by conside-ring the incident waves due to point sources where the corresponding scattered waves are measured on a closed, smooth curve inside the scatterer-object. From a mathematical point of view, our model is described by a mixed boundary value problem where the scattered field satisfies a Dirichlet-Robin mixed boundary condition on the boundary of the scatterer. We prove, for the direct scattering problem, well posedness in an appropriate Sobolev space setting and in particular we prove uniqueness, existence and stability of the solution as well. We also present similar results for the direct scattering problem by a non-homogeneous elastic medium with unknown buried obstacles. We also study the corresponding inverse scattering problems by introducing appropriate auxiliary integral operators in the form of single and double-layer potential from which a modified factori-zation method is established. In addition, we present and prove an inversion algorithm for the reconstruction of the boundary of the partially coated scatterer. Finally, we state useful remarks, conclusions and applications.