Το αντίστροφο πρόβλημα της εύρεσης του σχήματος κοιλοτήτων και εγκλεισμάτων στη στατιστική και δυναμική γραμμική ελαστικότητα

Abstract

In this thesis, the inverse problem of determining the position and the shape of an obstacle in static and dynamic linear elasticity is considered. ThTe problem is stated in R2 and the obstacle is a rigid body, a cavity or an inclusion in a homogeneous and isotropic elastic medium. The inverse problem is solved, for all types of obstacles, using the method of non linear integral equations proposed by Kress and Rundell for Laplace equation. This method is derived from the integral representation of the fields. The problem is equivalently transformed into a system of integral equations which is non linear with respect to the parametrized boundary and ill-posed. The linearization of the equations is performed using Frechet derivatives of the integral operators. To overcome the ill-posedness of the system we approximately solve it via Tikhonov regularization. The weakly singular, singular and strongly singular integral operators are computed using special collocation and quadrature rules. The numerical solution is achieved via an iterative algorithm based on an initial guess for the boundary. Numerical examples and reconstructions are given to illustrate the applicability of the method