On the Non-Linear Integral Equation Method for the Reconstruction of an Inclusion in the Elastic Body

Abstract

We apply the non-linear integral equation approach based on elastic potentials for determining the shape of a bounded object in the elastostatic two-dimensional domain from given Cauchy data on its boundary. The iterative algorithm is developed for the numerical solution of obtained integral equations. We find the Frechet derivative for the corresponding operator and show unique solviability of the linearized system. Full discretization of the system is realized by a trigonometric quadrature method. Due to the inherited ill-possedness in the system of linear equations we apply the Tikhonov regularization. The numerical results show that the proposed method gives a good accuracy of reconstructions with an economical computational cost.