An Invariant Method of Fundamental Solutions for the Cauchy Problem in Two-Dimensional Isotropic Linear Elasticity

Abstract

In this paper, we propose a numerical algorithm based on the method of fundamental solutions (MFS) for the Cauchy problem in two-dimensional linear elasticity. Through the use of the double-layer potential function, we give the invariance property for a problem with two different descriptions. In order to adapt this invariance property, we give an invariant MFS to satisfy this invariance property, i.e., formulate the MFS with an added constant and an additional constraint. The method is combining the Newton method and classical Tikhonov regularization with Morozov discrepancy principle to solve the inverse Cauchy problem. Some examples are given for numerical verification on the efficiency of the proposed method. The numerical convergence, accuracy, and stability of the method with respect to the the number of source points and the distance between the pseudo-boundary and the real boundary of the solution domain, and decreasing the amount of noise added into the input data, respectively, are also analysed with some examples.