Fundamental Solution Method for Periodic Plane Elasticity

Periodic elasticity problem is attractive subject and important from the theoretical and practical viewpoints. In this paper, we propose a fundamental solution method for elasticity problems of planes with one-dimensional periodic structure. The fundamental solution method [4, 15] is a numerical solver for partial differential equation problems and is widely used in science and engineering, especially used for potential problems, where the method is called the “charge simulation method”. This method approximates the solution by a superposition of the fundamental solutions of the differential operator and has the advantages that (i) it is easy to program, (ii) its computational cost is low and (iii) it achieves high accuracy such as exponential convergence under some conditions. However, the method had the weakness that it is difficult to apply the method to problems with spatial periodicity. The author overcame this weakness in the case of numerical conformal mappings [12] and in the case of Stokes flows [9, 10, 11]. In this paper, we present a fundamental solution method by modifying the ordinary fundamental solution method as in [13], whose approximate solution is given by a superposition of the displacement due to concentrated forces at discrete points, so that it is applicable to our problem with periodicity. We also show a numerical example of the presented method. The contents of this paper is as follows. In Section 2, we prepare some notations and formulate our problem. In Section