A necessary and sufficient BEM/BIEM for two-dimensional elasticity problems

Abstract

It is well known that the patch test is required for the finite element method (FEM). We may wonder whether we need any special test for the boundary element method (BEM). A sufficient and necessary boundary integral equation method (BIEM) to ensure a unique solution is our concern. In this paper, we revisit this issue for the interior two-dimensional (2-D) elasticity problem and investigate the equivalence of the solution space between the integral equation and the partial differential equation. Based on the degenerate kernel and the eigenfunction expansion, the range deficiency of the integral operator for the solution space in the degenerate-scale problem for the 2-D elasticity in the BIEM is analytically studied. According to the Fichera׳s idea, we enrich the conventional BIEM by adding constants and corresponding constraints. In addition, we introduce the concept of modal participation factor (MPF) to examine whether the adding term of rotation is required for interior simply-connected problems. Finally, two simple examples of degenerate-scale problems containing circular and elliptical boundaries subjected to various boundary conditions of the rigid body translation and rotation for 2-D elasticity problems are demonstrated by using the necessary and sufficient BIEM.