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

Abstract

In the development of finite element method (FEM), the patch test is required. 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 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 of the single-layer potential for the solution space in the degenerate-scale problem for the 2-D elasticity in the BIEM is analytically studied. Following 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 the rotation is required for interior simply-connected problems. Finally, a simple example of the degenerate-scale problem containing an elliptical boundary subjected to various boundary conditions of the rigid body translation and rotation for 2-D elasticity problems was demonstrated by using the necessary and sufficient BIEM.

Boundary Element and Augmented Lagrangian Methods for Contact Problem with Coulomb Friction

“…Consequently, the boundary variational formulation expresses in terms of boundary unknowns only. For contact problems, the key unknowns are displacement and stress on the contact boundary, which are considered as primary variables and can be obtained directly in the BEM [28][29][30]. erefore, the BEM seems to be the natural way for these problems [6][7][8][9][10][31][32][33][34][35][36][37].…”