Study on the double-degeneracy mechanism of BEM/BIEM for a plane elasticity problem with line segments

Abstract

There are four degenerate problems by using the BEM/BIEM. Only the degenerate scale and degenerate boundary may appear at the same time. This is called double degeneracy. A double-degeneracy mechanism of BEM/BIEM for the plane elasticity problem with line segments is studied, and the degenerate scale is analytically examined and numerically performed. Different from the past result with complex variables, we propose a new idea to deal with the problem, and obtain the analytical solution of the degenerate scale. This analytical derivation can clearly show why the BEM/BIEM suffer the degenerate scale in the plane elasticity problem with the line segment. Only rigid inclusion faces this problem instead of the crack due to the use of single-layer kernel. Double degeneracy of degenerate boundary and degenerate scale in the BEM are numerically examined. The double-degeneracy mechanism is clearly displayed through numerical results by showing the number of zero singular values in the influence matrix. Following the result of single line segment, we can extend to multiple line segments. Finally, the analytical and numerical results show consistency.