An adaptive element subdivision technique for evaluation of various 2D singular boundary integrals

In this paper, a unified algorithm is presented for the numerical evaluation of weakly, strongly and hyper singular boundary integrals with or without a logarithmic term, which often appear in two-dimensional boundary element analysis equations. In this algorithm, the singular boundary element is broken up into a few sub-elements. The sub-elements involving the singular point are evaluated analytically to remove the singularities by expressing the non-singular parts of the integration kernels as polynomials of the distance r, while other sub-elements are evaluated numerically by the standard Gaussian quadrature. The number of sub-elements and their sizes are determined according to the singularity order and the position of the singular point. Numerical examples are provided to demonstrate the correctness and efficiency of the proposed algorithm.