An adaptive element subdivision technique for evaluating 3D weakly singular boundary integrals

A general adaptive element subdivision technique is presented for the numerical evaluation of weakly singular integrals, which often appear in three-dimensional boundary element analysis equations. In this algorithm, the weakly singular boundary element is broken up into a few sub-elements through a sphere of decreasing radius. The sub-elements involving the singular point are evaluated numerically after using a coordinate transformation to remove the singularities, while other quadrilateral sub-elements and triangular sub-elements are evaluated numerically by the standard Gaussian quadrature and Hammer quadrature respectively. The number of sub-elements and their size are determined adaptively according to the position of the singular point. Numerical examples are presented for both planar and curved surface element. The results demonstrate our method can provide higher accuracy and efficiency than the conventional method.