Near singularity cancellation in weakly singular integrals of three-dimensional boundary element method

Abstract

This paper focuses on near singularity in weakly singular integrals of three-dimensional boundary element method. In traditional methods, only one directional transformation is introduced to remove the weak singularity. However, near singularity is still existed. Thus in our paper, serval methods for removal of weak singularity are introduced firstly. Then, angular and sigmoidal transformations are introduced to remove the near singularity in the circumferential direction combined with polar transformation. Generalized Duffy transformation is introduced to remove weak singularity in u direction, which also separates the near singularity in v direction. Based on this method, sinh transformation is constructed with the integrand of v. Finally, several numerical examples considering large top opening angle of the singular integration element and large ratio of the two edge length of the angle (the max edge length/ the min edge length) are proposed. Results demonstrate that generalized Duffy transformation coupled with sinh transformation in v direction is a best choose for weakly singular integrals.