A combined conformal and sinh–sigmoidal transformations method for nearly singular boundary element integrals

Abstract

Accurate and efficient evaluation of nearly singular integrals is a major concern in 3D BEM. Most existing widely-used non-linear transformations are only performed in radial direction. Actually, the near singularity may derive from three aspects: element shape, radial direction and angular direction. In this paper, a combined conformal and sinh–sigmoidal transformations method is proposed to evaluate nearly singular integrals arising in 3D BEM. The method can be decomposed into three steps: firstly, a conformal transformation is introduced to eliminate the shape effect caused by large aspect ratios and peak/big obtuse angles; secondly, the classical sinh transformation is applied in radial direction to cluster more Gaussian points towards the nearly singular point; finally, an improved sigmoidal transformation is utilized to rearrange Gaussian points in angular direction more reasonably. Extensive numerical examples including unit triangular element, elements with different aspect ratios, elements with different angles and curved triangular element are given to verify the robustness and competitiveness of presented method.