Calculation of 2D nearly singular integrals over high-order geometry elements using the sinh transformation

The accurate evaluation of nearly singular integrals plays an important role in the implementation of BEM. In general, these include evaluating the solution near the boundary or treating problems with thin domains, which are respectively named the boundary layer effect and the thin-body effect in BEM. Although many methods of evaluating two-dimensional (2D) nearly singular integrals have been developed in recent years with varying degrees of success, questions still remain. In this paper, we present an efficient strategy for numerical evaluation of 2D nearly singular integrals that arise in the solution of 3D BEM using eight-node second-order quadrilateral surface elements. The strategy is an extension of the sinh transformation, which is used to evaluate the 1D or 2D nearly singular integrals on simple geometry elements, such as usual linear or planar elements. Several numerical examples involving boundary layer effect and thin body problems in 3D potential problems are investigated to verify the proposed scheme, yielding very promising results.