An efficient scheme of calculating nearly singular integrals for the 3D BEM modeling of thin media

For engineering analysis of 3D problems, common difficulty to apply the boundary element method (BEM) is the so called "nearly singular integrals" that arise when the object is thin or the internal points of analysis are close to the boundary. In the present work, the local integration domain is sub-divided into 4 quadrants at the projection of the source point. By use of the FG-Squircular Mapping, the four quadrants are transformed to 4 quarter-discs for the integrations to be performed under the polar coordinates. As such, the singularity strength is reduced by one order. Thus, the Gauss points can be reasonably increased solely for the integration of the radial distance, while the other integration for the angular parameter remains regular. Such treatment greatly enhances the efficiency of the integration computation. To demonstrate the validity of all presented formulations, a few typical examples are presented to calculate the nearly singular boundary integrals for treating 3D problems of heat transfer as well as elastostatics.