Incremental improvements to the Telles third degree polynomial transformation for the evaluation of nearly singular boundary integrals

Telles(1,2), in a series of papers, developed a third-degree polynomial transformation which greatly improved the accuracy of weakly-singular and nearly-singular integral evaluation for a variety of common boundary integral kernel functions. The basic idea of the transformation was that the Jacobian of the transformation would "cancel out" in a sense the singularity or near singularity in the kernel function. The net effect was that the Gauss points were clustered within the element close to the singular or nearly-singular point. Through a least-squares error analysis, Telles determined an optimum small value for the Jacobian transformation at the closest point on the element to the field point. In this paper, two incremental improvements are made to the Telles transformation. First, the optimum small value for the Jacobian is determined based on both the minimum distance of the field point to the element and the particular quadrature rule being employed. This improvement allows a reduction in the number of Gauss points while maintaining a desired level of accuracy. Second, the transformation itself is modified so that the Gauss points are more evenly distributed within the element away from the singular or nearly-singular point for three dimensional problems. These incremental improvements are benchmarked through comparisons to semi-analytically evaluated integrals.