The Shape-Dependence and Near-Singularity of Singularity Cancellation Transformations for the Integral Equation Methods

The near-singularity and shape-dependence are two main topics in the singularity cancellation methods which are well-known for the treatments of weakly singular and near-singular integrals. The cancellation transformations for near-singular integrals are inefficient for the deformed triangles, and this phenomenon is referred to as shape-dependence in this paper. We first investigate the reason of these problems via the theoretical analysis with the numerical verification. Second, a new singularity cancellation method for weakly near-singular integral is proposed, which have fast and consistent convergence rate for both regular and irregular triangles. Some numerical results are presented to show the effectiveness of the theoretical framework and the proposed transformation.