On the Shape-Dependent Problem of Singularity Cancellation Transformations for Weakly Near-Singular Integrals

The singularity cancellation transformations are well-known for calculating weakly singular and near-singular integrals in integral equation solutions. However, some singularity cancellation methods suffer from the shape-dependent problem of inaccuracy and inefficiency for deformed triangles. By the theoretical analysis and numerical verification in this article, the relation between the near-singularity and shape-dependence of the singularity cancellation schemes is discussed. Moreover, a novel framework for devising cancellation transformations of weakly near-singular integrals is presented, which results in fast convergence for both regular and irregular triangular domains. Some numerical results are given to illustrate the validity of the theoretical framework and the efficiency of the proposed transformations.