Improved Double-Arctan Transformation for the Stable Evaluation of Nearly Hypersingular Integrals in SIE

The evaluation of singular or near-singular integrals is one of the key techniques in the integral equation method. Based on the previously proposed double-arctan transformation (DAT) for the nearly hypersingular integrals in curved surface modeling, this work presents new transformation methods to improve the accuracy and stability of DAT for the following two situations, i.e., the field point is: 1) extremely close to the source surface and 2) extremely close to the boundary of the source surface. The original DAT fails in these two situations and, hence, limits its applications. The reasons that cause such an instability are studied. Two additional transformations are introduced to make the DAT stable for these two cases. For situation I, the exponential transformation (ET) is adopted to optimize the distribution of the Gaussian quadrature points. For situation II, the shape-function transformation (SFT) is further proposed to remove the nearly angular singularity. We denote the newly developed DAT with ET and SFT as the improved DAT (IDAT) in this communication. The typical testing cases are provided to validate the proposed method.