Nearly Hypersingular Integrals by Double-Arctan Transformation in Higher Order Geometry Modeling

In order to directly utilize the dyadic Green's functions in surface integral equations (SIEs), a novel double-arctan transformation for nearly hypersingular integrals is proposed in this communication. This new transformation is flexible and applicable to nearly hypersingular integrals in the forms of RR/R3, R/R3, and 1/R3 over the curved surfaces by a fully numerical method. With the help of the sigmoidal transformation to improve the stability of this new singularly handling method, there results an efficient solution for the third-order near-singularity problems in SIEs. Moreover, the proposed method is also effective for the lower orders of the nearly singular integral kernels. With typical testing cases, the performance of this method is fairly evaluated, and its validity and stability is well demonstrated.