A projection height independent adaptive radial-angular-R² transformation for singular integrals

A new radial-angular-R² adaptive singularity cancellation transformation is proposed. This new transformation is flexible and applicable to singular integrals over triangular domains. When the height tends to zero and the observation point is in the plane of the source domain, the transformation remains stable and approaches the corresponding transformation within the source plane. Usually the Green's function and its gradient appear in the integral kernels leading to first order and second order singularities. The newly derived transformation formula provides an efficient solution for second order singular coupling integral kernels and the formula is in particular more efficient than alternative schemes for singular vector integral kernels. However, it is also effective for the lower order singular integral kernels. In electromagnetic boundary integral equation formulations, the proposed transformation is efficient for all types of singular integrals.