Super-hyper singularity treatment for solving 3D electric field integral equations

The super-hyper singularity treatment is developed for solving three-dimensional (3D) electric field integral equations (EFIE). EFIE usually takes two forms: one of which includes a super-hyper singular kernel generated by the double gradient of the Green's function. So far, there is no way to evaluate these super-hyper singular integrals needed for constructing the pertinent matrix equation. We apply the series expansion of the Green's function to the super-hyper singular kernel and derive closed-form expressions for their evaluations in the Cauchy principal-value sense. The derivation is based on the constant current approximation over a flat triangle patch, but it can be extended easily to higher-order approximations of the current. The scheme can be used to accurately calculate the near and self interaction terms in the impedance matrix for the method of moments (MoM), Nyström method or boundary element method (BEM).