A nystrom scheme based on cuboid elements for solving Volume Integral Equations
Summary form only given. Volume Integral Equations (VIEs) are indispensable for electromagnetic (EM) analysis by integral equation approach when inhomogeneous penetrable objects are involved. In the conventional method of moment (MoM) for solving the VIEs, unknown functions are expanded by a well-designed basis function like Schaubert-Wilton-Glisson (SWG) basis function, and a tedious testing procedure is required to transform the VIEs into matrix equations. Although the MoM is dominant, the Nyström method could be a good alternative because of its well-known merits. The objects are often meshed into tetrahedral elements for a good geometric approximation. Nevertheless, the objects take a cuboid shape in some applications like in packaging structures or in inverse scattering problems whose imaging domain is always a cubic box. Using cuboid elements to discretize the objects is a more favorable choice for convenience and efficiency in this situation. Also, it is easier to handle singular integrals over cuboid elements than tetrahedral elements because of their geometric characteristics. Albeit the Nyström method has been used for solving the VIEs with a tetrahedral discretization, it has not been employed based on the discretization with cuboid elements. We propose a Nyström scheme based on cuboid elements for solving the VIEs in this work. Unlike in the MoM, the matrix elements are not calculated by double numerical integrations involving scalar Green's function, but simply by sampling the dyad Green's function at observation points and source points when the distance between them is large. For near-interaction and self-interaction elements where an observation point is near or inside a source element, a polynomial, which is combined with the dyad Green's function, is used to interpolate unknown functions, leading to singular or badly-behaved integrals with different orders. The hypersingular integrals have been addressed in our previous work and weakly-singular integrals can be evaluated by Duffy's method. This work develops some analytical formulations for evaluating strongly-singular integrals based on a Cauchy-Principal-Value (CPV) sense. These formulations can be used to simplify the evaluation of matrix elements with strongly-singular integrands. Numerical examples are presented to demonstrate the scheme and good results have been obtained.