On the errors arising in surface integral equations due to the discretization of the identity operator

Abstract

Surface integral equations (SIEs) are commonly used to formulate scattering and radiation problems involving three-dimensional metallic and homogeneous dielectric objects with arbitrary shapes [1]–[3]. For numerical solutions, equivalent electric and/or magnetic currents defined on surfaces are discretized and expanded in a series of basis functions, such as the Rao-Wilton-Glisson (RWG) functions on planar triangles. Then, the boundary conditions are tested on surfaces via a set of testing functions. Solutions of the resulting dense matrix equations provide the expansion coefficients of the equivalent currents, which can be used to compute the scattered or radiated electromagnetic fields. In general, SIEs involve three basic operators, i.e., integro-differential K and T operators, and the identity operator I{X}(r) = X(r). Depending on the testing scheme and the boundary conditions used, there are four basic SIEs [2],[3], namely, the tangential electric-field integral equation (T-EFIE), the normal electric-field integral equation (N-EFIE), the tangential magnetic-field integral equation (T-MFIE), and the normal magnetic-field integral equation (N-MFIE). In the tangential equations, boundary conditions are tested directly by sampling the tangential components of the electric and magnetic fields on the surface. In the normal equations, however, electromagnetic fields are tested after they are projected onto the surface via a cross-product operation with the outward normal vector.