Fast Solution of Volume–Surface Integral Equations for Multiscale Structures

Abstract

Electromagnetic (EM) problems with conducting and penetrable media are formulated by volume–surface integral equations (VSIEs). Traditionally, the VSIEs are solved by the method of moments (MoM) with Rao–Wilton–Glisson (RWG) and Schaubert–Wilton–Glisson (SWG) basis functions. We develop a hybrid method to discretize the VSIEs where the surface integral equation (SIE) for the conducting part is discretized by the standard MoM, while the volume integral equations (VIEs) for the penetrable part are discretized by a point-matching scheme (PMS). The PMS can use unstructured or nonconforming meshes and make the integral kernels of VIEs be free of material parameters. Also, the VIEs can result in well-conditioned system matrices because they are the second kind of integral equations. For electrically large or complex problems with multiscale structures, we unite the hybrid method with the multilevel fast multipole algorithm (MLFMA) to accelerate the solutions. Since the MLFMA is sensitive to the conditioning of system matrices which the multiscale feature will deteriorate and is tedious to implement, using the VIEs can ameliorate the conditioning of system matrices and using the PMS can facilitate the implementation of MLFMA. The approach is demonstrated by several numerical examples and its good performance has been observed.