Higher-order Vsie-mom Formulation for Scattering by Composite Metallic and Dielectric Objects

A new higher-order method of moment (MoM) technique is presented for volume-surface integral equations (VSIE) for electromagnetic modeling of composite metallic and dielectric objects. The higher-order MoM scheme comprises higher-order hierarchical Legendre basis functions and an accurate representation of the object by higher-order curvilinear elements. Due to the orthogonal nature of the basis functions the continuity condition at the interface between metal and dielectric is satisfied explicitly. By utilizing the continuity condition the number of unknowns can be significantly reduced, especially for metal objects with thin dielectric coating. INTRODUCTION The coupled volume-surface integral equation (VSIE) is often employed for radiation and scattering analysis of composite metal and dielectric objects. In the VSIE formulation, microstrip antennas can be simulated taking into account the finite-size effects of the dielectric substrate. The VSIE is also suitable for modeling of scattering by metal objects with dielectric coating. Due to its volume integral equation part the VSIE treats inhomogeneous dielectric materials more accurately than the piece-wise constant approximation required by the standalone surface integral equation. Straightforward method of moment (MoM) solutions of integral equations become ineffective as the size and complexity of the object under investigation increases. Consequently, accelerated integral equation solvers, such as the fast multipole method (FMM/MLFMM), have been developed. These solvers, which are usually based on low-order basis functions (RWG, rooftop, or pulse), can operate on a huge number of unknowns demanding a moderate amount of computer memory. In this paper we suggest to reduce the number of unknowns by utilizing higher-order basis functions. Defined on rather large geometry elements these functions generally require much less unknowns per wavelength than low order basis functions. Here we use higher-order hierarchical Legendre basis functions [1] to solve the volume-surface integral equation. Previously, these functions have been applied to the analysis of metallic objects in free space [1] and in layered media [2] by the surface integral equations, and to the analysis of dielectric objects by the volume integral equations [3]. Being near-orthogonal the higher-order hierarchical Legendre basis functions allow a low condition number of the MoM matrix to be achieved. Moreover, since the electric charge is expanded in orthogonal functions, the continuity condition at the boundary between metal and dielectric can be enforced explicitly, which further reduces the number of unknowns. The reduction can be significant especially for metal objects with thin dielectric coating. FORMULATION The volume-surface integral equation comprises two coupled equations for a dielectric volume V and a perfectly electrically conducting (PEC) surface S S r r E r E r E V r r E r E r E r E s S s V i s S s V i ∈ − − = ∈ − − = ), ( ) ( ) ( ; ), ( ) ( ) ( ) ( (1) where ) (r E is the total electric field, ) (r E i is the incident electric field, ) (r E s V and ) (r E s S denote the electric field scattered by the unknown induced electric volume current density ) (r JV and the electric surface current density ) (r J S , respectively. It is more convenient to use the electric flux density ) (r D rather than ) (r JV as the unknown because the normal component of ) (r D is continuous across the boundary between two different dielectric materials. Higher-Order MoM. Following the higher-order discretization procedure, the geometry of the object is represented by higher-order curvilinear elements; hexahedra for dielectric volumes, and quadrilaterals for