A Fast Volume-surface Integral Equation Solver for Scattering Properties of NIMs

This paper presents a fast hybrid volume-surface integral equation approach for the computation of electromagnetic scattering from composed negative index media (NIM) such as split-ring-resonators (SRR) with wires. The volume electric field integral equation (EFIE) is applied to the dielectric region of this NIM, and the surface electric field integral equation is applied on the conducting surface. The method of moments (MoM) is used to discretize the integral equation into a matrix solution and adaptive integral method (AIM) is employed to reduce the memory requirement and CPU time for the matrix solution. The present approach is sufficiently versatile in handling scattering problems of composed NIMs, due to the combination of surface and volume electric field integral equations. Numerical results of calculating radar cross section (RCS) of such a NIM slab are finally presented to demonstrate the accuracy and efficiency of this technique. DOI: 10.2529/PIERS061003050010 NIM, which is also known as left-handed material (LHM) [1], presents dielectric constants (permeability and permittivity) simultaneously negative. Typical NIMs, such as SRR [2], are composed of dielectric body and conducting patches. Base on this feature, to investigate the scattering problem of such NIMs, we can employ volume EFIE to the dielectric region and surface EFIE on the conducting surface [3]. Previous researchers often use MoM [4] to discretize the integral equations. And AIM [5, 6, 3] has already been proved to be an efficient solver in reduce the memory requirement for storage and to speed up the matrix-vector multiplication in the iterative solver. For an SRR structure, in the dielectric region V , by taking the scattered field from both the surface current and volume current into consideration, the total electric field becomes: E(r) = E(r) + E V (r) + E sca S (r) (1) Since the tangential components of total electric field vanishes on conducting surface, we get: n̂× E(r) = −n̂× E V (r)− n̂× E S (r) (2) Equations (1) and (2) are known as the EFIE as the formulations involve only electric field. EFIE is suitable for open conducting surface. Inside the dielectric region V and on the surface of conducting body S, the incident wave induces volume current JV and surface current JS . The induced volume and surface currents will generate scattered EM field as following: E Ω (r) = −jk0η0AΩ(r)−∇ΦΩ(r), Ω = S or V (3) where the magnetic vector potential is defined as: