Surface-Volume-Surface Electric Field Integral Equation Research Articles

Analytic Solution of Surface–Volume–Surface Electric Field Integral Equation on Dielectric Sphere and Analysis of Its Spectral Properties

The exact solution of the surface–volume–surface electric field integral equation (SVS-EFIE) is presented for the problem of radiation in the vicinity of the homogeneous dielectric sphere. The solution is obtained via the Galerkin method of moments (MoM) utilizing rotational and irrotational complete sets of orthogonal vector spherical harmonics as basis and test functions according to the Helmholtz decomposition. In the case of radial or tangential electric dipole radiation, the electric field throughout the sphere evaluated via the analytic MoM solution of the SVS-EFIE is compared against the exact classical Mie series solution. The two are shown to agree to 12 digits of accuracy upon a sufficient number of basis/test functions taken in the MoM solution and the Mie series expansion. The exact solution confirms and validates the rigorous nature of the SVS-EFIE formulation. It also reveals the spectral properties of its individual operators, their products, and their linear combination, as well as the spectrum of the MoM impedance matrix. It is shown that, upon choosing basis and test functions in the Sobolev space <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{div}^{1/2}(S)$ </tex-math></inline-formula> and performing testing inner products evaluation in the space <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{div}^{-1/2}(S)$ </tex-math></inline-formula> , with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S$ </tex-math></inline-formula> being the surface of the sphere, the MoM impedance matrix features bounded condition number with increasing order of discretization similar with analogous exact MoM solution of the surface EFIE (S-EFIE) on the perfectly electrically conducting (PEC) sphere.

New Magnetic Current Based Surface-Volume-Surface Electric Field Integral Equation For Solution of Scattering Problems on Dielectric Objects

New Surface-Volume-Surface Electric Field Integral Equation (SVS-EFIE) based on magnetic current single source field representation (SVS-EFIE-M) is proposed for the problems of scattering on homogeneous non-magnetic dielectric objects. It's analytic MoM solution for a dielectric sphere is compared against the Mie series expansion to validate the exact nature of this new magnetic current based SVS-EFIE formulation. This analytic solution also allows to rigorously study spectral properties of the SVS-EFIE- M operators and to determine proper functional spaces in which MoM solutions of it should be sought in order to maintain bounded condition number of the pertinent matrix equation.

Surface–Volume–Surface EFIE Formulation for Fast Direct Solution of Scattering Problems on General 3-D Composite Metal–Dielectric Objects

A new formulation of the surface–volume–surface electric field integral equation (SVS-EFIE) combined with the regular EFIE is introduced for the electromagnetic analysis of the 3-D general composite structures of impenetrable metal and penetrable homogeneous dielectric objects. The proposed method introduces independent surface electric current densities on the boundary of each region which brings the benefit of the independent discretization of the regions according to their respective material properties and straightforward handling of the material junctions. This makes the algorithm more efficient and flexible for analysis of multiscale and large-scale composite structures in comparison with the traditional Poggio–Miller–Chang–Harrington–Wu–Tsai (PMCHWT) integral equation combined with the EFIE. The method of moment (MoM) solution of the new formulation is accelerated by the hierarchical ( $\mathcal H$ )-matrix framework. The numerical results are given to validate the new formulation. Also, the $\mathcal H$ -matrix acceleration of MoM solution of the new combined EFIE and SVS-EFIE formulation is shown to produce accurate results using only the small fraction of memory and total CPU time needed by conventional MoM.

$\mathcal {H}$-Matrix Accelerated Solution of Surface–Volume–Surface EFIE for Fast Electromagnetic Analysis on 3-D Composite Dielectric Objects

An efficient fast direct algorithm based on the hierarchical ( $\mathcal {H}$ -) matrices is presented for solution of the radiation problems on piecewise homogeneous dielectric objects using Method of Moment (MoM) discretization of the surface–volume–surface electric field integral equation (SVS-EFIE). The SVS-EFIE for the composite objects introduces independent surface electric current density on the boundary of each region. Therefore, different from the traditional Poggio–Miller–Chang–Harrington–Wu–Tsai formulation, in the SVS-EFIE, the object regions can be meshing independently according to their local properties which improves the flexibility and efficiency of the proposed method. It also makes the proposed algorithms appropriate for the analysis of both multiscale and large-scale composite structures. The numerical results from the proposed fast method are provided for the high-loss biological tissues from bioelectromagnetics applications and agree well with the analytical Mie series solution and commercial software. The CPU time and memory cost of the required $\mathcal {H}$ -matrix operations are analyzed in details and verified through several numerical experiments. The new computational framework allows for fast direct solution of 3-D radiation and scattering problems of moderate electrical size with $O(P^{\alpha } \log ^2 P)$ CPU time and $O(P^{\alpha } \log P)$ memory complexity, $P$ being the number of surface unknowns produced by the MoM discretization, and $1\leq \alpha \leq 1.5$ being a geometry-dependent parameter.

Surface-Volume-Surface EFIE for Electromagnetic Analysis of 3-D Composite Dielectric Objects in Multilayered Media

The surface-volume-surface electric field integral equation (SVS-EFIE) is generalized for the case of scattering problems on the composite nonmagnetic dielectric objects situated in planar nonmagnetic layered medium. The piece-wise homogeneous regions of the scatterer can be arbitrarily positioned with respect to the layers of stratification. The SVS-EFIE being a class of single-source integral equations is formed by restricting the surface single-source electric field representation in each distinct region of the scatterer through the volume-EFIE (V-EFIE) enforced on the boundary of that region for only the tangential component of the total field. As a result, the SVS-EFIE utilizes only the electric field dyadic Green's functions. This allows for its cast into the mixed-potential form using classical Michalski-Zheng's formulation and method of moments (MoM) discretization featuring easily computable integrals with singularities no stronger than 1/R, R being the distance from the source to the observation point in such integrals. The matrices of MoM discretization are represented inhierarchical form (as H-matrices) enabling solution of the scattering problems in multilayered media with O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α</sup> log N) CPUtimeand memory complexities, where α is a geometry-dependent constant ranging from 1 to 1.5 depending on the shape of the scatterer. While the MoM surface and volume meshes discretizing the regions of the scatterer are constructed to ensure that no mesh element crosses interfaces between the layers, the clusters of both the surface and volume elements in their respective recursive partitionings in the process of H-matrix construction are allowed to span multiple layers of the medium. Upon computation of the layered medium Green's function kernels with the discrete complex image method allowing clusters of elements to cross dielectric interfaces between the layers is shown to preserve compressibility of the corresponding H-matrix blocks.

Surface-Volume-Surface Electric Field Integral Equation for Solution of Scattering Problems on 3-D Dielectric Objects in Multilayered Media

Generalization of the surface-volume-surface electric field integral equation (SVS-EFIE) for the solution of electromagnetic scattering problems on 3-D dielectric objects embedded in multilayered media is proposed. While having only a single unknown surface current density on the boundary of the scatterer, the SVS-EFIE also features only electric field dyadic Green’s functions in its integral operators. This property in conjunction with Michalski–Zheng’s formulation of the multilayered media electric field Green’s function allows for the formulation of SVS-EFIE in the mixed potential form for the solution of the scattering problems in layered media. In the proposed method of moments (MoM) discretization scheme, the gradient and divergence operators associated with the electric field Green’s function are shifted to the basis and test functions of the discretized integral operators. As a result, the proposed formulation features no derivatives acting on the components of the layered media dyadic Green’s function, hence, substantially alleviating the numerical evaluation of the pertinent Sommerfeld integrals. The proposed MoM formulation scalarizes reaction integrals containing the multilayered media dyadic Green’s function through the use shape function-based definition of the basis and test functions. The resultant MoM integrals feature no singularities stronger than $1/R$ . The validation of the proposed SVS-EFIE formulation and its MoM discretization is performed through a comparison of the computed fields against the fields produced using commercial electromagnetic analysis software.

Formulation of Surface-Volume-Surface-EFIE for Solution of Three-Dimensional Scattering Problems on Composite Dielectric Objects

A novel formulation of the surface-volume-surface electric field integral equation (SVS-EFIE) is introduced for the solution of the scattering and radiation problems on the composite dielectric objects made of arbitrary number of piecewise homogeneous nonmagnetic dielectric regions. The regions composing the scatterer may or may not share common boundaries. Due to the fact that SVS-EFIE introduces the independent fictitious surface electric current densities on the boundary of each region, the proposed formulation allows for independent meshing of the scatterer regions according to its permittivity. As a result, the new SVS-EFIE formulation presents no complications in method-of-moments discretization at the material junctions. The proposed formulation exhibits the same CPU time and the memory complexities as traditional surface integral equation formulations when fields throughout the scatterer are to be computed. The new equation is validated through comparison of its solution against the Mie series and fields computed using the commercial software.