Surface Electric Field Integral Equation Research Articles

Multi-level Power Series Solution for Large Surface and Volume Electric Field Integral Equation

In this paper, we propose a new multi-level power series solution method for solving a large surface and volume electric field integral equation-based H-Matrix. The proposed solution method converges in a fixed number of iterations and is solved at each level of the H-Matrix computation. The solution method avoids the computation of a full matrix, as it can be solved independently at each level, starting from the leaf level. Solution at each level can be used as the final solution, thus saving the matrix computation time for full H-Matrix. The paper shows that the leaf level matrix computation and solution with power series gives as accurate results as the full H-Matrix iterative solver method. The method results in considerable savings time and memory savings compared to the H-Matrix iterative solver. Further, the proposed method retains the O(NlogN) solution complexity.

Multiresolution Subspace-Based Optimization Method for the Retrieval of 2-D Perfect Electric Conductors

Perfect Electric Conductors (PECs) are imaged integrating the subspace-based optimizationmethod (SOM) within the iterative multi-scaling scheme (IMSA). Without a-priori information on the number or/and the locations of the scatterers and modelling their EM scattering interactions with a (known) probing source in terms of surface electric field integral equations, a segment-based representation of PECs is retrieved from the scattered field samples. The proposed IMSA-SOM inversion method is validated against both synthetic and experimental data by assessing the reconstruction accuracy, the robustness to the noise, and the computational efficiency with some comparisons, as well.

Improving the Solution Accuracy of the Volume-Surface Integral Equation for the Open PEC Structures Embedded Within Dielectrics

In the solution of volume-surface integral equation (VSIE) for open perfect electric conductor (PEC) structures embedded within dielectric substrates, how to explicitly enforce the continuity condition (CC) of electric flux on the PEC–dielectric interfaces to improve the numerical accuracy is investigated. In the traditional VSIE formed by combining the surface electric field integral equation (EFIE) and volume integral equation (VIE), the fictitious surface charge, which does not exist physically, is accumulated on the PEC–dielectric interfaces. This will lead to the inaccuracy of numerical results, which can be eliminated by the CC. To correctly enforce the CC, the equivalent surface current distribution on both sides of open PEC should be distinguished. For this purpose, the combined field integral equation (CFIE) consisting of electric field and magnetic field integral equations is used to model the open PEC part. In the proposed approach, the number of surface unknowns needs to be doubled, which can be balanced by reducing the volumetric unknowns through the enforcement of CC. Furthermore, by fully considering the numerical symmetry between both sides of open PEC surface, the filling process of the matrix as well as the excitation vector in implementing the method of moments (MoMs) can be simplified. Numerical experiment demonstrates that the enforcement of CC can improve the VSIE accuracy obviously. For example, when calculating the frequency response of frequency selective surfaces (FSSs), the results obtained from the traditional VSIE often shift to higher frequencies, while the proposed approach gives reasonably good results.

Adaptive Multilevel Nonuniform Grid Algorithm for the Accelerated Analysis of Composite Metallic–Dielectric Radomes

An adaptive multilevel nonuniform grid (MLNG) algorithm is developed for the accelerated computation of fields radiated through composite metallic–dielectric radomes as well as antenna-radome interactions. The MLNG approach is applied to the mixed potential formulation of the coupled surface and volume electric field integral equations. The radome is decomposed into a hierarchy of subdomains (SDs) by an adaptive algorithm that closely follows the radome geometry, allowing significant savings in memory and CPU time. In the MLNG algorithm, only local generalized impedance matrices of the finest level SDs are evaluated. Far-zone potentials and fields are indirectly evaluated through a multilevel aggregation involving phase-and amplitude-compensated interpolation on nonuniform grids (NGs), requiring considerably fewer calculations as compared with the classical method of moments (MoM). The MLNG algorithm is incorporated in a preconditioned iterative solver. Accuracy as well as memory consumption and computation times of the algorithm are studied on realistic examples. The radome effect on the antenna input impedance and electric current density distribution is demonstrated. The method is validated by comparison with a commercial MoM software and shown to exhibit a computational complexity (CC) of O(NlogN), N being the number of unknowns.

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.

Hybrid Surface Electric Field Volume Magnetic Field Integral Equations for Electromagnetic Analysis of Heterogeneous Dielectric Bodies With Embedded Electrically Conducting Structures

In this article, hybrid surface electric field volume magnetic field integral equations (SEVMIE) for electromagnetic analysis of heterogeneous dielectric bodies with embedded perfect electrical conductor (PEC) structures are presented. These are then solved using a quasi-exact vector moment method employing mixed order edge-based divergence conforming basis functions to expand the current on the PEC surface and node-based linear Lagrangian basis functions to expand the magnetic field within the dielectric region. For an incident plane wave scattered by the combined structure, this procedure results in a linear matrix equation for the surface currents and the volume magnetic fields. The method is demonstrated for three spherical test cases and the results compared to those obtained by a semi-analytic Mie series expansion. The performance is compared to that of the well-known volume surface integral equation (VSIE) method and found to provide more accurate results with approximately one-third the number of degrees of freedom. The method is then demonstrated on the scattering cross section calculation of a heterogeneous composite structure. This new method enables the efficient and accurate analysis of composite structures with large dielectric regions.

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.

New Higher Order Method of Moments for Accurate Inductance Extraction in Transmission Lines of Complex Cross Sections

A new higher order (HO) method of moments is proposed for high accuracy extraction of the resistance and inductance matrices in the multiconductor transmission lines (MTLs) of complex cross sections. The computational framework is based on the numerical solution of a surface–volume–surface electric field integral equation of magnetostatics. In order to achieve an exponentially efficient reduction in the error of the solution, HO geometrical representation of the conductor cross sections is accompanied with the discretization of the unknown field quantities on the conductor boundaries and cross sections with 1-D and 2-D HO polynomial basis functions, respectively. The methodology allows for extraction of the network parameters in broad ranges of frequencies for which resistive and inductive contributions to the impedance matrix vary within a wide dynamic range. Comparison of numerically computed currents to the currents obtained analytically for canonical transmission line configurations is performed. Solutions of the extraction problem for MTLs with complex cross sections are compared against the finite-element method solutions to demonstrate the efficiency of the proposed methodology.

Surface–Volume–Surface Electric Field Integral Equation for Magneto-Quasi-Static Analysis of Complex 3-D Interconnects

A novel single-source surface–volume–surface integral equation is proposed for accurate broadband resistance and inductance extraction in 3-D interconnects. The new equation originates in the volume integral equation (VIE) traditionally used for magneto-quasi-static modeling of current flow in 3-D wires. The latter is reduced to a surface integral equation by representing the electric field inside each conductor segment as a superposition of cylindrical waves emanating from the conductor's boundary. As no approximation is utilized and all underlying boundary conditions and pertinent equations are satisfied in the reduction, the new integral equation is rigorously equivalent to the solution of the traditional volume electric field integral equation. The rigorous nature of the proposed single-source surface integral equation is corroborated numerically. In this paper, a detailed description of the method of moments discretization for the proposed surface integral equation is also presented. Numerical solution of the proposed surface integral equation for a 12-conductor bond-wire package is used to demonstrate the accuracy of the method and its computational benefits compared to the traditional solution based on the VIE.

Quadrilateral Barycentric Basis Functions for Surface Integral Equations

A framework for solving surface integral equation on quadrilateral, triangular, and mixed quadrilateral-triangular meshes is presented. The initial meshes are represented in terms of quadrilateral barycentric meshes (QBMs), which are obtained by partitioning each initial quadrilateral and triangle into four and three barycentric quadrilaterals, respectively. Quadrilateral barycentric basis functions (QBBFs) are defined on QBMs. The QBBFs serve a dual role. First, they allow defining primary basis functions (PBFs), which are well suited for representing surface currents on quadrilateral, triangular, and mixed meshes. Second, the QBBFs are used to define dual basis function (DBFs), which are natural for using in conjunction with Calderon multiplicative preconditioners (CMPs). These QBBFs, PBFs, and DBFs result in a substantial reduction in the number of nonzero elements in the sparse projection matrices for PBFs and DBFs as well as reduction of unknowns and quadrature points. When these PBFs and DBFs are used in CMPs, they reduce the number of iterations and eliminate the dense mesh breakdown of the surface electric field integral equation. Numerical examples demonstrate the efficiency of using the introduced QBBFs with associated PBFs and DBFs for solving electromagnetic surface electric field integral equations on quadrilateral, triangular, and mixed meshes.

Computation Model of the Reradiation Interference Protecting Distance Between Radio Station and UHV Power Lines

A three-dimensional computation model of UHVDC has been designed, which is used to offer higher accuracy in the assessment of reradiation interference protecting distance between radio station and UHV power lines. The model is excited by vertically polarized plane wave, and then the interference of requested point is calculated. Based on the wire and surface electric field integral equations of interference from power line, the equivalent principle and the reasons for increased deviation with increasing frequency of wire model are analyzed. In order to assure the continuity of induced current on steel tower and embody the structure of angle-steel effectively, three-dimensional surface model of UHVDC steel tower with and without auxiliary angle-steel based on triangular element are presented. With the RWG base function and Galerkin's method, Method of Moments is applied to calculate the interference of surface model from steel tower, and the results are compared with that of wire model. The results show the values drawn from wire model and surface model have the same change trend, and the error gradually increases with the increase of frequency. If 0.1 dB is selected as the allowable deviation value, the surface model should be used instead of wire model when frequency is over 16.7 MHz.

A Novel Skin-Effect Based Surface Impedance Formulation for Broadband Modeling of 3-D Interconnects With Electric Field Integral Equation

The problem of interconnect modeling embedded in multilayered substrate is initially formulated in terms of the volume integral equation (IE) with respect to 3-D conduction current density. One out of three degrees of freedom in volumetric current variation is then eliminated by approximating the current behavior over the coordinate normal to the conductor surface according to the skin-effect. The remaining two degrees of freedom in the volumetric current variation constitute the unknown current distribution on the conductor surface for which a governing surface electric field integral equation is obtained directly from the volume IE via restriction of the volumetric operator's range to the conductor surface. The resultant surface IE features a global to the conductor cross section surface impedance operator, which is shown to approximate the relationship between tangential electric and magnetic field components on the conductor surface. The proposed novel surface IE featuring multilayered media dyadic Green's function is amenable to various discretization schemes including the Rao-Wilton-Glisson method of moments. In this paper the latter is implemented by casting the scattered field operator into Michalski-Zheng's mixed-potential form in conjunction with enforcement of global relationships between the basis and testing functions in conductor cross sections according to the surface impedance operator. Numerical comparisons to alternative conductor loss models show that the method achieves volumetric solution accuracy within the framework of a boundary-element formulation.

A Calderón Multiplicative Preconditioner for Coupled Surface-Volume Electric Field Integral Equations

A well-conditioned coupled set of surface (S) and volume (V) electric field integral equations (S-EFIE and V-EFIE) for analyzing wave interactions with densely discretized composite structures is presented. Whereas the V-EFIE operator is well-posed even when applied to densely discretized volumes, a classically formulated S-EFIE operator is ill-posed when applied to densely discretized surfaces. This renders the discretized coupled S-EFIE and V-EFIE system ill-conditioned, and its iterative solution inefficient or even impossible. The proposed scheme regularizes the coupled set of S-EFIE and V-EFIE using a Calderón multiplicative preconditioner (CMP)-based technique. The resulting scheme enables the efficient analysis of electromagnetic interactions with composite structures containing fine/subwavelength geometric features. Numerical examples demonstrate the efficiency of the proposed scheme.

Investigation of low‐frequency behaviour of two surface integral full‐Maxwell formulations

PurposeTo apply two different integral formulations of full‐Maxwell's equations to the numerical study of interconnects in a low‐frequency range and compare the results.Design/methodology/approachThe first approach consists of a surface formulation of the full‐Maxwell's equations in terms of potentials, giving rise to a surface electric field integral equation. The equation, given in a weak form, is solved by using a finite element technique. The solenoidal and non‐solenoidal components of the electric current density are separated using the null‐pinv decomposition to avoid the low‐frequency breakdown. The second model is an extension of partial element equivalent circuit technique to unstructured meshes allowing the use of triangular meshes. Two systems of meshes tied by duality relations are defined on multiconductor systems. The key point in the definition of the equivalent network is to associate the pair primal edge/dual face to a circuit branch. Solution of the resulting electrical network is performed by a modified nodal analysis method and regularization of the outcoming matrix is accomplished by standard techniques based on the addition of suitable resistors.FindingsBoth the formulation have a regular behaviour at very low frequency. This is automatically achieved in the first approach by using the null‐pinv decomposition.Research limitations/implicationsSurface sources of fields.Originality/valueTwo different integral formulations of full‐Maxwell's equations for the numerical study of interconnects are compared in terms of low‐frequency behaviour.

A Coupled PEC-TDS Surface Integral Equation Approach for Electromagnetic Scattering and Radiation From Composite Metallic and Thin Dielectric Objects

The recently proposed surface electric field integral equation for thin dielectric sheets is combined with that for conductors to handle objects consisting of both dielectric and conducting materials. This leads to coupled conducting-dielectric surface integral equations which can easily be solved by the method of moments. This approach can handle both scattering and radiation problems such as coated objects, radome enclosed antennas and scatterers, and planar structures. It is observed that the normal electric field in a thin dielectric layer is indispensable when this dielectric layer resides in the vicinity of a conducting object. To validate this approach, couple numerical examples are examined. The results by this approach show agreement with other methods while it economizes on computation resource. It can be used to obtain rapid solutions in the initial phase of the design process. More importantly, it indicates the feasibility for microwave planar circuit applications

A parallel FFT accelerated transient field-circuit simulator

A novel fast electromagnetic field-circuit simulator that permits the full-wave modeling of transients in nonlinear microwave circuits is proposed. This time-domain simulator is composed of two components: 1) a full-wave solver that models interactions of electromagnetic fields with conducting surfaces and finite dielectric volumes by solving time-domain surface and volume electric field integral equations, respectively, and 2) a circuit solver that models field interactions with lumped circuits, which are potentially active and nonlinear, by solving Kirchoff's equations through modified nodal analysis. These field and circuit analysis components are consistently interfaced and the resulting coupled set of nonlinear equations is evolved in time by a multidimensional Newton-Raphson scheme. The solution procedure is accelerated by allocating field- and circuit-related computations across the processors of a distributed-memory cluster, which communicate using the message-passing interface standard. Furthermore, the electromagnetic field solver, whose demand for computational resources far outpaces that of the circuit solver, is accelerated by a fast Fourier transform (FFT)-based algorithm, viz. the time-domain adaptive integral method. The resulting parallel FFT accelerated transient field-circuit simulator is applied to the analysis of various active and nonlinear microwave circuits, including power-combining arrays.

A fast volume-surface integral equation solver for scattering from composite conducting-dielectric objects

This paper presents a fast hybrid volume-surface integral equation approach for the computation of electromagnetic scattering from objects comprising both conductors and dielectric materials. The volume electric field integral equation is applied to the material region and the surface electric field integral equation is applied on the conducting surface. The method of moments (MoM) is used to convert the integral equation into a matrix equation and the precorrected-FFT (P-FFT) method is employed to reduce the memory requirement and CPU time for the matrix solution. The present approach is sufficiently versatile in handling problems with either open or closed conductors, and dielectric materials of arbitrary inhomogeneity, due to the combination of the surface and volume electric field integral equations. The application of the precorrected-FFT method facilitates the solving of much larger problems than can be handled by the conventional MoM.

A stability analysis of a time marching scheme for the general surface electric field integral equation

We present a stability analysis of a mixed finite element approximation of the time dependent electric field integral equation which is used to compute the current induced on a perfect conductor by a transient electric field. The analysis shows that the solution has fast growing unstable modes that are due to a combination of the finite element basis functions and the integral quadrature rule. These modes cannot be removed by standard time averaging methods, but they can be eliminated by averaging in space appropriately.