Mixed-potential Formulation Research Articles

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.

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.

Multiscale Modeling of Electromagnetic Telemetry in Layered Transverse Isotropic Formation

Electromagnetic telemetry (EMT) systems are utilized for measurement while drilling to transmit data from borehole to the ground surface during drilling or vice versa. To help predict the successful deployment of the EMT system within complex formation and environmental noise, a reliable numerical algorithm is developed and applied to model the EMT system for directional drilling in layered transverse isotropic (TI) formation. In the telemetry system, the long and thin drill string having three-dimensional trajectory acts as an underground antenna that can be modeled as a thin wire excited by a delta gap voltage/current source near the drill bit. Using the thin wire assumption, the developed algorithm calculates the thin wire kernel in homogeneous uniaxial media with high accuracy to reduce the dimension and complexity of the problem. The mixed-potential form of layered media Green's function and electric field integral equation are used to model the electromagnetic interaction of the drill string and layered TI media. The effect of drilling fluid outside the pipe is modeled using the volumetric equivalence principle. The accuracy and efficiency of the new algorithm are demonstrated by several numerical examples.

A Low-Frequency EFIE-MLFMA Solver Based on Approximate Diagonalization of Green’s Function

The low-frequency fast multipole algorithm (LFFMA) based on the approximate diagonalization of Green’s function for the electric field integral equation (EFIE) is proposed for open and complex structures with high accuracy. Moreover, the multiresolution preconditioner is employed to the mixed potential form of EFIE with RWG basis functions for stability. Herein, a triangle-grouping scheme is adopted for the scalar potential so that the accuracy of the LFFMA can be guaranteed. Furthermore, the computational resources can be reduced largely since part of the radiation pattern of vector potential can share one of the scalar potentials by taking advantage of some mathematical deductions.

Exact Relationship Between the Locally Corrected Nyström Scheme and RWG Moment Method for the Mixed-Potential Integral Equation

The exact relationship between the Rao–Wilton–Glisson (RWG) method of moments (MoM) and the locally corrected Nystrom (LCN) method for the mixed-potential (MP) electric field integral equation (EFIE) is presented as an extension to our work where we established analogous exact relationship for solving the EFIE in its vector-potential (VP) form. It is shown that in order to achieve one such relationship for the MP EFIE, the first- and zeroth-order LCN methods must be, respectively, used for the discretization of the VP and scalar-potential terms of the MP EFIE. The resulting numerical scheme is a point-based RWG MoM discretization of the MP EFIE via the Nystrom method. Due to the MP formulation of the EFIE, the proposed method establishes notably higher accuracy compared to either RWG MoM or LCN discretizations of the EFIE in the VP form. The increased accuracy is attributed to the analytical cancellation of the line charge contributions in the MP formulation as opposed to numerical cancellation inherent in the VP formulation of the EFIE. The detailed study and explanations of the above cancellations is presented along with their impact on the accuracy of the respective schemes for both canonical and realistic scattering targets at different frequencies.

Efficient Analysis of Printed Structures of Arbitrary Shape on Coated Cylinders Via Spatial-Domain Mixed-Potential Green's Function

We present an approach for the method of moments (MoM) analysis of antennas of arbitrary shape printed on a cylindrical substrate. We propose several key components of a code to perform this analysis, with the explicit goal of adapting MoM codes devised for planar stratified structures. For that purpose, the background medium Green's function is expressed in the mixed-potential (MP) format to allow for efficient numerical integration; the spectral dyadic Green's function is first recast in the spectral counterpart of the spatial mixed-potential form. The convergence of the spectral-to-spatial transform is accelerated by extraction of spectral asymptotic functions and exploitation of the azimuthal periodicity. Several important issues of implementation, numerical stability, and accuracy are addressed with a view to incorporate the spatial MP Green's function into the MoM analysis of printed structures of arbitrary shape using RWG basis functions. The numerical results are in good agreement with those obtained by other methods and with measured data.

COMPREHENSIVE SOLUTION TO SCATTERING BY BIANOSOTROPIC OBJECTS OF ARBITRARY SHAPE

This paper presents a method of moments (MoM) solution for the problems of electromagnetic scattering by inhomogeneous three- dimensional bianisotropic scatterers of any shape. The electromagnetic response of bianisotropy has been described by the constitutive relations of the most general form composed of four 3 £ 3 matrices or tensors. The volume equivalence principle is used to obtain a set of mixed potential formulations for a proper description of the original scattering problem. Here, the total flelds are separated into the incident flelds and the scattered flelds. The scattered flelds are related to the electric and magnetic potentials which are excited by electric and magnetic bound charges and polarization currents. The body of the scatterer is meshed through the use of tetrahedral cells with face-based functions used to expand unknown quantities. At last, the Galerkin test method is applied to create a method of moments (MoM) matrix from which the numerical solution is obtained. Implemented in a MATLAB program, the numerical formulation is evaluated and verifled for various types of scatterers. The results are compared with those of previous work, and a good agreement is observed. Finally, a scattering from a two-layered dispersive chiroferrite sphere is presented as the most general example.

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.

Green's Function Calculation for a Line Source Exciting a 2-D Periodic Printed Structure

The electromagnetic field excited by an electric line source in the presence of a printed structure periodic along two directions with arbitrary geometry of the metallization is calculated. The Array Scanning Method (ASM) is adopted to express the field excited by the aperiodic source in terms of an integral superposition of suitable auxiliary Floquet-periodic fields. The latter are solutions of electric-field integral equations that are cast in a mixed-potential formulation in the unit cell and are numerically discretized with an efficient implementation of the method of moments in the spatial domain. The accuracy of the proposed approach is validated through an independent ASM code developed for metal-strip gratings and also against consolidated analytical models available for specific strip- and patch-based periodic structures.

Pseudo-gradient and Lagrangian boundary control system formulation of electromagnetic fields

This paper describes an electromagnetic field analogue of the classical Brayton–Moser formulation. It is shown that Maxwell's curl equations constitute a pseudo-gradient system with respect to a single electromagnetic mixed-potential functional and a metric defined by the constitutive relations of the fields. Besides its use for the generation of power-based Lyapunov functionals for stability analysis and Poynting-like flow balances, the electromagnetic mixed-potential formulation suggests a family of alternative variational principles. This yields a novel Lagrangian boundary control system formulation admitting nonzero energy flow through the boundary. The corresponding symplectic Hamiltonian system is still associated with the total electromagnetic field energy.

Electromagnetic Scattering from an Arbitrarily Shaped Three-dimensional Inhomogeneous Bianisotropic Body

This paper presents a numerical solution to scattering problems that involve 3D inhomogeneous bianisotropic scatterers of an arbitrary shape. The constitutive relations are assumed to be of the most general form and composed of four 3 £ 3 matrices or tensors. The problem is described through a mixed potential formulation. The electric and magnetic potentials are related to electric and magnetic bound charges and polarization currents and then to the electric and magnetic polarizations. The electric fleld and magnetic fleld integral equations are constructed. The method of moments technique is then applied to obtain a numerical solution to the problem. The volume of the scatterer is meshed by tetrahedral cells and face-based functions are applied to expand unknown quantities. The proposed formulation has been evaluated and verifled through examples of scattering by various chiral and gyrotropic scatterers illuminated by a plane electromagnetic wave. Numerical results of scattering from a chiroferrite sphere are also presented showing the ∞exibility of the proposed method.

A high resolution wave propagation scheme for ideal Two-Fluid plasma equations

Algorithms for the solution of the five-moment ideal Two-Fluid equations are presented. The ideal Two-Fluid model is more general than the often used magnetohydrodynamic (MHD) model. The model takes into account electron inertia effects, charge separation and the full electromagnetic field equations and allows for separate electron and ion motion. The algorithm presented is the high resolution wave propagation method. The wave propagation method is based on solutions to the Riemann problem at cell interfaces. Operator splitting is used to incorporate the Lorentz and electromagnetic source terms. To preserve the divergence constraints on the electric and magnetic fields two different approaches are used. In the first approach Maxwell equations are rewritten in their mixed-potential form. In the second approach the so-called perfectly hyperbolic form of Maxwell equations are used which explicitly incorporate the divergence equations into the time stepping scheme. The algorithm is applied to a one-dimensional Riemann problem, ion-acoustic soliton propagation and magnetic reconnection. In each case Two-Fluid physics described by the ideal Two-Fluid model is highlighted.

Review of singular potential integrals for method of moments solutions of surface integral equations

Abstract. Accurate evaluation of singular potential integrals is essential for successful method of moments (MoM) solutions of surface integral equations. In mixed potential formulations for metallic and dielectric scatterers, kernels with 1/R and r1/R singularities must be considered. Several techniques for the treatment of these singularities will be reviewed. The most common approach solves the MoM source integrals analytically for specific observation points, thus regularizing the integral. However, in the case of r1/R a logarithmic singularity remains for which numerical evaluation of the testing integral is still difficult. A recently by Yl¨a-Oijala and Taskinen proposed remedy to this issue is discussed and evaluated within a hybrid finite element – boundary integral technique. Convergence results for the MoM coupling integrals are presented where also higher-order singularity extraction is considered.

Semantics of dyadic and mixed potential field representation for 3-D current distributions in planar stratified media

The full spectral electric dyadic Green's function for three dimensional current distributions in planar stratified media can be obtained straight from Maxwell's equations. By following a physical reasoning analogous with the free space case but using general derivative relations for multilayered Green's functions, we derive a "basic" mixed potential form with a simple vector potential kernel but multiple scalar potential kernels, and also obtain the well established single scalar potential formulations with a dyadic vector potential kernel. Mixed potential forms are thus arrived at without the a priori introduction of scalar and vector potential, or choice of gauge condition. The nonuniqueness of the scalar potential kernel and the dyadic nature of the scalar and/or vector potentials are believed to be clarified by the proposed approach. A discussion of the different formulations focuses on physical meaning and numerical consequences for the solution of integral equations.

Electromagnetic scattering from an arbitrarily shaped three-dimensional homogeneous chiral body

The method of moments technique for analyzing electromagnetic scattering from an arbitrarily shaped three-dimensional homogeneous chiral body is presented based on the combined field integral equations. The body is assumed to be illuminated by a plane wave. The surface equivalence principle is used to replace the body by equivalent electric and magnetic surface currents. These currents radiating in unbounded free space produce the correct scattered field outside. The negatives of these currents produce the correct total internal field, when radiating in an unbounded chiral medium. By enforcing the continuity of the tangential components of the total electric and magnetic fields on the surface of the body, a set of coupled integral equations is obtained for the equivalent surface currents. The surface of the body is modeled using triangular patches. The triangular rooftop vector expansion functions are used for both equivalent surface currents. The coefficients of these expansion functions are obtained using the method of moments. The mixed potential formulation for a chiral medium is developed and used to obtain explicit expressions for the electric and magnetic fields produced by surface currents. Numerical results for bistatic radar cross sections are presented for three chiral scatterers - a sphere, a finite circular cylinder, and a cube.

Comparison of surface electric field and impedance matrix elements of layered media between electric field spatial Green's function and mixed potential

AbstractRecently an approach based on half‐space extraction technique was used to efficiently evaluate the electric field spatial Green's function on a surface. This Letter compares the results of this method with the mixed‐potential formulation in spatial domain. Both surface electric fields and impedance matrix elements for layered media are compared for source and observation‐point distances ranging from 0.1 free‐space wavelengths to 10 free‐space wavelengths. The results show two orders of magnitude improvement on the speed of convergence for Sommerfeld integration domain and better numerical stability. © 2002 Wiley Periodicals, Inc. Microwave Opt Technol Lett 34: 307–313, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.10445

Electromagnetic scattering by inhomogeneous dielectric bodies of revolution embedded within stratified media

A method of moments (MoM) solution for scattering by heterogeneous bodies of revolution (BOR) embedded within a multilayered environment is given. A modal volume integral equation (VIE) is formulated in the mixed potential form and solved with the use of the specialized basis functions.

Singularity treatment and high‐order RWG basis functions for integral equations of electromagnetic scattering

AbstractWe present numerical implementation of high order RWG basis functions for electromagnetic scattering for curved conductor surfaces with a procedure for treating the singularities of dyadic Green's functions in the mixed potential formulation of electromagnetic scattering. Copyright © 2001 John Wiley &amp; Sons, Ltd.

Multilayered Media Green's Functions for Mpie With General Electric and Magnetic Sources By the Hertz Potential Approach - Abstract

A complete set of three dimensional multilayered media Green's functions is presented for general electric and magnetic sources. The Green's functions are derived in the mixed potential form, which is identical with the Michalski-Zheng C-formulation. The approach applied in this paper is based on the classical Hertz potential representation. A special emphasis is on the formulation of the dyadic Green's functions GHJ and GEM. In these functions the derivatives due to the curl operator are taken in the spectral domain. This avoids the need of the numerical differentiation. Furthermore, it is found that the Hertzian potentials satisfy several useful duality and reciprocity relations. By these relations the computational efficiency of the Hertz potential approach can be significantly improved and the number of required Sommerfeld integrals can be essentially reduced. We show that all spectral domain Green's functions can be obtained from only two spectral domain Hertzian potentials, which correspond to the TE component of a vertical magnetic dipole and the TM component of a vertical electric dipole. The derived formulas are verified by numerical examples.