Method Of Moments Formulation Research Articles

Solution of volume‐surface integral equations using higher‐order hierarchical Legendre basis functions

The problem of electromagnetic scattering by composite metallic and dielectric objects is solved using the coupled volume‐surface integral equation (VSIE). The method of moments (MoM) based on higher‐order hierarchical Legendre basis functions and higher‐order curvilinear geometrical elements is applied to transform the VSIE into a system of linear equations. The higher‐order MoM provides significant reduction in the number of unknowns in comparison with standard MoM formulations using low‐order basis functions, such as RWG functions. Owing to the orthogonal nature of the higher‐order Legendre basis functions, the continuity condition at the interface between metal and dielectric can be satisfied explicitly, which further reduces the number of unknowns as well as improves the accuracy of the solution. Numerical results for a metallic sphere with dielectric coating show excellent agreement with the analytical Mie series solution. Scattering by more complex metal‐dielectric objects is also considered to compare the presented technique with other numerical methods.

Accurate evaluation of singular potential integrals in an asymptotic‐phase method of moments formulation

AbstractThe Khayat‐Wilton singularity cancellation technique has been adapted for computing the potential integrals involving the 3D Green's function and the linearly‐phased Rao‐Wilton‐Glisson (LP‐RWG) basis functions. The LP‐RWG basis functions are used to expand the induced currents in terms of current‐modes, which allows an efficient representation of the currents in large scatters using a fewer number of unknowns. The proposed procedure overcomes some drawbacks of other singularity schemes, such as the dependence of the triangular‐patch shape and the location of the observation points. In this sense, the proposed procedure is valid for singularities and “near” singularities on arbitrarily‐shaped triangular meshes. This significantly improves the reliability of the method of moments and related approaches in combination with the asymptotic phase formulation when dealing with arbitrary and automatically generated surface meshes. © 2007 Wiley Periodicals, Inc. Microwave Opt Technol Lett 49: 2189–2197, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.22680

Improved Solution of Tensor-Volume Integral Equation Using Mixed-Domain MoM with Polynomial Expansion

A new approach for solution of the Tensor-Volume Integral Equation (TVIE) using Galerkin-based moment method (MoM) for three-dimensional dielectric bodies is proposed. Two problems of plane wave scattering by a dielectric sphere and a thin-wire antenna in close proximity to a dielectric body are investigated. In both cases, cubic modeling is applied and a combination of entire-domain and sub-domain basis functions, including three-dimensional polynomial functions with different degrees is utilized for field expansion inside dielectric bodies. Power polynomial is adopted for this purpose and its property is discussed over the proposed mixed-domain MoM formulation. Numerical examples show that based on the proposed method, a relative fast algorithm and suitable accuracy are achieved compared with conventional MoM. The accuracy of the proposed method is verified by comparing it with the Mie theory, conventional MoM and the FDTD method.

A POSSIBLE REMEDY FOR THE OSCILLATIONS OCCURRING IN THIN-WIRE MoM ANALYSIS OF CYLINDRICAL ANTENNAS

Approximate, non-singular kernels are often used in moment-method formulations coping with thin-wire structures. Their use has important consequences, one of which is the appearance of oscillations in the computed currents when the number of sub-domain basis functions is sufficiently large. These oscillations are not due to round-off errors. In this paper, a smoothing procedure is used in conjunction with Galerkin's formulation with piecewise sinusoidal functions, which yields non-oscillating current distributions. Special attention is paid to the solutions over a wide range of discretization levels (number of basis/testing functions), in order to examine and illustrate the similarities and differences between results obtained with and without the proposed remedy. Finally, a comparison with results derived with the exact kernel is provided.

Analysing radio-frequency coil arrays in high-field magnetic resonance imaging by the combined field integral equation method

We present the combined field integral equation (CFIE) method for analysing radio-frequency coil arrays in high-field magnetic resonance imaging (MRI). Three-dimensional models of coils and the human body were used to take into account the electromagnetic coupling. In the method of moments formulation, we applied triangular patches and the Rao–Wilton–Glisson basis functions to model arbitrarily shaped geometries. We first examined a rectangular loop coil to verify the CFIE method and also demonstrate its efficiency and accuracy. We then studied several eight-channel receive-only head coil arrays for 7.0 T SENSE functional MRI. Numerical results show that the signal dropout and the average SNR are two major concerns in SENSE coil array design. A good design should be a balance of these two factors.

Numerical Analysis of Coupling Between Dielectric Image Guide and Microstrip

An integral equation method of moments formulation of the coupling between dielectric image guide and microstrip is studied. The formulation implements a surface equivalence using rooftop and traveling wave basis functions. The specific structure under consideration is the aperture coupled microstrip image guide transition. Analysis of the structure and typical results are compared to measurements on a simple prototype transition.

Efficient full-wave modeling of patch antenna arrays with new single-layer capacitive feed probes

This paper introduces a new variation on the capacitive feed probe for patch antennas on thick substrates. It consists of a small circular probe-fed capacitor patch that is situated next to the resonant patch. This configuration can bring about significant savings in terms of manufacturing cost, but also lends itself to a very efficient full-wave analysis. As such, the main focus of this paper is a spectral-domain moment-method formulation, which was specifically developed for the analysis of large, but finite, arrays of these antenna elements. Entire-domain and subdomain basis functions are combined in an efficient way to minimize the computational requirements, most notably computer memory. It is shown that, for general antenna array configurations, memory savings of more than 1000 times can be achieved when compared with typical commercial software packages where only subdomain basis functions are used. A number of numerical and experimental results are also included in order to verify the spectral-domain moment-method formulation and to illustrate various applications of the new antenna element.

Fast Multipole Acceleration of a MoM Code for the Solution of Composed Metallic/Dielectric Scattering Problems

Abstract. An existing method of moments (MoM) code for the solution of complex scattering bodies has been accelerated by means of a multilevel fast multipole method (MLFMM). We demonstrate the usage of this technique both for metallic structures (wires and surfaces) and for dielectric bodies (volume and surface equivalence principle). Aspects like the effect of the type of integral equation, preconditioning schemes, or iterative solution techniques are discussed. But also limitations are addressed, which are encountered when for instance attempting to model highly lossy dielectric bodies with a high permittivity. Several validation and application examples demonstrate the usefulness of this method, both with regard to the obtained accuracy, but also with respect to the potential saving in memory and run-time as compared to a standard MoM formulation.

Combining the multilevel fast multipole method with the uniform geometrical theory of diffraction

Abstract. The presence of arbitrarily shaped and electrically large objects in the same environment leads to hybridization of the Method of Moments (MoM) with the Uniform Geometrical Theory of Diffraction (UTD). The computation and memory complexity of the MoM solution is improved with the Multilevel Fast Multipole Method (MLFMM). By expanding the k-space integrals in spherical harmonics, further considerable amount of memory can be saved without compromising accuracy and numerical speed. However, until now MoM-UTD hybrid methods are restricted to conventional MoM formulations only with Electric Field Integral Equation (EFIE). In this contribution, a MLFMM-UTD hybridization for Combined Field Integral Equation (CFIE) is proposed and applied within a hybrid Finite Element - Boundary Integral (FEBI) technique. The MLFMM-UTD hybridization is performed at the translation procedure on the various levels of the MLFMM, using a far-field approximation of the corresponding translation operator. The formulation of this new hybrid technique is presented, as well as numerical results.

Fundamental properties of the field at the interface between air and a periodic artificial material excited by a line source

An efficient algorithm based on a moment-method formulation is presented for the evaluation of the field produced by a line source at the interface between an air superstrate and a one-dimensional-periodic artificial-material slab. The formulation provides physical insight into the nature of the fields via path deformation in the complex wavenumber plane. From an asymptotic analysis in the complex wavenumber plane it is found that the space wave produced by a line source consists of an infinite number of space harmonics that decay algebraically as x/sup -3/2/. Guided modes may also exist and be excited, including leaky modes.

Incorporation of linear-phase progression in RWG basis functions

A linear-phase term is incorporated into the triangular-type Rao–Wilton–Glisson (RWG) vector basis functions in the method of moments (MoM) formulation. The objective consists of including the dominant phase variation of the current density within the basis-functions formulation, thus allowing an efficient representation of the induced current on large bodies using fewer unknowns. Assuming a good estimation of the phase variation, the new MoM formulation with linearly-phased RWG basis functions is shown to greatly relieve the storage and solution time of the conventional MoM with RWG basis functions, while accurately reproducing the scattered fields of some chosen problems. © 2004 Wiley Periodicals, Inc. Microwave Opt Technol Lett 44: 106–112, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20560

Full-Wave Spectral Analysis and Design of Annular Patch Antenna With Electromagnetically Coupled Microstrip Feed Line

This paper presents a full-wave analysis, and the ensuing design considerations, of the annular patch antenna fed by an electromagnetically coupled microstrip line. The numerical analysis employs the spectral domain version of the integral equation, with the method of moments formulation; the basis functions on the ring patch are the eigenmodes of the corresponding magnetic wall cavity, and subsectional (rooftop) functions are employed on the access line. Both TM and TE modes are used, along with more-efficient "collective" solenoidal modes; mode sorting criteria are given to ensure uniform convergence. Efficient schemes are presented for the calculation of the spectral-domain reaction integrals, along with criteria to avoid calculation of negligible mode-to-mode couplings, based on the spectral analysis of the matrix condition number. Very good agreement is shown with measured data, and design considerations are given for matching the radiator.

Enhancement of the Numerical Stability of the Adaptive Integral Method at Low Frequencies Through a Loop-Charge Formulation of the Method-of-Moments Approximation

The performance of integral-equation-based fast iterative solution schemes such as the adaptive integral method (AIM) and the conjugate gradient (CG) fast Fourier transform (FFT) algorithm can be substantially improved at low frequencies if the electric current and electric charge densities are treated as two separate unknown quantities. The representation of the current density in terms of solenoidal expansion functions (loops) and the charge density in terms of pulse basis functions provides for an exact decomposition of the original electromagnetic (EM) boundary-value problem into its magnetostatic and electrostatic forms at zero frequency. This formulation allows for accurate EM modeling down to very low frequencies free of numerical instabilities, while the spectral properties of the matrix equation are substantially improved compared to the standard method of moments formulation in either loop-tree or loop-star basis. The AIM and the CG FFT algorithms can be appropriately adjusted to accommodate for the use of loop-charge basis functions, thus leading to efficient solvers with O(NlogN) solution complexity and O(N) memory requirements for two-and-one-half-dimensional and penetrable three-dimensional (3-D) structures. For general 3-D objects, the CPU time and memory of the algorithms scale as O(N/sup 1.5/logN) and O(N/sup 1.5/), respectively. The new implementation of the AIM is discussed and demonstrated through its application for the broad-band simulation of complex interconnect and electronic packaging structures.

Efficient Simulation of Rectangular Dielectric Resonators Using Volume MPIE-MoM Formulation With Combined Entire-Domain and Subdomain Basis Functions

An efficient method-of-moments (MoM) formulation using combined entire-domain and subdomain basis functions is proposed for simulating the resonant modes of a rectangular dielectric resonator in free space. An approximate analytical solution for the resonant mode of the DR is used as the entire-domain basis function, while a set of tetrahedral basis functions are used as subdomain basis functions. Since the main profile of the resonant mode can be well represented by the entire-domain basis function, only small number of subdomain basis functions are needed for further refinement of the representation. Compared with the MoM formulations using only subdomain or entire-domain basis functions, this method is much faster for the same accuracy. Numerical results are given and compared with other methods.

An Efficient MoM Formulation for Finite-by-Infinite Arrays of Two-Dimensional Antennas Arranged in a Three-Dimensional Structure

In strongly coupled antenna arrays, the behavior of the elements near the edge can exhibit very large deviations with respect to the infinite periodic array solution. Insight into these truncation effects can be obtained by simulating finite-by-infinite arrays. This paper describes an efficient method-of-moments (MoM) scheme for simulating such arrays. This scheme is capable of handling arrays of two-dimensional metallic antennas placed perpendicularly to the array plane, in lossless media. This formulation relies on the free-space Green's function related to arrays infinite in one direction only, with linear phase excitation. After extraction of its singular part, this function is tabulated. Then, the elements of the MoM impedance matrix are computed in the space domain, with the help of a limited number of integration points. The computation time needed for establishing the MoM system of equations and for solving it is comparable to the time needed in the linear array case. An extension of this formulation is also developed to study infinite-by-infinite arrays and semi-infinite arrays. The latter solutions also provide standard current distributions, which are used to obtain a fast approximate solution of the MoM system of equations. Simulation results are shown for broadband arrays, made of tapered slot antennas consisting of metallic plates.

Improving the convergence of the iterative solution of matrix equations in the method of moments formulation using extrapolation techniques

A strategy is presented for improving the convergence of iterative solvers when used for solving the matrix equations of the method of moments (MOM) formulation of EM scattering problems in a frequency sweep. Based on the use of an initial guess, which is derived via an extrapolation procedure involving the solution obtained at previous frequency steps, this procedure improves the convergence behaviour over the use of either a zero initial guess or the employment of the solution at the last frequency as the initial value for conjugate gradient (CG) iteration.

MNM: a technique for the iterative solution of matrix equations arising in the method of moments formulation

The advent of the fast multipole method (FMM) and other techniques for carrying out the matrix vector product in an efficient manner, in the context of the method of moments (MoM) formulation, has made it possible for us to take a quantum leap forward towards solving a class of large problems involving perfectly conducting scatterers. Among a plethora of different iterative algorithms available in the literature, the conjugate gradient (CG) and its variants are among the most widely used. The speed with which convergence to the correct solution is achieved in employing this iteration algorithm, is dependent upon the choice of the pre conditioner, as well as the initial guess. The authors focus on introducing a technique the (Maxwell and Markov technique, referred to herein as MNM) for choosing the initial guess. This can help reduce the number of iterations and, consequently the solution time, over the typical choice of a zero initial guess in the context of CG. The application of the method is illustrated via a number of numerical examples that combine the FMM with MNM, to derive the solution of representative electromagnetic scattering problems.

Complete surface current distribution in a normal-mode helical antenna using a Galerkin solution with sinusoidal basis functions

An investigation of the surface current distribution in a normal-mode helical antenna (NMHA) is reported. This enables precise prediction of the performance of NMHAs, since traditional wire-antenna simulations ignore important details, such as non-uniform and transverse current distributions. A moment-method formulation is developed, using two-dimensional basis functions to represent the total non-uniform surface current distribution over the surface of the wire of the helix. Piecewise-sinusoidal basis functions are employed in two normal directions, with an exact kernel formulation and application of Galerkin's solution method. The numerical solution of the singular integrals associated with self-impedance terms was computed with a very low relative error. The surface current distribution was computed for different helix geometries. It was found that the axially-directed component of the current distribution around the surface of the wire was highly non-uniform and that there was also a significant circumferential current flow due to inter-turn capacitance, both effects that are overlooked by standard filamentary current representations.

Parametric analysis of slot-loaded trapezoidal patch antennas

Slot-loaded trapezoidal patch antennas are analyzed in order to achieve a multifrequency operation. A moment method formulation based on new analytical entire domain basis functions is developed and compared to the one involving the Rao-Wilton-Glisson (1982) RWG triangular discretization. The numerical results obtained through the new efficient implemented code are compared to the measurements showing a good agreement. A parametric analysis of such antennas is also presented leading to useful design graphs showing the main antenna properties (resonance frequencies, cross-polarization levels, matching, etc.) as a function of the patch geometrical dimensions, symmetric collocation of the slots on the patch, and slot dimensions.

Beam steered reactively loaded printed microstrip loop antennas

Printed microstrip loop antennas loaded with reactive components are analysed to demonstrate the capability of beam steering by reactance adjustment. The geometry of the printed microstrip loop including reactive element loading is represented as a superquadratic function so that the current distribution, hence radiation pattern for various antenna loop shapes, can be calculated using a unified method of moment formulation. The effects of the loop shapes and dielectric substrates on the beam steering capability and input impedance characteristics are investigated. It is found that continuous beam steering can be achieved and that a compromise has to be made between the maximum beam steering and minimum input VSWR. A 1.0GHz experimental verification is provided, which demonstrates that beam steering of 24/spl deg/ can be achieved by control of a single reactive element, while low cross-polarisation (/spl sim/ -20 dB) and VSWR (/spl sim/ 1.7) can be simultaneously maintained.