Electric Field Integral Equation Formulation Research Articles

An Excitation-Aware and Self-Adaptive Frequency Normalization for Low-Frequency Stabilized Electric Field Integral Equation Formulations

The accurate solution of quasi-Helmholtz decomposed electric field integral equations (EFIEs) in the presence of arbitrary excitations is addressed: Depending on the specific excitation, the quasi-Helmholtz components of the induced current density do not have the same asymptotic scaling in frequency, and thus, the current components are solved for with, in general, different relative accuracies. In order to ensure the same asymptotic scaling, we propose a frequency normalization scheme of quasi-Helmholtz decomposed EFIEs which adapts itself to the excitation and which is valid irrespective of the specific excitation and irrespective of the underlying topology of the structure. Specifically, neither an ad-hoc adaption nor <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a-priori</i> information about the excitation is needed as the scaling factors are derived based on the norms of the right-hand side (RHS) components and the frequency. Numerical results corroborate the presented theory and show the effectiveness of our approach.

Electric Field Integral Equation Fast Frequency Sweep for Scattering of Nonpenetrable Objects via the Reduced-Basis Method

A reduced-basis approximation to solve for the electromagnetic scattering from nonpenetrable targets in a frequency band of interest is studied in detail. Perfect electric scatterers in homogeneous media are considered. Even though the frequency behavior can be complicated, the electromagnetic scattering does not arbitrarily change in the frequency band of interest. Thus, instead of using computationally inefficient, large-dimension numerical approximation scenarios, such as the method of moments, to solve for integral equations in the band of interest, a low-dimension reduced-basis space is identified to approximate the smooth evolution of the electromagnetic scattering itself as frequency changes. As a result, the frequency-parameter scattering problem is solved within a reduced-order model approach. The electric field integral equation formulation is addressed and properly selected electric currents are considered to span the reduced-basis space in the frequency band of interest. The involved frequency dependence in the integral equation due to the wave propagation phenomena described in the free-space Green’s function has traditionally limited the applicability of reduced-basis approximations for fast frequency sweep in integral equation methods in electromagnetics. In this article, we focus on this difficulty and propose a physics-based approach where the frequency-parameter dependence shows an affine behavior in the integral equation, without relying on rather complicated interpolation schemes. As a result, a fast frequency sweep for integral equation problems in electromagnetics is proposed. Finally, real-life applications will illustrate the capabilities of this methodology.

TM Electromagnetic Scattering from PEC Polygonal Cross-Section Cylinders: A New Analytical Approach for the Efficient Evaluation of Improper Integrals Involving Oscillating and Slowly Decaying Functions

The analysis of the TM electromagnetic scattering from perfectly electrically conducting polygonal cross-section cylinders is successfully carried out by means of an electric field integral equation formulation in the spectral domain and the method of analytical preconditioning which leads to a matrix equation at which Fredholm’s theory can be applied. Hence, the convergence of the discretization scheme is guaranteed. Unfortunately, the matrix coefficients are improper integrals involving oscillating and, in the worst cases, slowly decaying functions. Moreover, the classical analytical asymptotic acceleration technique leads to faster decaying integrands without overcoming the most important problem of their oscillating nature. Thus, the computation time rapidly increases as higher is the accuracy required for the solution. The aim of this paper is to show a new analytical technique for the efficient evaluation of such kind of integrals even when high accuracy is required for the solution.

Implementation of loop co‐tree basis functions for EFIE formulation at low frequency

In this study, a method of moments (MoM) of electric field integral equation (EFIE) formulation using the loop co-tree basis functions has been implemented, analysed, and applied to the electromagnetic solution of perfect electric conducting objects at low frequency. The EFIE with Rao-Wilton-Glisson (RWG) basis functions suffer from low-frequency instability. Even for an appropriate frequency, the EFIE with RWG basis functions is ill-posed. However, the matrix form of the EFIE with loop co-tree basis functions is well-conditioned when combined with an iterative solver such as a type of Krylov subspace methods. The approach is shown to be very effective for resolving both the low-frequency instability and the ill-condition of the MoM matrix with a generalised minimal residual method.

A Memory Saving Augmented EFIE With Modified Basis Functions for Low-Frequency Problems

A new augmented electric field integral equation (A-EFIE) formulation is discussed in the low frequency. In the new formulation, both the electric current and charge are considered as unknowns. Different from the conventional A-EFIE, a constraint on the normal component of the magnetic field is added to modifying the formulation of A-EFIE for improving the accuracy and a kind of constant nodal functions is introduced for expanding the electric charge. With the proposed method, besides avoiding the low-frequency breakdown, the storage of impedance matrix is reduced over 35% compared with the conventional A-EFIE, which is owing to the application of constant nodal functions associated with vertices. Numerical examples illuminate the validity and performance of the new A-EFIE with constant nodal functions.

A Generalized Reaction Theorem That Eliminates Internal Resonances in the Electric and Magnetic Field Integral Equations

This paper presents stable magnetic-field and electric-field integral equation formulations for exterior electromagnetic scattering by either penetrable or perfectly conducting closed bodies of general shape. The development relies on an extension of the conventional reciprocity theorem allowing the fields produced by some combination of sources in one environment to be connected with the fields generated by another combination of sources in a different environment. By the use of lossy metamaterials, the internal-resonance problem inherent to the original versions is eliminated, and thus the new formulations are amenable to a unique, highly accurate, and well-conditioned solution.

Gaussian Beam Scattering From a Deterministic Rough Metal Surface

A full-wave theory of plane wave scattering from rough surfaces called the correction current (CC) method was recently developed. This paper extends the CC method to the more general case of beam wave scattering. Gaussian beams are assumed incident upon a deterministic metal rough surface. Scatter patterns based on the CC method are calculated and compared to scatter patterns determined by the method of moments solution of an electric field integral equation formulation of the scatter problem. It was shown that good agreement between these two solution methods is obtained for scatter surfaces for which the CC method is shown to satisfy conservation of power and an accuracy criterion.

3-D Scattering From a PEC Target Buried Beneath a Dielectric Rough Surface: An Efficient PILE-ACA Algorithm for Solving a Hybrid KA-EFIE Formulation

An efficient hybrid KA-EFIE formulation is deployed to analyze the electromagnetic (EM) scattering from a 3-D perfectly electric conducting (PEC) object buried beneath a 2-D dielectric rough surface. In this approach, the electric and magnetic current densities on the rough surface are analytically obtained through the current-based Kirchhoff approximation (KA), whereas the electric current density on the buried object is rigorously determined by solving the electric field integral equation (EFIE) using the Galerkin’s method of moments (MoM) with Rao–Wilton–Glisson (RWG) basis functions. The KA-EFIE matrix system is then efficiently solved by the iterative propagation-inside-layer-expansion (PILE) method combined with the algebraic adaptive cross approximation (ACA). The current densities on the dielectric rough surface are thereafter used to handle the bistatic normalized radar cross-section (NRCS) patterns. The proposed hybrid approach allows a significant reduction in computation time and memory requirements compared to the rigorous Poggio–Miller–Chang–Harrington–Wu (PMCHW)-EFIE formulation which requires solving a large MoM matrix equation. Moreover, the hybridization of the ACA algorithm with the PILE method improves further the computational cost thanks to the rank-deficient propriety of the coupling matrices. To validate the hybrid approach, we compare its results with those of the rigorous PMCHW-EFIE approach.

A Numerical Method for Electromagnetic Scattering by 3-D Dielectric Objects Buried Under 2-D Locally Rough Surfaces

In this work, a numerical method is developed to solve a forward-scattering problem of three-dimensional (3-D) objects buried into two half-space media with two-dimensional (2-D) locally rough interface. The proposed method exploits the electric field integral equation (EFIE) formulation of the scattered field which involves the dyadic Green's function of the background medium. Since the required Green's function has no analytical expression (in general), buried object approach (BOA) is applied to calculate it numerically. Method of moments (MoM) is then used to solve the EFIE. The presented method is numerically validated by comparing it with finite difference time-domain (FDTD) method. Furthermore, an elaborative numerical analysis of the proposed method is presented to show its feasibility for various roughness profiles and buried objects.

Theory and Experiment of Spherical Aperture-Coupled Antennas

Rigorous mathematical method of moments (MoM) for analyzing spherical microstrip aperture-coupled antennas is presented in this paper by using dyadic Green's functions (DGF) in conjunction with magnetic field integral equation (MFIE) and electric field integral equation (EFIE) formulations. With the aid of linear Rao-Wilton-Glisson (RWG) triangular basis functions, spherical three-layer aperture-coupled patch antennas are analyzed. The effect of mutual coupling between slot and patch antenna is developed and the input impedance and radiation patterns of such an antenna are presented. A prototype of a spherical aperture-coupled antenna is fabricated and tested. The measurement results are compared to the results of the proposed method and a good agreement between the results is achieved. The measured efficiency and gain of this antenna at 3.15 GHz are 85% and 5.23 dB, respectively.

A DDM-Based Scheme for Analysis of Electromagnetic Scattering From Target and Rough Surface Composite Model

We present an optimized solution procedure for electromagnetic (EM) scattering problems of three-dimensional (3-D) target and rough surface composite model. By decomposing the composite surface into an open surface and a closed surface and applying a proposed domain decomposition method-based (DDM-based) scheme, we demonstrate a significant improvement in the iterative solution of these problems compared to the traditional electric field integral equation (EFIE) formulation.

Integral equation analysis and optimization of 2D layered nanolithography masks by complex images Green’s function technique in TM polarization

Analysis and optimization of diffraction effects in nanolithography through multilayered media with a fast and accurate field-theoretical approach is presented. The scattered field through an arbitrary two-dimensional (2D) mask pattern in multilayered media illuminated by a TM-polarized incident wave is determined by using an electric field integral equation formulation. In this formulation the electric field is represented in terms of complex images Green's functions. The method of moments is then employed to solve the resulting integral equation. In this way an accurate and computationally efficient approximate method is achieved. The accuracy of the proposed method is vindicated through comparison with direct numerical integration results. Moreover, the comparison is made between the results obtained by the proposed method and those obtained by the full-wave finite-element method. The ray tracing method is combined with the proposed method to describe the imaging process in the lithography. The simulated annealing algorithm is then employed to solve the inverse problem, i.e., to design an optimized mask pattern to improve the resolution. Two binary mask patterns under normal incident coherent illumination are designed by this method, where it is shown that the subresolution features improve the critical dimension significantly.

Low-Frequency Surface Integral Equation Solution by Multilevel Green's Function Interpolation With Fast Fourier Transform Acceleration

A fast low-frequency solver for surface integral equations is presented. Multilevel Lagrange interpolation of the pertinent homogenous space Green's functions is employed for the factorization of the integral operators. For high computational efficiency, the resulting multilevel convolution/translation operators are diagonalized by fast Fourier transform. A fast adaptive multilevel scheme of the interpolation procedure makes the approach also suitable for broadband applications. The method is investigated for the solution of electric field and combined field integral equations. For the electric field integral equation, the dyadic and mixed-potential integral equation formulations are considered and discussed. Furthermore, the impact of extrapolation errors on the approximation accuracy of the respective Green's functions is analyzed. In several numerical examples, excellent efficiency with respect to computation time and memory requirements as well as good accuracy of the obtained results is demonstrated.

Accelerated Source-Sweep Analysis Using a Reduced-Order Model Approach

This communication is concerned with the development of a model-order reduction (MOR) approach for the acceleration of a source-sweep analysis using the volume electric field integral equation (EFIE) formulation. In particular, we address the prohibitive computational burden associated with the repeated solution of the two-dimensional electromagnetic wave scattering problem for source-sweep analysis. The method described within is a variant of the Krylov subspace approach to MOR, that captures at an early stage of the iteration the essential features of the original system. As such these approaches are capable of creating very accurate low-order models. Numerical examples are provided that demonstrate the speed-up achieved by utilizing these MOR approaches when compared against a method of moments (MoM) solution accelerated by use of the fast Fourier transform (FFT).

A novel flexible GMRES with deflated restarted for efficient solution of electromagnetic problems

AbstractTo iterative solve large dense complex linear systems arising from electric field integral equations formulation of electromagnetic scattering problems, multilevel fast multipole algorithm (MLFMA) is built to speed up the matrix‐vector product (MVP) operations. This article presents a novel flexible generalized minimum residual with deflated restarted (FGMRES‐DR), which can greatly improve the convergence of those large dense complex linear systems. Based on physical mechanism of the scattering problems, in inner iteration of FGMRES‐DR, an iterative solution of the near‐field matrix in MLFMA is adopted. In outer iteration, deflation method which is purely algebraic is used to eliminate the influence on convergence because of the smallest eigenvalues. Numerical experiments indicate that FGMRES‐DR is very effective and can reduce the number of MVP compared with the FGMRES in large‐scale electromagnetic problems. © 2011 Wiley Periodicals, Inc. Microwave Opt Technol Lett 53:1360–1364, 2011; View this article online at wileyonlinelibrary.com. DOI 10.1002/mop.26019

Fast iterative solution of integral equation with matrix decomposition algorithm and multilevel simple sparse method

In order to efficiently solve large dense complex linear systems arising from the electric field integral equations formulation of electromagnetic problems, matrix decomposition algorithm–singular value decomposition (MDA–SVD) is used to accelerate the matrix–vector product (MVP) operation. Based on the symmetry of the impedance matrix, a modified multilevel simple sparse method (M-MLSSM) is proposed. Combining MDA and M-MLSSM, the authors obtain a sparse representation of the impedance matrix and construct a fast MVP operation. A perfectly electrically conducting sphere and the series-fed microstrip antenna array are simulated to demonstrate the validity of the proposed method. Moreover, numerical experiments demonstrate that combined MDA and M-MLSSM can accelerate the MVP operation and reduce memory requirements by at least half an order of magnitude compared to MDA–SVD.

A modified complex shifted preconditioner combined with sparse approximate inversion preconditioner for electromagnetic scattering

AbstractTo efficiently solve large, dense, and complex linear systems arising from electric field integral equations formulation of electromagnetic scattering problems, the multilevel fast multipole algorithm is used to accelerate the matrix–vector product operations. This article presents a modified complex shifted preconditioner (MCSP), which is combined with sparse approximate inversion (SAI) preconditioner can greatly improve the convergence of restart generalized minimal residual iterative method for large, dense, complex linear systems. Numerical experiments demonstrate that the MCSP combined with SAI preconditioner is very effective and can reduce the iteration number and the computational time significantly. © 2010 Wiley Periodicals, Inc. Microwave Opt Technol Lett 53:55–58, 2011; View this article online at wileyonlinelibrary.com. DOI 10.1002/mop.25651

Analytical Development of the Far-Zone Radiation Integral for an Arbitrary Planar Spiral Antenna

The planar spiral antenna's radiation integral is both conceptually and computationally complex. In the past, the far-field pattern has been found by either approximating the spiral's shape to make the computation tractable, or by using a numerical approximation. In this paper, the radiation integral for an arbitrary planar spiral antenna is now derived without approximating its shape, using a readily understandable differential-geometry analysis. This allows for physical insight into the radiation mechanisms and contributions from the various current components that are otherwise intractable using modern computational codes, and provides a significantly more efficient algorithm in cases when mutual coupling can be neglected. In addition, this analysis and physical insight leads to an effective spiral antenna design tool. Current numerical integration techniques are applied to find the far-zone electric field for an Archimedean spiral antenna using this development. The results are compared with a numerical solution of the electric-field integral-equation formulation to validate the analysis.

Efficient Wideband Electromagnetic Scattering Computation for Frequency Dependent Lossy Dielectrics Using WCAWE

This paper presents a model order reduction algorithm for the volume electric field integral equation (EFIE) formulation, that achieves fast and accurate frequency sweep calculations of electromagnetic wave scattering. An inhomogeneous, two-dimensional, lossy dielectric object whose material is characterized by a complex permittivity which varies with frequency is considered. The variation in the dielectric properties of the ceramic BaxLa4Ti 2+xO 12+3x in the <1 GHz frequency range is investigated for various values of x in a frequency sweep analysis. We apply the well-conditioned asymptotic waveform evaluation (WCAWE) method to circumvent the computational complexity associated with the numerical solution of such formulations. A multipoint automatic WCAWE method is also demonstrated which can produce an accurate solution over a much broader bandwidth. Several numerical examples are given on order to illustrate the accuracy and robustness of the proposed methods.

Shifted SSOR preconditioning technique for electromagnetic wave scattering problems

AbstractTo efficiently solve large dense complex linear system arising from electric field integral equations (EFIE) formulation of electromagnetic scattering problems, the multilevel fast multipole method (MLFMM) is used to accelerate the matrix‐vector product operations. The symmetric successive over‐relaxation (SSOR) preconditioner is constructed based on the near‐field matrix of the EFIE and employed to speed up the convergence rate of iterative methods. This technique can be greatly improved by shifting the near‐field matrix of the EFIE with the principle value term of the magnetic field integral equation (MFIE) operator. Numerical results demonstrate that this method can reduce both the number of iterations and the computational time significantly with low cost for construction and implementation of preconditioners. © 2009 Wiley Periodicals, Inc. Microwave Opt Technol Lett 51: 1035–1039, 2009; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.24254