Combined Field Integral Equation Formulation Research Articles

A Discontinuous Galerkin Combined Field Integral Equation Formulation for Electromagnetic Modeling of Piecewise Homogeneous Objects of Arbitrary Shape

We present a novel discontinuous Galerkin (DG) surface integral equation approach, based on the electric and magnetic current combined field integral equations (JMCFIE), for the electromagnetic analysis of arbitrarily shaped piecewise homogeneous objects. In the proposed scheme, nonoverlapping boundary surfaces and interfaces between materials can be handled independently, without any continuity requirement through multimaterial junctions and tear lines between surfaces in contact. The use of nonconformal meshes provides improved flexibility for computer-aided-design (CAD) prototyping and tessellation. The proposed formulation can readily address nonconformal multimaterial junctions, where three or more material regions meet. The continuity of the electric surface current across the junction contours is enforced by the combination of the boundary conditions implicit in the JMCFIE formulation and the weakly imposed interior penalty (IP) between the contacting surfaces within each region. This completely avoids the cumbersome junction problem, which no longer requires any special treatment. Numerical experiments are included to validate the accuracy and demonstrate the great versatility of the proposed JMCFIE-DG formulation for the management and solution of complex composite objects with junctions.

On the Higher Order Approximations for Efficient Computational Electromagnetics of the Surface Integral Equation

In this paper, we propose an efficient procedure to obtain loop and star basis functions useful to the surface integral equation. They are of any order, for 3-D curved surfaces, and they keep the solenoidal/nonsolenoidal splitting, improving the performance for low frequencies (near field). These bases, introduced in a former work and applied to the electric field integral equation (EFIE), herein are used in both the magnetic field integral equation (MFIE) and the combined field integral equation (CFIE) providing a good performance even at the low-frequency regime. For addressing the low-frequency performance, scale factors and normalization of the basis are applied to EFIE, MFIE, and CFIE formulations.

Augmented combined field integral equation for low frequency scattering analysis

In this study, a new augmented combined field integral equation formulation has been proposed to address the low-frequency breakdown problem. The proposed formulation is developed by adding the magnetic field integral operator to the conventional augmented electric field integral equation (AEFIE). In the meanwhile, a saddle point constraint preconditioning is also introduced to improve the convergence property of the system matrix. In some cases, the conventional AEFIE method cannot achieve the accurate solutions when the frequency is extremely low and a perturbation method is needed to tackle this accuracy problem. However, the new method is able to get the accurate solutions without any other strategy in the same cases mentioned above. Some numerical examples have been presented to demonstrate the validity of the proposed method.

Integral Equation Analysis of Multilayered Waveguide Bends Using Complex Images Green's Function Technique

In this paper, a new approach for analyzing multilayered waveguide bends using an integral equation technique is introduced. The equivalence principle is employed to form a combined field integral equation (CFIE). The use of complex images Green's functions in the method of moments solution of the developed CFIE formulation results in a highly accurate and fast solution of the problem and simultaneously, reduces computational complexity. The proposed method is applied to a general 2-D optical bend. The accuracy of the results is verified by comparison with the physical optics method.

A boundary integral equation domain decomposition method for electromagnetic scattering from large and deep cavities

Electromagnetic scattering analysis of large and deep cavities embedded in an arbitrarily shaped host body is of high interest to the engineering community. The objective of this work is to investigate an effective boundary integral equation domain decomposition method for solving the cavity scattering problems. The key features of the proposed work include: (i) the introduction of individual electric and magnetic traces as unknowns for each sub-region, (ii) the development of a multi-trace combined field integral equation formulation for decomposed boundary value problem, and (iii) the derivation of optimized multiplicative Schwarz preconditioning using complete second order transmission condition. The proposed method can be viewed as an effective preconditioning scheme for the integral equation based solution of the cavity scattering problems. The strength and flexibility of the proposed method will be illustrated by means of several representative numerical examples.

Fast Analysis of Electromagnetic Scattering From Three-Dimensional Objects Straddling the Interface of a Half Space

The electromagnetic scattering from 3-D perfectly electrically conducting (PEC) objects straddling the interface of a half-space is analyzed in this letter. The adaptive cross approximation (ACA) algorithm is adopted to enhance the computational efficiency. As the object is naturally separated into two parts by the interface of the half-space, two different mesh densities are required due to the contrast of the two background materials. Therefore, two multilevel tree structures are set up individually in the two regions. When the source and field points are in the same region, which corresponds to the primary terms and reflection terms, the implementation is similar to the free-space case. However, when considering the mutual coupling of the two regions, which corresponds to the transmitted terms, difficulties arise since the two trees are independent and the numbers of levels are, in general, different. In this case, a z-dependent grouping strategy is proposed to redefine the well-separated interactions and match the abrupt changes at the interface. Three clustering strategies are proposed and discussed carefully to account for the mutual coupling in the two different regions of the half-space. The combined field integral equation (CFIE) formulation is further applied to improve the convergence of the system. Several numerical examples are demonstrated to validate the proposed schemes.

Well‐conditioned boundary integral formulations for high‐frequency elastic scattering problems in three dimensions

We construct and analyze a family of well‐conditioned boundary integral equations for the Krylov iterative solution of three‐dimensional elastic scattering problems by a bounded rigid obstacle. We develop a new potential theory using a rewriting of the Somigliana integral representation formula. From these results, we generalize to linear elasticity the well‐known Brakhage–Werner and combined field integral equation formulations. We use a suitable approximation of the Dirichlet‐to‐Neumann map as a regularizing operator in the proposed boundary integral equations. The construction of the approximate Dirichlet‐to‐Neumann map is inspired by the on‐surface radiation conditions method. We prove that the associated integral equations are uniquely solvable and possess very interesting spectral properties. Promising analytical and numerical investigations, in terms of spherical harmonics, with the elastic sphere are provided. Copyright © 2014 John Wiley & Sons, Ltd.

Well-conditioned boundary integral equation formulations for the solution of high-frequency electromagnetic scattering problems

We present several versions of Regularized Combined Field Integral Equation (CFIER) formulations for the solution of three dimensional frequency domain electromagnetic scattering problems with Perfectly Electric Conducting (PEC) boundary conditions. Just as in the Combined Field Integral Equations (CFIE), we seek the scattered fields in the form of a combined magnetic and electric dipole layer potentials that involves a composition of the latter type of boundary layers with regularizing operators. The regularizing operators are of two types: (1) modified versions of electric field integral operators with complex wavenumbers, and (2) principal symbols of those operators in the sense of pseudodifferential operators. We show that the boundary integral operators that enter these CFIER formulations are Fredholm of the second kind, and invertible with bounded inverses in the classical trace spaces of electromagnetic scattering problems. We present a spectral analysis of CFIER operators with regularizing operators that have purely imaginary wavenumbers for spherical geometries—we refer to these operators as Calderón–Ikawa CFIER. Under certain assumptions on the coupling constants and the absolute values of the imaginary wavenumbers of the regularizing operators, we show that the ensuing Calderón–Ikawa CFIER operators are coercive for spherical geometries. These properties allow us to derive wavenumber explicit bounds on the condition numbers of Calderón–Ikawa CFIER operators. When regularizing operators with complex wavenumbers with non-zero real parts are used—we refer to these operators as Calderón-Complex CFIER, we show numerical evidence that those complex wavenumbers can be selected in a manner that leads to CFIER formulations whose condition numbers can be bounded independently of frequency for spherical geometries. In addition, the Calderón-Complex CFIER operators possess excellent spectral properties in the high-frequency regime for both convex and non-convex scatterers. We provide numerical evidence that our solvers based on fast, high-order Nyström discretization of these equations converge in very small numbers of GMRES iterations, and the iteration counts are virtually independent of frequency for several smooth scatterers with slowly varying curvatures.

Boundary Integral Equation Analysis of Spoof Localized Surface Plasmons Excited in a Perfectly Conducting Cylinder with Longitudinal Corrugations

The main purpose of this paper is to apply the boundary integral equation (BIE) method to the analysis of spoof localized surface plasmons (spoof LSPs) excited in a perfectly conducting cylinder with longitudinal corrugations. Frequency domain BIE schemes based on electric field integral equation (EFIE), magnetic field integral equation (MFIE) and combined field integral equation (CFIE) formulations are used to solve two-dimensional electromagnetic (EM) problems of scattering from the cylinder illuminated by a transverse electric plane wave. In this approach effects of spoof LSPs are included in the secondary surface current and charge densities resulting from the interaction between the plane wave and the cylinder. Numerical results obtained with the BIE schemes are validated by comparison with that of a recently proposed modal solution based on the metamaterial approximation.

Time domain CFIEs for electromagnetic scattering problems

Time domain integral equations complement other methods for solving Maxwell's equations by handling infinite domains without difficulty and by reducing the computational domain to the surface of the scatterer. In this paper we study the discretization error when convolution quadrature is used to discretize two new regularized combined field integral equation formulations of the problem of computing scattering from a bounded perfectly conducting obstacle.

Condition number estimates for combined potential boundary integral operators in acoustic scattering

We study the classical combined field integral equation formulations for time-harmonic acoustic scattering by a sound soft bounded obstacle, namely the indirect formulation due to Brakhage-Werner/Leis/Panic, and the direct formulation associated with the names of Burton and Miller. We obtain lower and upper bounds on the condition numbers for these formulations, emphasising dependence on the frequency, the geometry of the scatterer, and the coupling parameter. Of independent interest we also obtain upper and lower bounds on the norms of two oscillatory integral operators, namely the classical acoustic single- and double-layer potential operators.

TD-CFIE Formulation for Transient Electromagnetic Scattering from 3-D Dielectric Objects

In this paper, we present a time domain combined field integral equation formulation (TD-CFIE) to analyze the transient electromagnetic response from dielectric objects. The solution method is based on the method of moments which involves separate spatial and temporal testing procedures. A set of the RWG functions is used for spatial expansion of the equivalent electric and magnetic current densities, and a combination of RWG and its orthogonal component is used for spatial testing. The time domain unknowns are approximated by a set of orthonormal basis functions derived from the Laguerre polynomials. These basis functions are also used for temporal testing. Use of this temporal expansion function characterizing the time variable makes it possible to handle the time derivative terms in the integral equation and decouples the space-time continuum in an analytic fashion. Numerical results computed by the proposed formulation are compared with the solutions of the frequency domain combined field integral equation.

Surface integral equation formulations for solving electromagnetic scattering problems with iterative methods

In this paper, formulation of the surface integral equations for solving electromagnetic scattering by dielectric and composite metallic and dielectric objects with iterative methods is studied. Four types of formulations are considered: T formulations, N formulations, the combined field integral equation formulation, and the Müller formulation. By studying properties of the integral equations and their testing in the Galerkin method, “optimal” forms for each formulation type are derived. Numerical examples demonstrate that the developed new formulations lead to clear improvements in the convergence rates when the matrix equations are solved iteratively with the generalized minimal residual method. Both the Rao‐Wilton‐Glisson and Trintinalia‐Ling (TL) basis functions are used in expanding the unknown electric and magnetic surface current densities. In particular, the first‐order TL basis functions are required in the N formulations to maintain the solution accuracy when the surfaces include sharp edges.

Application of combined field Integral equation for electromagnetic scattering by dielectric and composite objects

Combined field integral equation (CFIE) is applied for computing electromagnetic scattering by arbitrarily shaped three dimensional dielectric and composite objects. The objectives of this paper are as follows. First, to present a CFIE formulation which can be used in the analysis of piecewise dielectric and composite metallic and dielectric objects with junctions. Second, to show that properly choosing the coupling coefficients of the equations the conditioning of the discretized matrix equation can be essentially improved and rapidly converging iterative solutions can be obtained even without preconditioning.

Combined Field Integral Equation Formulations for Electromagnetic Scattering From Convex Geometries

The properties of different forms of the combined field integral equation (CFIE) formulation of electromagnetic scattering from convex, perfect electrically conducting geometries are considered. Several difficulties encountered in the numerical implementation of the traditional CFIEs of electromagnetic scattering theory are discussed. An alternate form of the CFIE is introduced which is free of many of the deficiencies of traditional formulations for smooth, convex geometries. The improved numerical properties of the new formulation are illustrated with numerical examples.

Combined field integral equation for the analysis of scattering from 3D conducting bodies coated with a dielectric material

AbstractIn this paper, we present an analysis of electromagnetic scattering from arbitrarily shaped three‐dimensional (3D) conducting objects coated with dielectric materials. The integral equation treated here is the combined field integral equation (CFIE). The objective of this paper is to illustrate that only the CFIE formulation is a valid methodology in removing defects, which occur at a frequency corresponding to an internal resonance of the structure. Numerical results of radar cross sections (RCS) for coated conducting structures are presented and compared with other available solutions. © 2004 Wiley Periodicals, Inc. Microwave Opt Technol Lett 40: 511–516, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20019

Calculation of cfie impedance matrix elements with RWG and n x RWG functions

The method of moments (MoM) solution of combined field integral equation (CFIE) for electromagnetic scattering problems requires calculation of singular double surface integrals. When Galerkin's method with triangular vector basis functions, Rao-Wilton-Glisson functions, and the CFIE are applied to solve electromagnetic scattering by a dielectric object, both RWG and n/spl times/RWG functions (n is normal unit vector) should be considered as testing functions. Robust and accurate methods based on the singularity extraction technique are presented to evaluate the impedance matrix elements of the CFIE with these basis and test functions. In computing the impedance matrix elements, including the gradient of the Green's function, we can avoid the logarithmic singularity on the outer testing integral by modifying the integrand. In the developed method, all singularities are extracted and calculated in closed form and numerical integration is applied only for regular functions. In addition, we present compact iterative formulas for computing the extracted terms in closed form. By these formulas, we can extract any number of terms from the singular kernels of CFIE formulations with RWG and n/spl times/RWG functions.

Transient scattering from dielectric cylinders: E‐field, H‐field, and combined field solutions

In this work, the problem of transient scattering by arbitrarily shaped two‐dimensional dielectric cylinders is solved using the marching‐on‐in‐time technique. The dielectric problem is approached via the equivalence principle. Three different formulations, namely, the electric field integral equation formulation, the magnetic field integral equation formulation, and the combined field integral equation formulation are considered. Numerical results are presented for two cross sections, namely, a circle and a square, and compared with inverse discrete Fourier transform (IDFT) techniques. In each case, good agreement is obtained with the IDFT solution.

Application of the method of moments to acoustic scattering from fluid-filled cylinders using three different formulations

In this work, we present three different formulations, namely the pressure field integral equation formulation (PFIE), the velocity field integral equation formulation (VFIE), and the combined field integral equation formulation (CFIE) for solving acoustic scattering problems associated with two-dimensional fluid-filled bodies of arbitrary cross-section. In particular using the boundary conditions on the surface of the body, two equivalent problems, each valid for the outside and inside regions of the scatterer, are derived. By properly selecting the associated equations for these equivalent problems, the three different formulations are derived. The PFIE, VFIE, and CFIE are then solved by approximating the cylindrical cross-section by linear segments and employing the method of moments. Further, it is shown that the moment matrices generated by the PFIE and VFIE are ill-conditioned at resonant frequencies of the cylinder, whereas the CFIE generates a well-conditioned matrix at all frequencies. The solution techniques presented in this work are simple, efficient and applicable to truly arbitrary geometries. Numerical results are presented for certain canonical shapes and compared with other available data.

A hybrid finite element method for near bodies of revolution (EM scattering)

A genetic formulation for a hybrid finite element solution for three-dimensional electromagnetic scattering is given using the equivalent current approach. The major computational tasks involved in monostatic scattering calculations are analyzed and compared as a function of the method of implementing the near-field radiation condition, i.e. method of moments, model expansion, and body of revolution (BOR). A method utilizing a BOR formulation that addresses these computational issues is given. This BOR implementation utilizes Hermite cubic basis functions and a variable number of modes per basis function in order to achieve the greatest efficiency. The combined field integral equation formulation is used to eliminate nonphysical resonance of the mesh boundary. Examples are given showing the efficiency and accuracy of this BOR code by itself, and as part of this hybrid finite-element method.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>