Rao-Wilton-Glisson Research Articles

On the Dual Basis for Solving Electromagnetic Surface Integral Equations

A powerful technique for solving electromagnetic (EM) surface integral equations (SIEs) for inhomogenous objects by the method of moments (MoM) involves the well-known Rao-Wilton-Glisson (RWG) basis function to represent the electric current and, for field orthogonality and numerical stability reasons, a variation of the RWG basis known as the ntilde X RWG basis (where ntilde is a unit normal vector at the object surface) to represent the magnetic current. Though this combination provides a numerically efficient and effective solution that has been demonstrated on a variety of structures, one cannot feel entirely comfortable because of the presence of fictitious magnetic current associated with the modified basis. Chen and Wilton proposed a different, smoother basis in 1990 that avoids the fictitious line charges, but because of computational cost issues it has not been used beyond Chen's dissertation. Recently, this basis was rediscovered and has received considerable attention. Our work reexamines the dual basis, exploring issues not addressed by Chen and Wilton and showing accurate solutions for a variety of EM scattering structures.

Contamination of the Accuracy of the Combined-Field Integral Equation With the Discretization Error of the Magnetic-Field Integral Equation

We investigate the accuracy of the combined-field integral equation (CFIE) discretized with the Rao-Wilton-Glisson (RWG) basis functions for the solution of scattering and radiation problems involving three-dimensional conducting objects. Such a low-order discretization with the RWG functions renders the two components of CFIE, i.e., the electric-field integral equation (EFIE) and the magnetic-field integral equation (MFIE), incompatible, mainly because of the excessive discretization error of MFIE. Solutions obtained with CFIE are contaminated with the MFIE inaccuracy, and CFIE is also incompatible with EFIE and MFIE. We show that, in an iterative solution, the minimization of the residual error for CFIE involves a breakpoint, where a further reduction of the residual error does not improve the solution in terms of compatibility with EFIE, which provides a more accurate reference solution. This breakpoint corresponds to the last useful iteration, where the accuracy of CFIE is saturated and a further reduction of the residual error is practically unnecessary.

Numerical study of a novel MFIE formulation using monopolar RWG basis functions

Numerical character of a new magnetic field integral equation (MFIE) formulation solved by the method of moments (MoM) is studied. In this new formulation, monopolar Rao–Wilton–Glisson (RWG) functions are used as both the basis and the testing functions. Calculation details for the impedance matrix elements of this new formulation are presented with a form suitable for programming. Numerical results for electromagnetic scattering analysis of several small sharp-edged conducting objects show that the new formulation using the monopolar RWG basis functions is much more accurate than the traditional one using the RWG and the monopolar RWG basis functions.

Application of the combined RWG basis functions to the MFIE for the scattering analysis of small sharp-edged objects

We present a novel implementation of the magnetic field integral equation (MFIE) with the use of the combined Rao–Wilton–Glisson (RWG) basis and testing functions. Implementation details are outlined in the context of the method of moments. For a wide variety of small sharp-edged objects, we show that accuracy of the MFIE can be greatly improved with a combination parameter of about 0.2 for the combined RWG functions. This is achieved with relatively coarse discretization for the scatterers and with only minor modifications in the existing codes based on the RWG and n ×RWG functions.

Application of an accurate approach to calculate the almost singular double surface integrals of the MFIE

Accurate calculation of the almost singular double surface integrals is essential for successful solutions of the magnetic field integral equation (MFIE) by method of moments (MoM) using Rao–Wilton–Glisson (RWG) basis functions, especially for the scattering analysis of small conducting objects. In this paper, an accurate singularity extraction technique to avoid logarithmic singularity of the integration on the testing triangles is used for the MFIE. The numerical examples of several small sharp-edged conducting objects demonstrate that accuracy of the MFIE is improved with the use of the new technique. In addition, the MFIE converges much faster than the electric field integral equation (EFIE) when iterative equation solvers are employed, which means high efficiency and a suitable choice for very low frequency small sharp-edged conducting object problems using this new singularity extraction technique.

Application of a novel formulation to the MFIE using RWG basis functions in method of moments

We investigate the inaccuracy of the traditional magnetic field integral equation (MFIE) in the analysis of scattering of small conducting objects, with the use of Rao–Wilton–Glisson (RWG) basis functions. As a remedy to such problems, we propose to use a novel impedance matrix elements (IME) formulation of the MFIE, which is suitable to use RWG basis functions. Techniques to calculate the IME of the new formulation are outlined. It is shown that similar technique to compute the IME of the MFIE resulted from the use of curl-conforming basis functions can also be used here. Based on the examples of several relatively small conducting objects, it is demonstrated that significant improvement in the accuracy of the MFIE can be obtained for this new IME formulation with the use of RWG basis functions.

Effect of Wire Grid Covering Aperture on the Shielding Performance of Metal Rectangular Enclosure

Based on the method of moment (MOM), a new method is developed to evaluate the electromagnetic shielding effectiveness of the rectangular metal enclosure with aperture covered with wire grid. The wire grid is modeled as thin wire, and the currents on the wire segments, which are taken as unknowns, are modeled by triangular basic functions. In order to expand the unknown electric currents on the surface of the cavity, Rao-Wilton-Glisson (RWG) basic functions are used on the surface patches to model the concerned objects. In terms of the continuity of the tangential electric fields on the surface of conductive plane and thin wire, a set of equations can be established. From the equations set, the currents on the surface patches and wire grids can be obtained. The accurate shielding effectiveness of the rectangular enclosure of metal structure with apertures covered with wire grid can be determined. By comparing the calculation with measurement results, the method is validated. Finally, the shielding performance of a protection cell in a practical substation is analyzed as an application.

Improving the accuracy of the magnetic field integral equation with the use of a new impedance matrix element formulation

The electric field integral equation (EFIE) and the magnetic field integral equation (MFIE) are widely used in conjunction with method of moments for electromagnetic scattering analysis of three-dimensional conducting objects with closed surfaces. However, the MFIE suffers from an accuracy problem compared with the EFIE with the use of the Rao–Wilton–Glisson (RWG) basis function. This accuracy problem is more serious for objects with sharp edges or corners. To solve this problem, a new technique to compute the impedance matrix elements (IME) of the MFIE using an RWG basis function is presented here. Details to compute the IME and the advantage of this new formulation are displayed. In addition, the relationship between this new IME formulation and the formulation using the low-order curl-conforming basis function for the MFIE is given. Through the computation of the RCS of several relatively small sharp-edged conducting objects, it is shown that the accuracy of the MFIE can be greatly improved by the use of the new IME formulation.

Impedance Analysis of Printed Antenna on Three-Dimensional High-Permittivity Dielectric Substrate Using Mixed-Domain MoM

An integral equation approach with a new solution procedure using moment method (MoM) is applied for the computation of coupled currents on the surface of a printed dipole antenna and inside its high-permittivity three-dimensional dielectric substrate. The main purpose of this study is to validate the accuracy and reliability of the previously proposed MoM procedure by authors for the solution of a coupled volume-surface integral equations system. In continuation of the recent works of authors, a mixed-domain MoM expansion using Legendre polynomial basis function and cubic geometric modeling are adopted to solve the tensor-volume integral equation. In mixed-domain MoM, a combination of entire-domain and sub-domain basis functions, including three-dimensional Legnedre polynomial basis functions with different degrees is utilized for field expansion inside dielectric substrate. In addition, the conventional Rao-Wilton-Glisson (RWG) basis function is employed for electric current expansion over the printed structure. The accuracy of the proposed approach is verified through a comparison with the MoM solutions based on the spectral domain Green's function for infinitely large substrate and the results of FDTD method.

A Simple Strip Model in the Volume-Surface Integral Equation for Analysis of Arbitrary Probe-Fed Conformal Microstrip Antennas

A strip model is incorporated in the volume-surface integral equation (VSIE) to simplify the analysis of the probe-fed conformal microstrip antennas (CMSAs) with arbitrary shapes. The thin wire-surface junction of the feeding probe and the patch is transferred to the surface-surface junction that can be modeled by using Rao-Wilton-Glisson (RWG) basis functions easily. Thus, the use of complicated attachment mode basis function in the conventional VSIE is eliminated, which simplifies the calculation of the method-of-moments (MoM) matrix. The construction of RWG basis functions at the surface-surface junction is detailed as well. An effective remedy is also proposed to overcome the computation breakdown in the near-singularity treatment when the field point locates at the extension of one side of the source domain. Numerical results show the reliability and accuracy of the proposed method.

Using an accurate singularity extraction technique to the MFIE for the scattering analysis of small objects

The method of moments (MoM) solution on the magnetic field integral equation (MFIE) for electromagnetic scattering problems of conducting objects with Rao–Wilton–Glisson (RWG) basis functions requires the evaluation of almost singular double surface integrals over triangular subdomains. Calculating these integrals accurately is essential for the successful solution of the MFIE by MoM especially for small sharp-edged conducting objects. This paper studies the singularity of the MFIE and uses an accurate singularity extraction technique to avoid logarithmic singularity of the integration on the testing triangles for the MFIE. Based on the numerical examples of several small sharp-edged conducting objects, it is demonstrated that the accuracy of the MFIE is improved greatly with the use of the new singularity extraction technique.

Hierarchical Bases for Nonhierarchic 3-D Triangular Meshes

We describe a novel basis of hierarchical, multiscale functions that are linear combinations of standard Rao-Wilton-Glisson (RWG) functions. When the basis is used for discretizing the electric field integral equation (EFIE) for PEC objects it gives rise to a linear system immune from low-frequency breakdown, and well conditioned for dense meshes. The proposed scheme can be applied to any mesh with triangular facets, and therefore it can be used as if it were an algebraic preconditioner. The properties of the new system are confirmed by numerical results that show fast convergence rates of iterative solvers, significantly better than those for the loop-tree basis. As a byproduct of the basis generation, a generalization of the RWG functions to nonsimplex cells is introduced.

A Mesh-Adapted Closed-Form Regular Kernel for 3D Singular Integral Equations

The Green's functions employed in the method of moments (MoM) diverge when observation and source points coincide; this is at the origin of the difficulties in computing the MoM matrix entries, and in handling the near-field interactions in fast Fourier transform (FFT)-based fast methods and other sampling-based methods. In this paper, we show that this singularity can be avoided, and a modified regular Green's function can be used instead. This latter is obtained from the spectral representation of the usual Green's function via windowing of its spectrum; the width of the spectral window depends on the size of the mesh employed for discretizing the problem, so that the proposed regular Green's function is a mesh-adapted regular kernel. We address a general 3D problem; we relate the MoM reaction integrals to the 2D Fourier spectrum of the Green's function, that allows to discuss the necessary spectral bandwidth for the windowed Green's function. We employ a tapered window, and present a closed-form expression for the spatial Green's function. Numerical results are presented for 3D antenna and scattering problems discretized with Rao-Wilton-Glisson (RWG) functions, and for uniform and nonuniform meshing. They show that the proposed method yields accurate solutions also for the antenna input impedance. The meaning of the regularized Green's function is also discussed and put in perspective.

Analytical Evaluation of Transient Magnetic Fields Due to RWG Current Bases

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> A new analytical approach for obtaining the time samples of the magnetic field intensity due to an impulsively excited Rao–Wilton–Glisson (RWG) basis function is presented. The approach is formulated directly in the time domain. It is shown that the magnetic field is related to the arc segments formed by the intersection of the triangular patch of the RWG basis with the sphere that is centered at the observation point and that has a radius of <formula formulatype="inline"><tex>$R = ct$</tex></formula>, where <formula formulatype="inline"><tex>$c$</tex></formula> is the speed of light. In particular, the magnetic field can be expressed as the variations of two quantities with respect to <formula formulatype="inline"><tex>$R$</tex></formula>. The first quantity is the arc segment length, and the second quantity is the bisecting vector of the arc segment. Analytical representations of these quantities are presented. Contrary to previous studies, these representations do not require the calculation of the intersection points of the sphere with the boundaries of the bases. The validity of the obtained time domain formulae is demonstrated through comparison of the results with those obtained in the frequency domain by using numerical quadrature. Finally, it is demonstrated that the derived formulae yield closed-form expressions when convolved with piecewise polynomial temporal basis functions. </para>

Electromagnetic Scattering From Rough Surface Using Single Integral Equation and Adaptive Integral Method

An efficient algorithm for electromagnetic wave scattering from rough dielectric surfaces is developed. The algorithm is based on the single magnetic field integral equation (SMFIE) and the surface is discretized using Rao-Wilton-Glisson (RWG) triangular basis functions. The new feature of the algorithm is the application of the adaptive integral method (AIM) with SMFIE for speeding up the calculation. The developed new method utilizes the flexibility of RWG functions to model arbitrary rough surface and the speed of FFT, enabling accurate simulations over large surfaces.

Interaction of Electromagnetic Waves With 3-D Arbitrarily Shaped Homogeneous Chiral Targets in the Presence of a Lossy Half Space

The interaction of electromagnetic waves with an arbitrarily shaped three-dimensional (3-D) homogeneous chiral object located above a lossy half space is investigated using the method of moments (MoM) via the coupled mixed potential integral equations (MPIEs). Based on the surface equivalence principle, the equivalent surface electric and magnetic currents are used to replace the homogeneous chiral target in the presence of the half space. Two coupled MPIEs are developed for the unknown equivalent surface electric and magnetic currents by utilizing the continuity condition of the tangential total electric and magnetic field components on the chiral body's surface. The well-known Galerkin procedure with Rao-Wilton-Glisson (RWG) basis functions is applied to solve this problem. The spatial domain half-space Green's functions are obtained from the corresponding spectral domain Green's functions via the discrete complex image method (DCIM) combined with the generalized-pencil of function (GPOF) technique. The reciprocity theorem is employed to calculate the far-zone scattered field. Numerical results are presented for characterizing electromagnetic scattering by a 3-D arbitrarily shaped homogenous chiral object located above a lossy half space so as to demonstrate the accuracy and efficiency of the proposed technique.

Spectral Properties of the EFIE-MoM Matrix for Dense Meshes With Different Types of Bases

The relationship between the properties of the basis functions employed to discretize the electric field integral equation (EFIE) and the eigenvalue spectrum of the resulting method of moments (MoM) matrix is studied. We concentrate on dense meshes, i.e., on complex geometries with large number of unknowns per wavelength, and/or disparate mesh cells sizes, and/or low frequency. These are typical occurrences in antennas, packaging and microwave circuits. We show that if the basis functions have separated Fourier spectra, the diagonal MoM matrix entries are close to the eigenvalues, and explain it in terms of the Fourier spectrum of the Green's function. For such a basis, diagonal preconditioning drastically reduces the condition number. This does not happen with sub-domain basis functions like the Rao-Wilton-Glisson (RWG), or with their linear combination into loop-tree or loop-star bases that are employed to solve low-frequency problems. Finally we analyze the properties of a multiresolution (MR) basis formed by linear combinations of RWG, but whose functions possess some degree of (Fourier) spectral resolution. We show that there is still correspondence between matrix diagonal, Green's function (Fourier) spectrum, and matrix eigenvalues. Diagonal preconditioning of the MR-MoM matrix causes the eigenvalues to cluster around 1, as it happens with the MoM matrix for the magnetic field integral equation. This strongly impacts on the EFIE matrix condition number and the speed of convergence of iterative solvers.

Linear-Linear Basis Functions for MLFMA Solutions of Magnetic-Field and Combined-Field Integral Equations

We present the linear-linear (LL) basis functions to improve the accuracy of the magnetic-field integral equation (MFIE) and the combined-field integral equation (CFIE) for three-dimensional electromagnetic scattering problems involving closed conductors. We consider the solutions of relatively large scattering problems by employing the multilevel fast multipole algorithm. Accuracy problems of MFIE and CFIE arising from their implementations with the conventional Rao-Wilton-Glisson (RWG) basis functions can be mitigated by using the LL functions for discretization. This is achieved without increasing the computational requirements and with only minor modifications in the existing codes based on the RWG functions

Spaceborne Mesh Reflector Antennas With Complex Weaves: Extended PO/Periodic-MoM Analysis

Reflector antennas with complex mesh weave patterns are numerically analyzed by using extended physical optics (PO) combined with the periodic method of moments (MoM). The method applies the periodic MoM with Rao-Wilton-Glisson (RWG) basis and testing functions. It calculates the transmission coefficient matrix (T-matrix) of the realistic, complex weave pattern of the planar mesh, in order to calculate the modified PO current on the mesh reflector. When analyzed, the planar grid mesh is found to be compatible with the commonly used wire-grid model. The modified PO current on the locally planar mesh is implemented on the curved reflector surface for the diffraction analysis of the mesh reflector antennas. The far-field pattern of the offset parabolic antenna with grid mesh, modelled using periodic MoM, agreed well with those obtained from the wire-grid model. T-matrix calculated from the tabulated T-matrix data using the interpolation method is sufficiently accurate and compares well with the one calculated using the direct method, allowing users to choose one of the T-matrix calculation methods depending on the problem. The analysis of the mesh reflectors with two different complex weave patterns (single Satin and single Atlas at Ka band) is performed to exemplify the application of the extended PO/periodic MoM algorithm. It is shown that differences were observed in cross-polarization discrimination (XPD) for circular polarization operations depending on the weave patterns. This result demonstrates the usefulness of the proposed technique in accurate performance prediction of mesh reflector antennas with complex weave patterns

A Multiresolution System of Rao–Wilton–Glisson Functions

A multiresolution (MR) basis is described for the method of moments (MoM) analysis of a generic 3-D conductor discretized with triangular cells. The MR basis functions are constructed as linear combination of Rao-Wilton-Glisson (RWG) basis functions, thus allowing direct applicability on existing MoM codes. With respect to previous work by the authors, the generation of the MR functions is significantly simplified, yet keeping similar applicability and performance, in terms of conditioning of the MoM matrix, convergence of the iterative solvers, and possibility of sparsifying the MoM matrix by clipping. Moreover, with the proposed basis the basis-change matrix from RWG to MR is highly sparse at all levels; this allows a more efficient generation of the MR-MoM matrix