Volume Electric Field Integral Equation Research Articles

Efficient Solenoidal Discretization of the Volume EFIE for Electromagnetic Scattering From Dielectric Objects

A solenoidal basis discretization of the volume electric-field integral equation is developed that avoids the need for volumetric integrals. The resulting formulation produces a symmetric system, enables an efficient matrix generation, and requires the fewest unknowns of existing low-order tetrahedral-mesh approaches.

A More Scalable and Efficient Parallelization of the Adaptive Integral Method—Part I: Algorithm

A more effective parallelization of the adaptive integral method (AIM) is proposed for solving the 3-D volume electric field integral equation. The AIM computations are distributed among processes by employing two workload decomposition strategies: 1) a novel 2-D pencil decomposition of the auxiliary regular grid is used to parallelize the anterpolation, interpolation, and 3-D FFT-accelerated propagation steps; and 2) a balanced 3-D block decomposition of the scattering volume is used to parallelize the correction step. The scalability of the proposed parallelization method is investigated theoretically. To compare it with competing methods, the concepts of resource constraints, parallel efficiency constraints, scalability limits, and acceptable parallelization regions are introduced by using N-P plots, where N and P denote the number of unknowns and processes, respectively. Using these concepts, it is shown that the proposed parallelization of AIM has better weak and strong scalability compared to the traditional ones, i.e., it enables not only larger problems to be solved but also faster solution of a given problem by increasing P.

A Higher-Order Solution of Volume Integral Equation for Electromagnetic Scattering From Inhomogeneous Objects

A higher-order solution of volume integral equation is proposed for the analysis of electromagnetic (EM) scattering from inhomogeneous objects. The higher-order basis functions used in this solution are defined in curvilinear tetrahedron elements, in which the Lagrange interpolation polynomials are utilized to construct the basis functions for representing the unknown volume electric current density. The proposed higher-order solution is implemented with point matching for the method of moments (MoM) of the volume electric field integral equation (VEFIE). Several numerical examples are presented to validate the higher-order convergence and great efficiency to analyze the electromagnetic scattering from inhomogeneous objects.

FFT-Accelerated Analysis of Scattering From Complex Dielectrics Embedded in Uniaxial Layered Media

An efficient integral-equation method is presented for the analysis of electromagnetic scattering from arbitrarily shaped 3-D dielectric objects embedded in a single layer of a uniaxial planar-layered medium. The proposed method employs a mixed-potential volume electric-field integral equation (VEFIE) that permits the object of interest to be dispersive, anisotropic, inhomogeneous, and of arbitrary shape. The VEFIE is solved by an iterative frequency-domain method-of-moments procedure. The object is discretized by tetrahedral elements, and the procedure is accelerated by the adaptive integral method, which reduces the computational costs by enclosing the tetrahedral mesh with an auxiliary regular grid and performing anterpolation (mesh to grid), propagation (grid to grid), interpolation (grid to mesh), and near-zone correction (mesh to mesh) steps. The computationally dominant propagation step of the method is accelerated by decomposing the Green's functions into convolution and correlation terms in the stratification direction and using 3-D fast Fourier transforms to multiply the resulting propagation matrices with the necessary vectors. If the object of interest is meshed using <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX"> $N$</tex></formula> tetrahedra that are of a single length scale, then the setup time, memory requirement, and the iterative matrix-solution time (per iteration) of the proposed method scale as <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX"> $O(N)$</tex></formula> , <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$O(N)$</tex></formula> , and <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$O(N\log N)$</tex></formula> , respectively. Numerical results validate the method's accuracy and efficiency for various problems relevant to geophysical exploration.

An Electric Field Volume Integral Equation Approach to Simulate Surface Plasmon Polaritons

In this paper we present an electric field volume integral equation approach to simulate surface plasmon propagation along metal/dielectric interfaces. Metallic objects embedded in homogeneous dielectric media are considered. Starting point is a so-called weak-form of the electric field integral equation. This form is discretized on a uniform tensor-product grid resulting in a system matrix whose action on a vector can be computed via the fast Fourier transform. The GMRES iterative solver is used to solve the discretized set of equations and numerical examples, illustrating surface plasmon propagation, are presented. The convergence rate of GMRES is discussed in terms of the spectrum of the system matrix and through numerical experiments we show how the eigenvalues of the discretized volume scattering operator are related to plasmon propagation and the medium parameters of a metallic object.

New Single-Source Surface Integral Equations for Scattering on Penetrable Cylinders and Current Flow Modeling in 2-D Conductors

The traditional volume electric field integral equation (IE) used for solution of full-wave scattering problems on penetrable scatterers of arbitrary cross section and its magnetostatic counterpart commonly utilized for the resistance and inductance extraction problem are reduced to a novel derivative-free single-source surface IE. The reduction of volume to surface IE is based on representation of the electric field in the cylinder cross section in the form of a single-layer ansatz. Substitution of such surface based electric field representation into the volume IE reduces it to a surface IE with respect to the unknown surface current density. Since the new surface IE enforces exactly the field continuity at the material interfaces, the radiation condition as well as underlying Helmholtz equations both inside and outside the penetrable cylinder, it is rigorously equivalent to the solution of Maxwell's equations. The method of moments discretization of the new IE is shown to produce an error-controllable field approximation. Due to the presence of a product of surface-to-volume and volume-to-surface integral operators, the discretization of the novel surface-volume-surface IE requires both surface and volume meshes.

Volume Integral Equation Analysis of Thin Dielectric Sheet Using Sinusoidal Macro-Basis Functions

In this letter, we present an improvement over the conventional thin dielectric sheet (TDS) formulation for the analysis of thin dielectric sheets. Our focus is to address the problem of scattering from thin penetrable scatterers by developing a volumetric formulation based on the use of macro-basis functions. The electric fields produced by the source basis functions are derived directly, bypassing the summation of scalar and vector potentials. The latter results in a considerable time advantage in comparison to the conventional method of moments (MoM) solution of the volume electric field integral equation (V-EFIE), while the accuracy remains comparable. In contrast to the TDS formulation, both tangential and normal currents are employed so that problems with high and low permittivity, as well as those involving grazing incident angles, can be accurately analyzed. Several examples are reported that show excellent agreement with conventional techniques and demonstrate the effectiveness of the new approach.

Analysis of Volume Integral Equation Formulations for Scattering by High-Contrast Penetrable Objects

The volume integral equation method is applied in electromagnetic scattering from arbitrarily shaped three-dimensional inhomogeneous objects. The properties of the volume electric and magnetic field integral equations (VEFIE and VMFIE) are investigated. Numerical experiments show that if the Galerkin's method with the lowest mixed-order basis functions is used to discretize the equations the accuracy of the VMFIE can be significantly poorer than the accuracy of the VEFIE, in particular, for high-contrast objects at high frequencies. The accuracy of the VMFIE can be essentially improved with full first order (linear) basis functions. The linear basis functions are found to be useful also when a single volume integral equation is used to model a general scatterer where both permittivity and permeability differ from the background.

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 Calderón Multiplicative Preconditioner for Coupled Surface-Volume Electric Field Integral Equations

A well-conditioned coupled set of surface (S) and volume (V) electric field integral equations (S-EFIE and V-EFIE) for analyzing wave interactions with densely discretized composite structures is presented. Whereas the V-EFIE operator is well-posed even when applied to densely discretized volumes, a classically formulated S-EFIE operator is ill-posed when applied to densely discretized surfaces. This renders the discretized coupled S-EFIE and V-EFIE system ill-conditioned, and its iterative solution inefficient or even impossible. The proposed scheme regularizes the coupled set of S-EFIE and V-EFIE using a Calderón multiplicative preconditioner (CMP)-based technique. The resulting scheme enables the efficient analysis of electromagnetic interactions with composite structures containing fine/subwavelength geometric features. Numerical examples demonstrate the efficiency of the proposed scheme.

A Well-Conditioned Integral-Equation Formulation for Efficient Transient Analysis of Electrically Small Microelectronic Devices

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> A hierarchically regularized coupled set of time-domain surface and volume electric field integral-equations (TD-S-EFIE and TD-V-EFIE) for analyzing electromagnetic wave interactions with electrically small and geometrically intricate composite structures comprising perfect electrically conducting surfaces and finite dielectric volumes is presented. A classically formulated coupled set of TD-S- and V-EFIEs is shown to be ill-conditioned at low frequencies owing to the hypersingular nature of the TD-S-EFIE. To eliminate low-frequency breakdown in marching-on-in-time solvers for these coupled equations, a hierarchical regularizer leveraging generalized Rao–Wilton–Glisson functions is applied to the TD-S-EFIE; no regularization is applied to the TD-V-EFIE as it is protected from low-frequency breakdown by an identity term. The resulting hierarchically regularized hybrid TD-S- and V-EFIE solver is applicable to the analysis of wave interactions with electrically small and densely meshed structures of arbitrary topology. The accuracy, efficiency, and applicability of the proposed solver are demonstrated by analyzing crosstalk in a six-port transmission line, radiation from a miniature radio-frequency identification antenna, and, plane-wave coupling onto a partially-shielded and fully loaded two-layer computer board. </para>

Electromagnetic scattering by arbitrarily shaped stratified anisotropic media using the equivalent dipole moment method

The equivalent dipole moment method, which was used to model the isotropic media, is extended and applied to the analysis of the electromagnetic scattering characteristics of arbitrarily shaped multilayer electric anisotropic media in this work. The initial motivation to put forward this method is based on the intrinsic physical properties of the electric anisotropic media whose constitutive parameter permittivity is a tensor matrix that can be modeled as equivalent electric dipole moment. This method employs the method of moments to solve the electric field volume integral equation (VIE) formulated by discretizing the scattering body into tetrahedral volume elements, in which the electrical parameters are assumed constant in each element. Then the VIE is solved directly to obtain the scattered field. Numerical results are given to validate the accuracy and efficiency of this method. © 2010 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2010.

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.

Implicitly Restarted Gmres Fast Fourier Transform Method for Electromagnetic Scattering

The implicitly restarted generalized minimum residual method (IRGMRES) combined with the fast Fourier transform (FFT) technique is developed for solving three-dimensional (3-D) weak-form volume electric field integral equation of electromagnetic scattering problems. On several electromagnetic scattering problems, the resulted IRGMRES-FFT method converges two-three times faster than the conventional biconjugate gradient (BCG)-FFT method. Comparison with other Krylov-subspace iterative fast Fourier transforms methods also demonstrates the efficiency of the IRGMRES-FFT method.

An Improved Weak-Form BCGS-FFT Combined With DCIM for Analyzing Electromagnetic Scattering by 3-D Objects in Planarly Layered Media

Fast algorithms for electrically large objects buried in layered media are mainly hindered by two time-consuming processes. One is the table filling of Green's function, and the other is the solving of the impedance matrix equation. For the first, to accelerate the evaluation of the time-consuming Sommerfeld integral in the dyadic Green's function (DGF), the discrete complex image method (DCIM) is introduced to get a closed-form DGF. To further accelerate the calculation of DGF for the volume electric field integral equation (EFIE), DGF is split before applying DCIM. For the second, the iterative solver stabilized biconjugate gradient fast Fourier transform (BCGS-FFT) is combined with DCIM for solving the matrix equations. Meanwhile, the closed-form DGF enables the spherical-mean Green's function, which eliminates the singularity of Green's function. Numerical results show that the weaker singularity results in a faster and steadier convergence rate for iterative solvers

Fast iterative, coupled-integral-equation technique for inhomogeneous profiled and periodic slabs

A fast coupled-integral-equation (CIE) technique is developed to compute the plane-TE-wave scattering by a wide class of periodic 2D inhomogeneous structures with curvilinear boundaries, which includes finite-thickness relief and rod gratings made of homogeneous material as special cases. The CIEs in the spectral domain are derived from the standard volume electric field integral equation. The kernel of the CIEs is of Picard type and offers therefore the possibility of deriving recursions, which allow the computation of the convolution integrals occurring in the CIEs with linear amounts of arithmetic complexity and memory. To utilize this advantage, the CIEs are solved iteratively. We apply the biconjugate gradient stabilized method. To make the iterative solution process faster, an efficient preconditioning operator (PO) is proposed that is based on a formal analytical inversion of the CIEs. The application of the PO also takes only linear complexity and memory. Numerical studies are carried out to demonstrate the potential and flexibility of the CIE technique proposed. Though the best efficiency and accuracy are observed at either low permittivity contrast or high conductivity, the technique can be used in a wide range of variation of material parameters of the structures including when they contain components made of both dielectrics with high permittivity and typical metals.

A parallel FFT accelerated transient field-circuit simulator

A novel fast electromagnetic field-circuit simulator that permits the full-wave modeling of transients in nonlinear microwave circuits is proposed. This time-domain simulator is composed of two components: 1) a full-wave solver that models interactions of electromagnetic fields with conducting surfaces and finite dielectric volumes by solving time-domain surface and volume electric field integral equations, respectively, and 2) a circuit solver that models field interactions with lumped circuits, which are potentially active and nonlinear, by solving Kirchoff's equations through modified nodal analysis. These field and circuit analysis components are consistently interfaced and the resulting coupled set of nonlinear equations is evolved in time by a multidimensional Newton-Raphson scheme. The solution procedure is accelerated by allocating field- and circuit-related computations across the processors of a distributed-memory cluster, which communicate using the message-passing interface standard. Furthermore, the electromagnetic field solver, whose demand for computational resources far outpaces that of the circuit solver, is accelerated by a fast Fourier transform (FFT)-based algorithm, viz. the time-domain adaptive integral method. The resulting parallel FFT accelerated transient field-circuit simulator is applied to the analysis of various active and nonlinear microwave circuits, including power-combining arrays.

A fast volume-surface integral equation solver for scattering from composite conducting-dielectric objects

This paper presents a fast hybrid volume-surface integral equation approach for the computation of electromagnetic scattering from objects comprising both conductors and dielectric materials. The volume electric field integral equation is applied to the material region and the surface electric field integral equation is applied on the conducting surface. The method of moments (MoM) is used to convert the integral equation into a matrix equation and the precorrected-FFT (P-FFT) method is employed to reduce the memory requirement and CPU time for the matrix solution. The present approach is sufficiently versatile in handling problems with either open or closed conductors, and dielectric materials of arbitrary inhomogeneity, due to the combination of the surface and volume electric field integral equations. The application of the precorrected-FFT method facilitates the solving of much larger problems than can be handled by the conventional MoM.

RCS Computation of Composite Conducting-Dielectric Objects with Junctions using the Hybrid Volume-Surface Integral Equation

The hybrid volume-surface integral equation (VSIE) combined with the Method of Moments (MoM) is applied to the analysis of scattering from arbitrarily composite conducting-dielectric objects withconducting-dielectric or dielectric-dielectric junctions. The electric field volume integral equation is applied to the material region and the electric field surface integral equation is enforced over the conducting surface. Compared to the surface integral equation (SIE), the VSIE is more efficient and convenient since the formulation of the VSIE retains the same simple form regardless of the complexity of the object and materials, and no extra treatment is required when dealing withjunction problems.

A fast volume integral equation solver for electromagnetic scattering from large inhomogeneous objects in planarly layered media

A newly developed iterative method, the stabilized biconjugate gradient fast Fourier transform (BCGS-FFT) method is applied to simulate electromagnetic scattering from large inhomogeneous objects embedded in a planarly layered medium. In this fast solver, the weak-form formulation is applied to obtain a less singular discretization of the volume electric field integral equation. Several techniques are utilized to speed up the dyadic function evaluation. To accelerate the operation of the dyadic function on an induced current (i.e., the Green's operation), the function is split into convolutional and correlational components so that FFT can be applied. The CPU time and memory cost of this BCGS-FFT method is O(NlogN) and O(N), respectively, where N is the number of unknowns, significantly more efficient than the method of moments (MoM). As a result, this method is capable of solving large-scale electromagnetic scattering problems in a planarly layered background. A large-scale scattering problem in a layered medium with more than three million unknowns has been solved on a Sun Ultra 60 workstation with 1.2 GBytes memory.