Volume Electric Field Integral Equation Research Articles

A Time-Domain Volume Integral Equation Solver to Analyze Electromagnetic Scattering From Nonlinear Dielectric Objects

A time domain electric field volume integral equation (TD-EFVIE) solver is proposed for analyzing electromagnetic scattering from dielectric objects with Kerr nonlinearity. The nonlinear constitutive relation that relates electric flux and electric field induced in the scatterer is used as an auxiliary equation that complements TD-EFVIE. The ordinary differential equation system that arises from TD-EFVIE’s Schaubert-Wilton-Glisson (SWG)-based discretization is integrated in time using a predictor-corrector method for the unknown expansion coefficients of the electric field. Matrix systems that arise from the SWG-based discretization of the nonlinear constitutive relation and its inverse obtained using the Padé approximant are used to carry out explicit updates of the electric field and the electric flux expansion coefficients at the predictor and the corrector stages of the time integration method. The resulting explicit marching-on-in-time (MOT) scheme does not call for any Newton-like nonlinear solver and only requires solution of sparse and well-conditioned Gram matrix systems at every step. Numerical results show that the proposed explicit MOT-based TD-EFVIE solver is more accurate than the finite-difference time-domain method that is traditionally used for analyzing transient electromagnetic scattering from nonlinear objects.

Multi-level Power Series Solution for Large Surface and Volume Electric Field Integral Equation

In this paper, we propose a new multi-level power series solution method for solving a large surface and volume electric field integral equation-based H-Matrix. The proposed solution method converges in a fixed number of iterations and is solved at each level of the H-Matrix computation. The solution method avoids the computation of a full matrix, as it can be solved independently at each level, starting from the leaf level. Solution at each level can be used as the final solution, thus saving the matrix computation time for full H-Matrix. The paper shows that the leaf level matrix computation and solution with power series gives as accurate results as the full H-Matrix iterative solver method. The method results in considerable savings time and memory savings compared to the H-Matrix iterative solver. Further, the proposed method retains the O(NlogN) solution complexity.

Adaptive Multilevel Nonuniform Grid Algorithm for the Accelerated Analysis of Composite Metallic–Dielectric Radomes

An adaptive multilevel nonuniform grid (MLNG) algorithm is developed for the accelerated computation of fields radiated through composite metallic–dielectric radomes as well as antenna-radome interactions. The MLNG approach is applied to the mixed potential formulation of the coupled surface and volume electric field integral equations. The radome is decomposed into a hierarchy of subdomains (SDs) by an adaptive algorithm that closely follows the radome geometry, allowing significant savings in memory and CPU time. In the MLNG algorithm, only local generalized impedance matrices of the finest level SDs are evaluated. Far-zone potentials and fields are indirectly evaluated through a multilevel aggregation involving phase-and amplitude-compensated interpolation on nonuniform grids (NGs), requiring considerably fewer calculations as compared with the classical method of moments (MoM). The MLNG algorithm is incorporated in a preconditioned iterative solver. Accuracy as well as memory consumption and computation times of the algorithm are studied on realistic examples. The radome effect on the antenna input impedance and electric current density distribution is demonstrated. The method is validated by comparison with a commercial MoM software and shown to exhibit a computational complexity (CC) of O(NlogN), N being the number of unknowns.

3-D Full-Wave Inversion of Helicopter Transient Electromagnetic Data in Frequency Domain

Helicopter transient airborne electromagnetics (HTEM) has become a useful tool in mineral explorations and geological or environmental detection in recent decades. This article presents the frequency-domain 3-D full-wave inversion of the electromagnetic data recorded by a newly built HTEM system. In the forward process, the secondary magnetic field is calculated through the volume electric-field integral equation (EFIE). In the inversion process, the secondary field is first extracted from the total field measured by the HTEM system. Then the Born iterative method (BIM) is adopted to solve the nonlinear inverse scattering problem for the 3-D reconstruction of conductivity. It is first applied to the synthetic models to verify their effectiveness and accuracy. The effects of adjacent underground anomalies on the 3-D inversion performed in a local region within the long flight lines are studied and discussed. Then the BIM solver is used to invert for the underground anomalies using the field measured data recorded by the newly built HTEM system. The 2-D slices from the reconstruction are compared with the nomogram. It is found that the locations of the high-conductivity regions in the BIM results are consistent with locations of the peaks in the nomogram. The reconstructed profiles are also compared with the drilling data obtained near one flight line. The good agreement shows that the 3-D BIM inversion algorithm can be used to reconstruct the underground ore in HTEM surveys.

Multiparametric Electromagnetic Inversion of 3-D Biaxial Anisotropic Objects Embedded in Layered Uniaxial Media Using VBIM Enhanced by Structural Consistency Constraint

This article presents the 3-D inversion of biaxial anisotropic objects embedded in layered uniaxial media by the variational Born iterative method (VBIM). Although previously only the isotropic complex permittivity profile was reconstructed, here we retrieve all the six diagonal anisotropic permittivity and conductivity parameters simultaneously. In the forward model, the stabilized biconjugate gradient fast Fourier transform (BCGS-FFT) method is used to solve the volume electric field integral equation (EFIE). In the inversion model, VBIM is employed to minimize the cost function. In order to improve the inversion results, we propose a novel structural consistency constraint (SCC) applied to VBIM. The SCC is based on the Monte Carlo method and reduces the computational domain in the Born iteration. The major new contribution of this work is to combine the deterministic inversion algorithm VBIM with the stochastic Monte Carlo method to reconstruct all the diagonal anisotropic parameters of the scatterers. Several typical numerical models are simulated, and the results validate the efficiency, antinoise ability as well as the accuracy of VBIM-SCC for reconstructing 3-D biaxial anisotropic objects embedded in multilayered uniaxial media.

Higher order VIE method based on non‐conformal discretisation for EM scattering from anisotropic objects

This article presents a higher order method of moments (MoM) solution of the volume electric field integral equation (VEFIE) to model the electromagnetic (EM) scattering from anisotropic dielectric objects with arbitrary shape and inhomogeneity. This higher order solution is based on the higher order current modelling and higher order geometric modelling. Namely, the unknown electric flux density vector is discretised by the higher order hierarchical vector (HOHV) basis functions, and the geometric structure is discretised by the curved tetrahedral elements. Both kinds of higher order discretisation can reduce the number of mesh elements and the associated unknowns significantly. When the solution accuracy is on a comparable level, authors’ higher order VIE demands much less memory and CPU time than the conventional low-order volume integral (VIE). Moreover, authors’ scheme accommodates the non-conformal discretisation, because no divergence operator is imposed on the vectorial unknown. Numerical results are provided to demonstrate the accuracy, efficiency, and flexibility of the proposed approach.

Validation of a volume integral equation method for indoor propagation modelling

A full-wave indoor propagation model based on the two-dimensional (2D) volume electric field integral equation (VEFIE) is presented. The model is discretised with the method of moments (MoM). The linear system arising from the MoM discretisation is solved with an iterative solver accelerated by using the fast Fourier transform and a reduced operator. The 2D VEFIE is validated against the analytical Mie series solution for scattering from a dielectric cylinder before being applied to a typical indoor propagation environment and compared against measurements. It is also compared against current popular empirical models and ray tracing. The ability of the frequency domain VEFIE method to produce time domain information is demonstrated with an investigation of numerical considerations that must be made.

Application of a CMP to an EFIE–TDS surface integral equation

Convergence rates of coupled electric field integral equation (EFIE) and thin dielectric sheet (TDS) surface integral equation is improved using a Calderon multiplicative preconditioner (CMP) for efficient analysis of densely discretised structures comprising perfect electric conductors and thin dielectric layers. When relative permittivity of the thin dielectric layer is not high, the integral operator for the TDS tends to have fast convergence rate than that of the EFIE because the TDS is derived from a volume EFIE that is well-posed. However, the integral operator for the EFIE is both unbounded and ill-posed when it is applied to densely discretised surfaces. Thus, the coupled EFIE and TDS integral equation have a slow convergence rate in an iterative solver. The proposed method shows fast convergence rate by application of the CMP to the EFIE–TDS. As a result, an iteration number in the iterative solver is reduced in the proposed method.

Novel Single-Source Surface Integral Equation for Scattering Problems by 3-D Dielectric Objects

A new single-source integral equation is proposed for the solution of electromagnetic wave scattering problems. The traditional volume electric field integral equation is reduced to the new single-source surface integral equation by representing the electric field inside the scatterer as a superposition of spherical waves emanating from its boundary. Such new integral equation formulation has been previously developed for the scalar and vector cases of 2-D scattering problems. In this paper, the 3-D form of this new single-source surface integral equation for scattering on homogeneous nonmagnetic dielectrics is proposed. Detailed description of the method of moments (MoMs) discretization and its resultant matrices is presented. In order to validate the new integral equation formulation and verify the accuracy of its MoMs discretization, its solution is compared against the analytical Mie series solution and fields computed using the commercial electromagnetic analysis software.

Divergence-Conforming Constrained Basis Functions for Hexahedral Volume Elements

This communication presents an algebraic method for constructing arbitrary-order, divergence-conforming basis functions on hexahedral volume elements. The appropriate constraints within and on the boundaries of elements are provided. In particular, the handling of faces where quantities are discontinuous is discussed. The resulting bases are numerically characterized in terms of error convergence and system conditioning for a moment method discretization of the electric field volume integral equation for dielectric scatterers. Results show the accuracy of the proposed method as well as the low system matrix condition number that can be maintained as the basis order and mesh discretization are increased.

Taylor–Duffy Method for Singular Tetrahedron-Product Integrals: Efficient Evaluation of Galerkin Integrals for VIE Solvers

This paper presents an accurate and efficient technique for the numerical evaluation of singular six-dimensional (6-D) integrals over tetrahedron-product domains, with applications to the calculation of Galerkin matrix elements for discretized volume-integral-equation (VIE) solvers using Schaubert–Wilton–Glisson (SWG) and other tetrahedral basis functions. The proposed method extends the generalized Taylor–Duffy strategy—used to handle the singular triangle -product integrals arising in discretized surface-integral-equation formulations—to the tetrahedron-product case; it effects an exact transformation of a singular 6-D integral into a nonsingular lower dimensional integral, which may be evaluated by simple numerical cubature. The method is very general and may be applied—with the aid of automatic code generation facilitated by computer-algebra systems— to a wide variety of singular integrals arising in various volume-integral-equation (VIE) formulations with various types of tetrahedral basis functions, several examples of which are presented in this paper. To demonstrate the accuracy and efficiency of the method, it has been applied in the calculation of matrix elements for the volume electric-field integral equation discretized with SWG basis functions, where the method yields 12-digit or higher accuracy with a low computational cost—an improvement of many orders of magnitude compared to existing techniques.

Numerical Vector Green’s Function for S-Parameter Measurement With Waveport Excitation

We present a numerical vector Green’s function that directly links the object material property within the electric field volume integral equation (VIE) to the scattered S-parameters. This Green’s function is particularly useful for microwave inverse scattering applications in which S-parameter measurements are used to reconstruct the dielectric properties of unknown objects in an inhomogeneous background where the Green’s function does not have an analytic form. Unlike the dyadic Green’s function in a standard VIE, which relates the fields in the object domain (vector) to the scattered fields (vector), this vector Green’s function kernel relates the object domain fields (vector) to the measured S-parameters (scalar). Also this new vector Green’s function uses a waveport source method, which accurately accounts for the measured scattered S-parameters between antenna feeds of arbitrary impedance and modes. A detailed derivation is presented, and simulated and experimental examples are provided to validate this vector Green’s function for the prediction of scattered S-parameters.

New Vector Single-Source Surface Integral Equation for Scattering Problems on Dielectric Objects in 2-D

A new surface single-source integral equation (SSIE) is proposed for the solution of electromagnetic wave scattering problems in two dimensions. The traditional volume electric field integral equation (V-EFIE) is reduced to the new single-source surface integral equation by representing the electric field inside the scatterer as a superposition of cylindrical waves emanating from its boundary. While being rigorous in nature, the new single-source surface integral equation features half of the degrees of freedom compared with traditional surface integral equation formulations. It is amenable to derivative-free method of moments discretization as it features only electric-field-type of Green’s function instead of both electric and magnetic field Green’s functions.

New Single-Source Surface Integral Equation for Magneto-Quasi-Static Characterization of Transmission Lines Situated in Multilayered Media

We recently proposed a novel single-source integral equation (SSIE) for accurate broadband resistance and inductance extraction and current flow modeling in 2-D conductors. The new surface integral equation is advantageous compared with the traditional volume electric field integral equation (V-EFIE) used for the inductance extraction, since the unknown function is defined on the surface of conductors as opposed to the volumetric unknown current density in V-EFIE. The new SSIE is also more suitable for the solution of inductance extraction problems than the traditional surface integral equation formulations, as it features only a single unknown surface function as opposed to having the unknown equivalent electric and magnetic surface current densities. The new equation also features only the electric field Green’s functions unlike the previously known SSIE formulations. The latter property makes the new SSIE equation particularly suitable to the inclusion of the multilayered substrate effect into the inductance extraction model. This paper describes the generalization of the new SSIE formulation to the case of transmission line models embedded into the multilayered lossy substrates. This paper also shows how the matrix sparsity in the method of moments discretization of the novel integral equation can be exploited to accelerate its numerical solution and reduce associated memory use. This sparsity arises due to the skin-effect-based attenuation of the fields in conductors’ cross sections leading to vanishing levels of the matrix elements corresponding to the distant interactions. Typical examples of inductance extraction in complex interconnects situated in lossy substrate are considered to validate the proposed techniques against traditional approaches.

Parallel PWTD-Accelerated Explicit Solution of the Time-Domain Electric Field Volume Integral Equation

A parallel plane-wave time-domain (PWTD)-accelerated explicit marching-on-in-time (MOT) scheme for solving the time-domain electric field volume integral equation (TD-EFVIE) is presented. The proposed scheme leverages pulse functions and Lagrange polynomials to spatially and temporally discretize the electric flux density induced throughout the scatterers, and a finite-difference scheme to compute the electric fields from the Hertz electric vector potentials radiated by the flux density. The flux density is explicitly updated during time marching by a predictor–corrector (PC) scheme and the vector potentials are efficiently computed by a scalar PWTD scheme. The memory requirement and computational complexity of the resulting explicit PWTD-PC-EFVIE solver scale as $O({N_s}\,\log \,{N_s})$ and $O({N_s}{N_t})$ , respectively. Here, ${N_s}$ is the number of spatial basis functions and ${N_t}$ is the number of time steps. A scalable parallelization of the proposed MOT scheme on distributed-memory CPU clusters is described. The efficiency, accuracy, and applicability of the resulting (parallelized) PWTD-PC-EFVIE solver are demonstrated via its application to the analysis of transient electromagnetic wave interactions on canonical and real-life scatterers represented with up to 25 million spatial discretization elements.

Tunneling effects in electromagnetic wave scattering by nonspherical particles: A comparison of the Debye series and physical-geometric optics approximations

The accuracy of the physical-geometric optics (PG-O) approximation is examined for the simulation of electromagnetic scattering by nonspherical dielectric particles. This study seeks a better understanding of the tunneling effect on the phase matrix by employing the invariant imbedding method to rigorously compute the zeroth-order Debye series, from which the tunneling efficiency and the phase matrix corresponding to the diffraction and external reflection are obtained. The tunneling efficiency is shown to be a factor quantifying the relative importance of the tunneling effect over the Fraunhofer diffraction near the forward scattering direction. Due to the tunneling effect, different geometries with the same projected cross section might have different diffraction patterns, which are traditionally assumed to be identical according to the Babinet principle. For particles with a fixed orientation, the PG-O approximation yields the external reflection pattern with reasonable accuracy, but ordinarily fails to predict the locations of peaks and minima in the diffraction pattern. The larger the tunneling efficiency, the worse the PG-O accuracy is at scattering angles less than 90°. If the particles are assumed to be randomly oriented, the PG-O approximation yields the phase matrix close to the rigorous counterpart, primarily due to error cancellations in the orientation-average process. Furthermore, the PG-O approximation based on an electric field volume-integral equation is shown to usually be much more accurate than the Kirchhoff surface integral equation at side-scattering angles, particularly when the modulus of the complex refractive index is close to unity. Finally, tunneling efficiencies are tabulated for representative faceted particles.

Volume Integral Equation With Higher Order Hierarchical Basis Functions for Analysis of Dielectric Electromagnetic Scattering

An efficient method of moments (MoM) solution of volume electric field integral equation (V-EFIE) with a kind of higher order basis functions is presented to model scattering from a dielectric object with arbitrary shapes and inhomogenuity. This method is based on both higher order geometrical modeling and higher order current modeling, which is realized using a family of higher order hierarchical vector (HHOV) basis functions to expand the equivalent volume electric currents within curvilinear volume elements. This higher order scheme of MoM results in a significant reduction in meshing density and number of unknowns, leading to a remarkable reduction of memory and CPU time in comparison with conventional low-order MoM, while the accuracy of the solution remains at a comparable level. Numerical results agree with analytical solutions or with simulation results obtained by commercial software, demonstrating the accuracy, convergence, efficiency, flexibility, and adaptability of the new approach.

A Stable Marching On-In-Time Scheme for Solving the Time-Domain Electric Field Volume Integral Equation on High-Contrast Scatterers

A time-domain electric field volume integral equation (TD-EFVIE) solver is proposed for characterizing transient electromagnetic wave interactions on high-contrast dielectric scatterers. The TD-EFVIE is discretized using the Schaubert–Wilton–Glisson (SWG) and approximate prolate spherical wave (APSW) functions in space and time, respectively. The resulting system of equations cannot be solved by a straightforward application of the marching on-in-time (MOT) scheme since the two-sided APSW interpolation functions require the knowledge of unknown “future” field samples during time marching. Causality of the MOT scheme is restored using an extrapolation technique that predicts the future samples from known “past” ones. Unlike the extrapolation techniques developed for MOT schemes that are used in solving time-domain surface integral equations, this scheme trains the extrapolation coefficients using samples of exponentials with exponents on the complex frequency plane. This increases the stability of the MOT-TD-EFVIE solver significantly, since the temporal behavior of decaying and oscillating electromagnetic modes induced inside the scatterers is very accurately taken into account by this new extrapolation scheme. Numerical results demonstrate that the proposed MOT solver maintains its stability even when applied to analyzing wave interactions on high-contrast scatterers.

Electromagnetic Scattering by an Inhomogeneous Gyroelectric Sphere Using Volume Integral Equation and Orthogonal Dini-Type Basis Functions

The electromagnetic scattering by an inhomogeneous gyroelectric sphere is studied in this work. The solution is constructed using the well-known electric field volume integral equation (EFVIE). The kernel that incorporates the Green's function is expanded in spherical vector wave functions, and the unknown electric field and electric displacement is expanded in spherical vector wave functions using fully orthogonal Dini-type basis functions. The permittivity tensor of the spherical object supports a continuously varying radial inhomogeneity, while its permeability is that of free space. This approach allows a fully analytical computation of the unknown matrix coefficients in special cases of homogeneous permittivity tensors. The validation of our method is performed by comparisons with published results in the literature. New numerical results are given for various continuously varying permittivity tensors.

Surface–Volume–Surface Electric Field Integral Equation for Magneto-Quasi-Static Analysis of Complex 3-D Interconnects

A novel single-source surface–volume–surface integral equation is proposed for accurate broadband resistance and inductance extraction in 3-D interconnects. The new equation originates in the volume integral equation (VIE) traditionally used for magneto-quasi-static modeling of current flow in 3-D wires. The latter is reduced to a surface integral equation by representing the electric field inside each conductor segment as a superposition of cylindrical waves emanating from the conductor's boundary. As no approximation is utilized and all underlying boundary conditions and pertinent equations are satisfied in the reduction, the new integral equation is rigorously equivalent to the solution of the traditional volume electric field integral equation. The rigorous nature of the proposed single-source surface integral equation is corroborated numerically. In this paper, a detailed description of the method of moments discretization for the proposed surface integral equation is also presented. Numerical solution of the proposed surface integral equation for a 12-conductor bond-wire package is used to demonstrate the accuracy of the method and its computational benefits compared to the traditional solution based on the VIE.