Volume Integral Equation Solver 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.

On a Low-Frequency and Contrast-Stabilized Full-Wave Volume Integral Equation Solver for Lossy Media

In this article, we present a new regularized electric flux volume integral equation (D-VIE) for modeling high-contrast conductive dielectric objects in a broad frequency range. This new formulation is particularly suitable for modeling biological tissues at low frequencies, as it is required by brain epileptogenic area imaging, but also at higher ones, as it is required by several applications, including, but not limited to, deep brain stimulation (DBS). When modeling inhomogeneous objects with high complex permittivities at low frequencies, the traditional D-VIE is ill-conditioned and suffers from numerical instabilities that result in slower convergence and less accurate solutions. In this work, we address these shortcomings by leveraging a new set of volume quasi-Helmholtz projectors. Their scaling by the material permittivity matrix allows for the rebalancing of the equation when applied to inhomogeneous scatterers and, thereby, makes the proposed method accurate and stable even for high complex permittivity objects until arbitrarily low frequencies. Numerical results, canonical and realistic, corroborate the theory and confirm the stability and the accuracy of this new method both in the quasi-static regime and at higher frequencies.

A Butterfly-Accelerated Volume Integral Equation Solver for Broad Permittivity and Large-Scale Electromagnetic Analysis.

A butterfly-accelerated volume integral equation (VIE) solver is proposed for fast and accurate electromagnetic (EM) analysis of scattering from heterogeneous objects. The proposed solver leverages the hierarchical off-diagonal butterfly (HOD-BF) scheme to construct the system matrix and obtain its approximate inverse, used as a preconditioner. Complexity analysis and numerical experiments validate the O(N log 2 N) construction cost of the HOD-BF-compressed system matrix and O(N 1.5 log N) inversion cost for the preconditioner, where N is the number of unknowns in the high-frequency EM scattering problem. For many practical scenarios, the proposed VIE solver requires less memory and computational time to construct the system matrix and obtain its approximate inverse compared to a matrix-accelerated VIE solver. The accuracy and efficiency of the proposed solver have been demonstrated via its application to the EM analysis of large-scale canonical and real-world structures comprising of broad permittivity values and involving millions of unknowns.

Numerical Method for Solving Volume Integral Equations on a Nonuniform Grid

Numerical Method for Solving Volume Integral Equations on a Nonuniform Grid

A Fast Volume Integral Equation Solver With Linear Basis Functions for the Accurate Computation of EM Fields in MRI

A stable volume integral equation (VIE) solver based on polarization/magnetization currents is presented, for the accurate and efficient computation of the electromagnetic (EM) scattering from highly inhomogeneous and high contrast objects. We employ the Galerkin method of moments to discretize the formulation with discontinuous piecewise linear basis functions on uniform voxelized grids, allowing for the acceleration of the associated matrix-vector products in an iterative solver, with the help of FFT. Numerical results illustrate the superior accuracy and more stable convergence properties of the proposed framework, when compared against standard low-order (piecewise constant) discretization schemes and a more conventional VIE formulation based on electric flux densities. Finally, the developed solver is applied to analyze complex geometries, including realistic human body models, typically used in modeling the interactions between EM waves and biological tissue.

Noninvasive Estimation of Electrical Properties From Magnetic Resonance Measurements via Global Maxwell Tomography and Match Regularization.

In this paper, we introduce global Maxwell tomography (GMT), a novel volumetric technique that estimates electric conductivity and permittivity by solving an inverse scattering problem based on magnetic resonance measurements. GMT relies on a fast volume integral equation solver, MARIE, for the forward path, and a novel regularization method, match regularization, designed specifically for electrical property estimation from noisy measurements. We performed simulations with three different tissue-mimicking numerical phantoms of different complexity, using synthetic transmit sensitivity maps with realistic noise levels as the measurements. We performed an experiment at 7 T using an eight-channel coil and a uniform phantom. We showed that GMT could estimate relative permittivity and conductivity from noisy magnetic resonance measurements with an average error as low as 0.3% and 0.2%, respectively, over the entire volume of the numerical phantom. Voxel resolution did not affect GMT performance and is currently limited only by the memory of the graphics processing unit. In the experiment, GMT could estimate electrical properties within 5% of the values measured with a dielectric probe. This work demonstrated the feasibility of GMT with match regularization, suggesting that it could be effective for accurate in vivo electrical property estimation. GMT does not rely on any symmetry assumption for the electromagnetic field, and can be generalized to estimate also the spin magnetization, at the expense of increased computational complexity. GMT could provide insight into the distribution of electromagnetic fields inside the body, which represents one of the key ongoing challenges for various diagnostic and therapeutic applications.

Simulation of Electromagnetic Scattering of 3-D Inhomogeneous Biaxial Anisotropic Magnetodielectric Objects Embedded in Uniaxial Anisotropic Media by the Mixed-Order BCGS-FFT Method

This paper presents a volume integral equation (VIE) solver for the forward electromagnetic scattering of 3-D inhomogeneous biaxial anisotropic objects embedded in uniaxial anisotropic media. The optical axes of the objects can be rotated with arbitrary angles. The mixed-order basis functions are employed to discretize the VIE, i.e., the flux densities $(D, B)$ are expanded by the volumetric rooftop basis functions and the vector potentials $(A, F)$ are expanded by the second-order curl conforming basis functions. The weak form of the VIE is formulated by testing it using the same volumetric rooftop basis function and solved by the biconjugate gradient stabilized fast Fourier transform (BCGS-FFT) method. Several numerical simulations of different shapes anisotropic objects are performed and the results are compared with commercial software simulations to validate the accuracy and efficiency of the proposed solver based on different discretization schemes. The major new contribution of this paper is that not only the scatterer but also the background medium is magnetodielectrically anisotropic. Therefore, the dyadic Green’s function for the uniaxial anisotropic background medium is evaluated before solving the VIE.

Low-Complexity Direct and Iterative Volume Integral Equation Solvers With a Minimal-Rank ${\mathcal H}^2$-Representation for Large-Scale Three-Dimensional Electrodynamic Analysis

Linear complexity iterative and log-linear complexity direct solvers are developed for the volume integral equation (VIE) based general large-scale electrodynamic analysis. The dense VIE system matrix is first represented by a new cluster-based multilevel low-rank representation. In this representation, all the admissible blocks associated with a single cluster are grouped together and represented by a single low-rank block, whose rank is minimized based on prescribed accuracy. From such an initial representation, an efficient algorithm is developed to generate a minimal-rank ${\cal H}^2$-matrix representation. This representation facilitates faster computation, and ensures the same minimal rank's growth rate with electrical size as evaluated from singular value decomposition. Taking into account the rank's growth with electrical size, we develop linear-complexity ${\cal H}^2$-matrix-based storage and matrix-vector multiplication, and thereby an $O(N)$ iterative VIE solver regardless of electrical size. Moreover, we develop an $O(NlogN)$ matrix inversion, and hence a fast $O(NlogN)$ \emph{direct} VIE solver for large-scale electrodynamic analysis. Both theoretical analysis and numerical simulations of large-scale $1$-, $2$- and $3$-D structures on a single-core CPU, resulting in millions of unknowns, have demonstrated the low complexity and superior performance of the proposed VIE electrodynamic solvers. %The algorithms developed in this work are kernel-independent, and hence applicable to other IE operators as well.

The Mixed-Order BCGS-FFT Method for the Scattering of Three-Dimensional Inhomogeneous Anisotropic Magnetodielectric Objects

This paper presents the first FFT-based fast volume integral equation solver for inhomogeneous anisotropic magnetodielectric objects. The volume integral equations are formulated by employing mixed-order basis functions that expand the flux densities and vector potentials in the coupled field integral equations in terms of different sets of basis functions with different orders. Volumetric roof-top basis functions are used for flux densities whereas second-order curl conforming basis functions are used for vector potentials. A fast volume integral equations solver namely the BCGS-FFT method is then applied to accelerate the solution of this mixed-order weak-form formulation. Examples show that the mixed-order BCGS-FFT method has high accuracy compared to both analytical and numerical solutions. Examples also show the mixed-order BCGS-FFT method has high computational efficiency compared to commercial software.

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.

A Linear Complexity Direct Volume Integral Equation Solver for Full-Wave 3-D Circuit Extraction in Inhomogeneous Materials

An ${\cal H}^{2}$ -matrix based linear complexity direct matrix solution is developed for the volume integral equation (VIE) based broadband full-wave extraction of general 3-D circuits. Such circuits are in general electrically small or moderate, but contain arbitrarily shaped lossy conductors immersed in inhomogeneous dielectrics with ports located anywhere in the physical layout of the circuit. In the proposed direct solver, we first develop a well-conditioned VIE formulation without sacrificing the rigor and the advantages of the prevailing formulations. This formulation facilitates a robust direct solution of good accuracy even with a rank-1 representation. We then overcome the numerical challenge of solving the resultant highly unstructured system matrix mixed with both square and rectangular dense and sparse matrices by developing a fast linear complexity direct solution. This direct solution is capable of inverting dense matrices involving over 2 million unknowns in less than 1 h on a single CPU core running at 3 GHz. Numerical simulations of large-scale 3-D circuits and comparisons with state-of-the-art linear complexity iterative VIE solvers have demonstrated the accuracy, efficiency, and linear complexity of the proposed direct VIE solver.

Preconditioning of Periodic Fast Multipole Method for Solving Volume Integral Equations

Application of FMM to the volume integral equations for periodic boundary value problem for Maxwell's equations is investigated. We propose effective preconditioners for the discretized volume integral equations. We confirm the accuracy and high performances of the proposed method in several numerical examples including periodic problems near Wood's anomalies.

Fast Solutions of Volume Integral Equations for Electromagnetic Scattering by Large Highly Anisotropic Objects

Accurate analysis of electromagnetic problems including inhomogeneous or anisotropic structures requires solving volume integral equations (VIEs) in the integral-equation approach. When the structures are electrically large in dimensions or constitutively complicated in materials, fast numerical algorithms are desirable to accelerate the solution process. Traditionally, such fast solvers are developed based on the method of moments (MoM) with the divergence-conforming Schaubert-Wilton-Glisson basis function or curl-conforming edge basis function, but the basis functions may not be appropriate to represent unknown functions in anisotropic media. In this work, we replace the MoM with the Nyström method and develop the corresponding multilevel fast multipole algorithm (MLFMA) for solving large highly anisotropic problems. The Nyström method characterizes the unknown functions at discrete quadrature points with directional components and more degrees of freedom and it also allows the use of JM-formulation, which does not explicitly include material property in the integral kernels in the VIEs. These features, with its other well-known merits, can greatly facilitate the implementation of MLFMA for anisotropic structures. Typical numerical examples are presented to demonstrate the algorithm and good results have been observed.

ℋ 2 ‐matrix‐based fast volume integral equation solver for electrodynamic analysis

An ℋ2-matrix-based mathematical framework is introduced and further developed to reduce the computational complexity of the volume integral equation (VIE)-based analysis of electrodynamic problems. Numerical experiments have demonstrated a significant reduction in CPU time and memory consumption. A linear scaling with respect to matrix size in both CPU time and memory is achieved for small and medium-sized electrodynamic problems with an interpolation-based ℋ2-representation of the VIE operator in conjunction with a rank function for accuracy control. The proposed solver is applicable to arbitrarily shaped three-dimensional structures immersed in inhomogeneous materials.

Selected theoretical and numerical aspects of fast volume and surface integral equation solvers for simulation of elasto-acoustic waves in complex inhomogeneous media

Comparative analysis is considered of two fast FFT-matrix-compression based elasto-acoustic integral equation solvers, employing volumetric and surface formulations, and designed to analyze sound propagation inside a human head; in particular to examine mechanisms of energy transfer to the inner ear through airborne as well as non-airborn path-ways, and to assess effectiveness of noise-protection devices. Verification tests involving the fast surface and volume integral equation solvers are carried out comparing their predictions with those following from an analytical solution of field distribution in an elasto-acoustic layered sphere.Resultsiare presented of representative numerical simulations of acoustic energy transfer to the cochlea for a human head model containing a detailed geometry representation of the outer, middle, and inner ear. The geometry model consists of: (1) the outer surface of the skin surrounding the skull and containing (2) the outer ear represented by its exterior surface, the surface of the auditory canal, and the tympanic membrane modeled as a finite-thickness surface, (3) the middle ear consisting of the system of ossicles and supporting structures, (4) the skull described by its external surfaces and including (5) a set of surfaces representing the inner ear (boundaries of the cochlea, the vestibule, and the semi-circular canals). *This work is supported by a grant from US Air Force Office of Scientific Research.

On the Method of Moments Solutions for Volume Integral Equations With Inhomogeneous Dielectric Media

In the method of moments (MoM) for solving volume integral equations (VIEs), one usually approximates the unknown flux density with the Schaubert-Wilton-Glisson (SWG) basis function and has to assume a homogeneous material in each tetrahedral element. Also, the volume charge density is equivalently represented with a lower-order basis function, and there exist surface charges at the tetrahedral faces separating dissimilar media. These features could bring on certain numerical errors and inconvenience of implementation. In this letter, we suggest that the dyadic Green's function in the VIEs keep its original form without moving the del operator onto the basis and testing functions. In this way, the above problems can be removed although an efficient treatment for hypersingularity is required. With our technique of treating the hypersingularity, the scheme is feasible and shows certain merits as demonstrated in numerical examples.

A Full-Wave 2.5D Volume Integral Equation Solver for 3D Millimeter-Wave Scattering by Large Inhomogeneous 2D Objects

A two-and-a-half dimensional full-wave forward solver to compute the three-dimensional (3D) electromagnetic field scattered by an infinitely long inhomogeneous (lossy) dielectric cylinder with arbitrary cross-sectional shape under a given 3D time-harmonic illumination is presented. The relevant set of linear equations, obtained after performing a spatial Fourier transform of the fields along the axial direction and applying a method of moments discretization to the two-dimensional contrast source volume integral equation, is solved iteratively with a stabilized biconjugate gradient fast Fourier transform method. In this way, objects with cross-sectional dimensions of several to many wavelengths can be handled in a very fast way. Furthermore, a vectorial 3D Gaussian beam illumination, usually employed in active millimeter-wave (mm-wave) imaging systems, is implemented using a complex source Gaussian beam formulation. The validity of the method is proved by comparison with analytic results and with the results from a full-wave 3D solver. Finally, the method is applied to a millimeter-wave imaging example, showing the scattering from an object hidden under clothing on a human body.

APPLYING DIVERGENCE-FREE CONDITION IN SOLVING THE VOLUME INTEGRAL EQUATION

Applying divergence-free condition to volume integral equation solver will be discussed. Three schemes are available: basis reduction scheme, minimal complete volume loop basis set and expanded volume loop basis set. All of them will generate smaller matrix equations than the SWG basis. The first two schemes generate poorly-conditioned matrices that are hard to solve by iterative solvers. The expanded loop basis set is easier to solve iteratively in spite of the existence of a null space in the matrix. Moreover, the construction of the expanded loop basis set is much easier than the other two schemes.

A fast 2D volume integral-equation solver for scattering from inhomogeneous objects in layered media

The stabilized biconjugate gradient fast Fourier transform (BCGS-FFT) method is applied to simulate electromagnetic and acoustic scattering from inhomogeneous objects embedded in a layered medium in two dimensions. Two-dimensional layered-media Green's functions are computed adaptively by using Gaussian quadratures after singularity subtraction. The Green's function is split into convolutional and correlational components in order to apply the FFT so as to solve the scattering problem efficiently. The CPU time and memory cost of this BCGS-FFT method is O(N log N) and O(N), respectively, where N is the number of unknowns, which is significantly more efficient than using the method of moments (MoM). As a result, this method is capable of solving large-scale electromagnetic and acoustic scattering problems for inhomogeneous objects embedded in a layered medium with an arbitrary number of layers. © 2005 Wiley Periodicals, Inc. Microwave Opt Technol Lett 47: 128–134, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.21101

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.