Three-dimensional Maxwell Equations Research Articles

Convergence Analysis of Fully Discrete Finite Volume Methods for Maxwell's Equations in Nonhomogeneous Media

We will consider both explicit and implicit fully discrete finite volume schemes for solving three-dimensional Maxwell's equations with discontinuous physical coefficients on general polyhedral domains. Stability and convergence for both schemes are analyzed. We prove that the schemes are second order accurate in time. Both schemes are proved to be first order accurate in space for the Voronoi--Delaunay grids and second order accurate for nonuniform rectangular grids. We also derive explicit expressions for the dependence on the physical parameters in all estimates.

Electromagnetic Modeling of Thin-Film Bulk Acoustic Resonators

This paper introduces a novel technique for the electromagnetic analysis of thin-film bulk acoustic-wave resonators. The piezoelectric/acoustic linear equations are coupled to and solved together with Maxwell's equations. For the acoustic propagation, locally monodimensional behavior is assumed, while full-wave three-dimensional Maxwell's equations are rigorously solved for the complete circuit. The proposed approach is validated for single resonators by comparison with experimental data, providing very good agreement with reduced computational resources. Hence, a complete Agilent Technologies' nine-resonator passband filter is simulated in order to demonstrate a typical application to a real-world complex problem.

Electromagnetic Inversion of GPR Signals and Subsequent Hydrodynamic Inversion to Estimate Effective Vadose Zone Hydraulic Properties

We combine electromagnetic inversion of ground penetrating radar (GPR) signals with hydrodynamic inverse modeling to identify the effective soil hydraulic properties of a sand in laboratory conditions. Ground penetrating radar provides soil moisture time series that are subsequently used as input in the hydrodynamic inverse procedure. The technique relies on an ultrawide band (UWB) stepped frequency continuous wave (SFCW) radar combined with an off-ground monostatic transverse electromagnetic (TEM) horn antenna. Ground penetrating radar signal forward modeling is based on the exact solution of the three-dimensional Maxwell equations for describing free wave propagation and on linear systems in series and parallel for describing wave propagation in the antenna. Water flow in the sand is described by the one-dimensional Richards equation using the Mualem–van Genuchten parameterization. Both model inversions are formulated by the classical least-squares problem and are performed iteratively using advanced global optimization techniques. Compared with time domain reflectometry (TDR), results demonstrated the appropriateness of the GPR integrated approach to measure soil moisture remotely. In particular, the approach was found to be less sensitive to the inherent small-scale heterogeneities. Hydrodynamic inversion of soil moisture data led to hydraulic parameters agreeing reasonably well with direct measurements. The observed discrepancies were attributed to the different characterization scales and samples. The overall integrated approach offers great promise to map the effective hydraulic properties of the shallow subsurface at a high spatial resolution.

High-Order Accurate Methods for Time-domain Electromagnetics

We discuss the formulation, validation, and parallel performance of a high-order accurate method for the time-domain solution of the three-dimensional Maxwell's equations on general unstructured grids. Attention is paid to the development of a general discontinuous element/penalty approximation to Maxwell's equations and a locally divergence free form of this. We further discuss the motivation for using a nodal Lagrangian basis for the accurate and efficient representation of solutions and operators. The performance of the scheme is illustrated by solving benchmark problems as well as large scale scattering applications.

Parametric Oscillations in Photonic Crystal Slabs 3-D Time-Domain Analysis

Slab waveguides with a two-dimensional periodic distribution of the refractive index are proposed and investigated as optical microcavities for efficient parametric oscillations through four-wave mixing in isotropic materials. We carry out a case study based on the complete solution of three-dimensional Maxwell equations, including material dispersion and cubic nonlinear response.

Unconditionally-stable FDTD method based on Crank-Nicolson scheme for solving three-dimensional Maxwell equations

The approximate-factorisation-splitting (CNAFS) method as an efficient implementation of the Crank-Nicolson scheme for solving the three-dimensional Maxwell equations in the time domain, using much less CPU time and memory than a direct implementation, is presented. At each time step, the CNAFS method solves tridiagonal matrices successively instead of solving a huge sparse matrix. It is shown that CNAFS is unconditionally stable and has much smaller anisotropy than the alternating-direction implicit (ADI) method, though the numerical dispersion is the same as in the ADI method along the axes. In addition, for a given mesh density, there will be one value of the Courant number at which the CNAFS method has zero anisotropy, whereas the Crank-Nicolson scheme always has anisotropy. Analysis shows that both ADI and CNAFS have time step-size limits to avoid numerical attenuation, although both are still unconditionally stable beyond their limit.

A simple finite-difference frequency-domain (FDFD) algorithm for analysis of switching noise in printed circuit boards and packages

Simultaneous switching noise (SSN) compromises the integrity of the power distribution structure on multilayer printed circuit boards (PCB). Several methods have been used to investigate SSN. These methods ranged from simple lumped circuit models to full-wave (dynamic) three-dimensional Maxwell equations simulators. In this work, we present an efficient and simple finite-difference frequency-domain (FDFD) based algorithm that can simulate, with high accuracy, the capacity of a PCB board to introduce SSN. The FDFD code developed here also allows for simulation of real-world decoupling capacitors that are typically used to mitigate SSN effects at sub 1 GHz frequencies. Furthermore, the algorithm is capable of including lumped circuit elements having user-specified complex impedance. Numerical results are presented for several test boards and packages, with and without decoupling capacitors. Validation of the FDFD code is demonstrated through comparison with other algorithms and laboratory measurements.

A scalable HPF implementation of a finite‐volume computational electromagnetics application on a CRAY T3E parallel system

AbstractThe time‐dependent Maxwell equations are one of the most important approaches to describing dynamic or wide‐band frequency electromagnetic phenomena. A sequential finite‐volume, characteristic‐based procedure for solving the time‐dependent, three‐dimensional Maxwell equations has been successfully implemented in Fortran before. Due to its need for a large memory space and high demand on CPU time, it is impossible to test the code for a large array. Hence, it is essential to implement the code on a parallel computing system. In this paper, we discuss an efficient and scalable parallelization of the sequential Fortran time‐dependent Maxwell equations solver using High Performance Fortran (HPF). The background to the project, the theory behind the efficiency being achieved, the parallelization methodologies employed and the experimental results obtained on the Cray T3E massively parallel computing system will be described in detail. Experimental runs show that the execution time is reduced drastically through parallel computing. The code is scalable up to 98 processors on the Cray T3E and has a performance similar to that of an MPI implementation. Based on the experimentation carried out in this research, we believe that a high‐level parallel programming language such as HPF is a fast, viable and economical approach to parallelizing many existing sequential codes which exhibit a lot of parallelism. Copyright © 2003 John Wiley & Sons, Ltd.

A Nonoverlapping Domain Decomposition Method for Maxwell's Equations in Three Dimensions

In this paper, we propose a nonoverlapping domain decomposition method for solving the three-dimensional Maxwell equations, based on the edge element discretization. For the Schur complement system on the interface, we construct an efficient preconditioner by introducing two special coarse subspaces defined on the nonoverlapping subdomains. It is shown that the condition number of the preconditioned system grows only polylogarithmically with the ratio between the subdomain diameter and the finite element mesh size but possibly depends on the jumps of the coefficients.

An efficient finite-element time-domain method for the analysis of the coupling between wave and shielded enclosure

In this paper, we present an efficient model to study the coupling of an electromagnetic wave with two electronic equipment enclosures. The first one has two large apertures, and the second one presents a fine slot and two metallic plates. The three-dimensional Maxwell's equations are solved using a finite-element time-domain method with a mass lumping technique. The proposed model is shown to avoid numerical problems and decrease memory and CPU time. Numerical results obtained with or without lumping are presented and compared with those obtained by measurements.

A Nondiffusive Finite Volume Scheme for the Three-Dimensional Maxwell's Equations on Unstructured Meshes

We prove a sufficient CFL-like condition for the L2 -stability of the second-order accurate finite volume scheme proposed by Remaki for the time-domain solution of Maxwell's equations in heterogeneous media with metallic and absorbing boundary conditions. We yield a very general sufficient condition valid for any finite volume partition in two and three space dimensions. Numerical tests show the potential of this original finite volume scheme in one, two, and three space dimensions for the numerical solution of Maxwell's equations in the time-domain.

Application of physics-based two-grid method and sparse matrix canonical grid method for numerical simulations of emissivities of soils with rough surfaces at microwave frequencies

The simulations of emissivities from a two-dimensional (2D) wet soil with random rough surfaces are studied with numerical solutions of three-dimensional (3D) Maxwell equations. The wet soils have large permittivity. For media with large permittivities, the surface fields can have large spatial variations on the surface. Thus, a dense discretization of the surface is required to implement the method of moment (MoM) for the surface integral equations. Such a dense discretization is also required to ensure that the emissivity can be calculated to the required accuracy of 0.01 for passive remote sensing applications. It has been shown that the physics-based two-grid method (PBTG) can efficiently compute the accurate surface fields on the dense grid. In this paper, the numerical results are calculated by using the PBTG in conjunction with the sparse-matrix canonical grid method (SMCG). The emissivities are illustrated for random rough surfaces with Gaussian spectrum for different soil moisture conditions. The results are calculated for L- and C-bands using the same physical roughness parameters. The numerical solutions of Maxwell's equations are also compared with the popular H and Q empirical model.

A three-dimensional finite-volume solver for the Maxwell equations with divergence cleaning on unstructured meshes

A finite-volume scheme on unstructured meshes for the three-dimensional time-dependent Maxwell equations is presented. To avoid the increase of numerical errors caused by suppressing the information contained in Gauss' law as well as the divergence-free condition of the magnetic induction, a divergence cleaning step is added which does not require the solution of a Poisson equation. The elliptical constraints of the Maxwell equations is approximated by a hyperbolic condition, starting from the so-called Generalised Lagrange Multiplier Maxwell model. This results in a purely hyperbolic system that fits very well in the framework of high-resolution finite-volume schemes yielding an efficient and flexible parallel Maxwell solver for explicit field calculations in time domain on unstructured grids in three space dimensions. Simulation results obtained with this new approximation technique are presented and compared with analytical as well as with other methods.

Reflectionless Sponge Layers as Absorbing Boundary Conditions for the Numerical Solution of Maxwell Equations in Rectangular, Cylindrical, and Spherical Coordinates

A scaling argument is used to derive reflectionless sponge layers to absorb outgoing time-harmonic waves in numerical solutions of the three-dimensional elliptic Maxwell equations in rectangular, cylindrical, and spherical coordinates. We also develop our reflectionless sponge layers to absorb outgoing transient waves in numerical solutions of the time-domain Maxwell equations and prove that these absorbing layers are described by causal, strongly well-posed hyperbolic systems. A representative result is given for wave scattering by a compact obstacle to demonstrate the many orders of magnitude improvement offered by our approach over standard techniques for computational domain truncation.

Magnetization of Mesoscopic Superconducting Disks

Solutions of Ginzburg-Landau equations coupled with three-dimensional Maxwell equations reveal an intriguing magnetic response of small superconducting particles, qualitatively different from the two-dimensional approximation but in agreement with recent experiments. Depending on the radius and thickness, first or second order transitions are found for the normal to superconducting state. For a sufficiently large radius of the disk, several transitions in the superconducting phase are obtained which correspond to different angular momentum giant vortex states. The incorporation of the finite thickness in the calculation is crucial in order to obtain agreement with the position and the size of these jumps, and the line shape and magnitude of the magnetization curves.

Studies of the generalized Ohm's law

Within the generalized Ohm's law, we present a tranformation which it similar to a thermoelectronic transport problem in a 2D nonhomogeneous medium/ For a general class of films with any number of different components, the resulting equations can be decoupled by a universal transformation. For the purpose of studying the applicability of the generalized Ohm's law, we consider the problem of an electromagnetic plane-wave scattered from a film which, although materially homogeneous, has a corrugated surface, in the regime where the wavelength is larger than the inhomogeneity length scales. The optical response of such a film is calculated by applying the generalized Ohm's law, and indendently by solving the three-dimensional Maxwell's equations and invoking the physical boundary conditions at the surface (Rayleigh method).

Decoupling and testing of the generalized Ohm's law

The generalized Ohm's law is an approximation formulated for the description of optical properties of nonhomogeneous thin films beyond the quasistatic approximation, when the inhomogeneity length scale is still much smaller than the wavelength, but is not smaller than the skin depth. Within this approximation, we transform the problem to a form that is similar to a thermoelectric transport problem in a two-dimensional (2D) nonhomogeneous medium. We show that for a general class of nonhomogeneous films with any number of different components, the resulting equations can be decoupled by a universal transformation. This simplifies practical application of the theory by reducing the coupled-field problem in a nonhomogeneous 2D system to two uncoupled conductivitylike problems with the same microgeometry, for which extensive theory already exists. For the purpose of studying the applicability of the generalized Ohm's law, we consider the problem of an electromagnetic plane wave scattered from a film that, although materially homogeneous, has a corrugated surface in the regime where the wavelength is larger than the inhomogeneity length scales. The optical response of such a film is calculated by applying the generalized Ohm's law, and independently by solving the three-dimensional Maxwell's equations, invoking the physical boundary conditions at the surface (Rayleigh method). We compare results of the two approaches for several configurations and materials and thus learn about the range of parameter values for which the generalized Ohm's law can provide a good description of the optical properties of a thin film.

Type of phase transitions in a mesoscopic superconducting disc

The Ginzburg–Landau equations coupled with the three-dimensional Maxwell equations are solved for the disc geometry in order to explain recent magnetization experiments. In order to explain the experimental results on a 0.5 μm radius Al disc, we have to assume that the superconducting state stays in the lowest angular momentum giant-vortex state even in regions where it is not the lowest energy state.

A Comparative Study of Characteristic-Based Algorithms for the Maxwell Equations

Characteristic-based finite-difference and finite-volume schemes have been developed for solving the three-dimensional Maxwell equations in the time domain. A detailed eigenvector analysis for the Maxwell equations in a general curvilinear coordinate has also been completed to provide a basic framework for future finite-difference schemes. Although the basic concepts of the two algorithms are identical, the detailed formulations are vastly different for achieving split flux vectors according to the sign of the eigenvalues. A comparative study of these algorithms applied to an oscillating electric dipole is carried out to assess their relative merit for further development. In spherical coordinates, second-order windward numerical simulations of the radiating phenomenon are closely comparable in terms of accuracy and efficiency. These methods also demonstrate an ability to suppress reflected waves from the truncated boundary by a simple compatibility condition.

Characteristic-based, time-dependent Maxwell equation solvers on a general curvilinear frame

Four cell-centered, finite-volume characteristic-based procedures for solving time-dependent, three-dimensional Maxwell equations on a general curvilinear frame are developed and implemented. On a spherical coordinate system, these methods generate simulations or an oscillating electric dipole without wave reflection from the truncated computational domain. Through a comparative study, it is established that an upwind-biased third-order scheme demonstrates less dispersive and dissipative errors than second-order schemes