Maxwell's Equations In Time Domain Research Articles

Characterization of Surface–Borehole Transient Electromagnetic Response in Electrical Anisotropic Media

When encountering sedimentary rocks with obvious laminations or fracture development zones, the conductivity of the conductive medium in different directions will change significantly, and the subsurface medium will exhibit macroscopic conductivity anisotropy. To analyze the impact of electrical anisotropy on the surface–borehole transient electromagnetic exploration method, we used the finite element method to investigate the electrical anisotropy surface–borehole transient electromagnetic three-dimensional (3D) forward algorithm, in which we used a tetrahedral mesh to spatially discretize the time–domain Maxwell equation. Then, we discretized it using the second-order backward Eulerian difference method, and we obtained the fields through the PARDISO solver. The validity and correctness of the algorithm were verified through comparison of a one-dimensional (1D) anisotropic model, a complex three-dimensional (3D) isotropic model, and a three-dimensional (3D) anisotropic half-space model. A typical anisotropic geological model was constructed to analyze the effects of anisotropic strata and anomalies in the different principal axis directions on the surface–borehole transient electromagnetic response. The results show that the response of the anisotropic medium is related to the direction of the transmitting source, and the response pattern is complex and volatile. The electrical anisotropic anomaly does affect the amplitude, which should be given special attention when performing surface–borehole transient electromagnetic inversion interpretation.

Model order reduction for parameterized electromagnetic problems using matrix decomposition and deep neural networks

A non-intrusive model order reduction (MOR) method for solving parameterized electromagnetic scattering problems is proposed in this paper. A database collecting snapshots of high-fidelity solutions is built by solving the parameterized time-domain Maxwell equations for some values of the material parameters using a fullwave solver based on a high order discontinuous Galerkin time-domain (DGTD) method. To perform a prior dimensionality reduction, a set of reduced basis (RB) functions are extracted from the database via a two-step proper orthogonal decomposition (POD) method. Intrinsic coordinates of the high-fidelity solutions are further compressed through a convolutional autoencoder (CAE) network. Singular value decomposition (SVD) is then used to extract the principal components of the low dimensional coding matrices generated by CAE, and a cubic spline interpolation-based (CSI) approach is employed for approximating the dominating time- and parameter-modes of these matrices. The generation of the reduced basis and the training of the CAE and CSI are accomplished in the offline stage, thus the RB solution for given time/parameter values can be quickly recovered via outputs of the interpolation model and decoder network. In particular, the offline and online stages of the proposed RB method are completely decoupled, which ensures the validity of the method. The performance of the proposed CAE-CSI ROM is illustrated with numerical experiments for scattering of a plane wave by a 2-D dielectric disk and a multi-layer heterogeneous medium.

Optical investigation and computational modelling of BaTiO3 for optoelectronic devices applications

ABX3 perovskite-based materials have attracted research attention in various electronic and optoelectronic applications. The ability to tune the energy band gap through various dopants makes perovskites a potential candidate in many implementations. Among various perovskite materials, BaTiO3 has shown great applicability as a robust UV absorber with an energy band gap of around 3.2 eV. Herein, we provide a new sonochemical-assisted solid-phase method for preparing BaTiO3 thin films that optoelectronic devices can typically be used. BaTiO3 nano-powder and the thin film deposited on a glass substrate were characterized using physicochemical and optical techniques. In addition, the work demonstrated a computational attempt to optically model the BaTiO3 from the atomistic level using density functional theory to the thin film level using finite difference time domain Maxwell's equation solver. Seeking repeatability, the dispersion and the extinction behavior of the BaTiO3 thin film have been modeled using Lorentz-Dude (LD) coefficients, where all fitting parameters are listed. A numerical model has been experimentally verified using the experimental UV–Vis spectrometer measurements, recording an average root-mean-square error of 1.44%.

A non-intrusive model order reduction approach for parameterized time-domain Maxwell's equations

<p style='text-indent:20px;'>We present a non-intrusive model order reduction (NIMOR) approach with an offline-online decoupling for the solution of parameterized time-domain Maxwell's equations. During the offline stage, the training parameters are chosen by using Smolyak sparse grid method with an approximation level <inline-formula><tex-math id="M1">\begin{document}$ L $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M2">\begin{document}$ L\geq1 $\end{document}</tex-math></inline-formula>) over a target parameterized space. For each selected parameter, the snapshot vectors are first produced by a high order discontinuous Galerkin time-domain (DGTD) solver formulated on an unstructured simplicial mesh. In order to minimize the overall computational cost in the offline stage and to improve the accuracy of the NIMOR method, a radial basis function (RBF) interpolation method is then used to construct more snapshot vectors at the sparse grid with approximation level <inline-formula><tex-math id="M3">\begin{document}$ L+1 $\end{document}</tex-math></inline-formula>, which includes the sparse grids from approximation level <inline-formula><tex-math id="M4">\begin{document}$ L $\end{document}</tex-math></inline-formula>. A nested proper orthogonal decomposition (POD) method is employed to extract time- and parameter-independent POD basis functions. By using the singular value decomposition (SVD) method, the principal components of the reduced coefficient matrices of the high-fidelity solutions onto the reduced-order subspace spaned by the POD basis functions are extracted. Moreover, a Gaussian process regression (GPR) method is proposed to approximate the dominating time- and parameter-modes of the reduced coefficient matrices. During the online stage, the reduced-order solutions for new time and parameter values can be rapidly recovered via outputs from the regression models without using the DGTD method. Numerical experiments for the scattering of plane wave by a 2-D dielectric cylinder and a multi-layer heterogeneous medium nicely illustrate the performance of the NIMOR method.</p>

RI-CDI-FDTD method and program implementation for electromagnetic characteristics simulation of lossy Debye dispersive medium

Dispersive media refer to a class of natural substances, including living organisms, composite materials, plasma and water, with diverse applications in areas such as biomedicine, microwave sensing, electromagnetic protection, and stealth technology. In the pursuit of investigating the electromagnetic properties of these media, time-domain numerical methods, including finite difference in time domain (FDTD), finite element method, and time domain boundary integral equation method, have been widely utilized. Time-domain numerical methods are preferred to their frequency-domain counterparts owing to their ability to handle nonlinear and wideband problems, as well as various material properties. The FDTD method, in particular, is a highly adaptable, robust, and easy-to-use numerical method that directly solves the Maxwell equations while also simulating the reflection, transmission, and scattering of electromagnetic waves in complex dispersion media. Nonetheless, the traditional FDTD method suffers low computational efficiency arising from the Courant-Friedrichs-Lewy (CFL) stability condition. To solve the problem of low computational efficiency, a new method, the complying divergence implicit finite-difference time-domain (CDI-FDTD) method with a one-step leapfrog scheme, is introduced for lossy Debye dispersive media. The Maxwell equations in the frequency domain form a starting point, and the Fourier transform is utilized to transform the electromagnetic field components from the frequency domain to the time domain. To approximate the integral terms arising from the frequency-to-time domain transformation, a recursive integration (RI) method is employed. Subsequently, the time-domain Maxwell equations and auxiliary variables are discretized with a one-step leapfrog implicit scheme. The iterative formula of the RI-CDI-FDTD algorithm for lossy Debye dispersive media is then derived. The RI-CDI-FDTD method does not change the formulas of the traditional CDI-FDTD method while only requiring to add auxiliary variables for updating field components to the dispersive medium region. The numerical implementation is straightforward, and the electromagnetic modeling is flexible. Moreover, the unconditional stability of the RI-CDI-FDTD algorithm is proven by using the von Neumann method. Finally, some numerical examples are presented to demonstrate the effectiveness and efficiency of the proposed method. In conclusion, our work contributes a crucial numerical simulation tool to accurately modeling complex dispersive media while providing a systemic stability analysis method for time-domain numerical methods.

A novel time‐domain numerical methodology for the electromagnetic analysis of an H‐plane tee power divider

A time-domain numerical methodology for the transient analysis of multisection waveguide power dividers is proposed. Waveguide power dividers are essential components of microwave systems for high-power applications. A numerical strategy based on the 3D discontinuous Galerkin time-domain finite element method is used to estimate the electromagnetic propagation inside a power divider with three ports. The proposed approach leads to simulations that are fast, robust, and accurate, this being the result of a parallel local discretisation of the full-wave time-domain Maxwell's equations on unstructured tetrahedral meshes. The stability of the overall numerical scheme is obtained using an upwind numerical flux, an implicit second-order accurate time integration scheme, and Whitney's edge elements. Good agreements are obtained when the computed results are compared with other HFSS simulation results, and with measurement results found in the literature.

Divergence Error Based p-adaptive Discontinuous Galerkin Solution of Time-domain Maxwell's Equations

A $p$-adaptive discontinuous Galerkin time-domain method is developed to obtain high-order solutions to electromagnetic scattering problems. A novel feature of the proposed method is the use of divergence error to drive the $p$-adaptive method. The nature of divergence error is explored and that it is a direct consequence of the act of discretization is established. Its relation with relative truncation error is formed which enables the use of divergence error as an inexpensive proxy to truncation error. Divergence error is used as an indicator to dynamically identify and assign spatial operators of varying accuracy to substantial regions in the computational domain. This results in a reduced computational cost than a comparable discontinuous Galerkin time-domain solution using uniform degree piecewise polynomial bases throughout.

Automated adaptive TFSF method for solving time domain Maxwell's equations

AbstractThe total field–scattered field (TFSF) formulation is quite popular for solving Maxwell's equations in finite difference time domain (FDTD) and finite volume time domain (FVTD) methods. Here a fixed TFSF interface around the scattering body is employed to separate the total field (TF) cells and scattered field (SF) cells. This interface is selected based on experience as it is assigned prior to the solution, and, therefore, the solution accuracy may not be optimal. In the present work, a novel automated adaptive total field–scattered field (ATFSF) method is proposed for accurately computing the electromagnetic scattering by complex‐shaped metal objects excited by an external plane wave. In the proposed method, an algorithm is devised to dynamically identify the TF cells and the SF cells, to be solved using TF and SF methods, respectively, using intensity of electromagnetic (EM) wave. The algorithm is tested in a 2‐D spectral volume time domain (SVTD) framework for solving electromagnetic scattering, and the predicted surface current/radar cross section is compared against the SF, TF, and the conventional TFSF methods. It is seen that the present method improves the solution accuracy significantly.

Multiscale Approach and Analysis for Transient Simulation of Light Interaction With Nonlocal Metallic Nanostructure Arrays

Multiscale Approach and Analysis for Transient Simulation of Light Interaction With Nonlocal Metallic Nanostructure Arrays

Discontinuous Galerkin sparse grids methods for time domain Maxwell’s equations

In this paper, we present a time-explicit sparse grid discontinuous Galerkin method for solving the three-dimensional time-domain Maxwell equations. The conservation properties and convergence rates are established for different choices of numerical fluxes. The convergence rates are proved theoretically and then verified by several numerical examples. Even though our scheme does not preserve the divergence, but by implying the higher order polynomial, one can observe the same convergence rate as numerical solution. Several numerical tests are presented to validate these conclusions.

High-order, Dispersionless “Fast-Hybrid” Wave Equation Solver. Part I: O(1) Sampling Cost via Incident-Field Windowing and Recentering

This paper proposes a frequency/time hybrid integral-equation method for the time dependent wave equation in two and three-dimensional spatial domains. Relying on Fourier Transformation in time, the method utilizes a fixed (time-independent) number of frequency-domain integral-equation solutions to evaluate, with superalgebraically-small errors, time domain solutions for arbitrarily long times. The approach relies on two main elements, namely, 1) A smooth time-windowing methodology that enables accurate band-limited representations for arbitrarily-long time signals, and 2) A novel Fourier transform approach which, in a time-parallel manner and without causing spurious periodicity effects, delivers numerically dispersionless spectrally-accurate solutions. A similar hybrid technique can be obtained on the basis of Laplace transforms instead of Fourier transforms, but we do not consider the Laplace-based method in the present contribution. The algorithm can handle dispersive media, it can tackle complex physical structures, it enables parallelization in time in a straightforward manner, and it allows for time leaping---that is, solution sampling at any given time $T$ at $\mathcal{O}(1)$-bounded sampling cost, for arbitrarily large values of $T$, and without requirement of evaluation of the solution at intermediate times. The proposed frequency-time hybridization strategy, which generalizes to any linear partial differential equation in the time domain for which frequency-domain solutions can be obtained (including e.g. the time-domain Maxwell equations), and which is applicable in a wide range of scientific and engineering contexts, provides significant advantages over other available alternatives such as volumetric discretization, time-domain integral equations, and convolution-quadrature approaches.

An Accurate Three-Dimensional FDTD(2,4) Method on Face-Centered Cubic Grids With Low Numerical Dispersion

In this letter, a three-dimensional finite-difference time-domain (FDTD) method that is second-order in time and fourth-order in space on a face-centered cubic (FCC) grid, termed as FCC-FDTD(2,4), is proposed to solve the time-domain Maxwell's equations. The proposed method shows a much lower numerical dispersion error compared with the errors of other FDTD schemes, such as the FCC-FDTD(2,2) method, the FDTD(2,2) method, and the FDTD(2,4) method. This improvement in the numerical dispersion results from the FCC grid and the fourth-order scheme in space used in the proposed FCC-FDTD(2,4) method. Then, the Courant-Friedrichs-Lewy condition is given through spectral analysis, and an analytical numerical dispersion relation is obtained and comprehensively studied through numerical results with the FCC-FDTD(2,2) method, the FDTD(2,2) method, and the FDTD(2,4) method. Compared with the other three FDTD methods, the proposed method has much smaller numerical dispersion errors. Finally, the accuracy of the proposed method is further validated with other numerical examples.

Information Transfer by Near-Infrared Surface-Plasmon-Polariton Waves on Silver/Silicon Interfaces

Electronic interconnections restrict the operating speed of microelectronic chips as semiconductor devices shrink. As surface-plasmon-polariton (SPP) waves are localized, signal delay and crosstalk may be reduced by the use of optical interconnections based on SPP waves. With this motivation, time-domain Maxwell equations were numerically solved to investigate the transport of information by an amplitude-modulated carrier SPP wave guided by a planar silicon/silver interface in the near-infrared spectral regime. The critical-point model was used for the permittivity of silicon and the Drude model for that of silver. The signal can travel long distances without significant loss of fidelity, as quantified by the Pearson and concordance correlation coefficients. The signal is partially reflected and partially transmitted without significant loss of fidelity, when silicon is terminated by air; however, no transmission occurs when silicon is terminated by silver. The fidelity of the transmitted signal in the forward direction rises when both silicon and silver are terminated by air. Thus, signals can possibly be transferred by SPP waves over several tens of micrometers in microelectronic chips.

Unconditionally stable meshless method for solving 3D transient electromagnetic problems based on LOD scheme

One of the main problems of the conventional meshless methods in the time domain is their conditional stability. Up to now, some algorithms have been introduced to remove the stability condition, like alternating-direction-implicit (ADI) and locally one-dimensional (LOD) schemes. ADI meshless method has been investigated completely, but there is no report to exploit the full efficiency of the LOD meshless method. So, in this study, LOD scheme is employed to a meshless method for solving three-dimensional transient electromagnetic problems. The initial form of the proposed meshless method solves time-domain Maxwell's equations in three sub-steps, so it is called LOD3. LOD3 meshless method has first-order temporal accuracy. Moreover, LOD5 meshless method is introduced to upgrade temporal accuracy. Also, the results show that LOD5 meshless method is more accurate than ADI and LOD3 meshless methods. The accuracy of the proposed method is due to two main factors; Crank–Nicolson scheme and solving the equation in five sub-steps instead of one sub-step. Stability and accuracy of the proposed method are assessed through numerical experiment. Also, the unconditional stability of the proposed method is proved, analytically.

Analysis of time-domain Maxwell's equations in biperiodic structures

This paper is devoted to the mathematical analysis of the diffraction of an electromagnetic plane wave by a biperiodic structure. The wave propagation is governed by the time-domain Maxwell equations in three dimensions. The method of a compressed coordinate transformation is proposed to reduce equivalently the diffraction problem into an initial-boundary value problem formulated in a bounded domain over a finite time interval. The reduced problem is shown to have a unique weak solution by using the constructive Galerkin method. The stability and a priori estimates with explicit time dependence are established for the weak solution.

POD-based model order reduction with an adaptive snapshot selection for a discontinuous Galerkin approximation of the time-domain Maxwell's equations

In this work we report on a reduced-order model (ROM) for the system of time-domain Maxwell's equations discretized by a discontinuous Galerkin (DG) method. We leverage previous results on proper orthogonal decomposition (POD) [1,2], in particular for the wave equation [3], to propose a POD-based ROM with an adaptive snapshot selection strategy where the snapshots are produced by a high order discontinuous Galerkin time-domain (DGTD) solver. The latter is formulated on an unstructured simplicial mesh, and combines a centered scheme for the definition of the numerical fluxes of the electric and magnetic fields at element interfaces with a second order leap-frog (LF2) time scheme for the time integration of the associated semi-discrete equations. The POD-based ROM is established by projecting (Galerkin projection) the global semi-discrete DG scheme onto a low-dimensional space generated by the POD basis vectors. Inspired from the approach followed in [2,3], we derive error bounds for the POD-based ROM that is adapted to our particular modeling and discretization settings. The adaptive snapshot selection algorithm exploits the results of this analysis to measure the control error. A snapshot choosing rule aiming at keeping the error estimate close to a target selection error tolerance is proposed, which is similar to the standard rules found in adaptive time-stepping ordinary differential equations (ODEs) solvers. An incremental singular value decomposition (ISVD) algorithm is used to update the SVD on-the-fly when a new snapshot is available. The purpose of this adaptive selection strategy is to save memory without storing snapshots, while producing a smaller error. Numerical experiments for the 2-D time-domain Maxwell's equations nicely illustrate the performance of the resulting POD-based ROM with adaptive snapshot selection.

A Multiscale Approach Using Patches of Finite Elements for Solving Wave Propagation Problems in Microwave Discharge Plasma

We consider the development of an efficient numerical method for the simulation of microwave discharge plasmas. The method uses the idea of finite element patch and can deal with very disparate length scales of the plasma. In this paper, the time-domain Maxwell's equations, which are coupled with the plasma transport equations via the time-varying electron current density, are solved with a two-level Schwarz type algorithm based on a variational formulation of the standard Yee scheme. The patch of finite elements is used to calculate in an iterative manner the solution in the plasma region where a better precision is required. This numerical approach provides the Yee scheme with an efficient local-grid refinement capacity while preserving its stability. A numerical analysis shows its accuracy and computational efficiency on nested Cartesian grids. Simulation of a microwave breakdown in air under atmospheric pressure is then performed and results are discussed. We believe that both the inherent versatility with regard to the variational formulation and the efficiency of the proposed method can make it particularly suitable in modeling of microwave discharge plasmas by providing more insights of their nature and behavior.

Toward Information Transfer Around a Concave Corner by a Surface-Plasmon-Polariton Wave

As optical interconnects employing surface-plasmon-polariton (SPP) waves are expected to enable faster communication in integrated circuits, the time-domain Maxwell equations were solved using the finite-difference time-domain method to investigate the transfer of information via a pulse-modulated carrier SPP wave around a concave corner formed by two planar metal/air interfaces. The signal is launched along the first metal/air interface and received along the second metal/air interface. The corner angle affects the intensity and the duration of the received signal, but its shape is largely independent of the corner angle of the corner. Accordingly, the signal received around the concave corner is strongly and positively correlated with the transmitted signal, a promising result for SPP-wave-based optical interconnects. The energy of the received signal varies with the frequency of the carrier SPP wave. The received signal is better correlated with the transmitted signal when the carrier frequency is higher for a fixed value of the corner angle.

A Fully Fourth Order Accurate Energy Stable Finite Difference Method for Maxwell's Equations in Metamaterials

We present a novel fully fourth order in time and space finite difference method (FDM) for the time domain Maxwell's equations in metamaterials. We consider a Drude metamaterial model for the material response to incident electromagnetic fields. We consider the second-order formulation of the system of partial differential equations that govern the evolution in time of electric and magnetic fields along with the evolution of the polarization and magnetization current densities. Our discretization employs fourth-order staggering in space of different field components and the modified equation approach to obtain fourth-order accuracy in time. Using the energy method, we derive energy relations for the continuous models and design numerical schemes that preserve a discrete analogue of the energy relation. Numerical simulations are provided in one- and two-dimensional settings to illustrate fourth-order convergence as well as to compare with second-order schemes.

A high-order accurate scheme for Maxwell's equations with a generalized dispersive material model

A high-order accurate scheme for solving the time-domain Maxwell's equations with a generalized dispersive material model is described. The equations for the electric field are solved in second-order form, and a general dispersion model is treated with the addition of one or more polarization vectors which obey a set of auxiliary differential equations (ADE). Numerical methods are developed for both second-order and fourth-order accuracy in space and time. The equations are discretized using finite-differences, and advanced in time with a single-stage, three-level, space–time scheme which remains stable up to the usual explicit CFL restriction, as proven using mode analysis. Because the equations are treated in their second-order form, there is no need for grid staggering, and instead a collocated grid is used. Composite overlapping grids are used to treat complex geometries with boundary-conforming grids, and a high-order upwind dissipation is added to ensure robust and stable approximations on overlapping grids. Numerical results in two and three space dimensions confirm the accuracy and stability of the new schemes.