On the influence of the geometry on skin effect in electromagnetism

We consider the equations of electromagnetism set on a domain made of a dielectric and a conductor subdomain in a regime where the conductivity is large. Assuming smoothness for the dielectric–conductor interface, relying on recent works we prove that the solution of the Maxwell equations admits a multiscale asymptotic expansion with profile terms rapidly decaying inside the conductor. This skin effect is measured by introducing a skin depth function that turns out to depend on the mean curvature of the boundary of the conductor. We then confirm these asymptotic results by numerical experiments in various axisymmetric configurations. We also investigate numerically the case of a nonsmooth interface, namely a cylindrical conductor.     

Evolution of Probability Distribution in Time for Solutions of Hyperbolic Equations

We investigate the evolution of the probability distribution function in time for some wave and Maxwell equations in random media for which the parameters, e.g. permeability, permittivity, fluctuate randomly in space; more precisely, two different media interface randomly in space. We numerically compute the probability distribution and density for output solutions. The underlying numerical and statistical techniques are the so-called polynomial chaos Galerkin projection, which has been extensively used for simulating partial differential equations with uncertainties, and the Monte Carlo simulations.     

Quantum lattice gas approach for the Maxwell equations

We show that a quantum lattice gas approach can provide a viable means for numerically solving the classical Maxwell equations. By casting the Maxwell equations in Dirac form, the propagator may be discretized, and we describe how the accuracy relative to the time step may be systematically increased. The quantum lattice gas form of the discretization is suitable for implementation on hybrid classical-quantum computers. We discuss a number of extensions, including application to inhomogeneous media.      

分散系介电理论的数值计算方法

介电谱方法能够对体系非破坏连续监测了解体系有关动力学,结果特征和电的性质,在化学研究中发挥着很大的作用,分散系介电理论公式大多数是复杂的复数方程,解析时繁琐的数学推导,拟合及求解高次方程给介电研究带来很大困难,为了解决上述剖,本文利用C＋＋语言并引入复数类对Moaxwel-Wagne界面极化理论和Hanai理论的介电解析以及Cole-Cole 公式的模拟进行了程序化,讨论了Hanai理论曲线的影响因素, 同时把该程序应用于微乳状液及文献中球型分散体系的介电解析,结果表明该程序能有效,准确地根据体系的介电参数解析出体系的相参数,进一步分析可得到体系的结构特性,提高了介电谱方法定性和定量研究的水平。     

Iterative addition of parallel temperature effects to finite-difference simulation of radio-frequency wave propagation in plasmas

Accurate simulations of how radio frequency (RF) power is launched, propagates, and absorbed in a magnetically confined plasma is a computationally challenging problem that for which no comprehensive approach presently exists. The underlying physics is governed by the Vlasov–Maxwell equations, and characteristic length scales can vary by three orders of magnitude. Present algorithms are, in general, based on finding the constituative relation between the induced RF current and the RF electric field and solving the resulting set of Maxwell’s equations. These linear equations use a Fourier basis set that is not amenable to multi-scale formulations and have a large dense coefficient matrix that requires a high-communications overhead factorization technique. Here the use of operator splitting to separate the current and field calculations, and a low-overhead iterative solver leads to an algorithm that avoids these issues and has the potential to solve presently intractable problems due to its data-parallel and favorable scaling characteristics. We verify the algorithm for the iterative addition of parallel temperature effects for a 1D electron Langmuir by reproducing the solution obtained with the existing Fourier kinetic RF code aorsa (Jaeger et al., 2008).     

Computer programs for application of equations describing elastic and electromagnetic wave scattering from planar interfaces

MATLAB programs are presented which solve equations describing the scattering of plane elastic and electromagnetic waves from a planar interface separating homogenous, isotropic, and semi-infinite geologic media. The PSHSV program calculates and plots amplitude (reflection and refraction/transmission) coefficients, square root energy ratios, energy coefficients, and phase changes for elastic waves of P-, SH- or SV-type incident on an interface between elastic media. The EHEV program calculates and plots amplitude coefficients, square root energy ratios, energy coefficients, and phase changes for electromagnetic waves of EH- or EV-type incident on an interface between dielectric media. The applicability of the programs is demonstrated through the presentation of solutions (plotted as a function of incidence angle) obtained for geologic environments commonly encountered in seismic and ground penetrating radar applications.     

Electromagnetic waves in media with permittivity dispersion

A technique for the numerical simulation of electromagnetic wave propagation in materials with permittivity depending on the frequency is presented. The technique is based on numerical solutions of Maxwell equations with additional integral components in the bias current density. The technique to calculate the bias current density in dispersive media is represented and the corresponding modification of finite-difference scheme for Maxwell equations developed earlier is carried out. The electromagnetic pulse propagation in solid-fuel power systems is calculated.     

A lattice Boltzmann model for electromagnetic waves propagating in a one-dimensional dispersive medium

A first-order extended lattice Boltzmann (LB) model with special forcing terms for one-dimensional Maxwell equations exerting on a dispersive medium, described either by the Debye or Drude model, is proposed in this study. The time dependent dispersive effect is obtained by the inverse Fourier transform of the frequency-domain permittivity and is incorporated into the LB evolution equations via equivalent forcing effects. The Chapman–Enskog multi-scale analysis is employed to ensure that proposed scheme is mathematically consistent with the targeted Maxwell’s equations at the macroscopic limit. Numerical validations are executed through simulating four representative cases to obtain their LB solutions and compare those with the analytical solutions and existing numerical solutions by finite difference time domain (FDTD). All comparisons show that the differences in numerical values are very small. The present model can thus accurately predict the dispersive effects, and demonstrate first order convergence. In addition to its accuracy, the proposed LB model is also easy to implement. Consequently, this new LB scheme is an effective approach for numerical modeling of EM waves in dispersive media.     

High Spatial Order Energy Stable FDTD Methods for Maxwell’s Equations in Nonlinear Optical Media in One Dimension

In this paper, we consider electromagnetic (EM) wave propagation in nonlinear optical media in one spatial dimension. We model the EM wave propagation by the time-dependent Maxwell’s equations coupled with a system of nonlinear ordinary differential equations (ODEs) for the response of the medium to the EM waves. The nonlinearity in the ODEs describes the instantaneous electronic Kerr response and the residual Raman molecular vibrational response. The ODEs also include the single resonance linear Lorentz dispersion. For such model, we will design and analyze fully discrete finite difference time domain (FDTD) methods that have arbitrary (even) order in space and second order in time. It is challenging to achieve provable stability for fully discrete methods, and this depends on the choices of temporal discretizations of the nonlinear terms. In Bokil et al. (J Comput Phys 350:420–452, 2017), we proposed novel modifications of second-order leap-frog and trapezoidal temporal schemes in the context of discontinuous Galerkin methods to discretize the nonlinear terms in this Maxwell model. Here, we continue this work by developing similar time discretizations within the framework of FDTD methods. More specifically, we design fully discrete modified leap-frog FDTD methods which are proved to be stable under appropriate CFL conditions. These method can be viewed as an extension of the Yee-FDTD scheme to this nonlinear Maxwell model. We also design fully discrete trapezoidal FDTD methods which are proved to be unconditionally stable. The performance of the fully discrete FDTD methods are demonstrated through numerical experiments involving kink, antikink waves and third harmonic generation in soliton propagation.     

Simulating Plasma Microwave Diagnostics

Computational simulation of plasma diagnostics via microwave absorption has been successfully accomplished. This simulation capability is developed from solutions to a combination of the three-dimensional Maxwell equations and the generalized Ohm’s law in the time domain. As the simulation procedure developed, numerical results were obtained for a range of plasma transport properties including electrical conductivity, permittivity, and plasma frequency. The present results reveal the wave reflection at the media interface and substantial distortion of the electromagnetic field within a thin plasma sheet from a guided microwave. The present numerical simulation also accurately predicts the microwave blackout phenomenon as the wave propagates through a thick plasma sheet. The diffractions and refractions occurring at antenna apertures and passing through a plasma column are captured numerically. Finally, the numerical simulation has successfully duplicated a plasma diagnostic experiment in a hypersonic magneto-hydrodynamic channel.     

A numerical investigation on AC electrowetting of a droplet

We numerically analyze the AC electric field around a droplet placed on an insulator-covered electrode. The time-averaged effective electrical wetting tension, which is a function of AC frequency, is computed by integrating the Maxwell stress. The computed wetting tension is compared with the experimental result converted from the separately obtained contact-angle data. There is a good agreement between the two results at a low-frequency range and a qualitative agreement at a high-frequency range. Interestingly, the numerical results show that the electric-field strength decreases remarkably in the insulating layer near the TCL as the AC frequency increases. This decrease may account for the delay of the dielectric breakdown of an insulating layer in the AC case, which could be related to the contact-angle saturation phenomenon.     

Interior Penalty Discontinuous Galerkin Method for Maxwell’s Equations in Cold Plasma

In this paper, we develop an interior penalty discontinuous Galerkin (DG) method for the time-dependent Maxwell’s equations in cold plasma. Both semi and fully discrete DG schemes are constructed, and optimal error estimates in the energy norm are proved. To our best knowledge, this is the first error analysis carried out for the DG method for Maxwell’s equations in dispersive media.     

Numerical solution of eddy current problems in bounded domains using realistic boundary conditions

The aim of this paper is to analyze a finite element method to solve the low-frequency harmonic Maxwell equations in a bounded domain containing conductors and dielectrics, and using realistic boundary conditions in that they can be easily measured. These equations provide a model for the so-called eddy currents. The problem is formulated in terms of the magnetic field. This formulation is discretized by using Nédélec edge finite elements on a tetrahedral mesh. Error estimates are easily obtained when the curl-free condition is imposed explicitly on the elements in the dielectric domain.A multivalued magnetic scalar potential is introduced then to impose this curl-free condition. The discrete counterpart of this formulation leads to an important saving in computational effort. Problems related to the topology are also considered, more precisely, the possibility of having a non-simply connected dielectric domain is taken into account. Finally, the method is applied to solve two three-dimensional model problems: a test with a known analytical solution and the computation of the electromagnetic field in a metallurgical arc furnace.     

On the immersed interface method for solving time-domain Maxwell's equations in materials with curved dielectric interfaces

This paper deals with accurate numerical simulation of two-dimensional time-domain Maxwell's equations in materials with curved dielectric interfaces. The proposed fully second-order scheme is a hybridization between the immersed interface method (IIM), introduced to take into account curved geometries in structured schemes, and the Lax-Wendroff scheme, usually used to improve order of approximations in time for partial differential equations. In particular, the IIM proposed for two-dimensional acoustic wave equations with piecewise constant coefficients [C. Zhang, R.J. LeVeque, The immersed interface method for acoustic wave equations with discontinuous coefficients, Wave Motion 25 (1997) 237-263] is extended through a simple least squares procedure to such Maxwell's equations. Numerical results from the simulation of electromagnetic scattering of a plane incident wave by a dielectric circular cylinder appear to indicate that, compared to the original IIM for the acoustic wave equations, the augmented IIM with the proposed least squares fitting greatly improves the long-time stability of the time-domain solution. Semi-discrete finite difference schemes using the IIM for spatial discretization are also discussed and numerically tested in the paper.     

A multichannel sensor for parameters of layered media with one detective element

We propose detective element structures for a multichannel sensor based on a distributed long line with kinks distributed along its length. We find generalized relations between the eigenvalues of a resonator with an arbitrary number of kinks and the parameters of layered media with an arbitrary number of layers in order to form control systems from which, having measured the eigenfrequencies, we can find the positions of layer boundaries and their dielectric permittivity. We study structures and algorithms that ensure uniqueness of the solution for these systems in two- and three-layered media.     

A FDTD analysis on magnetized plasma of Epstein distribution and reflection calculation

A novel finite-difference time-domain (FDTD) method with recursive relationships among operators, developed for magnetized dispersive medium, is named shift operator FDTD (SO-FDTD) method. In this paper, the dielectric property of magnetized dispersive media is written as rational polynomial function, the relationship between D and E is deduced in time-domain. The high accuracy and efficiency of this method is verified by calculating the reflection and transmission coefficient of electromagnetic waves through a collisional plasma slab. As the electron density in plasma distributes as Epstein formula, the effect of the gradient coefficient σ and plasma thickness on the reflection coefficient is calculated. The results show that with different σ, the distribution between the reflection coefficient and the incidence frequency is same; but there is certain difference within some specific frequency range. And there are fewer effects of different thicknesses on the reflection coefficient under different conditions.     

Discontinuous Galerkin Approximations for Computing Electromagnetic Bloch Modes in Photonic Crystals

We analyze discontinuous Galerkin finite element discretizations of the Maxwell equations with periodic coefficients. These equations are used to model the behavior of light in photonic crystals, which are materials containing a spatially periodic variation of the refractive index commensurate with the wavelength of light. Depending on the geometry, material properties and lattice structure these materials exhibit a photonic band gap in which light of certain frequencies is completely prohibited inside the photonic crystal. By Bloch/Floquet theory, this problem is equivalent to a modified Maxwell eigenvalue problem with periodic boundary conditions, which is discretized with a mixed discontinuous Galerkin (DG) formulation using modified Nédélec basis functions. We also investigate an alternative primal DG interior penalty formulation and compare this method with the mixed DG formulation. To guarantee the non-pollution of the numerical spectrum, we prove a discrete compactness property for the corresponding DG space. The convergence rate of the numerical eigenvalues is twice the minimum of the order of the polynomial basis functions and the regularity of the solution of the Maxwell equations. We present both 2D and 3D numerical examples to verify the convergence rate of the mixed DG method and demonstrate its application to computing the band structure of photonic crystals.     

Event-based simulation of light propagation in lossless dielectric media

We describe an event-based approach to simulate the propagation of an electromagnetic plane wave through dielectric media. The basic building block is a deterministic learning machine that is able to simulate a plane interface. We show that a network of two of such machines can simulate the propagation of light through a plane parallel plate. With properly chosen parameters this setup can be used as a beam splitter. The modularity of the simulation method is illustrated by constructing a Mach–Zehnder interferometer from plane parallel plates, the whole system reproducing the results of wave theory. A generalization of the event-based model of the plane parallel plate is also used to simulate a periodically stratified medium.     

Augmented formulations for solving Maxwell equations

We consider augmented variational formulations for solving the static or time-harmonic Maxwell equations. For that, a term is added to the usual H (curl) conforming formulations. It consists of a (weighted) L² scalar product between the divergence of the EM and the divergence of test fields. In this respect, the methods we present are H (curl, div) conforming. We also build mixed, augmented variational formulations, with either one or two Lagrange multipliers, to dualize the equation on the divergence and, when applicable, the relation on the tangential or normal trace of the field. It is proven that one can derive formulations, which are equivalent to the original static or time-harmonic Maxwell equations. In the latter case, spurious modes are automatically excluded. Numerical analysis and experiments will be presented in the forthcoming paper [Augmented formulations for solving Maxwell equations: numerical analysis and experiments, in preparation].     

折叠偶极子阵列无芯标签极点特性分析

标签极点提取的准确性受多种因素影响,以折叠偶极子阵列无芯标签为研究对象,开展无耗电介质材料(厚度、相对介电常数)对其极点分布的特性分析。仿真结果表明,随着介质厚度及其相对介电常数的增大,谐振极点的衰减因子及谐振频率将呈现变小的趋势,极点分布以类S型曲线向坐标原点靠近。