An enhanced finite difference time domain method for two dimensional Maxwell's equations

An enhanced finite-difference time-domain (FDTD) algorithm is built to solve the transverse electric two-dimensional Maxwell's equations with inhomogeneous dielectric media where the electric fields are discontinuous across the dielectric interface. The new algorithm is derived based upon the integral version of the Maxwell's equations as well as the relationship between the electric fields across the interface. To resolve the instability issue of Yee's scheme (staircasing) caused by discontinuous permittivity across the interface, our algorithm revises the permittivities and makes some corrections to the scheme for the cells around the interface. It is also an improvement over the contour-path effective permittivity algorithm by including some extra terms in the formulas. The scheme is validated in solving the scattering of a dielectric cylinder with exact solution from Mie theory and is then compared with the above contour-path method, the usual staircasing and the volume-average method. The numerical results demonstrate that the new algorithm has achieved significant improvement in accuracy over other methods. Furthermore, the algorithm has a simple structure and can be merged into current FDTD software packages easily. The C++ source code for this paper is provided as supporting information for public access.     

Nonlinear stability of an axial electric field: Effect of interfacial charge relaxation

We introduce a nonlinear perturbation technique to third order, to study the stability between two cylindrical inviscid fluids, subjected to an axial electric field. The study takes into account the relaxation of electrical charges at the interface between the two fluids. At first order, a linear dispersion relation is obtained. Analytical and numerical results for the overstability and incipient instability conditions are given. For perfect dielectric fluids, the electric field has a stabilizing influence, while for leaky dielectric fluids, the electric field can have either a stabilizing or a destabilizing influence depending on the conductivity and permittivity ratios of the two fluids. At higher order, a nonlinear dispersion relation (nonlinear Ginzburg–Landau equation) is derived, describing the evolution of wave packets of the problem. For leaky dielectric fluids near the marginal state, a nonlinear diffusion equation (nonlinear incipient instability) is obtained. For perfect dielectric fluids, two cubic nonlinear Schrödinger equations are obtained. One of these equations to determine a nonlinear cutoff electric field separating stable and unstable disturbance, whereas the other is used to analyze the stability of the system. It is found that the nonlinear stability criterion depending on the ratio of permittivity, Such effects can only be explained successfully in the nonlinear sense, as the linear analysis unsuccessful to inform about them.     

On the formation of gap solitons in nonlinear electromagnetic metamaterials

A nonlinear (Kerr‐type) electromagnetic metamaterial, characterized by a double‐Lorentz model of its frequency‐dependent linear effective dielectric permittivity and magnetic permeability, is considered. The formation of gap solitons in the low‐ and high‐frequency band gaps of this metamaterial is investigated analytically. Evolution equations governing the gap solitons, of the form of a nonlinear Klein‐Gordon and a nonlinear Schrödinger equation, are obtained, and the structure of their solutions is discussed.     

Constructing approximate Green’s function for a vector equation for the electric field using the variational iteration method

In this work, we use He’s variational iteration method (VIM) to find approximate Green’s functions for a vector equation for the electric field with anisotropic dielectric permittivity and magnetic permeability. We present numerical examples which show that an approximate solution of an initial value problem (IVP) for a vector equation can be obtained by using these approximate Green’s functions.     

Self-Similarity and the Evaluation of Long-Time Nonhydrostatic Effects in Compositionally Driven Gravity Flows in Deep Surroundings

In the study of compositionally driven gravity currents involving one or more homogeneous fluid layers, it has been customary to adopt the hydrostatic assumption for the pressure field in each layer which, in turn, leads to a depth‐independent horizontal velocity field in each of these layers and significant simplifications to the governing equations. Under this hydrostatic paradigm, each layer will then have its motion governed by the well‐known reduced dimension shallow‐water equations. For the so‐called ‐layer or reduced gravity shallow‐water equations, similarity solutions for fixed volume gravity currents released in rectangular geometry have been found. Very few attempts have been made to evaluate contributions arising from the possible loss of hydrostatic balance in the context of the problems treated using the classic shallow‐water approach. Where such attempts have been pursued, they have usually been carried out in a time‐independent context or using layer‐averaged equations and very small amplitude disturbances. The vast majority of these studies into nonhydrostatic effects do not include any relevant numerical work to assess these effects. In this paper, we develop an approach for evaluating nonhydrostatic contributions to the flow field for bottom gravity currents in deep surroundings and rectangular geometry. Our approach makes no assumptions on the amplitudes of the disturbances and does not depend on layer‐averaging in the governing equations. We seek asymptotic expansions of the solutions to the Euler equations for a shallow fluid by using the small parameter δ², where δ is the aspect ratio of the flow regime. At leading order the equations enforce hydrostatic balance while those obtained at first order retain certain nonhydrostatic effects which we evaluate. Our method for evaluation of these first‐order contributions employs the self‐similar nature of the solution to the leading‐order equations in the new first‐order equations without any vertical averaging procedures being employed.     

A Stability Estimate for a Solution to a Three-Dimensional Inverse Problem for the Maxwell Equations

We consider the problem of determining dielectric permittivity and conductivity in the Maxwell equations. As additional information we prescribe the traces of the tangential components of the electromagnetic field on the lateral surface of a cylindric domain. We establish a stability estimate for a solution to the inverse problem and a uniqueness theorem.     

Nuclear energy conversion with stacks of graphene nanocapacitors

The efficiency of conventional techniques used to harvest energy in nuclear reactors lies around 35%. This limit exists, because the nuclear energy is converted to electrical energy via heat engines. We study an alternative approach where the kinetic energy of nuclear reaction products is directly converted into electric energy in a stack of charged capacitors with a gap size of 500 nm and graphene electrodes. Graphene is expected to be chemically and mechanically stable in high‐radiation environments, because its tensile strength of 130 GPa is very large, about 100 times larger than most metals. The dielectric strength of such nanocapacitors exceeds 1 GV/m, because avalanching is suppressed at small gap sizes. In a 1 GV/m electric field charged nuclear reaction products, such as 5.6 MeV alpha particles, come to rest in of a stack with 5000 nanocapacitors. We show that during the deceleration process more than 90% of kinetic energy of charged nuclear reaction products is converted to electric energy and stored as electric energy in the stack. Each stack is 2.5‐mm thick and produces a high‐voltage DC current. A device with a 1‐Ci²⁴¹Am source is expected to generate 22 mW of electric power.     

A stability estimate for a solution to an inverse problem of electrodynamics

We consider the integrodifferential system of equations of electrodynamics which corresponds to a dispersive nonmagnetic medium. For this system we study the problem of determining the spatial part of the kernel of the integral term. This corresponds to finding the part of dielectric permittivity depending nonlinearly on the frequency of the electromagnetic wave. We assume that the support of dielectric permittivity lies in some compact domain Ω ⊂ ℝ³. In order to find it inside Ω we start with known data about the solution to the corresponding direct problem for the equations of electrodynamics on the whole boundary of Ω for some finite time interval. On assuming that the time interval is sufficiently large we estimate the conditional stability of the solution to this inverse problem.     

Maxwell's equations in an unbounded structure

This paper is concerned with the mathematical analysis of the electromagnetic wave scattering by an unbounded dielectric medium, which is mounted on a perfectly conducting infinite plane. By introducing a transparent boundary condition on a plane surface confining the medium, the scattering problem is modeled as a boundary value problem of Maxwell's equations. Based on a variational formulation, the problem is shown to have a unique weak solution for a wide class of dielectric permittivity and magnetic permeability by using the generalized Lax–Milgram theorem. Copyright © 2016 John Wiley & Sons, Ltd.     

Regularity of weak solution to Maxwell's equations and applications to microwave heating

In this paper we first study the regularity of weak solution for time-harmonic Maxwell's equations in a bounded anisotropic medium Ω. It is shown that the weak solution to the linear degenerate system, , is Hölder continuous under the minimum regularity assumptions on the complex coefficients γ(x) and ξ(x). We then study a coupled system modeling a microwave heating process. The dynamic interaction between electric and temperature fields is governed by Maxwell's equations coupled with an equation of heat conduction. The electric permittivity, electric conductivity and magnetic permeability are assumed to be dependent of temperature. It is shown that under certain conditions the coupled system has a weak solution. Moreover, regularity of weak solution is studied. Finally, existence of a global classical solution is established for a special case where the electric wave is assumed to be propagating in one direction.     

Integrable couplings,bi‐integrable couplings and their Hamiltonian structures of the Giachetti–Johnson soliton hierarchy

On the basis of zero curvature equations from semi‐direct sums of Lie algebras, we construct integrable couplings of the Giachetti–Johnson hierarchy of soliton equations. We also establish Hamiltonian structures of the resulting integrable couplings by the variational identity. Moreover, we obtain bi‐integrable couplings of the Giachetti–Johnson hierarchy and their Hamiltonian structures by applying a class of non‐semisimple matrix loop algebras consisting of triangular block matrices. Copyright © 2014 John Wiley & Sons, Ltd.     

On the Diffraction by Biperiodic Anisotropic Structures

This article studies the scattering of electromagnetic waves by a nonmagnetic biperiodic structure. The structure consists of anisotropic optical materials and separates two regions with constant dielectric coefficients. The time harmonic Maxwell equations are transformed to an equivalent strongly elliptic variational problem for the magnetic field in a bounded biperiodic cell with nonlocal boundary conditions. This guarantees the existence of quasiperiodic magnetic fields in H 1 and electric fields in H (curl) solving Maxwell's equations. The uniqueness is proved for all frequencies excluding possibly a discrete set. The analytic dependence of these solutions on frequency and incident angles is studied.     

微通道液体流动双电层阻力效应

采用数值方法求解双电层的Poisson-Boltzmann方程和液体运动的Navier-Stokes方程,研究微通道双电层对压强梯度液体流动的阻力效应． 量纲分析表明,双电层阻力大小可以用一个无量纲的电阻力数表示．它与液体的介电系数、固体表面的zeta电位平方成正比,与液体的动力粘性系数、电导率以及微通道的宽度平方成反比．在计算流动诱导的流动电位势和电阻力时,提出电流密度平衡条件,可以消除传统电流平衡条件导致的固壁附近产生局部回流的不合理物理现象．还给出不同电阻力数的微通道流量、流量损失率、速度剖面的数值结果,合理解释了双电层对微通道液体流动的阻力效应．     

The variational approach to estimation of the dielectric permittivity of a unidirectional fibrous composite

Based on the dual variational formulation of an electrostatic problem in an inhomogeneous anisotropic dielectric, we construct two-side bounds for possible values of second rank tensor components for the dielectric permittivity of a unidirectional fibrous composite taking into account mutual positions of anisotropic reinforcing fibers in the plane perpendicular to their axes. We consider variants of arrangement of cross sections of fibers such that the composite is transversal-isotropic or orthotropic relative to the property of dielectric permittivity. Some estimates of the greatest error appearing in the choice of true values of each component of the dielectric permittivity tensor of the composite in the form of a half-sum of its boundary values are obtained. A successive improvement of two-side bounds for domains where the components of this tensor should lie is carried out. A quantitative analysis of calculation dependencies is presented.     

Time harmonic wave diffraction problems in materials with sign-shifting coefficients

Some electromagnetic materials present, in a given frequency range, an effective dielectric permittivity and/or magnetic permeability which are negative. We are interested in the reunion of such a “negative” material and a classical one. More precisely, we consider here a scalar model problem for the simulation of a wave transmission between two such materials. This model is governed by a Helmholtz equation with a weight function in the Δ principal part which takes positive and negative real values. Introducing additional unknowns, we have already proposed in Bonnet-Ben Dhia et al. (2006) [1] some new variational formulations of this problem, which are of Fredholm type provided the absolute value of the contrast of permittivities is large enough, and therefore suitable for a finite element discretization. We prove here that, under similar conditions on the contrast, the natural variational formulation of the problem, although not “coercive plus compact”, is nonetheless suitable for a finite element discretization. This leads to a numerical approach which is straightforward, less costly than the previous ones, and very accurate.     

Struktur der Grenzschicht zwischen einem feldfreien Plasma und einem Magnetfeld im Vakuum

Self-consistent solutions of Vlasov's equations describing a plane boundary layer without shear are given. The currents within the layer are due to free and trapped electrons. The ion current is neglected sincem _e ?m _i . The free electrons and the ions are assumed Maxwellian at infinity. For the trapped electrons a one parameter family of distribution functions is assumed. Profiles are computed for potentials, fields, densities and currents for different numbers of trapped electrons. Numerical solutions are obtained by applying the method of finite elements to a variational formulation of the problem.     

Equilibrium State of Inhomogeneous Plasma

We consider the problem of a self-consistent determination of an essentially inhomogeneous equilibrium state of classical plasma. The solutions of the stationary Vlasov–Poisson equations are constructed in the form of a localized transition layer that separates the domains of homogeneous plasmas with different equilibrium parameters. The layer can also transform into a local perturbation inside a homogeneous plasma. In both cases, the solution contains neither mass currents nor electric currents, and all electrodynamic and hydrodynamic quantities and their derivatives are continuous. The parameters of the adjacent domains uniquely determine the transition layer structure.     

The mathematical modeling of the electric field in the media with anisotropic objects

We present a numerical scheme for modeling the electric field in the media with tensor conductivity. This scheme is based on vector finite element method in frequency domain. The numerical computations of the electric field in the anisotropic medium are done. The conductivity of the anisotropic medium is positive defined dense tensor in general case. We consider the electric field from anisotropic layer, inclined anisotropic layer and some anisotropic objects in isotropic half-space.     

A numerical study of electromagnetic waves in periodic waveguides

Here are considered time‐harmonic electromagnetic waves in a quadratic waveguide consisting of a periodic dielectric core enclosed by conducting walls. The permittivity function may be smooth or have jumps. The electromagnetic field is given by a magnetic vector potential in Lorenz gauge, and defined on a Floquet cell. The Helmholtz operator is approximated by a Chebyshev collocation, Fourier–Galerkin method. Laurent's rule and the inverse rule are employed for the representation of Fourier coefficients of products of functions. The computations yield, for known wavenumbers, values of the first few eigenfrequencies of the field. In general, the dispersion curves exhibit band gaps. Field patterns are identified as transverse electric, TE, transverse magnetic, TM, or hybrid modes. Maxwell's equations are fulfilled. A few trivial solutions appear when the permittivity varies in the guiding direction and across it. The results of the present method are consistent with exact results and with those obtained by a low‐order finite element software. The present method is more efficient than the low‐order finite element method. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 490–513, 2014     

微通道周期流动电位势及电粘性效应

求解了双电层的Poisson-Boltzmann方程和流体运动的Navier-Stokes方程,得到在周期压差作用下,二维微通道的周期流动电位势,流动诱导电场和液体流动速度的解析解．量纲分析表明,流体电粘性力与以下3个参数有关:1) 电粘性数,它表示定常流动时,通道最大电粘性力与压力梯度的比;2) 形状函数,它表示电粘性力在通道横截面的分布形态; 3) 耦合系数,它表示电粘性力的振幅衰减特征和相位差．分析结果表明,微通道周期流动诱导电场、流动速度与频率Reynolds数有关．在频率Reynolds数小于1时,流动诱导电场随频率Reynolds数变化很慢．在频率Reynolds数大于1时,流动诱导电场随频率Reynolds数的增加快速衰减．在通道宽度与双电层厚度比值较小情况下,电粘性效应对周期流动速度和流动诱导电场有重要影响．