Three-dimensional Maxwell Equations Research Articles

A NUMERICAL METHOD FOR 3D TIME-DEPENDENT MAXWELL’S EQUATIONS IN AXISYMMETRIC SINGULAR DOMAINS WITH ARBITRARY DATA

In this article, we propose to solve the three-dimensional time-dependent Maxwell equations in a singular axisymmetric domain with arbitrary data. Due to the axisymmetric assumption, the singular computational domain boils down to a subset of R2. However, the electromagnetic field and other vector quantities still belong to R3. Taking advantage that the domain is transformed into a two-dimensional one, by doing Fourier analysis in the third dimension, one arrives to a sequence of singular problems set in a 2D domain. The mathematical tools of such problems have been exposed in [4,19]. Here, we derive a variational method from which we propose an original finite element numerical approach to solve the problem. Numerical experiments are also shown to illustrate that the method is able to capture the singular part of the solution. This approach can also be viewed as a generalization of the Singular Complement Method to three-dimensional problem.

Efficient energy structure-preserving schemes for three-dimensional Maxwell's equations

Two energy structure-preserving schemes are proposed for Maxwell's equations in three dimensions. The Maxwell's equations are split into several local one-dimensional subproblems which successfully reduces the scale of algebraic equations to be solved. To improve the convergence rate in space and to keep the sparsity of the resulting algebraic equations, the spatial derivatives are approximated by high order compact method. Some key indicators, such as stability, energy structure-preserving and convergence of the schemes are investigated. To make the theoretical more persuasive, some numerical examples are shown. Numerical results are accord with the theoretical results. This provides a practical approach to construct efficient structure-preserving algorithms multidimensional Maxwell's equations.

An energy-stable finite element method for nonlinear Maxwell's equations

In this paper, we consider the three-dimensional (3D) nonlinear Maxwell's equations in the optical media characterized by linear Lorentz dispersion, nonlinear Kerr effect and delayed Raman scattering. Using Crank-Nicolson time discretization and special treatment of the nonlinearity, we propose a second order accurate (in time) finite element method for the nonlinear problem. Fully discrete energy stability analysis is established to show that the scheme is unconditionally stable. The nonlinear system is solved by Newton-Krylov method. In order to efficiently solve the linearized algebraic system in each Newton sub-iteration, we further develop an effective preconditioner by using suitable approximations to the nonlinearity and the auxiliary space preconditioning technique. Numerical experiments are provided to examine the accuracy, the energy stability, the parallel scalability and the robustness in preconditioning of our proposed scheme. We then apply the scheme to simulate the spatial soliton propagation in 3D nonlinear glass. We also demonstrate the performance of the scheme in handling complex geometry through simulating the wave scattering by a spherical hole in the nonlinear optical glass.

Parallel block preconditioners for virtual element discretizations of the time-dependent Maxwell equations

The focus of this study is the construction and numerical validation of parallel block preconditioners for low order virtual element discretizations of the three-dimensional Maxwell equations. The virtual element method (VEM) is a recent technology for the numerical approximation of partial differential equations (PDEs), that generalizes finite elements to polytopal computational grids. So far, VEM has been extended to several problems described by PDEs, and recently also to the time-dependent Maxwell equations. When the time discretization of PDEs is performed implicitly, at each time-step a large-scale and ill-conditioned linear system must be solved, that, in case of Maxwell equations, is particularly challenging, because of the presence of both H(div) and H(curl) discretization spaces. We propose here a parallel preconditioner, that exploits the Schur complement block factorization of the linear system matrix and consists of a Jacobi preconditioner for the H(div) block and an auxiliary space preconditioner for the H(curl) block. Several parallel numerical tests have been performed to study the robustness of the solver with respect to mesh refinement, shape of polyhedral elements, time step size and the VEM stabilization parameter.

Time reversal of surface plasmons

<p style='text-indent:20px;'>We study in this work the so-called "instantaneous time mirrors" in the context of surface plasmons. The latter are associated with high frequency waves at the surface of a conducting sheet. Instantaneous time mirrors were introduced in [<xref ref-type="bibr" rid="b3">3</xref>], with the idea that singular perturbations in the time variable in a wave-type equation create a time-reversed focusing wave. We consider the time-dependent three-dimensional Maxwell's equations, coupled to Drude's model for the description of the surface current. The time mirror is modeled by a sudden, strong, change in the Drude weight of the electrons on the sheet. Our goal is to characterize the time-reversed wave, in particular to quantify the quality of refocusing. We establish that the latter depends on the distance of the source to the sheet, and on some physical parameters such as the relaxation time of the electrons. We also show that, in addition to the plasmonic wave, the time mirror generates a free propagating wave that offers, contrary to the surface wave, some resolution in the direction orthogonal to the sheet. Blurring effects due to non-instantaneous mirrors are finally investigated.</p>

Ultra-weak variational formulation for heterogeneous Maxwell problem in the context of high performance computing

Electromagnetic simulations on large domains require a huge memory consumption. Domain decomposition methods, based on Trefftz methods, could be an answer to this issue. In this paper, we associate to heterogeneous three-dimensional Maxwell equations one variational formulation which can be obtained either by upwind fluxes or Riemann traces. We associate to this variational formulation an iterative Trefftz Krylov solver. The poor conditioning due to the use of plane wave basis functions is bypassed thanks to a compression strategy. Moreover, the developed iterative solver is accelerated thanks to a left preconditioner. The considered numerical cases illustrate the performance of this basis reduction, which leads to the consideration of an industrial case of more than 750 millions of degrees of freedom.

Macroscopic phase-matching mechanism for orbital angular momentum spectra of high-order harmonics by mixing two Laguerre-Gaussian vortex modes

The high-order harmonic generation (HHG) by the interaction of an intense vortex infrared laser and a gas medium has been employed to produce the extreme ultraviolet (XUV) light beam carrying a high value orbital angular momentum (OAM). Here we extend this approach to generate the XUV light with superimposed OAM modes by mixing two different Laguerre-Gaussian (LG) modes as the driving beam. By solving the three-dimensional Maxwell's wave equations of the high-harmonic field and calculating the coherence lengths of HHG, we reveal the complex nature of phase-matching conditions in the gas medium, especially their dependence on the azimuthal angle. We demonstrate that the OAM spectra of the HHG can be tailored by varying the harmonic order, the position of gas medium, and the energy ratio of two LG beams. We also uncover that only including short-trajectory contribution in the single-atom response could broaden the OAM spectrum of the macroscopic HHG because of the featured phase-matching characteristics caused by the atomic intrinsic phase. We anticipate this work to stimulate some interests in generating an XUV vortex beam with the tunable OAM spectrum through the gaseous HHG process as well as in separating different OAM modes of an XUV vortex beam by advanced techniques.

Efficient Energy Structure-Preserving Schemes for Three-Dimensional Maxwell's Equations

Efficient Energy Structure-Preserving Schemes for Three-Dimensional Maxwell's Equations

Bifurcation analysis of the eigenstructure of the discrete single-curl operator in three-dimensional Maxwell’s equations with Pasteur media

Abstract This paper focuses on studying the bifurcation analysis of the eigenstructure of the $\gamma $-parameterized generalized eigenvalue problem ($\gamma $-GEP) arising in three-dimensional source-free Maxwell’s equations with Pasteur media, where $\gamma $ is the magnetoelectric chirality parameter. The weakly coupled case, namely, $\gamma < \gamma _{\ast } \equiv $ critical value, the $\gamma $-GEP is positive definite, has been well studied by Chern et al. (2015, Singular value decompositions for single-curl operators in three-dimensional Maxwell’s equations for complex media. SIAM J. Matrix Anal. Appl., 36, 203–224). For the strongly coupled case, namely, $\gamma> \gamma _{\ast }$, the $\gamma $-GEP is no longer positive definite, introducing a totally different and complicated structure. For the critical strongly coupled case, numerical computations for electromagnetic fields have been presented by Huang et al. (2018, Solving three-dimensional Maxwell eigenvalue problem with fourteen Bravais lattices. Technical Report). In this paper, we build several theoretical results on the eigenstructure behaviour of the $\gamma $-GEPs. We prove that the $\gamma $-GEP is regular for any $\gamma> 0$, and the $\gamma $-GEP has $2 \times 2$ Jordan blocks of infinite eigenvalues at the critical value $\gamma _{\ast }$. Then we show that the $2 \times 2$ Jordan block will split into a complex conjugate eigenvalue pair that rapidly goes down and up and then collides at some real point near the origin. Next it will bifurcate into two real eigenvalues, with one moving towards the left and the other to the right along the real axis as $\gamma $ increases. A newly formed state whose energy is smaller than the ground state can be created as $\gamma $ is larger than the critical value. This stunning feature of the physical phenomenon would be very helpful in practical applications. Therefore, the purpose of this paper is to clarify the corresponding theoretical eigenstructure of the three-dimensional Maxwell equations with Pasteur media.

Numerical simulation of multi-carrier microwave breakdown in air-SF6 mixtures

In this paper, the multi-carrier microwave breakdown in air-SF6 mixtures is analyzed with the spectral-element time-domain method. In this process, a three-dimensional multi-physics model coupling Maxwell's equations with electron fluid equations is established. The tail-erosion and frequency shift phenomenon caused by multi-carrier microwave breakdown in different proportions of air-SF6 mixtures can be observed. Numerical results demonstrate that multi-carriers will make the wireless communication system sensitive to microwave breakdown. On the other hand, the different proportions of air-SF6 mixtures can obviously improve the breakdown threshold to inhibit the multi-carrier microwave breakdown, which is favorable to the propagation of multi-carrier microwaves in air. Meanwhile, increasing the pressure can suppress the frequency shift phenomenon, and the proportion of different SF6 in the mixed gas has little effect on the frequency shift of the transmitted wave. Our research provides theoretical guidance for comprehensively exploring the characteristics and physical mechanism of multi-carrier microwave breakdown in air-SF6 mixtures and is helpful for protection of microwave devices and plasma application.

An Inverse Random Source Problem for Maxwell's Equations

This paper is concerned with an inverse random source problem for the three-dimensional time-harmonic Maxwell equations. The source is assumed to be a centered complex-valued Gaussian vector field ...

Numerical Scheme for Electromagnetic Scattering on Perturbed Periodic Inhomogeneous Media and Reconstruction of the Perturbation

We consider the electromagnetic scattering of nonperiodic incident fields on locally perturbed periodic inhomogeneous media. We model the scattering problem by the full vector-valued three-dimensional time-harmonic Maxwell's equations. The assumptions prevent the classical method of reducing the problem to one cell such that the problem has to be considered in an unbounded domain. We provide a Bloch--Floquet transform--based numerical scheme to solve the full vector-valued scattering problem in reasonable time without artificial boundary conditions. Moreover, we show the improvement of the convergence rate by suitable variable transforms inside the inverse Bloch--Floquet transform and discuss some examples of variable transforms to improve the convergence rate of the algorithm for smaller wave numbers. We then consider the inverse problem of reconstructing the local perturbation of the permittivity and use the numerical method for the direct problem and an inexact Newton method to show some examples of the quality of the reconstruction.

The bi-Lebedev scheme for the Maxwell eigenvalue problem with 3D bi-anisotropic complex media

This paper focuses on studying the eigenstructure of generalized eigenvalue problems (GEPs) arising in the three-dimensional source-free Maxwell equations for bi-anisotropic complex media with a 3-by-3 permittivity tensor ε>0, a permeability tensor μ>0, and scalar magnetoelectric coupling constants ξ=ζ̄=ıγ. The bi-Lebedev scheme is appealing because it preserves the symmetry inherent to the Maxwell eigenvalue problem exactly and because full degrees of freedom of electromagnetic fields at each grid point are taken into account in the discretization. The resulting GEP has eigenvalues appearing in quadruples {±ω,±ω̄}. We consider two main scenarios, where γ<γ∗ and γ>γ∗ with γ∗ as a critical value. In the former case, all the eigenvalues are real. In the latter case, the GEP has complex eigenvalues, and we particularly focus on the bifurcation of the eigenstructure of the GEPs. Numerical results demonstrate that the newborn ground state has occurred after γ=γ̃>γ∗, and the associated eigenvector has an exotic phenomenon of localization. Moreover, the Poynting vectors of the newborn eigenvector not only are concentrated in the material but also display exciting patterns.

Imaging of bi-anisotropic periodic structures from electromagnetic near-field data

Abstract This paper is concerned with the inverse scattering problem for the three-dimensional Maxwell equations in bi-anisotropic periodic structures. The inverse scattering problem aims to determine the shape of bi-anisotropic periodic scatterers from electromagnetic near-field data at a fixed frequency. The factorization method is studied as an analytical and numerical tool for solving the inverse problem. We provide a rigorous justification of the factorization method which results in the unique determination and a fast imaging algorithm for the periodic scatterer. Numerical examples for imaging three-dimensional periodic structures are presented to examine the efficiency of the method.

Improved 3D electromagnetic analytical model for planar induction heater with consideration of transverse edge effects

PurposeThis paper aims to propose a new 3D electromagnetic model to compute translational motion eddy current in the conducting plate of a novel linear permanent magnet (PM) induction heater. The movement of the plate in a DC magnetic field created by a PM inductor generates induced currents that are at the origin of a heating power by Joule effect. These topologies have strong magnetic end effects. The analytical model developed in this work takes into account the finite length extremity effects of the conducting plate and the reaction field because of induced currents.Design/methodology/approachThe developed model is based on the combination of the sub-domain’s method and the image’s theory. First, the magnetic field expressions because of the PMs are obtained by solving the three-dimensional Maxwell equations by the method of separation of variables, using a magnetic scalar potential formulation and a magnetic field strength formulation. Then, the motional eddy currents are computed using the Ampere law, and the finite length extremity effects of the conducting plate are taken into account using the image’s method. To analyze the accuracy of the proposed model, the obtained results are compared to those obtained from 3D finite element model (FEM) and from experimental tests performed on a prototype.FindingsThe results show that the developed analytical model is very accurate, even for geometries where the edge effects are very strong. It allows directly taking into account the finite length extremity effects (the transverse edge effects) of the conducting plate and the reaction field because of induced currents without the need of any correction factor. The proposed model also presents an important reduction in computation time compared to 3D finite element simulation, allowing fast analysis of linear PM induction heater.Practical implicationsThe proposed electromagnetic analytical model can be used as a quick and accurate design tool for translational motion PM induction heater devices.Originality/valueA new 3D analytical electromagnetic model, to find the induced power in the conducting plate of a novel translational motion induction heater has been developed. The studied heating device has a finite length and a finite width, which create edge effects that are not easily considered in calculation. The novelty of the presented method is the accurate 3D analytical model, which allows finding the real power heating and real distribution of the induced currents in the conducting plate without the need to use correction factor. The proposed model also takes into account the reaction field because of induced currents. In addition, the developed model improves an important reduction in the computation time compared with 3D FEM simulation.

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.

On longitudinal electromagnetic stirring in the continuous casting of steel blooms

Recent work highlighting an anomaly in the modelling of rotary electromagnetic stirring (EMS) in the continuous casting of round steel billets is extended to the case of longitudinal stirring for rectangular blooms. An earlier, still often-cited, model forms the basis of the current analysis, which uses asymptotic methods on the three-dimensional (3D) Maxwell equations and demonstrates how the earlier result for the components of the Lorentz force is but a particular case of a more general form. Time-dependent 3D computations using finite-element methods are also performed to verify the validity of the asymptotic analysis, and the relevance of the results to modulated EMS is noted.

Convergence of an AEFEM for time-harmonic Maxwell equations with variable coefficients

An Adaptive Edge Finite Element Method (AEFEM) for three-dimensional indefinite time-harmonic Maxwell equations with variable coefficients is proved to be convergent and arbitrary order Nédélec edge elements are considered. A posteriori upper bound, quasi-orthogonality and the contraction of the error estimator are provided to prove the contraction of the sum of the energy error and the scaled error estimator between two consecutive adaptive loops. Numerical experiments are presented to support the theoretical results.

Convergence Analysis of the PML Method for Time-Domain Electromagnetic Scattering Problems

In this paper, a perfectly matched layer (PML) method is proposed to solve the time-domain electromagnetic scattering problems in three dimensions effectively. The PML problem is defined in a spherical layer and derived by using the Laplace transform and real coordinate stretching in the frequency domain. The well-posedness and the stability estimate of the PML problem are first proved based on the Laplace transform and the energy method. The exponential convergence of the PML method is then established in terms of the thickness of the layer and the PML absorbing parameter. As far as we know, this is the first convergence result for the time-domain PML method for the three-dimensional Maxwell equations. Our proof is mainly based on the stability estimates of solutions of the truncated PML problem and the exponential decay estimates of the stretched dyadic Green's function for the Maxwell equations in the free space.

Numerical Solution of the Three-Dimensional Time-Harmonic Maxwell Equations by DG Method Coupled with an Integral Representation

Numerical Solution of the Three-Dimensional Time-Harmonic Maxwell Equations by DG Method Coupled with an Integral Representation