Novel high-performance element in the electromagnetic finite-element method——node-edge element

It is known in the computational electromagnetics （CEM） that the node element has a relative wellconditioned matrix, but suffers from the spurious solution problem; whereas the edge element has no spurious solutions, but usually produces an ill-conditioned matrix. Particularly, when the mesh is over dense, the iterative solution of the matrix equation from edge element converges very slowly. Based on the node element and edge element, a node-edge element is presented, which has no spurious solutions and better-conditioned matrix. Numerical experiments demonstrate that the proposed node-edge element is more efficient than now-widely used edge element.

3-D Numerical Simulation of the Electromagnetic Dam of Twin Roll Casting using Edge Element Method

The 3-dimension numerical simulation study on the electromagnetic dam used in the twin roll caster has been performed by using the edge element method. It was found that the materials and structures of the roll collars have great influence on the distribution of the magnetic flux density, eddy current density and the electromagnetic force in the molten pool. The conductive collars make the magnetic flux density decreased in the molten pool, but it also makes the magnetic force more uniformly, and the force in the low part of the molten pool where needs greater force have increased some what. The conductive collars make the EMD device more effective than the nonconductive collars.

CONVERGENCE OF ADAPTIVE EDGE ELEMENT METHODS FOR THE 3D EDDY CURRENTS EQUATIONS

We consider an Adaptive Edge Finite Element Method （AEFEM） for the 3D eddy currents equations with variable coefficients using a residual-type a posteriori error estimator. Both the components of the estimator and certain oscillation terms, due to the occurrence of the variable coefficients, have to be controlled properly within the adaptive loop which is taken care of by appropriate bulk criteria. Convergence of the AEFEM in terms of reductions of the energy norm of the discretization error and of the oscillations is shown. Numerical results are given to illustrate the performance of the AEFEM.

OPTIMAL ERROR ESTIMATES FOR NEDELEC EDGE ELEMENTS FOR TIME-HARMONIC MAXWELL'S EQUATIONS

In this paper, we obtain optimal error estimates in both L^2-norm and H（curl）-norm for the Nedelec edge finite element approximation of the time-harmonic Maxwell＇s equations on a general Lipschitz domain discretized on quasi-uniform meshes. One key to our proof is to transform the L^2 error estimates into the L^2 estimate of a discrete divergence-free function which belongs to the edge finite element spaces, and then use the approximation of the discrete divergence-free function by the continuous divergence-free function and a duality argument for the continuous divergence-free function. For Nedelec＇s second type elements, we present an optimal convergence estimate which improves the best results available in the literature.

Experimental and Numerical Analysis of Surface Magneto-Hydrodynamic Propulsion Induced by NdFeB Magnets

The so-called surface Magneto-hydro-dynamic(MHD)propulsion relies on the Lorentz force induced in weak electrolyte solutions(such as seawater or plasma)by NdFeB Magnets.The Lorentz force plays an important role in such dynamics as it directly affects the structures of flow boundary layers.Previous studies have mainly focused on the development of such boundary layers and related fluid-dynamic aspects.The main focus of the present study is the determination of electromagnetic field distributions around the propulsion units.In particular dedicated experiments and numerical simulations(based on the finite volume method)are conducted considering a NACA0012 airfoil immersed in seawater.The results show that,along the propulsion unit,the field strength undergoes a rapid attenuation in the direction perpendicular to the wall.

NON SPURIOUS SPECTRAL-LIKE ELEMENT METHODS FOR MAXWELL'S EQUATIONS

In this paper, we give the state of the art for the so called “mixed spectral elements” for Maxwell＇s equations. Several families of elements, such as edge elements and discon-tinuous Galerkin methods （DGM） are presented and discussed. In particular, we show the need of introducing some numerical dissipation terms to avoid spurious modes in these methods. Such terms are classical for DGM but their use for edge element methods is novel approach described in this paper. Finally, numerical experiments show the fast and low-cost character of these elements.

Developing Finite Element Methods for Simulating Transformation Optics Devices with Metamaterials

In this paper,we first develop the mathematical modeling equations for wave propagation in several transformation optics devices,including electromagnetic concentrator,rotator and splitter.Then we propose the corresponding finite element time-domain methods for simulating wave propagation in these transformation optics devices.We implement the proposed algorithms and our numerical results demonstrate the effectiveness of our modeling equations.To our best knowledge,this is the first work on time-domain finite element simulation carried out for the electromagnetic concentrator,rotator and splitter.

Analysis and Application of Single Level, Multi-Level Monte Carlo and Quasi-Monte Carlo Finite Element Methods for Time-Dependent Maxwell’s Equations with Random Inputs

This article is devoted to three quadrature methods for the rapid solution of stochastic time-dependent Maxwell’s equations with uncertain permittivity,perme-ability and initial conditions.We develop the mathematical analysis of the error estimate for single level Monte Carlo method,multi-level Monte Carlo method,and the quasi-Monte Carlo method.The theoretical results are supplemented by numerical experiments.

An Iterative Two-Grid Method of A Finite Element PML Approximation for the Two Dimensional Maxwell Problem

In this paper,we propose an iterative two-grid method for the edge finite element discretizations(a saddle-point system)of Perfectly Matched Layer(PML)equations to the Maxwell scattering problem in two dimensions.Firstly,we use a fine space to solve a discrete saddle-point system of H(grad)variational problems,denoted by auxiliary system 1.Secondly,we use a coarse space to solve the original saddle-point system.Then,we use a fine space again to solve a discrete H(curl)-elliptic variational problems,denoted by auxiliary system 2.Furthermore,we develop a regularization diagonal block preconditioner for auxiliary system 1 and use H-X preconditioner for auxiliary system 2.Hence we essentially transform the original problem in a fine space to a corresponding(but much smaller)problem on a coarse space,due to the fact that the above two preconditioners are efficient and stable.Compared with some existing iterative methods for solving saddle-point systems,such as PMinres,numerical experiments show the competitive performance of our iterative two-grid method.

High-Order Low Dissipation Conforming Finite-Element Discretization of the Maxwell Equations

In this paper,we study high order discretization methods for solving the Maxwell equations on hybrid triangle-quad meshes.We have developed high order finite edge element methods coupled with different high order time schemes and we compare results and efficiency for several schemes.We introduce in particular a class of simple high order low dissipation time schemes based on a modified Taylor expansion.

Band Structure Calculations of Dispersive Photonic Crystals in 3D using Holomorphic Operator Functions

We propose a finite element method to compute the band structures of dispersive photonic crystals in 3D.The nonlinear Maxwell’s eigenvalue problem is formulated as the eigenvalue problem of a holomorphic operator function.The N´ed´elec edge elements are employed to discretize the operators,where the divergence free condition for the electric field is realized by a mixed form using a Lagrange multiplier.The convergence of the eigenvalues is proved using the abstract approximation theory for holomorphic operator functions with the regular approximation of the edge elements.The spectral indicator method is then applied to compute the discrete eigenvalues.Numerical examples are presented demonstrating the effectiveness of the proposed method.

PARALLEL AUXILIARY SPACE AMG FOR H(curl) PROBLEMS

In this paper we review a number of auxiliary space based preconditioners for the second order definite and semi-definite Maxwell problems discretized with the lowest order Nedelec finite elements. We discuss the parallel implementation of the most promising of these methods, the ones derived from the recent Hiptmair-Xu （HX） auxiliary space decomposition [Hiptmair and Xu, SIAM J. Numer. Anal., 45 （2007）, pp. 2483-2509]. An extensive set of numerical experiments demonstrate the scalability of our implementation on large-scale H（curl） problems.

Local Multigrid in H(curl)

We consider H（curl, Ω）-elliptic variational problems on bounded Lipschitz polyhedra and their finite element Galerkin discretization by means of lowest order edge elements. We assume that the underlying tetrahedral mesh has been created by successive local mesh refinement, either by local uniform refinement with hanging nodes or bisection refinement. In this setting we develop a convergence theory for the the so-called local multigrid correction scheme with hybrid smoothing. We establish that its convergence rate is uniform with respect to the number of refinement steps. The proof relies on corresponding results for local multigrid in a H^1 （Ω）-context along with local discrete Helmholtz-type decompositions of the edge element space.

NUMERICAL LOCALIZATION OF ELECTROMAGNETIC IMPERFECTIONS FROM A PERTURBATION FORMULA IN THREE DIMENSIONS

This work deals with the numerical localization of small electromagnetic inhomogeneities. The underlying inverse problem considers, in a three-dimensional bounded domain, the time-harmonic Maxwell equations formulated in electric field. Typically, the domain contains a finite number of unknown inhomogeneities of small volume and the inverse problem attempts to localize these inhomogeneities from a finite number of boundary measurements. Our localization approach is based on a recent framework that uses an asymptotic expansion for the perturbations in the tangential boundary trace of the curl of the electric field. We present three numerical localization procedures resulting from the combination of this asymptotic expansion with each of the following inversion algorithms： the Current Projection method, the MUltiple Signal Classification （MUSIC） algorithm, and an Inverse Fourier method. We perform a numerical study of the asymptotic expansion and compare the numerical results obtained from the three localization procedures in different settings.

ADAPTIVITY IN SPACE AND TIME FOR MAGNETOQUASISTATICS

This paper addresses fully space-time adaptive magnetic field computations. We describe an adaptive Whitney finite element method for solving the magnetoquasistatic formulation of Maxwell＇s equations on unstructured 3D tetrahedral grids. Spatial mesh re- finement and coarsening are based on hierarchical error estimators especially designed for combining tetrahedral H（curl）-conforming edge elements in space with linearly implicit Rosenbrock methods in time. An embedding technique is applied to get efficiency in time through variable time steps. Finally, we present numerical results for the magnetic recording write head benchmark problem proposed by the Storage Research Consortium in Japan.

THREE-DIMENSIONAL NUMERICAL LOCALIZATION OF IMPERFECTIONS BASED ON A LIMIT MODEL IN ELECTRIC FIELD AND A LIMIT PERTURBATION MODEL

From a limit model in electric field obtained by letting the frequency vanish in the time-harmonic Maxwell equations, we consider a limit perturbation model in the tangential boundary trace of the curl of the electric field for localizing numerically certain small electromagnetic inhomogeneities, in a three-dimensional bounded domain. We introduce here two localization procedures resulting from the combination of this limit perturbation model with each of the following inversion processes： the Current Projection method and an Inverse Fourier method. Each localization procedure uses, as data, a finite number of boundary measurements, and is employed in the single inhomogeneity case; only the one based on an Inverse Fourier method is required in the multiple inhomogeneities case. Our localization approach is numerically suitable for the context of inhomogeneities that are not purely electric. We compare the numerical results obtained from the two localization procedures in the single inhomogeneity configuration, and describe, in various settings of multiple inhomogeneities, the results provided by the procedure based on an Inverse Fourier method.

Adaptive hp-FEM with Arbitrary-Level Hanging Nodes for Maxwell’s Equations

Adaptive higher-order finite element methods(hp-FEM)are well known for their potential of exceptionally fast(exponential)convergence.However,most hp-FEM codes remain in an academic setting due to an extreme algorithmic complexity of hp-adaptivity algorithms.This paper aims at simplifying hpadaptivity for H(curl)-conforming approximations by presenting a novel technique of arbitrary-level hanging nodes.The technique is described and it is demonstrated numerically that it makes adaptive hp-FEM more efficient compared to hp-FEM on regular meshes and meshes with one-level hanging nodes.

STATIC REGIME IMAGING OF LOCATIONS OF CERTAIN 3D ELECTROMAGNETIC IMPERFECTIONS FROM A BOUNDARY PERTURBATION FORMULA

We are concerned, in a static regime, with a three-dimensional bounded domain of certain an imaging approach of the locations in electromagnetic imperfections. This approach is related to Electrical Impedance Tomography and makes use of a new perturbation formula in the electric fields. We present two localization procedures, from a Current Pro- jection method that deals with the single imperfection context and an inverse Fourier process that is devoted to multiple imperfections configurations. These procedures extend those that were described in our previous work, since operating for a broader class of settings. Namely, the localization is additionally performed for certain purely electric imperfections, as established from numerical simulations.

Uniform Convergence of Adaptive Multigrid Methods for Elliptic Problems and Maxwell’s Equations

We consider the convergence theory of adaptive multigrid methods for secondorder elliptic problems and Maxwell’s equations.The multigrid algorithm only performs pointwise Gauss-Seidel relaxations on new degrees of freedom and their“immediate”neighbors.In the context of lowest order conforming finite element approximations,we present a unified proof for the convergence of adaptive multigrid V-cycle algorithms.The theory applies to any hierarchical tetrahedral meshes with uniformly bounded shape-regularity measures.The convergence rates for both problems are uniform with respect to the number of mesh levels and the number of degrees of freedom.We demonstrate our convergence theory by two numerical experiments.

Thin Layer Models for Electromagnetism

We present a review on the accuracy of asymptotic models for the scattering problem of electromagnetic waves in domains with thin layer.These models appear as first order approximations of the electromagnetic field.They are obtained thanks to a multiscale expansion of the exact solution with respect to the thickness of the thin layer,that makes possible to replace the thin layer by approximate conditions.We present the advantages and the drawbacks of several approximations together with numerical validations and simulations.The main motivation of this work concerns the computation of electromagnetic field in biological cells.The main difficulty to compute the local electric field lies in the thinness of the membrane and in the high contrast between the electrical conductivities of the cytoplasm and of the membrane,which provides a specific behavior of the electromagnetic field at low frequencies.