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The dimmed outlines of phenomenal things all into one another unless we put on the merge focusing-glass of theory, and screw it up some­ times to one pitch of definition and sometimes to another, so as to see down into different depths through the great millstone of the world James Clerk Maxwell (1831 - 1879) For a long time after the foundation of the modern theory of electromag­ netism by James Clerk Maxwell in the 19th century, the mathematical ap­ proach to electromagnetic field problems was for a long time dominated by the analytical investigation of Maxwell's equations. The rapid development of computing facilities during the last century has then necessitated appropriate numerical methods and algorithmic tools for the simulation of electromagnetic phenomena. During the last few decades, a new research area "Computational Electromagnetics" has emerged com­ prising the mathematical analysis, design, implementation, and application of numerical schemes to simulate all kinds of relevant electromagnetic pro­ cesses. This area is still rapidly evolving with a wide spectrum of challenging issues featuring, among others, such problems as the proper choice of spatial discretizations (finite differences, finite elements, finite volumes, boundary elements), fast solvers for the discretized equations (multilevel techniques, domain decomposition methods, multipole, panel clustering), and multiscale aspects in microelectronics and micromagnetics.



Gauged Current Vector Potential and Reentrant Corners in the FEM Analysis of 3D Eddy Currents

The nodal finite element realization of the T — Ω method involving a gauged current vector potential, T, is shown to yield erroneous results if applied to 3D eddy current problems with reentrant corners in the conducting region. The reason for the problem is pinpointed to be the implicit gauging of the vector potential. A remedy of using no gauge in elements around the reentrant corners is suggested.
Oszkár Bíró, Kurt Preis

Finite Elements for the Time Harmonic Maxwell’s Equations

We review the time harmonic Maxwell’s system and its approximation via the finite element method. The problem under consideration is strictly related to the so-called interior Maxwell’s eigenproblem.
Standard nodal (Lagrangian) elements are known to provide useless results on general meshes. Special two-dimensional meshes have been shown to give good results, but the use of them is not recommended. The use of a penalty strategy with nodal elements has been proved to give wrong results for domains with singularities. Some special schemes, which make use of nodal elements, circumvent this problem; one of them is described in this paper.
On the other hand the so-called edge elements represent the natural choice. A new proof of convergence for a method based on edge elements is summarized.
Daniele Boffi

Trace Theorems on Non-Smooth Boundaries for Functional Spaces Related to Maxwell Equations: an Overview

We study tangential vector fields on the boundary of a bounded Lipschitz domain in ℝ3. Our attention is focused on the definition of suitable Hilbert spaces over a range of Sobolev regularity which we try to make as large as possible, and also on the construction of tangential differential operators. Hodge decompositions are proved to hold for some special choices of spaces which are of interest in the theory of Maxwell equations.
Annalisa Buffa

Applications of the Mortar Element Method to 3D Electromagnetic Moving Structures

This paper deals with the modelling, the analysis and a numerical approach for the simulation of the dynamical behavior of a three-dimensional coupled magneto-mechanical system such as a damping machine. The model is based on the electric formulation of the eddy currents problem for the electromagnetic part and on the motion equation of a rotating rigid body for the mechanical part.
For the approximation, the magnetic system is discretized in space by means of edge elements and the sliding mesh mortar element method is used to account for the rotation. In time, a one step Euler method is used, implicit for the magnetic and velocity equations and explicit for the rotation angle. The coupled differential system can then be solved with an explicit procedure.
Here, we analyse the well-posedness of the continuous problem and give some details on its discretization.
Annalisa Buffa, Yvon Maday, Francesca Rapetti

Numerical Stability of Collocation Schemes for Time Domain Boundary Integral Equations

Time domain boundary integral formulations of transient scattering problems involve retarded potential integral equations (RPIEs). Collocation schemes for RPIEs are often unstable, having errors which oscillate and grow exponentially with time. We describe how Fourier analysis can be used to analyse the stability of uniform grid schemes and to show that the instabilities are often very different from those observed in PDE approximations. We also present a new stable collocation scheme for a scalar RPIE, and show that it converges.
Penny Davies, Dugald Duncan

hp-Adaptive Finite Elements for Maxwell’s Equations

This is a progress report on our current work on hp-adaptive finite elements for Maxwell’s equations. I recall the main definitions [2], and show how the recent progress on the ftp interpolation error estimates [3] has led to a fully automatic ftp adaptivity based on the idea of minimizing the ftp-interpolation error for a reference solution corresponding to a globally ftp-refined grid [5]. Critical to the implementation of these ideas is a our new data structure for ftp discretizations [4], supporting anisotropic refinements, and the calculation of prolongation operator for multigrid operations.
Leszek Demkowicz

Coupled Calculation of Eigenmodes

In many technical applications the electromagnetic eigenmodes — frequency spectrum and field distributions - of rf-components are to be determined during the design process. There are numerous cases where the studied component is too complex to allow for a detailed enough simulation on usual servers. One way out of this situation is domain decomposition and parallelization of the field simulation. Yet, this demands for a parallelized solver. In our approach, we combine the use of commercial single processor-based software for the field simulation with a tool based on scattering parameter description. The studied component is decomposed in several sections. The scattering matrices of these sections are computed in time domain for instance with a FDTD field solver. A linear system is set up to compute the eigenfrequencies of the complete system and the field amplitudes at the internal ports common to a pair of sections. With the knowledge of these amplitudes the fields of the eigenmodes can be computed with help of a frequency domain field solver. This approach is denoted as Coupled S-Parameter Calculation (CSC). Some advantages of this procedure are the possibility of easy exploitation of symmetries in the studied components and the use of very different granularities in discretization of the single sections. This paper presents the method, its validation using a standard eigenmode solver and applications in the field of accelerator physics. Special attention is given to the eigenmodes of structures with slight deviations from rotational symmetry.
H.-W. Glock, K. Rothemund, U. van Rienen

Boundary Element Methods for Eddy Current Computation

This paper studies numerical methods for eddy current problems in the case of homogeneous, isotropic, and linear materials. It provides a survey of approaches that entirely rely on boundary integral equations and their conforming Galerkin discretization. The pivotal role of potentials is discussed, as well as the topological issues raised by their use. Direct boundary integral equations and the so-called symmetric coupling of the integral equations corresponding to the conductor and the non-conducting regions is employed. It gives rise to coupled variational problems that are elliptic in suitable trace spaces. This implies quasi-optimal convergence of Galerkin boundary element schemes.
Ralf Hiptmair

A Simple Proof of Convergence for an Edge Element Discretization of Maxwell’s Equations

The time harmonic Maxwell’s equations for a lossless medium are neither elliptic or definite. Hence the analysis of numerical schemes for these equations presents some unusual difficulties. In this paper we give a simple proof, based on the use of duality, for the convergence of edge finite element methods applied to the cavity problem for Maxwell’s equations. The cavity is assumed to be a general Lipschitz polyhedron, and the mesh is assumed to be regular but not quasi-uniform.
Peter Monk

The Time-Harmonic Eddy-Current Problem in General Domains: Solvability via Scalar Potentials

The eddy-current problem for the time-harmonic Maxwell equations in domains of general topology is solved by introducing a scalar “potential” for the magnetic field in the insulator part of the domain. Indeed, since in general the insulator Ω I is multiply-connected, the magnetic field differs from the gradient of a potential by a harmonic field. We rewrite the problem in a two-domain formulation, in term of a scalar magnetic potential and a harmonic field in Ω I . Then the finite element numerical approximation based on this two-domain formulation is presented, using edge elements in the conductor and nodal elements in the insulator, and an optimal error estimate is proved. An iteration-by-sub domain procedure for the solution of the problem is also proposed.
Ana Alonso Rodríguez, Paolo Fernandes, Alberto Valli

Finite Element Micromagnetics

The development of advanced magnetic materials such as magnetic sensors, recording heads, and magneto-mechanic devices requires a precise understanding of the magnetic behavior. As the size of the magnetic components approach the nanometer regime, detailed predictions of the magnetic properties becomes possible using micromagnetic simulations. Micromagnetics combines Maxwell’s equations for the magnetic field with an equation of motion describing the time evolution of the magnetization. The local arrangement of the magnetic moments follows from the complex interaction between intrinsic magnetic properties such as the magnetocrystalline anisotropy and the physical/chemical microstructure of the material.
This paper reviews the basic numerical methods used in finite element micromagnetic simulations and presents numerical examples in the field of soft magnetic sensor elements, polycrystalline thin film elements, and magnetic nanowires.
Thomas Schrefl, Dieter Suess, Werner Scholz, Hermann Forster, Vassilios Tsiantos, Josef Fidler

Finite Integration Method and Discrete Electromagnetism

We review some basic properties of the Finite Integration Technique (FIT), a generalized finite difference scheme for the solution of Maxwell’s equations. Special emphasis is put on its relations to the Finite Difference Time Domain (FDTD) method, as both algorithms are found to be computationally equivalent for the special case of an explicit time-stepping scheme with Cartesian grids. The more general discretization approach of the FIT, however, inherently includes an elegant matrix-vector notation, which enables the application of powerful tools for the analysis of consistency, stability, and other issues. On the implementation side this leads to many important consequences concerning the basic method as well as all kinds of extensions.
Thomas Weiland


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