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2010 | Buch

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Eddy Current Approximation of Maxwell Equations

Theory, algorithms and applications

verfasst von: Ana Alonso Rodríguez, Alberto Valli

Verlag: Springer Milan

Buchreihe : MS&A

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Inhaltsverzeichnis

Frontmatter

1. Setting the problem

Abstract

In this chapter, starting from the classical Maxwell equations, we describe and motivate the problem we are going to consider.

Ana Alonso Rodríguez, Alberto Valli

2. A mathematical justification of the eddy current model

Abstract

The aim of this chapter is to analyze in which sense the eddy current model is a proper approximation of the full Maxwell system. As explained in the previous chapter, the eddy current problem is a simplified model derived from the full system of Maxwell equations by neglecting the displacement currents, namely, the term ιωεE. Therefore it can be seen either as the low electric permittivity limit or as the low-frequency limit of the full Maxwell system. The analysis is mainly based on the E-based formulation of Maxwell equations obtained by eliminating the magnetic field.

Ana Alonso Rodríguez, Alberto Valli

3. Existence and uniqueness of the solution

Abstract

The proof of the existence and uniqueness of the solution to problems (1.22), (1.20) and (1.24) is quite similar. In this chapter, followingAlonso Rodríguez et al. [11], we mainly focus on the magnetic boundary value problem (1.22), adding in Section 3.5 a few comments on the electric boundary value problem(1.20) and the no-flux boundary value problem (1.24).

Ana Alonso Rodríguez, Alberto Valli

4. Hybrid formulations for the electric and magnetic fields

Abstract

The classical approaches to Maxwell and eddy current equations are often based on the introduction of a magnetic vector potential and an electric scalar potential, the latter being used only in the conducting region, or on the use of a magnetic scalar potential in the insulating region (see, e.g., Jackson [137], Silvester and Ferrari [227]). We present these formulations in Chapters 6 and 5 respectively.

Ana Alonso Rodríguez, Alberto Valli

5. Formulations via scalar potentials

Abstract

As we have already remarked in the preceding chapters, a specific feature of eddy current problems is the presence of differential constraints acting in the non-conducting part of the domain: namely, curl H_I=J_e,I in Ω_I and div (ε_IE_I)=0 in Ω_I

Ana Alonso Rodríguez, Alberto Valli

6. Formulations via vector potentials

Abstract

Motivated by the fact that the magnetic induction B=μH is divergence-free in Ω, a classical approach to the Maxwell equations and to eddy current problems is based on the introduction of a vector magnetic potential A such that curl A=μH. Often, this is also accompanied by the use of a scalar electric potential VC in the conductor Ω_C, satisfying iwA_C+grad V_C=−E_C (see Silvester and Ferrari [227]; for the engineering literature, see, e.g., Chari et al. [79], Biddlecombe et al. [47], Morisue [180], Bíró and Preis [49]).

Ana Alonso Rodríguez, Alberto Valli

7. Coupled FEM-BEM approaches

Abstract

In this chapter we focus on some procedures for solving eddy current problems that are based on a strategy which couples the finite element method (FEM) and the boundary element method (BEM). This kind of coupling allows the numerical approximation of the solution in unbounded domains, a typical situation in electromagnetism. The boundary element method is used for the approximation in the complement of a bounded domain: either the conductor Ω_C or else an artificial computational domain Ω, containing Ω_C but in general not very large. Instead, in the bounded domain the solution is approximated using the finite element method. Compared with the formulations presented in the previous chapters, the coupled FEM—BEM approaches compute the FEM approximation of the solution in a smaller region (say, the conductor), not required to be so large that the use of homogeneous boundary conditions is justified. This can be done because the BEM method takes into account the behaviour of the solution in the external region.

Ana Alonso Rodríguez, Alberto Valli

8. Voltage and current intensity excitation

Abstract

In many electromagnetic phenomena it is useful to couple formulations in terms of electrical circuits with more general formulations based on Maxwell equations or on some reduced model like the eddy current system. On the common interface between the two models, the boundary data for the domain where the eddy current model is considered are current intensities or voltages.

Ana Alonso Rodríguez, Alberto Valli

9. Selected applications

Abstract

In this chapter we present some real-life problems that can be modeled by the eddy current equations. In some of these examples the time-harmonic eddy current system is used for numerical simulations, and a rich bibliography on the subject is available. However, we also include some applications where, to our knowledge, the eddy current model has not yet been used. We believe that it could be a more accurate description than the ones actually employed, and, using theme thod proposed in this book, it should be suitable for numerical simulations.

Ana Alonso Rodríguez, Alberto Valli

Backmatter

Titel: Eddy Current Approximation of Maxwell Equations
verfasst von: Ana Alonso Rodríguez
Alberto Valli
Copyright-Jahr: 2010
Verlag: Springer Milan
Electronic ISBN: 978-88-470-1506-7
Print ISBN: 978-88-470-1505-0
DOI: https://doi.org/10.1007/978-88-470-1506-7

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