Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben

2018 | Buch

Physical Framework and Models Kapitel lesen Erstes Kapitel lesen

Mathematical Foundations of Computational Electromagnetism

verfasst von: Franck Assous, Patrick Ciarlet, Simon Labrunie

Verlag: Springer International Publishing

Buchreihe : Applied Mathematical Sciences

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

Einloggen, um Zugang zu erhalten
insite
SUCHEN

Über dieses Buch

This book presents an in-depth treatment of various mathematical aspects of electromagnetism and Maxwell's equations: from modeling issues to well-posedness results and the coupled models of plasma physics (Vlasov-Maxwell and Vlasov-Poisson systems) and magnetohydrodynamics (MHD). These equations and boundary conditions are discussed, including a brief review of absorbing boundary conditions. The focus then moves to well‐posedness results. The relevant function spaces are introduced, with an emphasis on boundary and topological conditions. General variational frameworks are defined for static and quasi-static problems, time-harmonic problems (including fixed frequency or Helmholtz-like problems and unknown frequency or eigenvalue problems), and time-dependent problems, with or without constraints. They are then applied to prove the well-posedness of Maxwell’s equations and their simplified models, in the various settings described above. The book is completed with a discussion of dimensionally reduced models in prismatic and axisymmetric geometries, and a survey of existence and uniqueness results for the Vlasov-Poisson, Vlasov-Maxwell and MHD equations.

The book addresses mainly researchers in applied mathematics who work on Maxwell’s equations. However, it can be used for master or doctorate-level courses on mathematical electromagnetism as it requires only a bachelor-level knowledge of analysis.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Physical Framework and Models
Abstract
The aim of this first chapter is to present the physics framework of electromagnetism, in relation to the main sets of equations, that is, Maxwell’s equations and some related approximations. In that sense, it is neither a purely physical nor a purely mathematical point of view. The term model might be more appropriate: sometimes, it will be necessary to refer to specific applications in order to clarify our purpose, presented in a selective and biased way, as it leans on the authors’ personal view. This being stated, this chapter remains a fairly general introduction, including the foremost models in electromagnetics. Although the choice of such applications is guided by our own experience, the presentation follows a natural structure.
Franck Assous, Patrick Ciarlet, Simon Labrunie
Chapter 2. Basic Applied Functional Analysis
Abstract
To measure data and solutions spatially, we recall a number of useful definitions and results on Lebesgue and standard Sobolev spaces. Then, we introduce more specialized Sobolev spaces, which are better suited to measuring solutions to electromagnetics problems, in particular, the divergence and the curl of fields. This also allows one to measure their trace at interfaces between two media, or on the boundary. Last, we construct ad hoc function spaces, adapted to the study of time- and space-dependent electromagnetic fields.
Franck Assous, Patrick Ciarlet, Simon Labrunie
Chapter 3. Complements of Applied Functional Analysis
Abstract
We complement the classic results of Chap. 2 in two directions. In the first part, we review some recent results on the traces of vector fields, and especially the tangential trace of electromagnetic-like fields. In the second part, we focus on the extraction of potentials for curl-free and/or divergence-free fields and consequences. In this chapter, Ω is an open subset of \(\mathbb {R}^3\) with boundary Γ.
Franck Assous, Patrick Ciarlet, Simon Labrunie
Chapter 4. Abstract Mathematical Framework
Abstract
We first introduce basic notions on Banach and Hilbert spaces. Afterwards, we recall some well-known results, which help prove the well-posedness of the various sets of equations we study throughout this book.
Franck Assous, Patrick Ciarlet, Simon Labrunie
Chapter 5. Analyses of Exact Problems: First-Order Models
Abstract
In this chapter, we devote our attention to establishing mathematical properties concerning the electromagnetic fields that are governed by the time-dependent Maxwell equations. For that, we investigate a number of physical properties of the electromagnetic fields exhibited in Chap. 1, using the mathematical tools introduced in Chaps. 2, 3 and 4. We focus mainly on four items.
Franck Assous, Patrick Ciarlet, Simon Labrunie
Chapter 6. Analyses of Approximate Models
Abstract
In this chapter, we specifically study the approximate models that we derived from Maxwell’s equations. We refer to Chap. 1 for the models, and we rely on the mathematical tools introduced in Chaps. 2, 3 and 4. Unless otherwise specified, we consider complex-valued function spaces. Constants that are independent of the data, but that may depend on the domain or on the parameters defining the model, are generically denoted by C, C 0, C 1, etc. We provide incremental proofs for the well-posedness of the static, quasi-static and Darwin models, in the sense that solving the quasi-static models relies on the solution of static problems, whereas solving the Darwin models relies on the solution of static and quasi-static problems.
Franck Assous, Patrick Ciarlet, Simon Labrunie
Chapter 7. Analyses of Exact Problems: Second-Order Models
Abstract
This chapter is devoted to an alternative, second-order formulation of the Maxwell’s equations. We rigorously justify the process we outlined in Sect. 1.​5.​3. This new formulation is especially relevant for computational applications, as it admits several variational formulations, which can be simulated by versatile finite element methods. Our attention will be focused on three issues: equivalence of the second-order equations with the original, first-order equations studied in Chap. 5, the well-posedness of the new formulation and the regularity of its solution, as we did in that chapter. We also study how to take into account the conditions on the divergence of the fields, incorporating them explicitly at some point in the variational formulations. To these ends, we shall again rely on the mathematical tools introduced in Chaps. 2, 3 and 4, as well as on the specific properties of the spaces of electromagnetic fields introduced in Chap. 6.
Franck Assous, Patrick Ciarlet, Simon Labrunie
Chapter 8. Analyses of Time-Harmonic Problems
Abstract
In this chapter, we specifically study the time-harmonic Maxwell equations. They derive from the time-dependent equations by assuming that the time dependence of the data and fields is proportional to \(\exp (-\imath \omega t)\), for a pulsation ω ≥ 0 (the frequency is equal to ω∕(2π)). When the pulsation ω is not known, the time-harmonic problem models free vibrations of the electromagnetic fields. One has to solve an eigenproblem, for which both the fields and the pulsation are unknowns. On the other hand, when ω is part of the data, the time-harmonic problem models sustained vibrations. Generally speaking, we refer to this problem as a Helmholtz-like problem, for which the only unknown is the fields.
Franck Assous, Patrick Ciarlet, Simon Labrunie
Chapter 9. Dimensionally Reduced Models: Derivation and Analyses
Abstract
In this chapter, we consider some special situations in which the three-dimensional (3D) Maxwell equations can be reformulated as two-dimensional (2D) models. More precisely, the computational domain boils down to a subset of \(\mathbb {R}^2\), with respect to a suitable system of coordinates (cylindrical, spherical, cartesian). Nevertheless, the electric and magnetic fields, and other vector quantities, still belong to \(\mathbb {R}^3\). Under suitable symmetry assumptions, one gets a single set of 2D equations or, equivalently, a single 2D variational formulation. In the general case, the electromagnetic field would be the solution to an infinite set of 2D equations, or variational formulations, obtained by Fourier analysis.
Franck Assous, Patrick Ciarlet, Simon Labrunie
Chapter 10. Analyses of Coupled Models
Abstract
In this chapter, we analyse the coupled models introduced in Sect. 1.​3, namely the Vlasov–Poisson, Vlasov–Maxwell and magnetohydrodynamics (MHD) systems. They are basic models of charged particle, plasma and conducting fluid physics. We present the useful mathematical tools, and a variety of existence and uniqueness results for several types of solution.
Franck Assous, Patrick Ciarlet, Simon Labrunie
Backmatter
Metadaten
Titel
Mathematical Foundations of Computational Electromagnetism
verfasst von
Franck Assous
Patrick Ciarlet
Simon Labrunie
Copyright-Jahr
2018
Electronic ISBN
978-3-319-70842-3
Print ISBN
978-3-319-70841-6
DOI
https://doi.org/10.1007/978-3-319-70842-3

Neuer Inhalt

    Bildnachweise