Alonso Rodriguez / Valli Eddy Current Approximation of Maxwell Equations

Theory, Algorithms and Applications
1. Auflage 2010
ISBN: 978-88-470-1506-7
Verlag: Springer Milan
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 4, 355 Seiten

Reihe: MS&A

This book deals with the mathematical analysis and the numerical approximation of eddy current problems in the time-harmonic case. It takes into account all the most used formulations, placing the problem in a rigorous functional framework.

Since more than ten years the authors have performed researches in the field of numerical approximation of partial differential equations, especially the finite element approximation of problems in electromagnetics. They have published many papers on numerical approximation of partial differential equations on some of the most important journal devoted to this topic. The second author has written two other books on the subject: A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer, 1997 (second edition); A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations, Oxford University Press, 1999

Autoren/Hrsg.

Alonso Rodriguez, Ana

Valli, Alberto

Inhaltsverzeichnis

1;Title Page;4
2;Copyright Page;5
3;Preface;6
4;Table of Contents;12
5;1 Setting the problem;15
5.1;1.1 Maxwell equations and time-harmonic Maxwell equations;15
5.2;1.2 Eddy currents and eddy current approximation;18
5.3;1.3 Geometrical setting and boundary conditions;22
5.4;1.4 Harmonic fields in electromagnetism;24
5.5;1.5 The complete eddy current model;29
6;2 A mathematical justification of the eddy current model;34
6.1;2.1 The E-based formulation of Maxwell equations;34
6.2;2.2 The eddy current model as the low electric permittivity limit;38
6.3;2.3 The eddy current model as the low-frequency limit;40
6.3.1;2.3.1 Higher order convergence;43
7;3 Existence and uniqueness of the solution;48
7.1;3.1 Weak formulation, existence and uniqueness for the magnetic field;49
7.2;3.2 Determination of the electric field;51
7.3;3.3 Strong formulation for the magnetic field;56
7.3.1;3.3.1 The Faraday equation for the “cutting” surfaces;59
7.3.2;3.3.2 Suitability of other formulations;61
7.4;3.4 Existence and uniqueness for the complete eddy current model;64
7.5;3.5 Other boundary conditions;65
8;4 Hybrid formulations for the electric and magnetic fields;71
8.1;4.1 Hybrid formulation using the magnetic field in the insulator;72
8.2;4.2 A saddle-point approach for the EC /HI formulation;74
8.2.1;4.2.1 Finite element discretization;79
8.3;4.3 A saddle-point approach for the H-based formulation;88
8.4;4.4 Hybrid formulation using the electric field in the insulator;90
8.5;4.5 A saddle-point approach for the HC /EI formulation;95
8.5.1;4.5.1 Finite element discretization;99
8.5.2;4.5.2 Some remarks on implementation;104
8.5.3;4.5.3 Numerical results;109
8.6;4.6 A saddle-point approach for the E-based formulation;116
9;5 Formulations via scalar potentials;123
9.1;5.1 The weak formulation in terms of HC and ?I;124
9.2;5.2 The strong formulation in terms of HC and ?I;129
9.2.1;5.2.1 A domain decomposition procedure;131
9.3;5.3 The formulation in terms of EC and ??I;132
9.3.1;5.3.1 A domain decomposition procedure;136
9.4;5.4 Numerical approximation;137
9.4.1;5.4.1 The determination of a vector potential for the density current Je,I;138
9.4.2;5.4.2 Finite element approximation;140
9.5;5.5 The finite element approximation of EI;152
10;6 Formulations via vector potentials;158
10.1;6.1 Formulation for the Coulomb gauge and its numerical approximation;159
10.1.1;6.1.1 The weak formulation;165
10.1.2;6.1.2 Existence and uniqueness of the solution to the weak formulation;172
10.1.3;6.1.3 Numerical approximation;176
10.1.4;6.1.4 Numerical results;181
10.1.5;6.1.5 A penalized formulation for the electric field;188
10.2;6.2 Formulation for the Lorenz gauge and its numerical approximation;191
10.2.1;6.2.1 Decoupled weak formulations and alternative gauge conditions;194
10.2.2;6.2.2 Well-posed formulations based on the Lorenz gauge;199
10.2.3;6.2.3 Weak formulations and positiveness;202
10.2.4;6.2.4 Numerical approximation;205
10.3;6.3 Other potential formulations;206
11;7 Coupled FEM–BEM approaches;216
11.1;7.1 The (AC, VC ) ? ?I formulation;218
11.2;7.2 The (AC, VC ) ? ?? weak formulation;220
11.3;7.3 Existence and uniqueness of the weak solution;224
11.4;7.4 Stability as ? goes to 0;227
11.5;7.5 Numerical approximation;229
11.5.1;7.5.1 The non-convex case;232
11.6;7.6 Other FEM–BEM approaches;232
11.6.1;7.6.1 The code TRIFOU;232
11.6.2;7.6.2 An approach based on the magnetic field HC;235
11.6.3;7.6.3 An approach based on the electric field EC;241
12;8 Voltage and current intensity excitation;246
12.1;8.1 The eddy current problem in the presence of electric ports;247
12.1.1;8.1.1 Hybrid formulations in term of EC and ??I;249
12.1.2;8.1.2 Formulations in terms of HC and ??I;259
12.1.3;8.1.3 Formulations in terms of TC and ??I;261
12.1.4;8.1.4 Finite element approximation;265
12.1.5;8.1.5 Numerical results;269
12.2;8.2 Voltage and current intensity excitation for an internal conductor;274
12.2.1;8.2.1 Variational formulations;278
13;9 Selected applications;286
13.1;9.1 Metallurgical thermoelectrical problems;286
13.1.1;9.1.1 Induction furnaces;287
13.1.2;9.1.2 Metallurgical electrodes;290
13.2;9.2 Bioelectromagnetism: EEG and MEG;297
13.3;9.3 Magnetic levitation;304
13.4;9.4 Power transformers;309
13.5;9.5 Defect detection;314
14;Appendix;319
14.1;A.1 Functional spaces and notation;319
14.2;A.2 Nodal and edge finite elements;323
14.2.1;A.2.1 Grad-conforming finite elements;324
14.2.2;A.2.2 Curl-conforming finite elements;327
14.3;A.3 Orthogonal decomposition results;331
14.3.1;A.3.1 First decomposition result;331
14.3.2;A.3.2 Second decomposition result;334
14.3.3;A.3.3 Third decomposition result;336
14.4;A.4 More on harmonic fields;337
15;References;340
16;Index;353