Valli Eddy Current Approximation of Maxwell Equations

Theory, Algorithms and Applications
1. Auflage 2010
ISBN: 978-88-470-1506-7
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 347 Seiten

Autoren/Hrsg.

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