Eddy Current Approximation of Maxwell Equations

by Alonso Rodriguez, Ana; Valli, Alberto

ISBN13: 9788847015050

ISBN10: 8847015057

Format: Hardcover

Pub. Date: 2010-08-30

Publisher(s): Springer Nature

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Summary

This book deals with the mathematical analysis and the numerical approximation of eddy current problems in the time-harmonic case. All the most used formulations are taken into account, placing the problem in a rigorous functional framework. Nodal or edge finite elements are used for approximation. A detailed analysis of each formulation is presented, focusing on the well-posedness and on the efficiency of numerical schemes. A particular attention is devoted to the topology of the physical domain. Some specific applications to real-life problems are described. The reader will also find a complete presentation of some recent and new results on the subject.

Setting the problem	p. 1
Maxwell equations and time-harmonic Maxwell equations	p. 1
Eddy currents and eddy current approximation	p. 4
Geometrical setting and boundary conditions	p. 8
Harmonic fields in electromagnetism	p. 10
The complete eddy current model	p. 15
A mathematical justification of the eddy current model	p. 21
The E-based formulation of Maxwell equations	p. 21
The eddy current model as the low electric permittivity limit	p. 25
The eddy current model as the low-frequency limit	p. 27
Higher order convergence	p. 30
Existence and uniqueness of the solution	p. 35
Weak formulation, existence and uniqueness for the magnetic field	p. 36
Determination of the electric field	p. 38
Strong formulation for the magnetic field	p. 43
The Faraday equation for the "cutting" surfaces	p. 46
Suitability of other formulations	p. 48
Existence and uniqueness for the complete eddy current model	p. 51
Other boundary conditions	p. 52
Hybrid formulations for the electric and magnetic fields	p. 59
Hybrid formulation using the magnetic field in the insulator	p. 60
A saddle-point approach for the E_C/H_I formulation	p. 62
Finite element discretization	p. 67
A saddle-point approach for the H-based formulation	p. 76
Hybrid formulation using the electric field in the insulator	p. 78
A saddle-point approach for the H_C/E_I formulation	p. 83
Finite element discretization	p. 87
Some remarks on implementation	p. 92
Numerical results	p. 97
A saddle-point approach for the E-based formulation	p. 104
Formulations via scalar potentials	p. 111
The weak formulation in terms of H_C and ¿_I	p. 112
The strong formulation in terms of H_C and ¿_I	p. 117
A domain decomposition procedure	p. 119
The formulation in terms of E_C and ¿^*_I	p. 120
A domain decomposition procedure	p. 124
Numerical approximation	p. 125
The determination of a vector potential for the density current J_e,I	p. 126
Finite element approximation	p. 128
The finite element approximation of E_I	p. 140
Formulations via vector potentials	p. 147
Formulation for the Coulomb gauge and its numerical approximation	p. 148
The weak formulation	p. 154
Existence and uniqueness of the solution to the weak formulation	p. 161
Numerical approximation	p. 165
Numerical results	p. 170
A penalized formulation for the electric field	p. 177
Formulation for the Lorenz gauge and its numerical approximation	p. 180
Decoupled weak formulations and alternative gauge conditions	p. 183
Well-posed formulations based on the Lorenz gauge	p. 188
Weak formulations and positiveness	p. 191
Numerical approximation	p. 194
Other potential formulations	p. 195
Coupled FEM-BEM approaches	p. 205
The (A_C, V_C)-¿_I formulation	p. 207
The (A_C, V_C)- ¿_¿ weak formulation	p. 209
Existence and uniqueness of the weak solution	p. 213
Stability as ¿ goes to 0	p. 216
Numerical approximation	p. 218
The non-convex case	p. 221
Other FEM-BEM approaches	p. 221
The code TRIFOU	p. 221
An approach based on the magnetic field H_C	p. 224
An approach based on the electric field E_C	p. 230
Voltage and current intensity excitation	p. 235
The eddy current problem in the presence of electric ports	p. 236
Hybrid formulations in term of E_C and ¿_I^*	p. 238
Formulations in terms of H_C and ¿_I^*	p. 248
Formulations in terms of T_C and ¿_I^*	p. 250
Finite element approximation	p. 254
Numerical results	p. 258
Voltage and current intensity excitation for an internal conductor	p. 263
Variational formulations	p. 267
Selected applications	p. 275
Metallurgical thermoelectrical problems	p. 275
Induction furnaces	p. 276
Metallurgical electrodes	p. 279
Bioelectromagnetism: EEG and MEG	p. 286
Magnetic levitation	p. 293
Power transformers	p. 298
Defect detection	p. 303
Appendix	p. 309
Functional spaces and notation	p. 309
Nodal and edge finite elements	p. 313
Grad-conforming finite elements	p. 314
Curl-conforming finite elements	p. 317
Orthogonal decomposition results	p. 321
First decomposition result	p. 321
Second decomposition result	p. 324
Third decomposition result	p. 326
More on harmonic fields	p. 327
References	p. 331
Index	p. 345
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Eddy Current Approximation of Maxwell Equations

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