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The Finite Element Method for Electromagnetic Modeling
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The Finite Element Method for Electromagnetic Modeling
Edited by Gérard Meunier
First published in France in three volumes by Hermes Science/Lavoisier entitled “Electromagnétisme et éléments finis Vol. 1, 2 et 3” First published in Great Britain and the United States in 2008 by ISTE Ltd and John Wiley & Sons, Inc. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 6 Fitzroy Square London W1T 5DX UK
John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA
www.iste.co.uk
www.wiley.com
© ISTE Ltd, 2008 © LAVOISIER, 2002, 2003 The rights of Gérard Meunier to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Cataloging-in-Publication Data Electromagnétisme et éléments finis. English The finite element method for electromagnetic modeling / edited by Gérard Meunier. p. cm. Includes bibliographical references and index. ISBN: 978-1-84821-030-1 1. Electromagnetic devices--Mathematical models. 2. Electromagnetism--Mathematical models. 3. Engineering mathematics. 4. Finite element method. I. Meunier, Gérard. TK7872.M25E4284 2008 621.301'51825--dc22 2007046086 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN: 978-1-84821-030-1 Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire.
Table of Contents
Chapter 1. Introduction to Nodal Finite Elements . . . . . . . . . . . . . . . . Jean-Louis COULOMB
1
1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1. The finite element method . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. The 1D finite element method . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1. A simple electrostatics problem . . . . . . . . . . . . . . . . . . . . . . 1.2.2. Differential approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3. Variational approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4. First-order finite elements . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5. Second-order finite elements . . . . . . . . . . . . . . . . . . . . . . . 1.3. The finite element method in two dimensions . . . . . . . . . . . . . . . . 1.3.1. The problem of the condenser with square section. . . . . . . . . . . 1.3.2. Differential approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3. Variational approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4. Meshing in first-order triangular finite elements . . . . . . . . . . . . 1.3.5. Finite element interpolation . . . . . . . . . . . . . . . . . . . . . . . . 1.3.6. Construction of the system of equations by the Ritz method . . . . . 1.3.7. Calculation of the matrix coefficients . . . . . . . . . . . . . . . . . . 1.3.8. Analysis of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.9. Dual formations, framing and convergence . . . . . . . . . . . . . . . 1.3.10. Resolution of the nonlinear problems. . . . . . . . . . . . . . . . . . 1.3.11. Alternative to the variational method: the weighted residues method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4. The reference elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1. Linear reference elements . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2. Surface reference elements. . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3. Volume reference elements . . . . . . . . . . . . . . . . . . . . . . . . 1.4.4. Properties of the shape functions . . . . . . . . . . . . . . . . . . . . . 1.4.5. Transformation from reference coordinates to domain coordinates . 1.4.6. Approximation of the physical variable . . . . . . . . . . . . . . . . .
1 1 2 2 3 4 6 9 10 10 12 14 15 17 19 21 25 42 44 45 47 48 49 52 53 54 56
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1.4.7. Numerical integrations on the reference elements 1.4.8. Local Jacobian derivative method . . . . . . . . . 1.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6. References . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 2. Static Formulations: Electrostatic, Electrokinetic, Magnetostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Patrick DULAR and Francis PIRIOU 2.1. Problems to solve . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1. Maxwell’s equations . . . . . . . . . . . . . . . . . . . . 2.1.2. Behavior laws of materials . . . . . . . . . . . . . . . . . 2.1.3. Boundary conditions . . . . . . . . . . . . . . . . . . . . 2.1.4. Complete static models . . . . . . . . . . . . . . . . . . . 2.1.5. The formulations in potentials. . . . . . . . . . . . . . . 2.2. Function spaces in the fields and weak formulations . . . . 2.2.1. Integral expressions: introduction. . . . . . . . . . . . . 2.2.2. Definitions of function spaces . . . . . . . . . . . . . . . 2.2.3. Tonti diagram: synthesis scheme of a problem . . . . . 2.2.4. Weak formulations . . . . . . . . . . . . . . . . . . . . . 2.3. Discretization of function spaces and weak formulations . 2.3.1. Finite elements . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2. Sequence of discrete spaces . . . . . . . . . . . . . . . . 2.3.3. Gauge conditions and source terms in discrete spaces. 2.3.4. Weak discrete formulations . . . . . . . . . . . . . . . . 2.3.5. Expression of global variables. . . . . . . . . . . . . . . 2.4. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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60 63 66 66
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70 70 71 71 74 75 82 82 82 84 86 91 91 93 106 109 114 115
Chapter 3. Magnetodynamic Formulations . . . . . . . . . . . . . . . . . . . . Zhuoxiang REN and Frédéric BOUILLAULT
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3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Electric formulations . . . . . . . . . . . . . . . . . . . . 3.2.1. Formulation in electric field . . . . . . . . . . . . . 3.2.2. Formulation in combined potentials a - \ . . . . 3.2.3. Comparison of the formulations in field and in combined potentials . . . . . . . . . . . . . . . . . 3.3. Magnetic formulations . . . . . . . . . . . . . . . . . . . 3.3.1. Formulation in magnetic field . . . . . . . . . . . . 3.3.2. Formulation in combined potentials t - I . . . . . 3.3.3. Numerical example . . . . . . . . . . . . . . . . . . 3.4. Hybrid formulation . . . . . . . . . . . . . . . . . . . . . 3.5. Electric and magnetic formulation complementarities 3.5.1. Complementary features . . . . . . . . . . . . . . . 3.5.2. Concerning the energy bounds . . . . . . . . . . . 3.5.3. Numerical example . . . . . . . . . . . . . . . . . .
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117 119 119 120
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121 123 123 124 125 127 128 128 129 129
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3.6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
133 134
Chapter 4. Mixed Finite Element Methods in Electromagnetism . . . . . . Bernard BANDELIER and Françoise RIOUX-DAMIDAU
139
4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Mixed formulations in magnetostatics. . . . . . . . . . . . . . . . . . . . . 4.2.1. Magnetic induction oriented formulation . . . . . . . . . . . . . . . . 4.2.2. Formulation oriented magnetic field . . . . . . . . . . . . . . . . . . . 4.2.3. Formulation in induction and field . . . . . . . . . . . . . . . . . . . . 4.2.4. Alternate case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Energy approach: minimization problems, searching for a saddle-point. 4.3.1. Minimization of a functional calculus related to energy . . . . . . . 4.3.2. Variational principle of magnetic energy . . . . . . . . . . . . . . . . 4.3.3. Searching for a saddle-point . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4. Functional calculus related to the constitutive relationship . . . . . . 4.4. Hybrid formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1. Magnetic induction oriented hybrid formulation . . . . . . . . . . . . 4.4.2. Hybrid formulation oriented magnetic field. . . . . . . . . . . . . . . 4.4.3. Mixed hybrid method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5. Compatibility of approximation spaces – inf-sup condition . . . . . . . . 4.5.1. Mixed magnetic induction oriented formulation . . . . . . . . . . . . 4.5.2. Mixed formulation oriented magnetic field . . . . . . . . . . . . . . . 4.5.3. General case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6. Mixed finite elements, Whitney elements. . . . . . . . . . . . . . . . . . . 4.6.1. Magnetic induction oriented formulation . . . . . . . . . . . . . . . . 4.6.2. Magnetic field oriented formulation . . . . . . . . . . . . . . . . . . . 4.7. Mixed formulations in magnetodynamics. . . . . . . . . . . . . . . . . . . 4.7.1. Magnetic field oriented formulation . . . . . . . . . . . . . . . . . . . 4.7.2. Formulation oriented electric field . . . . . . . . . . . . . . . . . . . . 4.8. Solving techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.1. Penalization methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.2. Algorithm using the Schur complement . . . . . . . . . . . . . . . . . 4.9. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
139 140 141 144 146 147 147 147 149 151 154 154 154 156 157 157 158 160 160 161 162 163 164 164 167 167 168 171 174
Chapter 5. Behavior Laws of Materials . . . . . . . . . . . . . . . . . . . . . . Frédéric BOUILLAULT, Afef KEDOUS-LEBOUC, Gérard MEUNIER, Florence OSSART and Francis PIRIOU
177
5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Behavior law of ferromagnetic materials . . . . . . . . . . . . 5.2.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2. Hysteresis and anisotropy . . . . . . . . . . . . . . . . . . 5.2.3. Classificiation of models dealing with the behavior law 5.3. Implementation of nonlinear behavior models . . . . . . . . .
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177 178 178 179 180 183
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5.3.1. Newton method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2. Fixed point method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3. Particular case of a behavior with hysteresis . . . . . . . . . . . . . . 5.4. Modeling of magnetic sheets . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1. Some words about magnetic sheets. . . . . . . . . . . . . . . . . . . . 5.4.2. Example of stress in the electric machines . . . . . . . . . . . . . . . 5.4.3. Anisotropy of sheets with oriented grains . . . . . . . . . . . . . . . . 5.4.4. Hysteresis and dynamic behavior under uniaxial stress . . . . . . . . 5.4.5. Determination of iron losses in electric machines: nonlinear isotropic finite element modeling and calculation of the losses a posteriori . . . . . 5.4.6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5. Modeling of permanent magnets . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2. Magnets obtained by powder metallurgy . . . . . . . . . . . . . . . . 5.5.3. Study of linear anisotropic behavior . . . . . . . . . . . . . . . . . . . 5.5.4. Study of nonlinear behavior . . . . . . . . . . . . . . . . . . . . . . . . 5.5.5. Implementation of the model in finite element software . . . . . . . 5.5.6. Validation: the experiment by Joel Chavanne . . . . . . . . . . . . . 5.5.7. Conductive magnet subjected to an AC field . . . . . . . . . . . . . . 5.6. Modeling of superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2. Behavior of superconductors . . . . . . . . . . . . . . . . . . . . . . . 5.6.3. Modeling of electric behavior of superconductors . . . . . . . . . . . 5.6.4. Particular case of the Bean model. . . . . . . . . . . . . . . . . . . . . 5.6.5. Examples of modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
183 187 191 192 192 192 194 200 209 215 216 216 216 218 220 223 224 225 226 226 227 230 232 237 240 241
Chapter 6. Modeling on Thin and Line Regions . . . . . . . . . . . . . . . . . Christophe GUÉRIN
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6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Different special elements and their interest . . . . . . . . . . . . . . . 6.3. Method for taking into account thin regions without potential jump . 6.4. Method for taking into account thin regions with potential jump . . . 6.4.1. Analytical integration method . . . . . . . . . . . . . . . . . . . . . 6.4.2. Numerical integration method . . . . . . . . . . . . . . . . . . . . . 6.5. Method for taking thin regions into account . . . . . . . . . . . . . . . 6.6. Thin and line regions in magnetostatics . . . . . . . . . . . . . . . . . . 6.6.1. Thin and line regions in magnetic scalar potential formulations. 6.6.2. Thin and line regions in magnetic vector potential formulations 6.7. Thin and line regions in magnetoharmonics . . . . . . . . . . . . . . . 6.7.1. Solid conducting regions presenting a strong skin effect . . . . . 6.7.2. Thin conducting regions . . . . . . . . . . . . . . . . . . . . . . . .
245 245 249 250 251 252 255 256 256 257 257 258 265
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6.8. Thin regions in electrostatic problems: “electric harmonic problems” and electric conduction problems . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9. Thin thermal regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 7. Coupling with Circuit Equations . . . . . . . . . . . . . . . . . . . 277 Gérard MEUNIER, Yvan LEFEVRE, Patrick LOMBARD and Yann LE FLOCH 7.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Review of the various methods of setting up electric circuit equations . 7.2.1. Circuit equations with nodal potentials . . . . . . . . . . . . . . . . . 7.2.2. Circuit equations with mesh currents. . . . . . . . . . . . . . . . . . . 7.2.3. Circuit eqautions with time integrated nodal potentials . . . . . . . . 7.2.4. Formulation of circuit equations in the form of state equations . . . 7.2.5. Conclusion on the methods of setting up electric equations . . . . . 7.3. Different types of coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1. Indirect coupling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2. Integro-differential formulation . . . . . . . . . . . . . . . . . . . . . . 7.3.3. Simultaneous resolution . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4. Establishment of the “current-voltage” relations . . . . . . . . . . . . . . 7.4.1. Insulated massive conductor with two ends: basic assumptions and preliminary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2. Current-voltage relations using the magnetic vector potential . . . . 7.4.3. Current-voltage relations using magnetic induction . . . . . . . . . . 7.4.4. Wound conductors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.5. Losses in the wound conductors . . . . . . . . . . . . . . . . . . . . . 7.5. Establishment of the coupled field and circuit equations . . . . . . . . . . 7.5.1. Coupling with a vector potential formulation in 2D . . . . . . . . . . 7.5.2. Coupling with a vector potential formulation in 3D . . . . . . . . . . 7.5.3. Coupling with a scalar potential formulation in 3D . . . . . . . . . . 7.6. General conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 8. Modeling of Motion: Accounting for Movement in the Modeling of Magnetic Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . Vincent LECONTE 8.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. Formulation of an electromagnetic problem with motion 8.2.1. Definition of motion . . . . . . . . . . . . . . . . . . . 8.2.2. Maxwell equations and motion . . . . . . . . . . . . . 8.2.3. Formulations in potentials . . . . . . . . . . . . . . . . 8.2.4. Eulerian approach . . . . . . . . . . . . . . . . . . . . . 8.2.5. Lagrangian approach . . . . . . . . . . . . . . . . . . .
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277 278 278 279 280 281 283 284 285 285 285 285 286 286 287 288 290 291 292 292 303 310 317 318 321 321 322 322 325 329 335 338
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8.2.6. Example application. . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3. Methods for taking the movement into account . . . . . . . . . . . . . 8.3.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2. Methods for rotating machines . . . . . . . . . . . . . . . . . . . . 8.3.3. Coupling methods without meshing and with the finite element method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.4. Coupling of boundary integrals with the finite element method . 8.3.5. Automatic remeshing methods for large distortions . . . . . . . . 8.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 9. Symmetric Components and Numerical Modeling . . . . . . . . Jacques LOBRY, Eric NENS and Christian BROCHE
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9.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2. Representation of group theory . . . . . . . . . . . . . . . . 9.2.1. Finite groups . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2. Symmetric functions and irreducible representations 9.2.3. Orthogonal decomposition of a function. . . . . . . . 9.2.4. Symmetries and vector fields . . . . . . . . . . . . . . 9.3. Poisson’s problem and geometric symmetries . . . . . . . 9.3.1. Differential and integral formulations . . . . . . . . . 9.3.2. Numerical processing . . . . . . . . . . . . . . . . . . . 9.4. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1. 2D magnetostatics . . . . . . . . . . . . . . . . . . . . . 9.4.2. 3D magnetodynamics . . . . . . . . . . . . . . . . . . . 9.5. Conclusions and future work . . . . . . . . . . . . . . . . . 9.6. References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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369 371 371 374 378 379 384 384 387 388 388 394 403 404
Chapter 10. Magneto-thermal Coupling . . . . . . . . . . . . . . . . . . . . . . Mouloud FÉLIACHI and Javad FOULADGAR
405
10.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2. Magneto-thermal phenomena and fundamental equations 10.2.1. Electromagentism . . . . . . . . . . . . . . . . . . . . . 10.2.2. Thermal . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.3. Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3. Behavior laws and couplings . . . . . . . . . . . . . . . . . 10.3.1. Electrmagnetic phenomena . . . . . . . . . . . . . . . . 10.3.2. Thermal phenomena . . . . . . . . . . . . . . . . . . . . 10.3.3. Flow phenomena . . . . . . . . . . . . . . . . . . . . . . 10.4. Resolution methods . . . . . . . . . . . . . . . . . . . . . . . 10.4.1. Numerical methods . . . . . . . . . . . . . . . . . . . . 10.4.2. Semi-analytical methods . . . . . . . . . . . . . . . . . 10.4.3. Analytical-numerical methods . . . . . . . . . . . . . . 10.4.4. Magneto-thermal coupling models . . . . . . . . . . .
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405 406 406 408 408 409 409 409 409 409 409 410 411 411
Table of Contents
10.5. Heating of a moving work piece . 10.6. Induction plasma . . . . . . . . . . 10.6.1. Introduction . . . . . . . . . . . 10.6.2. Inductive plasma installation . 10.6.3. Mathematical models . . . . . 10.6.4. Results . . . . . . . . . . . . . . 10.6.5. Conclusion . . . . . . . . . . . 10.7. References . . . . . . . . . . . . . .
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xi
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413 417 417 418 418 426 427 428
Chapter 11. Magneto-mechanical Modeling . . . . . . . . . . . . . . . . . . . Yvan LEFEVRE and Gilbert REYNE
431
11.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2. Modeling of coupled magneto-mechancial phenomena. . . . . . . . . . 11.2.1. Modeling of mechanical structure . . . . . . . . . . . . . . . . . . . . 11.2.2. Coupled magneto-mechanical modeling . . . . . . . . . . . . . . . . 11.2.3. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3. Numerical modeling of electromechancial conversion in conventional actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.1. General simulation procedure . . . . . . . . . . . . . . . . . . . . . . 11.3.2. Global magnetic force calculation method. . . . . . . . . . . . . . . 11.3.3. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4. Numerical modeling of electromagnetic vibrations . . . . . . . . . . . . 11.4.1. Magnetostriction vs. magnetic forces . . . . . . . . . . . . . . . . . . 11.4.2. Procedure for simulating vibrations of magnetic origin . . . . . . . 11.4.3. Magnetic forces density. . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.4. Case of rotating machine teeth . . . . . . . . . . . . . . . . . . . . . . 11.4.5. Magnetic response modeling . . . . . . . . . . . . . . . . . . . . . . . 11.4.6. Model superposition method . . . . . . . . . . . . . . . . . . . . . . . 11.4.7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5. Modeling strongly coupled phenomena . . . . . . . . . . . . . . . . . . . 11.5.1. Weak coupling and strong coupling from a physical viewpoint . . 11.5.2. Weak coupling or strong coupling problem from a numerical modeling analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.3. Weak coupling and intelligent use of software tools . . . . . . . . . 11.5.4. Displacement and deformation of a magnetic system . . . . . . . . 11.5.5. Structural modeling based on magnetostrictive materials . . . . . . 11.5.6. Electromagnetic induction launchers . . . . . . . . . . . . . . . . . . 11.6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
431 432 433 437 442 442 443 444 447 447 447 449 449 452 453 455 458 459 459 460 461 463 465 469 470 471
Chapter 12. Magnetohydrodynamics: Modeling of a Kinematic Dynamo . Franck PLUNIAN and Philippe MASSÉ
477
12.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.1. Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
477 477
xii
The Finite Element Method for Electromagnetic Modeling
12.1.2. Maxwell’s equations and Ohm’s law . . . . . . . . . . 12.1.3. The induction equation . . . . . . . . . . . . . . . . . . 12.1.4. The dimensionless equation . . . . . . . . . . . . . . . 12.2. Modeling the induction equation using finite elements . . 12.2.1. Potential (A,I) quadric-vector formulation . . . . . . 12.2.2. 2D1/2 quadri-vector potential formulation . . . . . . . 12.3. Some simulation examples. . . . . . . . . . . . . . . . . . . 12.3.1. Screw dynamo (Ponomarenko dynamo) . . . . . . . . 12.3.2. Two-scale dynamo without walls (Roberts dynamo). 12.3.3. Two-scale dynamo with walls . . . . . . . . . . . . . . 12.3.4. A dynamo at the industrial scale. . . . . . . . . . . . . 12.4. Modeling of the dynamic problem . . . . . . . . . . . . . . 12.5. References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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481 482 483 485 485 488 491 491 495 498 502 503 504
Chapter 13. Mesh Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yves DU TERRAIL COUVAT, François-Xavier ZGAINSKI and Yves MARÉCHAL
509
13.1. Introduction . . . . . . . . . . . . . . . . . . . . . 13.2. General definition . . . . . . . . . . . . . . . . . . 13.3. A short history . . . . . . . . . . . . . . . . . . . . 13.4. Mesh algorithms . . . . . . . . . . . . . . . . . . . 13.4.1. The basic algorithms. . . . . . . . . . . . . . 13.4.2. General mesh algorithms . . . . . . . . . . . 13.5. Mesh regularization . . . . . . . . . . . . . . . . . 13.5.1. Regularization by displacement of nodes . 13.5.2. Regularization by bubbles . . . . . . . . . . 13.5.3. Adaptation of nodes population . . . . . . . 13.5.4. Insertion in meshing algorithms . . . . . . . 13.5.5. Value of bubble regularization. . . . . . . . 13.6. Mesh processer and modeling environment. . . 13.6.1. Some typical criteria. . . . . . . . . . . . . . 13.6.2. Electromagnetism and meshing constraints 13.7. Conclusion . . . . . . . . . . . . . . . . . . . . . . 13.8. References . . . . . . . . . . . . . . . . . . . . . .
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Chapter 14. Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jean-Louis COULOMB
547
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509 510 512 512 512 518 526 526 528 530 530 531 533 533 534 541 541
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14.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.1. Optimization: who, why, how? . . . . . . . . . . . . . . . . 14.1.2. Optimization by numerical simulation: is this reasonable? 14.1.3. Optimization by numerical simulation: difficulties. . . . . 14.1.4. Numerical design of experiments (DOE) method: an elegant solution . . . . . . . . . . . . . . . . . . . . . . .
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xiii
14.1.5. Sensitivity analysis: an “added value” accessible by simulation . 14.1.6. Organization of this chapter . . . . . . . . . . . . . . . . . . . . . . . 14.2. Optimization methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.1. Optimization problems: some definitions . . . . . . . . . . . . . . . 14.2.2. Optimization problems without constraints . . . . . . . . . . . . . . 14.2.3. Constrained optimization problems . . . . . . . . . . . . . . . . . . . 14.2.4. Multi-objective optimization . . . . . . . . . . . . . . . . . . . . . . . 14.3. Design of experiments (DOE) method. . . . . . . . . . . . . . . . . . . . 14.3.1. The direct control of the simulation tool by an optimization algorithm: principle and disadvantages . . . . . . . . . . . . . . . . . . . . . 14.3.2. The response surface: an approximation enabling indirect optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.3. DOE method: a short history . . . . . . . . . . . . . . . . . . . . . . . 14.3.4. DOE method: a simple example . . . . . . . . . . . . . . . . . . . . . 14.4. Response surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.1. Basic principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.2. Polynomial surfaces of degree 1 without interaction: simple but sometimes useful . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.3. Polynomial surfaces of degree 1 with interactions: quite useful for screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.4. Polynomial surfaces of degree 2: a first approach for nonlinearities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.5. Response surfaces of degrees 1 and 2: interests and limits . . . . . 14.4.6. Response surfaces by combination of radial functions. . . . . . . . 14.4.7. Response surfaces using diffuse elements . . . . . . . . . . . . . . . 14.4.8. Adaptive response surfaces. . . . . . . . . . . . . . . . . . . . . . . . 14.5. Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5.1. Finite difference method . . . . . . . . . . . . . . . . . . . . . . . . . 14.5.2. Method for local derivation of the Jacobian matrix . . . . . . . . . 14.5.3. Steadiness of state variables: steadiness of state equations . . . . . 14.5.4. Sensitivity of the objective function: the adjoint state method . . . 14.5.5. Higher order derivative . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6. A complete example of optimization. . . . . . . . . . . . . . . . . . . . . 14.6.1. The problem of optimization . . . . . . . . . . . . . . . . . . . . . . 14.6.2. Determination of the influential parameters by the DOE method . 14.6.3. Approximation of the objective function by a response surface . . 14.6.4. Search for the optimum on the response surface . . . . . . . . . . . 14.6.5. Verification of the solution by simulation . . . . . . . . . . . . . . . 14.7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.8. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
550 551 551 551 553 559 560 562 562 563 565 565 572 572 573 573 574 576 576 577 579 579 579 580 581 583 583 584 584 585 587 587 587 588 588
List of Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
595
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 1
Introduction to Nodal Finite Elements
1.1. Introduction 1.1.1. The finite element method The finite element method, resulting from the matrix techniques of calculation of the discrete or semi-discrete mechanical structures (assembly of beams), is a tool for resolving problems with partial differential equations involved in physics problems. We will thus tackle this method accordingly because it is useful in modeling mechanical, thermal, neutron and electromagnetic problems [ZIE 79], [SIL 83], [DHA 84], [SAB 86], [HOO 89]. The aim of this chapter is to present the principles of this method which have become essential in the panoply of the engineer. For this presentation, we will only deal with electrostatics. Indeed, this field has a familiar formulation in scalar potential, particularly suitable for the presentation of nodal finite elements, which will be the only ones discussed here. We will develop two examples of increasing complexity which are manageable “by hand”, 1D in a first part and 2D in a second. As it is very close to physical considerations, the variational approach will most of the time be favored. However, the more general method of weighted residues will also be presented. In our examples, we will see how to solve the problems at issue, but also how, using the obtained fields, to extract more relevant information.
Chapter written by Jean-Louis COULOMB.
2
The Finite Element Method for Electromagnetic Modeling
In the third and last part, we will present the concept of a reference element and the principles that make it possible to pass from the local coordinates to the domain coordinates. We will see that beyond the possibility of handling curvilinear elements, which is quite convenient for the discretization of manufactured objects, this technique leads to a general tool for working with geometric deformations. 1.2. The 1D finite element method 1.2.1. A simple electrostatics problem In order to present the finite element method, we propose, initially, to implement it on a simple 1D electrostatics example, borrowed from [HOO 89]. We will first formulate this problem in its differential form, then in its variation form. This form of integral will enable us to introduce the concept of first-order finite elements and then second-order finite elements. We thus consider the problem of Figure 1.1 where two long distant parallel plates of 10 m are: one with the electric potential of 0 V and the other with the potential of 100 V. Between the two plates, the density of electric charges and the dielectric permittivity are assumed to be constant. This problem could represent a hydrocarbon storage tank in which we wish to know the distribution of the electric potential. The lower plate corresponds to the free surface of the liquid, the upper plate to the ceiling of the tank and the intermediate part to the electrically charged vapors. x
f
v=100V
x+dx
* :
x
s
H U
10m
f
0 v=0V Figure 1.1. The cloud of electric charges between the two plates
Introduction to Nodal Finite Elements
3
1.2.2. Differential approach The physical and geometric quantities varying only according to one direction, this problem is 1D in the interval x [0, 10] and the electric field E and electric flux density D = HE vectors have only one non-zero component Ex and Dx. Let us consider a parallelepipedic elementary volume of constant section s in the direction perpendicular to x and of length dx. The flux of the electric density vector, leaving its border *, and the internal electric charge to its volume : are respectively:
³³ D.d* >Dx ( x dx ) Dx ( x )@.s
[1.1]
³³³ Ud: U .s.dx
[1.2]
*
:
The Gaussian electric law implies the equality of these two integrals, which gives, for the electric flux density, the following differential equation:
dDx dx
[1.3]
U
This equation is specifically one of Maxwell’s equations: divD
[1.4]
U
applied to a 1D problem in which the variations in the orthogonal directions to the x axis are zero. On the terminals of the domain, the boundary conditions are expressed in terms of electric potential v(0) = 0 V and v(10) = 100 V. It is thus judicious to specify the problem entirely in terms of v which is connected to the electric field by the relation dv E x grad v , which, in our 1D case, gives E x . The equation and the dx boundary conditions governing the distribution of the electric potential are thus d ª dv º H » dx «¬ dx ¼ v 0 v 100
U
for x[0, 10] for x = 0 for x = 10
[1.5]
4
The Finite Element Method for Electromagnetic Modeling
In our case, the electric permittivity is constant, which simplifies the equation and becomes
d 2v dx
2
U , H
v ( 0)
0,
v (10)
100
[1.6]
This problem has the following analytical solution v( x )
U 2H
Uº ª x 2 «1 »10 x 2 H¼ ¬
[1.7]
the knowledge of which will be useful for us when evaluating the quality of the solution given by the finite element method, which we will present below. 1.2.3. Variational approach
In fact, the finite element method does not directly use the previous differential form, but is based on an equivalent integral form. For this reason we will develop the variational approach which here is connected to the internal energy of the device. This approach is based on a functional (i.e. a function of the unknown function v(x)) which is extremal when v(x) is the solution. The functional, called coenergy for reasons which will be explained later, corresponding to electrostatics problem [1.5] is 2
Wc ( v )
1 10 ª dv º 10 H « » dx ³0 Uv dx ³ 0 2 ¬ dx ¼
[1.8]
We will show that, if it exists, a continuous and derivable function vm(x) which fulfills the boundary conditions vm(0) = 0 and vm(10) = 100 and which makes functional [1.8] extremal is also the solution of problem [1.5]. For that, let us consider a function v(x) built on the basis of vm(x) as follows v( x )
vm ( x ) DM ( x )
[1.9]
where D is an unspecified real number and M(x) is an arbitrary continuous and derivable function which becomes zero at the boundary of the domain (M(0) = 0 and M(10) = 0). By construction, function v(x) automatically verifies the boundary conditions v(0) = 0 and v(10) = 100.
Introduction to Nodal Finite Elements
5
The introduction into [1.8] of this function v(x) defines a simple function of D 2
Wc (D )
1 10 ª d º 10 H « >vm DM @» dx ³0 U >vm DM @dx ³ 0 2 ¼ ¬ dx
[1.10]
Note that, by assumption, for D = 0 this function is extremal. Let us now express the increase of Wc with respect to its extremum, 2
Wc (D ) Wc (0)
1 10 ª dM º 10 dv dM 10 dx D ³0 UMdx [1.11] D 2 ³0 H « » dx D ³0 H m dx dx 2 ¬ dx ¼
The integration by parts of the second integral gives 10
ª dvm º 10 dvm dM 10 d ª dvm º ³0 H dx dx dx «H dx M » ³0 dx «H dx »Mdx ¬ ¼ ¬ ¼0
dv dM
d
dv
ª 10 10 m mº ³0 H dx dx dx ³0 dx «H dx »Mdx ¬ ¼
[1.12]
because the arbitrary function M(x) is zero on the boundaries of the domain. We thus obtain for the increase of the functional 2
Wc (D ) Wc (0)
½ 1 10 ª dM º 10 d ª dv º D 2 ³0 H « » dx D ³0 ® «H m » U ¾Mdx 2 ¬ dx ¼ ¯ dx ¬ dx ¼ ¿
[1.13]
This polynomial of the second-degree is extremum for D = 0, therefore the coefficient of D must be zero. This coefficient is an integral, to be zero whatever the arbitrary function M(x), and it is necessary that the weighting coefficient of this function becomes zero for any X d ª dvm º H U dx «¬ dx »¼
0
x >0, 10@
[1.14]
which corresponds precisely to equation [1.5], which we want to solve. Therefore, if function vm(x) exists, it is indeed the solution of the specified problem. Moreover, the coefficient of D2 being positive, the extremum is a minimum. The result that we have just obtained is a particular case of a proof that is much more general of the calculus of variations. Equation [1.14] is in fact the Euler
6
The Finite Element Method for Electromagnetic Modeling
equation of functional [1.8], and could thus have been obtained directly by application of a traditional theorem. 1.2.4. First-order finite elements
In order to present the finite element method, we introduce several concepts shown in Figure 1.2. First of all, in the field of study, we define nodes at the positions x1 = 0, x2 = 10/3, x3 = 20/3 and x4 = 10. The electric potentials v1, v2, v3 and v4 at these nodes are called nodal values. Two of these nodal values, v1 = 0 and v4 = 100, are already known thanks to the boundary conditions, while two others, v2 and v3, will have to be determined by application of the finite element method. v
v4= 100
100 v2= ? v3= ?
v1 = 0
x x1=0
x2=10/3
x3=20/3
x4=10
Figure 1.2. Subdivision of the domain into three first-order finite elements
We thus define a subdivision of the domain into finite elements [x1, x2], [x2, x3] and [x3, x4] on which we apply an interpolation for the electric potential. We choose the linear interpolation (order 1) which is the simplest of the interpolations ensuring the continuity of the potential and its derivability per piece, as that is required by the variational approach. On the element [xi, xi+1], this gives for the potential v( x )
x x x x vi i 1 vi 1 i xi 1 xi xi xi 1
[1.15]
Introduction to Nodal Finite Elements
7
and for its gradient dv dx
vi 1 vi xi 1 xi
[1.16]
In order to determine the unknown nodal values v2 and v3, we will use functional [1.8], into which we will introduce the function v(x) defined in [1.15] per piece on each finite element. We will then obtain a function of the only two unknown factors. The extremality conditions of this function will be the equations defining these unknown factors. The subdivision of the domain allows the integral giving the functional to be expressed in a sum of integrals on the finite elements Wc
x3
x2
10
³
³
x1
0
³
x4
x2
Wc 1 Wc 2 Wc 3
³
[1.17]
x3
The elementary contribution Wc i of the element [xi, xi+1] is written Wc i
Wc i
1 2
xi 1
³
xi
2
xi 1 ªv v º ª x x x x º vi 1 i H « i 1 i » dx ³ U «vi i 1 » dx xi xi 1 ¼ ¬ xi 1 xi ¼ ¬ xi 1 xi x i
1 >vi 1 vi @2 1 U >vi 1 vi @>xi 1 xi @ H 2 xi 1 xi 2
[1.18]
The integral thus becomes Wc
1 >v2 v1 @2 1 H U >v2 v1 @>x2 x1 @ 2 x2 x1 2 1 >v v @2 1 H 3 2 U >v3 v2 @>x3 x2 @ 2 x3 x2 2 1 >v v @2 1 H 4 3 U >v4 v3 @>x4 x3 @ 2 x4 x3 2
[1.19]
8
The Finite Element Method for Electromagnetic Modeling
The stationarity conditions of Wc, with respect to the two unknown variables v2 and v3, lead to the following two equations wWc wv2
H
v2 v1 1 v v 1 U >x2 x1 @ H 3 2 U >x3 x2 @ x2 x1 2 x3 x2 2
0
wWc wv3
H
v3 v2 1 v v 1 U >x3 x2 @ H 4 3 U >x4 x3 @ x3 x2 2 x4 x3 2
0
[1.20]
To go numerically further, we arbitrarily fix the ratio between the electric permittivity and the density of electric charges
U H
1
[1.21]
We obtain the system of two equations with two unknown variables according to 3v2 3v3 10 5 10 3 3v2 3v3 100 10 5 3
[1.22]
which has the solution v2 v3
400 9 700 9
[1.23]
In Figure 1.3, we can evaluate the quality of the approximation obtained. The interpolation by first-order finite elements is not very far away from the reference solution. It is even exact at the nodes of the grid. In fact, this coincidence is related to the simplicity of the problem taken as an illustration and will not be found in more realistic applications. Here, the exact solution is a second-degree polynomial, whose average behavior is perfectly represented on each piece by linear interpolations. In order to improve the solution, we have two strategies. The first consists of decreasing the size of the finite elements; it is called the h method by reference to the diameter of the elements which is often denoted h. The second consists of increasing the order of the finite elements; it is denoted the p method because p is
Introduction to Nodal Finite Elements
9
often used to represent the order of the approximation. It is this second strategy which we will implement below.
v3
0 x Figure 1.3. Exact solution in continuous line and solution by first-order finite elements in dotted lines
1.2.5. Second-order finite elements
We now decide to implement the second-order elements. In order to simplify our work to the maximum, we define a minimal subdivision of the domain, i.e. three nodes at the positions x1 = 0, x2 = 5, x3 = 10 having the three nodal values v1, v2, and v3 and defining only one second-order finite element [x1, x2, x3]. The nodal values on the limits are v1 = 0 V and v3 = 100 V. Only the internal nodal value v2 is to be determined by the finite element method. On the single finite element, the electric potential is interpolated by v( x )
v1
>x2 x @>x3 x @ v >x3 x @>x1 x @ v >x1 x @>x2 x @ [1.24] >x2 x1 @>x3 x1 @ 2 >x3 x2 @>x1 x2 @ 3 >x1 x3 @>x2 x3 @
and its gradient by dv dx
v1
2 x x 2 x3 2 x x3 x1 2 x x1 x2 [1.25] v3 v2 >x2 x1 @>x3 x1 @ >x3 x2 @>x1 x2 @ >x1 x3 @>x2 x3 @
10
The Finite Element Method for Electromagnetic Modeling
The introduction of these approximations into functional [1.8], the integration then the application of the stationarity condition with respect to v2, led to the equation 2H v 2
H v1 H v 3 25U
[1.26]
which, for the numerical values selected previously v1 = 0, v3 = 100 and U/H = 1 results, for the unknown nodal value, in v2 = 125/2, which is the good value. Figure 1.4 shows the exact solution and the second-order finite elements solution. These are exactly superimposed. Indeed, the exact solution [1.7] is a second-degree polynomial, which is precisely the type of approximation implemented in the second-order finite element method. Here again, this coincidence is only related to the simplicity of the concerned problem. In more complex applications, we will no longer find such perfect solutions.
0
Figure 1.4. The exact solution and the second-order finite elements are exactly superimposed
1.3. The finite element method in two dimensions 1.3.1. The problem of the condenser with square section
We will again be interested in a problem of electrostatics, but this time of a 2D nature, in order to handle a more realistic example of implementation of the finite element method. We will find the differential then the variational forms of this type of problem, with the associated boundary conditions. We will present the general
Introduction to Nodal Finite Elements
11
concepts of domain meshing and finite element interpolation. We will explain the Ritz method and we will implement it to find an approximate solution to the problem. Lastly, we will see how to take advantage of this solution to obtain local and global information that is more explicit than a simple set of nodal values. The studied device is a condenser whose cross-section is represented in Figure 1.5 and whose depth h is very large in front of the section dimensions. y
H=HrH0 U=0
8
P4
4
P1
0V P3
P2
100V x 0
4
8
Figure 1.5. Cross-section of the long condenser
This condenser is composed of two overlapped conductors of square sections, one with the electric potential of 100 V and the other with the potential of 0 V. Taking into account the high dimension of the condenser in the direction perpendicular to the xOy plane, the 2D study of the device in its cross-section will give a very good idea of its global behavior. In fact, we are interested here in the capacitor of this condenser, which we will obtain by using the finite element method. For this purpose, we will initially determine the distribution of the electric potential within the dielectric, assumed to be perfect, placed between the two electrodes.
12
The Finite Element Method for Electromagnetic Modeling
1.3.2. Differential approach
The Maxwell’s equations, representative of the distribution of the electrostatic field in the dielectric, are (Gauss law)
[1.27]
curlE 0
(Faraday law in static mode)
[1.28]
D HE
(constitutive law of the dielectric material)
[1.29]
divD
U
where D is the electric flux density vector, E the electric field vector, U the density of electric charges and H the permittivity of the dielectric. The introduction of v, the electric scalar potential, such that E
[1.30]
grad v
automatically solves the second Maxwell’s equation since the rotational of a gradient is systematically zero. By combining the first and third equations, we obtain the partial differential equation of the electric potential div>H grad v @ U
[1.31]
which, in the reference frame xOy, is written w ª wv º w ª wv º H «H » wx «¬ wx »¼ wy ¬ wy ¼
U
[1.32]
and in the particular case of a constant electric permittivity and of a density of electric charges equal to zero w 2v wx
2
w 2v wy 2
'v
0
[1.33]
However, for the sought generality, we will use expression [1.31] in the rest of this presentation. To go further in the definition of the problem, we should specify the field of study and the boundary conditions. We could take the whole cross-section of the dielectric of Figure 1.5 as field of study, with v = 0 V on the external edge and
Introduction to Nodal Finite Elements
13
v = 100 V on the internal edge. However, the presence of several symmetries allows the zone of study to be considerably reduced, and thus the efforts of calculation. Indeed, we have just to calculate the solution in the eighth [P1, P2, P3, P4] of the domain (see Figure 1.6), then to reconstitute, thanks to symmetries, the distribution of the electric potential in all the dielectric.
y
P4
P3
0V
8 wv wn
0
4 P1
div>H grad v @ U
100V
wv wn
0
P2
x 0
4
8
Figure 1.6. Reduction of the field of study thanks to symmetries
With a partial differential equation such as [1.31], of elliptic type, and in order to specify the problem clearly, it is necessary to impose conditions on all the limits of the field of studies, either on the state variable v, called the Dirichlet condition, or on its normal derivative
wv , called the Neumann condition. We already know that wn
v = 100 V on the edge P1P2 and that v = 0 V on the edge P3P4. It remains to define the conditions on the rest of the border. On the axes of symmetry P2P3 and P4P1, the field has a particular direction: it is tangential. In fact, no electric flux crosses these parts of the border. Mathematically it means that the normal component of the induction is zero Dn = 0, i.e. a zero normal component of the field En = 0 and thus
that the homogenous Neumann condition will take as conditions on these limits.
wv wn
0 on the electric potential, which we
14
The Finite Element Method for Electromagnetic Modeling
1.3.3. Variational approach
The functional of coenergy of the previous differential equation which generalizes that given in [1.8] to the 2D case is ª H >grad v @2
³³ « «¬
2
º Uv » dxdy »¼
[1.34]
This first functional would be well adapted to the specified problem; however, we would rather use the following expression Wc ( v )
grad v
³³ ª«¬ ³0
D dE Uv º hdxdy »¼
[1.35]
This second functional is more general because it is able to handle a possible nonlinearity in the constitutive law D(E), and the presence of the depth h of the device makes it homogenous to an electrical energy. Let us check that the continuous and derivable function vm(x,y) which satisfies the boundary conditions vm = 100 V on P1P2 and vm = 0 V on P3P4 and which makes functional [1.35] stationary, is a solution of equation [1.31] and also satisfies wvm wn
0 on P2P3 and P4P1P2P3.
For this purpose, starting from vm(x,y), we build the function v x, y
v m x , y Gv x , y
[1.36]
where Gv(x,y) is a continuous and derivable function, zero on the Dirichlet type boundaries which play the role of an unspecified infinitesimal variation around the balance function vm(x,y). By construction, v(x,y) always verifies the boundary conditions of the problem on P1P2 and P3P4. Let us introduce v(x,y) into functional [1.35] and express the variation
GWc
³³ > D grad Gv U Gv @hdxdy
[1.37]
By using the vector relation div>D Gv @
divD Gv D grad GV
[1.38]
Introduction to Nodal Finite Elements
15
then the divergence theorem, we obtain
GWc
³³ >divD U @Gv hdxdy ³ Dn Gv hd*
[1.39]
The second integral relates to the border of the domain which can be either of the Dirichlet type or Neumann type. On Dirichlet borders, the variations Gv are zero and the integrals disappear. It remains
GWc
³³ >divD U @Gv hdxdy
³ Dn Gv hd* Neumann
[1.40]
This quantity GWc expresses the variation of functional [1.35] around the stationary state Wc(vm). Therefore, it has to be zero whatever variation Gv, which implies at the same time divD U 0 in the domain and Dn 0 on the Neumann boundaries. We thus recognize the partial differential equation and the desired homogenous Neumann boundary conditions. Moreover, with the usual behavior laws D(E) (monotonous increasing), it can be shown that this stationarity corresponds to a minimum. 1.3.4. Meshing in first-order triangular finite elements
The first stage of the finite element method consists of subdividing the domain of study into elementary sub-domains. For the 2D domain, the simplest subdivision method consists of cutting out in triangles. Figure 1.7 represents such meshing which comprises Nn = 12 nodes n1, n2, …, n12 and Ne = 12 finite elements, e1, e2, …, e12. The fact that here the number of nodes is equal to the number of elements is fortuitous. This meshing complies with the rules known as conformity rules. Thus, the elements do not overlap and two elements are neighbors, either by a common node, or by a common edge which they then share entirely. The details on the positions and the nodal values of the nodes are indicated in Table 1.1. The nodes located on border P1P2 have an electric potential fixed at 100 V and the nodes on P3P4 have a potential of 0 V. On the other hand, the potentials of the other nodes, internal or located on the Neumann borders, are unknown a priori and will thus have to be determined by the finite element method. The connectivities, the electric permittivities and the densities of electric charges of the finite elements are given in Table 1.2. For example, the nodes of the triangle e1 are the three nodes D = n2, E = n1, J = n5 in the prescribed order, its relative permittivity is equal to 1 and its density of electric charges is zero.
16
The Finite Element Method for Electromagnetic Modeling
0
Figure 1.7. Meshing of the domain in triangular finite elements
n1
n2
n3
n4
n5
n6
n7
n8
n9
n10
n11
n12
x
0
2
4
6
0
2
4
0
2
4
6
8
y
6
6
6
6
4
4
4
8
8
8
8
8
v
?
?
?
?
100
100
100
0
0
0
0
0
Table 1.1. Positions and nodal values of meshing nodes
e1
e2
e3
e4
e5
e6
e7
e8
e9
e10
e11
e12
D
n2
n5
n3
n6
n4
n9
n1
n10
n2
n11
n3
n12
E
n1
n6
n2
n7
n3
n8
n2
n9
n3
n10
n4
n11
J
n5
n2
n6
n3
n7
n1
n9
n2
n10
n3
n11
n4
Hr
1
1
1
1
1
1
1
1
1
1
1
1
U
0
0
0
0
0
0
0
0
0
0
0
0
Table 1.2. Relative connectivities, permittivities and densities of charges of the elements
Introduction to Nodal Finite Elements
17
1.3.5. Finite element interpolation
One of the characteristics of the finite element method relies on the way the interpolations of functions, which are defined per piece, i.e. per finite element, are built. Thus, let us consider the triangle of nodes nD, nE, nJ of nodal values vD, vE, vJ. The simplest of the interpolations for the electric potential, compatible with the constraints of continuity and derivability per piece imposed by physics and the variation approach, is a linear interpolation in x and y, of type v ( x, y )
[1.41]
a bx cy
The unknown coefficients a, b and c depend on the nodal values and on the shape of the triangle. The previous formula, expressed at each of the nodes of the element, must turn over the corresponding nodal value, which gives the system of three equations with three unknown variables vD vE vJ
ª vD º a bxD cyD « » a bx E cy E i.e., «v E » « vJ » a bxJ cyJ ¬ ¼
ª1 xD « «1 x E «1 xJ ¬
yD º ªa º » y E ».«« b »» yJ »¼ «¬ c »¼
[1.42]
For a triangle of a non-zero surface, the 3x3 matrix is invertible, which gives ªa º «b» « » «¬ c »¼
ª1 xD « «1 x E «1 xJ ¬
1
yD º ª vD º » « » y E » . «v E » yJ »¼ «¬ vJ »¼
[1.43]
The introduction of a, b and c in interpolation [1.41] leads to the formula v ( x, y )
vD M D ( x, y ) v E M E ( x, y ) vJ M J ( x, y )
[1.44]
where the functions MD(x,y), ME(x,y) and MJ(x,y) have as an expression
MD ( x , y ) M E ( x, y ) MJ ( x, y )
> xE yJ xJ yE yE yJ x xJ xE y @ / 2S > xJ yD xD yJ yJ yD x xD xJ y @ / 2S > xD yE xE yD yD yE x xE xD y @ / 2S
[1.45]
in which the determinant of the matrix is twice the surface of the triangle 2S
x E yJ xJ y E xJ yD xD yJ xD y E x E yD
[1.46]
18
The Finite Element Method for Electromagnetic Modeling
These three functions are called the shape functions of the finite element and have the following characteristic properties
MD ( xD , yD ) 1 MD ( xE , y E ) 0 MD ( xJ , yJ ) 0 M E ( xD , yD ) 0 M E ( x E , y E ) 1 M E ( xJ , yJ ) 0 MJ ( xD , yD ) 0 MJ ( x E , y E ) 0 MJ ( xJ , yJ ) 1 MD ( x, y ) M E ( x, y ) MJ ( x, y ) 1
(x,y) in the element
[1.47] [1.48]
Figure 1.8 shows the variation of the shape function defined on element e4 and associated with node n3. M3 (x,y)
y
M 1
v6 e4 0
v7
v3
x
Figure 1.8. Shape function defined on element e4 and associated with node n3
Let us now consider a node in the domain. The shape function of the domain, associated with this node, is defined per piece in the following way. For an element which has this node, the shape function of the domain is the shape function of this node on the element. For elements which do not have this node, the shape function of the domain is zero. From the construction of the shape functions on each element, the interpolation on an edge depends only on the nodal values of these edges. For an edge common to two elements, the interpolation is identical, whether it is seen by the first element or by the second one. This property ensures the continuity of the function in passing from one element to another. The shape functions of the domain are thus, by construction, continuous on all the domain and derivable per piece. Moreover, they preserve the properties listed in [1.47] and [1.48] which, in a more general way, are written
M i ( xi , yi )
1
M j zi ( xi , yi )
0
Nn
¦M j x, y 1
j 1
[1.49]
Introduction to Nodal Finite Elements
19
As an illustration, the shape function of the domain M3(x,y) of node n3, common to the triangles e3, e4, e5, e9, e10 and e11 is represented in Figure 1.9. It is a pyramid with a hexagonal base which is worth 1 at node n3 and 0 at all the other nodes. M3 (x,y)
M
y
1
0
x
Figure 1.9. Shape function of the domain associated with node n3
Ultimately, these functions, weighted by the nodal values, make it possible to interpolate the electric potential in the domain by the formula v ( x, y )
Nn
¦ v j M j x, y
[1.50]
j 1
1.3.6. Construction of the system of equations by the Ritz method
We have an interpolation function of electric potential [1.50] parameterized by the nodal values v1, v2, …, v12. For Dirichlet boundaries, as for everywhere else, the electric potential is interpolated linearly. Thus, if we fix the nodal values v5, v6 and v7 to 100 V, all this part of the border will be at 100 V. Similarly, if we impose the nodal values v8, v9, …, v12 at 0 V, all the corresponding borders will be at this potential. With the help of these few constraints, our interpolation function systematically verifies the Dirichlet conditions of the problem. Let us separate the nodal values into two groups. The first group comprises the NL = 4 nodal values v1, v2, v3 and v4 which are still unknown, whereas the second corresponds to Nn–NL = 8 nodal values v5, v6, …, v12 fixed a priori thanks to the Dirichlet boundary conditions. In order to obtain an approximation of the solution, we will determine the unknown nodal values by using the Ritz method. This method consists of
20
The Finite Element Method for Electromagnetic Modeling
introducing the interpolation function of the electric potential into functional [1.35], which then becomes a simple function of the unknown nodal values Wc ( v1,..., v N L )
ª
Nn
grad ¦ v j M j x , y
³³ « ³0
j 1
«¬
ª ³³ « U «¬
º D dE » hdxdy »¼
Nn
º ¦ v j M j x, y »h dxdy »¼ j 1
[1.51]
The combination of the unknown nodal values which makes the functional Wc stationary will be regarded as the best possible choice within the meaning of the Ritz method and will thus be retained. This stationarity leads to the NL following necessary conditions wWc wvi
³³ D. grad M i h dxdy ³³ U M i hdxdy
0 for i = 1, 2, …, NL
[1.52]
This is the system of the NL equations of which the resolution gives the NL unknown nodal values. In this case, the electric flux density is a linear function of the electric field and thus of the gradient of the electric potential, which allows the following additional developments Nn
³³ H ¦ v j grad M j . grad M i hdxdy ³³ U M i hdxdy 0 j 1
Nn
¦ v j ³³ H grad M j . grad M i hdxdy ³³ U M i hdxdy 0
[1.53]
j 1
By assigning M ij Ri
³³ H grad M j . grad M i hdxdy ³³ U M i hdxdy
[1.54]
equation [1.53] becomes Nn
¦ v j M ij Ri 0
j 1
[1.55]
Introduction to Nodal Finite Elements
21
The separation of the unknown nodal values and the fixed values NL
¦ v j M ij
Ri
j 1
Nn
¦ v j M ij j N L 1
for i = 1, 2, …, NL
[1.56]
lead to the matrix system SV
[1.57]
Q
where S is a square symmetric matrix of dimension NL*NL and of coefficient Sij M ij , V is the column vector of the NL unknown nodal values, and Q is the column vector of the NL sources with Qi
Ri
Nn
¦v j j N L 1
M ij .
The resolution of this matrix system will be able to directly give the required nodal values. 1.3.7. Calculation of the matrix coefficients
We are interested here in the effective construction of linear system [1.57] starting from information which we have on the meshing of the domain in finite elements. In fact, coefficients [1.54], at the origin of this system, are integrals of the domain which it is natural to express as sums of integrals on the sub-domains which are the finite elements
M ij
³³
...
domain
Ri
³³
domain
...
Ne
¦ e 1
sub domain e
Ne
¦ e 1
...
³³
³³
...
sub domain e
Ne
¦M
e ij
[1.58]
e 1
Ne
e i
¦R
[1.59]
e 1
with for each finite element e M ije
³³ H grad M j . grad M i hdxdy
[1.60]
e
Rie
³³ U M i hdxdy e
[1.61]
22
The Finite Element Method for Electromagnetic Modeling
We know that the shape functions of the domain Mi(x,y) are zero on the elements which do not have node ni. Consequently, summation [1.59] relates in fact only to the elements which share this node. In the same way, summation [1.58] relates only to the elements which have, at the same time, node ni and node nj
¦
M ij
M ije
[1.62]
e having ni and n j
Ri
¦
[1.63]
Rie
e having ni
Only the Mij coefficients corresponding to nodes belonging to the same element are non-zero. This property explains the sparse character of the finite element matrices. Let us now consider a triangle e, having for nodes nD, nE, nJ. Shape functions [1.45] are linear, their gradient is constant, the integral terms of [1.60] and [1.61] are thus very easy to integrate and have as expressions M ie j ri
hH xi ' xi" x j ' x j" yi ' yi" y j ' y j" 2 2S hUS 3
[1.64] [1.65]
with (i,i’,i") = (D,E,J), (E,J,D), (J,D,E) and (j,j’,j") = (D,E,J), (E,J,D), (J,D,E). Thanks to formulae [1.64] and [1.65], we would be able to evaluate the elementary contributions to the Mij and Ri coefficients defined in [1.62] and [1.63], then determine the coefficients of matrix S and second member Q defined in [1.57]. However, in practice, to reduce the calculation times, we factorize the calculations by elements, and we directly assemble matrix S and second member Q without making explicit coefficients Mij and Ri. Moreover, those whose line index corresponds to a Dirichlet value are not used at the time of the resolution. First of all, let us consider factorization by element. For a triangle, there are 3x3 coefficients [1.64] and 3 coefficients [1.65]. It is judicious to gather these calculations, to perform the intermediate operations only once, such as the access to the data and the calculation of the surface. For example, for element e1 of nodes n2, n1, n5, the application of formulae [1.64] and [1.65] provides the following results gathered in a large matrix Nn * Nn and in a large vector Nn
Introduction to Nodal Finite Elements
ª 2 1 « 1 1 « « . . « . « . « 1 0 « hH « . 2 « . « « . « . « « . « . « «¬ .
M e1
.
.
. . .
. . .
1 . 0 . .
.
.
1
.
.
.
.
.
. . . . . .
.º » » » » » » » » R e1 » » » » » » » » . ¼»
ª1º «1» «» «.» «» «.» «1» «» 2hU « . » 3 «.» «» «.» «.» «» «.» «.» «» ¬« . ¼»
23
[1.66]
After each basic calculation, the results are accumulated, while following the rules defined in [1.57], in matrix S and in second member Q which temporarily play the roles of accumulator, reserved initially for Mij and Ri. A loop on all the elements thus allows the components of the linear system to be obtained. However, there is still a penalizing aspect to this process. To factorize the calculations, we do so element by element, which is a good thing, but we store the intermediate results in a large matrix and in a large, almost empty vector [1.66]. It is much more judicious to gather the results in an elementary sub-matrix and subvector of respective size 3x3 and 3 for a triangle.
me
e ªmDD « e «mED « e m ¬« JD
e mDE
e mEE e
mJE
e º mDJ » e e mEJ » and r » e mJJ ¼»
ª rDe º « e» « rE » «re » ¬J ¼
[1.67]
For example, element e1 has as an elementary sub-matrix and sub-vector
m
e1
ª 1 1 0 º hH « 1 2 1»» and r e1 2 « «¬ 0 1 1 »¼
ª1º 2 hU « » 1 3 «» «¬1»¼
[1.68]
24
The Finite Element Method for Electromagnetic Modeling
The passage of indices 1,2,3 of [1.68] to the indices of the total arrangement e1 D,E,J of [1.66] is given by the connectivity of the element. Thus, coefficient m22
e
corresponds to coefficient M 111 , because the second node of the triangle e1 is n1. Let us notice by the way that, in accordance with formula [1.60], matrices me and M are symmetric. Moreover, the sum of the gradients of the form functions being zero according to [1.49], the sum of the coefficients of any line or column of these matrices is zero. e
Finally, we will retain the following algorithm for the construction of matrix S and second member Q which we retain ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
Sm0 Qm0 For e m 1 to Ne ~ Calculation of the sub-matrix m and the sub-vector r ~ For k m 1 to 3 ~ ~ i m Connectivity [k,e] ~ ~ if ni is free then ~ ~ ~ Q[i] m Q[i]+r[k] ~ ~ ~ For l m 1 to 3 ~ ~ ~ ~ j m Connectivity [l,e] ~ ~ ~ ~ If nj is free then ~ ~ ~ ~ ~ S[i,j] m S[i,j]+m[k,l] ~ ~ ~ ~ Otherwise ~ ~ ~ ~ ~ Q[i] m Q[i]–m[k,l]*V[j]
The application of the previous algorithm on the example of the condenser leads to the linear system 0 º ª v1 º ª 4 2 0 « 2 8 2 0 » «v » « ».« 2 » « 0 2 8 2» « v3 » « »« » 0 2 4 ¼ ¬ v4 ¼ ¬ 0
ª100 º «200» « » «200» « » ¬ 0 ¼
[1.69]
Introduction to Nodal Finite Elements
25
which has the solution ª v1 º «v » « 2» « v3 » « » ¬ v4 ¼
ª 48,89 º « 47,78» « » «42,22» « » ¬ 21,11 ¼
[1.70]
Linear system matrix [1.69] is also symmetric and, we will admit it, defined positive. Lastly, as was already stated, it is sparse. The collection of all these properties is very favorable, because this authorizes the use of very powerful methods of resolution of linear systems. 1.3.8. Analysis of the results
We now have a complete set of nodal values, some of which were known from the beginning thanks to the Dirichlet conditions, while others were obtained by resolution of the system of equations [1.52], or, in the linear case, by solving the system of linear equations [1.57]. These nodal values represent an approximate value of the electric potential at the nodes of the meshing. Their knowledge is indeed important, but not necessarily easy to understand. The expert, who is interested in the characteristics of the studied condenser, would certainly like to have local information, such as the potential or the field in any point, or global, as the charges stored on the electrodes or the capacity [COU 85]. We will see how, starting from the nodal values of the scalar electric potential, it is possible to deduce all information relating to the operation of the device. 1.3.8.1. Electric potential in an unspecified point The first of the physical quantities that can be very easily obtained in a point of coordinates (x,y) starting from the meshing and nodal values is the electric potential. For this purpose, it is enough to determine the element which contains the point, then to apply the interpolation formula which is the basis of the finite element method v ( x, y )
¦ v j M j x, y
[1.71]
26
The Finite Element Method for Electromagnetic Modeling
1.3.8.2. Electric field vector in an unspecified point It is also very easy to determine the electric field vector in this point by E ( x, y )
[1.72]
¦ v j grad M j
In first-order triangles, the electric potential having a linear variation in the space, the electric field is constant on each element. 1.3.8.3. Electric flux density vector in an unspecified point With regards to the electric flux density vector, it is obtained starting from the electric field and from the constitutive law of the material D( E ( x, y )) here D( x, y )
D( x, y )
[1.73]
HE ( x, y )
1.3.8.4. Electric energy It is sometimes useful to evaluate global information, such as the electric energy stored in the condenser. The latter corresponds to the injection of energy necessary to introduce the space charges and the electrode charges. We will assume that these operations were carried out in two stages. On the basis of an initial state where there is no charge and a uniformly potential zero, the first stage consists of introducing the space charges, the potential of the electrodes remaining zero. The second stage corresponds to the adjustment of the electrode potentials, the space charges remaining unchanged. During the first stage, the introduction of the charge Gq into an infinitesimal volume :q, of border *q, in which the potential created by all the other charges already present in the domain is worth v, requires the energy [DUR 64]
GWe
[1.74]
v Gq
According to Gauss’s law, the charge is equal to the outgoing flux of the infinitesimal volume and to the entering flux in its complementary
G We
v
³ > GDn @hd*
*q
³ > vGDn @hd*
[1.75]
*q
By applying the divergence theorem to complementary volume, we obtain
G We
³³
domain : q
div > vG D @ hdxdy
v³ > vG D @ hd * n
border
[1.76]
Introduction to Nodal Finite Elements
27
By noting that the potential or the flux variations are zero on the borders of the domain, we obtain
GWe
³³
domain:q
ª¬gradvG D vdiv>G D@º¼ hdxdy
³³ > EG D vGU@ hdxdy
[1.77]
domain:q
Outside the infinitesimal volume, the variations of charge density are zero
G We
³³ > EG D @ hdxdy
domain :q
³³ > EG D@ hdxdy ³³ > EG D @ hdxdy
domain
[1.78]
:q
An opposite treatment to that used above would show that the interaction energy in :q is zero. The variation of electric energy due to the introduction of the charge into the infinitesimal volume is thus
G We
[1.79]
³³ > EG D @ hdxdy
domain
The successive introduction of all the space charges leads, for the first stage, to the increase of energy
We 1 We 0
³³ ª¬« ³
domain
D1
0
EdD º hdxdy ¼»
[1.80]
where D1 is the field produced only by the internal charges, the electrodes being at potential 0 V. During the second stage, the injection of electric energy depends on potentials Vk of each electrode and on their charge variation GQk
GWe
[1.81]
¦VkGQk
Gauss’s law on the electric charges also applies to their variation. Thus, the charge variation on each electrode is equal to the opposite of the flux variation entering into the electrode
GWe ¦Vk
v³
electrodek
G Dnhd* ¦
v³
electrodek
vG Dnhd*
v³ vG D hd* n
[1.82]
border
The successive transformations above are possible because the potential is uniform on each electrode, the edges of electrodes belong to the border of the
28
The Finite Element Method for Electromagnetic Modeling
domain, and the flux variation is zero on the other parts of the border. The application of the theorem of the divergence gives
G We
div > vG D @ hdxdy
³³
domain
³³ > EG D vGU @ hdxdy
[1.83]
domain
With invariant space charges, the electric energy brought during stage 2 is D
³³ ª«¬ ³D1 E dD º»¼ hdxdy
We 2 We 1
[1.84]
The electric energy stored during the sequence of the two stages is thus
³³ ª«¬ ³
We 2 We 0
domain
D
0
EdD º hdxdy »¼
[1.85]
In the linear case, this gives the familiar expression We
D.E
[1.86]
³³ 2 hdxdy
An approximation of this stored electric energy can be obtained by integration on the elementary sub-domains. When the elements are first-order triangles, vectors E and D are uniform on the element and the density of energy is itself uniform. The elementary integral is thus the product of the density of energy by the surface of the element. Table 1.3 recapitulates the values of the stored electric energy for each triangle and a depth h of 1 m. e1
e2
e3
e4
e5
e6
e7
e8
e9
e10
e11
e12
Total
we 5.78 6.03 6.10 7.38 8.36 5.28 5.05 5.05 4.01 3.94 1.97 0.98 59.93 Table 1.3. Energy by triangle in nJ for a depth h = 1 m
The summation on the 12 triangles gives 59.93 nJ and represents the electric energy stored in 1/8 of the device. It is advisable to multiply the figure by 8 to obtain the total energy, which leads to We = 479.4 nJ.
Introduction to Nodal Finite Elements
29
1.3.8.5. Electric coenergy In a dual way, by considering this time that the state of the device is controlled by the voltage, we can define the variation of a quantity, called coenergy, homogenous with an energy, and depending on the charges and on variations of potential. During the first stage, the introduction of a space charge q into the infinitesimal volume :q produces a variation of potential Gv. The coenergy is defined by the variation
GWc
[1.87]
qGv
The successive introduction of all the space charges results, after some processes similar to those carried out on the energy, in the increase
Wc 1 Wc 0
³³ ª«¬ ³
domain
E1
0
DdE U v1 º hdxdy »¼
[1.88]
where E1 and v1 are the field and the potential produced only by the internal charges. During the second stage with invariant space charges, the variation of coenergy due to evolutions on the electrodes is defined by
GWc
[1.89]
¦ QkGVk
which leads to partial and total increases of the coenergy E
Wc 2 Wc 1
³³ ª«¬ ³E1 DdE º»¼ hdxdy ³³ >U >v v1 @@hdxdy
Wc 2 Wc 0
E ³³ ª«¬ ³0 DdE Uv º»¼ hdxdy
[1.90]
While comparing [1.90] with [1.35], we note that this increase in coenergy is the value taken by the functional when the exact solution is introduced there, i.e. when this functional is stationary. It is not a coincidence, because, chronologically, it is the knowledge of the coenergy expression that allowed the expression of the functional to be proposed. The mathematical step of minimization of the functional corresponds, in fact, to the search of a state with minimum energy.
30
The Finite Element Method for Electromagnetic Modeling
In the case of a linear law of behavior D(E), the coenergy expression is Wc
DE
ª º ³³ « 2 Uv » hdxdy ¬ ¼
[1.91]
1.3.8.6. Flux of the electric induction vector We are also interested in the flux of the electric induction vector through a surface. Thus, if L is the trace of surface in the 2D domain and h is its depth, this flux is calculated by numerical integration Qc
³ Dn hdL
[1.92]
L
where the induction vector itself is given by [1.73] at the points of integration. 1.3.8.7. Electric charges accumulated on an electrode In fact, the most frequent flux calculation takes place on the equipotential electrodes, because it then represents the accumulated electric charges. The previous formula could of course be applied there. However, there is an energy method prolonging the variation approach which is much more accurate and does not require any local evaluation [COU 85]. We know that the electric coenergy of system [1.90] is equivalent to functional [1.35] when this energy is at its minimal value. When the potentials of the electrodes vary, these two quantities evolve in an identical way. The identification of the variations, defined by [1.89] for the coenergy, and by
GWc
¦
wWc GVk wVk
[1.93]
for the functional, indicates that Qk
wWc wVk
[1.94]
We thus have an indirect way to determine this charge, by calculating the derivative of the functional with respect to the potential of the electrode. In fact, in the finite element approximation, the functional is controlled by the nodal values. The Dirichlet nodal values are themselves controlled by the potentials
Introduction to Nodal Finite Elements
31
of the electrodes, while the others are in the balance state. The derivative of the functional then becomes
wWc wVk wWc wVk
N N wW c
wvi 1 wvi wVk
¦
i
NL
¦ i 1
wWc wvi wWc wvi wWc wvi ¦ ¦ wvi wVk i electrode k wvi wVk i other electrodes wvi wVk
[1.95]
When the functional is at its extremal value, according to [1.52], the NL first derivatives are zero, which eliminates the first summation. Only the summation corresponding to electrode k remains since on the others the nodal values are attached to a different potential. Finally, the entering flux through this electrode is simply
Qk
wWc i electrode k wvi
¦
[1.96]
This algorithm offers several advantages. On the one hand, it is very simple to implement, since it re-uses coefficients already calculated during the resolution of the problem. In addition, it is in practice more accurate than the algorithm based on numerical integration of the flux evaluated locally on the border. Indeed, this algorithm is based on the same variational bases as those used for the resolution of the problem. In fact, the field distributions obtained being only approximations of the solution, two mathematically equivalent algorithms (boundary integration or domain integration) can differ significantly when they are transposed in the discretized situation. It is thus often judicious to wonder about the most appropriate algorithm for the discretization used! 1.3.8.8. Influence coefficients of one electrode on another: linear case The influence coefficient Ckl is the ratio of the electric charge variation Qk stored on the electrode k, according to the variation of potential Vl on electrode l, the other electrodes being maintained at constant potentials. Ckl
wQk wVl
[1.97]
32
The Finite Element Method for Electromagnetic Modeling
In the case of a linear problem, the dielectric permittivity being constant, these coefficients are constant whatever the voltages. It is thus very easy to determine them by using as many calculations of electric field as there are electrodes. For each calculation, it is enough to consider the densities of electric charges of the domain to be zero and to consider the potentials on all the electrodes to be zero except on electrode l where the potential is fixed at a non-zero value, for example 1 V. The accumulated electric charges Qk on an electrode k are then calculated by formula [1.96] and the influence coefficients are obtained by Ckl
Qk Vl
[1.98]
The electric charges accumulated on the electrodes l counterbalance the electric charges accumulated on all the other electrodes, which results in the equality Cll
[1.99]
¦ Ckl k zl
In the even more specific case (although very usual) of a linear problem simply comprising two electrodes with potentials V1 and V2, the effects of potential variations on one or other of the electrodes are identical, but with opposite signs. Only one influence coefficient, its capacity C, then characterizes the condenser C
C11
C22
C12
C21
Q1 V1 V2
Q2 V2 V1
[1.100]
In this case, this capacity is also directly calculable from [1.86] C
2We
V2 V1 2
[1.101]
The capacity of our condenser, corresponding to energy of 479.4 nJ for a potential difference of 100 V, is of 95.89 pF. 1.3.8.9. Influence coefficients of one electrode on another: nonlinear case When the problem is nonlinear, it is still possible to determine the influence coefficients. However, these coefficients are no longer constant, because they depend on the point of polarization of the device. We are thus dealing with incremental influence coefficients.
Introduction to Nodal Finite Elements
33
For this purpose, let us start again with definition [1.97] of an influence coefficient and formula [1.96] giving the charge on the electrode k
Ckl
w wVl
ª wW wv º «¦ c i » «¬ i wvi wVk »¼
w 2Wc wvi wW w ª wvi º ¦ c « » w w w i Vl vi Vk i wvi wVl ¬ wVk ¼
¦
[1.102]
The second summation disappears because node i is either free or on an wvi wWc 0 and the second electrode. In the first case = 1 or 0. We obtain wVk wvi
Ckl
¦¦ i
j
wvi w 2Wc wv j wVk wv j wvi wVl
[1.103]
When node j is on electrode l, derivative
wv j
is worth 1 and, when this node is wVl on another electrode, the derivative is worth 0. When the nodal value is free, its wWc variation is determined by noting that, according to [1.52], the quantities are wvi equal to zero, therefore with zero variations, whatever Vl is, which results in w ª wWc º « » wVl ¬ wvi ¼
w 2Wc wv j j wv j wvi wVl
Nn
¦
Nn w 2Wc wv j w 2Wc wv j ¦ j 1 wv j wvi wVl j N L 1 wv j wvi wVl
NL
¦
0 [1.104]
These equations, written for all the free nodal values, constitute the following system of NL linear equations with NL unknown variables
w 2Wc wv j ¦ j 1 wv j wvi wVl NL
w 2Wc for i = 1, 2, …, NL ¦ j electrode l wv j wvi
[1.105]
Its resolution supplements the set of the derivatives. This process also provides wvi the . Everything is then available for the calculation of the coefficient of wVk influence sought by formula [1.103], which can be reorganized in order to factorize the incremental charges.
34
The Finite Element Method for Electromagnetic Modeling
The matrix of this system, which is square and symmetric, is called the tangent matrix, because it also appears in the resolution of the system of nonlinear equations [1.52] by the Newton-Raphson method. For a linear problem, the tangent matrix is identical to matrix S of linear system [1.57]. 1.3.8.10. Sensitivity analysis to the physical and geometric parameters At the time of determining the coefficients of influence, we have considered a quantity resulting from the calculation (an electric charge of electrode) and we have calculated from it the derivative with respect to a parameter of the problem (an electrode potential). In a phase of study or sizing, the need is much broader. It can be, indeed, necessary to know the influence on a local quantity (potential, field, induction, etc.) or global (load, force, etc.), of a physical parameter (an electrode potential, the permittivity, the charges density, etc.) or geometric (a position, a dimension, a form, etc.). The objective of this section is to show how it is possible to determine the sensitivity of a quantity with respect to an unspecified continuous parameter [GIT 89]. These sensitivities are extremely useful, during the phase of design of a device, for its manual sizing or its automatic optimization. 1.3.8.10.1. Sensitivity analysis Let us consider a problem, similar to [1.52], resulting from the Ritz or Galerkine method, where the NL unknown values v1, v2, …, of the state variable, for example the free nodal potentials, result from the solution of NL linear or nonlinear equations written for a fixed value of a parameter p, for example, an electrode potential
^
` ri ^v1 p ,v2 p ,...,vN p , p`
ri v1, v2,...,vNL , p
L
0 for i = 1, …, NL [1.106]
Let us also consider a quantity f, for example, a charge on an electrode, resulting from a post-processing of the state variable
f
^
` f ^v1 p , v2 p ,...,vN p , p`
f v1, v2 ,...,vN L , p
L
[1.107]
This function depends on the parameter, directly if it appears in its expression and indirectly via the nodal values. The derivative
df of f with respect to p is a dp
very useful indicator of sensitivity, either for a human operator, or for an optimization algorithm. In order to evaluate such a derivative, two methods are available: the approximation by finite differences and the exact differentiation.
Introduction to Nodal Finite Elements
35
1.3.8.10.2. Sensitivity by finite differences The sensitivity calculation by the finite difference method consists of replacing the derivative by the approximation
df f p 'p f p | dp 'p
[1.108]
by using two answers calculated for values close to the parameter. This technique has the advantage of being very simple to implement. However, it has the drawback of very often leading to inaccurate results since the difference in two close quantities having errors leads to a larger relative error. In fact, there is a contradiction between the requirement of taking a very small shift 'p, in order to approach the mathematical definition of the derivative, and the need to have the two tests sufficiently distant in order to minimize the impact of inaccuracies on each of them. A trade-off for the step is proposed by [GIL 83] 'popt
2
H num
[1.109]
f " ( p)
where f"(p) is an approximation of the second derivative of f and Hnum is an estimate of the error made during a calculation of f. 1.3.8.10.3. Sensitivity by exact differentiation The determination of the sensitivity by exact differentiation consists of expressing the derivative on the basis of the partial derivatives with respect to the nodal values and the parameter
df dp
NL
wf dv j wf wp 1 wv j dp
¦
j
wf wV
T
.
dV wf dp wp
The vector of nodal derivative values
dV dp
[1.110]
ª dv j º « » is obtained by noting that ¬ dp ¼
residues [1.106] preserve zero values, which are therefore constant whatever the p, which implies that each of their derivatives is zero
dri dp
NL
wri dv j wri wp 1 wv j dp
¦
j
0 for i = 1, 2, …, NL
[1.111]
36
The Finite Element Method for Electromagnetic Modeling
These linear equations represent a matrix system of matrix
second member
wR wp
wR wV T
ª wri º « » and of «¬ wv j »¼
ª wri º « wp » , of which the solution provides the derivatives of ¬ ¼
nodal values
wR dV . wV T dp
wR wp
[1.112]
Within the framework of the finite element method, this matrix is already known since it is the matrix of the linear systems for a linear problem and the tangent matrix at the solution in the case of a nonlinear problem. By injecting the derivative of the nodal values in [1.110], we obtain the required sensitivity. It is the step followed for the determination of the coefficients of influence between two electrodes. 1.3.8.10.4. Sensitivity by adjoint states Very often, it is not only one parameter but several which are involved in a sizing process. Their influence on f can, of course, be obtained by repeating, as many times as necessary, the previous step. However, for each of them, it is necessary to solve a large linear system [1.112], indeed by preserving the same matrix, but each time with a second different member. We will see that it is possible to significantly reduce the calculation time by using the adjoint state method. In fact, the combination of [1.110] and [1.112] give for the sensitivity
df dp
wR 1 wR wf wV T wV T wp wp wf
/T
wR wf wp wp
[1.113]
where we have introduced vector /, which gathers the adjoint states of the nodal value derivatives. This vector is obtained by solving the matrix system
wRT / wV
wf wV
whose matrix is the transpose of that of [1.112].
[1.114]
Introduction to Nodal Finite Elements
37
The adjoint state vector depends only on f. When there are fewer functions to derive than parameters, the calculation of adjoint states is, in terms of a number of linear systems resolutions, more advantageous than that of the direct influences. 1.3.8.10.5. Higher order derivation The previous derivation process can be applied to any variable resulting from post-processing, including sensitivity. It is thus possible to obtain successively as many higher order derivatives as desired, either with respect to only one parameter, or with respect to several [GUI 94]. These derivatives can be used for construction of a Taylor or Padé development. 1.3.8.11. Forces, torques, stiffness Let us now consider a device comprising a mobile electrode with respect to the remainder of the domain: the pallet of an electrostatic contactor, the rotor of an electrostatic motor, etc. (Figure 1.10). Starting from the distribution of the potential, it is possible to determine the force or the torque exerted on this mobile conductor using the following algorithms: the integration of the Maxwell tensor [CAR 59], the application of the theorem of virtual work by finite differences or the application of virtual work by exact differentiation [RAF 77], [COU 83], [COU 84]. 1.3.8.11.1. Integration of the Maxwell tensor The force and the electric torque acting on a rigid body can be obtained by integration of the Maxwell tensor using the following process: – choice of an arbitrary surface S in the vacuum (or an equivalent medium) including only the rigid body; – calculation of the force by the integration
F
D.E
ª º ³³ « D.n E 2 n» dS ; ¼ S¬
[1.115]
– calculation of the torque by the integration
C
D.E
ª º ³³ « D.n r u E 2 r u n » dS ; ¼ S¬
[1.116]
where n is the normal unit vector on surface S, r is the radius vector of the current point of S compared to the axis of rotation, E is the electric field and D is the electric induction (electric flux density). Similar expressions exist in magnetism.
38
The Finite Element Method for Electromagnetic Modeling
V1 F S F V2
Figure 1.10. Force between two electric conductors
In theory, the choice of surface S is arbitrary. In practice, since the fields obtained by finite elements are only approximations, it is necessary to take precautions to obtain the best possible results. Either the surface S will cross the elements in their “medium”, where the approximation is the best, or the solution will be smoothed carefully. With the virtual work method that we will present below, these particular treatments are not necessary. 1.3.8.11.2. Virtual work The expressions of the force and the electric torque acting on a rigid part are also accessible by application of the virtual work theorem. During a virtual displacement Gu, the variation GWe of the internal energy of the device is equal to the electric energy brought by the external sources connected to the electrodes plus the work of the force –Fu.
¦VkGQk FuGu GWe ¦
wWe wW GQk e Gu wQk wu
[1.117]
In a first experiment, the device is supposed to be isolated electrically from the external world. Along each electrode k, the electric charges can vary, but their sum remains constant, which results in GQk = 0. The same experiment can be carried out for the study of the torque CZ for a rotation GT around an axis Z. From this it results that force and torque are the partial derivatives of the internal electric energy with respect to the displacements, the electric state variable (the electric charges of the electrodes) being kept constant
Introduction to Nodal Finite Elements
Fu CZ
wWe wu wWe wT
39
[1.118]
[1.119]
In a second experiment, the device is connected to electric voltage sources which maintain as constant the potentials Vk of the electrodes. This time, during a small displacement, electric charges GQk are brought or withdrawn from the electrodes. Let us note that the internal charges remain attached to the matter and are not concerned with these variations. The preceding energy assessment always remains valid. However, it is judicious to reveal here the system coenergy because for this second experiment, it is the potential which is state variable. By withdrawing, on both sides of assessment [1.117], the variations GQk of the potential products by charges, we obtain
¦VkGQk ¦G VkQk FuGu GWe ¦G VkQk ¦QkGVk FuGu
GWc
¦
wWc wW GVk c Gu wVk wu
[1.120]
[1.121]
From this it results that force and torque are the partial derivatives of the electric coenergy with respect to the displacements, the electric state variable (electric electrode potentials) being maintained as constant
Fu
CZ
wWc wu wWc wT
[1.122]
[1.123]
In the numerical field, it is possible to calculate two approximations of the coenergy for two positions close to the moving part, then to apply the finite difference method
'Wc 'u 'Wc CZ | 'T Fu |
[1.124]
[1.125]
40
The Finite Element Method for Electromagnetic Modeling
However, this procedure is expensive, since it requires two resolutions, and inaccurate. Indeed, this procedure expresses the difference in two close quantities tainted with error. Within the framework of the finite element method, the exact differentiation is more economic and more precise than the finite difference method. It consists, as with the integration of the Maxwell tensor, of extracting an approximation from the force or torque starting by using only one resolution of the state variable (in the electric potential). Here is the principle illustrated on the coenergy formulation that we have used throughout this chapter. The finite element resolution of problem [1.52] results in the best possible combination of the degrees of freedom for the selected approximation. The introduction of the obtained values into expression [1.35] gives an approximation of the system coenergy. This integral is the sum of the finite element integrals. Each elementary integral depends on the coordinates of the nodes, either by the integral term, or by the terminals. We are thus able to determine the derivative of an approximation of the coenergy with respect to the position of the rigid body and thus obtain an approximation of the force or torque. With potentials of electrodes maintained constant, the finite element approximation of the coenergy becomes a function of the only displacement u, either directly via the positions of the nodes, or indirectly via free nodal values.
Wc
Wc [v1 u , v2 u ,..., v N L u , u] Wc u
[1.126]
The exact derivative of the approximation gives an approximation of the force
Fu
dWc du
NL
wWc dv j wWc wu 1 wv j du
¦
j
[1.127]
However, according to [1.52], the partial derivatives with respect to the nodal values are zero and we thus obtain
Fu
wWc wu
[1.128]
which is the partial coenergy derivative when all the nodal values are maintained constant.
Introduction to Nodal Finite Elements
41
Figure 1.11. Virtual deformation of the meshing
Let us now consider the virtual displacement of the moving part in the meshed domain of Figure 1.11. This rigid part is included by a deformable material. The virtual deformation can, indifferently, be distributed on a layer of elements (as in the figure) or on several layers to introduce a type of average. The system coenergy is the sum of elementary integrals which are all expressed according to the coordinates of the geometric nodes of the elements. During the virtual deformation with constant nodal values of the potential, only the integrals on the deformed elements vary, either by the integral term or by the shape of the element. The exact derivation of these integrals with respect to the displacement is thus possible and provides a method of calculation of the force and torque. The influence of virtual displacement is taken into account thanks to the derivatives of the nodal coordinates. For example, for a virtual displacement Gu in the direction of the axis y, the derivatives of the fixed nodes are zero, whereas the derivatives of the coordinates of the nodes attached to the moving part are
dxi du dyi du
0 [1.129]
1
The buffer nodes between fixed nodes and mobile nodes, if they exist, have intermediate and progressive derivative values.
42
The Finite Element Method for Electromagnetic Modeling
For triangles, derivations with respect to the coordinates of the nodes are very simple to implement [COU 84] and lead directly by integration on the deformed elements to the same result as the integration of the Maxwell tensor “in the middle” of the elements. However, these specific derivations will not be developed here because we privilege the local Jacobian derivative method, which we will present at the end of the chapter and which is adapted to all types of elements and formulations. Forces and torques can be derived with respect to any other physical or geometric parameter. In particular, the derivations with respect to the positional parameters of the mobile body correspond to the stiffness [COU 83]. Lastly, the virtual work method is well adapted to the study of internal forces in deformable materials [REN 92] or in magnetized mediums [DEM 99]. 1.3.9. Dual formulations, framing and convergence
Instead of choosing the functional of coenergy [1.35], we could have chosen a functional of energy, inspired by [1.85], with the electric vector potential U as a state variable. These dual approaches, by coenergy and energy functionals, result in approximate solutions which frame the exact solution [REN 95]. We will give here only one simple illustration of this behavior in our 2D example. In a 2D case and in the absence of space charge, uz the component perpendicular to the xOy plane of the vector potential (the only useful one in 2D) has an equation similar to that of the scalar potential. However, the edges of the electrodes are homogenous Neumann borders, whereas the axes of symmetry are equipotential borders. Their difference expresses the electric flux crossing the device, i.e. charge Q of the electrodes which, for this formulation, must be given a priori. Table 1.4 shows the evolution of the obtained capacitors by successive resolutions on increasingly fine meshing. The first meshing, comprising 12 triangles, is that in Figure 1.7. The second meshing is obtained by subdividing each triangle of the previous meshing into four. The subdivisions are reiterated until the finest meshing with 3,072 elements. The first and the second lines of the table indicate the sequence numbers or order n of the grids and their respective number of triangles. The third and fourth lines give the capacitors Cc n Ce n
2Wc n / V 2
and
Q 2 / 2We n calculated from the energies Wc n and We n obtained on these
grids by means of the coenergy and energy functionals. As the base of the functions of approximation is enriched with each new meshing, the minima of the functionals,
Introduction to Nodal Finite Elements
43
and thus the energies, decrease with each subdivision. A capacitor decreases and the other increases. The two series tend, in a monotonous way, towards the same limit which they surround on two sides. The speed of convergence is low because of the point effect existing at angle P2. This phenomenon causes a singularity on the solutions which the finite element approximations follow with difficulty. In fact, the refinement strategy of meshing adopted here is not very effective. With a given number of elements, instead of subdividing in a uniform way in the entire domain, it would have been preferable to mesh more finely around the singular point. A denser meshing around P2, in 2,253 triangles of the second-order, has allowed the framing to be determined C = 90.489 pF r 0.001 pF. n
1
2
3
4
5
Nb triangles
12
48
192
768
3,072
Cc n
95.89
92.51
91.27
90.79
90.61
Ce n
86.31
88.74
89.78
90.20
90.37
91.098
90.625
90.521
90.497
90.491
90.972
90.606
90.518
90.496
90.491
Cc ex n
90.535
90.499
90.492
Ce ex n
90.490
90.489
90.489
Cc n Ce n
2
Cc n * Ce n
Table 1.4. Capacitors (pF) as a function of the number of subdivisions
The fifth and sixth lines of the table indicate the arithmetic and geometric capacitor averages. They are close to the exact value. It is also possible to extrapolate the results of the successive subdivisions. For example, the convergence hypothesis C hn | C 0 DhnE , with hn being the diameter of the triangles varying in 1 / 2 n , C(0) the required limit and D E two unknown coefficients, leads to the formula of following extrapolation Cex n
Cn
(Cn Cn 1 ) 2 Cn 2Cn 1 Cn 2
[1.130]
The two last lines show the results of these extrapolations requiring three consecutive meshing processes. They are also close to the limit.
44
The Finite Element Method for Electromagnetic Modeling
1.3.10. Resolution of the nonlinear problems
The finite element method leads to the fundamental system of equations [1.52]. When the problem is linear, which is the case in the example covered, the equations of this system are themselves linear according to the nodal values, which can then be obtained directly by solving matrix system [1.57]. However, in real devices, nonlinearities appear in the constitutive relations D(E) of materials. It is thus necessary to solve system [1.42] by means of an adapted method. Among all the available methods, the Newton-Raphson method is often used in finite elements. It concerns the generalization of the tangent method, making it possible to solve from one nonlinear equation with an unknown variable to the resolution of a system R(X) = 0 of NL equations R
T
½° ¾ °¿
with
^v1, v2 ,..., vN `T . It is an iterative method which, starting from an
NL unknown X
initial vector X
° wW wW c, c ,..., wWc ® v v w w wv N L °¯ 1 2
L
( 0)
^
` , builds a series of vectors X
( 0) T v1( 0) , v2( 0) ,..., v N L
(k)
which, if the
conditions of convergence are met, tends towards the solution. In order to go from iteration to the following, let us start with the first order Taylor development centered on X (k):
R X ( k ) 'X ( k ) | R X ( k )
where
wR ( k ) wX T
wR wX
T
(k )
'X ( k )
[1.131]
ª w 2Wc º « » is the tangent matrix of the system at iteration k and ¬« wv j wvi ¼»
'X(k) is a vector of the increments of the unknown variables. In order to try to cancel the first member, this vector is selected so that the second member of [1.131] is zero. It is thus the solution of the linear system wR ( k ) wX T
'X ( k )
R X (k )
[1.132]
The vector at the following iteration is obtained by X ( k 1)
X ( k ) 'X ( k )
[1.133]
Introduction to Nodal Finite Elements
45
The iterative process is stopped when the increment, which is sufficiently small, verifies
'X ( k ) X ( k 1)
d relative precision
[1.134]
In practice, this method converges well in some iterations (roughly less than 10) when the constitutive laws are monotonous and are not too stiff. When difficulties of convergence appear, it is usual to introduce an under-relaxation at the time of the passage from one iteration to another by X ( k 1)
X ( k ) Z 'X ( k )
[1.135]
The value of the coefficient of under-relaxation Z, ranging between 0 and 1, can be selected empirically or determined by an algorithm, for example by minimizing
the norm of R X ( k ) Z 'X ( k ) [FUJ 93]. The fixed point method [OSS 99], which is certainly slower but very robust, can be used as an alternative when the Newton-Raphson method does not converge. 1.3.11. Alternative to the variational method: the weighted residues method
We have, at the time of our presentation, favored the variational approach, because, based on energy concepts, it allows the integral form to be interpreted and sometimes to be re-used in post-processing. However, this approach is based on a functional that is sometimes unknown or even does not exist. In this case, passing from the differential form to an integral form is still possible, thanks to the weighted residues method which is also called the Galerkine method. In order to present the weighted residues method, let us again use the differential form of the 2D electrostatics problem in Figure 1.6 divD U
Dn
0
0
v V1
0 and v V2
100
in : (the domain)
[1.136]
on *N (the axes of symmetries)
[1.137]
on *D (the electrodes 1 and 2)
[1.138]
46
The Finite Element Method for Electromagnetic Modeling
where vector D(x,y) is defined by the sequence E grad v and D D( E ) , with v(x,y) a continuous scalar function, complying with [1.138] and, moreover, derivable once everywhere and twice per piece. The introduction of the solution into the first members of equations [1.136] and [1.137] would give a uniformly zero result. On the other hand, any other function would produce a non-zero residue. This observation is the basis of the weighted residues method whose idea is to measure and to zero the residue corresponding to v(x,y), by means of the functional r v, w
³³ >divD U @ w d: ³ Dn w d* :
[1.139]
*N
where w(x,y) is a continuous function which has a value of zero on the border *D, called a weighting function or test function. If, among all the possible functions v(x,y), there is one, vm(x,y), such that r vm , w 0 , whatever w(x,y), then this function, which by assumption already checks [1.138], also checks [1.136] and [1.137], is thus the solution of the problem. In the finite element context, the study is limited to functions v(x,y) defined, as in the variational approach, by [1.50] v ( x, y )
Nn
¦ v j M j x, y
[1.140]
j 1
where the M j x, y are predefined shape functions and the vj are the nodal values, known initially only on the Dirichlet boundaries. The NL degrees of freedom vj unknown are determined by solving a system of NL equations [1.139] written for NL independent test functions wi(x,y) r v, wi
0 for I = 1, 2, …, NL
[1.141]
These test functions are arbitrary. Two different sets of functions will lead to different sets of nodal values and therefore to different approximate solutions. They can themselves be defined by finite elements. In addition, it is usual to re-use, as test functions, the proper shape functions of function v. It is possible to stop going further by saying that it is now enough to solve (linear or nonlinear) system [1.141] to obtain the missing degrees of freedom. However, this has the drawback of leading, even in the simplest case of a linear problem with
Introduction to Nodal Finite Elements
47
identical shape functions and test functions, to non-symmetric matrices, which is penalizing in terms of memory storage and computing time. This situation is improved by using some additional arrangements carried out on functional [1.139]. Indeed, if a chosen test function w is not only continuous, but also derivable per piece, an integration by parts leads to r v, w
³³ > D grad w U w@ d:
[1.142]
:
This form is more suitable than writing the system of equations [1.141]. With it, only the derivability per piece of the shape functions is required. In addition, it leads, for a set of test functions identical to the set of the shape functions, to a system identical to that obtained by the Ritz method which has good symmetry properties. In conclusion, the Galerkine method offers a systematic approach to go from a differential form to an integral form that can be treated by discretization in finite elements. It has a broad applicability. For instance, the Ritz method can be considered as a particular case of the Galerkin method. In particular, the separation between shape functions, for the approximation of the solution, and weighting functions, for the evaluation of its residue, introduces additional possibilities compared to the Ritz method. This faculty will be made profitable to solve equations having stability problems, such as the equations with transport terms [MAR 92]. 1.4. The reference elements
In the previous sections, we presented two applications of the finite element method. We have seen that this method is based on a meshing of the used domain for the definition, per piece, of the shape functions whose adequate weighting ensures the approximation of the solution. In the first 1D example, we have used segments which are the only finite elements available in this space. In the second example, we have used triangles, but we could also have meshed the domain in quadrangles. Triangles are often used in 2D, because they are suitable for an automatic meshing. However, quadrangles are appreciated, because they are particularly appropriate for the discretization of narrow zones or involving particular physical phenomena. In addition, it is possible to mix, within the same meshing, different types of elements, provided the continuity conditions necessary for the functions of approximation are fulfilled. The developments carried out in two dimensions are extended in an obvious way to three dimensions. In this case, the space would have been discretized in elements of tetrahedral, hexahedral, prismatic or pyramidal volume.
48
The Finite Element Method for Electromagnetic Modeling
We thus have several topologies of 1D, 2D and 3D elements. For each topology, we can exploit the quality of the approximation. In the 1D example, we have successively used two polynomial approximations of first order and second order. In the 2D example, we have implemented triangles with first-order polynomial interpolation. We could also have used second order or a higher order or have mixed the orders by using transition elements. In practice, whatever the dimension of space, the first-order elements are by far the easiest to implement and thus, for this reason, are the most used. However, the elements of higher orders have some relevance. On the one hand, at a given calculation cost, they lead to an approached solution that is more accurate than the first-order elements, and on the other hand, they allow the curvilinear elements to be managed, which is quite useful for the discretization of objects having non-plane surfaces. For these reasons, the secondorder elements which seem to be a good trade-off between easy implementation and accuracy are also often used. The elements of higher orders, for example of orders 3 to 6 [SIL 83], allow high quality approximations. As we have seen in the first example, the construction of polynomial functions of an unspecified degree in a 1D sub-domain does not raise any particular difficulties. In the second example, we have defined the first-order shape functions in triangles by using identification. To go up in order, we could use the same process, but will prefer to use another, more powerful, process based on the concept of 1D, 2D and 3D reference elements, which we will now develop [ZIE 79]. 1.4.1. Linear reference elements
In a 1D space, the only reference elements available are the segments. On a segment, a point 3 is localized using a normalized local coordinate [ varying between –1 and +1. 1.4.1.1. First-order segment Figure 1.12 shows the first-order Lagrange reference element. It has two nodes, located at its terminals, of coordinates [1 = –1 and [2 = +1. Intrinsic functions associated with the terminals of the element are defined by the following Lagrange polynomials of degree 1
O1 1 [
1[ 2
O 21 [
1[ 2
[1.143]
Introduction to Nodal Finite Elements
49
Figure 1.12. First-order linear element
Figure 1.13. Second-order linear element
1.4.1.2. Second-order segment The rise in order is possible. As an example, Figure 1.13 represents the reference element which, in addition to the terminal nodes [1 = –1 and [2 = +1, has a middle node of coordinate [3 = 0. This element has as an intrinsic function the following Lagrange polynomials of degree 2
[ . [ 1 O1 2 [ 2
[ . [ 1 O 22 [ 2
O 32 [ 1 [ 2
[1.144]
1.4.2. Surface reference elements
In a 2D space, the most current reference elements are triangles and quadrangles which we will discuss below. 1.4.2.1. First-order triangle In a triangle, a point 3 is localized by means of two local coordinates, [ and \, called surface coordinates, which can be defined be the following ratios of surfaces
[
S[ S[ S\ S9
\
S\ S[ S\ S9
[1.145]
In order to make the role of the nodes symmetric, the third ratio S] ] 1 [ \ is also introduced. S[ S\ S9
50
The Finite Element Method for Electromagnetic Modeling
Figure 1.14. First-order triangular element
For an interior point 3 or on the edge of the triangle, the coordinates check simultaneously the three relations 0 d[ d1
0 d\ d 1
0 d] d1
[1.146]
Figure 1.14 represents the first-order triangular reference element. It has three nodes, which have as local coordinates (1,0), (0,1) and (0,0). The intrinsic functions associated with these three nodes are
O1 1 [ ,\ [
O 21 [ ,\ \
O3 1 [ ,\ ]
[1.147]
1.4.2.2. Second-order triangle Figure 1.15 shows the second-order triangular Lagrange reference element. Compared to the previous element, it has three additional nodes, in the middle of its edges, of coordinates (0.5, 0.5), (0, 0.5) and (0.5, 0). Its intrinsic functions are
O1 2 [ ,\ [ . 2[ 1 O 22 [ ,\ \ . 2\ 1 O 2 [ ,\ ] . 2] 1 3
O 42 [ ,\ O 2 [ ,\
4[\
[ ,\
4][
5 O 62
4\]
[1.148]
Elements of an even higher order can be encountered, up to order 6 [SIL 83]. Additionally, elements with different orders on each one of its edges can be used as a transition between different order elements.
Introduction to Nodal Finite Elements
51
Figure 1.15. Second-order triangular element
1.4.2.3. First-order quadrangle In a quadrangle, a point 3 is also localized using two normalized coordinates [ and \, each varying between –1 and +1. Figure 1.16 represents the first-order quadrangular reference element. It has four nodes, which have as local coordinates (+1, +1), (–1, +1), (–1, –1) and (+1, –1). The intrinsic functions associated with these four nodes are
O1 1 [ ,\ O 1 [ ,\
1 [ 1 \ / 4 1 [ 1 \ / 4 2 O 31 [ ,\ 1 [ 1 \ / 4 O 41 [ ,\ 1 [ 1 \ / 4
Figure 1.16. First-order quadrangular element
[1.149]
52
The Finite Element Method for Electromagnetic Modeling
1.4.2.4. Second-order quadrangle The rise to second-order can be complete, with the element with 9 nodes (4 nodes, 4 middles of edges and the center), or incomplete with the element with 8 nodes (4 nodes and 4 middles of edges) [ZIE 79]. We will present here only the latter, shown in Figure 1.17, which is in current usage. Here are some of its intrinsic functions
O1 2 [ ,\ O 2 [ ,\ 2
1 [ 1 \ 1 [ \ / 4 1 [ 1 \ 1 [ \ / 4
...
O5 2 [ ,\ O 2 [ ,\ 6
1 [ 1 \ / 2 1 [ 1 \ / 2 2
[1.150]
2
...
Figure 1.17. Incomplete second-order quadrangular element
1.4.3. Volume reference elements
In a 3D space, the main reference elements are tetrahedrons with 4, 10 or more nodes, hexahedrons with 8, 20, 26 or 27 nodes, prisms with 6, 15 or 18 nodes and possibly pyramids [COU 97] for the connection between the elements with triangular facets and the elements with quadrangular facets. As an illustration, Figures 1.18 and 1.19 represent some first and second-order 3D elements.
Figure 1.18. Some first-order 3D reference elements
Introduction to Nodal Finite Elements
53
Figure 1.19. Some second-order 3D reference elements
1.4.4. Properties of the shape functions
On each reference element, the shape functions have been built in order to always guarantee a certain number of fundamental properties. First of all, it is the nodal functions that comply with
Ol ; l 1
Ol ; k z l 0
[1.151]
where ;l is the vector of the local coordinates of node l. Then, they verify the property of unit partition, i.e. Element nodes
¦
Ol ; 1
[1.152]
l
where ; is the vector of the unspecified coordinates of a point in the element. Lastly, they verify the localization property on the edges of the element. Thus, on an edge of a surface or volume element, or on a facet of a volume element, the form functions of the nodes not belonging to this edge or this facet are uniformly zero, which results in
OlEdge ; Edge 0
Edge nodes
¦ l
Ol ; Edge 1
[1.153]
where ;Edge represents the unspecified coordinates of a point of the considered edge. In other words, the elements are organized in such a way that any facet of a volume reference element is a surface reference element, and any edge of a surface reference element is a linear reference element. Thus, two elements sharing an edge or a facet have on their common support the same local shape functions.
54
The Finite Element Method for Electromagnetic Modeling
1.4.5. Transformation from reference coordinates to domain coordinates
We have just reviewed some linear, surface and volume reference elements. Each element topology has its reference space, with its own local coordinates. On a given topology, several elements are definable, each one having nodes. The intrinsic shape functions are polynomial (except for the pyramid) associated with these nodes. The degree of these polynomials defines the order of the element. In our electrostatic 1D and 2D examples and, in fact, in all uses of the finite elements, the domain integrals appear. We will thus study the transformation between reference coordinates and domain coordinates on which these integrals are defined. When meshing the domain, each finite element, in practice, is defined by its reference element and by the geometric positioning of its nodes in the discretized domain. Let us consider an element with L nodes. Each node L has as a vector of domain coordinates Xl, as a vector of local coordinates ;l and as a function of local form Ol(;). The transition between the reference coordinates ; and the domain coordinates X is thus simply defined by X ;
L
¦ X l Ol ;
[1.154]
l
This definition is satisfactory, since a function of form Ok(;) takes the value 1 at node l, the value 0 at other nodes and interpolates the coordinates elsewhere. If the element is not too distorted, the previous formulae are invertible and define, in an implicit way, the reverse transformation. In fact, we will see that in the process of resolution by finite elements, this reverse transformation does not need to be explicit. This geometric transformation can be applied to all element types, the starting space being of a size equal to or lower than the arrival space. For example, for a triangular element of which nodes nD, nE and nJ are positioned in XD = [xD,yD]T, XE = [xE,yE]T and XJ = [xJ,yJ]T, the transition between the local coordinates ; = [[,\]T and the domain coordinates X = [x,y]T is defined by x [ ,\
xD O1 1 [ ,\ x E O 21 [ ,\ xJ O 21 [ ,\
y [ ,\
yD O1 1 [ ,\ y E O 21 [ ,\ yJ O 21 [ ,\
[1.155]
Introduction to Nodal Finite Elements
55
For a second-order element, having intermediate nodes on edges nD n E, nE n J, and nJ n D, this would give x
xD O1 2 [ ,\ ... xJ O 32 [ ,\ xDE O 42 [ ,\ ... xJD O 62 [ ,\
y
yD O1 2 [ ,\ ... yJ O 32 [ ,\ yDE O 42 [ ,\ ... yJD O 62 [ ,\
[1.156]
If the intermediate points are not exactly in the middle of the edges, the transformation is nonlinear [ZIE 79]. Curvilinear elements are built in this way, as is illustrated in Figure 1.20. Their use allows a domain with curves to be meshed with a reduced number of elements, compared to a discretization in rectilinear elements. For example, to mesh the quadrant of Figure 1.21, four curvilinear elements are sufficient. 1
3
2
[
3 NE
y ND
NDE
T[
P x
Figure 1.20. Curvilinear linear element
y
x
Figure 1.21. A quadrant discretized by means of curvilinear elements
The point P is an image in the domain of a point 3 of the reference element. When only one of the local coordinates varies, for example [, the point P moves
56
The Finite Element Method for Electromagnetic Modeling
along a curve (Figure 1.20). In this point, the vector of the derivatives T
ª wx wy º « w[ w[ » is tangent to the curve and its amplitude corresponds to a metric ¬ ¼ change for this coordinate. T[
It is convenient to gather all the components of the tangent vectors in only one matrix G, called the Jacobian matrix, obtained by derivation of [1.154]. For example, in the case of a transition from 2D to 3D, G has as an expression
G
ªT[T º « T» «¬T\ »¼
ª wx « w[ « « wx «¬ w\
wy w[ wy w\
wz º w[ » » wz » w\ »¼
ª wOl º « w[ » ¦ « wO » . >xl l « l» «¬ w\ »¼ L
yl
zl @
[1.157]
Generally, G is obtained by
G
ª wX T º « » «¬ w; »¼
L
¦ grad ;Ol . X lT
[1.158]
l
In the particular case of a rectilinear triangle of nodes nD, nE and nJ, used in the 2D electrostatic example, this gives simply
G
ª xD xJ «x x ¬ E J
yD yJ º yE yJ »¼
[1.159]
1.4.6. Approximation of the physical variable
Starting from the geometric position of the nodes of an element, we have seen how to go from local coordinates to domain coordinates. To go further, we will now describe the approximation of the physical variables. A physical scalar and continuous variable is interpolated, according to a procedure similar to that used for the coordinates, by using the nodal values and the intrinsic functions of the reference element. Thus, the electric potential could be defined as in [1.154] by L
v ; ¦ vl Ol ; l
[1.160]
Introduction to Nodal Finite Elements
57
However, for an element, there is no obligation to have the same type of approximation for the geometry and the physical variable. This situation occurs naturally when it is necessary to improve the quality of the solution by increasing the order of the approximation without modifying the meshing. For example, for the triangular element whose geometry will always be defined by the position of the three nodes, the electric potential could be interpolated in the first simulation by vD O1 1 [ ,\ v E O 21 [ ,\ vJ O 31 [ ,\
v [ ,\
[1.161]
then in a second one by v
vD O1 2 [ ,\ ... vJ O 32 [ ,\ vDE O 42 [ ,\ ... vJD O 62 [ ,\ [1.162]
The geometry is defined on a reference element, whereas the physics is interpolated on another reference element of the same topology. In addition, other physical variables could be interpolated with their own order or even according to different diagrams (hierarchical nodal elements, edge elements, facet elements or volume elements [BOS 83]) from the nodal interpolation, the only interpolation presented in this chapter. For the physical variable, we thus replace [1.160] by K
v ; ¦ vkMk ;
[1.163]
k
where Mk(;) are the intrinsic functions Ok(;) corresponding to the selected reference element. Formula [1.154] interpolates on the basis of the coordinates of the geometric nodes, whereas [1.163] interpolates on the basis of the nodal variables vk associated with the physical nodes. When the geometric and physical nodes coincide, the approximation is known as isoparametric. When the number of physical nodes is a sub- or an upper-set of the geometric nodes, it is called sub- or upper-parametric. In order to be able to give the integral’s finite elements, it is also necessary to have the expression of the gradient of the physical variable gradX v. In the reference space, the expression of the local gradient grad; v is a combination of the gradients of the local form functions
grad; v
K
¦ vk grad; Mk k
[1.164]
58
The Finite Element Method for Electromagnetic Modeling
The local gradients and the domain gradients are connected by the relations
grad; v
G gradX v
gradX v
G1grad; v
[1.165]
in which the Jacobian matrix G and its inverse on the right G–1 intervene
G1
ª w;T º « » «¬ wX »¼
and
G G1
I;
[1.166]
Table 1.5 gives the forms of the Jacobian matrices, of their determinant and their inverse on the right, for various combinations of dimensions of the reference space and the calculation domain. In the particular case of a rectilinear triangular of nodes nD, nE and nJ, in a 2D domain, this gives
G1
yD yJ º xD xJ »¼
1 ª yE yJ « det G ¬ xE xJ
[1.167]
with
detG
xD xJ yE yJ xE xJ yD yJ
[1.168]
As a summary, in a point 3 of coordinates ; in the reference element, we have at our disposal formulae, giving in its image P the domain, the coordinates, the Jacobian matrix, the determinant of this matrix, its inverse on the right, the physical variable and its gradient. Consequently, thanks to the change of coordinates [1.154], an integral on a finite element e can be replaced by an integral on the corresponding reference element '. For example, for an integral on a volume element, this gives
³³³ f ^X , U >X @, v>X @, gradX v>X @,...` dX e
³³³ f ^X ; , U >X ; @, v>X ; @, gradX v>X ; @,...` det G ; d; '
[1.169]
Introduction to Nodal Finite Elements
>x
>x@ G=
>[ @
det G =
G–1 =
G=
ª[ º «\ » ¬ ¼ det G =
G–1 =
>T[ @ T
ª wx º « w[ » ¬ ¼
T[
>T[ @
ª wx « w[ ¬
T
T[
>x wy º w[ »¼
>T[ @ T
det G
det G2 wy º w[ » » wy » w\ »¼
1 N\ det G
ª wx « w[ « « wx «¬ w\
ªT[T º « T» «¬T\ »¼
T[ u T\
>
N[\
N[
@
G=
wz º w[ » » wz » w\ »¼
T[ u T\
N[\ u T[ det G2
ªT[ « T» «T\ » « T» «¬T] »¼
ª wx « « w[ « wx « w\ « wx « ¬« w]
wy w[ wy w\ wy w]
@
wz º » w[ » wz » w\ » wz » » w] ¼»
T[ u T\ .T]
det G = G–1 =
wy w[ wy w\
>T\ u N[\ Tº
ª[ º «\ » « » «¬] »¼
wz º w[ »¼
>T[ @ 2
ª wx « w[ « « wx «¬ w\
ªT[T º « T» «¬T\ »¼
wy w[
T[
>T[ @ 2
y z@
ª wx « w[ ¬
ª wy º « w[ » « » « wx » «¬ w[ »¼
N[
>T[ @ det G
y@
59
>T\ u T]
T] u T[
T[ u T\
det G
Table 1.5. Expression of the Jacobian matrices, their determinant and their inverse on the right for various arrival and starting spaces
@
60
The Finite Element Method for Electromagnetic Modeling
which, in a more concise way, gives
³³³ f dX ³³³ f det G d; e
[1.170]
'
The flux of vector D through a surface element is, by introducing the normal vector N[\ T[ u T\ ,
³³ D.dS ³³ D.N[\ d[d\ e
[1.171]
'
Lastly, the circulation of vector E along a linear element is
³ E.dL ³ E.T[ d[
e
[1.172]
'
1.4.7. Numerical integrations on the reference elements
The coefficients of the equations to be solved are finite element integrals. In the case of coefficients [1.60] and [1.61], we could calculate these integrals analytically, because the selected finite elements were rectilinear triangles with linear interpolation of the variable and uniform physical property. This analytical process can be extended to triangles and rectilinear tetrahedrons of higher orders [SIL 83] or within the framework of the parametric elements thanks to integrals [1.170], [1.171] and [1.172]. However, non-polynomial variations are sometimes present in integral terms. They can be of physical origin, introduced for example by the nonlinearity of a constitutive law D(E) if E varies on the element, or of geometric origin in the case of curvilinear elements. The analytical integration is thus more problematic and a numerical integration must be considered. Numerical integration consists of replacing the integral of the function by a sum of weighted samples. There are a large number of numerical methods of which the best known are the rectangles method, the trapezoids method, the Simpson method and the Gauss method. We will more particularly be interested in the latter methods, also called numerical Gaussian quadratures, which are very often used in finite elements because they are the most effective for a given number of calculation points.
Introduction to Nodal Finite Elements
61
To explain the principle of this method, let us take an integral on a linear finite reference element, in which function f represents the whole integrating term, including the Jacobian determinant 1
³ f [ d[
[1.173]
1
Order
Polynomial integrated exactly
NG
pg
[g
1
f [ a0 a1[
1
{2}
{0}
3
f [ a0 a1[ ... a3[ 3
2
{1, 1}
f [ a0 a1[ ... a5[ 5
3
…
…
5
…
{
{
5 8 5 , , } 9 9 9 …
{
1 1 , } 3 3 3 3 } , 0, 5 5 …
Table 1.6. First- and third-order numerical Gaussian quadratures on the segment [–1, +1]
The approximation of the integral is given by 1
NG
1
g 1
³ f [ d[ | ¦ pg f [ g
[1.174]
where NG is the number of Gauss points on the reference segment [–1,+1], [g the coordinates of the Gauss points and pg their respective weights. Table 1.6 gives some normal choices for these values with their order of accuracy. For example, the formula with 1 point 1
³ f [ d[ | 2 f 0
[1.175]
1
which is very simple to implement and is exactly the rectangle method, is of first order because it is exact for the integrating polynomial terms of a degree lower than or equal to 1.
62
The Finite Element Method for Electromagnetic Modeling
Another example, the Gaussian quadrature with two points 1
§
1·
§
1·
³ f [ d[ | f ¨¨ 3 ¸¸ f ¨¨ 3 ¸¸ 1 © ¹ © ¹
[1.176]
which integrates a third-order polynomial exactly, is as accurate as the Simpson formula, which requires 3 points. In addition to their effectiveness, these formulae are numerically stable, because they use points located strictly inside the reference elements and with positive weightings. On the same basis, the integrals on the triangular, quadrangular, tetrahedral, hexahedral and prismatic reference elements are calculable, with the desired order, using numerical Gaussian quadratures
³³³ f [ ,\ ,] d[d\d] |
NG
¦ pg f [ g ,\ g ,] g
[1.177]
g 1
For example, for a triangle, 1 Gauss point is required for order 1, 3 for order 3 and 7 for order 5. For a tetrahedron 1, 4 and 15 points are needed. For quadrangles, hexahedrons and prisms, the positions and weightings can be obtained by crossing the formulae on a segment or a triangle. For these elements, more economic formulae are also available [ZIE 79], [STR 71]. The pyramids have rational shape functions which need to be handled in a certain way [HAM 56]. In practice, the difficulty of choice regarding numerical integration arises. This order depends at the same time on the coordinate interpolation and on the approximation of the physical variables. The 2D example of electrostatics, which we have handled using first-order triangular finite elements by analytical integration, could have been handled by numerical integration on isoparametric elements with three nodes. For the triangular reference element, the geometric and physical interpolations being linear, the Jacobian determinant is constant, the potential with linear variation and its gradient being constant. The elementary contributions to integrals [1.52] and [1.54] are firstdegree polynomials and could thus have been integrated numerically by a first-order formula, i.e. by a sample of an integrating term taken at the barycenter of each triangle and weighted by their surface.
Introduction to Nodal Finite Elements
63
For the same problem of electrostatics, but specified on the meshing in secondorder isoparametric elements in Figure 1.21, the numerical treatment would have been different. In these elements, the variation of the electric potential is seconddegree and its gradient is first-degree in triangles and second-degree in quadrangles. From the point of view of physics, the integrating term is thus of second or fourthdegree according to the type of element. From the geometric point of view, there is no general conclusion, because the variation of G and its determinant depends on the shape of each element. However, it is usual to make the assumption that the element has a quasi-uniform deformation and thus we do not add any additional requirement due to the geometric variation. Ultimately, on triangles we would take a third-order formula with 3 points and on quadrangles a fifth-order formula with 9 points. 1.4.8. Local Jacobian derivative method
The concept of the reference element enabled us to widen the applicability of the method to elements of various orders and shapes, and in particular to curvilinear elements. We will see that this concept can also be useful, even in the case of traditional rectilinear elements, to evaluate the influence of a geometric variation on calculated variables [COU 83], [NGU 99]. 1.4.8.1. Independence of reference elements with respect to the parameters On each finite element e, the integrals are expressed according to the local coordinates in the place of the domain coordinates, using one of the formulae [1.170], [1.171] or [1.172]. For example, in 2D, for J the domain or sub-domain integral of function f, on a depth h, gives J
¦ ³³ f h dxdy ¦ ³³ f det G hd[d\ e e
[1.178]
e '
This transformation of coordinates makes the terminals of integration completely independent of any parameter, which considerably simplifies the derivations. The derivative can thus be expressed indifferently by integration on the reference coordinates
wJ wu
ªw > f @det G f w >det G @ º»hd[d\ w wu u ¼ '¬
¦ ³³ « e
[1.179]
or, after inverse transformation, by integration in the domain
wJ wu
ªw > f @ f 1 w >det G @ º»hdxdy det G wu ¬ wu ¼
¦ ³³ « e e
[1.180]
64
The Finite Element Method for Electromagnetic Modeling
In fact, this handling makes it possible to express the influence of parameter u on the integrating term and on the deformation of the domain. The latter is intrinsic to the element, whereas the former is particular to each integrating term. We will see in the next example that when the integrating term itself depends on a deformation parameter, the local Jacobian derivative method is usable again. 1.4.8.2. Example of derivation of an integrating term: force by virtual work Let us consider the virtual work method by exact differentiation introduced earlier. Let us also consider the developments on the derivation of the coenergy in the virtually deformed elements. We interrupted it because we were waiting for a general method of taking into account the deformation. The local Jacobian derivative method is precisely the tool that we will use. Within the framework of the electrostatic formulation which served as an example in this chapter, the force is given by equation [1.128] which expresses the derivative of the coenergy with respect to the displacement, with constant nodal values. To simplify, we assume that the part concerned with the virtual deformation does not have any electric charge. The integrating term is thus simply f x, y
E T ³0 D dE
[1.181]
whose derivative with respect to the displacement, the law D(E) being assumed to be unaffected by the deformation, is wf wu
DT
wE wu
[1.182]
According to interpolations [1.165], the electric field E is dependent on the nodal values. In its expression, only the inverse of the Jacobian matrix is likely to vary when parameter u varies, since the nodal values are maintained and the local gradients of the form functions are independent. The derivative of the field is thus wE wu
w > grad X v @ wu
> @
w 1 G grad ; v wu
[1.183]
The derivative of the inverse of the Jacobian matrix could be obtained by derivation of its expression given in Table 1.5. However, we prefer to use the constant G G–1 = I; which we derive as follows: G
> @
w 1 w G >G @ G 1 wu wu
0
[1.184]
Introduction to Nodal Finite Elements
65
When matrix G is square, G–1 is also the inverse on the left, which leads to
> @
w 1 w G 1 >G @ G 1 G wu wu wE w G 1 >G @ E wu wu
[1.185] [1.186]
By posing
G ' G 1 det G
[1.187]
we obtain a first expression of the force in the direction u
Fu
ª ¦ ³³ « DT G' e
'¬
wG w det G E E ³0 DdE wu wu
º » hd[d\ ¼
[1.188]
Then, on the basis of the property
I ; det G
G' G
[1.189]
the derivative of the determinant is written
I;
w >det G @ wu
w >G'@ G G' w >G @ wu wu
[1.190]
which, after some arrangements, gives us a second expression
Fu
ª '¬
¦ ³³ « D T e
wG ' w det G D G E ³0 EdD wu wu
º »hd[d\ ¼
[1.191]
In the particular case where D is proportional to E, the two integrals, representing the energy and coenergy densities, are identical. In this case, the average of the two expressions gives a third expression which is more symmetric
Fu
1ª
wG '
wG
º
ª º ¦ ³³ « DT « G G' » E »hd[d\ wu ¼ ¼ ¬ wu e ' 2¬
[1.192]
66
The Finite Element Method for Electromagnetic Modeling
1.5. Conclusion
We have presented the finite element method on electrostatic examples. This very general method is also applicable to static or dynamic problems of electromagnetism and to multi-physical problems, in particular electromechanical and thermoelectric problems appearing in actuators, sensors and electromechanical devices of all sizes. In this presentation, we have focused on nodal elements because they are appropriate both to the interpolation of geometry and to the state variable (the electric potential) selected in this chapter. There are other interpolation possibilities which will be selected according to the characteristics of the fields to interpolate. Thus, the continuous scalar functions, such as the scalar potentials or the temperature, are interpolated naturally with nodal elements. The vector fields, such as electric or magnetic fields, requiring on the interface between two mediums a continuity of their tangential component with a possible discontinuity of their normal component, are processed with edge elements. The vector fields, such as electric or magnetic inductions, with continuity of the normal component and possible discontinuity of their tangential component, are interpolated with facet elements. Lastly, the scalar functions that are continuous per piece, such as charge densities, are interpolated naturally by the volume elements [BOS 93]. Lastly, for a given element topology and a given mode of interpolation, it is also possible to exploit the order of the elements, either by defining for each order families of different functions (as was done in this chapter for first and second-order nodal elements), or by adding to each rise in order additional functions to the lower order family. This is the concept of hierarchical elements, which is particularly suitable for adapting the quality of interpolation locally [ZIE 79]. 1.6. References [BOS 93] BOSSAVIT A., Électromagnétisme en vue de la modélisation, Springer-Verlag, Paris, 1993. [CAR 59] CARPENTER C.J., “Surface-integral methods of calculating forces on magnetized iron parts”, The Inst. of Elec. Eng., Monograph no. 342, pp. 19-28, 1959. [COU 83] COULOMB J.L., “A methodology for the determination of global electromechanical quantities from finite element analysis and its application to the evaluation of magnetic forces, torques and stiffness”, IEEE Transactions on Magnetics, vol. 19, no. 6, pp. 2514-2519, 1983.
Introduction to Nodal Finite Elements
67
[COU 84] COULOMB J.L., MEUNIER G., “Finite element implementation of virtual work principle for magnetic or electric force and torque computation”, IEEE Transactions on Magnetics, vol. 20, no. 5, pp. 1894-1896, 1984. [COU 85] COULOMB J.L., MEUNIER G., SABONNADIÈRE J.C., “Energy methods for the evaluation of global quantities and integral parameters in a finite element analysis of electromagnetic devices”, IEEE Transactions on Magnetics, vol. 21, no. 5, pp. 18171822, 1985. [COU 97] COULOMB J.L., ZGAINSKI F.X., MARÉCHAL Y., “A pyramidal element to link hexahedral, prismatic and tetrahedral edge finite elements”, IEEE Transactions on Magnetics, vol. 33, no. 2, pp. 1362-1365, 1997. [DEM 99] DE MEDEIROS L.H., REYNE G., MEUNIER G., “About the distribution of forces in permanent magnets”, IEEE Transactions on Magnetics, vol. 35, no. 3, pp. 1215-1218, 1999. [DHA 84] DHATT G.J.C., TOUZOT G., Une présentation de la méthode des éléments finis, Editions Maloine, 1984. [DUR 64] DURAND E., Électrostatique, 3 volumes, Masson et Cie, Paris, 1964. [FUJ 93] FUJIWARA K., NAKATA T., OKAMOTO N., MURAMATSU K., “Method for determining relaxation factor for modified Newton-Raphson method”, IEEE Transactions on Magnetics, vol. 29, no. 2, pp. 1962-1965, 1993. [GIL 83] GILL P.E., MURRAY W., SAUNDERS M.A., WRIGHT A.H., “Computing forward difference intervals for numerical optimization”, Siam J. Sci. Stat. Comput., vol. 4, pp. 310-321, 1983. [GIT 89] GITOSUSASTRO S., COULOMB J.L., SABONNADIÈRE J.C., “Performance derivative calculations and optimization process”, IEEE Transactions on Magnetics, vol. 25, no. 4, pp. 2834-2839, 1989. [GUI 94] GUILLAUME PH., MASMOUDI M., “Computation of high order derivatives in optimal shape design”, Numerische Mathematik, vol. 67, pp. 231-250, 1994. [HAM 56] HAMMER P.C., MARLOWE O.J., STROUD A.H, “Numerical integration over simplexes and cones”, Math. Tables and Other Aids to Computation, vol. 10, pp. 130-137, 1956. [HOO 89] HOOLE S.R.H., Computer-aided Analysis and Design of Electromagnetic Devices, Elsevier, 1989. [MAR 92] MARECHAL Y., MEUNIER G., “Modélisation par une méthode éléments finis tridimensionnelle des courants induits dus au mouvement”, RGE, no. 292, pp. 39-46, February 1992. [NGU 99] NGUYEN T.N., COULOMB J.L., “High order FE derivatives versus geometric parameters: implantation on an existing code”, IEEE Transactions on Magnetics, vol. 35, no. 3, pp. 1502-1505, 1999. [OSS 99] OSSART F., IONITA V., “Convergence de la méthode du point fixe modifiée pour le calcul de champ magnétique avec hystérésis”, Eur. Phys. J. AP, vol. 5, pp. 63-69, 1999.
68
The Finite Element Method for Electromagnetic Modeling
[RAF 77] RAFINEJAD P., Adaptation de la méthode des éléments finis à la modélisation des systèmes électromécaniques de conversion d’énergie, PhD Thesis, Grenoble, 1977. [REN 92] REN Z., RAZEK A., “Local force computation in deformable bodies using edge elements”, IEEE Transactions on Magnetics, vol. 28, no. 2, pp. 1212-1215, 1992. [REN 95] REN Z., “A 3D vector potential formulation using edge element for electrostatic field computation”, IEEE Transactions on Magnetics, vol. 31, no. 3, pp. 1520-1523, 1995. [SAB 86] SABONNADIÈRE J.C., COULOMB J.L., La méthode des éléments finis: Du modèle à la CAO, Hermes, 1986. [SIL 83] SILVESTER P.P., FERRARI R.L., Finite Elements for Electrical Engineers, Cambridge University Press, 1983. [STR 71] STROUD A.H., Approximate Calculation of Multiple Integrals, Prentice Hall, 1971. [ZIE 79] ZIENKIEWICZ O.K., La méthode des éléments finis, McGraw-Hill, 1979.
Chapter 2
Static Formulations: Electrostatic, Electrokinetic, Magnetostatics
Introduction In this chapter, based on the general form of Maxwell’s equations, the various static formulations (electrostatic, electrokinetic and magnetostatics) are presented. In addition to the homogenous boundary conditions, a special focus is dedicated to the boundary conditions which impose global variables of the flux or circulation type. Indeed, in many electromagnetic systems, we often have to solve a problem where a condition of the flux or potential difference type of a vector field acts as a source term. Subsequently, the source fields and the concepts of scalar potential and vector potential are introduced. These “mathematical beings” implicitly verify one of the equations of the system to be solved. Their use notably simplifies the problem and makes it possible, in certain cases, to introduce source terms naturally as well as global boundary conditions. It should be noted that the corollary of the use of potentials is the need to impose a gauge condition to have the uniqueness of the solution. The various formulations in potentials being defined, the construction of function spaces is then covered. These spaces will accommodate the scalar and vector fields which constitute the solutions of the problem. This then leads to Tonti diagrams which are an illustration of the series of function spaces. In the case of electrokinetic formulations, the weak formulations, in scalar and vector potential, are developed using the Green formulae. Chapter written by Patrick DULAR and Francis PIRIOU.
70
The Finite Element Method for Electromagnetic Modeling
After having presented the various formulations in the continuous domain, we focus on the discretization of function spaces and weak formulations elaborated. The finite elements and the nodal, edge, facet and volume basis functions with which discrete spaces are associated are thus presented. A geometric interpretation of these functions is made in the case of tetrahedrons. It is then shown that the scalar and vector fields, introduced into the formulations, can be expressed via these basis functions. The properties of discrete spaces are presented on the basis of the incidence concept. This concept allows discrete operators, which are the equivalents of vector operators in the continuous domain, to be introduced. The series of discrete function spaces are then integrated into discrete Tonti diagrams. Emphasis is placed on the gauge conditions and the calculations of source fields using tree techniques. The concept of facet trees is thus introduced. Then, the various weak formulations, in scalar and vector potentials, for the three problems of static electromagnetism are developed in discrete form. Lastly, a method is proposed to impose global variables. As an illustration, the specific case of electrokinetics with a scalar potential formulation is considered. 2.1. Problems to solve 2.1.1. Maxwell’s equations The whole of the classical electromagnetic phenomena is governed by Maxwell’s equations. These equations constitute a system of partial differential equations which link the magnetic phenomena to the electric phenomena, and which unify all the principles of electromagnetism. In continuous mediums, these equations are as follows [VAS 80, FOU 85]: curl h = j + wt d,
[2.1]
curl e = – wt b,
[2.2]
div b = 0,
[2.3]
div d = U,
[2.4]
where h is the magnetic field (A/m), b is the magnetic induction (T, i.e. Tesla), e is the electric field (V/m), d is the field of electric displacement or electric flux density (C/m2), j is the conduction current density (A/m2) and U is the electric charge volume density (C/m3).
Static Formulations
71
When the studied phenomena are invariants in time, the time derivatives become zero in Maxwell’s equations and a decoupling between magnetic and electric phenomena appears. The study of the electric phenomena is the object of electrostatics and electrokinetics, and that of the magnetic phenomena is the object of magnetostatics. 2.1.2. Behavior laws of materials It is important to add to Maxwell’s equations the relations which express the properties of materials, i.e. the behavior laws or constitutive relations [VAS 80]. Without them, systems [2.1]-[2.4] would be indeterminate. These relations are: b = P (h + m),
[2.5]
d = H e,
[2.6]
j = V e,
[2.7]
where P0 is the magnetic permeability of the vacuum (H/m), H is the electric permittivity, V is the electric conductivity (:–1m–1) and m (F/m) is the magnetization vector (A/m). In the following sections, for ferromagnetic materials, we will use the relation b = μ h. Rigorously speaking, P, H and V have a tensorial character and their value is not constant (saturation, hysteresis, dependency with respect to temperature, function of the position in space, etc.). Relations [2.5], [2.6] and [2.7] are called the magnetic, dielectric and local Ohm behavior laws. 2.1.3. Boundary conditions 2.1.3.1. Homogenous conditions Adequate boundary conditions have to be given on the boundary of the study domain : in order to ensure the uniqueness of the solutions. They can be, according to the problem considered, relevant to the tangential components of e and h, and to the normal components of d, j and b. At the boundary * of global domain :, we consider certain frequently encountered boundary conditions, which are homogenous conditions.
72
The Finite Element Method for Electromagnetic Modeling
For the electric quantities, on complementary surface portions *e and *d (or *j) of *, possibly non-connected (of several supports), the following conditions are defined: n e~*e = 0, n. d~*d = 0 or n. j~*j = 0.
[2.8-9-10]
For the magnetic quantities, on complementary surface portions *h and *b of *, possibly non-connected, the following conditions are defined: n h~*h = 0, n. b~*b = 0.
[2.11-12]
Such homogenous boundary conditions on the fields take place for reasons that are either physical (conditions at the infinite or associated idealized materials; this is, for example, the case in [2.8] and [2.11] respectively for perfect conducting and magnetic materials, i.e. of infinite conductivity and permeability) or relating to symmetry (fixing the direction of the fields). 2.1.3.2. Interface conditions of fields In the transition from one medium to another, the electromagnetic fields undergo discontinuities and are consequently not differentiable. However, it is possible to derive conditions of transmission of fields [DAU 87]. Let us consider a surface 6 between two continuous mediums, the sub-domains :1 and :2 (Figure 2.1). We do not make any assumption concerning the properties of these two mediums in order to obtain completely general relations. The normal n to 6 is oriented from :1 towards :2. Values of a field on both sides of 6 on the mediums :1 and :2 are respectively designated using indices 1 and 2; for example, h1 = h(x) for x :1 and h2 = h(x) for x :2. Charge and current densities Us and js respectively can be concentrated on the surface 6. This is, for example, the case for the surface of a perfect conductor, i.e. whose conductivity is infinite, or when the frequency of the source term is infinite.
Figure 2.1. Surface 6 between two continuous mediums :1 and :2
Static Formulations
73
Equations [2.1]-[2.4] can be integrated on volumes or surfaces including portions of the surface 6. The application of the divergence theorem or the Stokes theorem then implies the following transmission conditions or interface conditions: n (h2 – h1)~6 = js, n. (b2 – b1)~6 = 0,
n (e2 – e1)~6 = 0, n. (d2 – d1)~6 = Us.
[2.13-14] [2.15-16]
These are relative either to the tangential component, or to the normal component of the fields. They require that the normal component of b and the tangential component of e are continuous while crossing 6. On the other hand, if Us and js are different from zero, the normal component of d and the tangential component of h are discontinuous. The components which do not appear in [2.1316] are discontinuous. We will generally consider the case where Us and js are zero, i.e., we will not carry out the limits V o f or Z o f. The tangential component of h and the normal component of d are then continuous. NOTE.– the decomposition of a vector h according to its tangential and normal components in a point of a surface is given by: h = (n. h) n + (n h) n ; its tangential component is thus (n h) n, but here we shall refer in general directly to the vector n h which is orthogonal to it and which has the same norm, while remaining tangent to the surface. 2.1.3.3. Imposed global quantities of flux and circulation types In addition to the local boundary conditions, such as those previously defined, global conditions on the fields, via functionals of the flux and circulation types, can be imposed. Regarding fluxes, these can concern the total electric charge Q, the current intensity I and the magnetic flux <. Regarding circulations, these can concern the electromotive force or potential difference V and the magnetomotive force ). These fluxes, through surfaces *i located on the boundary of the study domain, and circulations, along curves Ji belonging to the domain, are defined by:
³*i n d ds Qi , ³*i n j ds Ii , ³*i n b ds
³J i e dl Vi , ³J i h dl ) i ,
[2.17-18-19] [2.20-21]
where n is the field of unit vectors normal to *i and * oriented towards the outside of :.
74
The Finite Element Method for Electromagnetic Modeling
2.1.4. Complete static models 2.1.4.1. Electrostatics model Electrostatics consists of the study of the space distribution of electric field e due to the distribution of electric charges. A distribution of charges in a perfect conductor can be, as we will see, defined using an electric scalar potential. The electrostatic model applied to study domain :, of boundary *, is characterized by the following differential equations, behavior law and boundary conditions: curl e = 0, div d = U, in :, d = H e, n e~*e = 0, n. d~*d = 0, with * = *e *d.
[2.22-23] [2.24] [2.25-26]
The global constraints that can be defined are related to total charge Q [2.17] and potential difference V [2.20], whose ratio defines a capacitance. 2.1.4.2. Electrokinetics model Electrokinetics consists of the study of the space distribution of current density j in the conducting materials. The electrokinetics model applied to the study domain :, of boundary *, is characterized by the following differential equations, behavior law and boundary conditions: curl e = 0, div j = 0, in :, j = V e, n e~*e = 0, n. j~*j = 0, with * = *e *j.
[2.27-28] [2.29] [2.30-31]
The global constraints that can be defined are related to current I [2.18] and potential difference V [2.20], whose ratio defines the inverse of a resistance. 2.1.4.3. Magnetostatics model Magnetostatics consists of the study of the magnetic phenomena in stationary conditions. The magnetic field is then invariant in time and is only due to imposed steady currents (j) or to permanent magnets (residual/remanent induction br). The equations to be considered result from Maxwell’s equations dealing with magnetic phenomena for which the time derivatives are zeroed. They are: curl h = j, div b = 0, in :,
[2.32-33]
Static Formulations
75
and the magnetic behavior law has to be added to it [2.5] b = br + P h,
[2.34]
with eventually P = P(h) for nonlinear materials. The boundary conditions considered are: n h~*h = 0, n. b~*b = 0, with * = *h *b.
[2.35-36]
The global constraints that can be defined are related to the magnetic flux < [2.19] and to the magnetomotive force ) [2.21], whose ratio defines an inductance. In order not to burden some of the subsequent developments, the term br in [2.34] will in general be omitted. 2.1.5. The formulations in potentials Whatever the problem to be tackled, whether electrostatics, electrokinetics or magnetostatics, we have to seek two vector fields defined by a divergence and a rotational, and boundary conditions. Moreover, these vector fields are linked by a behavior law. If we seek a vector field which complies only with one partial derivatives equation and the associated boundary condition, the field thus obtained is called an admissible field. In order to be the solution of the problem, this field must generate, via the behavior law, a second field having to comply with the other equation and the other boundary condition. In the following sections, we will see that the introduction of the concept of vector or scalar potentials is equivalent to proposing, under certain conditions, an admissible field. This is all the more true if the domain is simply connected with a connected boundary. Indeed, based on the theory of function spaces, we then show that the gradient image space is equal to the kernel of the rotational image space. In the same way, the rotational image space is equal to the kernel of the divergence. 2.1.5.1. Connectivity/connexity problems A simply connected domain is such that any continuous and closed line can be deformed continuously until it is reduced to a point. For example, a sphere is a simply connected domain. On the other hand, a torus is non-simply connected or multiply connected. The concept of connexity also applies to the boundaries of domains.
76
The Finite Element Method for Electromagnetic Modeling
A multiply connected domain can be made simply connected by the introduction of cuts. It is the same for a boundary using connections called links or chimneys. For example, we represent in Figure 2.2 a multiply connected domain : with an external boundary *1. In order to make the domain and the boundary simply connected, a cut 6 and a chimney / were introduced.
:
*
* 6
/
Figure 2.2. Example of multiply connected domain and boundary with the associated cut and link
2.1.5.2. Concept of source fields From Maxwell’s equations, we introduced three static problems. If we consider the case of electrostatics or magnetostatics, we directly see a source term appearing in the equations which corresponds respectively to the volume density of charges and to the current density. For these equations, the associated admissible field will be called a source field. It is pointed out that a vector field is completely defined if its rotational, divergence and boundary conditions are fixed. Under these conditions, we conceive that, for an equation with the partial derivatives, there is an infinite number of admissible fields and thus, an infinite number of source fields. Various methods can be considered to calculate them. However, in certain particular cases, we can easily find an analytical solution. Let us take an example of electrostatics: volume distribution of charges U contained between two infinite planes located, as shown in Figure 2.3a, at x = 0 and x = a. This example becomes a 1D problem and we represent in Figures 2.3b and 2.3c respectively the distribution of charges U and a trivial solution which complies with, for the component according to x of the source electric flux density ds, equation [2.23] (dsy = dsz = 0). We easily see in this figure that div ds = U for 0 d x d a and div ds = 0 for x < 0 and x > a. Lastly, if we add to component dsx a positive or negative constant, the proposed solution is always acceptable. There is thus an infinite number of source fields which comply with this equation.
Static Formulations
z
dsx
U
y
aUv
U
x
0
a
(a)
77
0
a
(b)
0
a
(c)
Figure 2.3. Problem studied (a); distribution of the volume density of charges (b); evolution of the component according to x of the source electric flux density (c)
While dealing with the case of electrostatics, let us denote by ds1 and ds2 two fields linked by the relation ds1 = ds2 + curl J where J indicates a vector field defined on a simply connected domain : with a connected boundary. These two fields are assumed to be solutions of equation [2.23]. To achieve uniqueness of the solution, it is necessary to impose, in addition to the boundary conditions, a constraint or gauge condition on ds. Let us consider the particular case where : is a homogenous medium. Equation [2.22] can then be written curl ds = 0. If this condition is imposed, we fix the rotational ds. The solution is thus unique. If in addition ds complies with the boundary conditions on *e and *d, then the solution obtained is a physical field which complies with the equations of the problem. For magnetostatics considerations, in the presence of a current density, the source term is introduced by equation [2.32]. In this case, there is an infinite number of source fields, which we will denote by hs, which comply with this equation. As for the electrostatics problem, it is possible, for certain applications, to find an elementary analytical solution. Let us now consider the general case and let us denote by hs1 and hs2 two fields defined with an uncertainty of a gradient by: hs1 = hs2 + grad D,
[2.37]
where D represents an unspecified scalar function defined on a simply connected domain :. The two fields are the solutionof equation [2.32]. In order to make the solution unique, we must impose a constraint or gauge condition. The most well known is the Coulomb gauge which imposes the divergence, that is to say div hs = 0. Other gauges can be used. As an example, we will refer to the gauge of [ALB 90]. hs. w = 0,
[2.38]
where w represents a vector field whose field lines connect all the points of the domain and do not close.
78
The Finite Element Method for Electromagnetic Modeling
Let us take two points P and Q pertaining to the domain : with P z Q and let us calculate the circulation of equation [2.37] on a contour K connecting P and Q. We then obtain:
³ h s1 .dl ³ h s2 .dl D Q - D P .
K
[2.39]
K
By taking as support for contour K a field line of w, the integrals of equation [2.39] are equal to zero. Under these conditions DQ = DP = 0, which imposes grad D = 0, thus the uniqueness of the solution. If the domain : is a homogenous medium, equation [2.33] is written div h = 0. The problem to solve is to calculate a field which verifies equation [2.32] with the Coulomb gauge and the boundary conditions on *h and *b. For the determination of the source field, in addition to the solution which consists of seeking an analytical expression for simple geometries, various methods are proposed. Among the most used, we can quote, in magnetostatics, the Biot and Savart formula. Whatever the encountered problem, the choice of the method will depend on the complexity of the domain but also on the numerical technique used to solve the equations. 2.1.5.3. The formulations in scalar potential Let us consider the electrostatics problem defined in section 2.1.4.1. Based on equation [2.22], we can introduce an electric scalar potential v such that: e = – grad v.
[2.40]
We can note that the electric field thus defined represents, for equation [2.22], an admissible field. In the above relation, the potential v is not unique. Indeed, if we consider the potentials v1 and v2 defined with an uncertainty of a constant k such as v1 = v2 + k, they lead to the same value of the electric field e. In order to have a unique solution, it is necessary to impose a constraint or a gauge condition on v. In practice, it is generally obtained using the boundary conditions. By grouping equations [2.23] and [2.24], we obtain the equation: div H grad v = – U,
[2.41]
which, taking into account the boundary conditions, should be solved on the whole domain. Condition [2.25], for the electric field, is written for the formulation in scalar potential. v~*e = v0 = constant (Cte).
[2.42]
Static Formulations
79
Moreover, this condition can be directly linked to relation [2.20] which imposes the circulation of the electric field along a path. Let us suppose that this path connects two boundaries denoted *A and *B. Since we need to impose a potential difference, it is possible to choose arbitrarily v = 0 on boundary *A of the domain and v = Vi on *B. For the electric flux density, since d = – H grad v, relation [2.26] is stated in the form: n. grad v~*d = 0.
[2.43]
In the case of electrokinetics, the formulation of the problem in scalar potential is obtained by using the same approach as for electrostatics. Equations [2.27], [2.28] and [2.29] then lead to the equation: div V grad v = 0,
[2.44]
where v represents the electric scalar potential. For this formulation, the boundary conditions on *e are stated as in the case of electrostatics and allow a potential difference to be imposed. In the same way as for the electric flux density, the boundary condition on the current density [2.31] is written: n. grad v~*j = 0.
[2.45]
For magnetostatics equations, in the general case, the introduction of the formulation in scalar potential is not immediate. Let us first suppose that the domain : is simply connected and the source field hs, with its associated boundary conditions (n hs~*h = 0), known. The magnetic field h can be put in the form of: h = hs – grad M,
[2.46]
which complies well with [2.32]. M denotes the magnetic scalar potential defined with an uncertainty of a constant. In a multiply connected domain, a scalar potential can be multi-valued, which requires the introduction of cuts aiming at making the domain simply connected [EMS 83] and consequently, the potential single-valued. It should be noted that in the static problems, the scalar potential is never multi-valued. In magnetostatics, it is the source field hs that entirely accounts for the effect of the known current sources (which differs from magnetodynamics where induced currents can be present).
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The Finite Element Method for Electromagnetic Modeling
By grouping equation [2.46] with relations [2.33] and [2.34], we can write: div (μ (hs – grad M = 0.
[2.47]
For the magnetic field, condition [2.35], i.e. n h~*h = 0, will be verified if: M~*h = M0 = Cte.
[2.48]
As for the electric scalar potential, the boundary condition can be connected to a global quantity. In this case, it is the variable )i defined by relation [2.21] which then corresponds to a scalar potential difference. If Ji links two boundaries *A and *B, on which n h = 0, we will impose M~*A = 0 and M~*B = )i. 2.1.5.4. The vector potentials To solve the equations of electrostatics, we can, as indicated previously, introduce the concept of a source field. Let us denote ds a source electric flux density which verifies [2.23] with associated boundary conditions. In the general case, by considering the source term known, the electric flux density will be written: d = ds + curl p,
[2.49]
where p represents the electrostatic vector potential with which we must associate, in addition to the boundary conditions, a gauge condition. It can be the Coulomb gauge. By grouping equations [2.22], [2.23] and [2.24], the problem to be solved is written: curl( 1 (ds curlp) 0 , H
[2.50]
with boundary conditions that take the form: n . d s *d
0 , n ( ds + curl p)~*e = 0, n p~*d = 0.
[2.51-52-53]
To impose global boundary conditions, we replace, in expression [2.17], the electric flux density by its expression given by [2.49]. In the case of electrokinetics, the formulation in vector potential is obtained directly from [2.28] while posing: j = curl t,
[2.54]
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81
where t represents the electric vector potential. By grouping [2.27], [2.29] and [2.54], we obtain: curl( 1 curl t) 0 .
V
[2.55]
To have a unique solution, in addition to the boundary conditions, it is necessary to gauge the vector potential. For this formulation, based on relations [2.30] and [2.31], the new boundary conditions are stated in the following way: n curl t~*e = 0, n t~*j = 0.
[2.56-57]
In the presence of a boundary condition of global quantity type defined by equation [2.18], we replace in this expression the current density by its expression given on the basis of the electric vector potential. For the magnetostatics problem, on the basis of equation [2.33], we introduce the concept of magnetic vector potential a such as: b = curl a.
[2.58]
This potential is not unique and, in order to have a unique solution, we have to impose a gauge condition (Coulomb or relation [2.38]). By grouping equations [2.32] and [2.34] (by omitting the term br), and taking into account [2.58], we have: curl Q curl a = j,
[2.59]
where Q represents the magnetic reluctivity (Q = 1/μ). With this formulation, the boundary conditions [2.35] and [2.36] become: n curl a~*h = 0, n a~*b = 0.
[2.60-61]
We can also introduce a global boundary condition while replacing in [2.19] the magnetic induction by relation [2.58]. After transforming the equation thus obtained, the boundary condition can be expressed in the form of an integral of contour.
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The Finite Element Method for Electromagnetic Modeling
2.2. Function spaces of the fields and weak formulations 2.2.1. Integral expressions: introduction In order to simplify the integral expressions which will be used in subsequent sections, we define the following notations relative to integrals on volume : and on surface *: ( u , v ):
³: u v d: ,
( u , v ):
³: u v d: ,
u , v !*
³* u v d* ,
u , v !*
³* u v d* ,
with u, v, u and v defined on : and * such that these integrals would have a sense. In general, they can be defined in Sobolev spaces [DAU 87b], of scalar and vector fields, i.e.: H1(:) = { u L2(:) ; wx u, wy u, wz u L2(:) }, H1(:) = { u L2(:) ; wx u, wy u, wz u L2(:) }. The establishment of weak formulations associated with the considered problems of partial derivatives, and on which is based the finite element method, leads us to consider two formulae, known as Green formulae. These are: ( u, grad v ) + ( div u, v ) = < v, n. u >*, ґ u H1(:), ґ v H1(:),
[2.62]
( u, curl v ) – ( curl u, v ) = < u n, v >*, ґ u, v H1(:),
[2.63]
established from the vectorial analysis relations u. grad v + v div u = div(v u), u. curl v – curl u. v = div(v u), integrated on the domain :, with the application of the divergence theorem for obtaining the surface integral terms. 2.2.2. Definitions of function spaces Generally, we have to solve, in a domain :, differential equations utilizing particular differential operators: the gradient, the rotational and the divergence. Such
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equations govern the space distribution of vector fields (magnetic field, electric field, vector potential, etc.) or scalars (scalar potential). The domain : is an open and bounded set of the Euclidean space, generally in three-dimensions, whose elements are called points. This set can be connected, i.e. of only one support, or non-connected. Its boundary w: is noted *. We will define a mathematical structure able to accommodate such equations. They will be mainly the operators and their domains of definition. The latter are function spaces of scalar and vector fields defined on : which should be characterized in a precise way, in order to enable them to accommodate the considered fields. We consider a structure composed of four function spaces and three differential operators [BOS 88]. The four spaces are two copies of L2(:) and two copies of L2(:); L2(:) is the space of scalar fields of integrable square on : and L2(:) is the space of the vector fields whose square of the Euclidean norm is integrable on :. They are denoted Ep, p = 0, 1, 2, 3. The three operators are the gradient (grad), the curl and the divergence (div). They are linear and unbounded differential operators. Their domains of definition are defined in a restrictive way in the sense that they are subspaces of L2(:) and L2(:), for which given boundary conditions have to be satisfied. The operators thus depend on part of *u of the boundary :. The domains of these three operators, gradu, curlu and divu (the index u could be omitted in the following sections), are respectively: Eu0(:) = { u L2(:) ; grad u L2(:), u~*u = Cte },
[2.64]
Eu1(:) = { u L2(:) ; curl u L2(:), n u~*u = 0 },
[2.65]
Eu2(:) = { u L2(:) ; div u L2(:), n. u~*u = 0 }.
[2.66]
The domains of the operators have been built in a way to satisfy the relations: gradu Eu0 Eu1,
curlu Eu1 Eu2 ;
i.e. cod(gradu) dom(curlu) and cod(curlu) dom(divu) ; we have Eu3
cod(divu).
This is indeed the case since, thanks to the introduced boundary conditions, we have the implications: u~*u = Cte n grad u~*u = 0, n u~*u = 0 n. curl u~*u = 0.
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The Finite Element Method for Electromagnetic Modeling
Thus, the operators “bind” the function spaces Eup, p = 0, 1, 2, 3, between them in order to form the series denoted: gradu curlu divu Eu 0 o Eu1 o Eu 2 o Eu 3 .
Generally, we have the inclusions: gradu Eu0 ker(curlu),
curlu Eu1 ker(divu),
but in the case where : and *u are connected and simply connected, these inclusions become equalities and the sequence is said to be exact, i.e. gradu Eu0 = ker(curlu),
curlu Eu1 = ker(divu).
The fields u and v (respectively u and v) highlighted in [2.62] (respectively [2.63]) belong to function spaces such as previously defined Eu0, Eu1, Eu2 and Eu3, included in the spaces H1(:) and H1(:)). For each of these spaces corresponds a particular succession of spaces, which lead us to define two sequences of distinct spaces, each sequence being associated with particular fields and particular boundary conditions. The operators intervening in each Green formula, pertaining to each sequence, are said to be adjoint to one another [MOR 53]. 2.2.3. Tonti diagram: synthesis scheme of a problem The two sequences defined in the previous section are characterized by four pairs of function spaces, in duality, and three pairs of operators, mutually adjoint, organized according to two structures. Gathering these two structures can be done thanks to a diagram called the Tonti diagram [BOS 88, BOS 91] (Figure 2.4).
Figure 2.4. Tonti diagram representing the two structures in duality
This global structure can accommodate a large variety of models with partial derivatives. The boundary conditions are taken into account at the level of the definition of the differential operators of the sequence. It has to be noted that the
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Tonti diagram schematized previously can be generalized to the problems of temporal evolution by adding a third dimension to it. The Tonti diagrams of the three considered static problems are represented in Figures 2.5 to 2.7. We analyze hereafter the diagram of the magnetostatics problem; the analysis of the other diagrams can be carried out by analogy. For the magnetostatics problem, the fields h, j and b can be accommodated by a mathematical structure resulting from the defined double structure (the indices u and v of function spaces will here be h and b, i.e. the indices of the complementary boundaries where the boundary conditions are given [2.35-36]). It is a question of seeking h in Eh1 and b in Eb2 verifying curl h = j and div b = 0, j being in Eh2. The defined function spaces are adapted to these fields. They constitute the domains of definition of the operators that can be applied to them. In addition, they take into account boundary conditions [2.35-36]. Note that the physical constraints of finite energy are also assured. The equations, the behavior laws and the boundary conditions of the magnetostatics problem can then be grouped together in Figure 2.7. The fields h, j and b are laid out at the suitable levels of the diagram and the equations which connect them, as well as the magnetic behavior law, are then highlighted. The equations are to be read “vertically” on the two sides of the diagram and can be regarded as being purely geometric. The behavior law should be read “horizontally”. It defines the physical properties of the matter and is thus not a geometric concept. Potentials are also introduced. The field h indeed can be, under certain conditions, derived from a magnetic scalar potential M and the field b can be derived from a magnetic vector potential a. These potentials have a position predetermined in the Tonti diagram: they are placed naturally downstream from the fields which can derive from it, i.e. on the lower floor in the corresponding sequence. The source field hs being part of the definition of h [2.46] also has its place in the diagram.
Figure 2.5. Tonti diagram of the electrostatics problem
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The Finite Element Method for Electromagnetic Modeling
Figure 2.6. Tonti diagram of the electrokinetics problem
Figure 2.7. Tonti diagram of the magnetostatics problem
2.2.4. Weak formulations 2.2.4.1. General principle: weak formulations of the electrokinetics problem In order to illustrate the concept of weak formulation, we consider the electrokinetics problem, limited to the domain :, for which equations are [2.27-2829], i.e. curl e = 0, div j = 0, j = V e, and for which the boundary conditions on complementary portions *e and *j are [2.30-31], i.e. n e~*e = 0, n. j~*j = 0. This initial form of the problem is that which we considered until now and constitutes its strong formulation. Let us consider the Green formula of the grad-div type in : [2.62] applied to field j and to scalar field v’ to be defined, i.e. (j, grad v’): + (div j, v’): = < n. j, v’ >*, ґ v’ Ee00(:),
[2.67]
where Ee00(:) is a space of type [2.64] where v~*e = 0. The last term of equation [2.67] is then reduced < n.j, v’ >*j and becomes zero if condition [2.31] is introduced there. Similarly, the second term of this equation becomes zero when equation [2.28] is introduced there. Equation [2.67] is then reduced to: (j, grad v’): = 0, ґ v’ Ee00(:).
[2.68]
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It is this form that is called weak formulation (here from equation [2.28]) [BRE 83]. We established it on the basis of a Green formula but we can now consider it as a form posed a priori for then deducing the information it contains. Actually, the weak formulation [2.68] contains at the same time equation [2.28] and boundary condition [2.31]. Indeed, by applying to it the Green formula of the grad-div type [2.62], we obtain: (div j, v’): = < n. j, v’ >*, ґ v’ Ee00(:).
[2.69]
This equation is verified for any function v’ Ee00(:), called a test function, and thus in particular for any function v’ of zero trace on *. From this it results that (div j, v’): = 0 for any function v’ of this kind and consequently that div j = 0 in :, i.e. equation [2.28] is checked. Thus, equation [2.69] is reduced to
* = 0, and by now considering all the functions v’ Ee00(:) without restriction, which can thus vary freely on *j, it finally follows that n.j~*j = 0, i.e. condition [2.31] is checked. From this it results that the weak formulation [2.68] involves the verification, with weak direction, from equation [2.28] and from boundary condition [2.31]. At the discrete level, this verification with the weak direction will correspond to an approximation. The higher the dimension of the space of test functions, the better the approximation. It is possible to obtain even more information from the weak formulation, particularly with regard to the conditions of transmission which appear on the interior surfaces to :. Let us then consider two sub-domains :1 and :2 of : separated by an interface 6 (Figure 2.8). : n
:1 6
:2
Figure 2.8. Interface 6 between two sub-domains :1 and :2
Let us apply the Green formula of the grad-div type [2.62] to the fields j and v’ successively in the two sub-domains :1 and :2. By considering the fact that div j = 0 in :, and hence particularly in :1 and :2, then by summing the obtained relations, we finally obtain the relation: (j, grad v’):1:2 = < n. (j1 – j2), v’ >6 + < n. j, v’ >w(:1:2), ґ v’ Ee00(:), [2.70]
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The Finite Element Method for Electromagnetic Modeling
where j1 and j2 represent the field j on both sides 6 in the respective domains :1 and :2. By considering the test functions v’ of support :1:2 and zero on w(:1:2), there remains from [2.70] the well-known condition of transmission n. (j1 – j2)~6 = 0, which thus appears checked with the weak formulation [2.68]. Note that the first term of [2.70] is zeroed thanks to equation [2.28]: the integration domain :1 :2 can indeed be extended to : thanks to the selected test functions. Finally, while carrying in [2.68] the behavior law [2.29] and by using definition [2.40] of the scalar potential v Ee0(:), we have: (V grad v, grad v’): = 0, ґ v’ Ee00(:),
[2.71]
which is the electrokinetic formulation in scalar potential. This formulation contains problem [2.27-31] in its totality. The potential v is the unknown factor and the other fields can be deduced from v thanks to the equations which remained in strong form. It appears that the potential v belongs to the same space as the test functions or at least to a space Ee0(:) which is parallel to it, i.e. where the boundary condition of v on *e is not necessarily homogenous i.e. v~*e = Cte. In general, the boundary condition arising from an integral term in the weak formulation (here, n.j~*j = 0) is called a natural condition (Neumann condition), whereas the condition expressed in a function space that is directly used for the expression of the unknown factor and the test function (v~*e = Cte in Ee0(:)) is called an essential condition (Dirichlet condition). Note that taking into account one natural non-homogenous boundary condition, here n.j~*j = n.js~*j, would result in extending [2.68] in the form: – (j, grad v’): + < n. js, v’ >*j = 0, ґ v’ Ee00(:),
[2.72]
and then, [2.71] would become: (V grad v, grad v’): + < n. js, v’ >*j = 0, ґ v’ Ee00(:).
[2.73]
A weak formulation can be regarded as a system of an infinite number of equations with an infinite number of unknown variables. We will see in subsequent sections how it is possible to approximate such a problem in order to allow its numerical resolution. This approximation will constitute the discretization phase. Note that it is reducing the number of test functions to a finite value which is responsible for the approximate character of the solution; the solution of a weak formulation being, at the continuous level, the same as that of the associated strong formulation.
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While dealing with the electrokinetics problem, a weak formulation can be established at the beginning of equation [2.27], i.e. curl e = 0, by considering the Green formula of the curl-curl type [2.63]. We have: (e, curl t’): = 0, ґ t’ Ej1(:).
[2.74]
By carrying in this equation the behavior law [2.29] and by using definition [2.54] of the electric vector potential t Ej1(:), we obtain the electrokinetic weak formulation in electric vector potential, (V–1 curl t, curl t’): = 0, ґ t’ Ej1(:).
[2.75]
2.2.4.2. Weak formulations of the considered problems The weak formulations of the studied static problems are given hereafter, in terms of scalar and vector potentials. In order to introduce the concept of global constraints within the weak formulations, the natural boundary conditions are considered non-homogenous. The fields ds, es, js, hs and bs, involved in these formulations are known fields a priori defined in volume or in surface; the fields defined in volume are source fields whereas those defined in surface relate to the natural boundary conditions.
Electrostatics formulation in electric scalar potential (H grad v, grad v’): + < n. ds, v’ >*d – ( U, v’ ): = 0, ґ v’ Ee00(:).
[2.76]
Electrostatics formulation in electric vector potential (H–1 curl p, curl p’): + (H–1 ds, curl p’): + < n es, p’ >*e = 0, ґ p’ Ed01(:).
[2.77]
Electrokinetics formulation in electric scalar potential (V grad v, grad v’): + < n. js, v’ >*j = 0, ґ v’ Ee00(:).
[2.78]
Electrokinetics formulation in electric vector potential (V–1 curl t, curl t’): + < n es, t’ >*e = 0, ґ t’ Ej01(:).
[2.79]
Magnetostatics formulation in magnetic scalar potential (– P grad M, grad M’): + (P hs, grad M’): – < n. bs, M’ >*b = 0, ґ M’ Eh00(:).
[2.80]
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The Finite Element Method for Electromagnetic Modeling
Magnetostatics formulation in magnetic vector potential (P–1 curl a, curl a’): + < n hs, a’ >*h = (js, a’):s, ґ a’ Eb01(:).
[2.81]
Volume source terms of equations [2.76] and [2.81] can be also written in the respective forms (making use of source fields): (U, v’): = (div ds, v’): = (ds, – grad v’):,
[2.82]
(js, a’):s = ( curl hs, a’ ):s = (hs, curl a’):s.
[2.83]
2.2.4.3. Global constraints Let us reconsider the electrokinetics problem in order to introduce the treatment of the global constraints. For the formulation in scalar potential, an essential condition can fix the potential at a constant value on a surface *i. This surface can be associated with a perfect current supply conductor. The constant value of the potential, if it is unknown, constitutes what we call a floating potential. It is then a question of fixing a condition on the current crossing this surface, or of defining a relation potential – current in the case of a coupling with an external circuit. In both cases, we will have global variables of potential and current types, which are closely dependent. Taking into account the potential (global, i.e. constant), when it is known, is not difficult to handle since it can be done directly by an essential condition. On the other hand, when the current Ii through a surface *i is fixed by a condition of the form of [2.18], it has to be noted that, for a constant and unitary function v’ on *i, the surface term < n.j, v’ >*i in [2.78] is indeed equal to current Ii [DUL 98]. Such a function will thus have to be able to be defined as a basis function of the unknown v, in particular at the discrete level. If the potential and the current are both unknown, it is a question, so that the problem is correctly posed, of taking into account a relation binding these two variables. This is the case within the framework of a coupling with external circuits. For the vector potential formulation, these specific global variables can still be considered. Nevertheless, the essential constraint concerns the current by fixing the circulation of the vector potential, i.e.
³*i n j ds ³*i n curl t ds v³w*i t dl Ii , by application of the Stokes theorem (w*i is the contour of *i).
[2.84]
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91
Constraint [2.20], consisting of imposing a potential difference, is associated with the term < n es, t’ >*e in [2.79] and is thus described as natural; the potential difference thus appears as the dual global variable of the current. Surface *e to be considered is the lateral surface of the conductor, i.e. of type *j noted *e’. We can show that the surface term < n es, t’ >*e’ is then, for a function t’ of unit circulation along a contour w*i of any surface *i, equal to a potential difference [DUL 98]. Indeed, for n t’ = – n grad c’, with c’ defined on *e’, made simply connected by the introduction of a cut Ji and undergoing a unit discontinuity 'c’ through cut Ji, we have: n es , t ' ! *e ' n es , grad c' ! *e ' grad c'e s , n ! *e ' curl(c' es ), n ! *e ' c' curl es , n ! *e ' .
Then, by using the Stokes formula for the first integral and knowing that the second integral becomes zero for a well defined field es, we finally obtain: n es , t ' ! *e '
³w*
e'
c' es dl
³J
i
'c' es dl
³J
i
e s dl
Vi .
[2.85]
By analogy, terms < n. ds, v’ >*d in [2.76], < n es, p’ >*e in [2.77], *b in [2.80] and < n hs, a’ >*h in [2.81] will be, for test functions (and integration surfaces) judiciously selected, respectively an electric charge, an electromotive force, a magnetic flux and a magnetomotive force [DUL 98].
2.3. Discretization of function spaces and weak formulations 2.3.1. Finite elements A finite element is defined by the triplet (K, PK, 6K) where: – K is a domain of space called a geometric element (generally of a simple form, for example a triangle or a quadrangle, in 2D, and a tetrahedron, a hexahedron or a prism, in 3D); – PK is a function space of finite dimension nK, defined on K; – 6K is a set of nK degrees of freedom represented by nK linear functionals Ii, 1 d i d nK, defined on the space PK and with values in IR.
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The Finite Element Method for Electromagnetic Modeling
Moreover, it is necessary that an unspecified function u PK can be determined in a single way thanks to the degrees of freedom of 6K. This last condition defines the unisolvance of the finite element (K, PK, 6K) [DAU 88]. The role of a finite element is to interpolate a field in a function space of finite dimension, and this, locally, and generally in a space field of simple topology, called a geometric element. Several finite elements can be defined on the same geometric element. We can then speak, under certain conditions, of mixed finite elements. Figure 2.9 illustrates various spaces which intervene in the definition of a finite element; the definition of the subset of points N K is associated with that of the functional. f PK
K = dom(f)
N
cod(f) f(x)
x
IR
I i(f)
Figure 2.9. Spaces associated with a finite element
For the most frequently used and most well known finite elements, the degrees of freedom are associated with the K nodes and the functionals Ii are reduced to functions of K coordinates; they are called nodal finite elements. However, the definition presented here is more general thanks to the freedom left in the choice of the functionals. Other forms of functionals will be defined in the following sections. They could be, in addition to nodal values, integrals along segments on surfaces and volumes. These forms can be well adapted to various types of fields to interpolate. It is a question of noticing that the subset NK (Figure 2.9) can then be brought back respectively to a point, a segment, a surface or a volume. The interpolation of a function u, in the space PK and on K, is given by the expression:
u K ( x)
nK
¦ a i p i ( x) , i 1
u K PK , x K ,
[2.86]
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93
where the nK coefficients ai of basis functions pi PK can be given by solving the linear system: I j (u )
nK
¦ a i I j (pi ) ,
j 1, ..., n K ,
i 1
provided that the function u is sufficiently regular so that the Ij(u), j = 1, …, nK, exist. This solution is simplified to the maximum if we define the functionals so that: I j (p i )
Gij , 1 d i, j d n K ,
where Gij is the Kronecker symbol, i.e. Gij = 1 if i = j and Gij = 0 if i z j. The matrix of the system is then the unit matrix and the solution is: aj
I j (u ) , j 1, ..., n K .
In this case, the interpolation uK PK is expressed by: u K ( x)
nK
¦ I j ( u ) p j ( x) ,
u K PK , x K ,
[2.87]
j 1
where the coefficients Ij(u) = Ij(uK), 1 d j d nK, are called degrees of freedom. 2.3.2. Sequence of discrete spaces 2.3.2.1. Shape functions (nodal, edge, facet, volume) The construction of finite element spaces especially requires a space discretization, or mesh, of the studied field. We work in 3D and use for that three types of geometric elements: tetrahedrons, hexahedrons and prisms with a triangular base; the exploitation in 2D is done without difficulty. Subsequently, functions or vector fields associated with various geometric entities with the mesh (nodes, edges, facets and volumes) are defined. These constitute bases for approximation spaces Wi, i = 0 to 3, which are discrete analogs of spaces Ei, i = 0 to 3.
Notations and definitions Let us consider the mesh of a field, carried out by assembling geometric elements which can be tetrahedrons, hexahedrons and prisms with a triangular base, as is shown in Figure 2.10. These elements are called volumes and their vertices
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The Finite Element Method for Electromagnetic Modeling
constitute the nodes. The sets of nodes, edges, facets and volumes of this mesh are denoted N, A, F and V, respectively. In a mesh, two unspecified geometric elements have in common either a facet, an edge, or a node, or are disjoined. We indicate the ith node of the mesh by ni. If there is no possible ambiguity, in the following, a node ni will be located only by its index i, with the proviso of defining its membership to the set of nodes, i.e. iN.
Figure 2.10. Assembling different geometric elements
The edges and the facets can be defined by ordered sets of nodes. Thus, we indicate an edge by aij, a triangular facet by fjkl, and a quadrangular facet by fjklm. Let us now consider an edge aij whose node j belongs to a facet but not node i. If this facet is triangular and composed of nodes j, k and l, it will be noted fjkl. However, it can be defined, for a given element, by the edge aij and will be denoted f i j . As an example, Figure 2.11 illustrates a triangular facet fjkl which can also be located with notation f i j . Thereafter, {j, i } will designate the set of nodes of a defined facet, for a given element, by edge aij.
l j
f ij
k
i Figure 2.11. Definition of a triangular facet with notation f i j
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2.3.2.1.1. The nodal functions We call a nodal function wni(x) a function which is equal to 1 for x = xi (with xi coordinates of node i), which varies continuously in the elements having node i in common and which is equal to zero in the other elements without undergoing discontinuity [DHA 81], i.e. Gij , i, j N,
w n i (x j )
[2.88]
where Gij represents the Kronecker symbol. With the nodal functions thus defined, we can check in each point of the study domain the relation:
¦ w n i (x) 1 .
[2.89]
i N
In order to reduce the notations, the nodal functions will be noted, thereafter, wn or wni. The set of functions wn, n N , generates the spaced denoted W0. 2.3.2.1.2. The edge functions In the general case, to edge ak,n of a mesh and for a given element, we associate the vectors fields w a k, n defined by [DUL 94]. w a k, n
w n k grad(
¦ w n i ) w n n grad( ¦ w n j ) .
i ^k, n `
^ `
[2.90]
j n, k
This vector field is zero in all the non-adjacent elements with edge ak,n. The space of vector fields generated by wa, a A , will be denoted W1. 2.3.2.1.3. The facet functions Let us take a triangular facet which will be denoted fk,l,n and the set of associated nodes {k, l, n}, {q1, q2, q3} or, as we defined previously, ^k , n` , ^q i , q i r 1 `. With this definition, in the general case, we associate with a facet of an unspecified element (tetrahedron, hexahedron or prism) the vector field [DUL 94]. wf
nf
a f ¦ w q c grad( c 1
¦
¦
w n i ) grad( wn j ) , i^q c , q c 1 ` j^q c , q c 1 `
[2.91]
where nf represents the number of nodes of facet f, and af is a coefficient which is equal to 2 for nf = 3 and 1 for nf = 4. In order to have a rotation in the indices, q n f q 0 and q 1 q n f 1 will be considered. This vector field is zero for all the
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The Finite Element Method for Electromagnetic Modeling
non-adjacent elements with facet f. The set of the vector fields wf, f F , generates the space W2. 2.3.2.1.4. The volume functions With a geometric element of the mesh we associate the scalar function wv defined in the following way: wv
1 vol(v)
,
[2.92]
where vol(v) represents the volume of the element considered. This function is zero for all the elements other than element v. W3 is denoted as the space generated by the functions wv, v V . 2.3.2.1.5. Geometric interpretation In the previous sections, we have introduced the four basis functions relating to the finite elements. For the nodal and volume functions, geometric interpretation does not present a major difficulty. However, this is not the case for the edge and facet. Also, in the following section, we will carry out a geometric interpretation of their vector field [DUL 94, DUL 96]. Although the study can be undertaken in the same way, in the case of hexahedrons or prisms, in order to simplify matters, we will limit ourselves to the case of tetrahedrons. Let us consider the tetrahedron of Figure 2.12 where the nodes are noted k, l, m and n and the nodal approximation function is noted w n i , i{k, l, m, n}. At any point x, belonging to the facet made up of the nodes k, l, m, we have, on the basis of property of the nodal functions [2.89], the relation:
¦
i ^k, l, m`
w ni
1.
[2.93]
Taking into account the notations presented above, the set {k, l, m} of the nodes of the facet could also be defined by: {k, n }, {l, n } or {m, n }. Let us take the case where the nodes of the facet are defined by edge n, k which gives {k, n }. A vector field is now introduced which is associated with this facet and defined by the gradient of equation [2.93]. Defined as u k, n , it has as an expression: u k, n
grad ¦
i^k, n `
w ni .
[2.94]
Given equation [2.93], this vector field is collinear to the normal direction of the facet. Consequently, its circulation is equal to zero on the three edges.
Static Formulations
97
n
k l
m
Figure 2.12. Studied tetrahedron: definition of the nodes, edges and facets
If expression [2.94] is weighted by the nodal function wnk, a vector field is obtained whose circulation is zero on all the edges of the mesh except for edge ak,n. An equivalent expression to that of relation [2.94] allows the vector field u n, k to be expressed whose circulation is also equal to zero on the edges of the facet made up of nodes {n, m, l}. This expression, weighted by the nodal function w n n , led to a vector field whose circulation is non-zero only on edge ak,n. By writing a linear combination of these two vector fields, we find the basic expression of the edge functions [2.90]. Taking into account equation [2.89], the edge functions are expressed, for tetrahedrons, in the form [BOS 88]: w ak,n
w nk grad w n n w nn grad w nk .
[2.95]
It can be shown that the non-zero circulation on edge ak,n is equal to 1. It was seen that by using expression [2.94] it is possible to define an orthogonal field to a facet. For a given element, let us consider the vector product of two fields built in this way. A third vector field is then obtained, directed tangentially with respect to the normal direction of the two facets. In the case of the tetrahedron of Figure 2.12, if the vector product of the fields u n, m and u n, l is considered, the flux of the vector field thus obtained is zero on the facets defined by nodes {n, k, l} and {n, m, k}. Moreover, if the field obtained is weighted by the nodal function wnk, being given its properties, the flux through the facet made up of the nodes {n, l, m} is also zero. We have thus built a vector field whose flux, through all the facets, is equal to zero except for facet {k, m, l}. It is possible to build, by circular permutation, two other vector fields having this property. The sum of these three vector fields leads, for tetrahedrons, to the basis facet functions. A coefficient of 2 allows the flux through the facet considered to be normalized to 1. Relation [2.91] is then obtained. Lastly, taking into account properties [2.89] of the nodal functions
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The Finite Element Method for Electromagnetic Modeling
and considering the facet of the tetrahedron in Figure 2.12, made up of nodes {k, l, m} [BOS 88], the following expression is obtained: wf
2(w n k gradw n m gradw n l w n l gradw n k gradw n m
[2.96]
w n m gradw n l gradw n k ).
This equation can be obtained from the general relation given by expression [2.91]. 2.3.2.1.6. Proprieties of basis functions In addition to the properties already exposed in the previous section, we will present the continuity properties of the basis functions. The nodal functions wn are continuous through the facets of the elements. This property can be checked using relation [2.89]. With regard to the edge functions wa, it is the tangential component which is continuous through the facets of the mesh. For the functions of facets wf, it is the perpendicular component which is preserved at the interface between two elements. Lastly, the volume functions wv are discontinuous from one element to another. The set of the main properties of the basis functions is summarized in Table 2.1. Functions wn wa wf wv
Properties
Continuity
Generated space
continuous
W0
³ w b . dl Gab
wa n
W1
³ w g . ds Gfg
wf. n
W2
³ w edW G ve
discontinuous
W3
w n i (x j )
Gij
a
f
v
Table 2.1. Properties of basis functions
2.3.2.1.7. Decomposition of physical variables Starting from the properties of the basis functions, summarized in Table 2.1, the physical variables relating to Maxwell’s equations as well as the scalar and vector potentials which result from this can be decomposed in spaces Wi, i = 0 to 3.
Static Formulations
99
Thus, the scalar potentials v and Mѽ will be decomposed in W0. As an example, we can write, for the magnetic scalar potential, M
¦ w n Mn ,
[2.97]
n N
where Mn represents the value of the magnetic scalar potential to node n of the mesh. The vector fields h and e, as well as the vector potentials hs, p, t and a, will be decomposed in the space W1. Under these conditions, to illustrate this decomposition, there will be for the magnetic field h: h
¦ wa ha .
[2.98]
aA
In this expression ha represents the circulation of the magnetic field on the edge a of the mesh. For the vector fields j, b, d and the potential ds, the decomposition will be performed in the space W2. The current density thus discretized can be written: j
¦ wf
jf ,
[2.99]
f F
where the terms in jf represent the flux of the current density through the facets of the mesh. Lastly, volume density of load U will be decomposed in W3 and we will have: U
¦ w v Uv ,
[2.100]
vV
where Uv represents the integral on the element v of the volume density of load. 2.3.2.2. Properties of discrete spaces 2.3.2.2.1. Incidence matrices In this section we will build incidence matrices which will enable us to apply the gradient, curl and divergence operators to the functions belonging respectively to spaces, W0, W1 and W2. For this study, we will define as a preliminary the concept of incidence [BOS 91]. The incidence of a node n on an edge a will be denoted i(n, a). It will be equal to 1 if the node is the end of the edge, –1 if it is the origin and 0 if the node does not belong to the edge. In the same way, we can introduce the incidence i(a, f) of an edge on a facet. It is equal to r1 if the edge belongs to the facet and 0 in the contrary case. The sign depends on the orientation of the edge, arbitrarily defined, with
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The Finite Element Method for Electromagnetic Modeling
respect to that built by circular permutation of the nodes of the facet. Lastly, the incidence i(f, v) of a facet on a volume is also defined. If the facet belongs to the element, the incidence is 1 and 0 if it does not belong to the element. f1 = ^n1, n2, n3 `
n4
f2 = ^n1, n3, n4 ` f2
a3 n1
a6
a1
f4 n3
a2 f1
f3 = ^n1, n4, n2 `
f3
f4 = ^n2, n4, n3 `
a5 a1 a4
n2
Figure 2.13. Definition of nodes, edges and facets of a tetrahedron
To clear up these definitions let us take the case of the tetrahedron of Figure 2.13. The nodes are numbered from n1 to n4, the edges from a1 to a6 and the facets from f1 to f4. The edges are oriented arbitrarily in the direction of the node of lower index (origin) towards the node of higher index (end). The facets are defined by their normal whose orientation is arbitrarily chosen while following the increasing numbering of the nodes. As an example we have, for the node n2 and the edge a4, i(n2, a4) = –1, and for the edge a6 and facets f2, i(a6, f2) = 1. These various incidences allow the construction of three matrices: edges-nodes GAN, facets-edges RFA and volumes-facets DVF whose elementary terms are defined by: Ga,n = i (n, a), a A and n N,
[2.101]
Rf,a = i (a, f), f F and a A,
[2.102]
Dv,f = i (f, v), v V and f F.
[2.103]
The three matrices thus defined are respectively of AuN, FuA and VuF. dimension. As an illustration, we represented in Tables 2.2, 2.3 and 2.4, again for the tetrahedron of Figure 2.13, the incidence matrices GAN, RFA and DVF respectively.
Static Formulations
Ga,n
n1
n2
n3
n4
a1
–1
1
0
0
a2
–1
0
1
0
a3
–1
0
0
1
a4
0
–1
1
0
a5
0
–1
0
1
a6
0
0
–1
1
Table 2.2. Example of matrix GA,N for the tetrahedron
Rf,a
a1
a2
a3
a4
a5
a6
f1
1
–1
0
1
0
0
f2
0
1
–1
0
0
1
f3
–1
0
+1
0
–1
0
f4
0
0
0
–1
+1
–1
Table 2.3. Example of matrix RF,A for the tetrahedron
Dv,f
f1
f2
f3
f4
v
1
1
1
1
Table 2.4. Example of matrix DV,F for the tetrahedron
101
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The Finite Element Method for Electromagnetic Modeling
2.3.2.2.2. Equivalent discrete operators In this section we will show that the incidence matrices can be used as discrete operators. Let us take the case, in electrostatics, of electric scalar potential v. Decomposed in W0, it is written in the form: v
¦ w n vn .
[2.104]
nN
The electric field decomposed in W1 is written e
¦ w a ea .
[2.105]
aA
However, it is shown that [BOS 91] gradw n
¦ G a, n w a .
[2.106]
aA
Under these conditions, by introducing the potential into the expression of the electric field, we obtain: e
¦ w a ea ¦ ¦ G a,n w a v n .
aA
[2.107]
nN aA
This relation highlights the existing link between the circulation of the electric field along the edges of the mesh and the nodal values of the electric scalar potential. A well-known property is thus found. From this relation, the vector of circulations of the electric field is expressed according to the vector of the nodal values of the scalar potential in the matrix form: ea = – GAN vn.
[2.108]
It is deduced from this equation that matrix GAN is the discrete equivalent of the operator gradient. Moreover, it can be shown that the gradient of a function belonging to W0 is included in W1, thus: grad (W0) W1.
[2.109]
It should be noted that this inclusion becomes an equality in the case of simply connected domains.
Static Formulations
103
Let us now consider a vector field belonging to W1 and let us take, for example, the case of the magnetic field h. This is written in the form:
¦ waha .
h
[2.110]
aA
The current density which is expressed by the rotational of h is decomposed in W2 and is written:
¦ w f jf .
j
[2.111]
f F
However, we have the following attribute: ¦ R f,a w f .
curl w a
[2.112]
f F
Hence, by introducing the expression of the magnetic field, we obtain:
j
F
¦ w f jf f
¦ ¦ R f,a w f h a .
[2.113]
aA f F
This expression shows us that the vector of the discrete values of the current density is expressed as a function of matrix RFA and of the discrete values of the magnetic field, thus: jf = RFA ha.
[2.114]
It is hence deduced that the facets-edges incident matrix RFA is the discrete equivalent of the rotational operator. Moreover, starting from equation [2.112], we have: curl (W1) W2.
[2.115]
On the other hand, since in the continuous field we have curl(grad v) = 0, we can check the property: RFA GAN = 0.
[2.116]
Now let us consider the divergence of the current density which we know is equal to zero. It is expressed in the form: div j
¦ div w f jf
f F
0 .
[2.117]
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The Finite Element Method for Electromagnetic Modeling
However, as for the other operators, there is a relation between the divergence operator and the volumes-facets incidence operator, that is to say: div w f
¦ D v,f w v .
[2.118]
vV
While replacing in equation [2.117], we obtain: div j
¦ ¦ D v,f w v jf 0 .
[2.119]
f F vV
This expression can be written in the following matrix form where DVF is the discrete equivalent of the divergence operator: DVF jf = 0.
[2.120]
We can also show that: div (W2) W3.
[2.121]
Lastly, the vector relation div curl u = 0 finds its equivalent in the discrete domain, thus: DVF RFA = 0.
[2.122]
From the three relations [2.106], [2.112] and [2.118], it is shown that spaces Wi, i = 0 to 3, form a discrete sequence represented in Figure 2.14. grad curl div o W1 o W 2 o W3 W0
Figure 2.14. Sequence of discrete spaces Wi
Lastly, in Table 2.5, for the various types of geometric elements, a summary of associated spaces, discrete operators and discrete variables are shown.
Static Formulations Type of element
Space W0
Discrete operator GAN
v, M
RFA DVF
h, e, a, hs, t, p
facets
W1 W2
volume
W3
_____
U
nodal edges
105
Discrete variables
d, j, b, ds
Table 2.5. Summary on discrete spaces
2.3.2.2.3. Discrete Tonti diagram The Tonti diagrams presented previously can contain discrete function spaces, such as defined, still divided into two sequences; the spaces Wui will be the discrete equivalents of Eui, with i = 0, 1, 2, 3. For the magnetostatics problem, for example, if we seek fields h Wh1 and b Wb2, where the spaces Whi and Wbj are the discrete equivalents of Ehi and Ebj, it is clear that the behavior law cannot be verified exactly. Indeed, the approximations Wh1 and Wb2 are generally different. Therefore, no relation of the type of the behavior law between elements of these sets can exist. In reality, since we discretize, an approximation error must appear: it will be located in the constitutive relations, which can only be satisfied “at best”, i.e. weakly. On the other hand, equations [2.32]-[2.33] could be satisfied exactly for the approximate fields if, however, those belong to approximation spaces. Now let us choose to consider the magnetostatics problem in two different ways. Firstly, let us take h in Wh1 and j in Wh2 so that equation [2.32] can be satisfied exactly, and satisfying exactly behavior law [2.34]. This is the equivalent of putting b in Wh1 and thus does only allow equation [2.33] to be weakly formulated. This method defines the h-conform formulation. Secondly, let us take b in Wb2 so that equation [2.33] can be satisfied exactly, and still satisfying exactly the behavior law [2.34]. This time, this is the equivalent of putting h in Wb2 and thus does only allow equation [2.32] to be weakly formulated. This method defines the b-conform formulation. Each one of these two methods has its advantages. The choice of one of them will be made according to the type of desired conformity. The h formulation allows the Ampere law to be satisfied exactly, whereas the b formulation allows the flux conservation law to be satisfied exactly.
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The Finite Element Method for Electromagnetic Modeling
2.3.3. Gauge conditions and source terms in discrete spaces 2.3.3.1. Gauge conditions As we indicated in section 2.1.5, in order to define a vector field in a single way, it is necessary to know its rotational, divergence and boundary conditions. If one of the two operators is not fixed, it is necessary to impose a gauge condition. In the following sections, we will see that, in the discrete domain, the non-uniqueness of the solution is highlighted using the incidence matrices and that it is possible to impose a gauge condition by using a tree technique. Subsequently, the simply connected domain is considered with a connected boundary. As an example, let us take the magnetostatics case and the source field hs defined by relation [2.46]. In a discrete form, we have: jf = RFA hsa,
[2.123]
with F < A. If the vector jf is known, this relation shows that there is an infinite number of solutions for hsa. The rank of matrix RFA being equal to A – (N – 1), it is possible to fix N – 1 arbitrary values of circulation, which are judiciously chosen. It should be recalled that the relation above imposes to hsa to verify the Ampere theorem on all facets of the mesh. In the same way, for the divergence operator, the rank of incidence matrix DVF is equal to F – (A – (N – 1)). Under these conditions, if we look for a vector field, in the discrete domain, defined only by its divergence, it will be necessary to fix A – (N – 1) values of its flux through facets. Lastly, it is to be noted that the rank of matrix GAN is equal to N – 1. It is thus verified that in order to solve a problem in scalar potential it is necessary to fix a value of the potential in a node of the mesh. In the case of the RFA matrix, to fix N – 1 circulations on the edges of the mesh, a tree technique is generally used. It should be recalled that, for a given mesh, the number of edges of the tree and co-tree is equal to N – 1 and A – (N – 1) respectively. Moreover, the tree allows all the nodes to be linked to each other without forming a closed loop. At this stage, we can note the existing analogy with the vector field w of gauge hs.w = 0. Consequently, to impose this gauge in the discrete domain, the circulation values are fixed at 0 on the edges of the tree. Lastly, for the boundary condition n hs |*h = 0, h = 0, it is enough to begin building the tree on *h before extending it to the whole domain. The same technique presented for the source field could be used for potential vectors p, t and a which are defined only by their rotational and their boundary
Static Formulations
107
conditions. As an indication for the mesh in Figure 2.10, we give in Figure 2.14 an example of a tree and co-tree.
Figure 2.14. Example of a tree and co-tree
With regard to the divergence matrix, let us consider the electrostatics case with the definition of the source electric flux density ds which is written, in the discrete domain, in the form: DVF dsf = Uv,
[2.124]
where dsf represents the vector of the flux values of the source electric flux density through the facets of the mesh and Uv the vector of the quantity of charges in the elements. Generally the distribution of charges is known and we seek to determine vector dsf. In order to make the system invertible, A – (N – 1) flux values through the facets must be imposed. However, for the rotational operator we fixed circulations on the edges of a tree. Under these conditions, we will build a tree of facets to fix degrees of freedom. To conclude such a construction, the links between facets and elements are transposed to the cases of links between edges and nodes [LEM 99]. Indeed, a facet connects two elements like an edge connects two nodes. It is then only necessary to create a graph where the elements will be nodes and the facets will be edges. Then, we build a tree and a co-tree. However, when transposed to the elements and the facets, the co-tree of the edges corresponds to the tree of facets and the tree of edges to the co-tree of facets. The procedure used is summarized in Figure 2.15.
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The Finite Element Method for Electromagnetic Modeling
elements
nodes
tree of edges
co-tree of facets
graph facets
edges
co-tree of edges
tree of facets
Figure 2.15. Construction procedure of the tree and the co-tree of the facets
In order to impose the boundary conditions of n. ds|*d = 0 type, as in the case of the source field hs, we start by building the tree on the facets belonging to *d, then extending it to the whole mesh. In conclusion, whether the vector field is defined by rotational or by divergence, it is possible to use a tree technique to impose a gauge in the discrete domain. 2.3.3.2. Discretization of source terms (local form) According to the problem to be solved, we saw in section 2.1.5 that the source terms could be the charge volume density, the current density or, when associated with potentials, the source field hs or induction ds. In this part, we will see how to discretize or calculate these various sources [LEM 98]. Concerning the charge volume density, there is no specific difficulty. For the discretization, the quantity of charges in each element is calculated. On the other hand, the discretization of the current density is much more difficult. In fact, the problem comes from the inductors which, very often, have complex forms. However, for obvious reasons of discretization, the mesh does not follow the shape of the inductor exactly. Moreover, the divergence of the current density is zero and this property must be preserved in the strong sense in the facet element space. Under these conditions, using its expression defined in the continuous domain, its transposition in the discrete domain is a delicate process. Various methods have been proposed in the literature. In the following sections, we present a technique based on the tree of facets. Let us consider an inductor of boundary *ind such as: *ind = *j *e. All the facets of boundary *ind except one, to avoid having a closed surface, will belong to the tree. On the facets belonging to *j we impose a flux equal to zero and on the facets of *e the flux of the current density. Concerning the facets of the tree located inside, the value jf is allocated to them. It is given by: jf
³ j . n ds ,
f
[2.125]
Static Formulations
109
where j represents the current density and n the normal of the facet defined in section 2.3.2.2.1. Then, we calculate by an iterative method the flux on the facets of the co-tree by imposing the zero divergence on each element. Various methods can be envisaged to determine the source field hs when the distribution of current in an inductor is known. However, as indicated in the previous section, it is possible to use the discretized current density if it is known. Initially, a tree of edges is built, on which a circulation equal to zero is imposed. Then, with an iterative method, a circulation on the edges of the co-tree is calculated while verifying the Ampere theorem. This method, which is simple and effective, allows the circulation of hsa on the edges of the mesh to be directly obtained without using finite element calculation. Knowing the discretized form of the distribution of the volume density of charges, the source vector ds of the electric flux density in the space of the facet elements can be obtained without difficulty. A tree of facets is built, on which a flux equal to zero is imposed. Then, a flux on the facets of the co-tree is calculated while verifying for each element Gauss’s theorem. As an illustration, for a coil with an iron kernel, we present in Figure 2.16a the distribution of the field of source vectors hsa obtained by a tree technique [LEM 99]. In Figure 2.16b, the rotational of the source field can be seen which corresponds to the distribution of the current density in the coil.
a)
b)
Figure 2.16. Distribution of the vector field hsa (a) and its rotational (b) [LEM 99]
2.3.4. Weak discrete formulations In this section, we will discretize the weak formulations presented in section 2.2.2. For the three static problems (electrostatics, electrokinetics and
110
The Finite Element Method for Electromagnetic Modeling
magnetostatics), we will develop the formulations in scalar and vector potential. The matrix form will be also introduced using the incidence matrices. First, we will consider the problems without global constraints. The source terms will be, according to the studied formulation, the charge volume density, an equipotential surface or a current density. Taking into account the boundary conditions, values of potentials can be fixed on a part of the boundary of the studied field (this is the case for an equipotential surface). These values then intervene as a source term in the formulation. However, in the developments presented below, they appear, for reasons of simplicity, in the support vector of the unknown terms. For the resolution, it will be necessary to rearrange the system of equations. 2.3.4.1. Formulations in scalar potential For the three static formulations of Maxwell’s equations, we will present the discretized form with a scalar potential approach. It should be recalled that, to assure the uniqueness of the solution, it is necessary to impose a constraint on the potential. In practice, this constraint can be obtained by fixing the potential in a node of the mesh. For the electrostatic formulation, the electric scalar potential v is introduced which is decomposed in the space of nodal elements W0. The charge volume density is decomposed in the space W3. Under these conditions, the discretized form of formulation [2.76], with its volume source term expressed according to [2.82], is written:
¦ ( H grad w n
jN
j
, grad w n i ) : v n j
¦ (w v
kV
k
, w n i ) : U v k , i N N *e ,
[2.126] where N*e represents the set of the nodes belonging to boundary *e on which scalar potential v is fixed by essential conditions. For the subsequent sections, the notation N’ = N – N*e will be used. By using property [2.106] of incidence matrix GAN, this equation can be put in the matrix form: t G AN' M AA G AN v n
S N'VU v ,
[2.127]
with, for the elementary terms of matrices MAA and SN’V, M al ,am
³ İ w a l . w a m dW and Sn l , v m ³ w n l w v m dW .
:
:
[2.128]
Static Formulations
111
As clarified above, in expression [2.127], the vector vn is of dimension N and includes in addition to the unknown factors the values of the scalar potential fixed on the boundary *e. These values are taken into account, as source term, by a rearrangement of the system to be solved. In the electrokinetics case, the weak discretized form of formulation [2.78] has as an expression:
¦ (V grad w jN
nj
,grad w ni ): vn j
0, i N N*e ,
[2.129]
with, for N*e, an equivalent definition to that of the electrostatic formulation. The matrix form is then deduced: t G AN' M AA G AN v n
0,
[2.130]
where matrix MAA is equivalent to that defined in electrostatics. In fact it is sufficient to replace the permittivity H by the conductivity V, thus: M a l ,a m
³ V wa . wa l
m
dW .
[2.131]
:
In the system of equations [2.130], the source terms are the scalar potential values fixed on boundary *e. For magnetostatics, the magnetic scalar potential is decomposed in the space of the nodal elements. As a source term, we use field hs decomposed in the space of edge elements W1. Under these conditions, formulation [2.80] takes the form:
¦ ( P grad w n
jN
j
, grad w n i ) : M n j
¦ (P w a
kA
k
, grad w n i ) : h sa k , i N N *h ,
[2.132] where N*h represents the set of the nodes on which an essential condition is imposed. We note N’ = N – N*h. The matrix form can then be written: t G AN' M AA G AN M n
t G AN' M AA h sa ,
[2.133]
112
The Finite Element Method for Electromagnetic Modeling
with, for matrix MAA, M a l ,a m
³ P wa . wa l
m
dW .
[2.134]
:
As an illustration, the main fields appearing in a magnetostatics formulation in scalar potential are represented in Figure 2.17 for an inductor and a magnetic core. The discontinuity of the source field hs calculated by a tree technique is apparent there, just as that of the associated magnetic scalar potential M, calculated by [2.132]. The resulting magnetic induction, obtained by a combination of these two fields, is characterized on the other hand by a continuous nature thanks to property [2.109].
a)
b)
c)
Figure 2.17. Distribution of magnetic field hs and tree of edges (a), of associated scalar potential M (b) and of resulting magnetic induction b = P h = P (hs – grad M) (c)
2.3.4.2. Formulations in vector potential For the formulations in vector potential p, t and a, the process is appreciably equivalent to that used for the formulations in scalar potential. Nevertheless, in order to ensure the uniqueness of the solution, it is necessary to impose a gauge condition; in this part, the tree technique will be used. In the electrostatics case, we introduced vector potential p and source electric flux density ds which are decomposed respectively in W1 and W2. Under these conditions, the weak formulation [2.77] is written:
(H1 curl w a j , curl w a i ): pa j (İ1 w fk , curl w a i ): dsfk , i A A '*d A tree ,
jA
kF
[2.135]
Static Formulations
113
where A’*d represents the set of co-tree edges belonging to boundary *d; Let us denote A’ = A – A’*d – Atree. Equation [2.135], written in matrix form, then takes the form: t R FA' M FFR FAp A
t R FA' M FFd sF
[2.136]
with, for matrix MFF, M f l ,f m
³İ
-1
w f l . w f m dW .
[2.137]
:
For electrokinetics, the electric vector potential t which is also decomposed in W1 is used. By using formulation [2.79], we obtain in the case of the discretized form:
¦( V jA
1
0, i A A '* j Atree ,
curl w a j ,curl w ai ): t a j
[2.138]
where A’*j represents the set of co-tree edges belonging to the boundary *j. Let us denote A’ = A – A’*j – Atree. The expression of the matrix form is: t R FA' M FF R FA t a
0,
[2.139]
with, for matrix MFF, M f l ,f m
³V
-1
w f l . w f m dW .
[2.140]
:
Lastly, for the magnetostatics formulation, the vector potential a as well as the source field hs are decomposed in W1. Under these conditions, the weak formulation [2.81], with its volume source term expressed according to [2.83], is written in discretized form:
( Q curl wai ,curl wa j ): a a j ( wa k ,curl wai ): hsa k , i A A '*b Atree ,
jA
kA
[2.141] where A’*b is also defined and represents the set of co-tree edges belonging to boundary *b. Let us denote A’ = A – A’*b – Atree. Under these conditions, the matrix form can be written: t R FA' M FFR FAa A
t R FA' M FFh sa ,
[2.142]
114
The Finite Element Method for Electromagnetic Modeling
with, for the elementary terms of matrices MFF and MFA, M f l ,f m
³ Q wf . wf l
m
dW and M f l , a m
³ wf . wa l
:
m
dW .
[2.143]
:
As is proposed in section 2.3.3, in order to impose the gauge condition, the use of a tree technique can be envisaged. However, if for solving a system of equations the conjugate gradient method is jointly used with a compatible formulation, then the problem is self-gauged [REN 96]. A formulation is said to be compatible if the matrix form of the required solution and the source term are in the kernel of the same discrete operator. Compatible forms for the discrete forms of equations [2.76] and [2.81] are obtained by expressing the volume source terms starting from the respective forms [2.82] and [2.83]. 2.3.5. Expression of global variables Taking the essential and natural global constraints into account was already considered at the continuous level. At the discrete level, it will be seen that it is useful to express the essential constraints explicitly in order to reveal the basis functions of approximation spaces with constraints. In addition to classical basis functions, which when used as test functions lead to the previously defined discrete forms, other basis functions, known as global, will lead to other equations involving global variables. In order to illustrate this procedure, the problem of electrokinetics is considered. For the formulation in electric scalar potential, the potential v is decomposed in W0, i.e. v
¦ vn w n .
[2.144]
nN
In order to reveal explicitly the constraints related to the floating potentials in [2.144], the set N of nodes of : is decomposed into two complementary subsets: the set of internal nodes to :, noted Nv, and the groups of nodes located on each electrode surface *i,f, f Cf, noted Nf ; the set of electrodes is noted Cf. Thus, considering that the potential is constant, of value vf, on each of these surfaces, [2.144] becomes: v
¦ n N
v
v n w n ¦ f C v f ¦ nN w n , f
f
[2.145]
Static Formulations
115
which can also be written: v
¦ nN
v n w n ¦ f C v f s f ,
v
[2.146]
f
where the functions sf, f Cf, constitute, with the functions wn associated with nodes of Nv, basis functions for the function space of potential. The functions sf, f Cf, are associated with each of the electrodes with a floating potential. Thus, each function sf is associated with a group of nodes, which constitutes a global geometric entity, whereas the nodes n Nv are elementary entities. The functions sf are defined, for a given electrode, as the sum of the nodal functions of all the nodes of the surface of this electrode, i.e. sf
¦ n N
f
wn .
[2.147]
This function sf is equal to 1 on electrode *i,f. Consequently, when it is used as test function v’ in [2.78], it leads to an equation, known as global, where the term < n.j, v’ >*i,f = < n.j, sf >*i,f is equal to the current Ii,f through this electrode. The global variables of circulation type are treated in a similar way, with global basis functions being sums of edge functions [DUL 98]. 2.4. References [ALB 90] A. ALBANESE and G. RUBINACCI, “Magnetostatics field computations in terms of two components vector potentials”, International Journal for the Numerical Methods in Engineering, Vol. 29, pp. 515-532, 1990. [BOS 88] A. BOSSAVIT, “Whitney forms: a class of finite elements for three-dimensional computations in electromagnetism”, IEEE Proceedings, Vol. 135, Pt. A, no. 8, pp. 493499, 1988. [BOS 91] A. BOSSAVIT, “Electromagnétisme en vue de la modélisation”, Collection Mathématiques et Applications, Springer-Verlag, 1991. [BRE 83] H. BREZIS, Analyse fonctionnelle, théorie et applications, Masson, Paris, 1983. [DAU 87] R. DAUTRAY, J.-L. LIONS, “Analyse mathématique et calcul numérique pour les sciences et les techniques”, Modèles Physiques, Vol. 1, Masson, Paris, 1987. [DAU 87B] R. DAUTRAY, J.-L. LIONS, “Analyse mathématique et calcul numérique pour les sciences et les techniques”, Transformations, Sobolev, Opérateurs, Vol. 3, Masson, Paris, 1987. [DAU 88] R. DAUTRAY, J.-L. LIONS, “Analyse mathématique et calcul numérique pour les sciences et les techniques”, Méthodes Intégrales et Numériques, Vol. 6, Masson, Paris, 1988.
116
The Finite Element Method for Electromagnetic Modeling
[DHA 81] G. DHATT, G. TOUZOT, Une présentation de la méthode des éléments finis, Maloine, 1981. [DUL 94] P. DULAR, J.-Y. HODY, A. NICOLET, A. GENON, W. LEGROS, “Mixed finite elements associated with a collection of tetrahedra, hexahedra and prisms”, IEEE Transactions on Magnetics, Vol. 30, no. 5, pp. 2980-2983, 1994. [DUL 96] P. DULAR, “Modélisation du champ magnétique et des courants induits dans des systèmes tridimensionnels non linéaires”, Collection des Publications de la Faculté des Sciences Appliquées, Liege University, 1996. [DUL 98] P. DULAR, W. LEGROS, A. NICOLET, “Coupling of local and global quantities in various finite element formulations and its application to electrostatics, magnetostatics and magnetodynamics”, IEEE Transactions on Magnetics, Vol. 34, no. 5, pp. 3078-3081, 1998. [EMS 83] C.R.I. EMSON, J. SIMKIN, “An optimal method for 3-D eddy currents”, IEEE Transactions on Magnetics, Vol. 19, no. 6, pp. 2450-2452, 1983. [FOU 85] G. FOURNET, Electromagnétisme à partir des équations locales, Masson, Paris, 1985. [LEM 98] Y. LE MÉNACH, S. ClÉNET, F. PIRIOU, “Determination and utilisation of the source field in 3D magnetostatics problems”, IEEE Transactions on Magnetics, Vol. 34, no. 5, pp. 2509-2512, 1998. [LEM 99] Y. Le MENACH, Contribution à la modélisation numérique des systèmes électrotechniques: prise en compte des inducteurs, PhD Thesis, L2EP, USTL, 1999. [MOR 53] PH. M. MORSE, H. FESHBACH, Methods of Theoretical Physics, Part I, McGraw-Hill Book Company, Inc., New York, 1953. [REN 96] Z. REN, “Influence of R.H.S. on the convergence behaviour of curl-curl equation”, IEEE Trans Mag, Vol. 32, pp. 655-658, 1996. [VAS 80] C. VASSALLO, Electromagnétisme classique dans la matière, Dunod, Paris, 1980.
Chapter 3
Magnetodynamic Formulations
3.1. Introduction In this chapter the problems of magnetodynamics in low frequency will be covered. This study concerns the problems of induced eddy currents in the G conductors. Thus, the volume electric charges U and the displacement currents w t d are omitted. Figure 3.1 shows a typical problem of eddy currents. It deals with the G j , of the distribution of calculation, under the excitation of a time-varying current 0 G G the magneticG field ( h or b ) in every point of the study domain : and of the density of current j in the study domain :c for any time higher than zero. Maxwell’s equations relating to this problem are: G G curl e Ct b (Faraday law)
[3.1]
G G curl h j (Ampere theorem)
[3.2]
with the constitutive relations of materials: K b
G
P h in : and
G j
G
V e in :c.
[3.3]
Equations [3.1]G and [3.2] involve in particular theG conservation laws of the to be solved magnetic flux div b 0 and the conduction current div j 0 . They are G G G G with the boundary conditions such that the fields n qe and nqh are imposed respectively on *e and *h. Chapter written by Zhuoxiang REN and Frédéric BOUILLAULT.
118
The Finite Element Method for Electromagnetic Modeling
n
*e :c P
V
:
P
Po jo *h
Figure 3.1. Eddy current problem
In a conducting region, it is possible to directly consider the field (electric or magnetic) as a working variable. In the finite element approximation, these fields can be approximated by nodal elements or edge elements. The edge elements have the characteristic of imposing between elements only the continuity of the tangential component of the field, whereas the nodal elements impose at the same time the tangential and normal continuity of the field. In a magnetic formulation, the discretization of the magnetic field by nodal elements can then be incorrect. Indeed, at the interface between two materials of various permeabilities, the normal component of the magnetic field is discontinuous. In order to resolve this difficulty, we can proceed in two ways. The first consists of choosing as unknown variables at the nodes of the elements (on the surface of separation) the normal component of induction bn and the two tangential components of the magnetic field (ht1,ht2). The second consists of working with continuous variables, i.e. potentials. For this purpose, it is necessary toGadd an additional unknown variable. The magnetic field G h can then be written t grad G I . The term gradI then allows the excess of continuity imposed by vector t [PRE 82], [BOU 90a], [NAK 88] to be corrected. This type of decomposition is particularly well adapted to represent study domains containing air since in this area the magnetic field can only shift by a scalar potential. By duality, in the electric formulation, the difficulty of the representation of the electric field by nodal elements arises when the devices comprise materials of various conductivities. The continuity of the normal component of the field makes it G G necessary to write electric field e in the form w t a gradv [BID 82]. The use of nodal elements thus results naturally in using potentials to solve the problems of magnetodynamics, which does not seem necessary in the case of edge Gelements. On G the other hand, nothing prohibits working with potential vectors ( a or t ) discretized by edge elements [CAR 77].
Magnetodynamic Formulations
119
In this chapter, we will present two electric and magnetic dual formulations and a hybrid formulation. For each dual formulation, we will give two directions: in field and in combined potentials. Their performances will be compared, then the complementary features of the two dual formulations will be discussed. 3.2. Electric formulations 3.2.1. Formulation in electric field G The electric field e will be taken as a working variable [REN 90a]. The solution in the weak sense of the Ampere theorem, by using the Whitney edge elements, leads to the following variational formulation: G Find e We1, such that: 1
G
G
G G
d
¨ curl e '¸ curl e d: dt ¨ e '¸ e d: :
:
d G G d e' j0 d: dt dt
³
:
³
G G G e'n u hd*
G 0 , e' We1
[3.4]
*h
G where n is the outgoing normal from domain :. Let us recall that the space WeG1, G G belonging to the domain of the curl operator includes boundary condition n u e 0 on G*e. This formulation ensures in the weak sense the current conservation G div j 0 , the divergence of e is implicitly defined in the conducting field V z 0 and is to be defined in the non-conducting field. G G With the formulation in e , obtaining the magnetic induction b then requires an G integration in time. It is then preferable to work with the primitive in the time of e , tG G i.e. a* edt . Formulation [3.4] becomes:
³
0
G Find a * We1, such that: 1
G
G
G G
G G
d
G G
G
¨ curl a "¸ curl a ' d: dt ¨ a "¸ a ' d: ¨ a "¸ j d: ¨ a "¸ nqh d* 0 , 0
:
G a" We1.
:
:
*h
[3.5]
120
The Finite Element Method for Electromagnetic Modeling
G The variable a * has the same unit as the magnetic potential vector, its rotational G G being equal to the magnetic induction: curl a * b . In a conducting domain V z 0, its divergence isGdefined and is equal to zero, because the formulation ensures in the G G weak sense div j 0 , thus div V a 0 . We then have div a 0 if the conductivity V is constant and different from zero. On the other hand, in a non-conducting field, G a * behaves as a classical magnetic vector potential which is defined subject to a G gradient field. Variable a * is called by some authors “the modified potential vector” [EMS 83].
3.2.2. Formulation in combined potentials a - \
G G Electric field e or its primitive a * can be expressed through the combination of G magnetic potential vector a and electric potential scalar v such that G G G G e w t a gradv where a* = a + grad \, where \ is the primitive of v in the G G G G G* time. Let us replace in equation [3.5] a" and a * by a" = a ' + gradG \ ' and G by a = G a +G grad \, which is similar to solving in the weak sense curl h j in : and div j 0 in :c. We have the following formulation: G Find a We1 and \ We0, such that: 1
G
G
G G
d
G
d
¨ curl a '¸ curl a d: dt ¨ a '¸ a d: dt ¨ a '¸ grad # # d: :
:
G G a' j0 d:
³
:
d dt
G
G
G
³ a' (n u h )d*
:
0
G a ' We1.
[3.6a]
*h
G
G G
d
³ V grad\ 'ad: dt ³ V grad\ 'grad\ d: ³ \ ' n j d*
:c
\' We0.
:c
0
*c
[3.6b]
The term of the boundary integral in G[3.6b] corresponds to the conductor border G G G *c. This term is zero for any \ ' since n. j 0 on *c. The condition n j 0 is then naturally imposed in this formulation which is not the case for formulations [3.4] and [3.5]. The approximation by Whitney elements ensures in the strong sense G the G tangential continuity of e and consequently the normal continuity of b . The G G resolution of the system gives circulations of a along edges a and the values of \ G at nodes \ . The physical quantities such as the circulations e of e (the
Magnetodynamic Formulations
121
G electromotive force) along edges and fluxes b of b through the facets are calculated by: e = – dt ( a + G \ ) and b = R a , where G and R are the discrete operators of grad and curl respectively.
NOTE 1.– the introduction of a scalar potential results in a more significant number of unknown variables. Compared to the previous formulation, the additional number of unknown variables is equal to the number of nodes in the conducting domain. G NOTE 2.– the potential vector a is not unique because its divergence is not specified. Its uniqueness is in general ensured by the Coulomb or Lorentz gauge applied by the penalty technique [CAR 77] [BRY 90], or by working with the GG G potential with two components while imposing a.w 0 , where w is a vector of arbitrary direction [ALB 90b]. We note that the system converges better without the explicit gauge condition with the conjugate gradient method. The iterative procedure implicitly imposes the divergence of the potential vector [REN 96c]. 3.2.3. Comparison of the formulations in field and in combined potentials
Since the gradient of the nodal elements is included in the space of edge elements (grad W0 W1), the formulation in field [3.5] and the formulation in potentials [3.6] are theoretically equivalent in the conducting domain :c. Nevertheless, the numerical behavior of the two formulations is very different. In order to illustrate this difference, we calculated by the two formulations the distribution of eddy currents in a conducting torus containing cracks. The currents are induced by a sinusoidal time varying magnetic field uniformly distributed in the space (Figure 3.2). The sinusoidal variation of time-quantity makes it possible to work with complex variables. The derivative with respect to time d/dt is then replaced by the complex term jZ, where Z is the electric pulsation. The system of equations is solved by the bi-conjugate gradient method. Taking into account the symmetry of the problem, only an eighth of the domain is modeled. The conducting part, the hole, the crack and a layer of the air around the torus are meshed by tetrahedral elements. Taking into account the open boundary, the finite element method is coupled in this example with the boundary integral method. A pure finite element modeling could also have been used.
122
The Finite Element Method for Electromagnetic Modeling
crack
conductor
20 mm 10 mm
b = B sin Zt B = 1T
crack 10 mm
crack : depth: 7.5 mm thickness: 0.5 mm
Figure 3.2. A conducting torus with cracks
100
residue
10-1
a* formulation a-v formulation
10-2 10-3 10-4 10-5 -6
10
10-7 10-8 10-9
0
50
100
150
200
250
300
Number of iterations
Figure 3.3. Comparison of the convergence of electric formulations in field and in potentials
In Figure 3.4 the distributions of the eddy currents obtained in a cut plane is shown. Figure 3.3 compares the behavior of the convergence of the two formulations. The results obtained show that the conditioning of the system is much better with the formulation in combined potentials [REN 96b] [FUJ 96] [REN 00]. It is therefore interesting to work with the formulation in combined potentials even if the number of unknown variables is higher.
Magnetodynamic Formulations
G
(a) Result of the formulation in field a *
123
G
(b) Result of the formulation in potentials a -v
Figure 3.4. Distribution of eddy currents in the torus (electric formulations)
3.3. Magnetic formulations 3.3.1. Formulation in magnetic field
Magnetic formulation is established by the resolution in the weak sense of Faraday’s law by taking the magnetic field h as a working variable: G Find h Wh1, such that:
1
G
G
G G
d
G
G G
¨ curlh '¸ curl h d: dt ¨ h '.h d: ¨ h '¸ (nqe ) d* 0 , :
:
*e
G h ' Wh1, G G where Wh1 takes into account the Dirichlet condition n u h
[3.7] G 0.
Formulation [3.7] applies directly in a conducting area :c. In the non-conducting sub-domains such as air or ferromagnetic materials, owing to the fact that G G curlh 0 , it must beG coupled with a formulation in scalar potential I. The coupling will be ensured by h grad I on the interface *c. In other words, the degrees of freedom at the edges h on surface *c in formulation [3.7] must be expressed by the degrees of freedom at nodes I [BOS 82] by using the relation: h = – GI .
[3.8]
124
The Finite Element Method for Electromagnetic Modeling
3.3.2. Formulation in combined potentials t - I
Since the scalar potential is used in the non-conducting domain, the formulation in combined potentials seems quite attractive for the connection of conducting and G non-conducting domains. In the conducting area the field h can be expressed by the combination of the current vector potential and the magnetic scalar potential: G t gradI . The weak formulation of Faraday’s laws and the derivation with respect to the time of the flux conservation contained in equation [3.1] implies: G Find t Wh1 and I Wh0, such that: G
1
G
G G
d
G
d
G
G G
¨ curl t '¸ curlt d: dt ¨ t '¸ t d: dt ¨ t '¸ grad * d: ¨ t '¸ (nqe ) d* 0 :c
:
:
*c
t' Wh1, d dt
[3.9a] G
d
³ PgradI 't d: dt ³ PgradI' gradI d:
:
:
d dt
G d P gradI 't0 d: dt
³
:
³
G G
I'(n b )d* 0 ,
*e
I' Wh0.
[3.9b]
G G G G where t0 is the source field due to the imposed current j0 (curl t0 j0 ) .
G It should be noted that in the non-conducting domain, h is expressed by –gradI. In order to avoid the multi-valued problem of I in the case of multi-connected conductors, it is necessary to introduce cut planes allowing potential jumps [KOT 87] [VER 87] [ROD 87] or to fill G conductivity. On G low G the holes by a materialGwith the interface *c of domains t – I and I, condition n u t 0 is imposed. This G conditionG allows natural continuity of the tangential component of h between domain t – I and domain I. Under this condition, it is not necessary to impose the condition of continuity [3.8] since it becomes natural. Moreover, the boundary integral on *c in [3.9a] is zero [BOU 90a]. G With Whitney edge elements, the tangential continuity of h and the normal G continuity of j are guaranteed. From the solutionsG of the system, that is to say t at the edges and I at the nodes, the circulations of h along edges h and the currents through facets j are given by: h = t – G I and j = R h , thanks to the properties of Whitney elements.
Magnetodynamic Formulations
125
G NOTE.– the uniqueness of vector potentials t requires a gauge. The explicit application of the gauge is not necessary when solving the system by an iterative method because the procedure implicitly imposes it, as in the case of the vector magnetic potential.
3.3.3. Numerical example
The previous example of Figure 3.2 is calculated respectively by the magnetic formulations in field and in potentials. In order to avoid the multi-valued problem of the scalar potential, the hole of the torus is filled by a conductor of low conductivity (conductivity 100 times smaller than that of the torus). While the distributions of the field obtained by the formulations in field and in potentials are appreciably the same, as is shown in Figures 3.5a and b, the convergence behavior of the solution is completely different (Figure 3.6). With the formulation in potentials, the system converges after 100 iterations, whereas with the formulation in field, about 1,000 iterations were required to reach the convergence. Moreover, the convergence of the formulation in field is degraded if the value of conductivity put in the hole is too low. Indeed, the convergence speed of this formulation slows down with the increased penetration depth of the problem. On the other hand, with the formulation in potentials, the convergence is just slightly sensitive to various parameters. It is therefore possible to conclude that a better conditioning of the system is obtained with the formulation in combined potentials.
G
(a) Results of the formulation in field h
G
(b) Results of the formulation in potentials t -I
Figure 3.5. Distribution of eddy currents in the torus (magnetic formulations s)
126
The Finite Element Method for Electromagnetic Modeling
100
residue
10-1
h-formulation t-I formulation
10-2 10-3 10-4 10-5 10-6 10-7 10-8 10-9
0
50
100
150
200
250
300
Number of iterations Figure 3.6. Convergence comparison of magnetic formulations in field and in potentials
On the other hand, the duality between the results obtained by the two formulations can be noticed straightaway. Results illustrated by Figures 3.4b and 3.5 show that the distribution of currents obtained by the magnetic formulation is more regular, especially at the crack corner. However, concerning the distribution of the magnetic field, as is shown in Figures 3.7a and b, it is the electric formulation which gives the best result.
G
(a) Result of the electric formulation a -v
(b) Result of the magnetic
G t -I
Figure 3.7. Distribution of magnetic field in a cut plane in the crack (comparison of the electric and magnetic formulations)
Magnetodynamic Formulations
127
3.4. Hybrid formulation
As was mentioned in the formulation in electric field, the uniqueness of electric field is ensured only in the conducting region. Its uniqueness outside of conductors requires imposing its divergence. On the other hand, the use of the modified vector potential requires using a gauge if the uniqueness is desired. In order to avoid this problem, it is possible to make it work in the non-conducting area with a magnetic quantity instead of an electric quantity. The magnetic field can be simply derived from the gradient of scalar potential. This hybrid formulation (a*-I) then has the advantage of leading to a reduction of the number of unknown variables [EMS 88]. The system of equations is obtained from equation [3.5] in conductor :c and from the primitive with respect to the time of equation [3.9b] in domain :–:c. We then have: G Find a * We1, such that: 1
G
G
¨ curl a '¸ curl a
d:
:
d 1 G G G G G a '¸ a d: ¨ a '¸ nq h d* 0 , a ' We . [3.10a] ¨ dt : *c
and I Wh0, such that: G G
G
³ P gradI' gradI d: ³ P gradI 't d: ³ I'(n b )d* 0
:
:
0,
*e
. I' Wh0.
[3.10b]
The coupling between the two formulations is carried out by the integrals on surface *c. By using the properties of the differential operators and the continuity of the tangential components of the magnetic field and the normal component of the induction, we have:
³
G G G a '.n u h d*
*c
G G
G G
³ n.(a 'u grad I a 'ut
0 ) d*
[3.11a]
*c
G G I ' ( n b ) d*
³
*c
G G n.(a u grad I ' )d*
³
[3.11b]
*c
NOTE.– the sign (–) in the second contour integral does not lead to a symmetric matrix system. It is possible to make it symmetric but the positive definite property of the system is then lost.
128
The Finite Element Method for Electromagnetic Modeling
3.5. Electric and magnetic formulation complementarities 3.5.1. Complementary features
Two dual formulations have been established: magnetic formulation [3.5] or [3.6] and electric formulation [3.7] or [3.9]. We already observed through the preceding example the complementary feature on the results of fluxes and currents obtained by the two formulations. Indeed, on the one hand, the magnetic formulation verifies the Ampere theorem in the strong sense, of the G because the use 1 Whitney elements W1 ensures the tangential continuity of h . Since curl W W2, G G then the normal continuity of j curl h is also verified. However, Faraday’s law, G G the tangential continuity of e and the normal continuity of b are verified only in the weak sense. On the other hand, the behavior is dual for the electric formulation. G G It is GFaraday’s law, the tangential continuity of e (or a * ) and the normal continuity G* of b curl a G which are verified in the strong sense. Whereas, the G tangential continuity of h , the Ampere law as well as the normal continuity of j are verified only in the weak sense. G G Numerically, with the magnetic formulation, we obtain “good results” for n u h G G G GG G (the mmf) and n. j (the current). The fields b and e are calculated using h and j G G G G with the help of constitutive laws b P h and e U j . The constitutive laws are GG G G verified but the continuity of n.b , of n u e and Faraday’s law are not verified. The magnetic flux and the electromotive force (emf) are then less accurate. With the electric formulation, we have the duality: the results are better for the emf and the magnetic flux than for the mmf and the current.
In order to obtain the best results for the magnetic and electric fields, it is wise to solve a given problem with the two dual formulations. That makes it possible to verify in the strong sense all Maxwell’s equations. The results of the two formulations are “complementary” and the errors due to the discretizations arise G G in this case in the form of the non-observance of the constitutive laws b P h and G G e U j , which can be expressed in a symmetric way: GG / (h ,b )
³³
:
GG / (e , j )
0
³³
:
h
e
0
G G b ( h ) dh d:
³³
0
:
G G j (e) de d:
³³
:
b
G G G G h (b) db d: b h d: ,
G G G e ( j) dj d: e j d: .
jG
0
³
[3.12]
:
³
:
[3.13]
Magnetodynamic Formulations
129
They are thus the Ligurian errors introduced in [RIK 88a]. The application of these expressions in each element of the grid gives indications of local errors made by numerical calculation. These constitute error estimators for the adaptive mesh refinement [LI 93]. 3.5.2. Concerning the energy bounds
Let us recall that in the case of a static problem, the property of the complementary energy bounds of the two dual formulations [HAM 76] has been shown (one in scalar potential and one in vector potential). The question which arises is: does this property of energy bounds appear in the case of a dynamic problem such as the problem of eddy currents? The answers are rather varied in the literature in connection with this subject. The positive answers are in [HAM 78] [HAM 89] [PEN 91], while the negative answers are in [RIK 88b] [BOS 92]. In a static problem, there are functionals whose stationary and extreme points (maximum or minimum) correspond to the solution of the problem. However, in a magnetodynamic problem, it is generally no longer the case. Penman [PEN 91] tried to show that the energy bounds are possible to establish at least in the case of 2D geometry. Unfortunately the given demonstration is not sufficiently convincing [LI 94]. Hammond [HAM 89] proposed finally being able to find a functional having an extreme point, a special variational method, by applying constraints to the variations of the quantities of the fields. Li’s work [LI 94] shows that these constraints cannot be fulfilled in the case of a discretization by the finite element method. Rikabi [RIK 88], on the other hand, showed that the energy bound is not defined because the decomposition of the Ligurian is in general not possible in a problem of eddy currents. Through a rigorous mathematical development, Bossavit [BOS 92] showed that in general the energy bounds cannot be established. In spite of a lack of theoretical demonstrations, some bounding phenomena are evident in the numerical results, provided that a good refinement of the meshing with respect to the penetration depth of the problem is achieved. We think that these bounding phenomena are rather numerical [LI 94]. 3.5.3. Numerical example
The studied example is a problem from the TEAM Workshop [TUR 88]. It is a question of calculating the eddy currents in a conducting hollow sphere put in a magnetic field initially uniform in the space and sinusoidal time-varying (Figure 3.8). There is an analytical solution to this problem. The frequency of the excitation field is 100 Hz. The penetration depth of skin effect is about 5 mm.
130
The Finite Element Method for Electromagnetic Modeling
The problem is solved by two dual formulations (coupled with the boundary integral method) with different meshing refinements. We examine two dual aspects of the numerical results. The first is the dual phenomenon of the two formulations on the numerical accuracy of the field for a given meshing. The second is the convergence of the energy results when the meshing is refined. z B0 sin(Z t) R2 P0
P0
P0 R1
V y
0
x R1 = 35 mm, R2 = 50 mm, B0 = 1 T,
V = 108 s/m, P = P 0
Figure 3.8. Conducting hollow sphere in a time-varying uniform field
The solution of the system directly gives the circulations of the field along the G edges (circulations of h (mmf) with a magnetic formulation and the circulations of G e (emf) with an electric formulation). Table 3.1 illustrates the average errors of circulations of the fields along the edges (with a meshing of 3,240 elements) with respect to analytical results. In order to evaluate the accuracy of the vector fields, we have calculated the current density and the flux density at the element barycenters from circulations of the fields along the edges (approximated by the basic functions of the Whitney elements W1 and their rotational). The average errors of flux densities and of current with respect to analytical results are given in Table 3.2. In these tables, the average errors are estimated by the following norm:
Hx
¦
N i 1
ana
xiana xical / max( xiana ) / N , where xi
cal
and xi are respectively the
analytical and numerical results (at edges or in the elements). N is the number of edges or elements.
Magnetodynamic Formulations
Formulation
Errors of circulations of e (emf) (%)
131
Errors of circulations of h (mmf) (%)
real
imaginary
real
imaginary
electric
2.08
1.67
-
-
magnetic
-
-
1.26
3.12
Table 3.1. Average errors of circulations of the fields along the edges
Formulation
Errors of current densities (%)
Errors of flux densities (%)
real
imaginary
real
imaginary
electric
7.53
8.64
2.69
8.43
magnetic
2.26
2.72
6.25
11.40
Table 3.2. Average errors of the current and flux densities at the element barycenters
Let us compare the errors reported in the two tables. We can see that the circulations of the electric field (emf) are correctly obtained by electric formulation, and the circulations of the magnetic field (mmf) are calculated accurately by magnetic formulation. Concerning the flux and current densities at the barycenters of the elements, the electric formulation provides a better solution of the magnetic flux density while the magnetic formulation gives good results of the current density. We can then observe a dual phenomenon on the accuracy of the field distribution. The best method to solve a problem is obviously to use the two dual formulations and to take the best solutions of each formulation.
The Finite Element Method for Electromagnetic Modeling analytical t-I formulation a-v formulation
55
Magnetic energy (J)
50 45 40 35 30 25 20
0
1,000
2,000
3,000
4,000
5,000
6,000
Number of elements
Figure 3.9. Convergence of the magnetic energy with the meshing refinement
40
analytical t-I formulation a-v formulation
35
Power losses (kW)
132
30 25 20 15 10 5
0
1,000
2,000
3,000
4,000
5,000
6,000
Number of elements Figure 3.10. Convergence of Joule losses with the meshing refinement
Magnetodynamic Formulations
133
Figure 3.11. Variation of the ratio penetration depth/size of the elements with the meshing refinement
At this stage, we examine the global energy results. The variations of magnetic energies and the Joule losses in the conducting sphere with a progressive meshing refinement are illustrated in Figures 3.9 and 3.10. In order to illustrate the meshing refinement with respect to the penetration depth, we have shown the variation of the rate (penetration depth/size of elements) according to the meshing refinement in Figure 3.11. It is noted that the complementary energy bounds exist if the size of the elements is rather small compared to the skin depth of the problem. With the electric formulation we obtain a upper bound for the Joule losses and a lower bound for the magnetic energy. With magnetic formulation, the phenomenon of the energy bounds is dual of that of the electric formulation. In our case, the energy results converge with the meshing refinement when the rate of the skin depth on the size of the elements is higher than 1.5. This existence of the numerical complementary feature on the energy bound deserves a thorough theoretical study. A possible explanation is that this phenomenon is due to the numerical discretization of the geometric space [LI 94]. 3.6. Conclusion
In the 3D calculation of the eddy current problems, thanks to the symmetry of Maxwell’s equations, we can obtain two complementary formulations: magnetic formulation and electric formulation.
134
The Finite Element Method for Electromagnetic Modeling
In the conducting region, the working variable can be the field or the combined vector-scalar potentials. From the point of view of the discretization by Whitney elements, the decomposition of the field in combined potentials does not seem necessary because the space of edge elements W1 includes the space of gradient of the nodal elements W0. In addition, the uniqueness of the potentials requires gauge conditions. However, the system of equations is better conditioned with the formulations in combined potentials. This results in a better convergence of the system even without explicitly imposing the gauge condition. Solving the electric and magnetic formulation at the same time allows all Maxwell’s equations to be verified and better results of fields to be obtained. The numerical error relates to the constitutive laws and leads to an error indicator for the adaptive mesh refinement. However, the existence of the energy bounds, an interesting property that we could have in the static problems, cannot be demonstrated in the case of an eddy current problem. Nevertheless, through numerical examples, a phenomenon of energy bounds is observed when the meshing refinement is reasonable with respect to the penetration depth. That can be due to the effects of the discretization of the geometric space. 3.7. References [ALB 90a] R. ALBANESE, G. RUBINACCI, “Analyses of three dimensional electromagnetic fields using edge elements”, IGTE symposium, Graz, Austria, October 1990. [ALB 90b] R. ALBANESE, G. RUBINACCI, “Formulation of eddy current problem”, IEEE Proc., vol. 137, Pt. A., 1990, pp. 16-22. [BID 82] C.S. BIDDLECOMBE, E.A. HEIGHWAY, J. SIMKIN, C.W. TROWBRIDGE, “Methods for eddy current computation in three dimensions”, IEEE Trans. Mag., vol. 18, no. 2, 1982, pp. 492-497. [BIR 89] O. BIRO, K. PREIS, “On the use of the magnetic vector potential in the finite element analysis of 3-D eddy currents”, IEEE Trans. Mag., vol. 25, 1989, pp. 3145-3149. [BOS 82] A. BOSSAVIT, J.C. VERITE, “A mixed FEM-BIEM method to solve 3D eddy current problems”, IEEE Trans. Mag., vol. 18, no. 2, 1982, pp. 431-435. [BOS 85] A. BOSSAVIT, “Two dual formulations of the 3-D eddy currents problem”, COMPEL, vol. 4, no. 2, 1985 pp. 103-116. [BOS 91] A. BOSSAVIT, “Complementarity in non linear magnetostatics: bilateral bounds on the flux current characteristic”, ISEF’91, Southampton, 1991. [BOS 92] A. BOSSAVIT, “Complementary formulations in steady-state eddy-current theory”, IEE Proc. A, vol. 139, no. 6, 1992, pp. 265-272.
Magnetodynamic Formulations
135
[BOU 90a] F. BOUILLAULT, Z. REN, A. RAZEK, “Calculation of 3-D eddy current problems by an hybrid T- : method”, IEEE Trans. Mag., vol. 26, no. 2, 1990, pp. 478481. [BOU 90b] F. BOUILLAUT, Z. REN, A. RAZEK, “Modélisation tridimensionnelle des courants de Foucault à l’aide de méthodes mixtes avec différentes formulations”, Revue de Physique Appliquée, vol. 25, July 1990, pp. 583-592. [BRY 90] C.F. BRYANT, C.R.I. EMSON, C.W. TROWBRIDGE, “A comparison of Lorentz gauge formulations in eddy current computation”, IEEE Trans. Mag., vol. 26, no. 2, 1990, pp. 430-433. [CAR 77] C.J. CARPENTER, “Comparison of alternative formulations of 3-dimensional magnetic field and eddy current problems at power frequencies”, IEEE Proc., vol. 124, no. 11, 1977, pp. 1026-1034. [COU 81] J.L. COULOMB, “Finite element three dimensional magnetic field computation”, IEEE Trans. Mag., vol. 17, 1981, pp. 3241-3246. [EMS 83] C. EMSON, J. SIMKIN, “An optimal method for 3D eddy currents”, IEEE Trans. Mag., vol. 19, 1983, pp. 2450-2452. [EMS 88] C. EMSON, C.W. TROWBRIDGE, “Transient 3D eddy current using modified magnetic vector potentials and magnetic scalar potentials”, IEEE Trans. Mag., vol. 24, no. 1, 1988, pp. 86-89. [FUJ 96] K. FUJIWARA, T. NAKATA, H OHASHI, “Improvement of convergence characteristic of ICCG method for the A-I method using edge elements”, IEEE Trans. Mag., vol. 32, no. 3, 1996, pp. 804-807. [GOL 94] N.A. GOLIAS, T.D. TSIBOUKIS, “Magnetostatics with edge elements: a numerical investigation in the choice of the tree”, IEEE Trans. Mag., vol. 30, no. 5, 1994, pp. 2877-2880. [HAM 76] P. HAMMOND, J. PENMAN, “Calculation of inductance and capacitance by means of dual energy principles”, IEEE Proc., vol. 123, no. 6, 1976, pp. 554-559. [HAM 78] P. HAMMOND, J. PENMAN, “Calculation of eddy currents by dual energy principles”, IEEE Proc., Pt. A., vol. 125, no. 7, 1978, pp. 701-708. [HAM 89] P. HAMMOND, “Upper and lower bounds in eddy current calculation”, IEE Proc., Pt. A, vol. 136, no. 4, 1989, pp. 207-216. [KAM 90] A. KAMEARI, “Calculation of transient 3D eddy current using edge elements”, IEEE Trans. Mag., vol. 26, no. 2, 1990, pp. 466-469. [KOT 87] P.R. KOTIUGA, “On making cuts for magnetic scalar potentials in multiply connected regions”, J. Appl. Phys., vol. 6, no. 8, 1987, pp. 3916-3918. [LI 93] C. LI, Modélisation tridimensionnelle des systèmes électromagnétiques à l’aide de formulations duales/complémentaires. Application au maillage auto-adaptatif, PhD Thesis, University of Paris XI, December 1993.
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The Finite Element Method for Electromagnetic Modeling
[LI 94] C. LI, Z. REN, A. RAZEK, “Complementarity between the energy results of H and E formulations in eddy current problems”, IEE Proc.-Sci. Meas. Technol., vol. 1, no. 1, 1994, pp. 25-30. [MAN 95] J. MANGE, Z.J. CENDES, “A generalized tree-cotree gauge for magnetic field computation”, IEEE Trans. Mag., vol. 31, no. 3, May 1995, pp. 1342-1347. [NAK 88] T. NAKATA, N. TAKAHASHI, K. FUJIWARA, Y. OKADA, “Improvements of the T- : method for 3-D eddy current analysis”, IEEE Trans. Mag., vol. 24, no. 1, 1988, pp. 94-97. [PEN 82] J. PENMAN, J.R. FRASER, “Complementary and dual energy finite element principles in magnetostatics”, IEEE Trans. Mag., vol. 18, no. 2, 1982, pp. 319-323. [PEN 87] J. PENMAN, M.D. GRIEVE, “Self-adaptive mesh generation technique for the finite element method”, IEE Proc., Pt. A. vol. 134, no. 8, 1987, pp. 634-650. [PEN 91] J. PENMAN, “Error bounds in dual and complementary eddy current systems”, ISEF’91, Southampton, 1991. [PRE 82] T.W. PRESTON, A.B.J. REECE, “Solution of three dimensional eddy current problems, the T-: method”, IEEE Trans. Mag., vol. 18, 1982, pp. 486-491. [PRE 88] T.W. PRESTON, A.B.J. REECE, P.S. SANGHA, “Induction motor analysis by time stepping techniques”, IEEE Trans. Mag., vol. 24, no. 1, 1988, pp. 471-474. [PRE 92] K. PREIS, I. BARDI, O. BIRO, C. MAGELE, G. VRISK, K.R. RICHTER, “Different finite element formulations of 3D magnetostatics fields”, IEEE Trans. Mag., vol. 28, no. 2, pp. 1056-1059, March 1992. [REN 88] Z. REN, J.C. VÉRITÉ, “Application of a new edge element in 3D eddy currents computation”, Int. Symp. on Electromagnetic Field, Beijing, China, October 1988. [REN 90a] Z. REN, F. BOUILLAUT, A. RAZEK, A. BOSSAVIT, J.C. VÉRITÉ, “A new hybrid model using electric field formulation for 3-D eddy current problems”, IEEE Trans. Mag., vol. 26, no. 2, March 1990, pp. 470-473. [REN 90b] Z. REN, A. RAZEK, “New technique for solving 3-D multiply connected eddy current problems”, IEE Proc., Vol. 137, Pt. A, no. 3, May 1990, pp. 135-140. [REN 96c] Z. REN, “Auto-gauging of vector potential by iterative solver-Numerical evidence”, 3rd Int. Workshop on Electric and Magnetic Fields, Liege, Belgium, May 1996. [REN 96d] Z. REN, A. RAZEK, “3D eddy currents computation: field or potential formulation?”, ICEF’96, Wuhan, China, October 1996. [REN 96b] Z. REN, “Influence of the R.H.S. on the convergence behaviour of the curl-curl equation”, IEEE Trans. on Mag., vol. 32, no. 3, May 1996, pp. 655-658. [REN 96a] Z. REN, A. RAZEK, “Computation of 3-D electromagnetic field using differential forms based elements and dual formulations”, Int. J. of Numerical Modelling, Electronic Networks, Devices and Fields, vol. 9, no. 1 and 2, 1996, pp. 81-98.
Magnetodynamic Formulations
137
[REN 00] Z. REN, A. RAZEK, “Comparison of some 3D eddy current formulations in dual systems”, IEEE Trans. on Mag., vol. 36, no. 4, July 2000, pp.751-755. [RIK 88a] J. RIKABI, C.F. BRYANT, E.F. FREEMAN, “An error based approach to complementary formulations for static field solution”, Int. J. Num. Meth. Eng., vol. 26, 1988, pp. 1963-1987. [RIK 88b] J. RIKABI, C.F. BRYANT, E.F. FREEMAN, “Error based derivation of complementary formulations for eddy current problems”, IEE Proc., Pt. A, vol. 135, no. 4, 1988, pp. 208-216. [ROD 83] D. RODGER, J.F. EASTHAM, “A formulation for low frequency eddy current solutions”, IEEE Trans. Mag., vol. 19, 1983, pp. 2443-2446. [ROD 87] D. RODGER, J.F. EASTHAM, “Multiply connected regions in the A-\ three dimensional eddy current formulation”, IEE Proc., Pt. A, vol. 134, no. 7, 1987, pp. 58-66. [SIM 80] J. SIMKIN, C.W. TROBRIDGE, “Three dimensional non-linear electromagnetic field computations, using scalar potentials”, IEE Proc., Pt. B, 127, no. 6, 1980, pp. 368374. [TUR 88] L.R. TURNER, K. DAVEY, C.R.I. EMSON, K. MYA, T. NAKATA, A. NICOLAS, “Problems and workshops for eddy current code comparison”, IEEE Trans. Mag., vol. 24, no. 1, 1988, pp. 431-434. [VER 87] J.C. VÉRITÉ, “Calculation of multivalued potentials in exterior regions”, IEEE Trans. Mag., vol. 23, no. 3, 1987, pp. 1881-1887.
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Chapter 4
Mixed Finite Element Methods in Electromagnetism
4.1. Introduction A formulation is considered “mixed” when it involves at least two unknown variables at the same point, for example, two fields, or one field and one potential. When the finite element method involves two unknown variables, one being defined within the elements and the other existing only on their interfaces – facets of 3D elements – then it involves a hybrid method. This type of formulation has been developed primarily for applications that require the determination of several quantities: constraints and displacements in structural mechanics, velocity and pressure in fluid mechanics. The value of mixed methods also derives from the fact that they can take full advantage of the benefit given by the variational approach: transposing some fundamental laws of physics to the discrete problem. A classical variational formulation, after discretization by finite elements, is the equivalent of seeking a configuration that minimizes functional calculus related to the discrete energy while mixed finite element methods can usually minimize the energy itself: complementary energy or Hellinger-Reissner energy in mechanics [QUA 97], and electromagnetic energy in the case of the Maxwell equations. In a sense, they provide a better description of physical laws.
Chapter written by Bernard BANDELIER and Françoise RIOUX-DAMIDAU.
140
The Finite Element Method for Electromagnetic Modeling
For electromagnetism, the interest of simultaneous determination of two quantities such as the magnetic field h, and the vector potential a, or the magnetic induction b and scalar potential ij is not obvious. The second unknown variable, which does not bring a priori any additional information, significantly increases the number of degrees of freedom and thus the calculation effort. However, in magnetostatics, it should be noted that only mixed formulations can deal with the fields as unknown variables. It is also observed that, in the literature devoted to numerical computation in electromagnetism, the part dedicated to mixed finite element methods is still evolving. Yet, their interest is now recognized: even if they bring greater complexity to the modeling and some additional calculation efforts that are not necessarily justified for a simple problem, they provide a better approximation of the main unknown variable – usually a field – in the case of complex geometries. The implementation of mixed finite element methods involves two main difficulties: the first difficulty concerns the fact that the different unknown variables must belong to compatible approximation spaces. However, this condition is automatically complied with when using “mixed finite elements”, for example, Whitney elements. The second difficulty concerns the fact that the linear equations matrix which is ultimately to be solved is indefinite. The conventional solution methods cannot be used and specific algorithms have to be introduced. Some of those most commonly used are presented below. In general, a mixed method is obtained by transforming a second order partial differential equation in a system of two first order equations. Expressing these two equations leads in the weak form to the mixed formulation. Hence, an unconstrained minimization problem is transformed into a constrained minimization problem and finally into a problem requiring the saddle point to be found. This approach, which we will discuss, does not seem the most straightforward for electromagnetism since the first order equations are immediately available: these are Maxwell equations. We therefore begin with the latter to establish various mixed formulations. We will show in a second stage that these formulations can also be obtained from an energy approach. 4.2. Mixed formulations in magnetostatics Now, let us consider a bounded domain : , with boundary * . The outgoing normal is noted n.
Mixed Finite Element Methods in Electromagnetism
141
The magnetostatic equations can be written: curl h
j
[4.1]
div b
0
[4.2]
in : , the current density j being imposed, with support in : . The magnetic permeability P , such that b P h is a function of space, the possible nonlinearities can be taken into account by an iterative Newton-Raphson method. In addition, the following boundary conditions are imposed according to the boundary: bn hun
[4.4]
0 on * h
such that * b * h h
[4.3]
0 on * b
* . Equation [4.1] can be replaced by:
s
[4.5]
h gradM
s
where h represents a “source” field such as curl h can be replaced by:
Ph
s
j . Similarly, equation [4.2]
[4.6]
curl a
where the uniqueness of a can be guaranteed by imposing div a a n 0 on * .
0 in : and
4.2.1. Magnetic induction oriented formulation Writing [4.5] in its weak form, we can obtain: ³:Q b bc d:
s
³: h bc d: ³: gradM bc d:
where bc is a test field whose proprieties will be subsequently indicated and where 1 Q . By integrating by parts, it follows:
P
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The Finite Element Method for Electromagnetic Modeling
³:Q b bc d: ³: M div bc d: ³* M bc n d*
s
³: h bc d:
The integral on the border * is zero. It is in fact canceled on * b because of boundary condition [4.3] which is strongly imposed by taking b n 0 , and thus bc n 0 on * b . It is also cancelled on * h since condition [4.4] can result in s zeroing the reduced potential M on * h , which implies that the field h also checks [4.4]. s
Note that this is still possible, even if h is not the “real” source field calculated with the law by Biot and Savart but differs by a gradient. It is always possible to s define a h checking: curl h
s
s
j h u n
[4.7]
0 on * h
Still equation [4.2] needs to be written in the weak form in order to obtain the second variational equation of the formulation, and finally the mixed formulation is as follows: s
2
Given j or h checking [4.7], find the couple (b M ) H b u L (: ) checking:
°³ Q b bc d: ³ M div bc d: ³ h s bc d:bc H b : : : ® 2 c div M d 0 M c L (: ) : b °¯ ³:
[4.8]
with the following definitions of functional calculus spaces: Hb
{v H (div : ) v n
[4.9]
0 on * b }
The main unknown variable is the magnetic induction b , hence the qualifier of oriented formulation b . The second unknown variable M can, as will later be shown, be interpreted as a Lagrange multiplier. It is now easy to see that this mixed problem is equivalent – in the sense of distributions – to the initial problem. Firstly, it is clear that the second equation of [4.8] is the same as [4.2]. Then, for the first equation of [4.8], let us take bc in 3
{D (:)} , therefore zero on * and undertake an integration by parts.1 We obtained:
1
1 We suppose here that M belongs to the functional calculus space H ( : ) which will be
defined subsequently in [4.19], such as grad M being added square and the integration by parts being admissible.
Mixed Finite Element Methods in Electromagnetism
³:Q b bc d: ³: gradM bc d: ³: h
s
3
bc d:bc {D (: )}
143
[4.10]
which implies [4.5] in the sense of distributions. Now, let us return to the first equation of [4.8] but this time taking bc in H b , and therefore non-zero on * . After integrating by parts, we obtain:
³:Q b bc d: ³: gradM bc d: ³*M bc n d* ³: h
s
bc d:bc H b
However, taking [4.5] into account, deduced thanks to [4.10], there remains only:
³*M bc n d*
0bc H b
Boundary condition [4.3] being imposed heavily in the space defined H b , the integral is only performed on * h . Therefore, we have – weakly – M 0 on * h which implies [4.4]. Boundary condition [4.4] is thus included – in the weak sense – in the mixed formulation. Note 1: if we seek b in the form b curl a , by taking test fields bc curl a c , we find, by taking boundary conditions into account, the classical formulation of vector potential:
[4.11]
³:Q curl a curl ac d: ³: j ac d:ac H a with the following definitions: H (curl : ) Ha
2
3
2
3
{u {L (: )} curl u {L (: )} }
{u H (curl : ) u u n
0 on * b }
[4.12] [4.13]
the space H a by definition containing boundary condition [4.3]. Note 2: it is clear that the solution a of [4.11] is not unique, but curl a is unique. Given the equivalence raised in Note 1 between the mixed formulation [4.8] and the classical formulation of vector potential, we will take for granted the existence and uniqueness of solution [4.8], at least regarding b . It is possible to check that M is also unique, assuming that there are two solution couples (b M1 ) and (b M 2 ) for mixed problem [4.8]. It follows:
144
The Finite Element Method for Electromagnetic Modeling
³: (M1 M 2 ) div bc d:
0bc H b
By choosing bc such that div bc deduced.
M1 M 2 the equality of M1 and M 2 is thus
4.2.2. Formulation oriented magnetic field
Let us now write [4.6] in the weak form:
³
:
P h hc d:
³
:
curl a hc d:
which will become, after integration by parts:
³
:
P h hc d: ³ a curl hc d: ³ n u a hc d* 0 :
*
In view of boundary conditions [4.3] and [4.4], the integral over the border * is zero. In fact, on * b it is imposed a u n 0 which implies [4.3]. In addition, on * h , h u n 0 ; this condition [4.4], also verified by test field hc , will be strongly satisfied since we impose it in the definition of the space of admissible fields.
The writing of [4.1] in the weak form immediately provides the second variational equation, and finally we obtain the mixed formulation. Since j H 0 (div 0 :) , we find the couple (h a ) H h u H *h (div0 :) verifying:
³
P h h 'd: ³ a curl h ' 0
³
curl h a 'd: ³ j a ' 0 a ' H *h (div 0 , :)
:
:
:
h ' H h
[4.14a] [4.14b]
:
with the following definitions of functional calculus spaces. H (curl :) {u {L2 (:)}3 curl u {L2 (:)}3 }
[4.15]
Hh
[4.16]
{u H (curl :) u u n
0 on * h }
H *h (div 0 :) {u H (div :) div u
0 in : u n
0 on * h }
[4.17]
Mixed Finite Element Methods in Electromagnetism
The introduction of conditions div a 0 in : and a n ensure the uniqueness of a , as will be seen in Note 2.
145
0 on * h aims to
The main unknown variable is the magnetic field h, hence the qualifier of the oriented formulation h. The second unknown variable a , as for M in the oriented mixed formulation b, can be interpreted as a Lagrange multiplier. As for the formulation in (b M ) , we can show the equivalence between this problem and the mixed initial problem: starting from [4.14a], by taking hc in {D(:)}3 , hence zero on * , after an integration by parts, we find [4.6], in the sense of distributions. Given this result, starting from [4.14a] but taking this time hc in H h , hence non-zero on * , we obtain after integration by parts:
³
*
n u a hc
0hc H h
Boundary condition [4.4] being imposed strongly in the space defined H h , the integration is only carried out on * b . Thus, we obtain – weakly – n u a 0 on * , which implies [4.3]. Boundary condition [4.3] is hence included – in the weak sense – in the mixed formulation. Note 1: if we seek h in the form h h s grad M , taking test fields hc grad M c , we find, by taking the boundary conditions into account, the classical formulation of reduced scalar potential:
³
:
P (h s grad M ) grad M c d: 0M c H M
s where h s is a “source” field such that curl h [4.4]. The functional spaces are the following: H 1 (:) {u L2 (:) HM
{u H 1 (:) u
[4.18] j and verifies boundary condition
wu L2 (:)} wxi 0 on * h }
[4.19] [4.20]
Space H M thus contains in its definition boundary condition [4.4]. Note 2: the solution of [4.18] is unique if * h is not empty. Given the equivalence observed in Note 1 between mixed formulation [4.14] and the classical formulation of reduced scalar potential [4.18], we will take for granted the existence and uniqueness of the solution, at least regarding h . In order to establish that a is
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The Finite Element Method for Electromagnetic Modeling
also unique, we assume that there are two couples (h a1 ) and (h a2 ) solutions to mixed problem [4.14]. It follows:
³
:
(a1 a2 ) curl hc d:
0 hc H h
As a1 and a2 are in the space H *h (div0 :) , a field vector u exists such that curl u , with curl u n 0 on * h . In addition, the test field hc being taken in * h , they verify hc u n 0 on * h and hence curl hc n 0 on * h . It is thus possible to take hc u , which implies the equality of a1 and a2 . a1 a2
In this context, it may be noted that the uniqueness of a is heavily dependent on the properties of different spaces containing test fields and the solutions. We will return later, with regards to the discrete problem, to the choice of work spaces that cannot be made arbitrarily. Mixed formulations [4.8] and [4.14] have been suggested in [BOS 88]. Comparisons with non-mixed methods are given in [DUL 97]. 4.2.3. Formulation in induction and field
In the two previous formulations, the constitutive relationship between magnetic materials was checked sharply. It is also possible to integrate this behavioral law in the equations, by writing: b
P
h
grad Mˆ
[4.21]
curl aˆ
[4.22]
and b Ph
where Mˆ and aˆ are not the usual potentials. Mˆ is the total magnetic scalar potential b
P
h whereas aˆ is the magnetic vector potential from which b P h is derived.
By expressing [4.21] and [4.22] in variational form as well as Maxwell equations [4.1] and [4.2], we obtain the following formulation proposed in [ALO 98]: Given j H 0 (div 0 :) , find (b h Mˆ aˆ ) H b u H h u L2 (:) u H *h (div 0 :) verifying:
Mixed Finite Element Methods in Electromagnetism
1 ° ³: P b bc d: ³: h bc d: ³: Mˆ div bc 0bc H b ° °° ³ b hc d: ³ P h hc d: ³ aˆ curl hc 0hc H h : : ® : ° ˆ c b M Mˆ c L2 (:) div 0 ³: ° 0 ° ³: curl h aˆ c 0aˆ c H *h (div :) ¯°
147
[4.23]
the functional spaces being defined earlier. 4.2.4. Alternate case
We have presented several mixed formulations for the magnetostatic case. By following a similar approach, we can easily establish the mixed formulations for electrostatics. Furthermore, we have chosen to impose homogenous boundary conditions in order to simplify the presentation. The presence of non-homogenous boundary conditions would not bring any fundamental change. There is clearly a need to correctly define the functional spaces in which the solutions are sought and to add some boundary terms in the second members of variational equations. It is also possible to consider Maxwell equations in an unbound domain; the only boundary conditions thus being the nullity of fields at infinity. In this case, the area external to the domain : can be treated using an integral method. Mixed methods can be generalized without causing a major difficulty to the coupling of the finite elementsboundary integral [BAN 98a], [BAN 01]. We will now show how mixed methods can be obtained from an “energy approach” in terms of variational principle. 4.3. Energy approach: minimization problems, searching for a saddle point 4.3.1. Minimization of a functional calculus related to energy
Let us show that the solution to problems [4.1] and [4.2], with the associated boundary conditions – in our case [4.3] and [4.4] – is equivalent to solving a minimization problem. For this purpose, we introduce functional calculus: J (D )
1 Q _ curl D _2 d: ³ j D d: : 2 ³:
[4.24]
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and search for a H a defined by [4.13], such that J (a)
Now, let D
[4.25]
inf J (D )
D H a
a O a c . By deriving with respect to O the expression:
J (a O a c) 12³ Q _ curl a _2 d: O ³ Q curl a curl a c d: :
:
³ j a d: O ³ :
:
O2
Q _ curl a c _2 d: 2 ³: j a c d:
and by writing that the derivative is zero for each a c H a , we deduce the formulation of potential vector [4.11]. Equally, if a H a is the solution of variational equation [4.11], we have for each a c H a : J (a c) J (a )
1 Q _ curl(a c a) _2 d: ³ Q curl a curl(ac a) d: ³ j (ac a ) d: : : 2 ³: 1 Q _ curl(a c a ) _2 d: t 0 2 ³:
Therefore, for each a c which is not the solution to [4.11], we have J (a c) ! J (a ) , indicating that a is a solution to the minimization problem [4.25]. Problems [4.11] and [4.25] are clearly equivalent. Note 1: when functional calculus J is minimal, the contribution of the first integral of [4.24] is equal to the magnetic energy, and that of the second integral is double this energy. The minimum of J is thus the opposite of magnetic energy. Note 2: by using the same approach, we can easily show that the variational formulation of reduced scalar potential [4.18] is equivalent to the following minimization problem: find M H M defined by [4.20], such that K (M )
inf K (\ )
\ HM
with the following expression of functional calculus K: K (\ )
1 P _ grad\ _2 d: ³ P h s grad\ d: : 2 ³:
[4.26]
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4.3.2. Variational principle of magnetic energy
4.3.2.1. Energy in terms of induction Let us introduce functional calculus for magnetic energy I b written in terms of induction: Ib (E )
1 Q _ E _2 d: ³ h s E d: : 2 ³:
[4.27]
as well as the functional space: H b0
{u H b div u
0 in :}
[4.28]
where H b is defined by [4.9] and let us consider the following problem: Find b H b0 such that: I b (b)
Let E
[4.29]
inf I b ( E )
E H b0
b O bc . By writing that:
dI b (b O bc)|O dO
0
0
it is shown that solving minimization problem [4.29] is equivalent to finding b H b0 such that:
³
:
Q b bc d:
³
:
h s bc d: bc H b0
[4.30]
Just as before, we can verify that the converse is true: if b is the solution of [4.30], then I b (bc) I b (b) t 0 bc H b0 , indicating that b is the solution to a minimization problem. Note 1: variational equation [4.30] is such that it can be obtained by strongly imposing div b 0 in mixed formulation [4.8].
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Note 2: it is possible to check that [4.30] is equivalent to problem [4.1] and [4.2] associated with the boundary conditions [4.3] and [4]. In order to confirm this, it is sufficient to write bc curl a c in [30], with a c belonging to H a defined by [4.13]; we take a c in {D(:)}3 , and therefore as zero on * 2, and we integrate by parts: [4.1] is found in the sense of distributions. By thereafter taking a c in H a , the boundary condition [4.4] is deduced. As for [4.2] and boundary condition [4.3], they are clearly included in the definition of spaces H b0 and H a . Note 3: the minimum value reached by the functional calculus I b is equal to the opposite of magnetic energy.
4.3.2.2. Energy in terms of magnetic field Let us introduce the functional calculus of magnetic energy I h written in terms of field: I h (k )
1 ³ P _ k _2 d: 2
[4.31]
We define the following affine variety: H hj
{u H h curl u
j in :}
[4.32]
where H h is defined by [4.15], as well as the associated vector space: H h0
{u H h curl u
0 in :}
[4.33]
Let us consider the following problem: Find h H hj such that: I h ( h)
Let k
[4.34]
inf I (k )
k H hj
h O hc where hc H h0 and by writing that:
dI h (h O hc)|O dO
0
0
2 D (:) indicates the set of functions indefinitely differentiable and with compact support in
: . These functions are thus zero on * .
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It is shown that solving constrained minimization problem [4.34] is equivalent to finding h H hj such that
³
:
P h hc d: 0 for all hc H h0
[4.35]
Note 1: variational equation [4.35] is such that it can be obtained by strongly imposing curl h j in mixed formulation [4.14]. Note 2: it is possible to check that [4.35] is equivalent to problems [4.1] and [4.2] associated with boundary conditions [4.3] and [4.4]. In order to confirm this, it is sufficient to write hc grad M c in [4.35], with M c belonging to H M defined by [4.20]; we take M c in D(:) , and therefore as zero on * , and we integrate by parts: [4.1] is found in the sense of distributions. By thereafter taking M c in H M , boundary condition [4.3] is deduced. Equation [4.1] and boundary condition [4.4] are strongly satisfied thanks to the definition of spaces H hj and H M . Note 3: the minimum of the functional calculus I h is equal to the magnetic energy. 4.3.3. Searching for a saddle point
The drawback of formulations [4.30] and [4.35] is that the constraint included in the workspace – zero divergence in H b0 , rotational given in H hj – is not in practice easy to impose during the discretization process. The easiest way to achieve this is of course to introduce a potential. However, this solution, which leads to conventional formulations in potential, does not allow the physical fields to be conserved as unknown variables. It is also possible to impose a constraint by using a tree technique, the implementation of which is also quite difficult. Another way to impose a constraint of type div b 0 or curl h j consists of relaxing the constraint by introducing a Lagrange multiplier. Instead of imposing a relationship between the unknown variables, the functional calculus is modified. 4.3.3.1. Functional calculus in terms of induction Now let us consider again problem [4.29], but looking for b in H b instead of H b0 . The constraint div b 0 is no longer imposed by the workspace. It is thus possible to relax this constraint by modifying the expression of the functional calculus to be minimized. Instead of I b ( E ) , it will minimize: I b ( E ) sup
\ L2 ( : )
³\ div E d:
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If the functional calculus is introduced: Lb ( E \ )
1 Q _ E _2 d: ³ h s E d: ³ \ div E d: : : 2 ³:
The problem consists of searching for a couple (b M ) H b u L2 (:) such that: Lb (b M )
inf sup Lb ( E \ )
[4.36]
E H b \ L2 ( : )
For each couple ( E \ ) H b u L2 (:) , we obtain: Lb (b\ ) d Lb (b M ) d Lb ( E M )
which implies that the constraint div b taking \ \ W W div b , would result in: lim Lb (b\ )
W of
0 is verified. If this was not the case,
f
The couple (b M ) is called a “saddle point” or “col” as a result of the geometric representation of the functional calculus Lb in the form of a saddle surface [BRE 91], [GIR 86]. The unknown variable M is the Lagrange multiplier associated with the constraint div b 0 . Now, let E
b O bc . By writing that:
d Lb (b O bc M )|O dO
0
0
we obtain:
³
:
Q b bc d: ³ M div bc d: :
³
:
h s bc d: bc H b
which is identical to the first equation of [4.8]. Similarly, by considering \ M OM c , and by writing that: d Lb (b M OM c)|O dO
0
0
the second equation of [4.8] is deduced:
Mixed Finite Element Methods in Electromagnetism
³
:
153
0M c L2 (:).
div bM c d:
Finding the saddle point of the functional calculus Lb is therefore equivalent to solving mixed problem [4.8]. 4.3.3.2. Functional calculus in terms of field The same approach can be followed for problem [4.34]: we start over from [4.34], but by searching for h in H h instead of H hj . The constraint curl h j will be imposed by the functional calculus thanks to the introduction of a Lagrange multiplier a , which will be sought in H *h (div0 :) . This latter is the space to which curl h belongs. The functional calculus to be minimized becomes: I h (k )
sup D H *h (div0 : )
³ D ( j curl k ) d:
where I h is defined by [4.31]. If we introduce the functional calculus: Lh (k D )
1 P _ k _2 d: ³ D ( j curl k ) d: : 2 ³:
the problem will consist of searching for a couple (h a ) H h u H *h (div 0 :) such that: Lh (h a)
By letting k
inf
k H h
sup D H *h (div0 : )
Lh (k D )
[4.37]
h O hc and by writing that:
d Lh (h O hc a)|O dO
0
0
We find the first variational equation [4.14a] of the mixed oriented problem h . Similarly, by letting D a O a c , and by writing that: d Lh (h a O a c)|O dO
0
0
we find equation [4.14b]. The search for a saddle point of the functional calculus Lh is equivalent to solving mixed problem [4.14].
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4.3.4. Functional calculus related to the constitutive relationship
Let us introduce the functional calculus F (b h)
1 _ b P h _2 d: 2 ³:
which reflects the error made on the constitutive relationship b P h [ALO 98]. It is then possible to minimize this functional calculus by imposing Maxwell equations as constraints, through two Lagrange multipliers. It is then possible to seek a saddle point of the functional calculus: U ( E k \ D )
F ( E k ) ³ \ div E d: ³ D ( j curl k ) :
:
By writing that: U (b h Mˆ aˆ )
inf
sup
E H b k H h \ L2 ( : ) D H
*h
(div0 : )
U ( E k \ D )
mixed formulation [4.23] is deduced. 4.4. Hybrid formulations
The difference between the terms “mixed” and “hybrid” is not always clear in the literature. Here, a formulation is called hybrid if it is formally a mixed method, but one of the unknown variables exists only on the interfaces [QUA 97]. These interfaces can be, for example, the facets of a 3D grid. We will first build two examples of hybrid methods from mixed formulations [4.8] and [4.14], and then we will present a method that is both mixed and hybrid. 4.4.1. Magnetic induction oriented hybrid formulation
A subdivision of : into several domains :i is considered. This kind of subdivision can be obtained by performing the finite element meshing of : . In this case, each :i is a “triangulation” element. It is assumed that we are able to build up the space: n
H b0
{b {L2 (:)}3 div b
0 in :i i b n
0 on * b }
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155
which means that the divergence of b is zero inside each :i , but not on the interfaces where the normal component of b is liable to undergo a jump. A term dedicated to relaxing the continuity condition of the normal component of the induction on the interfaces should be added to the functional calculus I b introduced for [4.27] in section 4.3.2.1. The functional calculus considered is then M b ( E \ )
1 _ E _2 d: ³ h s E d: ¦ ³ E n\ dsi : w:i 2 ³: i
[4.38]
where w:i is the boundary of the domain :i . By proceeding as previously, it n emerges that the search for saddle point (b M ) H b0 u H M (:) such that: M b (b M )
infn sup M b ( E \ )
E H b0 \ H 1 ( : )
is equivalent to solving the following hybrid problem: n
Given h s , find the couple (b M ) H b0 u H M (:) verifying: Q b bc d: ¦ ³w:i M bc n dsi ° ³: i ® ° ¦i ³w:i b n M ' dsi ¯
³
:
n
h s bcd :bc H b0
0M ' H M .
[4.39]
The functional calculus space H M is defined by [4.20]. The unknown variable M is a Lagrange multiplier associated with the continuity constraints of b n at the interfaces level. Note 1: boundary integrals over the w:i only cover the “internal” interfaces. They have a zero value on * because of the boundary conditions imposed in the n
spaces H b0 and H M . Note 2: formulation [4.39] can be obtained directly from the Maxwell equations, by again considering the approach that was used in section 4.2.1 in order to establish [4.8], but by integrating on the union of interiors of :i – i.e. by excluding the boundaries of w:i – so that an integration by parts can subsequently be achieved. n Indeed, the divergence of fields of H b0 involves a Dirac mass linked to a jump of b n on the w:i and as a result, the integration by parts on the whole : is not admissible. Each integration by parts on :i thus shows a boundary integral on w:i .
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4.4.2. Hybrid formulation oriented magnetic field
It is possible to set up a hybrid method oriented h on the same model, by introducing the following affine variety: n
H hj
{h {L2 (:)}3 curl h
j in :i i h u n
0 on * h }
0 in :i i h u n
0 on * h }
[4.40]
and the associated vector space: n
H h0
{h {L2 (:)}3 curl h
It is thus assumed that we could build vector field spaces whose rotational is imposed inside each :i but not on the interfaces where the tangential component of h is likely to undergo a jump. Then a Lagrange multiplier dedicated to relaxing the jump zero condition is introduced. The functional calculus to be considered is: M h (k D )
1 ³ P _ k _2 d: ¦ ³w:i D (k u n) dsi 2 i n
We are thus led to seek the saddle point (h a ) H hj u H a such that M h (h a)
inf sup M h (k D ) n
k H hj D H a
which is equivalent to solving the following hybrid problem: n
Given j , find the couple (h a ) H hj u H a verifying n P h h ' d: ¦ a (h 'u n) dsi 0 h ' H h0 ³ w:i ° ³: i ® ° ¦i ³w:i (h u n) a ' dsi 0 a ' H a . ¯
[4.41]
The functional space H a is defined by [4.13]. The unknown factor a is a Lagrange multiplier associated with the continuity constraints of h u n on the level of the interfaces. Note: the boundary integrals on the w:i are zero on * because of the boundary n conditions imposed in the spaces H h0 and H a .
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157
4.4.3. Mixed hybrid method
It may be useful to facilitate a numerical solution to “hybridize” a mixed formulation. Let us take the example of formulation [4.8]. If we replace the space H b by space; n
Hb
{b {L2 (:)}3 div b L2 (:i )i b n
0 on * b }
so that the continuity of normal inductions is no longer satisfied on the boundaries w:i , it is then necessary to release this condition of continuity by introducing an additional Lagrange multiplier [ on each w:i . The following hybrid mixed formulation is then obtained [ARN 85], [BRE 91]: n
Given j or h s verifying [4.7], find the triplet (b M [ ) H b u L2 (:) u H 1 (:) verifying: Q b bc d: M div b ' ¦ ³: ³w:i [ b' n dsi ° ³: i ° ® ³: div b M 'd:=0 ° ° ¦i ³w:i b n [ ' dsi ¯
³
:
h s b ' d:
0
n
bc H b M c L2 (:)[ c H 1 (:)
whose unknown factors are b , M and [ . The unknown factor [ is the Lagrange multiplier associated with the continuity constraint of b n on the interfaces. It represents the reduced scalar potential on the interfaces, where M is not continuous. [ is searched for in the space H 1 (:) defined by [4.19] in order to ensure its continuity on the interfaces. However, only its value on the interfaces matters. n
The definition of H b clearly shows that the basic functions which will be adopted to interpolate b are defined inside each :i , the :i being, for example, elements of a meshing. During the assembly of the matrix, there will be no connection between :i and : j for i z j . The matrix associated with ³ Q b bc d: :i
is thus diagonal by blocks. 4.5. Compatibility of approximation spaces – inf-sup condition
There exist necessary and sufficient conditions so that a mixed variational problem is well stated. However, the systematic verification of all these conditions –
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especially for the continuous problem – is not always necessary from the practical point of view. It is sometimes quicker to show that the mixed problem is equivalent, at least with regards to the fundamental unknown variable, to a non-mixed problem already understood to be well stated. On the other hand, the only problem that actually needs to be solved is a discrete problem. In this case, an appropriate choice of different types of finite elements involved in the discretization can automatically ensure the compatibility of work spaces. If the finite elements are well chosen, it is therefore not in practice necessary to check the “inf-sup condition” for the discrete problem. We will not present here in any detail the general theorems which can be found in many applied mathematics works [BRE 91], [GIR 86], [QUA97], [ROB 91]. We will only give, by way of an example, some indications of the conditions to fulfill so that a discretized mixed problem is well stated, then we will state these conditions in general terms. 4.5.1. Mixed magnetic induction oriented formulation
Let us consider mixed formulation [4.8] as an example. After the discretization, it becomes: Find the couple (b M ) Sb u SM verifying: s c c c c ° ³: Q b b d: ³: M div b d: ³: h b d:b Sb ® °¯ ³: div bM c d: 0M c SM
[4.42]
where Sb and SM are finite dimension spaces, Sb checking boundary condition [4.3]. The Galerkin method will thus lead to a matrix system of the form: §A ¨ ©B
BT · § b · ¸ ¨ ¸ 0 ¹ ©M¹
§f · ¨ ¸ ©0¹
[4.43]
comprising as many equations as unknown variables. It is then sufficient to check that the solution is unique to be assured of the existence of this solution. Suppose that there are two couples of solutions (b1 M1 ) and (b2 M 2 ) in Sb and let b b1 b2 , M M1 M2 . It follows: c c c ° ³: Q b b d: ³: M div b d: 0 b Sb ® °¯ ³: div bM c d: 0 M c SM
[4.44]
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159
If we take b ' b in [4.44], we obtain:
³
:
Q _ b _2 d: 0
which implies the uniqueness of b . The matrix A of system [4.43] is thus (at least) invertible. We will see later that this matrix is generally required to also be definite positive, resulting in a condition known as coercivity that we will discuss at the end of this section. While taking into account the nullity of b , let us now take the unspecified bc in Sb . It becomes:
³
:
[4.45]
M div bc d: 0 bc Sb
A priori, we cannot conclude from [4.45] that M is zero. To be entitled to conclude the nullity of M , it is sufficient to ensure that the choice of bc , such that div bc M in [4.45], is possible. This requires a degree of compatibility between Sb and SM which is achieved when SM contains the divergences of fields of vectors belonging to Sb . If this compatibility exists, then
³
:
M div bc d : 0 bc Sb M
0
[4.46]
which means that the matrix BT of [4.43] is injective and that B is surjective.3 If we assume that Sb H (div :) and that SM L2 (:) – i.e. the approximation spaces are included in the work spaces of the continuous formulation – the compatibility condition is written: there exists E ! 0 independent from meshing such that: sup b cSb
³
:
M div bc d:
& bc &H (div: )
t E & M &L2 ( : ) M SM
[4.47]
which naturally implies [4.46]. This compatibility condition is called the “inf-sup” condition or Ladyzhenskaya-Babuska-Brezzi condition.
3 Obviously the matrix B is not injective, otherwise it would mean that the space Sb only
contains one field of vectors likely to be the solution for b . In our example, if B was injective, the only field of Sb with zero divergence would be the zero vector.
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4.5.2. Mixed formulation oriented magnetic field
For the mixed formulation oriented h , it is possible to follow the same reasoning. In order to ensure the uniqueness of a , the following is needed:
³
:
a curl hc d :
0 h ' a
0
For this purpose, it is sufficient to have the discretization space of a containing the rotational of admissible discretized magnetic fields. This condition is achieved by searching for h in a sub-space of finite dimension of H h – and therefore of H (curl :) – and for a in a sub-space of finite dimension of H *h (div 0 :) .4 It is
thus possible to write a compatibility condition for the mixed formulation oriented h similar to [4.47]. 4.5.3. General case
More generally, let us write a mixed problem discretized in the following canonical form, where all functional spaces are of finite dimension: Given f and g are two continuous linear forms – they have corresponding sources – find u U , p P solutions of: (u u c) b(u c p) f (u c)u c U ® b(u p c) g ( p c)p c P ¯
[4.48]
Formulation [4.48] leads to a system of equations similar to [4.43] except that the two sides are a priori non-zero. Let us consider the functional space: U0
{v U b(v p c)
0p c P}
as well as the following assumptions: – First assumption: there exists a constant D ! 0 , independent of the meshing, such that: a(u0 u0 ) t D & u0 &U2 u0 U 0
[4.49]
4 These different spaces have been defined during the establishment of the continuous formulation.
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161
This property is called coercivity on the space U 0 . It leads to a matrix A defined as positive in the matrix system similar to [4.43]. – Second assumption (inf-sup condition): there exists a constant E ! 0 independent of the meshing, such that: sup qP
b(v q ) t E & q &P q P & v &U
If these two assumptions are verified, then, problem [4.48] is well formed: the solution exists, is unique and depends continuously on the data. Moreover, the method of approximation is convergent – when we refine the meshing – and stable. We will now see that it is possible, in practice, to be assured that we are working on a well constructed problem without being constrained to check that the discrete inf-sup condition is satisfied, which is often a difficult mathematical task. 4.6. Mixed finite elements, Whitney elements
The literature of mixed finite elements includes “catalogs” in which we can choose a suitable discretization for each unknown factor of the formulation, so that the inf-sup condition is “automatically” satisfied. The best known are the RaviartThomas elements in 2D [RAV 77], the Nédélec elements in 3D [NED 80], [NED 86], as well as the Brezzi-Douglas-Fortin-Marini elements [BRE 85], [BRE 87]. Bossavit gave an expression of the Nédélec elements of degree 1 in terms of differential form [BOSS 93]. We thus talk about Whitney elements. We will illustrate our subject of the discretization of mixed formulations by using these elements whose definition and properties are largely known in the community of numerical calculations for electromagnetism [BOS 93]. We will recall them here. The calculation field : being meshed in tetrahedrons and its boundary * in triangles, we will adopt the following usual notations:
– W0 indicates the vector space generated by the basic functions of degree 1 associated with the vertices, affine per tetrahedron. These functions are the barycentric coordinates related to each top of the grid. They are interpolation functions of the Lagrangian nodal finite elements P1 . – W0 is a sub-space of finite dimension of H 1 (:) . – W1 indicates the vector space generated by the basic functions of degree 1 associated with the edges. It is a sub-space of finite dimension of H (curl :) .
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– W2 indicates the vector space generated by the functions of facets. It is a subspace of finite dimension of H (div :) . – W3 indicates the vector space generated by the constant functions by tetrahedron. It is a sub-space of finite dimension of L2 (:) . Moreover, we will denote by Wh1 the space of the fields of vectors confirming boundary condition [4.4] and Wb2 that of the fields of vectors checking boundary condition [4.3]. Lastly, let us recall a fundamental property of Whitney elements: grad W 0 W 1 curl W 1 W 2 div W 2 W 3
[4.50]
4.6.1. Magnetic induction oriented formulation
The mixed formulation oriented b is written, after discretization: Given h s , we find (b M ) Wb2 u W 3 by checking: s 2 Q b bc d: M div bc d: ³: ³: h bc d: bc Wb ° ³: ® 3 °¯ ³: div bM c d: 0 M c W
[4.51]
where Wb2 is the set of the fields of vectors of W 2 checking [4.3]. We should note that with the selected approximation spaces – Wb2 and W 3 – the inf-sup condition is met. Indeed, we have (see [4.46]):
³
:
M div bc d : 0 bc Wb2
with M W 3 if and only if M 0 , which is so thanks to the fact that div bc W 3 . The uniqueness of M is thus assured. As for the coercivity condition – see [4.49] – it does not pose any problem since the norm of the vector fields of W 2 whose divergence is zero is reduced to ³ _ b _2 d: . The coercivity is thus checked while :
taking as a constant D the smallest value of Q . Let us note finally that b is sought in W 2 , div b W 3 , as for M c . The second equation of [4.51] thus implies that [4.2] is met exactly in W 3 . The numerical solution will thus provide for b a field of vectors of W 2 with zero divergence, which can be written like the rotational of a field a where a W 1 . A priori, we can expect to obtain for b the rotational of the solution provided by the traditional formulation of vector potential [4.11] when a is interpolated using edge functions
Mixed Finite Element Methods in Electromagnetism
163
of W 1 . We can then speculate about the value of a mixed method that also requires the calculation of a Lagrange multiplier M , that is to say, a degree of freedom per tetrahedron. However, without the adoption of a gauge, the formulation of vector potential [4.11] led to an ill-posed problem and convergence is not assured.5 4.6.2. Magnetic field oriented formulation
The formulation oriented h is discretized in the following way using the Whitney elements. Given j W 2 and at zero divergence, find (h a ) Wh1 u W*2h (div 0 :) checking: P h h 'd: a curlh ' 0 ³: ° ³: ® ° ³: curl h a 'd: ¯
h ' W 1h
³
:
j a 'd: a ' W 2 *h (div0 , :)
[4.52]
with W*2h (div 0 :) {u W 2 div u
0 in : u n
0 on *}
Let us notice that, in a similar way to the mixed formulation oriented b , since the current source j is in W 2 and has zero6 divergence, the second equation of [4.52] implies that [4.1] is checked exactly in W 2 . A field h will thus be obtained in the form h s grad M , h s belonging to W 1 and M to W 0 . Thus, it could be thought that the mixed formulation brings only additional calculations – the fluxes of a through the facets of the grid – compared to the traditional formulation in reduced scalar potential [4.18]. However, the reduced potential allows the magnetic field of reaction to be obtained, which must be added to the source field in order to determine the total field. In the areas of strong permeability, that can cause an important loss of numerical accuracy. On the other hand, the mixed formulation straightforwardly provides the total magnetic field. Note: the nullity of the divergence of the functions of W*2h is not obtained
immediately: the divergence of a field of vectors is interpolated using the functions 5 Various numerical experiments showed that the formulation in vector potential without imposed gauge could provide correct results if we adopted a suitable discretization of the source current and if we solved the system using the method of the conjugate gradient [REN 96]. 1 6 It is possible to take j the rotational of a source field belonging to W calculated using the Biot and Savart law.
164
The Finite Element Method for Electromagnetic Modeling
of facets of W 2 as a constant quantity per tetrahedron, equal to the sum of the degrees of freedom – the fluxes through the facets – divided by the volume of the element. The nullity of the divergence on the whole grid is then obtained by imposing relations between the degrees of freedom using a tree technique. We can however note that if this condition is not imposed on the divergence, the resolution of the final linear system using an iterative conjugate gradient method converges towards the expected solution. 4.7. Mixed formulations in magnetodynamics
In this section, the mixed formulations in harmonic mode will be presented within the approximation framework of the quasi-stationary states. For this purpose, the majority of the notations that have been used in the previous sections will be preserved. A domain : is considered where a part denoted : c is made up of conducting materials, of conductivity V . Inside : there is also an inductor, denoted by : s , carrying a current j s at the pulsation Z . : c and : s are disjoined. Equations in : satisfy Faraday’s law: curl e
[4.53]
iZP h
as well as Ampere’s law: curl h V e j s
[4.54]
with j V e in : c , V being zero outside : c . Boundary condition [4.4] will be preserved and it is imposed: eu n
[4.55]
0 on *b
which implies [4.3]. It is recalled that * h *b
*.
4.7.1. Magnetic field oriented formulation
The same process as in section 4.2 for magnetostatics will be followed here for magnetodynamics: equation [4.53] is multiplied by a test field hc that meets [4.4], with an integration carried out on : , so an integration by parts is achieved; equation [4.54] is multiplied by a test field ec that meets [4.55] and an integration is carried out on : . The following formulation is then obtained, presented in [BOS 88]:
Mixed Finite Element Methods in Electromagnetism
c c ° ³: P h h d: ³: e curl h d: ® °¯ ³: V e ec d: ³: curl h ec d:
0,
³
:
165
[4.56]
j s ec d:
However, the second equation of [4.56] implies that e
1
V
curl h in :c . This
expression of e can be carried over in the first equation of [4.56] while making an integral appear on : c and an integral on its complementary : 5 : c 7. Moreover, it is supposed that there is no electric charge in : 5 :c , i.e. outside the conductors. The following mixed formulation is then obtained: Given j s , find the couple (h e) H h u H *e (div0 :) that meets: 1 ° ³: P h hc d: ³:c V curl h curl hc d: ³: 5 :c e curl hc d: ® ° curl h ec d: ³ c j s ec d: :5: ¯ ³: 5 :c
0,
[4.57]
hc H h ec H *b (div 0 :)
with H *e (div0 :) {u H (div :) div u
0 in : u n
0 on * e }
[4.58]
H h being defined by [4.15].
Mixed formulation [4.57] allows, in addition to the magnetic field all over : as well as the current via curl h in the conductors, the electric field outside the conductors to be calculated. However, this electric field only marginally meets Faraday’s law. It is easily shown, while proceeding as in section 4.2, that this formulation is equivalent to the starting equations. The treatment of the case where the conducting domain : c is not simply related does not pose any particular problem since we do not explicitly impose only curl h 0 in the non-conducting areas. If we now impose that curl h 0 outside the conductors, in the simply related case, by writing h h s grad M in [4.56], we obtain: s
7 It is the meeting of : and non-conductive parts : .
166
The Finite Element Method for Electromagnetic Modeling
P h hc d: iZ ³:5 :c P grad M grad M c d: ³:c e curl hc d: ° ³:c ° s ® iZ ³: 5 :c P h grad M c d: ° ° ³ c curl h ec d: ³ c V e ec d: 0 ¯ : :
[4.59]
Formulation [4.59] is mixed only in appearance, and there is no major value in using it in this form. Indeed, the unknown factor e can be expressed as a function of h using the second equation then carried over in the first one. We then obtain the well known non-mixed formulation in h M [BOS 93]: iZ ³ c P h hc d: iZ ³
: 5 :c
:
iZ ³
: 5 :c
P grad M grad M c d: ³
:c
s
P h grad M c hc H h hc
1
V
curl h curl hc d:
[4.60]
grad M c in : 5 : c
The discretization of [4.57] is carried out while following the rules stated in section 4.5. By using Whitney elements, h is taken in W 1 checking [4.4] and e in W 2 , with zero divergence, with zero normal component on * b . The electric field obtained will thus be with continuous normal components at the crossing of the grid facets. It is thus not possible to calculate the charges appearing at the border of the conductors. The discretization of [4.60] is obviously carried out using edge functions of W 1 for h , and nodal functions of W 0 for M . If we really want to discretize the “false” mixed formulation [4.59], it can be noted that the conditions to meet will be less severe than for a “true” mixed method. It has been seen already that the matrix of the system is very different, since it does not have a zero on the diagonal. Let us now show that the solution is unique by checking that only the couple (0,0) is a solution of the formulation without a second side. While taking hc h in the first equation of [4.59] and ec e in the second equation of [4.59] – the bar indicates the complex conjugate – we obtain: iZ ³ P _ h _2 d: ³ c V _ e _2 d: :
:
0
Thus, we find that the solution is unique – since V is strictly positive in :c – without having made an assumption on the discretization of e . It is thus not necessary to satisfy a condition of compatibility between the discretizations of h and of e . Besides, the numerical implementation of this formulation will give results different from those of the method in h M simply by adopting a non-standard discretization – for example while taking h and e in W 1 – in this case, neither
Mixed Finite Element Methods in Electromagnetism
167
Ampere’s law, nor Faraday’s law are satisfied exactly. We can then evaluate, after resolution, an error on Ampere’s law and on Faraday’s law and deduce an error indicator from it a posteriori [BAN 97], [BAN 98b]. 4.7.2. Formulation oriented electric field
Here we can proceed in a similar way to in the previous section but the integration by parts relates now to Ampere’s law rather than Faraday’s law. Another mixed formulation, announced in [BOS 88], is obtained: ° ° ® ° ° ¯°
³ ³
:
:
P h hc d: ³ curl e hc d: 0 hc H * (div0 :) e
:
V e ec d: ³ h curl ec d: :
³
s
:
s
j ec d: ec H e
[4.61]
where He
{u H (curl :) u u n
0 on * e }
H *e (div)0 :) being defined by [4.58]. With [4.61], there is still a “false” mixed formulation where we can eliminate the unknown variable h . The formulation in the electric field for magnetodynamics [BOS 90] is then obtained: iZ ³ c V e ec d: ³ :
:
1
P
curl e curl ec d:
iZ ³
:s
j s ecec H e
that we discretize by taking e in W 1 , with boundary condition [4.55]. Let us notice that, under these conditions, it is not possible to strongly impose div e 0 in : 5 :c . In the absence of charges in the areas where V is zero, the nullity of div e will have to be written weakly in the non-conducting areas by adding a
variational equation of the type
³
: 5 :c
e grad u ' d:
0 . The electric charges
appearing on the surface of the conductors can be approximated by evaluating the normal component of e on the external boundaries of the conductors. 4.8. Solving techniques
The mixed methods presented in the preceding sections – except for those described as “wrongfully” mixed in magnetodynamics – all lead, after discretization, to a linear system in the form:
168
The Finite Element Method for Electromagnetic Modeling
§ A BT · § p · ¨ ¸ ¨ ¸ © B 0 ¹ ©v¹
§f · ¨ ¸ ©g¹
[4.62]
whose matrix is infinite. It is obviously not possible to solve such a system using traditional methods such as Cholesky factorization or the conjugate gradient method. Some techniques allowing us to resolve this difficulty will be presented below. 4.8.1. Penalization methods
We will take as an example the mixed formulation oriented h for magnetostatics. The transposition of the method to any other mixed formulation will be immediate. Let us disturb equation curl h curl h H a
j slightly, so it becomes:
j
where H is a small positive parameter tending towards zero [GIR 86], [BAN 94]. The second variational equation of [4.14] becomes
³
:
curl h a c d: H ³ a a c d: :
³
:
j a c d:a c H *h (div0 :)
When H tends towards zero, we obviously find the initial system. However, for any non-zero H , there is an expression of a as a function of h which allows the unknown variable a to be eliminated in the first variational equation of [4.14a]. The formulation then becomes: Given j , find h H h such that:
³
:
P h h c d:
1
H
³
:
curl h curl hc d:
1
H
³
:
j curl hc d:hc H h
[4.63]
Let us note that variational problem [4.63] is equivalent to the following unconstrained minimization problem: Given j , find h H h such that: K H ( h)
inf KH (hc)
h cH h
Mixed Finite Element Methods in Electromagnetism
169
where 1 H P _ h _2 d: ³ _ curl h j _2 d: ³ : 2 2 :
K H ( h)
When H tends towards zero, this unconstrained minimization problem is equivalent to [4.34], the problem of minimization with constraint. The disturbance that we introduced into the equation is thus tantamount to introducing the constraint curl h j in the functional calculus of energy via a penalization [BRE 91]. Formulation [4.63] comprises only one unknown variable, the magnetic field, which will be discretized using Whitney edge functions pertaining to W 1 . A first advantage of the penalization technique is thus the significant reduction in the number of degrees of freedom. The saving achieved corresponds to the number of degrees of freedom related to the discretization of a . Before eliminating the unknown variable a , the system to be solved is in the form: § A BT · § h · ¨ ¸ ¨ ¸ © B HC ¹ ©a ¹
§0 · ¨ ¸ ©g¹
[4.64]
The elements of the matrix C have an expression of: ci j
³
:
w f i w f j d:
where w f i represents the basic function of W 2 associated with the facet i . The matrix C is obviously definite positive. We have: a
1
H
C 1 ( g Bh)
and after elimination of a ( A BT C 1 B)h
1
H
BT C 1 g
[4.65]
which is the system to be formally solved. Naturally, it is not necessary to actually build the matrix C and invert it. The discretization of [4.63] directly provides, using Galerkin’s method, the system to be solved. Yet the writing of this system in form [4.65] makes it possible to see that the matrix is symmetric and definite positive
170
The Finite Element Method for Electromagnetic Modeling
since it is written A BT C 1 B . This is a second advantage of the penalization technique: a traditional algorithm can be used to solve this system. The two advantages stated above are, however, accompanied by serious 1 drawbacks: the first is that the presence of the factor considerably degrades the
H
conditioning of the matrix. Another practical difficulty intervenes when we determine H . It must be small, but compared to what? And if it is too small, the term 1 in ends up numerically “crushing” its counterpart and the method becomes
H
inconsistent. It is thus necessary to carefully adjust the parameter H . We can specify what the choice of a small H means in practice, by noticing that 1
P 0H that
has the dimension of the square of a length. We will thus have to choose H so 1
P0H
can be large compared to a dimension characteristic of the system
studied [BAN 93]. An alternative to this method consists of directly disturbing system [4.62] by replacing it with: §A ¨ ©B
BT · § p · ¸ ¨ ¸ -H I ¹ © v ¹
§f · ¨ ¸ ©g¹
[4.66]
For the mixed oriented h formulation, this is tantamount to solving: ( A BT B ) h
1
H
BT g
which is specifically equivalent to [4.65] where the matrix C is replaced by the unity matrix [HAM 01]. For the mixed formulation oriented b , after introducing a disturbance into the continuous problem, the second equation of [4.8] becomes:
³
:
div bM c d: H ³ MM c d: :
0
The unknown variable M being interpolated using the basic functions of W 3 constant for each tetrahedron, the elements of the matrix C are written:
Mixed Finite Element Methods in Electromagnetism
ci j
³
:
171
wie wej d:
where wi is the function of W 3 associated with the element i . We obtain C I . The penalization on the linear system of equations gives, in this case, the same results as the penalization on the continuous problem. 4.8.2. Algorithm using the Schur complement
Another technique for solving system [4.62] consists of eliminating the principal unknown variable in order to preserve only the Lagrange multiplier. It is supposed that the two assumptions stated in section 4.5.3 are checked. The matrix A is then definite positive. It follows: p
A1 ( f BT v)
The system to be solved is then the following: B A1 BT v
B A1 f g
[4.67]
Let us suppose that the matrix A is symmetric, which is the case for all the mixed formulations that we have presented, then, B A1 BT also being definite positive, [4.67] can be solved by the conjugate gradient method. It is not necessary to calculate A1 explicitly. It is enough, for each iteration of the combined gradient method, to solve a linear system associated with the matrix A . This method is called the mixed Schur complement. We will show how it is implemented in the case of mixed formulations [4.8] and [4.14]. For the mixed formulation oriented h , the system obtained by Galerkin’s method is: § A BT · § h · ¨ ¸ ¨ ¸ 0 ¹ ©a ¹ ©B
§0 · ¨ ¸ ©g¹
We obtain: h
A1 BT a
and the system to be solved by the conjugate gradient is: B A1 BT a
g
[4.68]
172
The Finite Element Method for Electromagnetic Modeling
It is thus possible to use the following algorithm, which allows h and a 8 to be calculated simultaneously [BAN 98a]: Initialization: a0
0
h0
0
r0
B h0 g
p0
r0
g
Iteration n : Calculate xn by solving A xn qn
B xn
Un
& rn &2 (qn pn )
hn 1
hn U n xn
an 1
an U n pn
rn 1
rn U n qn
E n 1
& rn 1 &2 & rn &2
pn 1
rn 1 E n 1 pn
Stop when
BT pn
& rn 1 & d H given. & r0 &
8 In the algorithm, bold characters are no longer used to indicate the vectors of degrees of freedom.
Mixed Finite Element Methods in Electromagnetism
173
For the mixed formulation oriented b , the initial system is given by [4.43]. It follows: A1 f A1 BT M
b
and the system to be solved by the conjugate gradient method is: B A1 BT M
B A1 f
It is thus possible to build an algorithm similar to the previous one [BAN 01] in order to calculate b and M . Only the initialization will be different. It follows: Initialization for the method oriented b: 0
M0
Calculate b0 by solving Ab0 r0
B b0
p0
r0
f
The convergence speed of these algorithms can be improved by using the augmented Lagrangian technique which is tantamount to preconditioning the matrix B A1 BT . For the two mixed formulations that have just been presented above, the augmented Lagrangian technique comes in fact to add the term: r ³ curl h curl hc d: :
to the first variational equation of the mixed formulation oriented h , as well as the term r ³ div b div bc d: :
for the formulation oriented b . Let us highlight that, unlike what was stated in connection with the penalization methods, the augmented Lagrangian method does not degrade the conditioning of the matrix because the coefficient r does not need to be very large. It is however useful to adjust this parameter in order to obtain the fastest possible convergence [BAN 98a]. Let us specify finally that if it influences – in a significant way – the computing time, the augmented Lagrangian technique does not modify the results.
174
The Finite Element Method for Electromagnetic Modeling
In addition, it is possible to facilitate the resolution of the systems associated with the matrix A by proceeding to the hybridization of the formulation as indicated in section 4.4.3. A matrix A is then obtained diagonally for each block [ARN 85], [BRE 94], [QUA 97]. Both variants of the penalization method, the mixed Schur complement method for the formulation oriented h as well as for the formulation oriented b were compared on solving a nonlinear magnetostatic test problem: Problem 13 of the TEAM Workshop [NAK 90]. They have all yielded results rather better than conventional methods in scalar potential or vector potential, with substantively equivalent performances [BAN 98a], [HAM 01], [BAN 01]. 4.9. References [ALO 98] P. ALOTTO, F. DELFINO, P. MOLFINI, M. NERVI, I. PERUGIA, “A mixed face-edge finite element formulation for 3D magnetostatic problems”, IEEE Trans. Mag., 34 (5), p. 2445–2448, 1998. [ARN 85] D.N. ARNOLD, F. BREZZI, “Mixed and non-conforming finite element methods: implementation post-processing and error estimates”, Math. Modelling Numer. Anal., 19, p. 7–35, 1985. [BAN 93] B. BANDELIER, C. DAVEAU, F. RIOUX-DAMIDAU, “An h-formulation for the computation of magnetostatic fields. Implementation by combining a finite element method and a boundary element method”, J. Phys. III, 3 (5), p. 995–1004, 1993. [BAN 94] B. BANDELIER, C. DAVEAU, F. RIOUX-DAMIDAU, “A new h-formulation for nonlinear magnetostatics in R3, IEEE Trans. Mag., 30 (5), p. 2889–2892, 1994. [BAN 97] B. BANDELIER, F. RIOUX-DAMIDAU, “Formulation variationnelle à deux champs pour la magnétodynamique dans R3”, J. Phys. III, 7 (9), p. 1813–1819, 1997. [BAN 98a] B. BANDELIER, F. RIOUX-DAMIDAU, “Mixed finite element method for magnetostatics in R3”, IEEE Trans. Mag., 34 (5), p. 2473–2476, 1998. [BAN 98b] B. BANDELIER, F. RIOUX-DAMIDAU, “Mixed formulation of magnetodynamics in R3. A posteriori error”, IEEE Trans. Mag., 34 (5), p. 2664–2667, 1998. [BAN 01] B. BANDELIER, F. RIOUX-DAMIDAU, “A mixed B-oriented finite element method for magnetostatics in unbounded domains”, COMPUMAG International Conference, Evian, July 2001. [BOS 88] A. BOSSAVIT, “A rationale for edge-elements in 3-D field computations”, IEEE Trans. Mag., 24 (1), p. 74–78, 1988. [BOS 90] A. BOSSAVIT, “Le calcul des courants de Foucault en dimension 3, avec le champ électrique comme inconnue. I: Principes”, Journal of Appl. Phys., 25 (2), p. 189–197, 1990.
Mixed Finite Element Methods in Electromagnetism
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[BOS 93] A. BOSSAVIT, Electromagnétisme, en vue de la modélisation, Springer-Verlag, 1993. [BRE 85] F. BREZZI, J. DOUGLAS, L.D. MARINI, “Two families of mixed finite elements for second order elliptic problems”, Numer. Math., 47, p. 217–235, 1985. [BRE 87] F. BREZZI, J. DOUGLAS, R. DURAN, M. FORTIN, “Mixed finite elements for second order elliptic problems in three space variables”, Numer. Math., 51, p. 237–250, 1987. [BRE 91] F. BREZZI, M. FORTIN, Mixed and Hybrid Finite Element Methods, SpringerVerlag, 1991. [BRE 94] F. BREZZI, D. MARINI, “A survey on mixed finite element approximations”, IEEE Trans. Mag., 30 (5), p. 3547–3551, 1994. [DUL 97] P. DULAR, J.F. REMACLE, F. HENROTTE, A. GENON, W. LEGROS, “Magnetostatic and magnetodynamic mixed formulations compared with conventional formulations”, IEEE Trans. Mag., 33 (2), p. 1302–1305, 1997. [GIR 86] V. GIRAULT, P.A. RAVIART, Finite Element Methods for Navier-Stokes Equations, Springer-Verlag, 1986. [HAM 01] L. HAMOUDA, B. BANDELIER, F. RIOUX-DAMIDAU, “A perturbation technique for mixed magnetostatic problem”, IEEE Trans. Mag., September 2001. [NAK 90] T. NAKATA, N. TAKAHASHI, K. FUJIWARA, K. MURAMATSU, P. OLEWSKI, “Analysis of magnetic fields of 3D non-linear magnetostatic model (Problem 13)”, Proceedings of the European TEAM Workshop and International Seminar on electromagnetic fields analysis, Oxford, April 1990, p. 107-116. [NED 80] J.C. NÉDÉLEC, “Mixed finite elements in R3”, Numer. Math., 35, p. 315–341, 1980. [NED 86] J.C. NÉDÉLEC, “A new family of mixed finite elements in R3”, Numer. Math., 50, p. 57–81, 1986. [QUA 97] A. QUARTERONI, A. VALLI, Numerical Approximation of Partial Differential Equations, Springer-Verlag, 1997. [RAV 77] P.A RAVIART, J.M. THOMAS, “A mixed finite element method for second order elliptic problems. Mathematical aspects of the finite element method”, I. Galligani, E. Magenes (eds.), Lectures Notes in Math, 606, Springer-Verlag, 1977. [REN 96] Z. REN, “Influence of R.H.S. on the convergence behaviour of the curl-curl equation”, IEEE Trans. Mag., 32 (3), p. 655–658, 1996. [ROB 91] J.E. ROBERTS, J.M. THOMAS, “Mixed and hybrid methods”, in Handbook of Numerical Analysis, P.G. Ciarlet and J.L. Lions (eds.), Elsevier Science Publishers B.V., North Holland, 1991.
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Chapter 5
Behavior Laws of Materials
5.1. Introduction Solving Maxwell’s equations numerically by using the finite element method makes it possible to take into account material behaviors a priori unspecified (nonlinear, anisotropic, with or without hysteresis). For that purpose, adequate behavior models and implementing resolution algorithms for nonlinear problems are needed. It is to be noted that the electromagnetic behavior of materials in the field of electrical engineering remains only very approximately represented in software available on the market at the present time. This is particularly the case for magnetic materials. Therefore, we will specifically deal with these materials in this chapter. In addition, the behavior of superconductors will also be covered. The lack of behavior models actually used within the framework of the finite element method is partly explained by the fact that the microscopic phenomena at the origin of the macroscopic behavior of magnetic materials are complex and difficult to model. The transition from microscopic to macroscopic is not yet well understood. There is a real difficulty in finding models of magnetic behavior that achieve a good trade-off between accuracy and numerical simplicity for an effective integration in a software tool based on the finite element method. The “phenomenological” models traditionally suggested are in general quite far away Chapter written by Frédéric BOUILLAULT, Afef KEDOUS-LEBOUC, Gérard MEUNIER, Florence OSSART and Francis PIRIOU.
178
The Finite Element Method for Electromagnetic Modeling
from physical reality and the use of more realistic models of behaviors is still an object of research. The chapter starts by presenting the behavioral characteristics of magnetic materials: nonlinearity, anisotropy and hysteresis. Then two methods dealing with solving nonlinear problems are shown within the framework of the finite element method. Concrete examples of behavior models, dedicated to various types of problems or materials, are then studied: anisotropy of sheets with oriented grains, hysteresis and dynamic behavior of sheets, calculation of iron losses, behavior of the permanent magnets, and finally modeling the electric behavior of the superconductors. It should be noted that an exhaustive review of all the models of existing behavior is not our concern here. We are instead dealing with the analysis of some particular models. Our goal is to show which poses difficulties for an accurate modeling of the behavior of materials and which approaches can be used. Indeed, in this field, only a few models have actually proved effective. 5.2. Behavior law of ferromagnetic materials 5.2.1. Definitions A magnetic behavior law is defined as the macroscopic relationship that binds magnetization M to the local field H in any point of a material. This behavior law translates the fact that the magnetization of matter is modified under the effect of a magnetic field. In physics, this law is written in the form [5.1], where F(H) is the tensor of magnetic susceptibility. The quantities M and H are expressed in A/m.
M ( H ) = [F(H )]H
[5.1]
In electrical engineering, where the useful quantity is induction, representing the magnetic behavior of a material through expression [5.2], where P(H) is the tensor of magnetic permeability, is preferred.
B(H ) = [P( H )]H
[5.2]
Constitutive relation [5.3], where P0 is the permeability of the vacuum (4S.10-7 H/m), establishes the link between these two formalisms.
B(H) = P O (H + M (H ))
[5.3]
Behavior Laws of Materials
179
This relation is also written according to magnetic polarization J, expressed in Tesla. Often polarization and magnetization designations are wrongly confused. B(H) = PO H + J (H ) with J (H ) = P O M (H )
[5.4]
The tensors of magnetic permeability and reluctivity are bound by: [5.5]
[P(H )] = P O ([Id] [F(H )]). 5.2.2. Hysteresis and anisotropy
In general, the magnetic materials used in electrical engineering are of ferromagnetic or ferrimagnetic nature. Their behavior has simultaneously hysteretic and anisotropic features, in a way more or less marked depending on the materials. As an example, Figure 5.1 shows the behavior of a non-oriented grain sheet of completely traditional nuance. The left curves show two hysteresis cycles measured in the rolling and transverse directions (RD and TD respectively). It should be noted that the cycle is slightly more horizontal in direction TD. The right-hand side curves show the evolution of the field which it is necessary to apply to obtain a rotating induction of amplitudes 1.0 and 1.4 T respectively. If the sheet was “perfectly nonoriented” (that is isotropic) these curves would be circles. The anisotropy of the material appears clearly, and it is more marked than can be foreseen when looking at the loops in directions RD and TD. 1200
800
HDT (A/m)
400
0 -400
-800
cir 1T cir 1.4T
-1200 -1200
-800
-400
0
HDL (A/m)
400
Figure 5.1. Behavior of a sheet with non-oriented grains under unidirectional applied field (left) and rotating induction (right)
800
1200
180
The Finite Element Method for Electromagnetic Modeling
Nonlinearity, anisotropy and hystereses characterize the behavior of all magnetic materials, but in a more or less marked way according to the type of material considered. Finding a general model appropriate for all materials and under all operating conditions is unrealistic. The micromagnetism equations are always the same, but the dominant magnetization mechanisms (displacement of domain walls or rotation of magnetization) depends at the same time on material (composition, texture) and on the stress type which it undergoes (unidirectional or rotating field, of low or high amplitude). On the other hand, we can hope for more or less accurate models depending on the goal of the study (global sizing of a machine or deepening a particular local phenomenon). It is thus worthwhile considering the phenomenonkeys to take into account for material considered under the conditions of its use in order to choose a model adapted to the phenomena studied. 5.2.3. Classification of models dealing with the behavior law In the most general case, the behavior law M(H) is a hysteretic and anisotropic nonlinear vector relation. In practice, it is taken into account by fairly simple models according to the stress type to which the material is subjected and according to the desired degree of accuracy. The various types of models are listed below. 5.2.3.1. Linear models These are the most elementary models. The behavior law is given by: B(H) = [P] H + Jr
[5.6]
where [P] is the tensor of material permeability assumed to be constant, and where Jr is a residual polarization, zero for soft materials and independent of the field in the case of ideal permanent magnets (Figure 5.2). This model is only appropriate for the weak stress, i.e. for materials used in a weak field (soft ferrites, for example) or for certain parts of devices.
Behavior Laws of Materials
B
B
B = P.H
B = P.H+Br
H
H
(a)
181
(b) Figure 5.2. Linear approximation in the case of soft materials (a) and in the case of hard materials (b)
5.2.3.2. Nonlinear isotropic models A linear behavior model has a very limited domain of validity. Generally, for a realistic simulation, it is necessary to use at least a nonlinear isotropic model of behavior in order to take into account the saturation of material. Under this assumption, the vectors B and H are collinear and the vector relation B(H) can be rewritten as a scalar relation between B, amplitude of vector B and H, amplitude of vector H. This relation is put in the form: B(H) = P(H).H
[5.7]
The permeability of material P(H) is reduced to a scalar function of the amplitude of the applied field. In this approximation, the soft materials are modeled by their curve of the first magnetization, which is justified because the coercive force is very low. The permanent magnets are modeled by their major cycle, but it should be checked a posteriori that the operating point remains in the zone of reversibility of the cycle and that there is no demagnetization of the material (Figure 5.3).
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The Finite Element Method for Electromagnetic Modeling
B
B
B = P(H).H
H
B = P + .H+Br
H
Figure 5.3. Hysteresis neglected in soft materials and hard materials
The simplest way to treat this type of behavior is to describe the scalar relation B(H) point by point starting from experimental statements and to use a good interpolation technique. Using analytical functions or prediction models of the behavior does not improve the accuracy of the finite element simulation itself. It is only interesting if we do not have experimental data. 5.2.3.3. Nonlinear anisotropic models Under these assumptions, the behavior law is written as: B(H) = [P(H)] H
[5.8]
Permeability arises in tensorial form, of which each component depends on the field applied. There are simplified models, but their domains of validity are limited (decoupling of the behaviors in the various directions in the domain of very low fields, elliptic interpolation for materials whose anisotropy is low). In general, nonlinear anisotropic models are still under development. We will present one model for the case of oriented grains sheets. 5.2.3.4. Hysteretic models In this case, the behavior depends not only on the current value of the field applied, but also on its history. B(H) = B(H, history)
[5.9]
Straightforward hysteresis models exist only in the scalar case, i.e. if we restrict ourselves to the case where B and H are collinear (axis of anisotropy or constant direction for a material not oriented). Under these conditions, the Preisach model is considered as a reference.
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5.3. Implementation of nonlinear behavior models As was shown previously, the behavior law of magnetic materials is nonlinear. The finite element equation is thus itself nonlinear and cannot be solved directly. Solving a nonlinear vectorial equation is not always easy. Various techniques exist, whose convergence is never guaranteed. The main approaches are the Newton method and the fixed point method. 5.3.1. Newton method 5.3.1.1. Principle Let us consider a function F(X) defined in vector space Rn. The Newton method allows the zero of this function to be determined starting from its Taylor series development to the 1st order. In the vicinity of an unspecified point Xk, this development can be written: ª dF º F(X k 'X) = F(X k ) « T (X k ) » 'X ¬ dX ¼
[5.10]
The algorithm consists of building a succession of linear problems by canceling the Taylor series development in the vicinity of the solution obtained with the preceding iteration. Consequently, the Xk+1 solution of the iteration k+1 verifies F(Xk+1+ǻX) = 0 with ǻX = Xk+1 - Xk. The equation to be solved is thus: ª dF º «¬ dX T (X k ) »¼ 'X = -F(X k )
[5.11]
ª dF º ª dF º The matrix « T » having a general term « i » is called the Jacobian matrix of ¬ dX ¼ ¬« dX j ¼» the system. The F(Xk) vector is called a residue. The smaller the residue, the closer the approximate solution Xk is to the real solution.
The most common convergence criterion is based on the norm of the increment ¨X. It is estimated that the approximate solution is correct when its relative variation between two iterations goes down below a certain threshold Hr: convergence criterion 1: X k -X k-1 d H r Xk
[5.12]
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The Finite Element Method for Electromagnetic Modeling
Another, in principle more correct, possibility, consists of testing the norm of the residue in order to obtain an absolute measurement of the quality of the solution. The smaller the residue is, the better the approximate solution can be: convergence criterion 2: F(Xk ) d H
[5.13]
5.3.1.2. Application to the magnetostatic formulation in vector potential In the case of a magnetostatic problem with a formulation in vector potential A such as B = curl A, the functional calculus has as an expression:
F
³: ^³
B
0
`
[5.14]
H T .dB-J.A d:
with J being the density of current source. For a given meshing, the solution of the finite element problem is obtained by minimizing the derivative of F with respect to the unknown nodal variables Ai. Therefore, the zero of the function F/Ai is to be determined. This function is obtained by composing the derivative as follows: wF wA i
w
³: ®¯ wB ³
B
0
wwAB - wAw (J.A)¾¿½d:
H T .dB .
i
[5.15]
i
The functions of nodal approximation being noted Di, the potential A and induction B are written respectively: N
A=
¦ A .D i
N
i
and B =
i 1
¦ A .curl D i
i
[5.16]
i 1
and therefore the expression of their derivative wA wAi
Di and
wB wAi
curl.D i
[5.17]
While deferring to expression [5.15], we obtain:
wA wA i
³: {curl D
T i
.H-J. D i } d:
[5.18]
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185
The Jacobian matrix is once again derived, and thus it can be written:
ª w2F º « » «¬ wA i wA j »¼
°
³: ®°curl D
T i
.
¯
wH ½° ¾ d: wA j ¿°
[5.19]
By using the derivative of composed functions, the H/Aj term is written as: wH wA j
ª wH º wB «¬ wB T »¼ . wA j
ª wH º «¬ wB T »¼ . curl D j
[5.20]
from where we obtain the expression of the Jacobian: ª w2F º « » ¬« wA i wA j ¼»
³: ®¯curl D
T i
½ ª wH º . « T » . curl D j ¾ d: ¬ wB ¼ ¿
[5.21]
The system to be solved, at each iteration (k) in Newton, can finally be written: [J]k . ([A]k+1 - [A]k) = [R]k
[5.22]
where [R] and [J] are respectively the matrices of a general term: [R i ] - ³ {curl Di T .H(B) - J. Di } d: : [J ij ] -
³: ®¯curl D
T i
½ ª wH º . « T » . curl D j ¾ d: ¬ wB ¼ ¿
[5.23] [5.24]
The physical data necessary to establish the system are thus the H(B) curve for calculating the residue and the derivative [dH/dBT] for calculating the Jacobian. In a Cartesian reference, this tensor is given by expression [5.25]. It is called the incremental reluctivity tensor, as opposed to the classical reluctivity tensor.
ª wH º «¬ wB T »¼
ª wH x « wB « x « wH y « ¬« wBx
wH x º wBy » » wH y » » wBy ¼»
[5.25]
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The Finite Element Method for Electromagnetic Modeling
5.3.1.3. Application to the magnetostatic formulation in scalar potential For a problem that does not comprise a current source, it is possible to work in scalar potential such as H = -gradI The functional calculus is then expressed as: F
³: ³
H
0
[5.26]
B T .dH d:
In order to minimize this, it is necessary to cancel the derivative with respect to nodal unknown variables [dF/dIi]. By using the derivation of composed functions as well as the expression of H and I with respect to the nodal unknown variables and form functions Di, we obtain: ª wF º « » ¬ wIi ¼
³: {grad D
T i
.B(H )} d:
[5.27]
The Jacobian matrix can be written: ª w2F º « » ¬« wIi wI j ¼»
³:{grad D
T i
ª wB º . « T » . grad D j} d: ¬ wH ¼
[5.28]
The Newton method applied to this function gives the succession of the following linear problems: [J]k . ([I]k+1 - [I]k) = [R]k
[5.29]
where [I] is the vector of the unknown variables at meshing nods and where the residue vector [R] and the Jacobian matrix [J] have as their respective general terms: Ri
-³ {grad D i T .B(H )} d: :
J ij
³:{grad D
T i
ª wB º . « T » . grad D j} d: ¬ wH ¼
[5.30]
[5.31]
The nonlinear behavior of materials is introduced without any particular theoretical difficulty, starting from the curve of magnetization B(H) and the incremental permeability tensor [dB/dHT]. 5.3.1.4. Notes on the convergence process The Newton method converges quickly when the function F(X) satisfies certain conditions of monotony and when the iterative process starts from an initial point
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187
close to the solution. This is particularly the case for evolutionary equations, when the initial point is the solution calculated with the step of previous time. However, these conditions are not always met and thus this method may fail to succeed. A solution then consists of weakening the problem, i.e. to take for the solution: Xk+1 = Xk + D ¨X with 0 < D 1
[5.32]
The quality of the solution can be evaluated when the convergence criterion is satisfied by calculating an error which uses the norm of the residue, weighted by the norm of the source term. Generally it can be noted that the problems dealt within the formulation in vector potential converge more easily than those dealt with in the formulation in scalar potential, except in the particular case of problems including a closed magnetic circuit. The Newton method requires, for each iteration, the calculation and inversion of the Jacobian matrix of the problem. To limit the calculation burden, it is usual to not make this update systematically and to perform an evaluation every 5 or 10 iterations, for example. Indeed, it is important to note that the Jacobian gives a direction of research, but does not affect the quality of the calculated solution at all. This solution can in fact be estimated from the residue. 5.3.2. Fixed point method The Newton method has the advantage of converging quickly in the vicinity of the solution, but it is sometimes difficult to approach this vicinity. We would then prefer the fixed point method, also known as the Gauss-Seidel method. The convergence of this method is slower (linear convergence instead of quadratic), but it is more robust if certain parameters are chosen. 5.3.2.1. Picard-Banach fixed point theory The iterative fixed point method comes from the theorem of the same name. Let y = f(x) be an application of a complete space HS in itself, and let <x,y> and |x| denote the scalar product and the norm defined in this space. The function f is known as Lipschitzian if there is a finite real reality / such that: (x’,x”) SH2
]f(x’)-f(x”)]b /]x’-x”]
If in addition / is lower than 1, the function f is said to be contracting.
[5.33]
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The Finite Element Method for Electromagnetic Modeling
The function f is uniformly monotone if a strictly positive real O exists, such that: (x’,x”) SH2
p O < x’-x”,x’-x” >
[5.34]
As an example, for a given curve B(H), / corresponds to the maximum permeability and O to the minimum permeability, equal to the permeability of a vacuum. Let us consider x*, a point of space SH. x* is a fixed point of f if it satisfies: x* = f(x*). Fixed point theorem stipulates that if f is uniformly monotone and contracting, then there is a unique fixed point x*. This point is the limit of the series defined by xn+1 = f(xn), for any initial point x0. This sequence thus generates the algorithm for obtaining the fixed point. The convergence of this algorithm is linear, and therefore slow, but unconditional, even in the presence of points of inflection. The Picard-Banach theorem makes it possible to find, whenever it exists, the fixed point of a function which is solely Lipschitzian. Let us consider f a Lipschitzian function uniformly monotone, let us consider D a real number, and let us consider the function f2 defined by: f2(x) = x - D {f(x)-x}
[5.35]
It is possible to show [HAN 75] that the equation y = f(x) admits a single solution if f2 is contracting, which is the case if D belongs to the interval [0.2O//2]. According to the Picard-Banach theorem, this solution is the fixed point of the function f2 and it can thus be calculated by the iterative process of substitution xn+1 = f2 (xn), which converges whatever the initial point x0 is. 5.3.2.2. Application to finite element methods If we use the relations B = P0(H+M), the formulation in magnetic vector potential A of the magnetostatic equation is written: curl (Q0 curl A) = J + curl M(B)
[5.36]
After discretization by the finite element method, the problem consists of solving the following algebraic system: [S].A = Q(A)
[5.37]
with: Sij
³: Q
0
curl Di T .curl D j d:
[5.38]
Behavior Laws of Materials
Qi
³: (J + curl M(B)) D
i
d:
189
[5.39]
The solution of the problem is thus the fixed point of the nonlinear function GA(A) defined by: GA(A) = [S]-1 Q(A) . In the case of the formulation in scalar magnetic potential I and in the absence of current, the equation with partial derivatives can be written: div (grad I div M(H)
[5.40]
After discretization, the system to be solved is: [T].I = R(I)
[5.41]
with: T
Tij
³: grad D
Ri
³: div M(H) D
i
.grad D j d:
i
d:
[5.42]
[5.43]
The problem then consists of finding the fixed point of the nonlinear function GI(I) defined by: GI I >7@-1R I . After complex mathematical elaboration, the work presented in [HAN 75] shows that if the local behavior B(H) is given by a uniformly monotone Lipschitzian function, then the global functions GA and GI are Lipschitzians, continuously uniform. The Picard-Banach theorem then allows in each case contracting functions G2A and G2I to be built, admitting respectively the same fixed points GA and GI. The solutions [A*] and [I*] of these problems are thus respectively the limits of the series [An+1] = G2A[An] and [In+1] = G2I[In]. In practice, this new G2 function is obtained naturally by using the following fictitious constitutive relation: B = PPF ( H + MPF )
[5.44]
where PPF is a fictitious permeability on which the convergence of the algorithm depends. MPF is a source term, also fictitious, similar to the real magnetization and corrected at each iteration of the calculation in agreement with the model of behavior law B[H].
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The Finite Element Method for Electromagnetic Modeling
The permeability PPF is chosen in such a way as to make G2A and G2I contracting and to ensure the convergence of the iterative process. It is even possible to find an optimal value of PPF, i.e. a value which ensures a faster convergence of the process. These values of PPF depend on the permeability of modeled material, but also on the formulation used to calculate the linear problem. 5.3.2.3. Formulation in vector potential A For each iteration (n), the equation representing hysteretic material is:
§ 1 · curl ¨ .curl A (n ) ¸ curl M PF(n ) P © PF ¹
[5.45]
The value of MPF is calculated for each iteration and recalculated with respect to the induction B = curl A: M PF(n 1) =
1 (n ) B - H[B (n ) ] P PF
[5.46]
The convergence condition is written: PPF < 2 Pmin
[5.47]
and the speed of the convergence is maximum if:
P opt
2
P min .P max P min P max
[5.48]
After the convergence of the iterative process, the real magnetization is calculated starting from the behavior relation H[B] and the constitutive material relation B = P0 (H+M). 5.3.2.4. Formulation in scalar potential I For each iteration (n), the equation dealing with the hysteretic material is: div(PPF grad I n)) = div (PPF MPF(n))
[5.49]
The value of MPF is corrected for each iteration by applying the B[H] model to the H=-grad I field: M PF (n 1)
1 B[H (n ) ]-H (n ) P PF
[5.50]
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191
The convergence condition is written:
1 P PF ! P max 2
[5.51]
and the speed of the convergence is maximum if:
P opt
P min P max 2
[5.52]
The real magnetization is calculated starting from the behavior law B[H] and the constitutive relation of the material B = P0 (H+M). 5.3.2.5. Notes on the convergence process The convergence method can be very slow. It is thus necessary to be wary of a convergence criterion that is based on the solution variation between two iterations. This variation can be very small without making the current solution close to the real solution. It is thus preferable to choose a test on the norm of the residue, which is a more robust criterion. Convergence depends on the value of fictitious permeability PPF selected. An inadequate value can cause the divergence of the iterative process, or slow down convergence so much that the algorithm can become “stuck”. In the case of a uniform problem, the permeability Popt which allows a fast convergence should be close to the usual permeability of the material. In practice, the field and thus the permeability are never uniform. Thus, we cannot work with a value of PPF that is adapted to all the local points of operation. The convergence process becomes slower as the local values of permeability are dispersed. On the other hand, it is possible, during an evolutionary calculation, to choose with each time step an optimized value for the usual operating point. Except for particular cases, the number of iterations necessary for convergence is often about 100, but it should be stressed that only the second member of the equation is recalculated. These iterations are thus much less time-consuming than those of the Newton method, which requires the Jacobian matrix of the system to be recalculated and inverted. 5.3.3. Particular case of a behavior with hysteresis The problems comprising a hysteretic material are treated with the same algorithms as for traditional nonlinear problems. The only difference is that at each point where the behavior of material must be evaluated (generally, Gaussian points),
192
The Finite Element Method for Electromagnetic Modeling
it is necessary to calculate a curve of local magnetization, a function of the local history: B = f(H, local history). It is thus necessary to establish data structures which make it possible to memorize the history, according to the model of hysteresis used. We can a priori use one or another of the methods described above, but it seems more difficult to make the Newton method converge than the fixed point method. These convergence difficulties can be attributed to the point of inflection that exists in the hysteresis cycle, in the vicinity of the coercive field. We examined the various types of behavior, and then presented the two main methods dealing with nonlinear problems. We will now describe some particular models of behavior, developed for materials normally used for the construction of electric machines. Thus, we will successively present examples of models dedicated to sheets, to permanent magnets, and finally to superconductors. 5.4. Modeling of magnetic sheets 5.4.1. Some words about magnetic sheets [BRI 97] Magnetic sheets and more particularly the FeSi alloys play a primary part in the construction of the electric machines, thanks to the excellent compromise they offer between technical qualities and cost of material. The FeSi alloys are used in the form of thin sheets (thickness lower than 1 mm) in order to limit the development of the eddy currents in dynamic modes. The magnetic circuits of the electric machines are then made of stacking sheets cut out beforehand with selected dimensions. Two main categories of sheet exist. Those sheets with oriented grains (GO SiFe) corresponding to a very precise texture (Goss texture) optimize the magnetic properties in the rolling direction (excellent permeability, low losses) to the detriment of the properties in the other directions. These sheets have a very strong anisotropy and are used primarily for the construction of the magnetic circuits of transformers. On the contrary, the sheets with non-oriented grains (NO) have equivalent properties in all directions. They are used in usual rotating machines. 5.4.2. Example of stress in the electric machines In electric machines, all directions of the sheet plane are stressed. Moreover, in certain areas of the circuit, the flux is not unidirectional but rotates in the sheet plane without ever canceling itself. Figure 5.4 shows some examples of B trajectory obtained by 2D finite element simulation of a cage type asynchronous machine.
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193
Y
1 2
X 3
4
5
6
7
9 Rotor
Stator
8
a) 1
B1
1
B2
0.5
0.5
By (T)
By (T)
B3 0
-0.5
-1 -1
B5
B4
0
-0.5
-0.5
0
0.5
-1 -1
1
-0.5
0
Bx (T)
0.5
1
Bx (T)
1 1
B7
B9 B6
0.5
B8 0
By (T)
By (T)
0.5
-0.5
-1 -1
b)
0
-0.5
-0.5
0
Bx (T)
0.5
1
-1 -1
-0.5
0
0.5
1
Bx (T)
Figure 5.4. a) Detail of the asynchronous cage machine: localization of the points of observation; b) trajectory of induction B in various points of stator [SPO 98]
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The Finite Element Method for Electromagnetic Modeling
5.4.3. Anisotropy of sheets with oriented grains 5.4.3.1. A very marked anisotropy Sheets with oriented grains have a very anisotropic and complex magnetic behavior. They have two axes of easy magnetization (rolling direction RD and transverse direction TD) and an axis of difficult magnetization at 55° in the rolling direction. Figure 5.5 illustrates the complexity of this anisotropy. We have traced the trajectory of induction for a field of fixed direction (75° of the rolling direction) with an increasing amplitude
Figure 5.5. Polar representation of induction B in a sheet GO when a field H of increasing amplitude is applied to 75° direction RD, according to G-M.Fasching [FAS 64]
For low values of the field, induction is very strongly attracted by the rolling direction. Then, when the field increases, for a sufficient energy level, the induction tips up towards the transverse direction and only approaches the direction of the field very gradually, avoiding the difficult direction at 55°. It is interesting to note that the projection of B on H is a continuously increasing function, whereas the module of induction decreases slightly at the time of the swing of B towards the transverse direction. It is obvious that the data of the behaviors in the rolling and transverse directions is not enough to characterize these sheets completely. 5.4.3.2. The coenergy model [PER 94] Modeling the 2D behavior of magnetic sheets requires the determination of a relation between two vector quantities, which is not simple. Many models simplify the problem excessively: the axes separation model [NAK 75], the two axes model [HUT 84], the ellipse and the rocked ellipse model [DIN 83]. Other authors
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195
endeavored to build interpolations starting from experimental data, which requires measuring the behavior of material in many directions [WEG 76] [ENO 94]. An alternative method consists of using the magnetic coenergy W’ because it is a scalar quantity, and therefore easier to handle than the vector quantity B. The voluminal density of magnetic coenergy stored in material, noted w’, is defined by relation [5.53]. By simplifying, we will only speak about coenergy w’.
w'(H ) =
³
H
0
[5.53]
B(H )dH
In the case of a material without hysteresis, it can be shown that the vectorial relation B(H) is equivalent to the relation w’(H). Knowledge of the behavior B(H) makes it possible to calculate the relation w’(H) by integration. Conversely, knowledge of the relation w’(H) allows the behavior B(H) to be determined by derivation: B(H) = gradH w’(H)
[5.54]
The equivalence between B(H) and w’(H) has led P. Sivester and R. Gupta [SIL 91] to propose modeling the anisotropic behavior of sheets by the means of isocoenergy lines in the plan (Hx, Hy), Hx and Hy being the field components respectively in the rolling direction RD and the transverse direction TD. T Péra has completely achieved the idea: he proposed an analytical expression for the isocoenergy lines and set up the numerical tools necessary for the correct operation of the model. The experimental data used as inputs for the model are simply B(H) anhysteretic curves measured in directions RD and TD. These curves allow w’(H) to be determined in these directions. The iso-coenergy lines are constructed starting from phenomenological considerations on the behavior of material in the intermediate directions. In the plane (Hx, Hy), the iso-coenergy line corresponding to the value w’0 is modeled by expression [5.55], in which Hx0 and Hy0 are the intersections of the line respectively with axes RD and TD (Figure 5.6). n
§ Hx · § Hy · ¨ ¸ ¨ ¸ © Hx 0 ¹ © Hy0 ¹
n
1
[5.55]
The parameter n controls the shape of the iso-coenergy lines, a function of the coenergy level. With the low coenergy values, the anisotropy is very marked and the apparent difficult direction is close to 90°. That results in a line of iso-coenergy of rectangular form, with Hxo << Hyo. For high coenergy values, the difference between directions RD and TD is smaller and the direction of difficult magnetization
196
The Finite Element Method for Electromagnetic Modeling
approaches the crystallographic difficult direction at 54.7°. The lines of iso-coenergy swell. This change in form is taken into account by the parameter n, according to the coenergy level (Figure 5.6).
Figure 5.6. Representation of iso-coenergy lines in the plane (Hx,Hy)
The function w’(H) having been modeled, the behavior B(H) is calculated by deriving w’(H) with respect to the field. This process requires numerical precautions effectively explained in [PER 94]. The model was applied to sheets with traditionally oriented grains. In order to illustrate the anisotropy effectively, we show below the behavior expected by the model for an imposed induction of circular form and of amplitude 1.5 T. Figure 5.7.a shows the trajectory of the field. It highlights the ease of magnetization in the direction RD and the difficulty in the difficult direction at 55°. Figure 5.7b shows the phase shifting between the field and the induction as a function of the angle between the induction and the direction RD. It should be noted that phase shifting changes sign according to whether the induction is between direction RD and the difficult direction (B is behind H) or that the induction is between direction TD and the difficult direction (B is leading H).
Behavior Laws of Materials
10000
HTD (A/m)
D=(H,B) (degree) E °
8000
80
B=1.5 T
40
6000
0
4000
-40
2000 0
197
B=1.5 T 0
2000 4000 6000 8000 10000
HRD (A/m)
(a)
-80 0
20
40
60
80
E (degree)
(b)
Figure 5.7. Behavior in rotating induction: (a) trajectory of H in the plane (x,y); (b) phase shifting between H and B as a function of the angle between B and the rolling direction
Figure 5.8 compares the behaviors calculated and measured for a uniaxis induction imposed in various directions (RD, 30°, 60°, TD). The results are very satisfactory. It is confirmed that the behaviors in directions RD and TD are well restored (they are data of the model) and that the difficult direction is effectively predicted. Other comparisons confirm that the model gives a good qualitative restitution of the anisotropy, but indicate that it over-estimates this anisotropy. To improve the model, it would be necessary to reconsider the expression of the lines of iso-coenergy.
Figure 5.8. Comparison between model and experimental data for a uniaxis induction
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The Finite Element Method for Electromagnetic Modeling
5.4.3.3. Application to single phase transformer modeling The coenergy model has been implemented in finite element software in order to calculate magnetostatics in 2D. The formulation in scalar magnetic potential I was used, because it directly uses the relation B(H) which the model of coenergy gives naturally. The nonlinearity of the material behavior is treated by the Newton method, which requires us to derive the relation B(H) to calculate the tensor of incremental permeability [B/HT]. The anhysteretic behavior of the model ensures the symmetry of this tensor. The numerical calculation of a tensor [B/HT] requires precautions to obtain usable results. The numerical implementation of the model was tested in the case of simple problems. Then representative devices of the oriented grain sheet applications were simulated. There follows the results obtained for the distribution of flux in the yoke of a single-phase transformer operating at zero load. The goal of calculation is to highlight the effects of the anisotropy in the area of the seals between the columns and the yoke. Figure 5.9 shows the geometry as well as the boundary conditions of the problem under consideration. The current in primary winding cannot be taken into account directly. It is modeled by a potential difference ¨I=I1I0 between the borders of *1 and *0. This potential difference ¨I is equal to the circulation of the field between *0 and *1 and thus to the number of Amp turns in the primary winding. The yoke is meshed by 270 second-order triangular elements. The material was modeled with the coenergy model, then with an isotropic model whose curve B(H) is the behavior of the sheet with oriented grains according to the rolling direction.
Figure 5.9. Scheme of the modeled transformer
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199
The problem thus defined was solved for two values of ¨I. Significant convergence difficulties were encountered, for isotropic material as well as for anisotropic material. A very strong relaxation of the Newton method is necessary. Moreover, it was necessary to use an algorithm that requires a high calculation cost to determine, for each iteration, an optimal relaxation coefficient. Convergence is finally obtained for the two field levels. However, this convergence is more difficult to obtain in the case of oriented grains. For ¨I =10 Amps, the average induction in the yoke is 1.4 T. Figures 5.10 and 5.11 show that the distributions of the vectors B and H obtained for an isotropic sheet and for an anisotropic sheet are different. In the latter case, the induction remains very close to the rolling direction, even approaching the seal at 45°. The difference is more sensitive on the magnetic field, which presents a very clear phase shifting with the induction.
Figure 5.10. Field and induction for an isotropic material
Figure 5.11. Field and induction for the GO sheets
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The Finite Element Method for Electromagnetic Modeling
In order to quantify the influence of these local differences, the free energy, defined by the integral of the scalar product B.H on the yoke, was calculated for the two configurations. The results show an insignificant relative difference with regard to the numerical errors related to the finite element method itself. The local shifts do not translate significantly to the selected global quantity. The global model (finite elements and model of coenergy) was used to analyze other electric devices: synchronous generator, three-phase transformer seal, Epstein frame with test-tubes cut out in directions other than directions RD or TD. In all these studies, results are obtained, but significant convergence problems are encountered, which, in fact, significantly limits the use of the coenergy model. 5.4.3.4. Conclusion The coenergy model is a good model of behavior: starting from simple measurements made in directions RD and TD, it is able to correctly reproduce the vectorial behavior in the intermediate directions without complexity or excessive calculation cost. However, its use in structural calculations continues to demand caution because of serious convergence difficulties. We believe that it is not the behavior model which is the cause of these difficulties, but the behavior itself, with its significant instabilities specific to oriented grain sheets. 5.4.4. Hysteresis and dynamic behavior under uniaxial stress The previous study was focused on the anisotropy of sheets with oriented grains and neglected the hysteresis of material. This approach supposes implicitly that the distribution of flux in a transformer is more affected by the anisotropy than by the sheet hysteresis and that the hysteresis appears primarily by means of the losses which it generates. Under these conditions, the flux distribution in the sheet plane is calculated by using an anhysteretic behavior model. Then the local density of losses is assessed in the post-processing phase by a more or less empirical formula utilizing the induction level and frequency. We now change perspective: the study is restricted to the behavior of material under uniaxis request. In fact, we are more interested in the hysteresis and in the flux diffusion in the sheet thickness. The objectives are to model the material hysteresis in static operating mode, to predict the hysteresis in dynamic mode and to calculate its losses, evolution and frequency. 5.4.4.1. Dynamic behavior of sheets and losses The forecast of the losses in sheets remains a concern in electrical engineering because they are highly dependent on the shape of the excitation signal. This
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201
concern is reinforced today by the massive use of switched-mode power supplies which introduces higher order harmonics in currents and excitation voltages. This phenomenon causes increased eddy currents, and thus a degradation in the total behavior of the sheets that should be taken into account to know the limits of their use. Let us consider the traditional problem of the flux diffusion in a sheet of infinite width. The sheet is subjected to a field on the Hs(t) surface of fixed direction (Figure 5.12). The currents induced by the temporal variations of induction are opposed to the penetration flux in the sheet. The global behavior of the sheet is modified. The induction is not homogenous in the sheet, but the traditional measurement of the flux gives access solely to the average induction Bm, defined by: Bm
1 e Bx (y) dy 2e ³-e
[5.56]
In fact, it is this global behavior Bm(Hs) which we need to know. This global behavior allows us to determine the sheet performances under its conditions. When the stress frequency of material increases, the induced currents increase, causing a widening of the total hysteresis sheet cycle and a decrease in the apparent permeability. The frequency of sheet use is thus limited. y
2e
Jz(y)
Hs(t)
Bm B x(y) x
Hs
Figure 5.12. Diffusion of flux in a sheet of infinite width and apparent behavior
The analysis of the phenomena in the sheet thickness makes it necessary to solve the Maxwell’s equations associated with the behavior relations. For the problem considered, the flux distribution in the sheet is governed by diffusion equation [5.57] in which V is the electric conductivity of the material and the magnetic behavior law of the material B=f(H):
w 2 H x (y) wy 2
V
dBx (y) dt
[5.57]
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The Finite Element Method for Electromagnetic Modeling
We can solve this equation simply for linear behavior conditions, characterized by a constant permeability P. In general, numerical solutions are necessary in order to take into account the saturation and the hysteresis of the material. In steady state conditions, the average power P dissipated over one period T is calculated by integrating the Joule losses due to the induced currents j and the hysteresis losses (surface of the hysteresis loop described locally). It is shown in addition that these losses are given by the surface of the apparent cycle.
P
Pcf
1ª T V Uj2 d: dt º ³ ³ H.dB d:) = ³ H s .dB m »¼ : cycle T ¬« ³0 ³: T cycle 1 T Uj2 d: dt and Phys T ³0 ³:
1 H.dB d: T ³: ³cycle
[5.58]
[5.59]
This separation in losses by eddy currents Pcf and hysteresis losses Phys appears naturally when using the definition of the electromagnetic energy radiated through a surface (flux of the Poynting vector P=ExH) and does not have anything artificial. However, it is necessary to bear in mind that it acts as a purely formal separation, which does not correspond to an effective decoupling of the electric and magnetic phenomena. The induced current j depends on induction B(t) and reciprocally the field H is influenced by the induced current. To calculate the sheet losses following an unspecified excitation requires us to numerically solve the diffusion equation of the magnetic flux in the presence of hysteresis. It is thus necessary to have an appropriate hysteresis model. We will now present the two reference models of static hysteresis: the Preisach model and the Jiles-Atherton model. We will then show examples of dynamic cycles calculated by the finite element method and measured. The explicit calculation of the dynamic cycles not being possible in the case of complete machines, we will then present a calculation method of the losses a posteriori. 5.4.4.2. Preisach hysteresis model [MAY 91] The Preisach model is based on elementary hysteresis operators, bistable JDEcharacterized by their switching values D and E as well as their amplitude P(D,E) (Figure 5.13a). The basic idea is that any hysteretic behavior can be reproduced by a distribution of such operators, according to the formula:
M(H)
Ms ³
DtE
P(D, E) J DE (H) dD dE
[5.60]
Behavior Laws of Materials
203
The description of the cycle is contained in the function P(D,E), which gives the weight associated with the operator JDE. This function is generally represented by its isovalues in the plane (D,E) (Figure 5.13b). JDE
E
+1
E
D
H
D
-1
P DE
b)
a)
Figure 5.13. a) Preisach operator; b) Preisach density in the Preisach plane hysteresis being a dissipative phenomenon, D is always higher than E. In addition, the magnetization of the material saturates beyond a certain field Hsat, the Preisach density P(D,E) is zero for D>Hsat and E<-Hsat.
H
D1
M
D2
D1
H(t)
D2 t E2
E2
H
E1
E1 Figure 5.14. Memorizing and erasing of field extrema: D2 and E2 are erased as soon as the amplitude of H exceeds D2. This corresponds to re-closing the minor cycle described between D2 and E2: beyond D2, the behavior is the same as for the one without cycle
The Preisach model was the subject of numerous publications and adaptations. We will review here only the most important properties, from the point of view of the user: i) the model detects and records the extreme values of the field applied, but it also reproduces the erasing of the extrema stored by fields of larger amplitude (Figure 5.14);
204
The Finite Element Method for Electromagnetic Modeling
ii) the closed minor cycles corresponding to identical extreme fields are of smaller size and are superimposed by vertical adjustment (Figure 5.15); M
cycle 1
M2 M1
cycle 2
H
H+
H-
Figure 5.15. Closed minor cycles
iii) the identification of the model can be made on the basis of the inversion curves of order 1 (Figure 5.16). The behavior M(H) is expressed directly according to these data in the following way. Now, let (Di) and (Ej) represent a series of respectively increasing and decreasing and memorized minima and maxima and let ¨M(D,E) be the function defined by ¨M(D,E) = MD - MDE. For an initial state of negative saturation and a field applied H increasing at the moment of calculation, the magnetization is given by the formula: n
M(H) -M s ¦ ^'M(Di , Ei-1 ) - 'M(D i , Ei )` 'M(H, En )
[5.61]
i 1
We obtain similar formulae for the other possible initial and final conditions. The value of this form is that the model is simple to program. The drawback is that it is necessary to have a set of curves of a complete enough inversion to cover the surface of the cycle. M MD E D M DE
H
Figure 5.16. Definition of order 1 inversion curves
Behavior Laws of Materials
205
Properties i) and ii) completely define the application domain of the model: if a system with hysteresis demonstrates them, then we can represent it using the Preisach model. The magnetic materials correspond pleasingly with the first property, but not the second. This has led to various generalizations of the Preisach model, among them the “moving model” [LED 91]. The traditional Preisach model can be applied to systems not verifying the congruence property of the minor cycles. An error can then be made, which can be acceptable depending on the application. Inversion curves are not always available and other methods of identification, using experimental data that is easier to measure, exist. It is possible to use the centered cycles method, which behaves well when the material does not demonstrate the congruence of the minor cycles [BER 00]. The method by Biorci and Pescetti allows the distribution P(D,E) to be determined based on the major cycle and curve of the first magnetization. However, it is not adapted to all materials and thus should be used carefully [BIO 58]. Another approach consists of choosing the distributionP(D,E) of an analytical form for which the parameters are adjustable with respect to a global criterion (surface of the cycle or general appearance of the cycle) [ROU 96]. 5.4.4.3. The Jiles-Atherton model [JIL 86] This model is based on energy considerations. Irreversibility, at the origin of the hysteresis behavior, is introduced by means of the energy dissipated through a collision of the partitions Eacc(M). This energy is defined by relation [5.62], where k is a coefficient which characterizes the dissipation: E acc (M) P 0 k
³
M
0
dM
[5.62]
The energy really stored in the material corresponds to the energy that would be absorbed in the case of absence of hysteresis, minus the losses dissipated through the collision of the partitions. With anhysteretic magnetization being denoted Man, this is written: P0 M.dHe = P0 Man dHe - P0 k dM
[5.63]
In this expression, He represents the effective field in the material. It is the sum of the applied field and an exchange interaction field proportional to the magnetization: He = H + D.M The parameter D depends on the coupling between magnetic domains.
[5.64]
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The Finite Element Method for Electromagnetic Modeling
The differential form of equation [5.63] with the effective field of excitation He, allows M to be expressed which will be identified with the irreversible magnetization in the form:
M irr
M an - kG
dM irr dH e
[5.65]
In this expression, G is equal to ±1 according to whether the field H is increasing or decreasing. Replacing He with relation [5.64] we obtain:
dM irr dH
M an - M irr kG - D(M an - M irr )
[5.66]
The anhysteretic magnetization curve Man is represented by the modified Langevin function:
M an
§ a · § H DM · M s ¨ coth ¨ ¸¸ a © ¹ H DM ¹ ©
[5.67]
where Ms is the magnetization at the saturation of the material and a is a parameter of the model. In order to take into account the reversible phenomena, which represent the swelling of the Bloch, Jiles and Atherton partitions, we introduce the reversible magnetization Mrev and express the magnetization by the following relation: M = Mirr + Mrev
[5.68]
This reversible magnetization is assumed to be proportional to the difference between anhysteretic magnetization and irreversible magnetization. It then follows: Mrev = c (Man - Mirr)
[5.69]
where c is a coefficient of the model. From this expression, the reversible susceptibility is written: dM rev dH
§ dM an dM irr · c¨ ¸ dH ¹ © dH
The final expression of the total differential susceptibility is:
[5.70]
Behavior Laws of Materials
dM dH
§ M an - M irr dM an · (1 - c) ¨ +c ¸ k (M M ) dH ¹ G D an irr ©
207
[5.71]
This equation defines the Jiles-Atherton model. To determine the five parameters of the Jiles-Atherton model (Ms, a, D, k and c), it is necessary to determine the differential susceptibility in certain points of the major cycle, the first magnetization curve and the anhysteretic curve [JIL 92] [DEB 01]. The parameter identification procedure is a priori both simple and rigorous, but practical difficulties are often encountered because the determination of differential susceptibilities is always very sensitive to unavoidable measurement errors [CLE 00]. It should be noted in addition that, in the initial form presented here, the model does not allow modeling of closed minor cycles. The results can thus have some errors for certain excitation conditions. 5.4.4.4. Example of dynamic behavior calculated Once the quasi-static sheet hysteresis is modeled, the eddy current that develops in the dynamic mode is calculated using the finite element method. In the case of large width sheets subjected to a one-axis field, the problem is 1D: the variables differ only in the thickness of sheet. In the reference frame (x,y) of Figure 5.12, the flux diffusion equation is solved on the half-thickness sheet and is written:
-
w 2 H x (y, t) dB (y, t) V x 2 wy dt
0
[5.72]
The boundary conditions are:
wH x (y 0, t) 0 and Hx(y = e,t) = Hs(t) wy
[5.73]
The behavior of the material is given by the model B(H, history). Solving this problem stepwise over time is now possible, with the numerical methods being used. The Crank-Nicolson diagram allows the problem to be discretized in time. The finite element method ensures the space discretization. Finally, the fixed point method ensures the convergence of the nonlinear problem to be solved at each time step [HAN 75] [BOT 98]. The relation B(H) is replaced by the relation B = PPF(H+MPF), in which PPF is a fictitious permeability and MPF is a corresponding residual magnetization. The
208
The Finite Element Method for Electromagnetic Modeling
convergence of the method greatly depends on the value of PPF. This is why a new value is used at each time step, in order to achieve the best trade-off between a guaranteed convergence and a fast convergence. The iterative diagram is given by equation [5.74]. The index n and the upper script (s) correspond respectively to the time step tn and the iteration s of the fixed point method. At the time tn, the unknown variable is the field Hn(s) and the source term uses the fictitious magnetization Mn(s-1), recalculated with each iteration of the fixed point method according to formula [5.75]. The convergence criterion is based on the norm of the residue, which must be lower than 10-4.
w 2 H n (s) V (P PF ) n H n (S) - T 't wz 2
V (P PF ) n -1 {H n -1 (M PF )n -1} ... 't
... (1-T)
w 2 H n -1 V - (P PF ) n (M PF ) n (s-1) wz 2 't
[5.74]
(M PF ) n (s)
1 B[H n (s) ] - H n (S) (P PF ) n
[5.75]
This calculation principle was implemented with various hysteresis models. The figures below show examples of calculation results obtained with the internal variable hysteresis model developed in LMT-Cachan [GOU 98] [BIL 99]. The material considered is a FeSi alloy with non-oriented grains with a thickness of 0.5 mm. It is subjected to a triangular excitation field as a function of time. Various stress speeds dHs/dt are studied. Figure 5.17a shows the applied field measured at the surface for frequencies ranging from 5 to 500 Hz. For this last frequency, the wave form deviates from the setting, but this is without consequence because it is the surface field actually measured, which is used as an input of the calculation model. Figures 5.17b, 5.17c and 5.17d compare the measured and calculated cycles.
Behavior Laws of Materials
1.6
1000
5 Hz
500
Induction B (T)
Surface field Hs (A/m)
1500
0 -500 -1000 -1500
0
0.5
1
Time * frequency
5 Hz
a)
50 Hz
1.5 500 Hz
b)
0.8 0 -0.8 -1.6 -1000
0 -0.8
-1000
c)
0
500
Surface field Hs (A/m)
1000
500 Hz
50 Hz
Induction B (T)
Induction B (T)
0.8
-500
1.6
1.6
-1.6
209
-500
0
500
Surface field Hs (A/m)
1000
d)
0.8 0 -0.8 -1.6 -1000
-500
0
500
Surface field Hs (A/m)
1000
Figure 5.17. Hysteresis cycles at various frequencies: (a) field on the surface measured and used as input for the finite element calculation and a comparison between measured and calculated cycles (square marks) at (b) 5 Hz, (c) 50 Hz and (d) 500 Hz
The very good agreement between measured and calculated cycles proves the relevance of the finite element calculation model for the material considered. However, this model can be insufficient for other non-oriented grain materials [CHE 00] [ROU 96] and in the case of sheets with oriented grains. Indeed, in this type of material, the domain structure is modified when the frequency increases. It is then necessary to use a dynamic hysteresis model, which takes into account this effect [BER 91]. 5.4.5. Determination of iron losses in electric machines: nonlinear isotropic finite element modeling and calculation of the losses a posteriori
We showed how to calculate the iron losses in a sheet subjected to an excitation field applied in a fixed direction. In the case of an unspecified machine, it would be necessary to generalize the preceding step and to make a 3D calculation of the currents induced in the sheet by taking into account the material hysteresis and the anisotropy. This calculation should be made while in addition simulating a complete
210
The Finite Element Method for Electromagnetic Modeling
operation of the machine, i.e. by including the movement and the coupling with the circuit equations. Such a calculation is currently too large to be considered. This is why the losses are estimated a posteriori, by post-processing of the behavior calculated in the absence of hysteresis. The calculation of the losses a posteriori consists of solving the problem by considering the isotropic and anhysteretic nonlinear material allowing the temporal evolution of the induction to be locally determined. Thereafter, the density of losses is calculated in any point by using a model of losses or a dynamic hysteresis model. In both cases, it is essential that these models can reproduce the influence of the frequency and the wave form on the losses. This method supposes that neither hysteresis nor the induced currents significantly modify the distribution of B(t) in the magnetic circuit. The first point (negligible influence of the hysteresis) was verified for a transformer in the thesis of Olivier Deblecker: in steady state, the losses calculated by using a scalar or vectorial hysteresis model while processing with finite element calculation are appreciably the same as those estimated by applying these models a posteriori [DEB 01]. Results with a similar tendency were obtained by Mr. Repetto in the case of rotating machines [REP 01]. These results can be understood if the width of the cycle (about 100 A/m) is considered with respect to the amplitude of the excitation field (a few thousand A/m). On the other hand, neglecting the influence of the eddy currents on the operation point is more debatable, because the apparent permeability of the material varies greatly with the frequency. However, current computers do not have adequate capabilities to treat a complete machine geometry, or to take into account the layering of the magnetic circuit. Currently, machine losses can only be calculated using an a posteriori approach. We will describe a model of this type, developed within the framework of the theses of Christophe Cester and Thierry Chevalier and based on the use a posteriori of a dynamic hysteresis model [CES 96], [CHE 99]. 5.4.5.1. Dynamic scalar hysteresis model: LS (loss surface) model This model was developed following many studies on magnetic losses in trapezoidal induction in which the influence of the parameter dB/dt was highlighted [KED 86]. The model is based on this parameter. It supposes that the behavior of material is completely determined by the instantaneous value of the induction and its derivative. The model is then defined by a characteristic surface H(B, dB/dt) obtained in experiments by carrying out tests in triangular induction with fixed dB/dt and variable frequency in order to vary this parameter. Thus, for a B(t) signal of unspecified form and frequency, we do not carry out a frequency decomposition, but a temporal decomposition. We then associate with each couple (B(t), dB(t)/dt) the field H(t) provided by the surface. The corresponding hysteresis cycle is thus achieved. Figure 5.18 gives an example of this characteristic surface, obtained on an
Behavior Laws of Materials
211
iron sheet with non-oriented grains currently used in electrical machines. We clearly observe a discontinuity in the vicinity of the zero dB/dt which corresponds to the material quasi-static cycle.
Figure 5.18. Characteristic surface H(B,dB/dt) of the LS model
The model is based then on a decomposition of field H(B, dB/dt) in a static contribution Hstat(B) and a dynamic contribution Hdyn(B, dB/dt). Hstat describes the quasi-static behavior of material whereas Hdyn comprises all the dynamic effects generated by displacements of walls and the induced currents. The Hstat term is described by a simple model of scalar hysteresis; the Hdyn term is given by the surface Hdyn(B, dB/dt) extracted from the total surface by withdrawing the quasistatic field, then modeled by analytical functions The dynamic LS model was tested for different excitations, representative of the flux variations really undergone by material in the rotating machines: sinusoidal or trapezoidal induction of variable amplitude and frequency, sinusoidal induction charged with harmonic of variable amplitude, rank and phase, asymmetric induction comprising a DC of variable amplitude and a harmonic component of frequency and amplitude variables. In the majority of the cases tested, the losses estimated by the model deviate by less than 10% from the losses obtained in experiments. The curves of Figure 5.19 show an example of hysteresis cycles calculated and measured for two different stresses. The LS model satisfactorily reproduces the influence of the frequency and the form of stress.
212
The Finite Element Method for Electromagnetic Modeling 2
2
1.5
1.5
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
-1.5
-1.5
1
0.7
-2 -2000
-300
-2
-1500
-1000
-500
0
500
1000
1500
2000
a) B(t) sinusoidal at 50, 200 and 400 Hz
-2400
-1800
-1200
-600
0
600
1200
b) B(t): sinus 50 Hz + 11th harmonic
Figure 5.19. Measured and estimated hysteresis cycles
5.4.5.2. Coupling with finite element simulation Finite element simulation aims to determine the distribution of the temporal variation of the B(t) signal in the magnetic circuit of the machine. It is carried out in our case with the Flux2D¥ software. The resolution is carried out stepwise over time by integrating the movement of the rotor. Under these conditions, the behavior of the magnetic material is modeled by a nonlinear, isotropic and hysteresis free (anhysteretic) law B(H). The LS model is a scalar hysteresis model in which the directions of the magnetic field and induction vectors are assumed to be identical. However, this property is valid only if the material remains excited in the direction of one of these principal axes, rolling direction (RD) and perpendicular transverse direction (TD). As soon as it deviates from TD or RD, B and H are no longer in phase; we are dealing with vectorial behavior. In order to take these phenomena into account, curves B(H), used during simulations as well as the LS model, are established by considering mean characteristics of the material following directions RD and TD. In addition, a particular treatment is carried out for the 2D variations of B(t). In order to calculate the losses in each element, the vectorial signal B(t) is projected along two orthogonal axes of which one is parallel with the direction of the maximum amplitude of B(t) (Figure 5.20). The components B(t) and BA(t) are then used as model inputs for calculating H(t) and HA(t) and for deducing the
Behavior Laws of Materials
213
corresponding hysteresis cycles from them. The total losses are assumed to be the sum of the losses associated with each component. The choice of these reference axes is not arbitrary. It makes it possible to directly apply the scalar model in the case of a uniaxial induction and to be freed from the numerical errors which can be generated by the method.
Figure 5.20. Reference axes used for the decomposition of the signal B(t)
This assumption, which is based on the principle of superposition, is quite large and is not always valid. However, in the case of the sheet used for the simulated machine, the approximation remains correct. Indeed, the 2D wave forms B3(t) and B4(t) of Figure 5.4 were reproduced on a measuring platform of the “rotating field” type frame or RSST (rotational single sheet tester) to test this method. The calculated losses deviate by less than 10% from the experiment [SPO 98]. 5.4.5.3. Application to the calculation of induction machine losses The LS method was used to calculate the iron losses of two cage induction machines with different geometries, one of 4 poles and 4 kw, and the other of 2 poles and 5.5 kw [CES 96], [CHE 99]. These two industrial manufactured machines were equipped with coils allowing local measurements of flux in the teeth and the yoke of the stator. The rotors considered are with straight notches in order to conform with the 2D simulations carried out. Simulations were carried out in synchronism, at the point of the machine’s nominal operation taking into account or not the induced currents in the cage. First, the local behavior of the material in the machine was analyzed. The curves of Figure 5.21 compare the forms of B(t) obtained by simulation and measured in a tooth and in the yoke at the stator of the machine. A good agreement is observed. Failing to account for the hysteresis during the resolution of the problem is no longer punitive in this case.
214
The Finite Element Method for Electromagnetic Modeling
Figure 5.21. Wave forms measured and simulated (left: tooth – right: yoke)
The distribution of the density of the iron losses in the induction machine is represented in Figure 5.22 by a gray gradation. This figure highlights the saturation of the stator teeth but especially the existence of losses in the rotor even at synchronism. These losses account for approximately half of the stator losses and are due to a flux angular velocity of high frequency, generated by the passage of the rotor teeth in front of those of the stator.
Figure 5.22. Distribution of the density of iron losses in the stator and the rotor of a 5 kW induction machine, sinusoidal supply, no-load operation at synchronism
The estimated total losses were then compared with the experiment. A specific measuring platform and various rotor prototypes (smooth rotor, rotor without cage, etc.) were used to determine separately the iron losses for the stator and for the rotor.
Behavior Laws of Materials
215
Table 5.1 summarizes the results obtained for the two machines. This approach has allowed not only the iron losses for the stator and the rotor to be estimated, but also additional losses to be highlighted. They are due to Joule losses in the cage as well as to Joule losses probably due to short-circuits generated by manufacturing processes and located in the periphery of the rotor and on the contours of the teeth: machining and slot deformations. By considering all these phenomena, the approach developed has allowed us to estimate the losses with an error of 2% for the 4 kW machine and an error of 10% – close to the 5.5 kw machine. Cage type induction motor, 4 poles, 4 kW Rotor Losses in W
stator
Calculation
109
Measurement
117
Joule losses cage
Short circuit rotor surface
Short circuit rotor teeth
Iron losses rotor
6
15
5
45
71 60
total
180 177
Cage type induction motor, 2 poles, 5.5 kW Rotor Losses in W
stator
Calculation
129
Measurement
138
Joule losses cage
Short circuit rotor surface
Short circuit rotor teeth
Iron losses rotor
17
30
14
61
122 139
total
251 277
Table 5.1. Estimated and measured magnetic losses
The developed LS method allows an accurate and simple estimation of iron losses in the electric machines and any magnetic circuit of unspecified geometry in general. Nowadays, work is underway to supplement the validation of the dynamic hysteresis model by testing other materials and other excitation conditions as well as the validation of the global method by simulating other electric machine structures. 5.4.6. Conclusion
We presented the various aspects of sheet behavior that are necessary to consider for accurate modeling of these materials: anisotropy, hysteresis, dynamic mode and losses.
216
The Finite Element Method for Electromagnetic Modeling
We did not deal deeply with a fundamental question related to the calculation of the dynamic behavior of sheets, namely the distinction between induced currents losses and hysteresis losses. Indeed, the hysteresis losses are actually induced current losses which develop on a microscopic scale of the walls. In the proposed dynamic calculation, these microscopic currents are completely decoupled from the macroscopic currents and modeled by a hysteretic behavior. This complete decoupling of the microscopic and macroscopic scales are in fact carried out in materials with fine grains, but not in materials with coarse grains, like GO sheets in particular. The proposed modeling approach is then not inarguable, but the reader can refer to the work of G. Bertotti for a thorough discussion [BER 98]. Furthermore, we also did not speak about the influence of the mechanical state of materials on their magnetic behavior. However, it is a very important effect which results in an almost systematic degradation of the behavior of sheets when we shape them to build the magnetic circuit of a machine. The cutting and assembly processes generate local plastic deformations and stresses in the mass of the parts, which modifies the nominal behavior of the material and involves additional losses. The existence of these effects is known, but their modeling is for the moment far from being mastered. For more information on the influence of the mechanical state on the magnetic behavior, the reader can refer, among other references, to the work of C. Gourdin [GOU 98] [BIL 99]. 5.5. Modeling of permanent magnets 5.5.1. Introduction
In the majority of electromagnetic field computation software, the behavior of permanent magnets (SmCo5, NdFeB, ferrite) is approached by a linear model, based exclusively on the behavior of the magnet in the direction of easy magnetization. This model allows numerous devices to be satisfactorily simulated, but it is completely inappropriate to take the demagnetization of magnets into account. When this phenomenon is likely to occur (at the time of the assembly phase of magnets, for example), more powerful models should be used. We present such a model here, both nonlinear and anisotropic, and developed by Joel Chavanne as part of his work [CHA 88]. 5.5.2. Magnets obtained by powder metallurgy
Magnets obtained by powder metallurgy consist of single-crystal grains (a few tens of Nm) having a very strong uniaxial magnetocrystalline anisotropy. The coordinates of easy magnetization C of the grains are distributed around an average
Behavior Laws of Materials
217
direction x corresponding to the macroscopic axis of easy magnetization (Figure 5.23). Studies show that this distribution has a symmetry of revolution around this axis and that it can be represented by a Gaussian [GIV 85].
y
H TH T1
C M(H) x
z Figure 5.23. Distribution of C grain axes around a mean direction x
A second function, better suited to the development of models by analytical calculation, can be used [JAH 87]. This function is given by formula [5.76], in which T1 is the angle that forms the axis C of the particle with the macroscopic direction x and n is a parameter which controls the width of the distribution. Figure 5.24 shows the influence of this parameter. The typical values for the usual magnets are in the neighborhood of n = 7. For n = 0, we obtain an isotropic distribution: p(T1 )
2n 1 2n cos T1 2S
Figure 5.24. Examples of distribution of particles C axes orientation
[5.76]
218
The Finite Element Method for Electromagnetic Modeling
The grains are separated by an intermediate phase with very different magnetic properties. The exchange interactions between grains are reduced and these can be considered, at first approximation, as independent. The behavior of the magnet in a field H then results from the superposition of the individual behaviors of the constitutive grains. 5.5.3. Study of linear anisotropic behavior
Firstly, let us consider an isolated grain, subjected to a magnetic field H lying in an arbitrary direction TH from the x-direction. For small angular variation of the magnetization, using the coherent rotation model leads to the classical linear reversible model of the behavior of a single crystal [CHA 89]. In the coordinate system of the grain, this behavior is given by:
M
ªMs º ª0 0 0º Ms « » 0 1 0 » H «« 0 »» Ha « «¬ 0 »¼ «¬ 0 0 1 »¼
[5.77]
where Ms and Ha respectively represent the magnetization and the crystalline anisotropy field of the grain. Let us now consider all the grains assumed to have the same physical characteristics Ms and Ha. By taking into account the distribution function of the grains, it is possible to write M, after integration, in the form: M(H) = Mr + [Fr] H where Mr is the residual magnetization along the x-direction and such that:
Mr
Ms
2n 1 2n 2
[5.78]
Ms represents the macroscopic magnetization with saturation of the magnet. [Fr] is the tensor of susceptibility defined by:
[F r ]
ª 2 º 0 0 » « 2n+3 « » Ms « 2n+2 0 0 » » Ha « 2n+3 « » 2n+2 « 0 » 0 2n+3 »¼ ¬«
[5.79]
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219
In the case where the grains are perfectly oriented (n very high), we find the relation obtained for a monocrystal. An interesting result is that the ratio U between the parallel susceptibility and the transverse susceptibility is independent of Ms/Ha and allows the determination of n. We obtain:
U
2n 2 2
[5.80]
This model was used for the study of magnets systems with the assumption that the field applied is sufficiently weak to allow the linear approximation. In this way, we modeled – within the framework of the finite element method in 3D – a high field source with magnets of 4 T [BLO 99]. The modeling has allowed us to seek the nuances, the shape and the optimal directions of magnets making it possible to obtain a maximum field in the center of the structure (Figure 5.25). In postprocessing, the calculation of the field applied to the magnet allows us to establish whether or not there was demagnetization. In this light, simulations were carried out to determine “the good” sequence of assembly of the structure.
Figure 5.25. 4.5 Tesla in the hollow of the hand
We have noticed that the magnets in the center underwent strong rotations of magnetization because of a very high transverse field (twice the coercive force) and, although reversible, these phenomena are incorrectly described by the previous linear model.
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The Finite Element Method for Electromagnetic Modeling
5.5.4. Study of nonlinear behavior
The nonlinear behavior of the magnets depends directly on the mechanisms of coercivity and inversion of magnetization in the grains. The simplest model is the coherent rotation mechanism. The particle is supposed monodomaine and its behavior is controlled by the magnetocrystal anisotropy. This model envisages a coercive force equal to the anisotropy field, whereas the coercive force actually measured is the smaller of the two orders of magnitude. This difference is explained by the fact that the magnetization reversal is never a single domain mechanism. There is always a microstructural defect which causes the nucleation of a reverse domain. Then this domain propagates for a field quite lower than the anisotropy field because this mechanism is less expensive from the energy point of view (Figure 5.26).
M saturation
nucleation
propagation
Figure 5.26. Mechanism of reversal of magnetization in a grain
For the study of the influence of a field of an unspecified direction, it is useful to know the angular variation of the coercive field of the magnetization according to the angle T that makes H with the axis C of the grain. Two models are traditionally proposed. The first is the coherent rotation model, while the second supposes that only the projection of the field according to magnetization is active (Kondorsky’s model of trapping breakdown). Figure 5.27 compares the results of these two models. In addition, the measurements made on isolated SmCo5 particles show that the angular variation follows Kondorsky law [5.81]:
hc(T)
hc(0) cos T
[5.81]
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Figure 5.27. Angular variation of the coercive field: a) coherent rotation of magnetization, b) Kondorsky’s model of trapping breakdown
The use of the Kondorsky law and the function of angular distribution allows us to build a model which predicts the behavior of the magnet for a field in an unspecified direction starting from the following experimental data: M//(H), demagnetization curve in the direction of easy magnetization x and FA, presumably constant transverse susceptibility. For a field of amplitude H and forming an angle TH with respect to the x axis (Figure 5.23), the parallel and perpendicular components to H, respectively noted M1 and M2, are given by: M1(H,TH) = M//(H1) cos TH + FAH sin2 TH
[5.82]
M2(H,TH) = M//(H2) sin TH + FAH sin TH cos TH
[5.83]
where H1 and H2 are defined by:
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The Finite Element Method for Electromagnetic Modeling
H1 = H cos (D TH)
H2
1 H cos (E TH ) O
[5.84]
[5.85]
and D, E and O are given by: D(n)
§ 2 3 ·ҏ Arcos ¨¨ ¸¸ S © 2n 3 ¹
[5.86]
E(n)
§ 2 2 · Arcos ¨¨ ¸ S 2n 1 ¸¹ ©
[5.87]
O(n)
2n 3 2n 1
[5.88]
The behaviors measured and predicted by the model for fields applied in various directions have been compared, and for various types of permanent magnets (SmCo5, Ferrite, NdFeB). In all cases, an excellent agreement was obtained. As an example, Figure 5.28 shows the M1(H) curves obtained for a field applied in a fixed direction in the case of a ferrite magnet. We can note in particular that the proposed model is definitely better than that which would have consisted of carrying out a direct interpolation between the two curves M// and MA (case n = infinite).
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223
infinite
Figure 5.28. Curves of magnetization calculated and measured for a ferrite magnet
5.5.5. Implementation of the model in finite element software
Joel Chavanne’s model has been implemented to carry out 3D magnetostatic calculations using the finite element method. The previous relations have been simplified with the approximations O=1 and D E. This approximation consists of neglecting the demagnetization effects on the orthogonal component to Ox because of the small width of the distribution of axes C around axis x (assumption valid for n=7 for example). A formulation in scalar potential was used, which requires us to know the relation B(H) and the differential permeability tensor [B/HT]. The relation B(H) rises directly from the previous equations, specifically the expression B = Po [H+M(H)]. In the reference frame related to the magnet, the differential permeability tensor has as the expression:
ª wB º T ¬« wH »¼ R magnet
ª wBx « wH « x « 0 « « 0 « ¬
wBx wH y P 0P r A 0
wBx º wH z » » 0 » » P 0P r A » » ¼
[5.89]
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The Finite Element Method for Electromagnetic Modeling
with:
wBx wH x
P0 [cos(DTH ) cos TH D sin(DTH ) sinTH ] . [P r & (H1 ) - 1] P 0
[5.90]
wBx wH y
ª Hy P 0 [sin(DTH ) sin TH -D cos(DTH ) cosTH ] . «P r & (H1 ) « H y2 Hz2 ¬
º » [5.91] » ¼
wBx wH z
ª Hz P 0 [sin(DTH ) sin TH -D cos(DTH ) cosTH ] . «P r & (H1 ) « Hy2 Hz2 ¬
where P (H) r&
1 wB& (H) P 0 wH
º » » ¼
[5.92]
[5.93]
5.5.6. Validation: the experiment by Joel Chavanne
In order to validate the proposed model, Joel Chavanne created an interesting experiment: he approached two magnets (NdFeB, a ferrite) magnetized in opposite directions (Figure 5.29). It should be noted that the two magnets push each other away, then attract each other when they are very close to one another. This phenomenon is explained by reversals of magnetization in the ferrite magnet caused by the NdFeB magnet which has a much more rigid magnetization. The digital simulation by 3D finite elements allows this phenomenon to be accurately found. The measurement results of the fields outside the magnets are in perfect agreement with the simulation, which in addition shows a strong reversal rate of magnetization in the ferrite magnet.
Behavior Laws of Materials
NeFeB
225
Ferrite
? Figure 5.29. Approach of an NdFeB magnet and a ferrite magnet
5.5.7. Conductive magnet subjected to an AC field
The Joel Chavanne model was also used to model the behavior of a permanent magnet subjected to a pulsating magnetic field of sinusoidal type, and therefore in the presence of induced currents. This study was undertaken within the framework of European project MACCHARRACTEC [GOL 00]. The final objective is to determine the influence of the frequency and to deduce the intrinsic characteristic of the magnet based on measurements taken in variable mode. The direct problem is solved stepwise in time using a magnetodynamic formulation of type T-I (T denoting the electric vector potential, I the magnetic scalar potential). This formulation requires an expression of B(H) law and knowledge of the resistivity of the magnet. The results make it possible to find the average intrinsic hysteresis cycle obtained (the simulation also makes it possible to check the local behavior). Figure 5.30 shows that the hysteresis curve is very sensitive to the frequency of the applied field. If it increases, the differences in residual magnetization and the coercive field (compared to the static characteristic) become increasingly large.
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The Finite Element Method for Electromagnetic Modeling
Figure 5.30. Dynamic hysteresis curve simulated on a cylindrical NdFeB magnet
5.6. Modeling of superconductors 5.6.1. Introduction
Superconductors are characterized by remarkable electric and magnetic properties: below a certain critical temperature Tc, the electrical resistivity becomes zero. This phenomenon is accompanied by a perfect diamagnetism (no penetration of magnetic flux in the superconductor). The loss of resistivity and the absence of heating through the Joule effect allow very high current densities to be circulated with almost zero losses. The current density reaches 20 to 6,000 A/mm2, whereas the limit ranges between 1 and 15 A/mm2 in the traditional conductors. The possibility of transporting strong intensities under small overall dimensions requires us to reconsider the usual design limits of the electric material. However, the gains in volume and weight which the use of superconductors allows must be considered with respect to the cost of material and the requirement of a cryogenic environment to maintain the conductors below their critical temperature. 5.6.1.1. Various types of superconductors We can distinguish superconductors with low and high critical temperature (respectively LTS and HTS), the limit between the two being located at 30 K. There is a certain arbitrariness in this border. However, in practice this distinction is important because LTS superconductors must be cooled with liquid helium, whereas
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227
cooling with liquid nitrogen, which is much more economical, is sufficient for HTS superconductors. A more fundamental difference exists between the type I and type II superconductors. The first are perfectly conductive and diamagnetic below their critical temperature, whereas the second ones have a very low but non-zero resistivity and are only partially diamagnetic. We will reconsider these behavior differences in the subsequent sections. 5.6.1.2. Applications of superconductors Currently, LTS superconductors are especially used, primarily to produce electromagnets in high-energy physics or medical imagery. HTS superconductors, more recently discovered, do not yet have industrial or commercial applications. However, their relative ease of use and the progress made in their design make it possible to consider a use in electrical engineering. Transformers and superconductive engines with better performances compared to the traditional devices are studied. In addition, they are smaller and lighter. Other original applications are also being studied. They use two properties specific to the superconductors. The first relates to the phenomenon of levitation: a magnet placed above a superconductive chip enters by levitation in a position of stable equilibrium. This property can be used to create completely passive magnetic suspensions or for the storage of energy by inertia. The second uses the property of transition from the superconductive state towards a resistive state when the critical current is exceeded. This enables superconductor current limiters to be made. 5.6.2. Behavior of superconductors
5.6.2.1. Critical surface and variables
Figure 5.31. Critical surface delimiting the area where supraconductivity exists [BAI 98]
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The Finite Element Method for Electromagnetic Modeling
Supraconductivity exists only in a domain limited by three critical variables: the temperature, the magnetic field and the current density. These three variables are not independent of each other. The critical surface, defined in space (T, J, H), defines the domain where the material is a superconductor (Figure 5.31). As soon as we leave the domain, the material becomes strongly resistive and non-magnetic. There is a change in the electric behavior and the magnetic behavior. 5.6.2.2. Magnetic behavior For a temperature lower than the critical temperature, the superconductive state disappears when the applied magnetic field is higher than the critical magnetic field. The type I superconductors then make a transition from a perfectly diamagnetic behavior to a non-magnetic behavior and their magnetic permeability goes from P=0 to P=Po in a discontinuous way (Figure 5.32). The apparent diamagnetism of the type I superconductors is due to the fact that in the absence of resistivity, nothing limits the induced currents which oppose the flux penetration flow in the conductors. At the microscopic scale, the magnetic induction penetrates on a very low thickness OL, called the London length, on which currents develop known as shielding super currents. This length varies from 10 to 300 nm. At the macroscopic scale, the conductor behaves like a perfectly diamagnetic material. For type II superconductors, the transition from the perfect superconductive state to the resistive state is achieved between two critical fields Hc1 and Hc2. The material is then in a mixed state: it remains electrically superconductive but is no longer perfectly diamagnetic because the shielding is only partial. The penetration of the magnetic flux is made in a quantified form by a set of tubes called vortices. Each vortex transports the same quantum of flux of about 2.10-15 Wb. In practice, Hc1 is very small compared to Hc2 and the B(H) macroscopic behavior is effectively approached by a line with a slope Po (Figure 5.32). B
B
B = P0 H
B = P0 H
type I
type II
Hc
H
Hc1
Hc2
Figure 5.32. Magnetic behavior of type I and type II superconductors
H
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229
5.6.2.3. Electric behavior The third characteristic which limits the superconductive state is the critical current density Jc(H, T). The material limits its electric transport capacity to a critical current density Jc which corresponds to the maximum value beyond which a resistivity appears. This transition is found in the electric behavior law J(E). As for the magnetic properties, it is necessary to distinguish the type I and type II superconductors. In the type I superconductors, the current circulates only on the surface, in the London length. When the magnetic field, created on the surface by the current, reaches the value of the critical field, the state of superconductivity is destroyed. The transported current then corresponds to the critical current. The resulting J(E) law is represented in Figure 5.33.
j
Jx
j c(x) V en
0
e
jc(x)
Figure 5.33. Electric behavior law of a type I superconductor
The behavior of type II superconductors in the mixed state is more complex. When a current crosses the conductor, it creates a magnetic flux which penetrates the material in the form of a vortex. These vortices are subjected to Lorentz forces (JxB) which, in the absence of any obstacle (ideal material), act to cancel the total current. There is thus no possibility for the current to be transported. In practice, the material is characterized by anchor points which are opposed to the free reorganization of the vortices and allow a macroscopic current to circulate. According to Anderson and Kim [KIM 63], the vortices have a certain probability of jumping from one anchor point to another anchor point through thermal activation. In the presence of a Lorentz force, the jump direction becomes privileged, which produces a macroscopic current. Depending on the value of Lorentz forces, and thus of the current, we distinguish various phenomena (Figure 5.34). For low currents, the vortices move from one anchor point to another anchor point: this phenomenon is called TAFF (Thermally Activated Flux-Flow). For high currents, the phenomenon accelerates and we speak of “flux-creep”. Lastly, when the Lorentz
230
The Finite Element Method for Electromagnetic Modeling
forces become much higher than the anchoring forces, the vortices move freely (flux-flow). In all cases, the forces applied to the vortices involve a displacement of the vortices and thus a dissipation of energy. E (V/m) 1
thermally activated flux-flow
fluxcreep
flux -flo w
10- 5
10-10
J (A/m2) 106
108
1 010
Figure 5.34. Electric behavior law of a type II superconductor
5.6.3. Modeling of electric behavior of superconductors
5.6.3.1. Introduction The electromagnetic behavior of a device including superconductors is given by Maxwell’s equations associated with the behavior laws. The magnetic characteristic does not pose any modeling problem since the superconductor has a behavior similar to that of the vacuum. On the other hand, the electric behavior is nonlinear and various models have competing claims. We will briefly present them by order of increasing complexity, then we will speak about the implementation of these models in a computer code by the finite element method. 5.6.3.2. Some examples of models The Bean model (1962) [MAY 69] [BOS 93] [PRI 97] [MAS 97] is the most frequently used in engineering. This model postulates that the existence of an electric field E in any point of the sample induces in this point a current density with an amplitude Jc parallel to E. In mathematical terms, this model is written in the following way: If E
0, then J J c , otherwise J
Jc
[5.94]
This model supposes that the critical current density is independent of the value of magnetic induction B, which is not verified in experiments. In order to take this
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231
dependency into account, several models were proposed such as the Kim law [KIM 63] or the anisotropic Kim law [MAS 98]. This law is given: J c (B)
§ B· J c (0) ¨1 ¸ © B0 ¹
-1
[5.95]
Recently, some work [SYK 97] [VIN 00] has been carried out in the area of the power law, defined as follows: J
§ E· Jc ¨ ¸ © Ec ¹
N
[5.96]
This empirical law represents the “flux-creep” phenomenon effectively. For higher values of the exponent N, the power law leads to the Bean model. In general, depending on the materials and their manufacturing, parameter N varies between 5 and 30 in HTS superconductors and can go up to 100 in LTS type II superconductors [VIN 00]. Finally, there are laws that take into account the displacement and the anchoring of vortices [YOS 94]. This is what is called the “Flux-Flow-Creep” (FFC) model, in which two distinct analytical expressions are used in order to respectively model the flux-creep [5.97] and flux-flow [5.98] modes: E
§U J · § U0 · ¸¸ exp¨ 2 U c J c sinh ¨¨ 0 ¸ if 0<J<Jc © KT ¹ © kT J c ¹
E
UJ c U f J c ¨¨
· § J ¸¸ if J>Jc J ¹ © c
[5.97]
[5.98]
In these formulae, U0 is the anchoring potential, T the temperature, k the Boltzman constant and Uc and Uf the resistivities specific to the material. It has to be noted that the FFC model is the only one which takes into account the temperature. 5.6.3.3. Digital implementation in calculation using the finite element method Traditionally, the calculation of fields in superconductor materials is achieved using the finite element method. This calculation is carried out by solving a magnetodynamic problem with a nonlinear electric behavior law:
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The Finite Element Method for Electromagnetic Modeling
1 ¬ dJ and J = J(E) curl curlE dt ®
[5.99]
This calculation model is generally used in the case of 2D geometries. The electric field and the current density are then orthogonal to the study plane. In the case of the approximation of the behavior law by expressions [5.95] to [5.99], the Newton-Raphson algorithm can be used. The convergence is then reached more or less quickly according to the choice of state variables [VIN 00]. The magnetic field formulation [5.100] proves to have the best behavior in terms of convergence: curlE
d H
dt
and E
[5.100]
E ( J ) with J curlH
5.6.4. Particular case of the Bean model
The Bean model requires careful handling, because this behavior law is not differentiable, or even representable by a functional graph. It is thus not possible to use it directly. 5.6.4.1. Mathematical analysis of the Bean model We are interested here in the generalized Bean model [BOS 93] in the sense that it takes the transition at the normal state into account. The graph of this model is represented in Figure 5.35, where Jc, En and V respectively indicate the critical current density, the value of the electric field from which the material becomes conducting resistive and the electric conductivity of the superconductor in the normal state. j
Jx
jc(x) V 0
en jc(x)
Figure 5.35. Generalized Bean model
e
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233
We will formulate the behavior law by demonstrating the calculation of a functional convex U such as J w U(E) where w U(E) indicates the under-gradient at the point E of the functional calculus U [BOS 93]. Let us consider the case of a 2D problem. The current density J and the electric field E are normal to the study plane and have as components in point x respectively the algebraic values J(x) and E(x). The electric behavior law is verified if and only if the couple {E(x), J(x)} belongs to the graph Jx and if J(x) and E(x) are of the same sign. The difficulty is to formulate this membership to the graph while expressing, for example, J(x) as a function of E(x). For that, we set up the Ux function, presented graphically in Figure 5.36, in the following way:
U x ( E ( x))
if E x En J c x E x ° ® J x E x V E x E otherwise n °¯ c
[5.101]
Ux
jc(x) 0
en
e
Figure 5.36. The function Ux
The Ux(E(x)) value represents the surface delimited by the graph Jx, the x-axis and the vertical axis passing by the E(x) point. This geometric construction of the Ux functional calculus by surface calculation is quite general and can be applied whether Jx is a functional graph, linear or not. The monotony property of the graph Jx implies that the continuous function Ux is convex. It is then possible to calculate, in any point E(x) pertaining to the domain of Ux, the under-gradient defined by: w U x (E(x)) = {g IR / f' IR, U x (f' ) U x (E(x)) t g(f' E(x))}.
The calculation of the under-gradient for the Ux function above gives:
[5.102]
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The Finite Element Method for Electromagnetic Modeling
J x wU x E x ® ¯ > J c x , J c x @
if if
E x z 0 E x 0
[5.103]
We can thus express the behavior law by:
{E(x), J(x)} J x J(x) wU x (E(x))
[5.104]
This concept of under-gradient has the advantage of being able to express the behavior law while being based solely on the assumption of continuity and monotony of the graph Jx. Finally, the generalized Bean model can simply be written as: [5.105]
J wU(E) where U is defined by:
U(E)
³U
x
(E(x)) dx
[5.106]
Unlike expression [5.104], which is only a scalar relation between the values of the fields E and J in point x (local relation), relation [5.105] is functional since the variables E and J are functions of the current point x. In the current density J formulations, it is necessary to express the behavior law in the form of E(J) [PRI 96]. Ux*, the Fenchel conjugate of Ux [EKE 13], is then used, and the following equivalence:
{E(x), J(x)} J x E(x) wU x * (J(x))
[5.107]
In the formulations where both the electric field and the current density are the unknown variables of the problem [VAS 96], the behavior law is expressed based on the Ligurian ȁ, defined by [5.108], and of the equivalence [5.109]. It is then only necessary to seek the zeros of the Ligurian:
/ (E(x), J(x))
U x (E(x)) U x * (J(x)) E(x), J(x),
{E(x), J(x)} J x / (E(x), J(x)) 0.
[5.108]
[5.109]
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235
5.6.4.2. 2D finite element formulation with the Bean model The behavior law being now well formulated, the next step is to implement it in a computer code. By itself this law is not sufficient to determine the electromagnetic behavior of the superconductor. Indeed, for a given electric field E, the current density J verifying the law is not necessarily unique since, for example, there are several possible values for the current density J when the electric field E is zero. In order to clarify this uncertainty, we must consider the history of the superconductor by associating with behavior law [5.105] Maxwell’s equations for low frequencies. The vector form equation, in electric field E, coming directly from Maxwell’s equations, becomes in the case of a 2D analysis scalar equation [5.110], where E and J are scalars linked by the behavior law:
§ 1 · dJ - div ¨ grad E ¸ 0 dt © P0 ¹
[5.110]
Mathematically, problem [5.110] is known as the “Stefan problem” or as nonlinear diffusion [STE 89]. In order to solve it, we can use the implicit Euler scheme. We obtain:
§ 1 · (J n - J n -1 ) - 't div ¨ grad E n ¸ 0 © P0 ¹
[5.111]
where Jm and Em are the values of the current density and the electric field at the moment m't. We can then notice that Em achieves the minimum of the following functional calculus:
§ 't · F(E) = 2 U[E(x)] + ³ ¨ grad(E(x))2 2 J n -1 (x)E(x) ¸ dx © P0 ¹
[5.112]
It is then necessary to carry out at each time step a convex optimization. The functional calculus being strictly convex, the minimum is unique.
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The Finite Element Method for Electromagnetic Modeling
5.6.4.3. Proposed model _j_
_j_
J
JH
jn jc
jc
V 0
V en
_e_
ec
en
a)
_e_
b) Figure 5.37. a) Function graph J b) Function graph JH
Let us return to the behavior law, recalled in Figure 5.37a. Its implementation in a computer code using finite elements is facilitated if it is modified in a graph leading to a functional calculus Ux quadratic per pieces. The minimum of Ux can be found by traditional methods such as the Gauss-Seidel method. More precisely, the Bean model is approximated by a family of more regular and less rigid functions (Figure 5.37b), depending on parameters H1 and H2, and such that these approximation functions tend towards the graph of the behavior law of Figure 5.37a when H1 and H2 tend towards 0. The parameters H1 and H2 being strictly positive real, the quantities Jn, Ec and Jc are defined in the following way: Jn = V En
Jc = H1 Ec
Jc = (1 H2)Jn
[5.113]
The parameters En and Jn delimit the area in which the material is superconductive. Ec and Jc, are selected in such a way as to fit the experimental characteristic as well as possible. Indeed, we need only adjust the parameters H and H according to the studied material. It is possible to treat, with this model, the linear (H = H = 1), nonlinear or strongly nonlinear (H1 and H2 o 0) behavior. In addition to the advantages related to the freedom of choice of the model according to the parameters H1 and H2, the modeled characteristic reveals two regions of operation (|J| < Jc and Jc < |J| < Jn) which can be considered similar to the “flux-creep” and “fluxflow” regions.
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237
5.6.5. Examples of modeling We will now present some computation results, resulting from work completed at the Electrical Engineering Laboratory of Paris. The first two examples relate to the behavior of superconductive cables. The levitation of a permanent magnet above a superconductive chip is then treated. These calculations were made using the model that we have just presented. 5.6.5.1. Massive material plunged in a uniform field In this simulation, a massive superconductor with an infinitely long rectangular section is plunged into a uniform magnetic field. The applied field varies linearly per piece as a function of time. Figure 5.38 shows the evolution of the applied field as well as the distribution of the current density at instants t1, t2 and t3 indicated in the graph. The values of the physical parameters and in particular the maximum value of the source magnetic field were selected to obtain the total penetration of the magnetic field at the moment t2. We can note that for t1 and t3, two moments when the applied magnetic field takes the same values, the two distributions of the current density are not identical. That highlights a hysteresis phenomenon. The selected parameters make the graph J very stiff, so the problem is of “Stefan” type. That explains the appearance of a free border in the conductor, a very stiff transition between the part where the current density is zero and the part where |J| is worth Jc.
Figure 5.38. Applied magnetic field and profile of current densities
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The Finite Element Method for Electromagnetic Modeling
5.6.5.2. Superconductor cable In practice, superconductor cables are not massive. They consist of a multitude of small filaments of which the characteristic diameter is, in the case of LTS cables, in the order of a micron. Each filament is coated with a conducting shield. In practice, it is impossible to model a real cable with a discretization in connection with the dimension of the elementary conductor. It is then necessary to develop a homogenized model of the cable. Such a model was applied to a bundle of 49 superconductive filaments. This bundle is replaced by a single conductor modeled by a homogenized behavior law which is the weighted average of the behavior of the various components of the cable. In other words, if we note by Js the law J(E) in the superconductor and by Jn that of the normal matrix, the law in the homogenized compound is given by:
J
K J s (1 K) J n
[5.114]
where K indicates the proportion of superconductor in compound material. The validation of [5.114] is shown in [LEV 95] for maximum and continuously graphs. Figure 5.39 shows the distributions of the current density in the material calculated with and without this homogenization, within the framework of the Bean model. It is to be noted, firstly, that in both cases there are almost the same field penetration depths (Ha = 5 u 10+4 A/m). Secondly, it can be seen that the maxima of the current density are in a ratio of the same order of magnitude as the proportion of superconductor in the compound, with the result that the total current intensity is the same in the two devices.
Figure 5.39. Distributions of J in the superconductive compound (left) and in the homogenized conductor (right)
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239
However, the model is satisfactory only if it leads to the same values, not only for average losses, but also for the instantaneous losses. Figure 5.40 shows that the evolutions of the Joule losses over time are very close in both cases. +1
Pertes (x 10 Watt/m) Sans homogéneisation Avec homogénéisation
5,0 4,0 3,0 2,0 1,0 0,0 0,0
0,5
1,0
1,5
2,0
2,5
-2
Temps (u10 sec)
Figure 5.40. Time evolution of Joule losses, Bean model
This model of homogenization should be able to be applied to the case of OPIT cables in HTS superconductor, even if the filaments are larger (about 100 microns) [VIN 00]. 5.6.5.3. Interaction magnet superconductor This third example is intended to put forward the theoretical possibility of achieving an electromagnetic levitation by using jointly a magnet and a superconductive chip. Figure 5.41 shows the interaction force between a magnet and a cylindrical superconductive chip as a function of a composite magnetsuperconductor. The results of numerical simulation are compared with the results of actual measurements. The experiment consists of approaching the magnet with the superconductor then moving it away. The speed of the magnet remains the same during all its motion. The minimal distance between the magnet and the superconductor, for this figure, is 3 mm. The superconductor and the magnet have, respectively, a radius of 10.5 mm and 11 mm and a 10 mm thickness and 20 mm. The magnet is characterized by a magnetic polarization of 1.1 T and the superconductor with a critical density of current 90 A/mm2. The results of measurement come from the LEG based on a massive chip of a ceramic type (YBaCuO) provided by the CNRS-CRETA.
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The Finite Element Method for Electromagnetic Modeling
Figure 5.41. Interaction force between the magnet and the superconductor chip
Calculation shows that the interaction force is a force of repulsion. This force is not the same one in the approach and withdrawal phases. As for the cable, it represents a well-known hysteresis phenomenon in the medium of superconductivity. It is a global phenomenon which does not come from the behavior of materials, since those do not have any hysteresis. 5.7. Conclusion
What can we say after having swept a panorama as vast as that which we have just seen? Around 60 pages are quite insufficient to describe everything, and the objective of the book is not to make the reader a specialist in the modeling of materials. We have shown some examples of models in the hope of highlighting the difficulties of modeling the behavior of materials in our experience. In order to gain a deeper knowledge in each subject tackled, it is advisable to refer to the quoted texts, which contain a more precise description of the problems as well as offering a more complete bibliography. Currently, the increase in computing capabilities allows complex behavior models to be integrated in the simulation of devices close to real cases in terms of geometry. However, behavior models remain far from reality. Indeed, very often, all the aspects which we presented separately act simultaneously. Let us briefly consider the example of a GO sheet’s magnetic transformer circuit: the area of the joints in T where anisotropy, stress in the rotating field, and dynamic effects are all cumulate. Yet it is also in this area where the material mechanical state is most disturbed! Work thus continues towards more complete models.
Behavior Laws of Materials
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5.8. References [BAI 98] J. BAIXERAS, Les supraconducteurs, Application à l’électronique et à l’électrotechnique, Eyrolles, CNRS Editions, 1998. [BEA 62] C.P. BEAN, “Magnetization of hard superconductors”, Physical Review Letters, 8, 6, pp. 250-253, 1962. [BER 85] G. BERTOTTI, “Physical interpretation of eddy current losses in ferromagnetic materials. I. Theoretical considerations”, J. Appl. Phys., 57 (6), pp. 2110-2117, 1985. [BER 91] G. BERTOTTI, “Generalized Preisach model for the description of hysteresis and eddy current effects in metallic ferromagnetic materials”, J. Appl. Phys., 69 (8), pp. 46084610, 1991. [BER 98] G. BERTOTTI, Hysteresis in Magnetism, Academic Press, 1998. [BER 00] Y. BERNARD, Contribution à la modélisation de systèmes électro-magnétiques en tenant compte du phénomène d’hystérésis. Extension du modèle de Preisach adaptées au calcul du champ, Doctoral thesis, Orsay University (Paris-Sud), 2000. [BIL 99] R. BILLARDON, L. HIRSINGER, F.OSSART, “Magneto-elastic coupled finite element analyses”, Revue Européenne des Eléments Finis, vol. 8, no. 5-6, pp. 525-551, 1999. [BIO 58] G. BIORCI, D. PESCETTI, “Analytical theory of the behavior of ferromagnetic materials”, Il Nuovo Cimento, vol. VII, no. 6, pp. 829-842, 1958. [BLO 99] F. BLOCH, O. CUGAT, J.C. TOUSSAINT, G. MEUNIER, “Approches novatrices à la génération de champs magnétiques intenses: optimisation d’une source de flux à aimants permanents”, European Physical Journal, Applied Physics, vol. 5, pp. 85-92, January, 1999. [BOS 93] A. BOSSAVIT, “Sur la modélisation des supraconducteurs: Le ‘modèle de l’état critique’ de Bean, en trois dimensions”, J. Phys. III France, 3, pp. 373-396, 1999. [BOT 98] O. BOTTAUSCIO, M. CHIAMPI, L.R. DUPRÉ, M. REPETTO, M. VON RAUCH, J. MELKEBEEK, “Dynamic Preisach modeling of ferromagnetic laminations: a comparison of different finite element formulations”, J. Phys. IV France, 8, pp. Pr2-647-650, 1998. [BRI 97] P. BRISSONNEAU, Magnétisme et matériaux magnétiques, Hermes, 1997. [CES 96] C. CESTER, Etude des pertes magnétiques supplémentaires dans les machines asynchrones alimentées par onduleur à modulation de largeur d’impulsion, Doctoral thesis, de l’Institut National Polytechnique de Grenoble, Génie Electrique, 1996. [CHA 88] J. CHAVANNE, Contribution à la modélisation des systèmes statiques à aimants permanents, Doctoral thesis, de l’Institut National Polytechnique de Grenoble, September, 1988. [CHA 89] J. CHAVANNE, G. MEUNIER, “A new model for non-linear anisotropic hard magnetic material”, IEEE Trans. Magn., vol 25, no. 4, July, 1989. [CHE 99] T. CHEVALIER, Modélisation et mesure des pertes fer dans les machines électriques, application à la machine asynchrone, Doctoral thesis, de l’Institut National Polytechnique de Grenoble, Génie Electrique, 1999.
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[CHE 00] T. CHEVALIER., G. MEUNIER, A. KEDOUS-LEBOUC AND B. CORNUT, “Numerical computation of the dynamic behavior of magnetic material considering magnetic diffusion and hysteresis”, IEEE Trans. Mag., vol. 36, no. 4, pp. 1218-1221, 2000. [CLE 00] S. CLÉNET, J. CROS, I. HAOUARA, P. VIAROUGE, F. PIRIOU, “A direct identification method of the hysteresis model for the design of SMC transformers”, IEEE, Trans. Mag., vol. 36, pp. 3369-3466, 2000. [COL 87] B.D. COLEMAN, M.L. HODGDON, “On a class of constitutive relations for ferromagnetic hysteresis”, Arch. Rational Mech. Anal., 99, no. 4, pp. 375-396, 1987. [DEB 01] O. DEBLECKER, Contribution à la modélisation des champs magnétiques dans les systèmes comportant des milieux non linéaires et hystérétiques, Doctoral thesis, Faculté Polytechnique de Mons, 2000. [DEL 82] DEL VECCHIO, “Computation of losses in non oriented electrical steels from a classical viewpoint”, J. Appl. Phys., 53(11), pp. 8281-8286, 1982. [DEL 91] E. DELLA TORRE, “Existence of magnetization dependent Preisach models”, IEEE Trans. Magn., vol. 27, no. 4, pp. 3697-3699, 1991. [DEL 94] F. DELINCE, Modélisation des régimes transitoires dans les systèmes comportant des matériaux non linéaires et hystérétiques, Thesis, Faculté des Sciences appliquées de Liège, 1994. [DIN 83] A. DI NAPOLI, R. PAGGI, “A model of anisotropic grain-oriented steel”, IEEE Trans. Mag., vol. 19(4), pp. 1557-1561, 1983. [EKE 73] I. EKELAND, R. TEMAM, Analyse convexe et problèmes variationnels, Dunod Gauthier-Villars, Paris, 1973. [ENO 94] M. ENOKIZONO, S. KANAO, K. YUKI, “Permeability tensor of grain oriented steel sheet”, J.M.M.M., 1994. [FAS 64] G.M. FASCHING, H. HOFMAN, “Die feldvektoren B und H schwacher magnetfelder in anisotropen Eisenblechen”, Z. Ang. Phys., 17, pp. 244-247, 1964. [GIV 85] D. GIVORD, A. LIÉNARD, R. PERRIER DE LA BATHIE, P. TENAUD, T. VIADIEU, “Determination of the degree of crystallites orientation in permanent magnets by x-ray scattering and magnetic measurements”, Le Journal de Physique, Vol. 46, no. C6, p. 313, September 1985. [GOL 00] C. GOLOVANOV, G. MEUNIER, G. REYNE, R. GROSSINGER, E. WITTIG, M. TARABA, “Eddy current analysis in permanent magnet under pulsed field”, Studies in Applied Electromagnetics and Mechanics, vol. 18, IOS Press, pp. 81-84, 2000. [GOU 98] C. GOURDIN, Identification et modélisation du comportement électromagnétoélastique de structures ferromagnétiques, Doctoral thesis, Paris University VI, 1998. [HAN 75] I.F. HANTILA, Rev. Roum. Sci. Techn. – Electrochn. Et Energ. 20, pp. 397-407, 1975.
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[HAS 92] H. HASHIZUME, T. SUGIURA, K. MIYA, S. TODA, “Numerical analysis of electromagnetic phenomena in superconductors”, IEEE Trans. Magn., 28, 2, pp. 13321335, 1992. [HAU 97] A.O. HAUSER. “Calculation of superconducting magnetic bearing using a commercial FE-Program (ANSYS)”, IEEE Trans. Mag., vol. 33, no. 2, pp. 1572-1575, March, 1997. [HIE 95] P. HIEBEL, P. TIXADOR, X. CHAUD, “Lévitation magnétique par association d’aimants permanents et de supraconducteurs à haute température critique”, J. Phys., III France, 5, pp. 647-659, 1995. [HUT 84] D. HUTTENLOHER, H.W. LORENZEN, D. NUSHELER, “Investigation of the importance of the anisotropy of cold rolled electrical steel sheets”, IEEE Trans. Mag., vol. 20 (5), pp. 1968-1970, 1984. [JAH 87] J. JAHN, K. ELK, R. SHUMAN, J.M.M.M., 68, 335, 1987. [JIL 86] D.C. JILES, D.L. ATHERTON, “Theory of ferromagnetic hysteresis”, J.M.M.M., vol. 61, pp. 48-60, 1986. [JIL 92] D.C. JILES, J.B. TOELKE, M.K. DEVINE, “Numerical determination of hysteresis parameters for the modelling of magnetic properties using the theory of ferromagnetic hysteresis”, IEEE. Trans. Mag., vol. 28, pp. 27-35, 1992. [KED 86] A. KEDOUS, D. LEBOUC ET P. BRISSONNEAU, “Etude des pertes dans les tôles magnétiques soumises à des variations d’induction B(t) de forme trapézoïdale”, Rev. Phys. Appliquée, 21, pp. 269-275, 1986. [KIM 63] Y.B. KIM, C.F. HEMPSTEAD, A.R. STRNAD, “Magnetization and critical supercurrents”, Physical Review, 129, 2, pp. 528-535, 1963. [LEV 95] C. LEVILLAIN, P. MANUEL, P.G. THEROND, “Current, Induction Profiles and Hysteretic Losses in High-Tc Superconducting Tapes”, IEEE Trans. ASC-5, 2, pp. 705708, 1995. [MAI 69] A. MAILLEFERT, Contribution à l’étude de la pénétration macroscopique de l’induction magnétique dans les supraconducteurs de seconde espèce impurs, Doctoral thesis, d’état soutenue en novembre 1969 au sein du Laboratoire de Génie Electrique de Paris. [MAS 97] M. MASLOUH, F. BOUILLAULT, A. BOSSAVIT, J.C. VERITE, “From Bean’s model to the H-M characteristic of a superconductor: some numerical experiments”, IEEE Transactions on Applied Superconductivity, vol. 7, no. 3, pp. 3797-3801, September 1997. [MAY 91] I.D. MAYERGOYZ, Mathematical Models of Hysteresis, Springer-Verlag, New York, 1991. [NAK 75] T. NAKATA, Y. ISHIHARA, K. YAMADA, A. SASANO, “Non-linear analysis of rotating flux in the T-joint of a three-phase, three-limbed transformer core”, Proc. of the Soft Magnetic Materials 2 Conference, pp. 57-62, 1975. [PER 94] T. PÉRA, Lois d’aimantation anisotropes et non linéaires: modélisation et validation expérimentale, Doctoral thesis, l’Institut National Polytechnique de Grenoble, 1994.
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[PHI 95] D.A. PHILIPS, L.R. DUPRÉ, J.A.A. MELKEBEEK, “Magneto dynamic field computation using a rate dependent Preisach model”, IEEE Trans. Magn., vol. 30, no. 6, pp. 43774379, 1994. [PRI 96] L. PRIGOZHIN, “The Bean model in superconductivity: variational formulation and numerical solution”, J. Comp. Phys., 129, pp. 190-200, 1996. [PRI 97] L. PRIGOZHIN, “Analysis of critical-state problems in type-II superconductivity”, IEEE Transactions on Applied Superconductivity, vol. 7, no. 4, pp. 3866-3873, December, 1997. [ROU 96] L.L. ROUVE, Prise en compte du comportement fréquentiel des toles FeSi en modélisation électrotechnique, Doctoral thesis, l’Institut National Polytechnique de Grenoble, 1996. [SIL 91] P.P. SILVESTER, R.P. GUPTA, “Effective computational models for anisotropic soft B-H curves”, IEEE Trans. Mag., 27, p. 3804, 1991. [SPO 98] S. SPORNIC, Automatisation de bancs de caractérisation 2D des tôles magnétiques. Influence des formes d’ondes sur les mécanismes d’aimantation, Doctoral thesis, l’Institut National Polytechnique de Grenoble, Génie Electrique, 1998. [STE 89] J. STEFAN, “Über einige Probleme der Theorie der Wärmeleitung”, Sitz. Akad. Wiss. Wien, Mat.-Nat. Classe 98, pp. 473-484, 1889. [SYK 97] J.K. SYKULSKI, R.L. STOL, A.E. MAHDI, “Modelling HTc supraconductors and AC power loss estimation”, IEEE Trans. Mag., vol. 33, no. 2, pp. 1195-1198, March, 1997. [TIX 95] P. TIXADOR, Les supraconducteurs, Hermes, Collection matériaux, 1995. [TRE 99] E. DU TRÉMOLET DE LACHEISSERIE, Magnétisme, Presses Universitaires de Grenoble, 2 volumes, 1999. [VAS 96] F. VASILIU, F. BOUILLAULT, A. DEGARDIN, A. KREISLER, “Modelling of the electric characteristic of type-II superconductors by means of Ligurian minimization”, IEEE Trans. Magn., 32, 3, pp. 1144-1147, 1996. [VIN 00] E. VINOT, G. MEUNIER, P. TIXADOR, “Modeling superconductors with a power law”, Studies in Applied Electromagnetics and Mechanics, Non-Linear Electromagnetic Systems, vol. 18, pp. 179-182, 2000. [YOS 94] Y. YOSHIDA, M. USEKA, K. MIYA, “Magnetic field and Force analysis of high Tc superconductor with flux-flow and flux-creep”, IEEE Trans. Mag., vol. 30, no. 5, pp. 3503-3506, 1994. [WEG 76] P.T. WEGLER, “Computation of magnetic fields in non-linear anisotropic media with field dependent degree of anisotropy”, Proc. of Compumag, Oxford, pp. 168-176, 1976.
Chapter 6
Modeling of Thin and Line Regions
6.1. Introduction Some devices such as the tank and accessories of a transformer, ship hulls, airgaps in machines, contactors or magnetic recording heads, shielding, etc. are mainly made up of sheet or line type parts of thin air-gaps or cracks. Modeling these parts using traditional finite volume elements used in 3D software is tiresome, and even impossible. Moreover, the skin effect in ferromagnetic materials increases the difficulties of meshing eddy current problems in under sinusoidal conditions. To cope with these difficulties, it is possible to use special “shell elements” for the modeling of magnetic or thin conducting regions, “surface impedance” elements for the modeling of conducting regions having a low skin depth, etc. This chapter presents these special elements. 6.2. Different special elements and their interest We call a “thin region”, an region which has a low thickness compared to its other dimensions, and “neighbor regions”, the external regions which have a common border with the thin region. Electrotechnical devices are also made of line type parts such as clamping beams, bus bars of transformers, windings, etc.
Chapter written by Christophe GUÉRIN.
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Meshing thin regions and line regions using an automatic mesh generator, which generates tetrahedral elements in 3D (and triangular elements in 2D), result, because of the low thicknesses or weak sections, in a very significant number of elements or in elements which are too long. Some problems are difficult, even impossible, with current computing tools. The tetrahedral or triangular finite elements can have low accuracy when they are too long. An alternative to this difficulty of meshing the thin regions is the use of a mapped mesh generator or by extrusion which generates hexahedral or prismatic elements in 3D (quadrangular in 2D). The latter withstands a strong lengthening, which is not easy and takes time. The other way based on the use of an automatic mesh generator, consists of using special elements which allow the thin regions to be modeled by surfaces and line type regions to be modeled by lines. Thus, the description of the geometry and the mesh is largely simplified. The physical phenomena which occur inside these regions are taken into account in the integral formulation by surface or linear terms. The average surface (called * thereafter) which will describe a thin region will pass through the middle of this region.
Figure 6.1. Modeling of a thin region and a line type region with special elements
For some magnetoharmonic problems, the solid conductors are characterized by a strong skin effect. When the skin depth is small compared to the characteristic dimension of the conductor with a material with linear properties, the physical quantities such as the current or the magnetic field have a known exponential decay. The meshing of the conducting region with traditional volume elements must consist of elements which are smaller than the size of the skin depth. This situation will lead, for some problems, to a very high number of elements. Special surface elements, using the concept of surface impedance, which describe the surface of the conducting region, allow the exponential decay to be taken into account. They also allow the magnetic field to only be calculated on the surface and outside. The problems which are characterized with a low skin depth and which can require the use of such elements are, for example, problems of induction heating and problems relating to the transformers (tanks). Let us now consider a magnetoharmonic
Modeling of Thin and Line Regions
247
problem where there are thin regions of low thickness in which eddy currents flow. When the skin depth is lower than the thickness of the thin region and its material is linear, the variation of the quantities along the thickness is exponential. The common characteristic of special elements is to suppose known the variation of the physical quantities along the thickness of the thin region or the skin depth. We call “shell elements” the surface elements of a thin region to be described (see Figure 6.2).
Hyperbolic sinus variation (shell element)
Constant quality (shell element)
Figure 6.2. Various types of special surface elements ranked according to the variation of the physical quantities along the thickness
We can also rank the special elements according to types of unknown variables used by the associated formulations (fields, vector potentials, scalar, electric, magnetic). According to the type of physical problem that represents a thin region, and according to the unknown variable type, we have two types of shell elements: elements “without potential jump” and elements “with potential jump”. If the potential is constant through the thickness, the element is known as “without potential jump”, otherwise, it is known as “with potential jump” and the unknown variables are duplicated in each node of the element (see Figure 6.3). V3h V3
V1h V3b
V1 V1b V2h V2
Element without potential jump
V2b
Element with potential jump
Figure 6.3. Element without potential jump and element with potential jump
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The Finite Element Method for Electromagnetic Modeling
Let us take the example of a thin plate consisting of a ferromagnetic material of high permeability. The plate is surrounded by air. The induction and the field in this plate are mainly tangent since the flux is channeled by it. If the total scalar potential is used, the magnetic field is written H = –grad I, the equipotential surfaces are perpendicular to the surface of the plate. Thus, the potential at point A on a side of the plate will be equal to the potential at point B which is opposite (see Figure 6.4).
Figure 6.4. Constant potential I through the thin region
The modeling of the plate will require only one special element without a potential jump [BRU 91]. Let us note that in scalar potential, the continuity of the tangential component of the magnetic field is exactly assured by the nodal finite elements. It can be shown that the use of the magnetic vector potential requires an element with a potential jump for the modeling of the plate. On the other hand, for the dual problem, the thin air-gap of a magnetic circuit, where the induction is mainly normal, a special element with vector potential could be without jump and an element in scalar potential will necessarily be with potential jump [NAK 90]. The nodal finite elements exactly ensure the normal component of the induction in vector potential. We can extrapolate these observations for the other magnetostatic, magnetoharmonic, transient magnetic formulations, etc. which comprise various types of thin regions. They are, for example, a magnetic circuit air-gap, a ferromagnetic plate in the air, a conducting plate in the air, a crack of a low thickness slightly conducting in a conductor, etc. The following table indicates, for the potentials used in the thin regions and the neighboring regions, if the special element requires a potential jump or not. The use of special elements for the modeling of thin regions has other advantages: – thickness e of the thin region which can be changed without modifying the geometry or the mesh in order to carry out parametric studies easily according to this thickness e; – the time of calculation which is reduced compared to the use of traditional volume elements [NAK 90].
Modeling of Thin and Line Regions Magnetostatics Ferromagnetic thin sheet
B
Pe << P
Pe >> P
Conducting thin sheet
A
Formulations in B A, AV, A*, E
Jump
H
Ve >> V
Ve << V TIѽ H
Jump
Without jump
Crack
B
I, Ir
Formulations in H I, Ir, TIѽ H
Magnetodynamics
Air gap
H
249
Jump
Without jump AV, A*, E
Without jump
Without jump
Jump
Table 6.1. Element with jump and element without jump of the unknown variable. Pe, Ve: permeability and conductivity of the thin region, P, V: permeability and conductivity of the neighboring regions
6.3. Method for taking into account thin regions without potential jump The method presented here allows any type of thin region to be described with any nodal or edge approximation formulation by surface elements without potential jump [BRU 91]. This method is valid in the case of thin regions where the potentials are of constant thickness, i.e. in which the physical quantities, such as the magnetic field and current density, do not vary with the thickness. Let us denote by e, the thickness of the thin region. The process of obtaining the formulation for the thin region consists of decomposing the volume integrals of the volume formulation into a line integral along thickness e and a surface integral along surface *of the thin region.
³ : F d:
³ * ³ e F dz d*
[6.1]
where F is the integrating term of the integrals of the volume region formulation. As the potentials and the physical quantities are considered constant along the thickness, the linear integral is obvious. It is worth eF. Thus, the terms of the finite elements formulation for the thin region are obtained by transforming the terms of the formulation for volume elements into surface integrals, by multiplying them by thickness e and using the shape functions of surface elements instead of those of volume elements.
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The Finite Element Method for Electromagnetic Modeling
We take, as an example, the formulation for the total magnetic scalar potential. The linear system for this formulation for volume regions is written (see [BRU 91]):
>A@>I @ >C @ with > Aij @
>Ci @
[6.2]
³: grad wit >P @ grad w j d: , ³: grad wit B d:
where B is the induction. The matrix terms for the formulation of thin regions can be written: – for ª¬ Aij º¼ : – for ª¬Ci º¼ :
³ * e grad s wit ª¬ȝ º¼ grad s wj d * ;
³ *e grad s wi t B ds .
6.4. Method for taking into account thin regions with potential jump
The idea is to consider a surface element with potential jump as a prismatic element. We make the assumption that the potential is considered linear in the direction of the element thickness. Thus, quantities such as the magnetic field are supposed to be constant in the thickness. The prismatic element has a first order interpolation function along the thickness and any order along the other directions. We will integrate the interpolation functions along the thickness in order to obtain the formulation of an element with potential jump [SUR 86] [POU 93]. Two valid methods for a nodal approximation are successively presented [GUE 94a]. The formulation in total scalar potential is taken as an example for the application of these methods. In the first method, the integrals of the shape functions along the thickness are calculated in an analytical way before assembly and any numerical processing. In the second method, which is a more general approach, the integration is performed numerically, at the time of integration of the matrix terms in the matrix. We assume here that the elements with potential jump have their nodes duplicated. The duplicated nodes are at the same coordinates as those of the origin.
Modeling of Thin and Line Regions
251
6.4.1. Analytical integration method
We consider a reference coordinate system related to the element. Let us write x,y for the tangential curvilinear coordinates on average surface * of the element and z for the one normal on the surface of the element. We must consider curvilinear coordinates x,y,z of the real element expressed in the local reference coordinate system and not those of the reference element, in order to take into account the thickness of the element. Thickness e is assumed to be constant in each element. The magnetic scalar potential I is interpolated using the approximation functions of w’i of the prismatic element which has n’ = 2 n nodes: n'
¦ w'i x, y, z Ii
I
[6.3]
i 1
Functions w’i and w’i+n differ only by the terms according to z, which allows the Lagrangian functions wi(x,y) to be defined. These functions are the surface interpolation functions on average surface * and O1(z) and O2(z) which are the Lagrange interpolation functions in the thickness, i.e., those of a nodal line element of the 1st order with 2 nodes of length e. The potential is written: n
n
i 1
i 1
¦ wi x, y O1 ( z ) Ii1 ¦ wi x, y O2 ( z ) Ii 2
I
12 ez , O2 12 ez and Ii1
with O1
Ii ,Ii 2
[6.4]
Ii n , i >1,n@
Indices “1” and “2” refer respectively to sides “1” and “2” of the element with potential jump. Thin region : is described by the formulation in total scalar potential. The volume terms corresponding to the thin region are written in their discrete form: n'
ª
º ¼
¦ « ³ P grad wi 'grad w j ' I j d:»
j 1¬:
[6.5]
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The Finite Element Method for Electromagnetic Modeling
We are interested now in n first equations which correspond to the nodes on side “1” of the element. They are written:
°
n
³: P ® ¦ °¯ j
1
wwi wx
O1
wwi wy
O1
wi e
ª ww j ½ º½ ½ ww j O1 ° O2 ° «° »° ° «° wx »° ° ° wx ° «° ww j ° » °° ° ° ww j O1 ¾ I j1 ® O2 ¾ I j 2 » ¾ d: «® ° ° wy ° «° wy »° ° wj ° «° w j ° »° ° ° ° «° »° e ¿ ¯ ¬¯ e ¿ ¼ °¿
0
[6.6]
The other n equations which correspond to the nodes on side “2” of the element are obtained from the first n equations by permuting potentials Ij1 and Ij2. The volume integral on volume : is decomposed in a surface integral on * and an integral along z (F is the integrating term of [6.6]):
³: F d:
e/2
³* §¨© ³ e / 2 F dz ·¸¹d*
[6.7]
After integration and passage into the non-discretized integral form along z, the total formulation is thus written, using the surface gradient operators: eP
³:1 P1 grad wi gradI1 d:1 ³* 3 grad s wi grad s I1d* eP P P grad s wi grad s I 2 d* ³* wi I1 d* ³* wi I 2 d* 6 e e eP ³: 2 P 2 grad wi gradI 2 d: 2 ³* 3 grad s wi grad s I 2 d* eP P P ³* grad s wi grad s I1 d* ³* wi I 2 d* ³* wi I1 d* 6 e e
³*
[6.8] 0
[6.9] 0
The system of equations above is symmetric. The second equation can be deduced from the first by permuting indices “1” and “2”. 6.4.2. Numerical integration method
Unlike the previous method where an analytical integration was carried out, the integration along the surface and along the thickness is performed numerically, using the Gauss method, at the moment of integration of the matrix terms into the matrix, as for a volume element. We must consider the local coordinates u,v,w on the reference element. The approximation functions of the surface element with
Modeling of Thin and Line Regions
253
potential jump are calculated as for a prismatic element: the product of the approximation functions of a surface element without potential jump along * with those of a 2-node line element along the thickness is produced: wi (u,v,w) = wsj (u,v) . wƐk (w) , i >1 , n@, j >1 , ns@, k >1 , 2@ where wsj (u,v) (with j >1 , ns@) are the shape functions of the surface element, wƐk (w) (with k >1 , 2@) are those of the line element, ns is the number of nodes of the surface element and n = 2 ns is the number of nodes of the surface element with potential jump. The derivative of functions wi with respect to the local coordinates are written: w wi
w ws
wu
wu
wA ,
w wi
w ws
wv
wv
wA ,
w wi
ws
ww
w wA ww
[6.10]
We must now express the Jacobian matrix [J] 3 u 3 of the transformation of the real element with potential jump to the reference element. Let [J1] be the 2u3 matrix formed by the first two lines of [J], and [J2] be the 1u3 matrix formed by the third line of [J]. Matrix [J1] is calculated as follows:
>J1 @
ªw x « «w u «w x «w v ¬
wy wu wy wv
w zº » wu» w z» w v »¼
§ w wi ½ ¨° ° ¨° wu ° ¦ ¨ ® w w ¾ xi i° i 1¨ ° ¨° w v ° ¿ ©¯ n
yi
· ¸ ¸ zi ¸ ¸ ¸ ¹
[6.11]
Figure 6.5. Transformations of J1 and J2
[J1] is in fact the matrix of the transformation of the real surface element without potential jump, into the reference surface element without potential jump. [J2] is the
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The Finite Element Method for Electromagnetic Modeling
matrix of the transformation of the real line element into the reference line element (see Figure 6.5). Let nu and nv be two orthonorms, at coordinates (u, v) in reference coordinate system (O,u,v) of the reference element. Vector nu, respectively nv, is parallel to vector Ou, respectively Ov. In reference coordinate system (O,x,y,z) of the real element, they are two tangential orthogonal vectors on the surface of the surface element point (x (u,v), y (u,v), z (u,v)). Matrix [J1] is formed of the two transposed vectors nu and nv. Let nw be the normal unit vector on the surface. Matrix [J2] is formed by the transposed vector nw multiplied by the half thickness e of the thin region, as indicated below:
>J 2 @
ªw x « «¬ w w
w y ww
w zº » w w »¼
e 2
nTw
[6.12]
In [6.12] nWT is multiplied by e/2, as the length of the line element is worth e in the real reference coordinate system (O, x, y, z) and is worth 2 in the reference coordinate system of the reference element (O, u, v, w). The polynomial derivatives with respect to the global coordinates (x, y, z) are given by: wwi ½ ° ° ° wx ° ° wwi ° ® ¾ ° wy ° ° wwi ° ° ° ¯ wz ¿
grad wi
wwi ½ ° ° ° wu ° 1 w w ° ° JT ® i ¾ w v ° ° ° wwi ° °¯ ww °¿
> @
[6.13]
The general term Aij of the linear system matrix corresponding to the formulation in total scalar potential is written like an integral on the real element: ª
n
º
T ³³es ³el « ¦ P grad wi grad w j I j »des del
«¬ j
1
[6.14]
»¼
After passage of the global coordinates to the local coordinates, this term becomes: u 1
v 1
w 1
ª n
º
T ³ ³ ³ « ¦ P grad wi grad w j I j »det>J @dudvdw
u 1 v 1 w 1
«¬ j 1
[6.15]
»¼
The integration on the element is carried out using the Gaussian-quadrature method. The functions which are integrated along the thickness are second-order
Modeling of Thin and Line Regions
polynomials of the form
12 r ez 12 r ez .
255
Two Gauss points lead to an exact
integration of these functions, at least except for the numerical errors, the Gaussianquadrature method integrating exactly a polynomial of order 2m-1 with m points of integration. The method presented here is general and easy to implement. In fact, it applies to any formulation: the integration and the assembly of the surface element with potential jump are performed in the software in a similar way to the integration and the assembly of a traditional volume element. The difference in treatment between the two types of elements lies only in the retrieval of the interpolation functions and derivatives of these functions with respect to the coordinates: for a surface element with potential jump, the functions of the line element and those of the surface element are combined. 6.5. Method for taking thin regions into account
The method presented here is similar to the method described in section 6.3 for taking into account thin regions without potential jump. It makes it possible to describe any type of line region with any formulation with nodal or edge interpolation by line elements. This method is valid in the case of line regions where the potentials are constant in the section, i.e. in which the physical quantities, such as the magnetic field and current density, do not vary in the section. The method is deduced from the one described in section 6.3, by considering section s of the line region instead of thickness e of the thin region [BRU 91]. The formulation for the line region consists of breaking up the volume integrals of the volume formulation into a surface integral along section s and a linear integral along line O of the line region.
³ : F d:
³ O ³ s F dz d*
[6.16]
where F is the integrating term of the integrals of the formulation for volume regions. As the potentials and the physical quantities are considered constant along the thickness, the surface integral is obvious. It has a value sF. Thus, the terms of the finite element formulation for the line region are obtained by transforming the terms of the formulation for volume elements into line integrals, by multiplying them by section s and by using the shape functions of the line elements instead of those for volume elements. For example, for the formulation in total magnetic scalar potential of section 6.3, the matrix terms for the formulation for line regions are written:
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The Finite Element Method for Electromagnetic Modeling
– for ª¬ Aij º¼ : – for ª¬Ci º¼ :
³ O s gradA wit >ȝ @ gradA w j d O ; G ³O s gradA wi B d O .
6.6. Thin and line regions in magnetostatics 6.6.1. Thin and line regions in magnetic scalar potential formulations
6.6.1.1. Thin and line regions without potential jump Since the surface or line elements are without potential jump, we take into account only the surface gradients in thin or line regions without potential jump in magnetic scalar potential formulations. The magnetic scalar potentials make it possible to take into account the regions in which fields H and B are mainly tangent with the thin or line region. The permeable regions and regions with tangent magnetizations can then be taken into account [BRU 91]. The surface and line formulations without potential jump are obtained from the volume formulation thanks to the methods described in sections 6.3 and 6.5. 6.6.1.2. Thin regions with potential jump In thin regions with potential jump in scalar potential formulations, fields H and B can have any direction [GUE 94a]. Fields H and B must be constant in the thickness of the region. Such regions thus make it possible to describe thin air-gaps, permeable regions, as well as regions with magnetization of any direction. The surface formulation with potential jump is obtained from the volume formulation thanks to the method described in section 6.4. 6.6.1.3. Air-gap edges in magnetic scalar potentials When a magnetic circuit with a thin air-gap is described with magnetic scalar potentials, there is a potential jump on all the surface of the air-gap, which is prolonged in the air, beyond the edge. When this air-gap is described by a thin region with potential jump, this thin region must be prolonged in the air, until the potential jump is assumed to be negligible. On the edge of the thin region of the prolonged air-gap, the degrees of freedom on the two sides are then confounded (I1 = I2) [GUE 94a].
Modeling of Thin and Line Regions
257
6.6.1.4. Example: magnetic circuit with air-gap Air-gap
Figure 6.6. Magnetic circuit with thin air-gap in magnetic scalar potential (reduced in the air and the air-gap, total in the magnetic circuit). On the right: isovalues of induction on the vertical symmetry plane
6.6.2. Thin and line regions in magnetic vector potential formulations
6.6.2.1. Thin and line regions without potential jump Thin regions without potential jump in a magnetic vector potential formulation make it possible to take into account the surface currents (layers) and the thin airgaps, and the line regions to take into account the linear currents [NAK 90], [BRU 91]. The surface and line formulations without potential jump are obtained from the volume formulation thanks to the methods described in sections 6.3 and 6.5. 6.6.2.2. Thin and line regions with potential jump In thin regions with potential jump in nodal vector potential formulation, fields H and B can have any direction. Fields H and B must be constant in the thickness of the region. Such regions then make it possible to describe thin magnetic regions, thin air-gaps or surface currents (layers) [GUE 94a]. The surface formulation with potential jump is obtained from the volume formulation thanks to the method described in section 6.4. 6.7. Thin and line regions in magnetoharmonics
In magnetoharmonics, for eddy current problems, special elements are used in the following cases: – solid conducting regions where the skin effect is strong: the skin depth is much lower than characteristic dimension of the thin region;
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The Finite Element Method for Electromagnetic Modeling
– conducting thin regions. Several cases can arise depending on whether the skin depth is higher or lower than the thickness of the thin region; – conducting line regions; – slightly conducting or insulating thin regions in a solid conducting region. 6.7.1. Solid conducting regions presenting a strong skin effect
6.7.1.1. The surface impedance condition For a linear, homogenous and isotropic material, the skin depth in the conductors is calculated by:
G
2 / VZP
[6.17]
Meshing difficulties appear when skin depth G becomes smaller compared to the characteristic dimension of the solid conductors to be modeled. This situation occurs when either the frequency, permeability or resistivity is high. Surface impedance Zs connects the tangential component to the surface of the conductor of the magnetic field H to the tangential component of the electric field E by the following relation: nuE
Z s n u n u H
[6.18]
where n is the unit vector normal at the surface and outgoing from the conducting region. In order to obtain a first expression of the surface impedance, we must consider the problem of a plate with an infinite thickness subjected to a uniform sinusoidal field parallel with the side of the plate which is composed of a linear material. This one-dimensional problem is solved in [STO 74]. The complex impedance found is constant, i.e. independent of the value of the field. It will be called “surface impedance in the one-dimensional (or 1D) approximation”: Zs
Hs
1 j
Es
VG
[6.19]
Modeling of Thin and Line Regions
259
Figure 6.7. Plate of infinite thickness in a uniform field
This surface impedance associated with the finite element method is used on any geometries and not only plane ([KRA 88], for example). In 2D, a formulation in vector potential using the 1D approximation can be used [HOO 85]. In 3D, the magnetic scalar potential is generally used as state variable [GUE 94a], sometimes the magnetic vector potential A associated with the electric scalar potential V [LOU 95]. For the description of the regions outside the conductors, the method of boundary integrals can be used [KRA 88] [TAN 88]. In this case, only the surface of the conductor has to be meshed, but the rigidity matrix is full. The finite element method can also be used, which leads to a sparse band matrix [ROD 91], [GUE 94a]. 6.7.1.2. Validity and limitations of the surface impedance condition There is a limitation of topological order, which is related to the magnetic scalar potential. In fact, when the conducting region is non-simply connected, i.e. it comprises at least one hole, this potential cannot be used without specific processing. There are also limitations of a geometric nature, during the use of the expression of the surface impedance in the one-dimensional approximation. This expression is valid if the following conditions are checked: – G << L
L: characteristic length of the conductor;
– G << R
R: characteristic curvature radius of the conductor.
Moreover, the one-dimensional approximation is no longer valid when the problem consists of edges or corners, or when the radius of curvature is small compared to the skin depth. Surface impedance formulae modified for small curvature radii, for edges with 90° or for an edge of any T angle are proposed in [DEE 90] [JIN 93].
260
The Finite Element Method for Electromagnetic Modeling Strong features
Weak features
– Easy implementation
– Bad accuracy in the corners and edges if the one-dimensional value of the surface impedance is used
– Low CPU time cost
– Good accuracy when the skin depth is – Taking into account conductors which are small non-simply connected is impossible without – No volume meshing inside the conductors specific processing Table 6.2. Strong and weak features of the surface impedance method in magnetic scalar potential compared to a volume formulation H, E, AV, T-:
6.7.1.3. The surface impedance condition in magnetic scalar potential Several presentations of the surface impedance formulation in reduced scalar potential exist: [BOS 84], [TAN 88] or [KRA 88]. The formulation presented in this section is taken from [GUE 94a]. The materials must have an isotropic permeability and an isotropic linear conductivity.
Figure 6.8. Notations of the formulation
The quantities subscripted by “0” are the values of these quantities on surface * of conductor :c. Those subscripted by “s” are the quantities on * tangential to *. By considering the case of the plate of infinite thickness [STO 94], the variation of the quantities along direction z is exponential:
E z
E0 e
1 j
z
G ,
H z
H0e
1 j
z
G ,
J z
J0e
1 j
z
G
[6.20]
The tangential magnetic field is expressed with tangential source field Hjs and reduced scalar potential Ir:
Modeling of Thin and Line Regions
Hs
261
[6.21]
H js - grad s I r
The formulation in reduced potential, in the air region :, is recalled below:
³: P 0 grad w grad I r d: ³* wB n e d*
³* P 0 wH j n e d *
[6.22]
where ne is the outgoing normal of region :: ne = -n. The second term of [6.22] is transformed thanks to surface impedance relation [6.18] in order to take into account the conducting region. Faraday’s law curl E = -jZB makes it possible to express the normal of induction B on conductor surface *: Bn = -1/(jZ) curl En. Galerkine’s method on surface * is applied to this relation, which is then transformed thanks to the vector analysis formulae and Stokes’ theorem:
³* wB n d*
1 jZ
³* E u n grad wd*
1
³ wE dO jZ O
[6.23]
The third term of relation [6.23] is a linear integral on a contour O located on surface *. It is zero on the edge between two surface elements located on the conductor surface, to ensure the continuity of the tangential component at the edge of the electric field E. This term is also zero on a symmetry plane where the condition H u n’ = 0 is imposed (Ir = 0) and on a symmetry plane where the condition Hn’=0 is imposed (n’ is the normal to the symmetry plane). Expression (Eun) in the second term of [6.23] is expressed according to the tangential field on surface Hs, by using the surface impedance relation [6.18] and property n u (n u H s ) = (n H s )n (n n)H s : Eun
Es u n
Z s(n u H s ) u n
Z s
Hs
[6.24]
Relation [6.23] is thus written:
³* wB n d*
1
³ Z s grad s w H s d* jZ *
[6.25]
The final formulation is obtained by using [6.21] in [6.25], and [6.25] in formulation [6.22]:
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The Finite Element Method for Electromagnetic Modeling
³: P 0 gradw grad I r d: ³* P 0 wH j n e d*
1 jZ
1 jZ
³* Z s grad s w grad s I r d* [6.26]
³* Z s grad s w H j d*
The formulation in total potential is obtained by canceling the terms due to the source field Hj in the previous equation. The passage condition (H2 – H1) u n = K states that the surface current density K is due to the jump of the tangential component of the field through the layer of skin depth, where H1 = Hs and H2, which is the field inside the conductor, is assumed to be negligible with a certain depth. K and the surface density of Joule losses Pj are expressed according to field Hs calculated using potential Ir thanks to [6.21] [KRA 88] [GUE 94a]: K
n uǾ s ,
Pj
1
Re( Z s ) H s 2
2
[6.27]
6.7.1.4. Validity of the condition in the presence of permeable conductor When the permeability of the conducting region is sufficiently high and the frequency is not too high, the magnetic field in the air on conductor surface * is mainly normal on this surface. The numerical application of the surface impedance condition in reduced scalar potential gives bad results in this case. This problem of numerical inaccuracy is similar to the well-known permeable region problem described by the formulation in reduced potential in magnetostatics: the tangential component of the total field, which is weak, is the difference in two great numbers [PRE 92]. A solution to solve this problem consists of using the surface impedance condition in total scalar potential to describe the conductor, instead of reduced scalar potential, and including this conductor in an region of air in total scalar potential; the possible coils being in an region of air in reduced scalar potential. In [BOS 86], criteria are given to determine if the field is mainly tangential or normal on the surface of the conductors:
P0 G G !! and 1 , the field is mainly tangential; L P L P P G and 0 1 , the field is mainly normal. – if 0 L P P – if
Modeling of Thin and Line Regions
263
6.7.1.5. Taking into account magnetic nonlinearities with the surface impedance condition 6.7.1.5.1. Use of a step function B(H) curve Preston and Deeley have developed surface impedance type methods in 3D in scalar potential for strongly saturated materials [PRE 82], [DEE 79], [DEE 86]. They have used the traditional model by Agarwal and MacLean. In this calculation model of the losses in strongly saturated steel sheets, curve B(H) is a rung (see Figure 6.9 below) [MAC 54], [AGA 59]. Under these conditions, simple loss and surface impedance formulae are obtained analytically. Bo
H
Figure 6.9. B(H) curve in Agarwal’s model
More generally, there are two extreme cases: on the surface of the conductors, either the magnetic field is assumed to be sinusoidal, or the electric field is assumed to be sinusoidal. Actually, neither the electric field nor the magnetic field is sinusoidal, except in rare cases where it is possible to consider a 1D problem. The calculations carried out to obtain the surface impedance formulae (ZsH) in the case of the sinusoidal magnetic field are detailed in [MAC 54] and [AGA 59] and in [LOW 76] in the case of the sinusoidal electric field (ZsE). Z sH
with G
8
1
3S VG
2 j
and Z sE
2
2H 0
VZP ( H 0 )
VZB ( H 0 )
27S 3 1 § 4 · j¸¸ ¨¨1 3S ¹ 2 5 VG ©
[6.28]
where H0 is the peak tangent magnetic field and G is the penetration depth of the field for a nonlinear material. As the surface impedance depends on tangential field H0, it is necessary to perform iterations to adjust this surface impedance, by starting with a zero field and by taking, at iteration n, the field found in the previous iteration (so-called fixed point method). In practice, 4 to 6 iterations are required. It is preferable to use the induction corresponding to the real curve B(H) rather than induction B0 of the
264
The Finite Element Method for Electromagnetic Modeling
idealized step function curve. The results are accurate on a wider range of fields (from low to high fields). In order to ensure that the surface impedance formula is also valid in the first zone of curve B(H) (weak fields), the nonlinear and linear formulae can be weighted by a function of field H0 [FORD 94]. 6.7.1.5.2. Use of a 1D finite element model The method presented here is more accurate than the previous one [KRA 97] [AYM 97]. It uses the solution of a 1D problem and an energy equivalence for taking into account materials with a nonlinear characteristic B(H). The solved 1D problem is the conducting plate problem having infinite thickness, subjected to a magnetic field parallel to its surface and uniform, where the plate has a nonlinear B(H) characteristic. The imposed field is such that either the magnetic field is time varying sinusoidal, or the induction is sinusoidal. The 1D problem is solved step by step over time using finite elements. For an imposed magnetic field H1, impedance Zs is calculated according to the surface density of active power Ps and of reactive power Qs by formula [AYM 97]: Zs
2
dPs j dQs H12
[6.29]
Before the principal resolution, at the beginning of this one, a characteristic Zs(H1) is calculated. For this purpose, several impedances which correspond to various amplitudes of the imposed magnetic field H1 are calculated. We calculate points on curve Zs(H1) which allow all of curve B(H) to be described. Each calculated point corresponds to the resolution of the 1D problem with an amplitude H1. The curve is obtained by interpolation between each point of calculation. During the principal resolution, this curve Zs(H1) is directly used. As this impedance depends on the field on the surface of the conductor, iterations should be carried out, as for the method of the previous section, by the method known as the fixed point method. 6.7.1.6. Example: transformer with its tank This is about a 50 Hz three-phase, three-limb distribution transformer. The magnetic circuit which is laminated is described by an region with a constant permeability with the formulation in total scalar potential. The oil inside the tank, which has the same permeability as that in the air, is described by the formulation in reduced scalar potential. As the tank is made of magnetic steel, the skin depth, approximately 3 mm at 50 Hz, is much lower than the thickness of the tank (1 cm). The field on the external side of the tank, which is an almost perfect screen, is thus considered zero. The internal surface of the tank is described by the surface impedance condition in reduced scalar potential. Calculations were carried out first
Modeling of Thin and Line Regions
265
with a linear B(H) characteristic of the tank then with a nonlinear one. The presence of two symmetry planes makes it possible to describe only a quarter of the device.
Figure 6.10. On the left: quarter of the geometry of the transformer described in software FLUX3D (only the middle voltage (MV) and high voltage (HV) coil windings of the one of the three phases of the transformer are represented). On the right: arrows of the surface current density at a given instant, and isovalues of Joule losses in the tank (the darkest grays represent the highest losses), at rated conditions
6.7.2. Thin conducting regions
6.7.2.1. Formulations in magnetic scalar potential D. Rodger has proposed an interesting formulation for dealing with thin conducting regions in the case G >> e. It uses a scalar quantity “t” linked to the surface current density in the thin region and the magnetic scalar potential for the neighboring regions [ROD 87], [ROD 88], [ROD 92]. Nevertheless, the permeability of the thin region and the neighboring regions must be the same. We present in this section a more general formulation which allows the modeling of permeable thin regions while taking into account the skin effect in the thickness [GUE 94a]. This formulation, in magnetic scalar potential, requires surface elements with potential jump. The analytical solution of the conducting plate problem having a finite thickness subjected to transverse uniform fields is used [STO 74]. This analytical solution allows the surface impedances to be obtained, which will be used to obtain the formulation. This formulation was proposed in [KRA 90], [KRA 93] coupled with the boundary integral equations to take into account the neighboring regions. The formulation presented here was adapted for neighboring regions described by the finite element method [GUE 94a].
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The Finite Element Method for Electromagnetic Modeling
The only validity condition of this formulation is the one which states that the region must be thin: e << L where L is a characteristic length of the thin region. The materials must have a linear and isotropic permeability and conductivity. 6.7.2.1.1. Equations in the thin region
Figure 6.11. Notations
Let us take H1s and H2s as the tangential magnetic fields on both sides of the thin region :. The analytical solution of the one-dimensional problem of the finite thickness plate subjected to imposed fields H1s and H2s is given in [STO 74]. The expression of the field is written: 1 ª H1s sh §¨ ae az ·¸ H 2 s sh §¨ ae az ·¸º» ¹ ¹¼ © 2 © 2 shae «¬
H s ( z)
with a
1 j
G
[6.30]
.
Current density J is tangential to the thin region. Ampere’s law curl H = J allows the expression of J to be obtained by deriving the expression of H above with respect to z: J ( z)
n1 u
wH s z wz
a sh ae
n1 u ª« H1s ch§¨ ae az ·¸ H 2 s ch§¨ ae az ·¸º» © 2 © 2 ¹¼ ¹ ¬
[6.31]
The electric fields E1 on the “side 1” surface of the plate (E1 = E(z/2) = UJ(z/2)), and E2, on side “2” (E2 = E(–z/2) = UJ(–z/2)), can thus be written: E1
n1 u E H 2 s D H1s
[6.32]
E2
n2 u E H1s D H 2 s
[6.33]
with the complex surface impedances D
a
V thae
and E
a
V shae
.
Modeling of Thin and Line Regions
267
Relations [6.32] and [6.33] bind surface feature electric fields E1 and E2 to H1s and H2s, tangential magnetic fields on both sides of the thin region :. It is noted that relation [6.33] can be obtained from relation [6.32] by permuting indices “1” and “2”. 6.7.2.1.2. Formulation finite elements in reduced scalar potential The following proof is similar to that in section 6.7.1.3, concerning the surface impedance condition, but differs by the thin region which comprises two borders instead of only one. Thereafter, we will focus on side “1” of the thin region. The formulation in reduced scalar potential in neighboring region :1 can be written:
³:1 grad w P1 grad I1 d: ³* wB n1 d*
[6.34]
³* P1 wH j n1 d*
The second term of [6.34] is transformed thanks to relations [6.32] and [6.33]. Thus, this term and the one corresponding to region : allow regions : and : to be coupled and the thin conducting region to be taken into account. We use relation [6.23] which is valid on both borders of thin region :. On the border of side “1”, this relation is written:
³* wB1 n1 d*
1 jZ
³* E1 u n1 grad wd*
1
[6.35]
³ wE1 dO jZ O
The line integral term disappears with the boundary conditions, and between 2 adjacent elements for the same reasons as in section 6.7.1.3. The tangential fields H1s and H2s are expressed according to the tangential source field Hjs and of the reduced potential, by: H1s = Hjs - grads I1,
By using the electric field expression [6.32] and n u (n u H s ) = (n H s )n (n n)H s , relation [6.35] becomes:
³* wB1 n1 d*
1
[6.36]
H2s = Hjs - grads I2
³ grad s w DH1s EH 2 s d* jZ *
the
propriety
[6.37]
The formulation in the thin region represented by the relation above, can now be coupled with the one of the formulation of the neighboring region :, in order to ensure the continuity of the normal component of induction B ((B2–B1)n1 = 0) at
268
The Finite Element Method for Electromagnetic Modeling
the crossing of the interface. Having replaced relations [6.36] and [6.37] in [6.34], we obtain the final formulation:
³:1 P1 grad w grad I1 d:1
1
jZ 1
jZ
³* E grad s w grad sI 2 d*
1 jZ
³* D grad s w grad sI1 d* ³* wP1 H j n1d *
³* D E grad s w H j d *
[6.38]
Equation [6.38] corresponds to side “2” of the thin region. It is necessary to associate with it the second equation which corresponds to side “1”, which is deduced from [6.38] by permuting indices “1” and “2”. The total scalar potential formulation is obtained by canceling the terms due to source field Hj in equation [6.38]. The passage condition (H2 – H1) u n = K allows the surface current impedance to be expressed as a function of the scalar potential. K
n1 u grad s I1 I 2
[6.39]
The surface density of Joule losses is expressed according to the tangential components of the magnetic fields on the two sides of the thin region H1s and H2s, by [GUE 94a]:
P
1
1 §¨ 2¨
©
H1s
VG
2
2· H 2 s ¸¸ sh 2J sin 2J H1s H 2*s H1*s H 2 s shJ cos J chJ sin J ¹ ch 2J cos 2J
e and, the surface density of reactive power is expressed according to G magnetic fields H1 and H2 by [GUE 94a]:
where J
Q
1
VG
1 2
§ ¨¨ H1 ©
2
2· H 2 ¸¸ sh 2J sin 2J H1 H 2* H1* H 2 chJ sin J s h J cos J ¹ ch 2J cos 2J
Modeling of Thin and Line Regions
269
Unlike the Joule losses, the reactive power is dependent on all the components of the fields. In fact, the Joule losses are calculated by integral 1/2V |J|² where current density J is tangential and is expressed according to H1s and H2s, whereas the reactive power is calculated by integral 1/2 P |H|² and is expressed according to the total magnetic field H. It can be shown that the variation of the normal component Hn(z) has the same variation as the tangential component Hs(z) given by [6.30]. In the extreme case where skin depth G becomes very small in comparison to thickness e, impedance D tends towards surface impedance ZS and coupling impedance E between the 2 sides becomes zero. Formulation [6.37] corresponds, in this case, to two conditions of surface impedance [6.26] on the two faces of the thin region, which decouples the 2 sides. Inversely, in the extreme case where skin depth G becomes very large in comparison to thickness e, impedances D and E tend towards 1/(Ve) and the power in the thin region becomes mainly resistive and the two sides are strongly coupled. 6.7.2.1.3. Edges of thin regions, line air-gaps and holes The condition I – I = constant on a line imposes that current density K is tangential with this line. Thus, the line currents are the equipotentials of I– I. When a thin region comprises an edge, this edge is a line current on which the condition “I – I = constant” must be imposed. In fact, the current density is characterized with a conservative flux in the thin region and only exists in this region. The constant is taken as equal to zero on the edge, which corresponds to confounding the degrees of freedom on the two sides: I = I [ROD 87]. Applying this condition I = I on a line allows insulating line air-gaps on a thin region to be described, for example, in the case of two jointed conducting plates, insulated between them. It is often the case of stator sheets of turbo-alternators, to reduce the losses by eddy currents. These line air-gaps must touch the edge of the thin conducting region, otherwise connectivity problems may occur [GUE 94a]. The magnetic scalar potentials can lead to problems of regions which are nonsimply connected when a thin region has holes. A solution consists of describing these holes as a material of low conductivity [ROD 87].
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The Finite Element Method for Electromagnetic Modeling
6.7.2.1.4. Examples: transformer with its tank, conducting disk
Figure 6.12. 100 MVA three-phase, three-limb transformer under overloaded conditions. Tank modeled using surface elements taking into account the skin effect along the thickness. 6 HV and LV coils are not represented. On the left, view of surface meshing of the magnetic core and the tank. On the right, lines and cones representing the surface current density in the tank
Figure 6.13. Conducting disk with a line air-gap on a diameter, subjected to a vertical sinusoidal uniform field. On the right, lines and cones representing the surface current density
6.7.2.1.5. Composite shells The shields used in EMC (electromagnetic compatibility) are often made of several joined sheets of various materials. It is possible to take into account these associations of two or more layers with the formulation for thin conducting regions in magnetic scalar potential described in section 6.7.2.1 [ABA 01]. Let us consider two plates. Between these two plates, the tangential component of the magnetic field is conserved, thus the magnetic scalar potential I between these two plates is
Modeling of Thin and Line Regions
271
conserved. Let us take Iң and I as the potentials on the external sides of the plates. The potential I between the two plates is expressed by an affine combination of potentials Iң and I. Once Iҏ is suppressed in the initial matrix which is dependent on the unknown variables Iң, I and I, a system function of only Iң and I, which corresponds to a plate equivalent to the first two, is obtained. The method can be extended to a shield of more than two plates, by calculating the system which corresponds to a plate equivalent to two joined plates, then by calculating the system which corresponds to the equivalent plate coupled to another plate, then by continuing in a recursive way. 6.7.2.2. AV formulation 6.7.2.2.1. Thin regions and line regions without potential jump in AV formulation The AV formulation without potential jump allows thin conducting regions to be described when the skin depth is significantly larger than the thickness of the thin region. The neighboring regions must be described by a formulation in potential vector A: A formulation or AV formulation. When a thin conducting region is described by potentials A and V, the thin region is much more conducting than the neighboring regions, and the skin depth is large in comparison to thickness e, it can be shown that the current density is constant in the direction of this thickness [NAK 90]. Therefore, potentials A and V are constant in the thickness direction, the current is tangential on the surface of the thin region and the induction is normal in this surface. The validity conditions of this formulation are thus e << L, V >> Vext and G >> e, where L is a characteristic length of the thin region, e its thickness, V its conductivity, Vext the conductivity of the external region and G the skin depth in the thin region. AV formulation also allows conducting line regions to be described when the skin depth is significantly larger than the dimensions of the line region section. The surface formulation and the line formulation without potential jump are obtained from the volume formulation thanks to the methods described in sections 6.3 and 6.5. 6.7.2.2.2. Thin regions with potential jump in AV formulation In thin regions with potential jump in AV formulation, quantities H, B, J and E can have any direction. The skin depth must be larger than the thickness of the thin region. Under these conditions, quantities H, B, J and E are assumed to be constant in region thickness. Such regions thus allow slightly conducting thin regions in very conducting volume regions to be described, as well as thin conducting regions in the air, etc. The surface formulation with potential jump is obtained using the volume formulation thanks to the general method described in section 6.4.
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The Finite Element Method for Electromagnetic Modeling
6.7.2.3. Other formulations for thin conducting regions The formulation for thin conducting region proposed by O. Biro uses as state variables scalar quantity t in the thin region and potential vector A in the external regions. In these regions, the potential vector is used to accept the non-simply connected regions [BIR 92]. Z. Ren has used a surface element without potential jump in electric field E with edge elements [REN 90]. The external regions are taken into account by an integral method. 6.8. Thin regions in electrostatic problems, “electric harmonic problems” and electric conduction problems
For electrostatic problems, “electric harmonic problems” and electric conduction problems the state variable used is generally the electric potential. For these applications, it is possible to perform a reasoning similar to that described in section 6.2 on magnetostatics, with the magnetic scalar potentials, for thin sheets and airgaps. For electrostatic problems and “electric harmonic problems”, the surface formulation with potential jump makes it possible to describe, for example, the thin cracks located in the dielectric of capacitors. For electrostatic problems, the surface charge densities can be described by thin regions with or without potential jump. For “electric harmonic problems”, the thin conducting regions with high permittivity surrounded by a vacuum can be described for the simulation of pollution on insulators. 6.9. Thin thermal regions
In thermal problems, the state variable used is generally the temperature. The surface formulation without temperature jump allows very good heat thin conducting regions to be described, i.e. having a great thermal conductivity compared to the medium where they are, for example, metal thin sections in the air, etc. In these regions, the heat flow must mainly be tangential. The surface formulation with temperature jump allows any type of thin region to be described, for example, the existing thin layers of low thermal conductivity in sandwich structures constituted by the power electronic components on their radiator. The surface densities of heat can also be described by thin regions with or without temperature jump.
Modeling of Thin and Line Regions
273
6.10. References [ABA 01] ABAKAR A., “Modélisation tridimensionnelle de systèmes électromagnétiques comportant des régions filaires et des régions minces: application en CEM 50 Hz à des dispositifs EDF”, PhD Thesis of INP Grenoble, April 2001. [AGA 59] AGARWAL P.D., “Eddy current losses in solid and laminated iron”, Trans. AIEE, vol. 78, Part II, 1959, pp. 169-179. [AYM 97] AYMARD N., “Etude des phénomènes magnétodynamiques pour l’optimisation de structures 3D de chauffage par induction à partir du code TRIFOU et d’essais sur prototypes”, PhD Thesis, Ecole doctorale des Sciences pour l’Ingénieur, Nantes, November 1997. [BIR 92] BIRO O., PREIS K., RICHTER K.R., HELLER R., KOMAREK P., MAURER W., “FEM calculation of eddy current losses and forces in thin conducting sheets of test facilities for fusion reactor components”, IEEE Trans. Mag., vol. 28, no. 2, March 1992, pp. 1509-1512. [BOS 84] BOSSAVIT A., “Impédance d’un four à induction: cas où l’effet de peau dans la charge est important”, EDF, Bulletin de la Direction des Etudes et Recherches, C series, no. 2, 1984, pp. 71-77. [BOS 86] BOSSAVIT A., “Stiff problems and boundary layers in electricity: a mathematical analysis of skin-effect”, BAIL Conference, Novossibirsk, July 1986. [BRU 91] BRUNOTTE X., “Modélisation de l’infini et prise en compte de régions magnétiques minces. Application à la modélisation des aimantations de navires”, PhD Thesis, INP Grenoble, December 1991. [DEE 79] DEELEY E.M., “Flux penetration in two dimensions into saturating iron and the use of surface equations”, IEE Proc, vol. 126, no. 2, February 1979, pp. 204-208. [DEE 86] DEELEY E.M., “Surface impedance methods for linear and nonlinear 2-D and 3-D problems”, Eddy Current Seminar, 24-26 March 1986, RAL 86-088, Chilton, pp. 101109. [GUE 94a] GUÉRIN C., “Détermination des pertes par courants de Foucault dans les cuves de transformateurs. Modélisation de régions minces et prise en compte de la saturation des matériaux magnétiques en régime harmonique”, PhD Thesis, INPG Grenoble, September 1994. [GUE 94b] GUÉRIN C., TANNEAU G., MEUNIER G., BRUNOTTE X., ALBERTINI J.-B., “Three dimensional magnetostatic finite elements for gaps and iron shells using magnetic scalar potentials”, IEEE Trans. Mag., vol. 30, no. 5, September 1994, pp. 2885-2888. [GUE 95] GUÉRIN C., TANNEAU G., MEUNIER G., NGNEGUEU T., “A shell element for computing eddy currents in 3D. Application to transformers”, IEEE Trans. Mag., vol. 31, no. 3, May 1995, pp. 1360-1363. [HOO 85] HOOLE S.R.H., CARPENTER C.J., “Surface impedance for corners and slots”, IEEE Trans. Mag., vol. 21, no. 5, September 1985, pp. 1841-1843.
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[JIN 93] JINGGUO W., LAVERS J.D., “Modified surface impedance boundary conditions for 3D eddy current problems”, IEEE. Trans. Mag., vol. 29, no. 2, March 1993, pp. 1826-1829. [KRA 88] KRÄHENBÜHL L., “Surface current and eddy-current 3D computation using boundary integral equations techniques”, 3rd International Symposium, IGTE, Gratz, 2728 September 1988. [KRA 90] KRÄHENBÜHL L., “A theory of thin layer in electrical engineering; application to eddy-current calculation inside a shell using the BIE software PHI3D”, 4th International Symposium, IGTE, Gratz, 10-12 October 1990. [KRA 93] KRÄHENBÜHL L., MULLER D., “Thin layers in electrical engineering. Example of shell models in analysing eddy-currents by boundary and finite element methods”, IEEE Trans. Mag., vol. 29, no. 2, March 1993, pp. 1450-1455. [KRA 97] KRÄHENBÜHL L., et al., “Surface impedance, BIEM and FEM coupled with 1D non-linear solutions to solve 3D high frequency eddy current problems”, IEEE Trans. Mag., vol. 33, no. 2, March 1997, pp. 1167-1172. [LOU 95] LOUAI F.Z., “Modèles magnétodynamiques d’éléments finis pour structures tridimensionnelles de chauffage par induction”, PhD Thesis, Ecole Doctorale des Sciences pour l’Ingénieur, Nantes, 1995. [LOW 76] LOWTHER D.A., WYATT E.A., “The computation of eddy current losses in solid iron under various surface conditions”, Compumag Conference, Oxford, 1976, no. SW7. [MAC 54] MAC LEAN W., “Theory of strong electromagnetic waves in massive iron”, Journal of Applied Physics, vol. 25, no. 10, October 1954, pp. 1267-1270. [NAK 90] NAKATA T., TAKAHASHI N., FUJIWARA K., SHIRAKI Y., “3D magnetic field analysis using special elements”, IEEE Trans. Mag., vol. 26, no. 5, September 1990, pp. 2379-2381. [POU 93] POULBOT V., “Contribution à l’étude des champs électriques très basses fréquences en milieu océanique”, PhD Thesis, INPG, Grenoble 1993. [PRE 82] PRESTON T.W., REECE A.B.J., “Solution of 3 dimensional eddy current problems: the T-: method”, IEEE Trans. Mag., vol. 18, no. 2, 1982, pp. 486-491. [PRE 92] PREIS K., BARDI I., BIRO O., MAGELE C., VRISK G., RICHTER K.R., “Different finite element formulations of 3D magnetostatic fields”, IEEE Trans. Mag., vol. 28, no. 2, March 1992, pp. 1056-1059. [REN 90] REN Z., RAZEK A., “A coupled electromagnetic mechanical model for thin conductive plate deflection analysis”, IEEE Trans. Mag., vol. 26, no.5, 1990, pp. 16501652. [ROD 87] RODGER D., ATKINSON N., “3D eddy currents in multiply connected thin sheet conductors”, IEEE Trans. Mag., vol. 23, no. 5, September 1987, pp. 3047-3049. [ROD 88] RODGER D., ATKINSON N., “Finite element method for 3D eddy current flow in thin conducting sheets”, IEE Proc., vol. 135, Part A, no. 6, July 1988, pp. 369-374.
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275
[ROD 91] RODGER D., LEONARD P.J., LAI H.C., HILL-COTTINGHAM R.J., “Surface elements for modelling eddy currents in high permeability materials”, IEEE Trans. Mag., vol. 27, no. 6, November 1991, pp. 4995-4997. [ROD 92] RODGER D., LEONARD P.J., LAI H.C., “Interfacing the general 3D A-I method with a thin sheet conductor model”, IEEE Trans. Mag., vol. 28, no. 2, March 1992, pp. 1115-1117. [STO 74] STOLL R.L., The Analysis of Eddy-currents, 1974, Clarendon Press, Oxford. [SUR 86] SURANA K.S., PHILLIPS R.K., “Three dimensional curved shell finite elements for heat conduction”, Computers and Structures, vol. 25, no. 5, 1987, pp. 775-785. [TAN 88] TANNEAU G., “Surface eddy currents in ‘TRIFOU’ when the skin depth is thin”, IEEE Trans. Mag., vol. 24, no. 1, 1988.
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Chapter 7
Coupling with Circuit Equations
7.1. Introduction The first 2D and 3D field computation software did not allow coupling with the external electric circuit. However, this coupling quickly becomes essential as soon as a simulation of the actual operation of a device is needed. In certain cases, it is possible and even interesting to characterize a device by a set of static or dynamic simulations without resorting to the coupling. However, this technique can quickly become tiresome, even impossible, in other cases, taking into account the great number of simulations that must be performed for a complete characterization (linked to saturation, to movement, to multiple sources, etc.). This chapter proposes, without being exhaustive, some examples of coupling with circuit equations in two and three dimensions. After a short review of the various methods for setting up an equation of the electric circuits and of the various possible types of coupling, we will establish the relations allowing the current and the voltage to be linked within the framework of the finite element formulations. Coupling techniques themselves will then be developed successively for two then for three dimensions. If the use of the vector potential is essential in two dimensions, the coupling technique, although very general, proves more disputed in three dimensions where formulations based on scalar potential have appeared. We will show that it is possible to develop very general coupled formulations with magnetic scalar potential.
Chapter written by Gérard MEUNIER, Yvan LEFEVRE, Patrick LOMBARD and Yann LE FLOCH.
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The Finite Element Method for Electromagnetic Modeling
7.2. Review of the various methods of setting up electric circuit equations We will very briefly present the various methods of setting up electric circuit equations. Let us consider a circuit made up of electric elements such as resistors, inductors and capacitors, described by an electric behavior, i.e. a relation between the voltage across an electric element and the current passing through it. The components can be of a “source” or “passive” type. The sources can be real or ideal and will be represented by the voltage, the current and the internal resistance. The passive components are resistances, inductances, mutual inductances and capacitors. In order to be able to represent the diodes and the switches, we can define nonlinear or controlled resistances. Component
Notation
Equation
Voltage source
u
u = u(t)
Current source
i
i = i(t)
Resistance
r
u = r.i
Inductance
l
u = l.di/dt
Mutual inductance
m
u = m.di/dt
Capacitor
c
i = c.du/dt
Nonlinear resistance
rnl
u = rnl.i
Controlled resistance
rT
u = rT .i
Table 7.1. Examples of components of electric circuits
7.2.1. Circuit equations with nodal potentials The structure of the circuit is represented by an oriented graph. It is composed of n nodes and b branches. Kirchhoff’s current law expresses the conservation of charges. In each node, the sum of the currents is equal to zero: b
¦ n ij .i j 0 i 1,2,...n
[7.1]
j 1
where nij is worth r 1 according to the orientation of branch j with respect to node i, and 0 if branch j is not connected to node i. In order to obtain a system of independent equations, we choose a node of the circuit (node 0) for which the conservation of the currents is a linear combination of the equations of the other nodes. We can write the system in the matrix form:
Coupling with Circuit Equations
N.I = 0
279
[7.2]
where N is a matrix of dimension (n-1, b) for the fundamental currents of the circuit. We define n-1 potential Vi where Vi represents the voltage drop between node i and node 0. We obtain: U = Nt.V
[7.3]
The current of branch i is calculated using voltage u and relation i=f (u) characterizing the component. If the circuit comprises only resistances ri, capacitors ci and sources which are independent from current Is, we have: G n .V C n .
dV I sn dt
0
[7.4]
where Gn = N.G.Nt with Gii = (ri)-1 and Gij = 0 if i is different from j Cn = N.C.Nt with Cii = ci and cij = 0 if i is different from j Isn = N.Is The matrices G and C are diagonal and the matrices Cn and Gn are symmetric. 7.2.2. Circuit equations with mesh currents Kirchhoff’s voltage law expresses the fact that the sum of the branch voltages of a closed loop is equal to zero: b
¦ m ij .u j 0 i 1,2,...c
[7.5]
j 1
where mij is worth r 1 according to the orientation of branch j with respect to mesh i, and 0 if branch j is not connected to mesh i. The system of equations representing the electric circuit is obtained by decomposition of the passive network into c independent meshes (c=b+n-1). Kirchhoff’s voltage law is written in the matrix form: M. U = 0
[7.6]
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The Finite Element Method for Electromagnetic Modeling
In each loop, we define a mesh current with an arbitrary orientation. The mesh currents form the vector {Im}={Im1, Im2, ....., Imc}t and the branch currents are a linear combination of the mesh currents, by taking account of the orientation of the branch with respect to the orientation of the meshes: I = Mt. Im
[7.7]
The state variables for the calculation are the mesh currents. The system of equations is established by applying the second Kirchhoff law in each independent loop. If the circuit comprises only resistances, inductances and sources independent of voltage Us, we obtain: R m .I m L m .
dI m dt
Em
[7.8]
with: Rm = M.R.Mt with Rii = ri and Rij = 0 if i is different from j Lm = M.L.Mt with Lii = li and Lij = 0 if i is different from j Em = M.Us The matrices R and L are diagonal and the matrices Rm and Lm are symmetric. 7.2.3. Circuit equations with time integrated nodal potentials This method allows inductances, resistances and capacities to be taken into account, while allowing the discontinuity of the voltage on the switches. We define a potential < such that: t
Ȍk
³V
k
dt
[7.9]
t 0
The vector of branch voltages is given by: U
Nt.
dȌ dt
>@
Coupling with Circuit Equations
281
The establishment of the linear system consists of writing the relations of the current conservation in the n-1 nodes on the basis of the integrated nodal potentials. We obtain: B n .Ȍ G n .< C n .
d 2< I sn dt
0
[7.11]
with Bn = N.B.Nt with Bii = (li)-1 and bij = 0 if i is different from j Gn = N.G.Nt with Gii = (ri)-1 and gij = 0 if i is different from j Cn = N.G.Nt with Cii = ci and Cij = 0 if i is different from j Isn = N.Is In order to impose a voltage source Us between the nodes a and b, it is usual, if the symmetric character of the matrix needs to be kept, to eliminate a variable from the system. We can for example express
Ȍa
Ȍb
³ Us dt
[7.12]
t 0
7.2.4. Formulation of circuit equation in the form of state equations In the state equations method, a circuit is described as an oriented graph comprising oriented edges binding a certain number of nodes. Each edge comprises only one electric component which is associated with two electric quantities, the voltage on its terminals and the current which crosses it (Va, Ia). From this graph, a subset of edges of the circuit, called a tree, connecting all the nodes without forming meshes, is extracted. The edges belonging to the tree are called branches of the tree, the other edges are called links. A particular tree called the normal tree is chosen. In this tree, voltages Vbc at the capacitor terminals in the branches and the currents flowing through the inductances in the links are the circuit’s state variables. The branches of the normal tree are preferably all the voltage sources (Vbe, Ibe), then resistances (Vbr, Ibr) and inductances (Vbl, Ibl), except the current sources. Consequently, the links are preferably all the current sources (Vmj, Imj), then inductances (Vml, Iml), resistances
282
The Finite Element Method for Electromagnetic Modeling
(Vmr, Imr) and capacitors (Vmc, Imc), except the voltage sources. The construction of the normal tree is carried out using the Welsh algorithm. A fundamental mesh is defined as a mesh containing only one link, the other edges of the mesh being branches. Each fundamental mesh is identified with the link which it contains. By writing that the sum of the terminal voltages of a fundamental mesh is zero, we obtain the first topological relation binding the terminal voltages of the branches to the terminal voltages of the links: Vmc ½ ° ° °° Vmr °° ® ¾ ° Vml ° ° ° °¯ Vmj °¿
ª s1 « « s5 « « s9 « «¬s13
s2 s6
s3 s7
s 4 º Vbe ½ ° »° s8 » °°Vbc °°
s10 s14
s11 s12 » ° Vbr ° »° ° s15 s16 ¼» ¯° Vbl ¿°
»®
[7.13]
¾
In the same way, a fundamental cut is defined as a cut containing only one branch; the other edges of the cut are links. Each fundamental cut is identified with the branch which it contains. By writing that the sum of the currents in the edges of a fundamental cut is zero, we obtain the second topological relation which binds the currents in the branches to the currents in the links: Ibe ½ ° ° °°Ibc °° ® ¾ ° Ibr ° ° ° °¯ I bl °¿
ª s1t « « st « 2t « s3 « «¬ s4t
s5t
s9t
s6t s7t s8t
s10t s11t s12t
s13t º Imc ½ »° ° s14t » °° Imr °° »® ¾ s15t » ° Iml ° »° ° s16t »¼ °¯ Imj ¿°
[7.14]
The “si” are blocks of matrices made up of 0, and 1 or of -1. They are calculated using the Welsh algorithm. In order to establish the state equations, it is necessary to take into account the constitutive relations of the passive elements of the circuit. These relate to the voltage-current relations at the resistance terminals, capacitors and inductances in the branches or in the links: Vbr
R b .I br
I bc
Vmr
R m .I mr
I mc
d Vbc dt d C m . Vmc dt
Cb .
Vbl Vml
d d I bl M bm . I ml dt dt d d L m . I ml M mb I ml dt dt
Lb.
[7.15]
Rb and Cb are respectively the diagonal matrices of resistances and capacitors in the branches of tree, Rm and Cm, in the links. Lb is a square matrix of the proper
Coupling with Circuit Equations
283
inductances and the mutual inductances between the branches inductances of the tree and Lm that of links inductances. Mbm is the matrix of the mutual inductances between branches inductances and links inductances and Mmb its transpose. From these constitutive relations of the circuit elements and the two topological relations describing the fundamental meshes and the fundamental cuts, we obtain the circuit’s state equations in the form:
d X A .X B .E C . d E e e e e e dt e dt e
[7.16]
The vector Xe, called a state vector, is formed by the voltage Vbc at the capacitor terminals in the branches and the currents Iml in the link inductances. The vector Ee, called an input vector, is formed by voltages Vbe of the voltage sources in the branches and of the currents Imj of the current sources in the links: Xe
°Vbc ½° ® ¾ ¯° Iml ¿°
Ee
°Vbe °½ ® ¾ ¯° Imj ¿°
[7.17]
An unspecified vector Ye of electric quantities of the circuit, which we call an output vector, can be easily expressed as a function of the state vector and of the input vector: Ye Fe .Xe G e .E e H e . d E e dt
[7.18]
Matrices Fe, Ge and He, as well as Ae, Be and Ce, are calculated on the basis of the topological relations and of the constitutive relations of circuit elements. This type of relation will be used thereafter to establish the coupling equations with the electromagnetic structure. Indeed, we can take as an output vector the electric quantities associated with line coils or with massive conductors of the electromagnetic structure interacting with the circuit. 7.2.5. Conclusion on the methods of setting up electric equations The nodal potential method allows current sources, resistances and capacitors to be taken into account. Taking into account the voltage sources results in a relation between the two potentials at the source terminals. The inductances can be taken into account by introducing their currents as variables of the problem. However, the electric potential at the inductance terminals can be discontinuous in time.
284
The Finite Element Method for Electromagnetic Modeling
The mesh currents allow voltage sources, resistance and inductances to be naturally taken into account. The method requires the implementation of an algorithm for the choice of the tree graph and the fundamental meshes (Welsh algorithm). The opening of a switch involves the cutting of one or several meshes. Lastly, accounting for capacitors generally requires us to introduce their voltages as variables. The integrated potentials over time method is adapted to the analysis of the electric circuits composed of current sources, of inductances, of resistances and of capacitors. We can take into account the voltage sources via linear combinations and the presence of switches does not add any additional difficulty. Another significant advantage is that this type of potential is continuous over time and that the conditioning of the system of equations does not depend on the numbering of the nodes and branches. In the circuit equations method in the state variables form, the assembly of the matrices intervening in the state equations system is relatively complex. However, the advantage of the state equations method is that it allows circuits, including all component types and all source types, whether current sources or voltage sources, to be automatically taken into account. In conclusion, two methods particularly hold our attention because of their generality: integrated potentials over time and state equations. In the following section we will show, within the framework of the 2D formulations, coupling examples with the mesh method, the integrated potentials and the state equations. In 3D, we will only present couplings using the integrated potential over time. 7.3. Different types of coupling We have to jointly solve two matrix systems: one established during discretization by finite elements and the other generated by establishing electric circuit equations. The system resulting from discretization by finite elements is generally of much greater size and results in a sparse matrix which is in general symmetric, definite positive. Three types of coupling can be considered: – an indirect coupling; – an integro-differential formulation; – a simultaneous resolution.
Coupling with Circuit Equations
285
7.3.1. Indirect coupling In this case, the two systems of equations are solved successively. The advantage comes from the fact that we can use software adapted to each type of equation. Moreover, the finite element matrix preserves its properties (sparse character, conditioning). The major drawback is that this method requires a large number of iterations particularly when the problems have strong physical couplings. In 3D, the costs can quickly become prohibitive. In conclusion, this coupling is only interesting compared to the other when the reaction of a system is weak. 7.3.2. Integro-differential formulation This method of direct coupling consists of eliminating the circuit equations and in preserving as unknowns, only those resulting from the discretization by finite elements. The main advantage of this method is that it leads to a well conditioned symmetric matrix involving only one type of unknown. On the other hand, the matrix of the global system of equations loses its sparse character because the elimination of the circuit unknown factors causes a coupling of the nodes of the conducting areas (an example of integro-differential formulation is given during the presentation of the coupling 2D vector potential with mesh equations). In 3D, this technique can prove to be very expensive in terms of memory storage and computation time. 7.3.3. Simultaneous resolution As in the previous case, we jointly solve the equations, but without making eliminations. Simply, the final matrix combines the two linear systems by introducing coupling terms. The main advantage of this technique is that it preserves the sparse character of the matrix system. However, conditioning can be affected by this coupling. In certain cases, normalization coefficients on the variables are necessary, even essential. Lastly, it is valuable to preserve the symmetric character of the equation systems, which is generally achievable. 7.3.4. Conclusion In conclusion, we will adopt simultaneous resolution because this method allows a strongly coupled resolution of the equations while keeping the sparse character of the matrices.
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The Finite Element Method for Electromagnetic Modeling
7.4. Establishment of the “current-voltage” relations 7.4.1. Insulated massive conductor with two ends: basic assumptions and preliminary relations Let us consider a portion of a conductor insulated and crossed by a current. The two ends *a and *b are scalar electric isovalues such that u = va – vb. Current i can enter and exit only by these ends. With these assumptions, we will look for the relation that links voltage u to current i.
*b
i *a u Figure 7.1. Massive conductor
First, let us consider an unspecified current of 1 A, ingoing through *a and outgoing through *b, with jo(x,y,z) the corresponding current density. A particular current density jo can be obtained by an electrokinetic resolution of the problem. However, any solution jo complying with div(jo) = 0 and jo.n = 0 apart from the surfaces *a and *b is also appropriate. Now, denoting by v the electric scalar potential, we will show that, independently of the current i which crosses the conductor:
³ j .grad v dȍ o
-u
[7.19]
:c
Considering the identity div(jov) = jo.gradv + v div(jo) and taking into account the fact that we have div(jo) = 0, we can write:
³ j .grad v dȍ ³ div ( j v) dȍ o
ȍc
o
[7.20]
ȍc
We transform the right-hand side integral into a surface integral by applying the divergence theorem. We obtain:
Coupling with Circuit Equations
[7.21]
³ j .grad v dȍ ³ v j .n dī o
o
ȍc
287
īc
By noting that jo.n = 0 apart from *a and *b surfaces, we finally obtain:
³ j .grad v dȍ
va
o
ȍc
³ j .n dī v ³ j .n dī o
īa
b
o
-u
[7.22]
īb
7.4.2. Current-voltage relations using the magnetic vector potential In the more general case, the electric field e is linked to the magnetic induction b according to the Maxwell-Faraday equation curl(e) = -db/dt. In the conducting media, a particular relation links current density j to electric field e. Since curl(e) = -db/dt, we can write e in the form e = -da/dt – gradv, where v represents the electric scalar potential and a the magnetic vector potential. The current-voltage relation then takes the form: u
da
³ j .e dȍ ³ j . dt dȍ o
o
ȍc
[7.23]
ȍc
In the particular but rather common case where the electric conductivity of the conductor is independent of e (it can be defined by a conductivity V such that j = V e), and where the selected solution jo is the electrokinetic solution, we can write current-voltage relations in the form (more usual): u
rc i
da
³ j . dt dȍ o
[7.24]
ȍc
where rc represents the resistance of the conductor obtained with the electrokinetic solution. This relation is obtained by noting that jo = -V grad(vo) where vo is the electrokinetic solution. Thus, we have: jo.e = -V grad(vo).e = -grad(vo).j
[7.25]
By noting that grad (vo).j = div(vo.j) since div (jo) = 0 and by applying the divergence formula, we finally obtain:
288
The Finite Element Method for Electromagnetic Modeling
³ j .e dȍ o
- v oa
ȍc
³ j.n dī - v ³ j.n dī ob
īa
u o .i
rc .i
[7.26]
īb
During the treatment of the 2D problems in the presence of massive conductors, the current-voltage relation is obtained by integrating the current density j which, in 2D, is perpendicular to the study plane. In addition, grad(v) is uniform on the section of the conductor which makes it possible to write the voltage in the form u = -grad(v) /L, where L is the length of the 2D study domain. The total current i crossing a massive conductor is finally written (by noting that j = Ve and e = -da/dt – grad(v)): i - ı
³
Sc
da u dS ı dS L dt
[7.27]
³
Sc
where Sc represents the surface of the conductor trace in the 2D study plane and a is the normal component of the magnetic vector potential with respect to the study plane. We will find in [LOM 93] that the current-voltage relation is similar for an axisymmetric problem. 7.4.3. Current-voltage relations using magnetic induction We will establish below a relation dedicated to the use of the magnetic scalar potential. Indeed, it is important to establish such a relation because of the value of using the magnetic scalar potential when solving 3D problems. Since we chose jo such that div (jo) = 0, there exists an electric potential to such that jo= curl(to). By noting that div (a x to) = to.curl(a) - a.curl(to), we can write: db
da
³ j . dt dȍ ³ t . dt dȍ ³ (a x t o
ȍc
o
ȍc
o ).n dī
[7.28]
īc
Let us now consider a simply connected definition domain for to, :o, which includes :c (see Figure 7.2), blocking the volume enclosed by the conducting :c. In other words, the conductor domain :c is entirely included in :o and it is not possible to go around the conductor without crossing the domain :o. Under these conditions, the choice of to can be made so that to x n = 0 on the border *o of the domain :o, while complying with curl(to) = jo in :o. An example of such a definition is given in [KLA 92], [DRE 94]. It should be noted that if we limit the integration domain :o to :c, we cannot find a solution that complies with to x n = 0 on *c. Indeed, the integral of a closed contour c around the conductor should
Coupling with Circuit Equations
289
comply with relation ³c to.dl = 1, which is incompatible with previous condition applied on *c. By integrating on the box including :o, with to x n = 0 on *o, the surface integral disappears and we obtain [MEU 03], [BIR 04]: u
db
[7.29]
³ j .e dȍ ³ t . dt dȍ o
o
ȍc
ȍo
In the case of a domain restricted by symmetries, we can still keep the previous expression if it complies with the conditions to x n = 0 on symmetries with a tangent field and to. n = 0 on symmetries with a normal field, u then representing the terminal voltage of the portion of the modeled conductor.
ȍo
ȍc
Figure 7.2. Example of a box including a conductor
This relation will enable us to couple the field equations with the electric circuit equations in the case of formulations using the magnetic scalar potential. It should be noted that, unlike the previous case (using the magnetic vector potential a), the integral is no longer limited to the only conductor :c, but to a box :o containing it and that it is necessary to determine a electric vector potential to, checking to x n = 0 on the boundary of this box. In the case where conductivity is isotropic and independent of the electric field e, we obtain (in a similar way to the case of the relation using the magnetic vector potential) the current-voltage relation in the form (rc is electrokinetic resistance): u
rc .i
db
³ t . dt dȍ o
:o
[7.30]
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The Finite Element Method for Electromagnetic Modeling
7.4.4. Wound conductors In the particular case of a wound conductor, it is usual to carry out a homogenization: we make the assumption that the size of wires is sufficiently small to avoid numerically solving the problem of non-uniform distribution of the current in each elementary conductor. In practice, the current density j is expressed directly on the basis of the total current and of the geometric characteristics of winding. We can define a space function job(x,y,z) characterizing the winding which links the average current density j to the current crossing the turns (j = job.i). Let us take as an example job = (ns/S).WW in the case of a coil with a cross-section S having ns turns regularly wound and W the unitary vector tangent to the wires. If we supply our conductor with a DC current of 1A, the real current density jo which circulates in the wires is linked to job by the relation job = O. jo where O represents the filling factor (Od1); in general, O can depend on x,y,z. The relation that we previously established for an insulated conductor can be applied to our coil and, by applying it to the overall length of the wire and by supposing that the conductivity is independent of the electric field e, we obtain: u
rb .i
³j
:c
o
da dȍ rb.i dt
³j
:b
ob
da dȍ dt
[7.31]
:b represents the volume of the coil (:c the useful volume of conductor) and rb the resistance of the coil. This can be calculated by: rb
1
³ıj
o
2
:c
dȍ
1
³ ıȜ j
ob
2
dȍ
[7.32]
:b
In the case of a coil with cross-section Sb having ns turns, we have job = (ns/Sb).t and thus rb = (1/VO). (ns/Sb)2 Vb where Vb represents the volume of the coil. We obtain: u
rb.i
³
:b
n s .IJ da dȍ S b dt
[7.33]
In 2D, the vector potential has only one component perpendicular to the study domain and W is collinear to a. Under these conditions, the finite element domain’s contribution to the voltage drop is written:
Coupling with Circuit Equations
u
rb i L
n s da dS b dt
³S
Sb
291
[7.34]
L represents the length of the study domain. Lastly, the relation using the magnetic induction is written, in the case of a wound conductor: u
rb i
³t
:o
o
db dȍ dt
[7.35]
where to is the electric vector potential which complies with curl(to) = job. 7.4.5. Losses in the wound conductors The previous relations, in the case of wound conductors, do not take into account the additional losses caused in the conductors when the latter are subjected to variable magnetic fields (proper or external). This assumption is admissible if the dimension of the wires is small compared to the thickness of skin. However, in a large number of applications, the additional losses have a very significant contribution compared to the electrokinetic losses, without altering or putting in doubt homogenization with the magnetic field calculation. The importance of taking these losses into account can be easily understood when for example we must correctly calculate the current of a coil during a voltage supply. In the case of an elementary conductor subjected to a uniform external field and of a sinusoidal time-dependent form (in particular the contribution due to the other elementary conductors), we can prove that this conductor can be represented as an insulator of complex permeability [MAT 92], [MOR 98]. The latter can be calculated analytically in the case of cylindrical or rectangular conductors. By applying a technique of homogenization (between conductors and insulators), it is then possible to determine an equivalent tensorial permeability of the wound conductor. The introduction of this permeability, when solving the field equations, finally makes it possible to take into account losses that are caused by an external field on each elementary conductor (proximity losses). They are expressed locally in the form: p
*
³ jȦ h . b d:
[7.36]
:b
Lastly, losses in a proper field should also be taken into account (i.e. the losses caused to itself by each elementary conductor) [MOR 98].
292
The Finite Element Method for Electromagnetic Modeling
7.5. Establishment of the coupled field and circuit equations 7.5.1. Coupling with a vector potential formulation in 2D 7.5.1.1. Presentation of field equations Now we will consider the case of a study field including air, ferromagnetic areas and conducting areas (wound or massive). The displacement currents are neglected. To simplify the case we will suppose that the properties of the materials constituting these areas are expressed in scalar form: Q to represent the magnetic behavior (h = Q b) and V to represent the electric behavior (j = V e). We will see, in particular in 3D, how to formulate a problem without restriction on these behavior laws. In the insulating areas or with an imposed current density (absence of induced currents) the field equations to be solved are those of magnetostatics: curl h = jb
[7.37]
div b = 0
[7.38]
and
where jb represents the average current density in the coils. Equation div b = 0 allows the introduction of the vector potential a (b=curl a). The equation to be solved in the insulating areas, or with an imposed current density, is written: curl (Ȟ curl a) j b
[7.39]
In the conducting areas, electric field e is linked to magnetic induction b by the equation: curl e
db dt
[7.40]
which introduces the electric scalar potential v such that: e
da grad v dt
[7.41]
Coupling with Circuit Equations
293
The equation to be solved in these areas is written:
curl (Ȟ curl a) ı
da ı grad v dt
0
[7.42]
In a 2D case, only the component along the axis of invariance (which is normal to the study domain) of magnetic vector potential a, the density of current jb and electric potential gradient v are considered. With this assumption the gauge condition div a = 0 is automatically checked. We indicate by L the length of the system along the axis of invariance. After discretization with nodal finite elements and by interpolating the normal component of vector potential an in the form an = 6 wi ani, where wi represents the nodal interpolation function, we obtain the differential equation system according to (integrating the conducting and non-conducting areas): S.A T.
d A C c .U c C b .I b dt
0
[7.43]
A = {an1, an2, …anN} represents the values of the vector potential at the N nodes of the mesh, Uc is the voltage on the massive conductors and Ib the current in the wire coils. Matrices S, T, Cc and Cb are obtained by the assembly of the elementary matrices. We have: Sij
L Q grad w i .grad w j dS
³
Tij
Sc
C b ik
L
³ı w w i
j
dS
Sck
³
Sck
n sk w i dS S bk
C c ik
³ı w
[7.44] i
dS
Sck
In addition, relations [7.34] between the currents Ib and the voltages Ub at the terminals of a wire coil can be put in the matrix form: Ub
R b .I b C bt .
dA dt
[7.45]
Rb is a diagonal matrix containing the resistances of the wire coils. Relations [7.27] between the currents Ic and the voltages Uc at the terminals of the massive conductors are put in the matrix form:
294
The Finite Element Method for Electromagnetic Modeling
Ic
G c .U c C ct .
dA dt
[7.46]
where Gc is a diagonal matrix containing the conductances of the massive conductors. In the particular case where the various quantities are sinusoidal, the previous equations can be written by using complex notation, d/dt being replaced by jZ. The solution of the problem can then be obtained in only one calculation, which is extremely valuable in terms of computing time. However, the use of this type of formulation is a priori rather restrictive: – the material properties of the areas treated by finite elements must be linear; – the coupled circuit components exclude any nonlinear behavior (diodes and switches in particular); – the finite element domain must be fixed; – the supplies must be sinusoidal. In practice, complex notation can in certain cases be used even when the previous assumptions are not all checked (with the help of some precautions and approximations). We can note, for example, the use of an “equivalent” behavior law b(h) in the case of nonlinear material, or, in the case of an induction machine, the modification of the rotor conductivity to simulate velocity. In this last case, the space harmonics due to the displacement are then neglected. 7.5.1.2. Coupling with the mesh equations We will first present this coupling in the case of a sinusoidal mode which is simpler than the general case in steps over time. With the mesh equations, we have to introduce the massive or wound conductors as branches belonging to the studied circuits. These branches have a particular relation u(i), as in the case of the circuit branches. The writing of the mesh equations (section 7.2.2) revealed the mesh currents Im as unknown factors. We will show how to couple these unknown factors with the magnetic vector potential, first in the case of wound conductors, then in the case of massive conductors. In the case of wound conductors, the field equation for these areas is written in matrix form S.A – Cb.Ib = 0 and readily reveals the coils currents Ib. We can link these currents to the mesh currents via a branch-mesh incidence matrix Mb. We write: Mbt.Im = Ib where Im represents the set of the mesh currents.
Coupling with Circuit Equations
295
The writing of the mesh equations, which consists of expressing the second Kirchhoff’s law (sum of voltages of the branches of a closed loop equal to zero), has as its expression, by taking into account the branches that the coils (section 7.2.2) constitute: Mb.Ub + Zm.Im = Em
[7.47]
The Zm matrix and the Em sources come from the external circuits. By noting that Ub = RbIb + jZCbt.A, the simultaneous resolution of the field and circuit equations leads us to the coupled symmetric system according to: ª S « «- M b .C bt ¬«
C b .M bt º A ½ Z R bm » ® ¾ » ¯I m ¿ - m jȦ ¼»
0 ½ ° E ° ®- m ¾ °¯ jȦ °¿
[7.48]
where
R bm
M b .R b .M bt .
It is possible to solve this system by eliminating the unknown Im factors. We thus write a integro-differential formulation: the elimination of Im is obtained by inverting the Zm+Rbm matrix and finally by solving a system which is put in the form: ( S jȦCb Z1.Cbt ǹ -Cb Ǽ
[7.49]
The matrix obtained is symmetric, and definite positive. The major disadvantage of this formulation is that it couples the unknowns on the nodes of the conductors, which results in losing the sparse character of the matrix ( C b Z 1.C bt term). Similarly to the wire conductor, the voltage Uc of the massive conductors is regarded as being that of a mesh branch. However, this time the field equation is written (S-jZT).A – Cc.Uc = 0 and reveals voltages Uc. We thus have as unknown factors: the terminal voltages of the massive conductors Uc, the mesh currents Im and the magnetic vector potential A. The current-voltage relation allows Uc and Im to be linked. By defining the branch-mesh incidence matrix Mc for the massive conductors, we thus obtain the following matrix system:
296
The Finite Element Method for Electromagnetic Modeling
ª «S - jZT « « - Ct c « « « 0 «¬
- Cc Gc jȦ Mc jȦ
º 0 » » A½ M ct » ° ° . ®U c ¾ jȦ » ° ° Z » ¯Im ¿ - m» jȦ »¼
½ ° 0 ° ° ° ® 0 ¾ ° E m ° ° jZ ° ¿ ¯
[7.50]
The second and third equations were divided by jZ in order to symmetrize the system. The elimination of the unknown factors Uc and/or Im leads to a integrodifferential system where the matrix loses its sparse character. Here, finally, is the global matrix system obtained (massive and wound conductors), in the case of a sinusoidal mode: ª «S jZ T -C c « « G t c « -Cc jȦ « « Mc t « -M b .Cb jȦ ¬
º -C b .M bt » » A½ » ° ° M ct » . ®Uc ¾ jȦ » ° ° Im Z R bm » ¯ ¿ - m » jȦ ¼
½ ° 0 ° °° °° ® 0 ¾ ° E ° ° m ° ¯° jZ ¿°
[7.51]
In the case of a stepwise over time system, the use of a time-dependent discretization method allows the coupled matrix system to be built. The use of an implicit method consists of writing the equations at the time t+Gt and in expressing the derivatives of unknown factors x by the relation: § dx · ¨ ¸ © dt ¹ t įt
x t įt x t Gt
[7.52]
Here, for example, is the coupling matrix system obtained in the case where the external circuits comprise only resistances and inductances (matrices Rm and Lm):
Coupling with Circuit Equations
1 ª « S įt T « t « -Cc «-M .C t « b b ¬
297
º » A ½ »° ° » ®Uc ¾ G c įt -M c įt -M c įt -(R m R bm ) įt L m » °¯ I m °¿t įt » ¼ -Cb .M b t
-Cc
0 ½ ° ° ® 0 ¾ °-Em įt ° ¯ ¿t įt
1 ½ - T.A ° ° įt ° ° t -Cc .A ® ¾ °-L I M .C .A ° b b ° m m ° ¯ ¿t
[7.53]
As an application, we propose the study of an induction machine of 18 KW, while taking into account the squirrel cage. The stator windings are of the wire type and are supplied by a three-phase system. The rotor is composed of massive conductors connected at their end by a squirrel cage. An adapted component (symbolized in Figure 7.4 by a squirrel) makes it possible to describe the latter by specifying the extreme end resistances and inductances. Because of symmetries, only one quarter of the motor is studied. The equiflux lines for the slip at 3.55% are as follows.
Figure 7.3. Equiflux for a slip of 3.55%
The circuit studied is represented in Figure 7.4.
298
The Finite Element Method for Electromagnetic Modeling
Figure 7.4. Circuit studied
The table below summarizes the results obtained from the various calculations carried out for two values of slip g. Measurement
Stator current in Amps (g = 1.31%) Torque in N.m (g = 1.31%) Cos M (g = 1.31%) Stator current in Amps (g = 3.55%) Torque in N.m (g = 3.55%) Cos M (g = 3.55%)
22.9 64.6 0.730 36.4 122 0.863
Software Case of magnetodynamics
Software Case of evolutionary magnetic
21.75
21.91
(5%)
(4.3%)
63.2
64
(2.2%)
(0.9%)
0.761
0.718
(4.2%)
(1.7%)
38.4
38.53
(5.5%)
(5.9%)
128.4
127.6
(5.2%)
(4.6%)
0.853
0.79
(1.2%)
(8.5%)
Table 7.2. Simulation results for the induction machine
Coupling with Circuit Equations
299
7.5.1.3. Coupling with time integrated nodal potentials We are in a situation comparable to that of the mesh equations, because the unknown factors in the circuit equation system are integrated potentials over time <. Hence, in the case of the massive conductors we can link the unknown Uc to < and keep only A and < as unknown factors, whereas in the case of coils we preserve the unknown A, Ib and <. Any elimination of unknown factors leads to a system which loses its sparse character. We introduce the node-branch incidence matrices Nb and Nc, allowing us to link Ub and Uc to the integrated nodal potential <. With a step similar to the one we presented for the coupling with the mesh equations, the contribution of the conductors (wound and massive) to the global equation system, in the case of a time-stepping formulation using an implicit Euler method, is as follows: 1 ª « S įt T « t « C b « 1 « N c .Cct ¬ įt
C b
R b .įt Nb
1 º Cc .N ct » A ½ įt » ° ° N bt . » ®I b ¾ » °Ȍ ° 1 G cn » ¯ ¿t įt įt ¼
1 ½ ° įt T.A ° ° ° t ® Cb .A ¾ °1 ° ° G cn .Ȍ ° ¯ įt ¿t
[7.54]
with Gcn = Nc.Gc.Nct. Indices c refer to the massive conductors and indices b to the wound conductors.
The contributions of the circuit equations are assembled in the third equation. The various circuit elements can be treated in a similar way to finite elements: the electric components behave like elements where the nodal potentials are the unknown factors of the system. Let us take the example of a conductance gk =1/rk: we denote by
g k .u k
Ne
> 1
g k (dȌ a dȌ b )
g k .N et .Ȍ k
[7.55]
with
1
@
t
and Ȍ k
^Ȍ a
Ȍb `
t
This current is involved in nodes a and b: – entering current to node a: +gk d
[7.56]
300
The Finite Element Method for Electromagnetic Modeling
The conductance element thus brings the following contribution to the global system equations: º ª1 « įt Gn k » ^Ȍ k `t įt ¼ ¬
1 ½ ® G nk .Ȍ k ¾ ¯ įt ¿t
[7.57]
with
>N e @ g k >N e @t
G nk
ª g k « g ¬ k
gk º g k »¼
[7.58]
As an application, we propose the modeling of a superconductive cable carried out within the framework of the European project BIG-POWA [STA 02]. The coupling is necessary since we are in the presence of a massive conductor supplied with current (and taking into account the nonlinearity of the superconductor, it is not equivalent to a voltage supply). The problem is solved step by step in time. At each time step the solution is obtained using a Newton-Raphson procedure. The type of matrix system to implement with this method is detailed in the case of the vector potential formulation in 3D (section 7.5.2).
f
é)
Figure 7.5. Simulation of a multi-filament superconductive cable
Figure 7.5 presents the geometry of an OPIT conductor with imposed current and external field (imposed by boundary conditions). The pictured ribbon includes 37 filaments. The distribution of the current density and, therefore, the losses could thus be calculated for various values of the current and the magnetic field (Figure 7.6). The results are in good agreements with experimental measurements
Coupling with Circuit Equations
301
I=Imax, B=Bmax
I=0, B=0
-50A/mm 2
0
50A/mm 2
100A/mm 2
Figure 7.6. Current distribution in a multi-filament conductor
7.5.1.4. Coupling with state equations State equations are useful when the circuits which supply an electromagnetic device have a very general structure able to contain all types of components (resistances, capacitors, inductances or mutual inductances) and all types of sources (voltage or current sources). The wire coils and the massive conductors of the electromagnetic devices belong to the circuit. We then take them into account as voltage sources of the circuit depending on the field. We have seen that the currents Ib and the voltage Ub at the terminals of the wire coils are linked to the values of the vector potentials of the grid nodes A by the matrix relation: Ub
R b .I b C bt .
d A dt
[7.59]
In the field equation discretized by finite elements, the variables that are involved are the currents Ib and not the total voltages Ub at the terminals of the wire coil. We can thus integrate the wire coils in the circuit as resistances Rb in series with voltage sources of values Uf, as functions of the field by the matrix relation: Uf
C bt .
d A dt
[7.60]
302
The Finite Element Method for Electromagnetic Modeling
In addition, the currents Ic and voltages Uc at the terminals of the massive conductors are linked to the values of the vector potentials of the grid nodes A by the matrix relation: Ic
G c .U c C ct .
d A dt
[7.61]
Unlike with wire coils, total voltages Uc intervene in the field equation and, at the circuit level, the massive conductors are required to be regarded as voltage sources of values Uc. Their resistances are not taken into account at the circuit level, but at the level of the field equations. By separating the proper sources of the circuit from the sources depending on the field related to the wire coils and to the massive conductors, the circuit equations can be written in the following way: d Xe dt
A e .X e B ep .E ep B ef .U f Bec .U c C ep .
d E ep dt
[7.62]
where Xc represents the state variables of the circuit, Eep the proper sources of the circuit, Uf the terminal voltages of wire coils, and Uc the terminal voltages of the massive conductors. J.F. Charpentier has demonstrated that the derivative of the voltage Uf and Uc do not intervene in the state equation [CHA 96]. On the level of the electromagnetic device, the field sources are the currents Ic in the massive conductors and the currents Ib in the wire coils. It is thus necessary to express these currents according to the variables of the circuit. In fact, it is sufficient to consider them as variables forming exit vectors of the circuit which we can thus express according to the state vectors and the entry vectors of the proper sources of the circuit: Ib
Fb .X e H b .E ep
Ic
Fc .X e H c .E ep
[7.63]
J.F. Charpentier has demonstrated that on the level of the circuit, the currents do not depend directly on the voltages Uc and Ub. In addition, the derivatives of the proper circuit sources do not intervene [CHA 96]. In the case of massive conductors and on the basis of the previous relations, we can establish the following relation: G c .U c C ct .
d A - Fc .X e dt
H c .E ep
[7.64]
Coupling with Circuit Equations
303
The following system of differential equations is obtained from the circuit’s state and coupling field equations: d A C c .U c - C b .Fb .X e C b .H b .E ep dt d G c .U c C ct . A - Fc .X e H c .E ep dt d d d X e A e .X e B eb .C bt . A Bec .U c B ep .E ep C ep . E ep dt dt dt
S.A T.
[7.65]
The second member of this first order differential equation system consists of the Eep sources specific to the circuit, real voltage or current sources. The unknown factors of the system are the values of the non-zero component of the vector potential at the grid nodes A, the state variables of the circuit Xc, the terminal voltages of massive conductors Uc and the currents in wire coils Ib. Example applications have been published. They can be found in the reference list associated with this chapter. 7.5.1.5. Conclusions on the couplings with the vector potential in 2D Coupling with the circuit equations in 2D has enabled simulation tools (software) to evolve significantly. Hence, these tools can simulate a large number of devices in real configurations (various supplies, short-circuit, etc). The proposed solutions: mesh equations, integrated nodal potential or state equations are generally satisfactory. Coupling with the circuit equations in 2D is also, to a certain extent, a step towards the 3D case: it makes it possible, for example, to take into account the effects of extreme ends of a squirrel cage rotor or the modeling of rotors with tilted slots. It is sufficient, in this last case, to couple several 2D finite element domains using circuit equations [DZI 00]. 7.5.2. Coupling with a vector potential formulation in 3D
As in 2D, the coupling equations with supply circuits will be different according to whether we deal with wire coils or massive conductors. 7.5.2.1. Coupling in the presence of wire areas In the case of wire coils, the main equation is that of the insulating areas and the principle of coupling is very similar to that carried out in 2D. The natural coupling involves the mesh equations (see section 7.5.1.2). Nevertheless, the integrated nodal
304
The Finite Element Method for Electromagnetic Modeling
potentials are of greater value because of their continuity. We thus present the coupling with the latter. After discretization with finite elements, we express the vector potential a in the form a = 6 wi.ai where wi represents adapted interpolation functions (see Chapter 2). Field equations [7.38] and [7.39], in the wire areas, are traditionally put in matrix form:
>S@^A` - >Cb @^I b `
0
with
[7.66] Cbik
³
wi .jobk d:
ȍ bk
In this expression, jobk characterizes the wound conductor k (see section 7.4.4) and we have jbk = jobk.Ibk. A significant difficulty is related to the description of these wire areas, because their geometries can be very complex (see for example Figure 7.7 where the rotor conductors can intermingle). When these wire coils cannot be described by a volume function of simple space, various approaches can be proposed. Thus, Thomas Dreher proposed [DRE 95] to integrate the wire coils via linear integrals, by representing the coil as a set of wire conductors.
Figure 7.7. Example of wound conductors
This approach is rather complex to implement because it requires calculating the intersections of the wire conductors with the finite elements. Another alternative consists of transforming the integral Cbik by introducing auxiliary potentials tok such that jok= curl tok. In this case, the integral Cbik is written:
Coupling with Circuit Equations
³ curl w .t
C b ik
i
d:
ok
ȍ bk
³ w . (n x t i
ok
) d*
305
[7.67]
ī bk
In order to eliminate the surface term, we have to choose a solution tok which checks tokxn = 0 on *bk. This is impossible to obtain if we limit the integration of tok to the conductor, since in this case we could no longer comply with the Ampère theorem. The solution consists of choosing an integration domain :ok which includes the conductor and on which it is then possible to find a solution “to” checking tokxn = 0. As before, this solution has the advantage of being able to describe the wound conductors independently of the meshing, tok being able to be determined by a preliminary inductive calculation as in the formulations using the magnetic scalar potential. Moreover, in the case of edge elements, this technique allows the matrix system to be made “compatible”, facilitating the resolution of the matrix system (without needing a gauge) [REN 96]. We present an implementation of their technique below, within the framework of a time-stepping formulation with a T-diagram and in the presence of nonlinearities. The use of the integrated nodal potentials results in keeping the unknown factors A, Ib and < (with mesh equations we would only have A and Im). These various unknown factors are calculated at the moment t+Gt by writing the various equations at the moment t+TGt. We obtain: x t șįt
ș x t įt (1 - ș) x t
§ dx · ¨ ¸ © dt ¹ t șįt
with 0 ș 1
[7.68]
x t įt x t įt
The Newton-Rapshon method consists of solving at each time step a nonlinear ª wR º system of equations R(Xt+Gt) = 0 by solving iteratively « » .^'X` ^R`t șįt . ¬ wX ¼ t șįt Within the framework of the use of the vector potential a, the finite element method results in solving the following nonlinear equations: Ra
³ curl w . h - w .j i
i
b
d:
[7.69]
:
For the insulating areas (jb = 0), the contribution of these areas to the global equation system arises in the matrix form:
306
The Finite Element Method for Electromagnetic Modeling
§ wR a · ¨ ¸ . ǻA © wA ¹
R a
R ai
i
[7.70]
with
³ curl w . h
t șįt
d:
:
wR a i wA j
ª wh º T curl w i . « » . curl w j dȍ ¬ wb ¼ t șįt :
[7.71]
³
As with field equations, we use the Newton-Raphson method for the circuit equations: in this case, the equations to be solved in each node are (Ik are the currents of the branches): R\ = 6Ik = 0
[7.72]
We calculate <iteratively by solving the system: § wR ȥ ¨ ¨ w< ©
· ¸. '< ¸ ¹
[7.73]
R ȥ
Let us take the example of a nonlinear conductance: we denote by
[7.74]
where Ik is given by relation [7.55]. In order to calculate the Jacobian, we determine: wI k wȌ a
w I k wU k . w U k wȌ a
wI k wȌ b
wI k wU k . wU k wȌ b
gdk įt
[7.75]
gdk įt
where gdk represents the dynamic conductance. The contribution of an element of a nonlinear conductance to the global matrix system is thus written in the form: ª1 tº « įt N e .g d k .N e »^ǻȌ k `t įt ¼ ¬
^N e .I k `t șįt
[7.76]
Coupling with Circuit Equations
307
with ª g d k « g ¬ dk
N e .g d k .N et
g d k º g d k »¼
[7.77]
Hence, we can treat the nonlinear behavior of a diode. The wound conductors link the vector potential a to the integrated electric potentials < through the current-voltage relation which links the currents Ibk of the coil to the potentials
³ curl w . h - w . ¦ j i
i
ob k .I b k
d:
:
Ruk
³j
ob k .
ȍ
R\k
da d dȍ rb k I b k - N et .Ȍ k dt dt
[7.78]
N e .I b k
The contribution of the conductor k to the system of equations is written: ª wR a « « wA « wR « uk « wA « « 0 «¬
wR a wIb k wR u k wI bk wR ȥk wI bk
º 0 » » ǻA ½ wR u k » ° ° » ®ǻI b ¾ wȌ » ° k ° » ¯ ǻȌ ¿ 0 » »¼
R a ½ ° ° ® R u k ¾ ° R ° ¯ ȥk ¿
[7.79]
Finally, having multiplied the second equation by TGt (to make it symmetric), the contribution of the wound conductors to the global system of equations is written: ª J « C t « b «¬ 0
with
Cb 2
ș įt R b ș Nb
0 º ș N bt »» 0 »¼
ǻA ½ ° ° .®ǻI b ¾ ° ǻȌ ° ¿ t įt t șįt ¯
Ra ½ ° ° ® ș įt R u ¾ ° R ° ȥ ¿ ¯
[7.80] t șįt
308
The Finite Element Method for Electromagnetic Modeling
ª wh º ș curl w i . « » . curl w j dȍ wb ¼ t șįt ¬ :
J ij
³
C b ik
ș
³
[7.81]
curl w i . t o k d:
ȍ ok
The various quantities (residues and matrix terms) have to be calculated at the moment t+TGt. As an application example, we present below the modeling of a variable reluctance machine (collaboration with the Moulinex company) supplied with an electronic circuit as shown in the figure below. A position sensor provides the control setting for the supply transistors. The diodes are modeled by nonlinear conductances since their switch-on instant is not known.
(
Figure 7.8. Geometry and supply circuit of variable reluctance motor
The modeling has required a step by step over time resolution while taking into account the motion. The results presented below have been obtained using nodal elements [DRE 94], [DRE 96].
Coupling with Circuit Equations
309
: MEASURE
Figure 7.9. Simulation results and experimental comparison
The simulations carried out allow the experimental currents to be found with a rather good accuracy. 7.5.2.2. Coupling in the presence of massive areas In the example using the vector potential a, it is usual to associate it with the electric scalar potential v and to simultaneously solve the two equations curl h = j and div j = 0. We obtain: curl (Ȟ curl a) ı div ( ı
da ı gradv 0 dt
da ı gradv) 0 dt
[7.82]
With this formulation, it is natural to couple the massive conductors with the electric potentials of the circuits by the introduction of boundary conditions on the equipotential surfaces of the massive areas (binding the electric potential va of the circuit and the nodal values v of the surface *a for example). The use of a preliminary electrokinetic solution vo (with a volt at the conductor terminals) allows, with the use of the modified vector potential and the use of edge elements [DUL 00], us to solve only the first equation (the second then being checked implicitly). We obtain: curl (Ȟ curl a*) ı
da * (v a - v b ) grad v o dt
[7.83]
This equation is linked to the circuits via potentials va and vb. This formulation requires only one vector unknown in the conducting medium (but requires a preliminary resolution to obtain vo).
310
The Finite Element Method for Electromagnetic Modeling
Hence, the implementation of the circuit coupling by using the nodal potentials or the time integrated nodal potentials < is easy. On the otherhand, as in 2D, a coupling with the mesh equations requires the introduction of an additional currentvoltage equation in association with an additional unknown variable: the current in the conductor [MEU 86]. We will not go into detail regarding implementation of these formulations here, instead, we will carry it out in the case of using the magnetic scalar potential. 7.5.2.3. Conclusions on the coupling with the magnetic vector potential in 3D As in 2D, the vector potential allows a natural coupling with the field equations. Indeed, the general information of the potential vector makes it possible on the one hand to readily express the current-voltage relation and on the other hand does not pose any problem in the presence of non-simply connected regions. However, the high cost of the vector potential formulations led the community to seek solutions based on the magnetic scalar potential. 7.5.3. Coupling with a scalar potential formulation in 3D
The particular value of the magnetic scalar potential is that with this method the insulating areas can be treated using a scalar unknown variable: this is far from being negligible in 3D, taking into account the size of the problems to be tackled. Moreover, these problems can be nonlinear and evolutionary, whereas methods based on scalar potentials (reduced or totals) have largely proved to be reliable in magnetostatics, in particular for the treatment of industrial problems. As in magnetostatics, the fundamental idea lies in the reduction of the magnetic field with respect to the current sources. However, unlike magnetostatics, the current sources can be the unknowns of the problem. The establishment of the current-voltage relation for the wire coils and the massive conductors will enable us to generalize this type of formulation in the presence of the electric circuit [BUI 94], [MEU 98]. We will show that it is possible to treat any type of problem (nonlinearities, nonsimply connected conducting regions, wire or massive coils). As a matter of fact, the formulations based on scalar potentials prove to be very powerful, particularly when compared to the formulations using magnetic vector potential. As we will see below, these formulations require pre-calculations in the form of vector potentials tok associated with each inductive coil (wire or massive).
Coupling with Circuit Equations
311
7.5.3.1. Coupling in the presence of wound conductors Now, let us consider a simply connected domain, composed of various parts of the wire coils. We suppose that these last are included in sub-domains :o where the magnetic field is expressed in the form: h = to - gradI
>@
which is a field that checks curl to = jb with jb the current density of the inductive coils. In practice, to is expressed as a linear combination of the inductors belonging to the same sub-domain. We write: to = 6 tokik
[7.85]
where tok is the field created by a current ibk of 1 Amp. The use of tok enables us to meet curl h = jb. The complete solution is obtained by solving the field equation div b = 0 associated with the circuit equations. The field equation uses the scalar unknown variables ) and the coil currents Ib. The coupling that uses the mesh currents makes it possible to keep only ) and Im as unknown variables. However, because of the interest presented by the integrated potentials in the time < (see section 7.2.5), we present below a coupling with the latter. It requires the use of three types of unknown variables ), Ib and <. Using the Newton-Raphson method, and in a similar manner so that for the case of the potential vector magnetic a (section 7.5.2.1), the contribution of a wire coil to the residues of the equations system is written: R ij i ( ), I b )
³ grad w . b dȍ i
ȍo
R u k () , I b , < )
³t
ok .
:o
R ȥ k (< )
db d d: rb k I b k N et .Ȍ k dt dt
[7.86]
Ne . Ibk
The derivation of these residues contributes to the global system of a set of coils. After making the system symmetric and within the framework of a time-stepping procedure, we obtain (implicit method, T=1):
312
The Finite Element Method for Electromagnetic Modeling
ª K « C t « o «¬ 0
C o G Nb
0 º ǻĭ ½ ° ° N bt »» . ®ǻI b ¾ 0 »¼ t įt ¯°ǻȌ ¿°t įt
R ij ½ ° ° ® įt R u ¾ ° R ° ȥ ¿ ¯
[7.87] t įt
with K ij
³ grad w
ȍ
C o il
i
ª wb º .« ». grad w j dȍ ¬ wh ¼ ª wb º
³ grad w . «¬ wh »¼ . t i
ol
[7.88]
dȍ
ȍo
G kl
įt rb k
³t
ok
ȍo
ª wb º . « » . t o l dȍ ¬ wh ¼
We have applied this formulation to the modeling of a current transformer. The electric circuit is composed of two coils (B1 and B2), an inductive coil with an imposed current and a coil in short-circuit (Figure 7.10). The air area which includes the supply coils has been chosen as domain :o (h = to-gradI . The ferromagnetic parts are treated in total scalar potential (h = -gradI) in order to guarantee a good calculation accuracy. It is thus advisable to choose functions tok which check tok x n = 0 at the air-iron interfaces (see section 7.4). It should be noted that in the presence of a closed magnetic circuit, the air does not constitute an including box at the coils and that it is then necessary to introduce a cut into the magnetic circuit. B2
R = 270 :
B1(1 :) B2
I=I0 sin(Zt)
B1
Figure 7.10. Geometry of a current transformer
Coupling with Circuit Equations
313
In the case of this transformer, the introduction of a surface air-gap, able to calculate and absorb the potential jump [LEF 01], has finally allowed the simulation of this device. The results (obtained with the FLUX3D software) are in good agreement with the experimentation (see Figure 7.11).
Figure 7.11. Induced current in the secondary winding
7.5.3.2. Coupling in the presence of massive conductors We present in this section an approach allowing the coupling of circuit equations in the case of massive conductors with a formulation using the magnetic scalar potential. For the conducting areas we choose to use a formulation of the type t-toIwhere t represents the electric vector potential. We will show that it is possible to propose a rather general formulation making it possible in particular to take into account: – the nonlinearities j(e) and b(h); – conductors with multiple inputs. In order to be free from the non-simply connected problems, we will seek a reduced solution compared to the total currents which circulate through the massive conductors. Let us consider a massive conductor with several inputs. We defined current paths so that all the inputs can be joined directly or indirectly. In the case of a system with N inputs, n-1 paths are necessary (at least). We can define an incidence
314
The Finite Element Method for Electromagnetic Modeling
matrix Nc which defines these paths. If we suppose that each path k carries a current ik, we have Nc.I = Ia where Ia represents the vector of n access current ia.
ia3
ia1 i1 i2 i3 ia2
ia4
Figure 7.12. Example of a massive conductor with several inputs
As an example, Figure 7.12 presents a massive conductor with 4 accesses. With the 3 selected currents ik the incidence matrix node-branches Nc is written:
Nc
ª1 « 1 « «0 « ¬0
0 1 1 0
0º 1 »» 0» » 1¼
[7.89]
Now, let us consider an elementary current of 1 Amp and the corresponding current density jok connecting two inputs (jok x n = 0 apart from these two accesses and tok.n = 0 on the accesses). Moreover, if this current density checks div jok = 0, we can show, as when establishing the current-voltage relation (see section 7.4.3), that uk
³j
:c
o k . e d:
d dt
³t
o k .b d:
[7.90]
:o
where tok is such that jok = curl tok in :o and tok x n = 0 on *o. We can for example choose, for the densities jok, the electrokinetic solutions or even those obtained via judiciously placed inductors. Voltages uk are connected to potentials < by the dȌ . relation U N c . dt
Coupling with Circuit Equations
315
The use of a magnetic scalar potential I in the air leads us, in order to comply with the Ampere theorem, to seek a solution reduced with respect to the currents ik. On the other hand, the presence of massive conducting areas requires the introduction of the electric vector potential t. Thus, we write, within the framework of a t-to-I formulation: h = t + 6tokik – grad Iin :c; h = 6tokik – grad Ielsewhere
> @
The current density j can be written: j = curl t + 6jokik
[7.92]
where curl t represents the difference in distribution of the currents induced with respect to initial selected solutions jo. We can choose t such that t x n = 0 on the surface of the massive conductor, which makes it possible to ensure j.n = 0. The accesses are generally located on symmetry planes where the magnetic field is dI 0 on the accesses to be tangential. This allows the conditions t.n = 0 and dn readily imposed. In a manner identical to the case of wire conductors, we can couple the field and circuit equations. With the mesh equations we obtain the unknown variables T, ) and Im. The choice of the integrated nodal potentials requires the use of four types of unknown variables T, ), Ib and < With the Newton-Raphson method, the contribution to the residues of the domain :o (including the massive conductor :c) is written: R ti
db
³ curl w .e w . dt i
i
d: (curl e - db/dt)
:c
R Ii
- grad w i . b dȍ (div b 0)
³
[7.93]
ȍo
Ruk
³
:o
R\k
db d: tok . dt
³
:c
dȌ jo k .e d: - N ct k . dt
- N c k .i k
where Nck represents the kth column of the incidence matrix Nc. The contribution to the system of equations of the domain :o is, after discretization with an implicit method (T=1) (the symmetry of the matrix system is obtained by multiplying the first and the third equation by Gt), finally written:
316
The Finite Element Method for Electromagnetic Modeling
ª L « E t « « Ft « ¬ 0
E K
F Co
C ot 0
G Nc
0 º ǻT ½ 0 »» °°ǻĭ °° .® ¾ N ct » °ǻIc° » 0 ¼ °¯ǻȌ °¿
įt R t ½ ° RI ° ° ° ® ¾ įt R u° ° ° Rȥ ° ¯ ¿ t įt
t įt
[7.94]
with: se ¯ Lij t ¨ curl wi . ¡ ° . curl wj ¡¢ sj °± ȍc E ij
sb ¯
¨ w . ¡¡¢ sh °°± . grad w i
j
sb ¯
¨ w . ¢¡¡ sh ±°° i
.wj dȍ
:o
d:
:o
se ¯ Fik t ¨ curl wi . ¡ °. jok ¡¢ sj °± ȍc
sb ¯
¨ w . ¡¡¢ sh °°± . t
ok
i
dȍ
[7.95]
:o
sb ¯ K ij ¨ grad w i . ¡ ° . grad w j d: ¡¢ sh °± ȍo Coik
sb ¯
¨ grad w . ¡¡¢ sh °°± i
. tok dȍ
:o
se ¯ G kl t .¨ jok . ¡ ° .jol d: ¡¢ sj °± :c
¨t
ok
:o
sb ¯ . ¡ ° .tol d: ¡¢ sh °±
ia3
ia1 i1
i4 i2 i3
ia2
ia4
Figure 7.13. Massive conductor with hole
In the case of holes in a massive conductor, the previous formulation can be directly used [MEU 03]. This non-simply connected problem, linked to the currents being able to negotiate the turn of the holes, is solved by introducing as many additional unknown variables ik as there are holes. Thus, in the example of Figure
Coupling with Circuit Equations
317
7.13, we have to define a path which makes the turn of the hole. As previously, the corresponding solutions jok can be obtained by inductors or even by resolutions in electrokinetics. In this last case, fictitious cuts (1 per path) may be required to carry out these resolutions, imposing an electric potential jump and hence obtaining the solutions jok. The global resolution of system [7.94] is carried out without taking possible cuts into account. Finally, it should be noted that the current-voltage relations are written in the form [7.96] and that they are not coupled directly with the unknown variables <:
³j
:c
ok . e
d:
d dt
³t
ok .b
d:
0
[7.96]
:o
7.5.3.3. Conclusions on coupling with a scalar potential formulation in 3D We have presented a very general formulation allowing coupling with circuit equations. Its value particularly lies in the fact that it uses the scalar potential in the non-conducting areas. It allows the use of a total potential in the ferromagnetic areas. The introduction of the electric vector potential makes it possible to treat the massive conductors even in the case of non-simply connected media. Moreover, this formulation enables us to account for thin or line areas using the magnetic scalar potential (reduced or total) [ABA 00], [ABA 01]. The principal drawback of this formulation is that it requires pre-calculations of potentials tok in the conducting areas (wound or massive). For this calculation various techniques can be used: analytical techniques [KLA 92], based on a jump of the potential reduced-total [LEF 01], or source field techniques [LEM 98]. 7.6. General conclusion
This chapter clarifies some examples for coupling finite element formulations with the external electric circuit in two and three dimensions. The choice of setting for the circuit equations is important in view of effectively treating any type of component. In this way, the integrated potential method over time and that of the generalized state equations offer powerful solutions. With regards to finite element formulations, the vector potential is adapted perfectly to coupling in 2D, whereas in 3D the use of the magnetic scalar potential offers an interesting and powerful alternative compared to the magnetic vector potential.
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The Finite Element Method for Electromagnetic Modeling
7.7. References [ABA 00] ABAKAR A., COULOMB J.L., MEUNIER G., ZGAINSKI F.X., GUERIN C., “3D modeling of thin wire and thin plate junction”, Conference Proceedings CEFC2000, Milwaukee, WI, USA, June 4-7, 2000. [ABA 01] ABAKAR A., MEUNIER G., COULOMB J.L., ZGAINSKI F.X., “3D modeling of thin wires interacting with plates. Extracting the singularity due to the loop wire self inductance”, The European Physical Journal – Applied Physics, EDP Sciences, Eur. Phys. J. AP. 14, 63-67, 2001. [ARK 87] ARKKIO A., “Analysis of induction motors based on the numerical solution of the magnetic field and circuit equations”, Helsinki Acta Polytecnica Scandinavica, Electrical Engineering Series, no. 59, 97 p, 1987. [BIR 04] BIRO O., PREIS K., BUCHGRABER G., TICAR I., “Voltage-driven coils in finiteelement formulations using a current vector and a magnetic scalar potential”, IEEE Trans-Mag, Vol. 40, no. 2, part 2, pp 1286-1289, 2004. [BUI 94] BUISSOU S., PIRIOU F., “Comparison between two formulations in terms of potential for the coupling of magnetic and electric circuit equations”, Science, Measurement and Technology, IEE Proceedings, vol. 141, issue 6, pp. 486-490, November 1994. [CHU 75] CHUA L.O., LIN P.M., Computer Aided Analysis of Electronic Circuits, PrenticeHall, 1975. [CHA 96] CHARPENTIER J.F., Modélisation des ensembles convertisseurs statiques machines électriques par couplage des équations du champ électromagnétique et du circuit électrique, PhD thesis, Institut National Polytechnique de Toulouse, October 1996. [CHA 97] CHARPENTIER J.F., LEFEVRE Y., PIQUET H., “Une méthode générale pour modéliser les convertisseurs statiques associés à des dispositifs électromagnétiques”, Journal de Physique III France 7, pp. 2225-2237, November 1997. [CHA 98] CHARPENTIER J.F., LEFEVRE Y., PIQUET H., “An original and natural method of coupling electromagnetic field with circuit equations”, IEEE Trans. on Mag., vol. MAG34, no. 5, pp. 2489-2492, 1998. [DAV 83] DAVAT B., LAJOIE-MAZENC M., HECTOR J., “Magnetic structure and feeding circuit modelling”, IEEE Trans. on Mag., vol. MAG-19, pp. 2471-2473, 1983. [DRE 94] DREHER T., Couplage de la méthode des éléments finis tridimensionnels avec une méthode d'analyse du circuit électrique: application à la modélisation des machines électriques tournantes, PhD thesis, INP Grenoble, 1994. [DRE 95] DREHER T., MEUNIER G., “3D line current model of coils and external circuits”, IEEE Trans. on Mag., vol. MAG-31, no. 3, pp. 1853-1856, 1995. [DRE 96] DREHER T., PERRIN-BIT R., MEUNIER G., COULOMB J.L., “A three dimensional finite element modelling of rotating machine involving movement and external circuit”, IEEE Trans. on Mag., vol. MAG-32, no. 3, pp. 1070-1073, 1996.
Coupling with Circuit Equations
319
[DUL 00] DULAR P, HENROTTE F., LEGROS W., “A general and natural method to define circuit relations associated with magnetic vector potential formulation”, IEEE Trans. on Mag., vol. MAG-36, no. 3, pp. 1630-1633, 2000. [DZI 00] DZIWNIEL P., BOUALEM B., PIRIOU F., DUCREUX J.P., THOMAS P., “Comparison between two approaches to model induction machines with skewed slots”, IEEE Trans. on Mag., vol. MAG-36, pp. 1453-1457, 2000. [KLA 87] KLADAS A., PIRIOU F., RAZEK A., “Résolution simultanée des équations magnétiques et électriques dans les systèmes électromagnétiques”, SEE Colloquium, Gifsur-Yvette, pp. 63-69, 13 March 1987. [KLA 92] KLADAS A., TEGOPOULOS J.A., “A new scalar potential formulation for 3D magnetostatics necessitating no source field calculation”, IEEE Trans. Mag., vol. MAG28, no. 2, pp. 1103-1106, 1992. [KON 81] KONRAD A., “The numerical solution of steady-state skin effect problems – an integrodifferential approach”, IEEE Trans. on Mag., vol. MAG-17, 1981. [KUO 97] KUO-PENG P., SADOWSKI N., BASTOS J.P.A., CARLSON R., BATISTELA N.J., LAJOIEMAZENC M., “A General Method for Coupling Static Converters with Electromagnetic Structures”, IEEE Trans. on Mag., vol. MAG-33, no. 2, pp. 2004-2009, 1997. [LEF 01] LE FLOCH Y., GUERIN CH., BOUDAUD D., MEUNIER G., BRUNOTTE X., A current transformer modeling, Team Workshop and Application Forum, Evian, 6 July 2001, Special Issue of COMPEL 2002. [LEM 98] LE MENACH Y., CLENET S., PIRIOU F., “Determination and utilization of the source field in 3D magnetostatic problems”, IEEE Trans. on Mag., vol. MAG-34, issue 5, pp. 2509-2512, 1998. [LOM 92] LOMBARD P., MEUNIER G., “A general method for electric and magnetic coupled problem in 2D and magnetodynamic domain”, IEEE Trans. on Mag., vol. MAG-28, pp. 1291-1294, 1992. [LOM 93] LOMBARD P., MEUNIER G., “A general purpose method for electric and magnetic combines problems for 2D, axisymmetric and transient systems”, IEEE Trans. on Mag., vol. MAG-29, pp. 1747-1740, 1993. [MAT 92] MATAGNE E., “Modélisation magnétique macroscopique des faisceaux de conducteurs”, Conference Proceedings NUMELEC 92, Grenoble, 17-19 March, 1992. [MEU 86] MEUNIER G., COULOMB J.L., “3D eddy current and external circuit”, Conference Proceedings “Eddy Current Seminar”, Oxford, March 1986. [MEU 98] MEUNIER G., TUAN L.H., MARECHAL Y., “Computation of coupled of 3D eddy current and electrical circuit using T-To-: formulation”, IEEE Trans. on Mag., vol. MAG-34, no. 2, pp. 3074-3077, 1998. [MEU 03] MEUNIER G., LE FLOCH Y., GUERIN C., “A non linear circuit coupled t-t0-f formulation for solid conductors”, IEEE-Trans-Mag, Vol. 39, no. 3., pp. 1729-1732, May 2003.
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The Finite Element Method for Electromagnetic Modeling
[MOR 98] MOREAU O., POPIEL L., PAGES J.L., “Proximity losses computation with a 2D complex permeability modelling”, IEEE Trans. on Mag., vol. MAG-34, no. 5, pp. 36163619, 1998. [NAK 82] NAKATA T., TAKAHASHI N., “Direct finite element analysis of flux and current distribution under specified conditions”, IEEE Trans. on Mag., vol. MAG-89, pp. 325330, 1982. [PIR 93] PIRIOU F., RAZEK A., “Finite element analysis in eletromagnetic systems accounting for electric circuit”, IEEE Trans. on Mag., vol. 29, pp. 1669-1675, March 1993. [REN 96] REN Z., “Auto-gauging of vector potential by iterative solver – numerical evidence”, Actes de la Conférence “Electric and Magnetic Fields”, EMF-96, pp. 119124, Liège, 1996. [SAD 93a] SADOWSKI N., Modélisation des machines électriques à partir de la résolution des équations du champ en tenant compte du mouvement et du circuit d’alimentation (logiciel EFCAD), PhD thesis, Institut National Polytechnique de Toulouse, January 1993. [SAD 93b] SADOWSKI N., CARLY B., LEFEVRE Y., LAJOIE-MAZENC M., “Finite elements simulation fed by current inverters”, IEEE Trans. on Mag., vol. MAG-29, no. 2, pp. 1683-1688, 1993. [SAL 86] SALON S.J., ISTFAN B., SABONNADIERE J.C., MEUNIER G., “Advances in numerical modelling techniques applicable to induction machinery”, Proc. Annexes of the Int. Conf. on Evolution and Modern Aspects of Induction Machines, Turin, pp. 1-4, 8-11 July, 1986. [SLA 01] SLAMA A., MAZAURIC V., MEUNIER G., MARECHAL Y., “On solving connectivity problems within modelling of massive conductors, COMPEL, vol. 20, January 2001. [SHE 85] SHEN D., MEUNIER G., COULOMB J.L., SABONNADIERE J.C., “Solution of magnetic fields and electrical circuits combined problems”, IEEE Trans. on Mag., vol. MAG-21, no. 6, pp. 2488-2291, 1985. [STA 02] STAVREV S., GRILLI F., DUTOIT B., NIBBIO N., VINOT E., KLUTSCH I., MEUNIER G., TIXADOR P., YANG Y., MARTINEZ E., “Comparison of numerical methods for modelling of superconductors”, IEEE Trans. on Mag., vol. MAG-38, March 2002. [TSU 93] TSUKERMAN I.A., KONRAD A., MEUNIER G., SABONNADIERE, J.C., “Coupled fieldcircuit problems: trends and accomplishments”, IEEE Trans. on Mag., vol. MAG-29, no. 2, pp. 1701-1704, March 1993.
Chapter 8
Modeling of Motion: Accounting for Movement in the Modeling of Magnetic Phenomena
8.1. Introduction Electromechanical conversion is an application in a large number of magnetic systems. Rotating machines, solenoids, contactor actuators, Thomson effects triggers, electromagnetic launchers, electromagnetic brakes; these are all devices capable of converting an electrical energy into a mechanical energy. Their operation is essentially transient. In general, they involve conductive moving parts immersed in a magnetic field which in turn often varies with time. These parts are then subjected to currents that are induced by the movement and variation of the magnetic field. These currents, known as eddy currents, can significantly degrade the operation of the device by dissipating a portion of the supplied energy, via the Joule effect. Sometimes, as in a device with Thomson effect, it is the very presence of these currents which ensures its operation. It is therefore essential for the design of such systems to accurately assess the induced effects in order to faithfully reproduce the electromechanical conversion. In order to perform such a modeling, simulation tools must be equipped with: – adapted formulations, Chapter written by Vincent LECONTE.
322
The Finite Element Method for Electromagnetic Modeling
– methods to deal with the distortions of geometry. As part of the finite element method, the latter point is a major challenge, since the calculations are carried out on a mesh, the geometry of which must therefore be adapted to the changes here. In this context, this chapter attempts to make an overview of the various formulations and techniques allowing the induced effects in devices with moving parts to be taken into account. 8.2. Formulation of an electromagnetic problem with motion 8.2.1. Definition of motion First, and before addressing the influence of movement on electromagnetic phenomena, here are some kinematic elements allowing the movement itself to be characterized. We are interested in the study of motion of rigid bodies (non-deformable). After some definitions, the general 3D motion is envisaged. We will then restrict ourselves to a movement which we know the trajectory of, as is the case for the traditional electromechanical systems 8.2.1.1. Some definitions In the following sections, and with no prejudice to the generality, only one moving part will be considered in the studied system. It is attached to the mobile '
of origin O , the fixed reference frame being designated by coordinate system . The transformation allowing the passage from the fixed reference mark to the mobile mark is defined by: x = x ' Px' ,
[8.1]
O
where P corresponds to a matrix of rotation and x
O'
is the position of the origin of
. By describing [8.1] with respect to time, we obtain:
v = v ' ȍ u x' . O
[8.2]
The movement of the mobile part is hence decomposed into a translation of
O'
and a rotation around an axis passing through O ' . ȍ indicates the rotation vector
Modeling of Motion
323
which has the axis of rotation as a direction and the angular velocity of rotation as a magnitude. Let U and U' be the respective expressions of the same vector in They are associated by the following relation:
and
PU' = U.
.
[8.3]
For the time derivative, there is the formula of the reference frame change: § wU' · dU = P¨ ȍ' u U' ¸. ¨ wt ¸ dt © ¹
[8.4]
8.2.1.2. General motion in 3D The most general movement in 3D can be described with six degrees of freedom, corresponding to the three coordinates of the centre of inertia of the considered solid and three angles that define the orientation of the device. coincides with the It is assumed that the origin of the reference frame device’s center of inertia. In addition, let us choose the axes of so that they coincide with the main axes of the solid. In order to describe their orientation with respect to the axes of , it is convenient to use the Euler angles ) , 4 and < [LAN 66]. The transformation P introduced by [8.1] can be defined by the following matrix:
sin)cos< cos)cos4sin< ª cos)cos< sin)cos4sin< « « « cos)sin< sin)cos4sin< sin)sin< cos)cos4cos< « « «¬ sin)sin4 cos)sin4
sin4sin< º » » sin4cos< » » » cos4 »¼
By applying the fundamental relation of the dynamic to a solid in motion, and by writing it in , we obtain the system:
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The Finite Element Method for Electromagnetic Modeling
§ dvxc · °m¨ dt : ycv zc : zc v yc ¸ = Fxc ¹ ° © ° ° § dv yc · : zc vxc : xc v zc ¸¸ = Fyc ®m¨¨ ¹ ° © dt ° ° § dvz c · : xcv yc : ycv xc ¸ = Fzc °m¨ ¹ ¯ © dt
[8.5]
Three other equations are obtained by writing the theorem of kinematic '
momentum applied in O :
d: xc ° I xc dt ( I z c I yc ): yc : z c = *xc ° ° ° d: yc ( I xc I zc ): zc : xc = *yc ® I yc dt ° ° ° d: z c °¯ I zc dt ( I yc I xc ): xc: yc = *zc
[8.6]
where m is the mass of the solid and I x ' , I y ' and I z ' are its main inertia momentum. When the components of the forces Fxc ,
Fyc and Fzc applied to a solid are known, as well as their momentum with respect to Oc , the six previous equations allow us to solve the unknowns v xc , v yc , v z c , : xc , : yc and : zc which correspond respectively to the components of the velocity and the angular velocity of the solid expressed in . It is possible thereafter to return easily to the degrees of freedom describing the position of the mobile. In practice, such a calculation is needed in applications such as magnetic sustentation. 8.2.1.3. Movement with one degree of freedom In most electromechanical systems, the movement of parts is guided and follows determined trajectories. It is then reduced to a much simpler problem where only one degree of freedom is required to describe the movement. The guiding of mechanical parts also induces contacts and frictions which appear in the equation of motion.
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325
– Case of translation All the points of the solid have the same velocity v which is deduced from the solution of:
m
dv O1v O2 v 2 = Fext dt
[8.7]
Fext is the projection on the translation axis of the total external force applied to the mobile, including the contribution of magnetic efforts. The coefficients O1 where
and O2 correspond respectively to dry and viscous frictions. It is often difficult to evaluate them in practice. – Case of rotation Let (' ) be the rotation axis. The angular velocity at which the mobile rotates is the solution of:
I where
d: O1: O2 : 2 = *ext dt
[8.8]
*ext is the resultant on (') of external torques applied on the mobile.
8.2.2. Maxwell equations and motion 8.2.2.1. Definition of an electromagnetic problem with motion The type of problem we are trying to solve includes fixed parts in the reference . They include source currents, magnets, frame of the laboratory identified by magnetic and conductive parts (Figure 8.1). In order to simplify the presentation, but without prejudicing the wider scheme, only the moving part regarded as rigid and conductive and to which is attached the reference frame is considered.
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The Finite Element Method for Electromagnetic Modeling
* ’
js
PV
O’
Br
PV
Figure 8.1. Definition of the problem type to be solved
The mathematical equations that describe a low frequency electromagnetic system are Maxwell equations in which the displacement currents are neglected. In the fixed reference frame , they are expressed in the following manner: curl H
div B curl E
J Js
[8.10]
0
[8.9]
wB wt
[8.11]
Here, the source currents described by J s and induced currents J are distinguished. These equations must be complemented by the laws of material behavior. The field and the magnetic induction are related by the formula: B = P ( H ) H.
[8.12]
The dependency in H of the magnetic permeability P allows nonlinear materials to be considered. On the other hand, permanent magnets are characterized by their residual induction, denoted by B r . For this type of material [8.12] is replaced by: B = P H Br .
[8.13]
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327
The Drude model, also called “billiard balls” [ROB 79, ASC 76], allows us to interpret Ohm’s Law, and concludes with the proportionality between the drift speed of conduction electrons and the electric field seen from the conductor. This means that the relation of proportionality binding the electric current to the electric field should be written in a reference frame in which the conductor is at rest. Thus, we have:
J V E for a motionless conductor
[8.14]
J'
[8.15]
V E' for a conductor in motion 1
where V is the electric conductivity of the material in S m . It will be seen later how, for a conduction in motion, Ohm’s law is expressed in a fixed reference frame. For a complete description of the problem, the previous equations must be associated with the boundary conditions and with passage relations of fields at interfaces. At the edges of the domain, the following boundary condition applies: B n = 0 on *.
[8.16]
At the interface between two media with different physical properties and with the absence of surface currents, we have: (B1 B 2 ) n
0,
( H1 H 2 ) u n ( J1 J 2 ) n (E1 E 2 ) u n
0, 0, 0.
[8.17] [8.18] [8.19] [8.20]
The formulation of the problem must take into account the equations and the behavior laws of materials, as well as the boundary conditions, including the conditions at the fields’ interfaces. 8.2.2.2. Form of Maxwell equations in a mobile reference frame Our goal now is to discover what form the Maxwell equations will take in the and what are the transformations to be applied to E, B, H and J reference frame to . It is known that Maxwell equations are preserved in order to go from when applying the Lorentz transformations [ROS 68]. For a uniform movement of
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The Finite Element Method for Electromagnetic Modeling
translation with low velocities (& v & c) , the Lorentz transformation for fields takes the simplified form: E ' = E + v u B , J' = J
[8.21]
B' = B, H' = H
[8.22]
where v = v ( x, t ) . In our case can be in rotation with respect to . It will be seen later that the acceleration produced by this rotation can be neglected. Nevertheless, the transformation must take into account the position of the axes in the change of coordinates. It is written: PE' = E v u B, PJ ' = J PB' = B, PH ' = H
[8.23]
With these transformations, equations [8.9], [8.10] and [8.11] take the same and . However, by applying these transformations to [8.15], Ohm’s form in law for a conductor in motion changes shape when expressed in : J = V E v u B
[8.24]
Now, let us evaluate the effect produced by an acceleration of the conductor. When the metal is accelerated, an additional inertia acts on the electrons of dv where m is the conduction [LAN 69]. The force it exerts on an electron is m dt mass of the electron. It is equivalent to the action on a charge e of an electric field m dv . The electric field responsible for the conduction thus which is equal to e dt m dv . becomes E e dt Equation [8.11] for the accelerated conductor can thus be written: curl E
wB wt
m e
§ dv · ¸ © dt ¹
curl ¨
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329
By using [8.2] and by further development we obtain [LAN 69]: curl E
wB wt
2m d : e
dt
We note here that only a variable rotation over time affects the Maxwell-Faraday equation. The previous equation can also be put in the form: curl E
2m · § :¸ ¨B wt © e ¹ w
In order to make B and
:=
2m ȍ of the same order or magnitide, we should have: e
1 § eB · ¨ ¸ 2© m ¹
To have an impact under an induction of 1T, the conductor must rotate at the speed : = 8.8 1010 rd/s , which is not possible for a piece of electromechanical transducer! Therefore, for the study of induced currents in traditional electrotechnical structures, the choice of the reference frame does not influence the form of Maxwell’s equations. The expression of Ohm’s law in a conductor is on the other hand simpler in a reference frame attached to this one. Thus, in general, it is easier to adopt a Lagrangian description, which consists of treating each part of the problem in a reference frame in which it is at rest. The magnetic phenomena are then observed from the material point eventually in movement. 8.2.3. Formulations in potentials In order to address the problem of eddy currents, the fields H and E can be used. By supplementing these fields with scalars potentials, the formulations H- I [BOS 88, KET 98] or E-\ [BOS 90] are obtained. Without directly using the fields, potentials can be introduced to solve the Maxwell equations. Two major formulation families are distinguished in the literature: – those based on the magnetic vector potential A and the potential scalar electric V;
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The Finite Element Method for Electromagnetic Modeling
– those associating the electric vector potential T with the magnetic scalar potential I [BIR 93]. Let us see for both cases how the terms associated with the conductor displacement speed appear. 8.2.3.1. Formulation AV-A 8.2.3.1.1. Equations to solve To comply with [8.10] and [8.11], the magnetic vector potential A and the electric potential V are introduced so that: B
curl A,
E=
wA wt
grad V .
By using [8.9], for the fixed non-conductive parts, only the unknown A is required: curl (
1
P
curl A )
J s curl (
1
P
B r ).
[8.25]
In the conductive part, that may be in motion, Ohm’s law is added [8.24]: curl (
1
P
§ wA · v u curl A grad V ¸ 0. © wt ¹
curl A ) V ¨
[8.26]
8.2.3.1.2. Uniqueness of the solution In order to ensure the uniqueness of the potential A, a gauge is either added or left out, depending on the type of elements used in the finite element formulation of the problem. – Case of nodal elements Traditionally the Coulomb gauge is used, which consists of imposing div A = 0. To do so, the most common approach is to introduce this gauge in equation [8.26] §1 · using the penalty term: grad ¨ div A ¸ . After the addition of this term, the zero P © ¹ divergence of the current, which could be obtained directly from equation [8.26], is
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331
no longer assured. It must then be imposed explicitly. Hence, in the conductors, equation [8.26] is replaced by the system formed by: 1 curl ( curl A) P
§1 · grad ¨ div A ¸ P © ¹
[8.27]
§ wA · V ¨ v u curl A grad V ¸ © wt ¹
0
§ § wA ·· div ¨ V ¨ v u curl A grad V ¸ ¸ ¹¹ © © wt
0.
and [8.28]
In order to impose a gauge throughout the domain, there is an additional need to impose some boundary conditions and interface conditions not explained here [BIR 89]. – Case of edge elements In this case, an explicit gauge is not necessary, provided we check the compatibility matrix system ensuring zero divergence of the density of sources currents Js [REN 96]. Another method consists of imposing A u = 0 where u is a field of vectors whose lines of field are not closed. The construction of u is based on a tree of mesh edges allowing all nodes to be linked without forming cycles [ALB 90b, ALB 90a]. 8.2.3.1.3. Boundary and interface It remains for all of the boundary and interface conditions [8.16] to [8.20] to be taken into account. Let us see how movement is involved in the writing of these conditions. Figure 8.2 shows the type of interface considered between two media having different physical characteristics. 1 2
P V1
n
P V2
Figure 8.2. Passage between two physical domains with different characteristics
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The Finite Element Method for Electromagnetic Modeling
On the edges of the domain, [8.16] is assured by imposing A u n = 0 on * . Thus, indeed: B n = div( A u n ) = 0 on *.
In order to guarantee the continuity of the perpendicular component of the induction [8.17], it is enough to impose A u n continuously at the interface, so that: (B1 B 2 ) n = div
A1 A 2 u n = 0,
The preservation of the tangential component of electric field [8.20] is provided by the continuity of V and that of A u n. To show this, we develop [8.20]: (E1 E 2 ) u n = =
w A1 A 2
wt w A1 u n A 2 u n
wt grad (V1 V2 ) u n.
u n grad (V1 V2 ) u n
A1 A 2 u
wn wt
Let us assume that only 1 medium is in movement, the change in direction of the wn boundary is written: = ȍ u n and we obtain: wt (E1 E 2 ) u n =
w A1 u n A 2 u n
A1 u n A 2 u n u : wt grad (V1 V2 ) u n.
Conditions [8.18] and [8.19] can be imposed only by the weak formulation of the problem. They are written respectively
Q 1 curl A1 Q 2 curl A 2 u n = 0, and
§ wA1 · v u curl A1 grad V1 ¸ n © wt ¹ § wA 2 · V 2 ¨ v u curl A 2 grad V2 ¸ n = 0. w t © ¹
V1 ¨
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333
8.2.3.2. Formulation T - I 8.2.3.2.1. Equations to solve In the quasi-static approximation, the zero divergence of induced currents in the conductors allows the vector potential T to be defined such that: [8.29]
J = curl T.
Ampère’s law [8.9] on the other hand implies the existence of potential such that:
I
[8.30]
H = T grad I .
By using [8.11], the equation to be solved in the conductor, that may be in motion is obtained: w §1 · curl T v u P (T grad I ) ¸ ȝ T grad I = 0. wt ©V ¹
curl ¨
[8.31]
For air, the flux conservation is solved in the form: div( P0 grad I ) = 0 . 8.2.3.2.2. Uniqueness of the solution The methods allowing the problem of the uniqueness of T to be dealt with are similar to those used for A. – Case of nodal elements The uniqueness of the potential T can be obtained by adding a penalty term in [8.31]. With the addition of this term, the zero divergence of induction is no longer implicit and should then be expressed explicitly. For the conductors, [8.31] has to be replaced by:
§1 · §1 · curl T v u P ( T grad I ) ¸ grad ¨ div T ¸ ©V ¹ ©P ¹
curl ¨
P
w wt
T grad I
= 0
[8.32]
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The Finite Element Method for Electromagnetic Modeling
and div P (T grad I ) = 0.
[8.33]
– Case of edge elements As for A, T u = 0 can be imposed on the edges of a tree built on the mesh. 8.2.3.2.3. Boundary and interface conditions What was strongly imposed in the formulation AV - A is here only carried out in a weak manner and vice versa. The continuity of the component perpendicular to the current is assured by imposing the continuity of T u n. Indeed, in this case: ( J1 J 2 ) n = div
T1 T2 u n = 0.
For the conservation of the tangential component of the magnetic field, the potential I has to be added to the continuity of T u n. It follows: ( H1 H 2 ) u n = (T1 T2 ) u n grad (I1 I2 ) u n = 0.
The boundary condition of the domain [8.16] can only be carried out weakly: P 0 grad (I ) n = 0 on *.
The integral formulation will also have to weakly implement [8.17] and [8.20], which are written:
§1 · ¨ curl T1 v u P1 (T1 grad I1 ) ¸ u n © V1 ¹ § 1 · ¨ curl T2 v u P 2 (T2 grad I2 ) ¸ u n = 0 ©V2 ¹ and
P1 (T1 grad I1 ) n P 2 (T2 grad I2 ) n = 0.
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335
8.2.3.2.4. Definition of source currents As presented so far, the formulation T- I does not allow a non-zero source current to be imposed. Indeed, at the conductor/air interface, the condition T u n = 0 is imposed, which involves: I = ³ J dS = ³ curl T dS = v³ T dl = 0
In order to describe the source currents, a vector potential T0 can be introduced so
that
³ curl T0 dS = I .
The
magnetic
field
can
then
be
written
H = T0 T grad I in the conductors and H = T0 grad I elsewhere. The
equation to solve in the conductors becomes: w §1 · curl T v u P ( T grad I ) ¸ P T grad I = wt ©V ¹ §1 · wT curl ¨ curl T0 v u P T0 ¸ ȝ 0 wt ©V ¹
curl ¨
[8.34]
In the air, the flux conservation is then written: div( P 0 grad I ) = div T0 . It can be seen that the potential T0 appears as a source. It should be determined by a calculation beforehand. The use of the scalar potential I makes this formulation more economical than AV-A, which requires three unknowns in the non-conductive parts. Nevertheless, the use of the potentials I and T0 introduce, in some configurations, connectivity problems, which can make the implementation of the formulations T - I rather delicate [LUO 97]. 8.2.4. Eulerian approach This approach consists of describing all phenomena from the reference frame , i.e. the movement of parts and changes in fields is observed from the vantage of the laboratory. For the parts not involved in the movement, no further development is needed. In the following sections, we will focus on the formulation to be used in the conductive parts in motion.
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The Finite Element Method for Electromagnetic Modeling
8.2.4.1. General case Let us consider equation [8.26] again (the gauge problem is not considered here) and let us transform the terms associated with the speed of movement in order to better identify them. To do this, let us consider the following vector identity: grad ( A v ) = v u curl A v grad A A u curl v A grad v
By calculating the the two last terms using [8.2], we obtain: v u curl A = v grad A A u ȍ grad ( A v )
By using this relation and by posing V ' = V A v , equation [8.26] can be rewritten: curl (
1
P
§ dA '· A u ȍ grad V ¸ = 0. © dt ¹
curl A ) V ¨
[8.35]
The same type of developments in T- I transform [8.31] to obtain:
§1 · § d (T grad I ) · curl T ¸ P ¨ ȍ u (T grad I ) ¸ = 0 dt ©V ¹ © ¹
curl ¨
[8.36]
In [8.35] and [8.36], two types of terms related to the motion are obtained: – The first is a total derivative (or particulate); it measures changes in a material point of the body in motion. This can be evaluated by finite differences. In general, the simplest way to calculate it is to move the mesh with the moving part. – The second shows directly a rotational speed. It will be seen throughout the following sections that such a term may be the source of numerical instabilities [MAR 91]. Generally, when it is impossible to simplify [8.35] or [8.36], the Eulerian description is not used in practice. In this case, the better suited Lagrangian approach is preferred. However, in some cases, presented hereafter, it is possible to significantly reduce the calculating time by using the Eulerian approach. 8.2.4.2. Case of geometric invariance with the speed A special case appears when the section of the conductor perpendicular to the direction of the speed is invariant during motion. In other words, it occurs when the boundary of the part in motion does not change with the displacement, which may
Modeling of Motion
337
be expressed by v n = 0 at each point of the boundary (see Figure 8.3). In such a situation, a fixed mesh can be used for the entire geometry, in which the mobile part is displaced [ROD 90]. v
:
Figure 8.3. Example of geometries invariant with speed
In addition, when speed is constant over time, the influence of the movement can be measured by solving a problem that is not time-dependent. Indeed in this case equation [8.26] is written: curl (
1
P
curl A ) V v u curl A grad V = 0.
[8.37]
The equivalent formulation for T- I is written:
§1 · §1 · curl T v u P (T grad I ) ¸ = curl ¨ curl T0 v u P T0 ¸ [8.38] ©V ¹ ©V ¹
curl ¨
This kind of formulation is designated in the literature as a “formulation with velocity term” or “transport term”. Let us emphasize the fact that here it concerns magnetostatic formulations for which one solving process is sufficient to describe the motion. 8.2.4.3. Discretization – stability problems The formulations developed above lead, after discretization, to the construction of a matrix system whose solution provides the values of the potentials on the nodes or on the edges of the finite element mesh. The Galerkine method, which consists of taking the weighting functions to be identical to the interpolation functions, is a classic means of obtaining such a discretization. Unfortunately, applied to the formulation with a transport term, this method leads to an unstable solution containing non-physical oscillations that grow with speed. The application of the Galerkine method with first degree Lagrange polynomials on a 1D problem can establish a “magnetic Peclet” number that measures the unstable nature of the numerical solution: Pe = PV vh
338
The Finite Element Method for Electromagnetic Modeling
where h is the length of the elements used. With such elements, oscillations appear starting from Pe t 2 . Values of P , V and v being determined by the problem, the use of a good mesh allows a non-oscillating solution to be obtained. The equations encountered here are similar to those obtained from the study of thermal convection-conduction phenomena and are quite close to the Navier-Stokes equations in fluid mechanics. As for these areas, the quality of the solution can be improved by using the so-called “Petrov-Galerkine” methods. These methods consist of using weighting functions that are different from interpolation functions and better adapted to the physical solution. They are of two types: – the polynomial decentered weighting functions [HEI 77]; – the exponential weighting functions [TEL 83]. 8.2.5. Lagrangian approach In this approach, the phenomena are observed from each material point. We have previously seen that Maxwell equations took the same form both in the fixed reference frame and in the moving reference frame attached to the part in motion. Hence, each part of the problem is described in a reference frame in which it is at rest, i.e. for the fixed parties and for the moving part. When there are n bodies in motion, then n moving reference frames have to be considered. The most coherent technique with this approach consists of attaching a mesh to each moving part and effecting the subsequent displacements. It can be seen that a technique is required to tune the solutions obtained for different parts of the mesh, whether mobile or fixed. This point will be developed in section 8.3. With such a description, any speed term is discarded. Indeed, expressed in [8.26] is written:
§ wA ' '· curl ( curl A ) V ¨ grad V ¸ = 0. ¨ wt ¸ P © ¹ 1
'
,
[8.39]
where A' is the vector potential whose coordinates are expressed in . The equation obtained has the same form as the problem of currents induced in a motionless part treated in .
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339
8.2.5.1. Space discretization When we approximate a vector field on a finite elements mesh by nodal elements, each one of its components is reached by a scalar interpolation. Thus, for the magnetic potential vector A calculated at the point M, we write: A = ¦D i A i i
where Ai are the nodal values of the potential at the nodes of the mesh element in which point M is located. When a Lagrangian description is used, a difficulty appears on the level of the elements which touch the fixed part/moving part border. For such elements, all the unknown variables at the nodes are not expressed in the same reference frame. In the example of Figure 8.4, the interpolation of A in the considered element is written: A = D1 A1 D 2 PA'2 D 3 PA'3
where P is the transformation in [8.1].
A’2 A’3 Mobile A1
Air
Figure 8.4. Interpolation of the unknown variable in the element touching the moving part/fixed part border
Such a difficulty does not appear when edge elements are used. Indeed, such elements naturally imply a Lagrangian description of the problem. In this case the approximation of A is written: A = ¦Wi Ai . i
where the degree of freedom Ai is a scalar and represents the circulation of A along the edge i. There is thus no question of a reference frame for the unknown variables,
340
The Finite Element Method for Electromagnetic Modeling
since the movement is directly taken into account by the shape functions Wi , which move simultaneously with the mesh. 8.2.5.2. Time discretization In the general case of electromagnetic transient modes, an evolution problem must be solved (Figure 8.5) that can be dealt with using a time-stepping procedure (Figure 8.6).
Figure 8.5. Definition of an evolutionary problem
Figure 8.6. Description of a time-stepping procedure for an evolutionary electromechanical solution
At each time step, the magnetic equations are solved, to which electric circuit equations describing the supply of the system can be strongly coupled. In most cases, the kinematic equations of motion [8.7] or [8.8] are weakly coupled. Thus, they are explicitly solved at the end of each time step, by considering that the force or the magnetic torque is constant between two steps. This choice of implementation
Modeling of Motion
341
is the one most often encountered, because of its simplicity. It can also be justified by the significant difference between the electric time constant and the mechanics time constant. If this is not the case, a strong coupling should be considered, although this is more complex to achieve [REN 94]. In order to obtain the temporal evolution of the vector of unknown variables X, we must solve a system of the type: MX T
dX = F with dt
X(t = 0) = X0 .
[8.40]
Traditionally, a T-scheme is used to discretize X over time; its derivative being approximated by finite differences: X = T Xt 't (1 T ) Xt ,
dX Xt 't Xt ; . 't dt
– In the linear case, at each time step, we solve: T· § §T · ¨ T M ¸ Xt 't = ¨ (1 T )M ¸ Xt F. 't ¹ © © 't ¹
– In the nonlinear case, the iterative Newton-Raphson method is used, where the following system is solved at each iteration: S 'X = R ,
where R is the residual vector that should be zeroed. ǻX is the increment of the solution where S is defined by:
Sij =
wRi (t T't ) . wX j (t 't )
In our case, we have: R = MX T
T dX F and S = T M 't dt
[8.41]
This time-stepping procedure corresponds to the solution of a system of differential equations by the Euler method. Based on the value of T, the scheme used is called: explicit (T = 0), implicit (T = 1) or Crank-Nicolson (T = 1/2).
342
The Finite Element Method for Electromagnetic Modeling
Here, the implicit scheme is the easiest to implement. Indeed, in order to calculate the potential at the moment t 't , this scheme does not require knowledge of the solution at the moment t in the non-conducting parts, which is not the case if T z 1 . This can be seen easily in the linear case by reconsidering equation [8.40]. If the writing of the system is limited to the non-conducting parts, we obtain:
T MXt 't = (T 1)MXt F.
[8.42]
Thus, if a procedure for re-meshing air is used with T z 1 , the solution Xt must be projected on the new mesh. Such an interpolation between two meshes is cumbersome to implement and it is a source of numerical noise. In the conducting parts, which are considered to be rigid in this study, it is sufficient to keep the meshing from one step to another. A second advantage of the implicit Euler scheme is that it is unconditionally stable. However, its lack of accuracy may cause some difficulties when choosing the time step to deal with stiff problems. In this case, a prediction-correction algorithm allows an adaptation of the time step [VAS 90]. Other methods, such as the RungeKutta implicit type of s steps (sDIRK), seem particularly well suited to the stiff problems [NIC 96, CAM 98]. They are of higher orders and allow an adaptation of the time step. This may, however, prove to be a more costly calculation than a simple implicit Euler method. 8.2.6. Example application Using a simple example, let us detail a way to discretize a problem of eddy currents with movement, until the matrix system is obtained. In what follows, in order to simplify the presentation, only conductors, which may or may not move, are considered. 8.2.6.1. Equations to be solved We are assuming an invariance of the geometric and magnetic phenomena according to a direction e z allowing the problem to be treated in two dimensions, as suggested in Figure 8.7. Traditionally in this case, a formulation of type AV - A is used.
Modeling of Motion
343
B
ez J
A
Figure 8.7. Description of a problem handled in 2D
Let us now note the simplifications made by the 2D approximation: – The magnetic vector potential can be manipulated as a scalar quantity because in such a problem A = Ae z . A direct consequence is that: div A = 0 . It is thus enough to impose the value of A at a point in the domain to ensure its uniqueness. On the other hand, the movements that can be considered are in the calculation plane and therefore it follows that A u ȍ = 0 . – As shown in Figure 8.8, the currents are considered according to e z . We then notice that div J = 0 . Moreover, the electric potential V is constant on the surface of conductors in any plan parallel to the study plan. I
S
L
Figure 8.8. Definition of a conductor with depth L 1
The electric potential variation along a conductor of length L is written: grad V =
wV wz
ez =
'V L
ez .
[8.43]
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The Finite Element Method for Electromagnetic Modeling
Taking all these observations into account, equation [8.26] can be written in a simplified form: curl (
1
§ dA 'V · e z ¸ = 0. L © dt ¹
[8.44]
curl A ) V ¨
P
Generally, the considered conductors in movement are massive and are not connected to an electric circuit. An equation imposing a zero total current in the conductor’s S section must be added: I=
§
³ ¨© V S
dA dt
V
'V L
· ¹
e z ¸ ds = 0.
[8.45]
8.2.6.2. Discretization In order to obtain [8.44] in an integral form, the weighted residual method is used. The discretization is then determined thanks to the Galerkin method. This consists of choosing the interpolation functions of the unknown variables as weighting functions. Here, the unknown variable to be discretized is a scalar, and a nodal approximation is used. Let us denote by Ai the unknown variables at nodes and by D i the corresponding interpolation functions. We consider the resolution of a nonlinear problem with the Newton-Raphson method as proposed in section 8.2.5.2. For each node i, there is a residue to be zeroed in the form: Ri =
1
§
³ ¨© grad Į P grad A VD i
S
dA i
dt
VD i
'Vk
· ¸ ds L ¹
[8.46]
The index k is the number of the conductor to which the node i belongs. For the unknown 'Vk , the expression which gives an additional residue to be zeroed per conductor is used: rk =
§
³ ¨© V S
dA dt
V
'Vk
· ¸ ds L ¹
[8.47]
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For a nonlinear material, P depends on A. Let us determine the matrix contribution of the conductor k to the final system; after symmetrization, we obtain: ª S <11> « « «S < 21> ¬
S <12> º ª 'A º ª R º »« » = « ». »« » « » S < 22> »¼ «¬ ' ('\ k ) »¼ «¬ rk »¼
[8.48]
with:
VD iD j · § 1 grad grad D D i j ¨ ¸ ds ³S © 't ¹ P
<11>
= LT
<12>
= S ji
Sij
Sij
<22>
Sij
<21>
=
³
S
V L't
=
³
§
³© §
dA
ds
1
P
grad A VD i
dA dt
VD i
d '\ k ·
¸ ds ¹
dt
V d '\ k ·
³ ¨© V dt L S
't
ds
Ri = L ¨ grad D i S
rk =
VD i
S
dt
¸ ds ¹
Due to reasons relating to the symmetry of the matrix system, the use of the t
electric potential integrated over time \ defined by \ = ³ Vdt is preferred here. 0
It was assumed that the mesh of the conductor in movement was attached to it. dA can be easily assessed by the finite difference method by writing: The term dt A (t 't ) Ai (t ) dA = ¦D i i i dt 't
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The Finite Element Method for Electromagnetic Modeling
8.3. Methods for taking the movement into account 8.3.1. Introduction During the temporal simulation of a problem with movement, the geometry changes with the position of the moving parts. Thus, the mesh on which the calculations are based must be adapted to these changes. As shown in section 8.2.5, the most appropriate and natural choice is to keep the same mesh for the nondistorted aspects of the problem. However, it is necessary to manage the air-gap distortions and more generally the surrounding air. In the particular case of electrical machines or 2D linear motions, fairly straightforward re-meshing techniques exist and are of a constant number of nodes. For more general movements or any mobile forms, large distortions of the air zone appear and other methods should be considered. One strategy consists of using for the air a method that does not require any meshing and coupling it with the finite elements used in other parts of the device. A new approach is to treat the air by means of so-called meshless methods [HÉR 00]. More typically it is possible to use boundary integrals for the air [SAL 85, KUR 98, NIC 96]. This well known method is very broad and robust, although the high calculation cost has so far limited its use to simple cases. A last method, which consists purely of finite elements, is to use a procedure for automatic re-meshing of the air. This latter must then be able to change the place and the number of nodes, as well as the connection between them, i.e. the topology of the mesh. This section presents the various methods for taking the movement mentioned here into account, focusing on techniques allowing large distortions to be handled.
8.3.2. Methods for rotating machines Numerous methods have been proposed in the literature for treating motion in 2D rotating machines. What makes this case quite simple is the fact that an air-gap whose form is invariant during the movement can always be defined. This feature provides access to several techniques for linking the solution obtained in the stator with the solution for the rotor.
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Figure 8.9. Definition of a macro-element for handling the movement of a rotating machine
We can take advantage of the simple geometry of the air-gap and of the frequency of the phenomena in order to find an analytical solution linking the unknown variables of the rotor and the stator. Indeed, if the unknown variable used is the magnetic vector potential A, it is sufficient to solve 'A = 0 on the geometry defined in Figure 8.9. The analytical solution obtained allows a special finite element called a macro-element [RAZ 82] to be defined. The major drawback of this method is the calculation time, which is longer than for a conventional finite element system. The matrix system obtained is in fact a large broadband. This is due to the macro-element that has a matrix contribution whose size is given by the number of nodes defining it. Another method consists of defining a meshed band using a single layer of elements (see Figure 8.10a) and reconnecting the rotor and the stator nodes progressively with the rotation [DAV 85].
(a)
(b)
Figure 8.10. (a) Automatically re-meshed band; (b) sliding surface with meshing connection
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The Finite Element Method for Electromagnetic Modeling
As proposed in [GOL 98] and [BUF 99], the connection can be made not through a band to remesh but on a sliding surface (see Figure 8.10b). During the movement, the nodes of the stator and the rotor do not coincide on the surface. This generates so-called non-conforming approximations that can be addressed through the use of Lagrange multipliers.
8.3.3. Coupling methods without meshing with the finite element method In 1991 the fundamental principles of a new method of numerical simulation without meshing were set forth: the diffuse element method [NAY 91]. Since then, this has given birth to a family of meshless methods [ARM 96]. Recently, the work of C. Hérault [HÉR 00] has shown the implementation of one such method, called HP-Clouds, for the simulation of electromagnetic systems and proposed to couple it with the finite element method in order to deal with problems with movement. Let us here give a quick overview of this type of coupling. 8.3.3.1. Principle of the meshless method: HP-Clouds Similarly to the finite element method, this method allows an approximation of continuous quantities in the space to be generated. The approximation is supported by a cloud of N nodes to which nodal values are attached. The quantity u at coordinates x is approximated by: N
u~ (x) = ¦Ii (x)ui
[8.49]
i =1
The difference with the FEM. comes from the definition of the function form
Ii .
For FEM, these are defined on the basis of the meshing elements. For the HPClouds method, for each node i a zone of influence represented by a ball of radius ri is defined and the expression of the associated form function can be expressed by:
Ii (x) =
Wi (x xi ) N
[8.50]
¦W j (x x j ) j =1
This so-called Shepard function is defined on the basis of weighted functions Wi , such that: Wi ( x) = w( x ) if & x x i &d ri and Wi ( x) = 0 if not
[8.51]
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We then have the choice of the function w that will determine the regularity of the approximation. Infinitely derivable approximations on the entire computational domain can very easily be obtained. This is an advantage with respect to the FEM which, at first order, provides an approximation that is only continuous. The recovery of the computational domain by balls is necessary for the approximation to be valid. The quality of the latter will then depend on the number and the arrangement of nodes as well as the size of the balls. In addition, it can be noted that the HP-Clouds approximation is noninterpolating, i.e. that: u (xi ) z ui
[8.52]
The nodal values then lose their physical character, making the definition of boundary conditions more complex.
Figure 8.11. Application example for the coupling between HP-Clouds and the FEM
8.3.3.2. Coupling with the finite element method The regularity of the approximation as well as the lack of meshing are two key advantages when taking movement into account. However, it is necessary to adapt the cloud of nodes to the distortions that undergo the computational domain during the movement. However, this reorganization is far less traumatic for the regularity of the temporal solution than a re-meshing in the case of the FEM. This coupling thus seems to be well suited to this type of problems since it allows the benefits of both methods. Regarding the finite elements, the nonlinearities of materials, the eddy currents and the coupling with electric circuits can be taken into account. On the other hand, the method without meshing offers much more distortion freedom for the area of concern.
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The Finite Element Method for Electromagnetic Modeling
The form of the matrix terms obtained by the two methods is the same; the changes are only in the integration domains: for the FEM, the integration is carried out on elements, while for the HP-Clouds method, the integration is carried out on intersections of balls. This stage of integration is a challenge for the HP-Clouds method for which the high order shape functions as well as the non-trivial form of integration domains lead to a high cost in calculation time. Another delicate point in the coupling is the way the two approximations are connected. The difficulty arises because of the non-interpolating approximation character of the HP-Clouds. Linear combinations must then be entered into the with system. For the point k of coordinates xk having as a nodal value uk FEM
regards to the finite element, we can thus write: N
¦Ii (x k )uiHP Clouds = ukFEM
[8.53]
i =1
Although very promising results have already been achieved thanks to this technique, difficulties remain to be overcome in order to make the method more competitive with respect to older and better controlled methods.
8.3.4. Coupling of boundary integrals with the finite element method The boundary integrals method has been used for a long time in the field of mechanics [MAC 88]. Its performance also helps when dealing with problems in electromagnetism [KRA 83]. Its coupling with the finite element method, called subsequently BEM-FEM coupling (BEM stands for boundary element method), has been the subject of many studies. One of the main reasons for its use is the inclusion of infinity in the open problems [SAL 85, BRU 91, BOS 88]. It has also been used to address problems with movement [BOU 88, NIC 96, KUR 98]. Keeping in mind that the area treated with boundary integrals does not require any mesh, this technique makes it possible to take large distortions of air into account. 8.3.4.1. Application of the second Green’s identity to magnetic vector potential Let us consider the domain : and the boundary * of Figure 8.12.
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351
* M
P :
Figure 8.12. Domain treated by boundary integrals
The second Green’s identity applied to scalar functions u and v is written: wu · § wv ³: u 'v v'u d : = ³* ¨ u v ¸ d * wn ¹ © wn
[8.54]
On the other hand, the Green’s functions associated with the Laplacian operator 1 1 ln MP in 2D and GM ( P ) = on : are defined by: GM ( P ) = in 3D. 2S 4S MP In the air, the magnetic vector potential A is at zero Laplacian. If equality [8.54] is applied to each of its components using the Green’s functions, we obtain:
wA · § wG c( M ) A( M ) = ³* ¨ A M GM d* wn wn ¸¹ ©
[8.55]
where c( M ) is a coefficient which corresponds to the angle under which M sees * and which can generally be calculated in the form: c( M ) = ³*
wGM d* wn
[8.56]
It is thus possible to calculate A at any point of : by knowing only the value of A and of its normal derivative on the boundary *. 8.3.4.2. Discretization and finite element coupling In order to simplify the presentation of this section, we consider magnetostatic coupling; the coupling in transient conditions does not introduce any further difficulty. Only a boundary mesh of the BEM domain is required. In our case, it corresponds to the trace of the mesh of the finite element domain. In order to
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The Finite Element Method for Electromagnetic Modeling
discretize the integral equations, a Galerkin method can be used. For this purpose, equation [8.55] is weighted with test functions Wi and can be written [REN 88]: § wA · · § wG G ³*c Wi A d * = ³* ¨ ³*Wi ¨ A ¸ d* ¸ d* w wn ¹ ¹ n © ©
[8.57]
: FEM : BEM *
: FEM
: FEM
Figure 8.13. Problem handling the FEM-BEM coupling
Another possibility is to use a collocation method, which consists of writing equation [8.55] at nodes. The variational method is more accurate but it is also more expensive, since it requires a double integration [REN 88]. For the discretization of the integral equations, let us work in 2D and choose the collocation. In this case, we write for each node i : wA · § wG ci Ai = ³* ¨ A G ¸ d* wn ¹ © wn
[8.58]
Because of its continuity, it is preferable to use the quantity H t rather than in order to achieve the coupling [BRU 91]. By using a nodal interpolation for A H t , the discretization of the integral equations gives: [DG ][ A j ] [G ][H t ] = 0 j
with DGij = ³*D j
wA wn and
[8.59]
wG d * ciG ij and Gij = ³*D j P0 Gd * . wn
There is a lack of equations able to solve the degrees of freedom located on the boundary. The FEM part allows the necessary equations to be provided. This is done by writing the finite element formulation, in which the terms on the FEMBEM boundary are kept:
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1
§ ©
· ¹
353
[8.60]
³: FEM ¨ grad D i P grad A ¸ d : ³*D i H t d * = ³:FEM D i j d :. After discretization, we obtain:
[8.61]
[S][ A j ] [Q][H t ] = [J ], j
with Sij = and J i = ³:
§ ©
1
· ¹
³:FEM ¨ grad D i P grad D j ¸ d : , Qij = ³*D iD j d *
FEM
Di j d : .
Equations [8.61] and [8.59] form the matrix system to be solved. Figure 8.14 provides an outline of the situation. N int denotes the total number of nodes within the region : FEM and N fr the number of nodes on the boundary. Aint refers to unknown internal variables, and : FEM and Afr to unknowns located on the boundary.
Nint
0
Aint
=
FEM
Afr
Nfr Nfr
FEM
0
0
Ht
Figure 8.14. Matrix system obtained by directly coupling the BEM with the FEM
The matrix obtained is not symmetric and the blocks corresponding to the boundary integrals are full. This type of system takes much more time to solve than a diagonal dominant and symmetric system like the one generated by the FEM. The
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The Finite Element Method for Electromagnetic Modeling
size of the full blocks is directly dependent on N fr . In order to reduce the calculation time, it is thus interesting to choose a boundary that minimizes this number of nodes. 8.3.4.3. Difficulties related to the implementation If the coupling method is very attractive from a theoretical point of view, it is a little less so with respect to the implementation. This is partly due to the size and shape of the matrix system to be solved, but also because of some difficulties including the integration of terms. The following gives some ideas of these specific difficulties.
Integration 1 1 or 2 , r r where r is the distance between the observation point M and the point of integration P. When M becomes close or belongs to the element of integration, difficulties of numerical quadrature appear. In the past, many studies have been conducted to calculate such singularities. Two ways to deal with the problem can be listed:
The cores to be integrated comprise singularities of the type lnr ,
– extract the singularity and treat it analytically [HAG 98, ANC 79]; – use a modified Gaussian quadrature formula allowing a better distribution of the Gauss points around the singularity [HAY 90, HUB 97]. Analytical treatments are very accurate but expensive. The most appropriate solution seems to be that using a modified Gaussian formula.
Solving the matrix system Figure 8.14 suggests a simultaneous resolution of the FEM and BEM parties. Another technique, which generally leads to a shorter resolution time [MOR 90], is to proceed in two phases by inversing system [8.59]: [H t ] = [G ]1[DG ][ A j ], j
then by re-injecting in [8.61]:
[S] [Q][G ]
1
[DG ] [ A j ] = [J ]
The stiffness matrix R is then generally defined by [R ] = [Q][G ]1[DG ] . The matrix inversion required for its calculation can be very expensive if the number of nodes on the boundary is high. Otherwise, it depends only on the geometry. Thus,
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355
where there is no movement, it is not necessary to reassess the matrix and its calculation should therefore be carried out only once. In addition, it can be made symmetric. Indeed, in practice, it is a reasonable approximation to replace [R ] with ([R ] t[R ]) / 2 [MOR 90, REN 88].
Some techniques are aimed at reducing the calculation time and storage needs by approximating the matrix terms corresponding to the remote interactions [KUR 01, BED 00]. On the contrary, we can cope with the size of the system by introducing computing means accordingly and by using for example the decomposition in domains and/or the parallelization of the calculation code [RIS 01].
8.3.5. Automatic remeshing methods for large distortions 8.3.5.1. Setting the scene Our goal here is to propose methods for remeshing the air at each time step. We also intend to assess the consequences of such remeshing on a time domain solution. Let us recall that when using an implicit Euler scheme in order to carry out a time domain resolution, the calculation of the solution of a time step does not require any knowledge of the solution in the air of the previous time steps (see section 8.2.5.2). This is advantageous because, when remeshing, it is not necessary to interpolate the solution of the previous time step on the new mesh. In what follows, we limit ourselves to meshes composed of simplexes, i.e. with triangles in 2D and tetrahedrons in 3D, which is very representative of the meshings used in the finite element calculation tools in electromagnetism. Moreover, it should be noted that only a part of the domain needs to be remeshed. Two strategies are then possible: – the addition of nodes on these boudaries is accepted without changing the rest of the mesh and then non-compliant elements are addressed; – the new mesh is forced to respect the boundaries already meshed. In what follows the state of the art automatic remeshing methods are presented. Some of the problems arising from such techniques are then highlighted. 8.3.5.2. Remeshing at constant number of nodes For low amplitude movements, an “elastic” meshing which repositions the nodes without changing their connectivity can be used. For instance, it only requires us to set the coordinates of nodes based on the position of the mobile part. This method provides a satisfactory solution in some simple cases [KAW 00]. Nevertheless, it
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The Finite Element Method for Electromagnetic Modeling
leads rather quickly to meshes with very distorted elements or to reversals of elements. In the particular case of linear displacements in 2D, a technique similar to the band used for the rotating machines can be used. That presented in Figure 8.15 can, without adding nodes, deal effectively with the translation of parts with specific forms [JAR 93].
Figure 8.15. Band translation for the linear movements in 2D
8.3.5.3. Methods for large distortions When the amplitude of motion generates too large a deformation of air, adjusting the position of the nodes is not enough; it is necessary to add topological operations on the mesh (reversal of edges and/or of facets). Some techniques are able to isolate mesh areas where these operations are necessary, and thus allow only small portions to be remeshed [DUV 96]. When such techniques are not available, a full remeshing of an area defined by the user is required. We then have relatively little control on the topological changes taking place on the mesh. In [TAN 98], the mobile and fixed parts are meshed separately and their meshes are intersected and reconnected for each new position. If the method seems rather general, it does not guarantee the quality of the mesh. The frontal methods seem particularly suited to meshing an area for which the boundary mesh is given. They start from the contour elements, while simplexes gradually build towards the interior of the geometry, thus forming a front. A lot of experience is required to master this type of mesh. Many heuristic rules are needed to generate new nodes of the mesh, as well as to solve problems that arise when multiple fronts meet [LOH 96, MOL 95].
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In the area of molding simulation, Coupez has presented a very original and powerful method for automatic remeshing [COU 91]. By topological operations on the mesh, he looks for a topology which contains boundary meshes and which makes optimal triangulation. His method is inspired by the work of Talon [TAL 89] and appears to be very powerful. There is also another approach used by Joe, as well as by Kettunen et al. [JOE 91, KET 95]. They decompose the geometry into convex elementary subsets. These are then meshed by a process using topological operations to maximize the solid angles of tetrahedrons. Finally, the most prevalent method consists of creating a mesh based on the Delaunay criterion [HER 82, GEO 91]. For this type of mesher, observing a boundary mesh is a difficulty. Indeed, the Delaunay criterion can generate a convex triangulation enveloping a set of nodes, and when the geometry to be meshed is not convex, the boundary discretization is not systematically observed. It is then necessary to force the meshing algorithm to include the boundary elements in the mesh. In 2D, this operation is quite simple and can be solved by reversing operations of edges [TAL 89]. In 3D, the process of recovery of boundary facets and edges can be difficult [GEO 91]. In practice, observing the boundary is a relatively easy constraint to deal with if the boundary is meshed fairly finely. 8.3.5.4. An original process The Delaunay criterion makes it possible to build a triangulation from a set of nodes, but does not derive any placement strategy from them. In a process of remeshing, the cloud of nodes should be adapted to the distortions of the portion to be remeshed. The simplest way to act in practice is to entirely regenerate it, which usually brings many changes in the mesh. In order to reduce this, it seems advantageous to have a technique that moves the nodes locally so as to follow the displacement of the moving object. The “meshing via bubble packaging”, also called “bubble meshing”, is a technique that allows for such control on the evolution of the cloud of nodes according to the geometry changes [LEC 01a]. The originality of the method relies on the use of balls centered at the meshing nodes. A physical model and an iterative process are used to describe the displacement of balls. These balls are moving towards equilibrium, leading to a satisfactory spatial arrangement for triangulation. Figure 8.16 shows the result obtained on a simple structure thanks to such a technique.
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The Finite Element Method for Electromagnetic Modeling
Figure 8.16. The meshing is suitable for the displacement of the mobile part
Note that the meshes obtained using these methods are of great regularity. In addition, the re-arrangement of nodes produced by the physical model tends to locate the topological operations to be carried out on the mesh. Nevertheless, a careful implementation is required to provide speed and robustness to this technique. 8.3.5.5. Problems of the small air-gaps Many devices include small air-gaps in one or several given positions of their mobile part. Their presence creates large distortions in the air and can cause, according to the mesher used, a significant change in the number of nodes of the mesh. It is therefore necessary to be careful with this type of situation.
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Figure 8.17. Meshing details of the electromagnet in open and closed position
Usually during the meshing process the user has very little control over the nodes generated internally. The only information that must be provided is the discretization of the boundary region to be meshed. The density of nodes required is then propagated within the geometry. The presence of small air-gaps during movement requires a fine mesh around them. When the air-gap widens, the mesher adds a large number of nodes in an area that does not require a fine discretization (see Figure 8.17). This phenomenon is even more amplified as the initial air-gaps are small. In the 3D case, this can lead to an “explosion” of the number of nodes. Note that this is the direct consequence of the choice that was made here, which consists of not changing the mesh of non-distortable parts of the device in order to simplify the estimation of the induced currents in these parts. If this constraint is not imposed, a solution consists of setting the density of nodes on the boundary according to the position of the mobile part. Note also that the management of the non-compliance of meshes makes it possible to circumvent the problem. To improve our control of the number of internal nodes, the mesher must be able to take into account user information indicating the desired density of nodes inside the area to remesh. Such control can be introduced rather simply in the meshing processes by balls mentioned above. These methods make it possible to control the number and size of balls and thus of the elements generated. Figure 8.18 shows what can provide such a method in the treatment of a problem with a small air-gap.
(a)
(b)
Figure 8.18. Control of node density brought about by the meshing by balls
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The Finite Element Method for Electromagnetic Modeling
Another strategy for dealing with this type of problem is to introduce, when the position of the mobile part allows it, special elements describing the small air-gaps [LEC 01b]. This last method, which is not detailed here, is still quite difficult to implement. 8.3.5.6. Remeshing and numerical noise Now we intend to show some numerical examples of automatic remeshing in a simple test case dedicated to highlighting the problems associated with this method. In addition, we compare the results with those obtained with the FEM-BEM coupling described previously.
Figure 8.19. Description of test case and used meshing
The structure serving as an example is depicted in Figure 8.19. Magnetostatic simulations are performed for several positions of the mobile part on the axis (Oz). At each change of position, the air part is remeshed automatically by an algorithm based on the Delaunay criterion, while observing the boundary meshing of different pieces of the device. The problem is treated with a total magnetic potential formulation. In Figure 8.20 the evolution of several quantities depending on the position is presented. Generally speaking, the results obtained with the FEM-BEM coupling are very smooth. However, some calculations made with the re-meshing method are strongly disturbed by the mesh change.
Modeling of Motion
Figure 8.20. Simulation results
361
362
The Finite Element Method for Electromagnetic Modeling
Number of unknown variables ( N i )
FEM-BEM
Remeshing
2,589
min. 2,493 max. 2,973
Number of terms ( Nt )
2,110,580
Non-zero matrix
min. 19,338 max. 23,158
Fill in rate ( N t / N i2 )
31%
0.3% (average)
Total time (6 positions)
2 h 36 min.
9 min. 30 s
Table 8.1. Statistics for the comparison between remeshing and FEM-BEM coupling in 3D
Table 8.1 highlights the significant cost differences between the two calculation methods in 3D. For this simple example, the FEM-BEM coupling is still usable with the chosen boundary discretization (see Figure 8.19). For problems representative of the actual devices, the limit of a current standard machine’s capabilities is quickly reached. Here we see that re-meshing is a usable and rewarding method from the perspective of calculation costs. However, attention must be paid to the numerical distance introduced by the meshing changes. In order to minimize this, we must be capable of producing a high-quality meshing of the distortable part at each time step, which, in 3D, requires special attention.
8.4. Conclusion The performance of modern computers makes digital simulations of transient modes more accessible. In addition, it is exigent for industry to equip the electrotechnical design software with tools intended to completely model mechanical converters. A faithful evaluation of the electromechanical performances is indeed necessary in order to arbitrate between various technologies or to optimize the conversion. Thus, it is important to develop models allowing the simulation of electromechanical transient modes, in which the magnetic, electric and mechanical aspects are taken into account. Taking movement into account is an important difficulty to surmount in such a modeling.
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In the case of a geometric invariance with respect to velocity, it is advantageous to use a Eulerian description of the phenomena which then leads to a magnetostatic formulation with transport term. This is a favorable method because of its low calculation time, but it is advisable to keep the digital instabilities brought by the velocity term under control. The majority of the problems with movement do not allow the use of such an approach because of the distortions of the geometry; a transient problem must then be solved. In this case, it is preferable to adopt a Lagrangian description which makes it possible to use the same equations as for a fixed body. For 3D problems, the edge interpolation is particularly well suited because it places us “automatically” in such an approach. Nevertheless, the use of nodal elements remains an alternative possibility. Effective and proven methods exist for rotating machines and for simple distortions in 2D which do not involve a modification of the number of nodes. For the most general case, two competing techniques can be quoted: – the method coupling the boundary integrals with the finite elements: it brings an elegant and robust solution; – remeshing methods, which are inexpensive and keep the finite elements character of the solution. A coupling with the HP-Clouds method was also presented but for the moment it lacks the refinement required to be a complete success. The generality and the quality of the solution brought by the FEM-BEM coupling make its use very worthwhile when treating electromechanical transients in 2D, but its very high cost in computer resources reduces its 3D use to the treatment of simple cases. In the near future, the remeshing methods therefore seem to be the best indicated for the treatment of the movement in 3D. Nevertheless, a good quality discretization has to be retained, in order to reduce the numerical noise introduced during the changes of meshing, which is rather a strong constraint in 3D.
8.5. References [ALB 90a] ALBANESE R., RUBINACCI G., “Analysis of three-dimensional electromagnetic field using edge elements”, Journal of Computational Physics, vol. 108, p. 236–245, 1990. [ALB 90b] ALBANESE R., RUBINACCI G., “Formulation of the eddy-current problem”, IEE Proceedings, vol. 137, no. 1, p. 16-22, 1990.
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The Finite Element Method for Electromagnetic Modeling
[ANC 79] ANCELLE B., Emploi de la méthode des équations intégrales de frontière et mise en oeuvre de la conception assistée par ordinateur dans le calcul des systèmes électromagnétiques, Thesis, Institut National Polytechnique de Grenoble, 1979. [ARM 96] ARMANDO DUARTE C., ODEN J., A review of some meshless methods to solve partial differential equations, Report, Texas Institute for Computational and Applied Mathematics, 1996. [ASC 76] ASCHCROFT N., MERMIN N., Solid State Physics, Holt Saunders International Editions, 1976. [BED 00] BEDENDORF M., “Approximation of boundary element matrices”, Num. Math., vol. 86, no. 4, p. 565-589, 2000. [BIR 89] BIRO O., PREIS K., “On the use of the magnetic vector potential in the finite element analysis of three-dimensional eddy currents”, IEEE Trans. Magn., vol. 25, no. 4, p. 3145-3149, July 1989. [BIR 93] BIRO O., PREIS K., RENHART W., VRISK G., RICHTER F., “Computation of 3D current driven skin effect problems using a current vector potential”, IEEE Trans. Magn., vol. 29, p. 1325-1332, 1993. [BOS 88] BOSSAVIT A., “Le calcul des courants de Foucault en trois dimensions, en présence de corps à haute perméabilité magnétique”, Revue de Physique Appliquée, vol. 23, p. 1147-1205, 1988. [BOS 90] BOSSAVIT A., “Le calcul des courants de Foucault en dimension 3, avec le champ électrique comme inconnue. I: principes”, Revue de Physique Appliquée, vol. 25, p. 189197, 1990. [BOU 88] BOUILLAULT F., RAZEK A., “Hybrid numerical methods for movement consideration in electromagnetic systems”, IEEE Trans. Magn., vol. 24, no. 1, p. 259261, January 1988. [BRU 91] BRUNOTTE X., Modélisation de l’infini et prise en compte de régions magnétiques minces. Application à la modélisation des aimantations des navires, PhD thesis, INP Grenoble, 1991. [BUF 99] BUFFA A., RAPETTI F., MADAY Y., “Calculation of eddy currents in moving structures by a sliding mesh-finite element method”, Proceedings of COMPUMAG, Sapporo, p. 368-369, 1999. [CAM 98] CAMERON F., PICHÉ R., FORSMAN K., “Variable step size integration methods for transient eddy current problems”, IEEE Trans. Magn., vol. 34, no. 5, p. 3319-3322, 1998. [COU 91] COUPEZ T., Grandes transformations et remaillage automatique, PhD thesis, Ecole Nationale Supérieure des Mines de Paris, 1991. [DAV 85] DAVAT B., REN Z., LAJOIE-MAZENC M., “The movement in field modeling”, IEEE Trans. Magn., vol. 21, no. 6, p. 2296-2298, 1985. [DUV 96] DUVAL B., Optimisation de maillages non structurés dans des géométries déformables, PhD thesis, University of Rouen, 1996.
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[GEO 91] GEORGE P., HECHT F., SATEL E., “Automatic mesh generator with specified boundary”, Computer Methods in Applied Mechanics and Engineering, vol. 92, p. 269288, 1991. [GOL 98] GOLOVANOV C., COULOMB J., MARÉCHAL Y., MEUNIER G., “3D mesh connection techniques applied to movement simulation”, IEEE Trans. Magn., vol. 34, no. 5, p. 3359-3362, 1998. [HAG 98] HAGHI-ASHTIANI B., Méthodes d’assemblage rapide et de résolution itérative pour un solveur adaptatif en équations intégrales de frontières destiné à l’électromagnétisme, PhD thesis, Ecole Centrale de Lyon, 1998. [HAY 90] HAYAMI K., “High precision numerical integration methods for 3D boundary element analysis”, IEEE Trans. Magn., vol. 26, no. 2, March 1990. [HEI 77] HEINRICH J., HUYAKORN P., ZIENKIEWICZ O., “An ‘upwind’ finite element scheme for two-dimensional convective transport equation”, Int. Jour. for Num. Meth. in Eng., vol. 11, p. 131-143, 1977. [HER 82] HERMELINE F., “Triangulation automatique d’un polyèdre en dimension N”, RAIRO, Analyse Numérique, vol. 13, no. 3, p. 211-242, 1982. [HÉR 00] HÉRAULT C., Vers une simulation sans maillage des phénomènes électromagnétiques, PhD thesis, INPG, 2000. [HUB 97] HUBER J., RUCKER W., HOSCHEK R., RICHTER K., “A new method for the numerical calculation of cauchy principal value integrals in BEM applied to electromagnetics”, IEEE Trans. Magn., vol. 33, no. 2, p. 1386-1389, 1997. [JAR 93] JARNIEUX M., GRENIER D., REYNE G., MEUNIER G., “F.E.M. computation of eddy current and forces in moving systems, application to linear induction launcher”, IEEE Trans. Magn., vol. 29, no. 2, p. 1989-1992, 1993. [JOE 91] JOE B., “Delaunay versus max-min solid angle triangulations for three-dimensional mesh generation”, International Journal for Numerical Methods in Engineering, vol. 31, p. 987-997, 1991. [KAW 00] KAWASE Y., YAMAGUCHI T., YOSHIDA M., HIRATA K., “3D finite element analysis of rotary oscillatory actuator using a new auto mesh method”, Proceedings of CEFC’2000 Milwaukee, p. 401, 2000. [KET 95] KETTUNEN L., FORSMAN K., “Tetrahedral mesh generation in convex primitives”, International Journal for Numerical Methods in Engineering, vol. 38, p. 99117, 1995. [KET 98] KETTUNEN L., FORSMAN K., BOSSAVIT A., “Formulation of the eddy current problem in multiply connected regions in terms of h.”, International Journal for Numerical Methods in Engineering, vol. 41, p. 935-954, 1998. [KRA 83] KRAHËNBÜHL L., La méthode des équations intégrales de frontière pour la résolution des problèmes de potentiel en électrotechnique et sa formulation axisymétrique, PhD thesis, Ecole Centrale de Lyon, 1983.
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[KUR 98] KURZ S., FETZER J., LEHNER G., RUCKER W., “A novel formulation for 3D eddy current problems with moving bodies using a Lagrangian description and BEMFEM coupling”, IEEE Trans. Magn., vol. 34, no. 5, p. 3068-3073, 1998. [KUR 01] KURZ S., RAIN O., RJASANOW S., “Application of the adaptative cross approximation technique for the coupled BE-FE solution of 3D electromechanical problems”, Proceedings of COMPUMAG, vol. III, Evian, p. 116-117, July 2001. [LAN 66] LANDAU L., LIFCHITZ E., Physique théorique: Mécanique, Moscow: Ed. Mir, 1966. [LAN 69] LANDAU L., LIFCHITZ E., Electrodynamique des milieux continus, Moscow: Ed. Mir, 1969. [LEC 01a] LECONTE V., HÉRAULT C., MARÉCHAL Y., MEUNIER G., MAZAURIC V., “Optimization of a finite element mesh for large air-gap deformations”, European Physical Journal AP, vol. 13, p. 137-142, 2001. [LEC 01b] LECONTE V., MAZAURIC V., MEUNIER G., MARÉCHAL Y., “Analysis of the dynamic behavior of a high sensitivity electromechanical relay with small air-gaps”, Proceedings of COMPUMAG, vol. IV, Evian, p. 10-11, July 2001. [LOH 96] LOHNER R., “Progress in grid generation via advancing front technique”, Engineering with Computers, vol. 12, p. 186-210, 1996. [LUO 97] LUONG H., Amélioration de la formulation en potentiel scalaire magnétique et généralisation au couplage entre équations de champ et de circuit électrique, PhD thesis, Institut National Polytechnique de Grenoble, 1997. [MAC 88] MACKERLE J., BREBBIA C., The Boundary Element Reference Book, Computational Mechanics, Springer-Verlag, 1988. [MAR 91] MARÉCHAL Y., Modélisation des phénomènes magnétostatique avec terme de transport: application aux ralentisseurs electromagnétiques, PhD thesis, Institut National Polytechnique de Grenoble, 1991. [MOL 95] MOLLER P., HANSBO P., “On advancing front mesh generation in three dimensions”, International Journal for Numerical Methods in Engineering, vol. 38, p. 3551-3569, 1995. [MOR 90] CARRON DE LA MORINAIS G., Contribution à la modélisation des phénomènes magnétodynamiques en 3 dimensions, PhD thesis, Institut National Polytechnique de Grenoble, 1990. [NAY 91] NAYROLES B., TOUZOT G., VILLON P., “La méthode des éléments diffus”, Compte rendu à l’Académie des Sciences, no. 313, Series II, 1991. [NIC 96] NICOLET A., DELINCÉ F., “Implicit Runge-Kutta Methods for Transient Magnetic Field Computation”, IEEE Trans. Magn., vol. 32, no. 3, p. 1405-1408, 1996. [RAZ 82] RAZEK A.A., COULOMB J., FELIACHI M., SABONNADIÈRE J., “Conception of an air-gap element for the dynamic analysis of the electromagnetic field in electric machines”, IEEE Trans. Magn., vol. 18, p. 655-659, 1982.
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[REN 88] REN Z., BOUILLAULT F., RAZEK A., VÉRITÉ J., “Comparison of different boundary integral formulations when coupled with finite elements in three dimensions”, IEE Proceedings, vol. 135, p. 501-507, 1988. [REN 94] REN Z., RAZEK A., “A Strong coupled model for analysing dynamic behaviors of non-linear electromechanical systems”, IEEE Trans. Magn., vol. 30, no. 5, p. 3252-3255, September 1994. [REN 96] REN Z., “Influence of the RHS on the convergence behavior of the curl-curl equation”, IEEE Trans. Magn., vol. 32, no. 3, p. 655-658, 1996. [RIS 01] RISCHMULLER V., KURZ S., RUCKER W., “Parallelization of coupled differential and integral methods in the framework of domain decomposition”, Proceedings of COMPUMAG, vol. IV, Evian, p. 202-203, July 2001. [ROB 79] ROBERT P., Traité d’électricité, vol. II, Matériaux pour l’Electrotechnique, Presses Polytechniques Romandes, 1979. [ROD 90] RODGER D., ALLEN N., LEONARD P., “An optimal formulation for 3D moving eddy current problems with smooth rotors”, IEEE Trans. Magn., vol. 26, no. 5, p. 23592363, September 1990. [ROS 68] ROSSER W., Classical Electromagnetism via Relativity: an Alternative Approach to Maxwell’s Equations, Butterworths, London, 1968. [SAL 85] SALON S., “The hybrid finite element – boundary element method in electromagnetics”, IEEE Trans. Magn., vol. 21, no. 5, p. 1040-1042, 1985. [TAL 89] TALON J., Génération et amélioration de maillages 2D et 3D pour éléments finis, PhD thesis, INPG, 1989. [TAN 98] TANI K., YAMADA T., KAWASE Y., “A new technique for 3D dynamic finite element analysis of electromagnetic problems with relative movement”, IEEE Trans. Magn., vol. 34, no. 5, p. 3371-3374, 1998. [TEL 83] TELLIAS-HASSON M., Résolution de l’équation de convection-diffusion et d’un modèle de circulations océaniques générales par les éléments finis, PhD thesis, Institut National Polytechnique de Grenoble, 1983. [VAS 90] VASSENT E., Contribution à la modélisation des moteurs asynchrônes par la méthode des éléments finis, PhD thesis, Institut National Polytechnique de Grenoble, 1990.
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Chapter 9
Symmetric Components and Numerical Modeling
9.1. Introduction The problems of electromagnetism often concern symmetric structures. Let us think for example of the stator and rotor of a rotating machine or of a transformer magnetic core. When the sources of excitation share some symmetries, the resolution is simplified by the application of appropriate boundary conditions. Any initial problem is then reformulated on a restricted domain which results in a substantial economy of calculations. This is the case in Figure 9.1a where a symmetric magnetic domain : with respect to a plane V is influenced by a source field Hs, also symmetric with respect to this plane. It is clear that a study restricted to the “symmetry cell” C is enough to solve the problem provided that the condition of a zero tangent field along V is imposed. However, such an approach is no longer valid whenever the source fields do not share the symmetry of the domains (Figures 9.1b and 9.1c). Nevertheless, it is still possible to take advantage of symmetries of a problem thanks to a linear decomposition similar to Fortescue’s method of symmetric components, well-known to electrotechnicians. The principle consists of reducing a given problem in a family of subproblems to be solved on a symmetry cell of the initial problem. The global solution is then obtained by superposition of the partial results and application of symmetry operations. The basis of the method is in the representation theory of finite groups [HAM 64, JAN 67, SER 71], a particularly important chapter of group theory. This branch of mathematics is encountered in many fields of physics, such as Chapter written by Jacques LOBRY, Eric NENS and Christian BROCHE.
370
The Finite Element Method for Electromagnetic Modeling
crystallography, mineralogy, quantum mechanics or organic chemistry. However, it is rarely used for numerical field calculation; despite this, let us refer to the theoretical and numerical experiments by various authors [ALL 92, BAL 82, BON 91, BOS 85, BOS 86, LOB 94, LOB 96a].
Figure 9.1. Geometric symmetries and excitation symmetries
We have no intention of giving the complete theory here. That would involve a lot of elaboration whereas the goal of this book is to present a synthesis of various aspects of finite element calculation. Instead, we state the main results by illustrating them using concrete examples. The interested reader can refer to specialized literature for further information [HAM 64]. Within the framework of this chapter, we start with a presentation of the finite group concept while focusing more particularly on symmetry groups. The study of symmetric functions will then be approached and finally lead to the orthogonal decomposition theorem, which is, within the framework of our study, the essential point of the theory. We will detail the application of this theorem to the Poisson problem, in particular its weak integral form for a numerical resolution using finite elements. Lastly, we will examine a series of applications in 2D magnetostatics and 3D magnetodynamics, with, as a support, the performances obtained compared to the traditional processing.
Symmetric Components and Numerical Modeling
371
9.2. Representation of group theory 9.2.1. Finite groups 9.2.1.1. Definitions We initially establish the concept of finite groups in a general context in order to focus thereafter only on symmetry groups. A finite group G is a finite set of elements provided with a noted law of composition “.” such that: – e G: if g G, e.g = g (existence of a neutral element); – if g G, g-1 G: g.g-1 = e (existence of an inverse); – if g and h G, g.h G (internal law and defined everywhere); – if f, g, h G, (f.g).h = f.(g.h) (associative law). If the law is (not) commutative, group G is known as (non-) commutative or (non-) abelian. The nG number of elements of group G is called the group order. NOTE.– we will omit from here on the symbol (.) of the law of composition and a product ff, for example, will be noted f2. A sub-group H of G is a part of G which also checks the properties of a group. 9.2.1.2. Symmetry groups The symmetry of an object : is described by the set of transformations or isometries g which preserve the distance between any pair of points and which bring it in coincidence with itself
if g : x l g x and g : y l g y
thus d( gx , gy ) d( x , y )
and: g := :
There are three types of fundamental transformations: – the rotation of a certain angle around an axis; – the mirror reflection with respect to a plane; – the translation of a certain step.
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The Finite Element Method for Electromagnetic Modeling
The final isometry exists only for bodies of infinite extension such as crystalline networks; this will not be further discussed here. Only rotations and reflections that generate symmetry groups of finite bodies remain. In fact, all these transformations leave at least one fixed point in which the various axes of rotation and existing planes of reflection are cut. For this reason these groups are also called punctual groups. We present below the most important groups. 9.2.1.2.1. Cyclic group Cn This group describes the rotational symmetry of an order n around an axis: Cn
^ e, r, r , r , ! , r ` 2
3
n 1
where r is the basic rotation of an angle 2S/n. The group is known as cyclic due to the obvious relation: rn+p = rp. It is abelian and its order is equal to n. The cyclic symmetry is frequent in practice (electric motor). 9.2.1.2.2. Dihedral group Dn This group is generated by a rotation of order n around an axis and n rotations s of order 2 around concurrent axes and perpendicular to the first: Dn
^ e, r, r , ! , r 2
n 1
, s, sr, sr 2, ! , sr n 1
`
In 2D, this is the symmetry of the regular polygon with n sides. The Dn group includes n rotations rk, of order k and n rotations srk, of order 2. The order of the group is 2n and Dn is non-abelian as soon as n > 2. This type of symmetry governs geometries such as the arrangement of skewed rotor slots of an induction machine. Let us note that the Cn group is a sub-group of Dn. 9.2.1.2.3. Cnv group The Cnv group is generated by rotation r of order n and n reflections around vertical planes srk passing through the axis of rotation. Cnv is isomorphous to Dn. 9.2.1.2.4. Cs group This group describes reflection s compared to a horizontal plane: Cs
^ e, s ` ,
n Cs
2
9.2.1.2.5. Ci group This group involves a polar symmetry i compared to a point. It is isomorphous to Cs.
Symmetric Components and Numerical Modeling
373
9.2.1.3. Symmetry cell In order to illustrate the various groups defined above, let us consider domain : of Figure 9.2 which illustrates non-abelian group D3.
Figure 9.2. Group D3
Isometries e, r, r2, s, sr and sr2 are respectively the unity, the two angle rotations equal to o2S/3 and three mirror reflections. D3 is non-abelian because, for example, sr z rs. The cyclic group C3 is a sub-group which describes only part of the isometries of the object (Figure 9.3). Figure 9.3 shows the geometry of the domain with the description of a symmetry cell C, sub-domain minimal from which domain : can be built by applying all the elements of group G to it.
(a)
(b)
Figure 9.3. Domain with group symmetry (a) C3 and (b) D3 – symmetry cell
Let:
:
*
g (C )
g G
and if g and h G : g ( C ) h( C ) The set of points {gx: g G} for every x of : is called orbit of x.
374
The Finite Element Method for Electromagnetic Modeling
9.2.2. Symmetric functions and irreducible representations 9.2.2.1. Actions of a group on functions Let us consider now a real or complex scalar function ȥ on a symmetric domain ȍ of group of isometries G. With each element g of G, we associate an operator Og which acts on the function ȥ according to:
Og \ (x)
.
\ ( g 1 x )
Og carries out on ȥ the same operation of symmetry as g on any point x so that the function ȥ is transformed into the function Ogȥ. To fix the ideas, let us consider a translation t of step a of a function ȥ of 1 in 1 (Figure 9.4):
Figure 9.4. Action of group Ot on a function of 1 in 1
t : x o tx
Ot \ (x)
xa
\ ( t 1 x )
\ (x a)
Operators g and Og are called actions of group G on the points and on the functions respectively. The fundamental properties of the Og operators are as follows: – g , h G , Og Oh
Ogh ,
– g G , Og 1 Og1 , – Oe
I (operator unity).
They thus form a finite group G’ isomorphous to G. Application O which associates any element g of G with an element Og of G’ and which checks the above properties is called linear representation of group G in the
Symmetric Components and Numerical Modeling
375
space of scalar functions ȥ. 9.2.2.2. Symmetric functions The group representation theory shows that there are, for a given group G, various “classes” of functions, characterized by some conditions of symmetry on domain ȍ. These classes are called the group irreducible representations. Their number k is specific to the group and they are affected by a degree equal or higher than the unity. With each representation Q (= 1, ..., k), of degree dQ, are associated dQ x dQ functions ȥij (Q) which are divided into dQ sets calledQ-symmetric:
\ (jQ)
^\ Q : i ( ) ij
1, ! , dQ
`
j 1, ! , dQ
These sets verify the conditions:
Og \ ij(Q)
dQ
¦ Dli(Q) (g) \ lj(Q)
g G
[9.1]
l 1
or, in an equivalent way and by noting the complex conjugate by *:
\ ij(Q) ( g x )
dQ
¦ Dil(Q) * (g) \ lj(Q) ( x )
g G
l 1
or also in a matrix form:
§ \1( Qj ) · ¨ ¸ ¨ . ¸ ¨ ¸ ( gx ) ¨ . ¸ ¨ \ (Q ) ¸ © dQ j ¹
§ \1( Qj ) · ¨ ¸ ¨ ¸ . D (Q ) * ( g ) ¨ ¸ ( x) ¨ . ¸ ¨ \ (Q ) ¸ © dQ j ¹
g G
In these relations, factors Dil(Q)(g) are complex numbers which determine the coefficients of square matrices D(Q)(g), of order dQ. In each irreducible representation Q there are thus as many matrices as isometries g. These matrices are tabulated and described in reference books [SER 71]. By language extension, they are also called irreducible representations. It is shown that the irreducible representations, of an abelian group, are equal in number to the order of the group and that they are all of one unity degree (k = nG, dQ = 1, Q = 1, …, k). Only the non-commutative groups present representations of higher degree. An important corollary is that only the knowledge of functions ȥij(Q) on a cell of symmetry C of domain ȍ is sufficient to build them on the whole domain.
376
The Finite Element Method for Electromagnetic Modeling
Let us illustrate these concepts using concrete examples. 9.2.2.2.1. Example 1 Initially, let us examine the case of the abelian group Cs which describes the mirror with respect to plane ı (Figure 9.1a). This group is characterized by two irreducible representations of degree 1 which we denote p and i and which are described by Table 9.1. e
V
p
1
1
i
1
-1
Table 9.1. Irreducible representations of group Cs
For any field ȍ described by this type of symmetry, there are thus classes of functions ȥ(p) and ȥ(i), associated with the representations p and i, whose symmetry conditions [9.1] are written, for g = ı (the conditions for g = e correspond to an obvious identity (unity)):
\ (p) (V x) \ (p) ( x) \ (i) (V x) \ (i) ( x) Representations p and i thus correspond to even and odd functions respectively on domain ȍ and group theory proves their uniqueness. 9.2.2.2.2. Example 2 The cyclic group abelian C3 is described by three irreducible representations which we will denote h, d and i of degree 1. Table 9.2 shows its description.
Symmetric Components and Numerical Modeling
e
r
r2
h
1
1
1
d
1
a2
A
i
1
a
a2
377
Table 9.2. Irreducible representations of group C3 (a = ej2ʌ/3)
The classes of associated functions verify the following symmetry conditions:
\ (h) (x) \ (h) (r x) \ (h) (r 2 x) , \ (d) (r x) a \ (d) (x) , \ (d) (r 2 x) a2 \ (d) (x) ,
\ (i) (r x) a2 \ (i) (x) , \ (i) (r 2 x) a \ (i) (x) . We find the zero (h), direct (d) and inverse (i) sequences usually used for the study of unbalanced three-phase systems. 9.2.2.2.3. Example 3 Let us now examine the case of the non-abelian group D3 which generates two irreducible representations of degree 1 and one of degree 2 (Table 9.3): Let ȥ(1), ȥ(2), {ȥ11(3) , ȥ21(3)} and {ȥ12(3) , ȥ22(3)} be the v-symmetric functions or couples which belong respectively to these classes. The representations of degree 1 provide the particularly simple conditions:
\ (1) (x) \ (1) (r x)
!
\ (1) (sr 2 x) ,
\ (2) (r 2 x) \ (2) (r x) \ (2) ( x) , \ (2) (sr 2 x) \ (2) (sr x) \ (2) (s x)
\ (2) ( x) ,
while the representations of degree 2 generate relations such as: (3) (3) (r x) a2 \11 (x) , \11
378
The Finite Element Method for Electromagnetic Modeling (3) (3) \11 (s x) \ 21 (x) , (3) (3) \ 22 (sr x) a2 \12 (x) .
r2
s
sr
sr2
e
r
1
1
1
1
1
1
1
2
1
1
1
-1
-1
-1
3
§1 0· ¨¨ ¸¸ ©0 1¹
§a 0 · ¨¨ ¸ 2¸ ©0 a ¹
§ a2 0 · ¸ ¨ ¨ 0 a¸ ¹ ©
§0 1· ¸¸ ¨¨ ©1 0¹
§ 0 a2 · ¸ ¨ ¨a 0 ¸ ¹ ©
§0 ¨¨ 2 ©a
a· ¸ 0 ¸¹
Table 9.3. Irreducible representations of D3 (a = ej2ʌ/3)
It will be noted that, in the case of representations of degree higher than 1, there is no function which is “symmetric by itself”. The symmetry is a collective and reciprocal feature of several indissociable functions. 9.2.3. Orthogonal decomposition of a function
The theorem of orthogonal decomposition is the main tool which allows symmetries in an electromagnetism problem to be better exploited. Following the developments seen in the previous point, it is stated as follows. THEOREM.– any real or complex function ȥ, defined on domain ȍ of symmetry group G is decomposable in symmetric functions ȥjj(Q) according to: k
\
dQ
¦ ¦ \ (jjQ)
[9.2]
Q 1 j 1
where (v = 1 with k, j = 1 with dv):
\ (jjQ)
Pjj(Q) \
dQ nG
¦ D(jjQ) (g)
Og \
[9.3]
g G
Operators Pjj(Q) are called projectors. “Partner” functions \ij(Q) (i j) which do not appear as symmetric components of function \ are necessary in symmetry relations [9.1] shown above. They are obtained from the initial function by similar relations:
Symmetric Components and Numerical Modeling
\ ij(Q)
Pij(Q) \
dQ nG
¦ Dij(Q) (g)
Og \
379
[9.4]
g G
For each irreducible representation Q, any function \ thus generates dQ2 linearly independent components \ij(Q) which is often called the Fourier component of \. The Fourier analysis of this function is thus performed. Expression [9.2] is the Fourier synthesis. This concerns the generalization of well-known techniques such as the evenodd decomposition of a function or the traditional Fortescue decomposition used in polyphase network analysis. 9.2.4. Symmetries and vector fields 9.2.4.1. Definitions In section 9.1.2, we have defined an action by group Og on the scalar functions. In fact, this concept can be generalized to vector fields [BOS 92]. This is of particular interest for us because we consider thereafter the symmetry problems with the formulation in H for the 3D eddy current problems. It is important above all to introduce the concept of tangent application. Let us consider a continuous application g of the affine Euclidean space A3 in itself: g : A 3 o A3 : x o g x
Let v be a vector of E3 – the vector space associated with A3 – placed at point x; the end point y of vector tv (t ) is defined by the “sum” (symbolic) y = x + tv. In addition, vector tv is obtained by the “difference” y - x. Let us now define the tangent application g*(x) which, for any vector v fixed in x, associates vector g*(x) v put in gx according to (Figure 9.5):
g (x) : E 3 o E 3 : v o g (x) v
lim
t o0
g (x t v) g (x) t
Figure 9.5. Definition of the tangent application g* (x)
380
The Finite Element Method for Electromagnetic Modeling
This concept enables us to widen the concept of group action of symmetry G towards the complex vector fields defined on E3. Let us call Og the operators associated with isometries of G and which transform any vector field H according to: Og H ( g x)
g ( x ) H ( x)
or
O g H ( x)
g ( g 1 x) H ( g 1 x)
The operators thus obtained are a generalization of those encountered for the scalar fields. Figure 9.6 illustrates the action of these operators in the cases of a rotation and a mirror reflection.
Figure 9.6. Group action on field H in the case of a rotation r and a mirror reflection
The theory established previously is extended easily to vector fields and the essential theorem of the orthogonal decomposition is found. The projectors of expression [9.4] are written:
Pij(Q )
dQ nG
¦ Dij(Q ) ( g )
Og
( i , j 1, ! , dQ )
g G
The Fourier synthesis of a field H takes the same form as that of expression [9.2], obtained for a scalar function: H ( x ) H (jjQ ) ( x ) Q, j
with H (jjQ ) P jj( Q ) H
[9.5]
Symmetric Components and Numerical Modeling
381
The concept of v-symmetric sets returns again which are here in the form, for fixed Q and j:
^H Q :i
H (jQ)
( ) ij
1, ! , dQ
`
j 1, ! , dQ
Components Hij(Q) verify the symmetry condition similar to [9.1]:
Og Hij(Q)
dQ
¦ Dli(Q) (g)
Hlj(Q)
g G
[9.6]
l 1
which can also be written: dQ
Hij(Q) ( g x )
¦ Dil(Q)* (g) g ( x) Hlj(Q) ( x )
g G
l 1
9.2.4.2. Orientation problems We have a coherent definition of a group action of symmetry on scalar as well as vector fields. Let us now examine the fundamental problem of the introduction of the traditional differential operators grad, curl and div. 9.2.4.2.1. Gradient operator If ȥ is a scalar function, it is shown easily that:
grad Og \ ( g x )
Og grad \ ( g x) g* (x) grad \ (x)
g G
In other words, the symmetry operators commutate with the gradient or, the gradient is invariant under the symmetry operations. Figure 9.7 illustrates this feature for a rotation r and for a mirror reflection s. Function ȥ and its gradient are represented in a symbolic form in the space \ 2 .
Figure 9.7. Gradient invariance under a group action
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The Finite Element Method for Electromagnetic Modeling
9.2.4.2.2. Curl operator Consider a field H, it comes to: curl O g H ( g x )
F 0 ( g ) . g* ( x) curl H ( x)
where F0 is a representation of degree 1 (thus irreducible) of group G such that:
F0 (g) =
+ 1 if g is a pure rotation, - 1 if g is pure mirror reflection.
The curl and symmetry operators thus commutate with a sign difference depending on whether the considered isometry g respects the orientation or not (Figure 9.8). This sign is specified by the quantity F0 (g) called the orientation character.
Figure 9.8. Rotational and group action
Let us pose J = curl H, if we consider a Q-symmetric set {Hij(Q)} resulting from field H, the curl of its components can not form another v-symmetric set. In fact, components Hij(Q) verify symmetry condition [9.6] and if the curl operator is applied to both sides, it becomes: curl H ij(Q ) ( g x )
dQ
¦F
0
( g ) Dil(Q )* ( g ) g ( x ) curl H lj(Q ) ( x )
l 1 dP
( )* il
¦ DP
( g ) g ( x ) J lj( P ) ( x )
l 1
J ij( P ) ( x )
where Jij(P) belongs to a P-symmetric set and thus to an irreducible representation P different a priori from Q (but of the same degree dP = dQ): the field symmetries Hij(Q) and Jij(P) are thus not necessarily of comparable nature because the even-odd character is modified in the presence of mirror reflection symmetries. For example, with a cyclic group Cn, made exclusively of rotations, the rotation operator does not modify the initial representation. On the other hand, if a horizontal reflection V is introduced, there is
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383
obviously a change of the type of even-odd representation. It is confirmed that, in the case of dihedral groups Dn, there is modification for the representations of degree 1 but not for those of degree 2. 9.2.4.2.3. Operator divergence Let J be a vector field, the divergence invariance is verified under symmetry operations:
div Og J ( g x )
Og div J ( g x)
One consequence is the invariance of the scalar Laplacian as 2 = div grad. This is also the same for the Laplacian vector because 2 = grad div – curl curl (double change of sign in the curl curl iteration!). 9.2.4.3. Integration on sub-varieties The integration of scalar or vector fields on sub-varieties of A3 (points i, lines Ȗ, surfaces ī, volumes ȍ) which have symmetries highlights that operators g and Og-1 (or Og-1) are adjunct for the integration. We thus obtain successively by definition from Og and from the invariance of the measurement of Lebesgue: – for the points:
³ gi I
I ( g i)
³ i Og
1
I;
– for the lines:
³ gJ
H ds
³J
Og 1 H ds ;
– for the surfaces:
³ g*
J n dī
³ * Og
1
J n dī ;
– for the volumes:
³ g:
f d:
³ * Og
1
f d: .
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The Finite Element Method for Electromagnetic Modeling
9.3. Poisson’s problem and geometric symmetries 9.3.1. Differential and integral formulations The rational exploitation of symmetries in an electromagnetism problem is based on the following principle. It is a question of substituting for the initial linear problem, defined on a space domain : and formulated for an unknown field \, a set of subproblems, similar to the first problems, but whose common domain of definition is this time a symmetry cell C. Once these problems are solved, the construction of the total solution \ is obtained by superposition thanks to symmetry operations. In order to illustrate this technique, we will consider the problem of Poisson’s equation, defined on a domain : of a symmetry group G (Figure 9.9):
2 \ f
on :
[9.7]
with the boundary conditions:
\ \0
on sȍȥ (bold lines) ,
sȥ 0 sn
on s: q
Figure 9.9. Poisson’s problem with symmetry
The term source f is an a priori unspecified function of domain :. Once the irreducible representations of group G are determined, the associated operators Pij(Q) are built. Let us apply them successively to the two members of Poisson’s equation:
Pij(Q) 2 \
Pij(Q) f ,
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385
we have seen that Og and Laplacian operators commutate. This will be the same with respect to Pij(Q). Consequently, we obtain:
2 \ ij(Q)
fij(Q)
where each component \ij(Q) is associated with source term fij(Q). For a given representation Q and a fixed value of j, it is necessary to solve a set dQ of problems coupled by constraints on 6. If we consider again the conclusions of section 9.1.3, problem [9.7] becomes equivalent to the following multiple problem (Figure 9.10):
Figure 9.10. Cell C of border wC = * 6
For Q = 1, ... k, j = 1, ... dQ , solve:
2 \ ij( Q ) fij( Q )
on C
i 1, ! , dQ
[9.8]
with the boundary conditions, on wC: \ ij( Q ) \ 0ij( Q )
s\ij( Q ) sn
0
Og \ ij(Q) ( x )
on * \ ,
on * q , dQ
¦ Dli(Q) (g) \ lj(Q) ( x )
x6 , g G : g x6
l 1
It is known that a resolution by finite elements requires a weak formulation of initial problem [9.7], hence:
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The Finite Element Method for Electromagnetic Modeling
Search \ V:
³:
grad \ grad \ c d:
f \ c d:
³:
\c Vc
[9.9]
where V = { \: \ = \0 on w:\ } and V’ = { \‘: \‘ = 0 on w:\ }. Let us remember that solution \ is the sum of symmetric components \jj(Q):
\
¦ \ jj(Q) ,
Q,j
the weak form associated with [9.8] is thus written:
³:
grad \ (jjQ) grad \ c d:
³:
f jj(Q) \ c d:
\c Vc
that we will note in a more compact form:
a : \ (jjQ) , \ c
L : f jj(Q) , \ c
\c Vc
[9.10]
If we now consider again the property of cell C:
:
* g (C) ,
g G
we can easily check that:
a: u, v
¦ a g C u, v 1
gG
¦ a C Og u , Og v
gG
If we apply this relation to the first member of [9.10], we obtain successively:
a : \ (jjQ ) , \ c
¦ a O C
g G
g
\ (jjQ ) , O g \ c
Symmetric Components and Numerical Modeling
a : \ (jjQ ) , \ c
dQ
¦ a §¨¨ ¦ D C
g G
©
dQ
§
(Q ) lj ( g )
l 1
nG dQ
©
g G
· ¸ ¹
\ lj(Q ) , O g \ c ¸
l 1
¦ a C ¨¨ \ lj(Q ) , ¦
387
· Dlj(Q ) ( g ) O g \ c ¸ ¸ ¹
dQ
¦ a C \ lj(Q ) , \ ljc (Q ) l 1
where:
\ ijc(Q )
Pij(Q ) \ c
Let us proceed similarly with the second member of L: (fjj(Q), \c), problem [9.9] is finally equivalent to the following:
For Q = 1, ... k, j = 1, ... dQ , search for the v-symmetric set { \ij(Q) 0 V(Q): i = 1,...,dQ }: dQ
dQ
l 1
l 1
¦ a C \ lj(Q ) , \ ljc (Q ) ¦ L C flj(Q ) , \ ljc (Q )
\ ljc (Q ) V c (Q )
[9.11]
V(Q) and Vc(Q) are defined in agreement with spaces V and Vc and supplemented by the condition on Ȉ of [9.8]. The set of forms [9.11] constitutes the weak forms associated with sub-problems [9.8]. 9.3.2. Numerical processing
The integral form [9.9] is the starting point of the numerical processing of the Poisson’s equation for a resolution by finite elements. It is known that it is initially necessary to proceed with a decomposition of the involved domain ȍ and to adopt an approximation for the quantities to be calculated. This technique results in the construction of a system of linear equations to be solved by a direct or iterative method. The vector-solution provides the distribution of the potential in all the interpolation nodes considered. In the case of multiple formulations as described in [9.11], it is a question of forming several systems of a reduced size compared to the initial problem. The successive resolution of these systems provides the partial solutions whose superposition leads to the effective distribution of the required fields. The size and the number of these systems are related on the order nG of the group and on the degrees dQ
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The Finite Element Method for Electromagnetic Modeling
of irreducible representations subjected to the study. If N is the number of nodes of the meshing of symmetry cell C, there is, for each irreducible representation Q, dQ systems of approximate size dQ u N where the symmetric “useful” and “partner” components are linked. The analysis in complexity of the assembly and resolution algorithms makes it possible to estimate the saving of calculation time brought by the exploitation of symmetries compared to the traditional processing. Ratios Gass and Gsol of the execution times for these two main phases are within the following limits [LOB 93] nG d Gass d
nG3 dQ3
¦ Q
and
Gsol
nGH dQH
¦
[9.12]
Q
where İ is the order of the resolution method (1.5 d İ d 3). Concerning the reduction of the memory capacity reserved for a software code with symmetry processing, it is appreciably proportional to nG. 9.4. Applications
In order to show the interest of symmetric decomposition, we apply the theory to concrete examples for which the performances in terms of gain of calculation time and memory capacity are measured. Beforehand, it is advisable to divide the problems into two categories depending on the nature of the equation system resulting from the discretization. The cases of symmetric and sparse systems which are typical of the problems solved by finite elements are today usually solved by iterative conjugate gradient techniques. The first example suggested is of this type. As it will be seen, the group theory breaks down these problems in a particular way. In the case of finite element – boundary element couplings, the systems are made of a non-symmetric composite structure (sparse block – dense block). We use the Gaussian technique while taking advantage of the sparse character of the matrix for the resolution even though iterative techniques are also applied to this type of system. The second example will illustrate this category of problems. 9.4.1. 2D magnetostatics
9.4.1.1. Resolution methods The system matrix resulting from a discretization by finite elements admits a variety of characteristics which condition the choice of resolution methods as well as their computer implementation: memory capacity, disk space, calculation time and accuracy.
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The iterative methods allowing the resolution of large, sparse, linear systems are gaining more and more popularity among the international scientific community. Until recently, direct resolution methods were preferred because of their robustness and their predictive behavior. However, the recent discoveries regarding iterative methods and the growing need for algorithms dedicated to the resolution of very large systems have caused a fast change of mentalities in favor of iterative techniques. In the field of the numerical calculation of electromagnetic fields, the conjugate gradient method will be naturally imposed for the resolution of positive definite, symmetric, sparse, linear systems. However, in the case of static problems, the use of symmetries leads to Hermitian matrices. In fact, the Q-symmetry conditions introduce, in the calculation of the system elements A x = b, the generally complex representation coefficients. However, an adequate ordering of the unknown variables allows a complex symmetry character of the finite element matrix to be preserved. The ICCG (incomplete Cholesky conjugate gradient) method is unfortunately not appropriate in the case of Hermitian matrices. However, a new algorithm can be derived from GMRES (generalized minimum residual) methods. It concerns the method of conjugate residues for which the successive residues vectors are A-orthogonal: rk
b Ax k
r k 1, Ar i
0
i [ 0, k ]
This algorithm requires memory storage of an additional vector and 2n more operations than the conjugate gradient algorithm, for an appreciably identical convergence. The conjugate residue method will be associated with a pre-conditioning technique adapted to the Hermitian matrices so as to accelerate the convergence properties. The chosen pre-conditioning is based on the Cholesky factorization LDLH (the superscript H indicates the conjugate transpose of matrix L). The problem occurs due to the fact that matrix L is not necessarily sparse even if matrix A is sparse. The incomplete Cholesky decomposition, denoted IC, thus consists of calculating elements lij of matrix L only at the places where the coefficients of matrix A are non-zero. This option benefits from the memory storage structure of the initial sparse matrix. If M is the preconditioning matrix, then the algorithm of the pre-conditioned conjugate residues (ICCR) is written:
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The Finite Element Method for Electromagnetic Modeling
1. Initialization: x 0 2. Calculate: r 0 b A x 0 , ~ r 0 M 1 r 0 , p 0 3. For k = 0, 1, 2, ..., until convergence, do: 4. Dk ~ r k, A ~ rk Apk , ~ pk 5.
x k 1
6.
r k 1 r k D k A p k ~ r k 1 ~ r k Dk ~ pk ~ Ek r k 1, A ~ r k 1
7. 8.
~ r 0 and ~ p0
M 1 A p 0
xk D k pk
~ r k,A ~ rk
~ r k 1 E k p k A p k 1 A ~ r k 1 E k A p k p k 1
9. 10.
~ p k 1 11. 12. End do
M 1 A p k 1
In addition, matrix A has a large proportion of zero elements because of the local nature of the nodal equations. This aspect benefits the storage by compact CSR (compressed sparse row) structuring and the calculation speed by taking into account only the non-zero coefficients. Only the non-zero elements are stored in memory in the form of a vector. 9.4.1.2. Numerical example We consider the application of the decomposition technique in symmetric components to the magnetostatic analysis of an induction motor with two poles and wound rotor. We present the savings in calculation time and memory capacity to which the rational exploitation of symmetries leads. The rotor presents 48 notches and the stator is assumed to be smooth (Figure 9.11). Study domain : consists of the air-gap and the rotor winding conductors of permeability P0 (:1), the stator and the rotor of relative permeability Pr = 1,000 (:2). It is supposed that only one phase is excited with a constant current density Js. The study of the magnetostatic equations leads to Poisson’s equation: 2 A
P J s
where A is the magnetic vector potential which is reduced to its component directed along the z axis in the case of this two-dimensional problem.
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391
Figure 9.11. Induction motor with wound rotor
Study domain : thus consists of two homogenous areas :1 and :2 of permeability P1 and P2 respectively. The continuity conditions of the potential vector and its normal derivative to the interface of the two mediums are written: A1 A 2
1 wA
P1 wn
1
1 wA P 2 wn
2
In addition, the excitation has an obvious symmetry with respect to the rotor so that Poisson’s problem can be solved on only one quarter of the geometry. The Neumann condition wA/wn = 0 is naturally introduced along the antisymmetry lines where the magnetic field is normal at every point. In addition, the Dirichlet condition A = constant will be applied on the external limits of the stator and on the lines of symmetry along which the magnetic field is tangential. The induction motor is characterized by 48 rotation isometries and 48 mirror reflection isometries. It is thus characterized by the dihedral non-abelian group D48 of order 96. This symmetry group presents 27 irreducible representations: 4 of order 1 and 23 of order 2. However, in the particular case of our study, the sources also present a certain character of symmetry so that only 12 representations of order 2 will be excited. This situation leads to the resolution of 24 sub-problems coupled by the Q-symmetry conditions on interface 6 of the symmetry cell (Figure 9.12). In addition, it can be (Q ) (Q ) shown that components A11 and A22 of the potential vector are conjugate complex in the case of a unitary representation. Therefore, there only remain 12 sub-problems defined on the symmetry cell of the domain.
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The Finite Element Method for Electromagnetic Modeling
Figure 9.12. Symmetry cell
The solutions obtained using a traditional approach and the rational use of symmetries are rigorously identical. Figure 9.13 illustrates the lines of A constant.
Figure 9.13. Flux lines on a quarter of the motor geometry
9.4.1.3. Gain in memory space The rational use of symmetries requires only the discretization of the symmetry cell. It can be easily understood that if nG is the order of the symmetry group characterizing the geometry of the field of study, then the cell will be nG times smaller than the complete field. All calculations are carried out on this cell only. The results are then extended to the whole field by application of Q-symmetry and the group operators. If it is considered that the majority of the memory capacity necessary for the resolution of Poisson’s equation is occupied by the matrix resulting from the discretization by the finite element method, the theoretical gain will be proportional to nG/(max dQ)2. 9.4.1.4. Gain in calculation time Symmetry processing decreases the calculation times significantly. We will assume again that the numerical resolution of the matrix system resulting from the discretization by finite elements concentrates the majority of CPU resources. If n is the dimension of the global system relating to the complete structure, the time necessary for the resolution is given by: T
O ( nD )
(1.5 d D d 3)
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393
where D is the complexity of the algorithm used. In the abelian case, the use of symmetries substitutes for the initial dimension problem n, nG sub-problems of common dimension n/nG. The calculation time T is reduced to Tsym: Tsym
ª § n O «nG ¨¨ « © nG ¬
Dº
· ¸¸ ¹
» » ¼
T D 1
nG
so that the theoretical gain will be proportional to nGD 1 . However, in the case of non-abelian groups, it is shown that there are generally approximately nG/4 coupled sub-problems of size dQ n/nG. For the dihedral groups, the gain will thus be given: D 1
§n · 2¨ G¸ © 2 ¹
Time (s)
Saving CPU time can thus be very significant according to the values taken by nG and D. The gain will be considerable in the case of a direct Gauss or Cholesky resolution method whose complexity is in n3. However, in the case of an iterative conjugate gradient resolution method whose complexity is in n log (n), the gain will be about nG0.5 in the case of the cyclic groups and about (2 nG0.5) in the case of the dihedral groups. Figure 9.14 compares the calculation times obtained by a traditional approach on a quarter of the geometry and by group theory. The asymptotic gain is 3.46 as only 12 of the 27 irreducible representations are excited. 100 90 80 70 60 50 40 30 20 10 0
FEM SYM D48
0
5000
10000
15000
20000
25000
30000
35000
Number of nodes
Figure 9.14. Comparison of the two methods
When the number of nodes reaches 20,000 units on a quarter of the geometry, the calculation times are considerably increased in the case of a traditional approach. This situation is due to the fact that, considering the size of the data, many swaps between
394
The Finite Element Method for Electromagnetic Modeling
the hard disk and the RAM are made necessary. Other tests were also carried out in the case of an absolutely unspecified excitation. The numerical results thus show a gain of 14 in agreement with the asymptotic gain of 13.86.
9.4.2. 3D magnetodynamics 9.4.2.1. Formulation Let us consider a symmetric conductive domain : plunged into space E as indicated on Figure 9.15. A winding excitation :s carries a known current density Js. The mediums involved are assumed to be linear. The so-called H formulation of the magnetodynamics is adopted [BOS 85] in which the unknown variables are the magnetic field H in : and a reduced potential ) in the air :c. In harmonic mode, the problem is formulated as follows: find H and M = ) on wȍ: H
H s grad )
³ U :
in :c
curl H . curl H c j Z P H . H c d : j Z P0
Hc, )c : Hc
v³ H w:
sn
I I c d *
grad )c in :c , M c )c on w:
0
[9.13]
where Hs is the excitation field which would exist in the absence of domain :. is an operator which links M to w)/wn. In order to solve problem [9.13], we adopt a mixed numerical scheme FEM-BEM [BOS 82]. The degrees of freedom are circulations he of H along the internal edges (edge elements) of a tetrahedral meshing of : and nodal values Mn of M on border w: (boundary elements).
Figure 9.15. General 3D magnetodynamics problem with symmetry
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395
Let us denote the symmetry group of : as G and symmetry cells of : and w: as C and *, respectively. If the orthogonal decomposition is applied, initial problem [9.13] is replaced by the family of the following sub-problems:
Q = 1,... k, j = 1, ... dQ , find the v-symmetric sets { ( Hlj(Q), M lj(Q) ): l = 1,...,dQ } such that:
dQ
¦ ³ U l 1
C
curl H lj(Q ) . curl H ljc(Q ) j Z P H lj(Q ) . Hclj(Q ) d : j Z P0
v³ H *
(Q ) sn lj
(Q ) lj
(Q ) lj
I
Ic lj
(Q )
d*
[9.14]
0
with the following constraints on internal border 6 of C:
O g H lj(Q )
dQ
¦
Dil(Q ) ( g ) H ij(Q ) and Og M lj(Q )
i 1
dQ
¦ Dil(Q ) ( g ) M ij(Q )
g G
i 1
[9.15] Test function H’lj(Q) and M lj(Q) and source term H s lj(Q) are obtained by application of adequate vector and scalar projectors respectively on fields H’, M’ and Hs. Effective fields H and M are finally obtained by superposition of partial fields. 9.4.2.2. Example 1: groups D2h and D4h The first example consists of a copper cube : in the vicinity of which passes an unlimited rectilinear thread-like conductor carrying a sinusoidal current Is, laid out so that source field Hs does not share any symmetry with the cube (Figure 9.16).
Figure 9.16. Proposed example
We will not exploit all the symmetry of the cube because of its complexity: the associated group, denoted Oh, comprises some 48 isometries and is characterized by
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The Finite Element Method for Electromagnetic Modeling
representations of degree 1, 2 and 3. We will limit ourselves to the sub-groups D2h and D4h, which are simpler and more normal than the cube group. Group D2h is described by Figure 9.17 where a stereogram and the description of a symmetry cell C of domain : are showed. It comprises 3 axes r, r’, r’’ of order 2, mutually perpendicular and a symmetry plane ı, merged with the X-Y plane. The group D2h is abelian so that it presents 8 irreducible representations of degree 1, shown again in Table 9.4. If we adopt group D2h, the initial problem, formulated on the domain :, is replaced by 8 similar sub-problems to be solved on the cell of symmetry C.
D2h = { e, r, r’, r’’, V, Vr, Vr’, Vr’’ } Figure 9.17. Symmetry elements and stereogram of group D2h
1 2 3 4 5 6 7 8
e 1 1 1 1 1 1 1 1
r 1 -1 1 -1 1 -1 1 -1
r’ 1 1 -1 -1 1 1 -1 -1
r’’ 1 -1 -1 1 1 -1 -1 1
V 1 1 1 1 -1 -1 -1 -1
Vr 1 -1 1 -1 -1 1 -1 1
Vr’ 1 1 -1 -1 -1 -1 1 1
Vr’’ 1 -1 -1 1 -1 1 1 -1
Table 9.4. Irreducible representations of D2h
Group D4h is non-abelian, it comprises 16 elements and has 10 irreducible representations (Table 9.5): 8 of degree 1 and 2 of degree 2. Figure 9.18 shows the symmetry elements by a stereogram as well as a symmetry cell of the studied domain.
Table 10.5. Irreducible representations of D4h
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397
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The Finite Element Method for Electromagnetic Modeling
We now substitute 12 partial problems for the original problem: 8 are associated with the representations of degree 1 and 4 are obtained from the two representations of degree 2. Let us remember that the last problems present coupling constraints and are approximately double the size of the first problems.
D4h = {e, r, r2, r3, s, sr, sr2, sr3, V, Vr, Vr2, Vr3} Figure 9.18. Symmetry elements and stereogram of group D4h
9.4.2.2.1. Numerical processing In order to obtain experimental values of the savings in terms of calculation time and memory capacity, we compare the performances of three algorithms. The first performs a traditional resolution, without any use of symmetry (coarse symmetry C1). The choice of the source makes any simplification by an intuitive reasoning impossible here and the problem must be treated on the complete domain, broken up into nt tetrahedrons. The two others relate to the groups D2h and D4h, the meshing of the associated symmetry cell preserves the tetrahedron density. Thus, we choose nt/8 tetrahedrons for the abelian case and nt/16 for the non-abelian case. It is interesting to compare the traditional resolution with the processes which roughly use the symmetries of the problem. In particular, is it advantageous to consider the case of non-abelian group D4h with its sub-problems of order 2, with respect to the abelian group D2h which leads to simpler problems although all of higher size? Let us recall that the Gaussian method with exploitation of the sparse character of the matrix was applied here for the resolution of the various assembled equation systems. For this purpose, we use a version of the software code MA28, from the Sparse Matrix Library of the NAG. This program, based on Gaussian elimination, is designed for solving sparse systems with real elements (double precision). Since our problem deals with complex equations, the source code was consequently modified. As an illustration, the real part of current density J relating to the process without symmetry of the cube problem is presented in Figure 9.19. Figures 9.20 and 9.21 show the real part of J for representations 2 and 7 of group D2h. We leave it to the reader to verify the conditions of proper symmetries in these particular cases.
Symmetric Components and Numerical Modeling
Figure 9.19. Current density, real part – global solution
Figure 9.20. Current density, real part Re (J(2))
399
400
The Finite Element Method for Electromagnetic Modeling
Figure 9.21. Current density, real part Re (J(7))
9.4.2.2.2. Measure of gains in terms of CPU time and memory The calculation times obtained were measured and the resulting gains are reported in the graph of Figure 9.22. The orders of magnitude are in agreement with the theoretical asymptotic values. Thus, orders of magnitude of nG2, i.e. 64, for the abelian case and nG3/3dv3= 170 in the non-abelian case are found [LOB 93]. Regarding the gain in terms of memory capacity, 2.8 for the case of D2h (nG/2 = 4) and 6.2 for the case of D4h (nG/2 = 8) are obtained. 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0
C1/ D2h
1000
1500
2000
2500
3000
3500
4000
C1D4h
4500
5000
5500
6000
Figure 9.22. Algorithms C1, D2h, D4h – comparison of global gains
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401
We check, through these results, the interest brought by the use of symmetries. Furthermore, the processing of a non-abelian case, although of a more complicated implementation with respect to the abelian case located before it in terms of order, is worth implementing. 9.4.2.3. Example 2: groups D28h The second example deals with the study of an induction motor with slitted solid iron rotor. It concerns a machine of 1 MW intended for high speed drives. The rotor currents are distributed here throughout the entire mass of the rotor so that the traditional machine theory does not apply directly. Figure 9.23 presents a simplified geometry of the motor. Rotor ȍ is assumed to be linear and homogenous in order to preserve material symmetry. The central part has 28 symmetric notches. The stator is of a traditional construction. It has 36 slots intended to accommodate a three-phase winding with two poles. The nominal voltage is of 1,600 V with a frequency of 250 Hz (nominal speed: 15,000 tr/min).
Figure 9.23. Induction motor with slitted solid iron rotor
Figure 9.24. Elements of symmetry of the dihedral group D28h
The symmetry of the rotor is described by non-abelian group D28h (order = 112) as illustrated in Figure 9.24. Table 9.6 shows their irreducible representations.
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The Finite Element Method for Electromagnetic Modeling e
rk
s
srk
V
Vrk
Vrk
1
1
1
1
1
1
1
1
2
1
1
-1
-1
1
1
k
-1
(-1)
-1
(-1)k+1
3
1
(-1)
4
1
(-1)k
5 to 17
§1 0· ¨¨ ¸¸ ©0 1¹
(h=Q-4)
§ a hk ¨ ¨ 0 ©
0 ·¸ a hk ¸¹
18
1
1
19
1
1
1
-1
-1
-1
(-1)k+1
21
1
(-1)k
(h=Q-21)
1
(-1)
(-1)
§ a hk ¨ ¨ 0 ©
a hk ·¸ 0 ¸¹
1
1
§1 0· ¨¨ ¸¸ ©0 1¹
§ 0 ¨ ¨ a hk ©
k
20
22 to 34
§0 1· ¨¨ ¸¸ © 1 0¹
k
0 ·¸ a hk ¸¹
§0 1· ¨¨ ¸¸ © 1 0¹
§ 0 ¨ ¨ a hk ©
a hk ·¸ 0 ¸¹
1
(-1)
1
(-1)k
§1 0· ¨¨ ¸¸ ©0 1¹
k
-1 k
§ a hk ¨ ¨ 0 ©
0 ·¸ a hk ¸¹
(-1)k (-1)k+1 § 0 ¨ ¨ a hk ©
a hk ·¸ 0 ¸¹
-1
-1
1
-1
-1
-1
-1
k+1
(-1)
(-1)k+1
-1
(-1)k+1
(-1)k+1
§ 1 0 · ¨¨ ¸¸ © 0 1¹
§ a hk ¨ ¨ 0 ©
0 ·¸ a hk ¸¹
§ 0 ¨ ¨ a hk ©
a hk ·¸ 0 ¸¹
Table 9.6. Irreducible representations of D28h (a=e j2S/28, 1 d k d 27)
In the case of a resolution with complete use of symmetries, there are ȈdQ = 60 subproblems to be solved on a symmetry cell of the motor (half of a half slit). However, the excitation of the machine is characterized by a partial symmetry compared to the machine, which implies that some of its symmetric components are zero. In fact, in this case, the stator currents form a positive balanced system of magnetomotive force F equal to: F
§ 3 ¨ ¨ 2 ¨ ¨ ©
¦ IM
D 6 k 1 t0
D
e
jD T
¦ IM
D 6 k 1 t0
D
e
jD T
· ¸ ¸ ¸¸ ¹
[9.16]
where ș is the angular position and IMĮ is a real factor which depends on the distribution of the stator winding. The first sum in expression [9.16] groups the fundamental term and the harmonics of the space which rotates at an angular velocity Z/D and a direct direction (Z is the stator pulsation) while the second sum relates to the negative sequence harmonics. If we break up F into its symmetric components with respect to group D28h, it appears that only 7 irreducible representations of degree 2 are non-zero (odd index Q from 5 to 17) so that there are 14 components F jj(Q) to be calculated. The 7 components F11(Q) relate to harmonics -h+28m and components F 22(Q) relate to harmonics h+28m where h is again shown in Table 9.6 and m is a positive or negative integer. The real part of the current density component J is plotted in Figure 9.25 for a slip s of 0.1%.
Symmetric Components and Numerical Modeling
403
Figure 9.25. Distribution of the current density (g = 0.1%) – real part
The traditional process would be limited to the use of the excitation symmetry, which relates to a quarter of the motor, i.e. geometry 28 times larger than in the case presented here. However, only one large system is to be built. Nevertheless, a gain of 400 in terms of CPU time is estimated by our method in the case of a Gaussian resolution. 9.5. Conclusions and future work We have presented an effective method for the numerical resolution of electromagnetism problems with a symmetric geometry, subjected to an unspecified constraint. It concerns a decomposition technique which consists of replacing an initial problem, a priori not geometrically reducible, by a family of sub-problems defined on a symmetry cell of the domain concerned. Group theory is the mathematical basis of the method. Within the framework of this book, we have presented a simplified approach in order to better highlight the practical application of the process. Our statement was illustrated by the numerical analysis of some magnetostatics and magnetodynamics problems. The interest of the decomposition method is justified by the gains obtained in terms of calculation times and memory capacity. Future research concerns, in particular: – the processing of nonlinearities, a technique has been developed which allows the problem to be bypassed and the first results are encouraging [LOB 96b]; – problems with partial symmetries, as in the case of studies of motors where the stator and rotor are generally of different symmetry groups.
404
The Finite Element Method for Electromagnetic Modeling
9.6. References [ALL 92] ALLGOWER E.L., BÖHMER K., GEORG K., MIRANDA R., “Exploiting symmetry in boundary element methods”, SIAM J. Numer. Anal., vol. 29, no. 2, 1992, pp. 534-552. [BAL 82] BALLISTI R., HAFNER C., LEUCHTMANN P., “Application of the representation theory of finite groups to field computation problems with symmetric boundaries”, IEEE Transactions on Magnetics, vol. 18, no. 2, 1982, pp. 584-587. [BON 91] BONNET M., “On the use of geometrical symmetry in the boundary element methods for 3D elasticity”, BETECH 91, pp. 185-200. [BOS 82] BOSSAVIT A., “A mixed FEM-BIEM method to solve 3-D eddy-current problems”, IEEE Transactions on Magnetics, vol. 18, no. 2, 1982, pp. 431-435. [BOS 85] BOSSAVIT A., “The exploitation of geometrical symmetry in 3-D eddy-current computation”, IEEE Transactions on Magnetics, vol. 21, no. 6, 1985, pp. 2307-2309. [BOS 86] BOSSAVIT A., “Symmetry, groups, and boundary value problems. A progressive introduction to noncommutative harmonic analysis of partial differential equations in domains with geometrical symmetry”, Comp. Meth. in Appl. Mech. and Engng., vol. 56, 1986, pp. 167-215. [BOS 92] BOSSAVIT A., “Symmetry in workshop problems”, Proceedings of the 3rd Int. Team Workshop, 1992, pp. 265-272. [HAM 64] HAMERMESH M., Group Theory, Addison-Wesley, 1964. [JAN 67] JANSEN L., BOON M., Theory of Finite Groups – Applications in Physics, NorthHolland, 1967. [LOB 93] LOBRY J., Symétrie et éléments de Whitney en magnétodynamique tridimensionnelle, PhD Thesis, Faculté Polytechnique de Mons, 1993. [LOB 94] LOBRY J., BROCHE C., “Exploitation of the geometrical symmetry in the boundary element method with the group representation theory”, IEEE Transactions on Magnetics, vol. 30, no. 1, 1994, pp. 118-123. [LOB 96a] LOBRY J., “Use of group theory in symmetric 3-D eddy-current problems”, IEE Proc.-Sci. Meas. Technol., vol. 143, no. 6, 1996, pp. 369-376. [LOB 96b] LOBRY J., BROCHE C., TRÉCAT J., “Symmetry and TLM method in nonlinear magnetostatics”, IEEE Transactions on Magnetics, vol. 32, no. 3, 1996, pp. 702-705. [SER 71] SERRE J.-P., Représentations linéaires des groupes finis, Hermann, 1971.
Chapter 10
Magneto-thermal Coupling
10.1. Introduction This chapter tackles magneto-thermal coupling problems and presents formulations, resolution methods, as well as the applications relating to induction heating (IH) of metallic or fluid (plasma) mediums. IH is produced thanks to currents induced in a driver subjected to a time varying magnetic induction field. In addition, the interaction between the eddy currents and the induction field produces electromagnetic forces which act on the material. Thus, in the case of a load made up of conductive fluid (plasma), the effects generated by such forces result in gas displacement. The expression of partial differential equations (PDE) describing the physical phenomena (electromagnetic, thermal, etc.) having a role in IH systems, is obtained from fundamental physics equations and material properties. In the case of electromagnetism, these are Maxwell’s equations and electric (conductivity, etc.) and magnetic (permeability, etc.) material characteristics. In the case of thermal phenomena, they are the laws of thermodynamics and the thermal properties (thermal conductivity, specific heat, etc.) of materials. When the load consists of plasma, the flow phenomena must be taken into account using the laws of fluid mechanics. The solution to these equations allows the field distribution to be obtained. Knowledge of these fields leads to, in addition to understanding the precise behavior Chapter written by Mouloud FÉLIACHI and Javad FOULADGAR.
406
The Finite Element Method for Electromagnetic Modeling
of the system (overheating point, zone of saturation, etc.), an accurate evaluation of the global variables (resistance, inductance, power, force, etc.). This resolution requires the use of numerical methods able to take into account the geometric complexity (system of sheet heating, system with cold crucible, etc.) as well as the nonlinearity of the PDE (electromagnetic, thermal, mechanical) and of their coupling (electric quantities dependent on temperature, etc.). The main methods generally used in solving the PDE which describe the fields in the nonlinear mediums are the finite difference method (FDM), the finite volume method (FVM) and the finite element method (FEM). The FEM is better adapted to the processing of systems presenting complex geometries. In mediums of simple geometry and with linear behavior, the equations can be solved using analytical methods (AM). More generally, the boundary integral method (BIM) applies to linear mediums able to have complex forms. In order to achieve economic and sufficiently accurate modeling, the coupling of various methods (FEM-AM, FEMBIM, etc.) can be required. In the following sections, we will initially present the magneto-thermal phenomena and their corresponding descriptions. Then, we will tackle the problems of behavior laws, and the formulation and resolution of the PDE. The final part of this chapter will be devoted to modeling two IH systems, i.e. the heating of a work moving piece and the production of an induction plasma. 10.2. Magneto-thermal phenomena and fundamental equations 10.2.1. Electromagnetism The laws of electromagnetism, allowing the eddy currents to be calculated, are related to magnetodynamics and are obtained from Maxwell’s equations (Ampere theorem, Faraday law, etc.) and the constitutive relationships of the medium (magnetization characteristic, Ohm’s law, etc.). From these laws, various formulations leading to the expression of PDE can be considered (field, potential, etc.) [LAV 93]. Among the systems which can be studied using 2D formulations, two main cases can be considered: – systems whose geometry is invariant by translation (along axis z); – axisymmetric systems whose geometry is invariant by rotation around axis Oz. In the first case, the currents circulate according to the longitudinal direction (Oz) and in the second case the currents only have components according to the
Magneto-thermal Coupling
407
orthoradial direction (T). In these cases, the magnetic vector potential has the same direction as the current. The PDE which describe the electromagnetic phenomena of systems presented above are given by the following expressions: – 2D Cartesian: V
wA wt
w wx
(Q
wA wx
)
w wy
(Q
wA wy
)
J
[10.1]
– axisymmetric case (2D cylindrical coordinates): V
wA wt
w
(
Q wrA
wr r wr
)
w
(
Q wrA
wz r wz
)
J
[10.2]
In these equations, J is the excitation current density, V represents electric conductivity and Q is the magnetic reluctivity. When the behavior of the system is linear (no magnetic saturation) and its supply is carried out by a sinusoidal source, a complex vectorial representation of electromagnetic fields can be used. In fact, the electromagnetic phenomena change quickly compared to the thermal phenomena which are characterized by much slower dynamics. Thus, for coupling with thermal problems, the study of the electromagnetic phenomena is undertaken in steady state conditions. In this case, equation [10.1] becomes: jZVA
w wx
(Q
wA wx
)
w wy
(Q
wA wy
)
J
[10.3]
In the nonlinear case, the formulation described by equation [10.3] remains valid when an equivalent dynamic reluctivity is determined [HAD 84]. Note 10.1. The velocity term does not appear in equation [10.3]. In fact, the moving velocity of the heated parts is generally sufficiently low, allowing us to neglect the currents induced by the movement compared to the currents due to the temporal variation of the inductor field. Note 10.2. In order to adapt the generator to the system, the source term of equation [10.3] must be expressed according to the supply voltage. Thus, the impedance of the system can be calculated [FEL 95], [FEL 96].
408
The Finite Element Method for Electromagnetic Modeling
10.2.2. Thermal The conservation equation which expresses the first thermodynamic principle is:
³ Mds ³ pdv ³ J
DT dv Dt
DT
wT
wT
Dt
wt
[10.4]
with vx
wT wx
vy
[10.5]
wy
The first term represents the power provided to the equipment involved. The second term is the power generated by the internal sources. The second part expresses the instantaneous variation of the internal power (temporal variation of temperature and displacement of the matter at the speed v). In these relations, T is the temperature, M and p are respectively surface and volume densities of power. b is the specific heat and v is the displacement speed of the matter. Thus, the PDE which governs the temperature in the 2D Cartesian case can be written as: UCp
DT Dt
w wx
(O
wT wx
)
w wy
(O
wT wy
)
p
[10.6]
In this equation, U is the density, Cp is the calorific capacity and O is thermal conductivity. The second part of [10.6] represents the average electric power density during a time period, and constitutes the principal element of magneto-thermal coupling. When the material consists of plasma, the coupling with the equation of fluid mechanics is achieved through the velocity term. The exchange conditions (convective, radiative, etc.) with the outside should be associated with the thermal PDE [FOU 91]. 10.2.3. Flow The PDE which govern the flow of the fluids (plasma) are obtained from the conservation laws (momentum, mass). The solution to such equations, which requires the knowledge of the magnetic force densities and those of the physical
Magneto-thermal Coupling
409
properties (density, viscosity), allows us to determine the velocity of the gas flow as well as the pressure gradient to which the fluid is subjected [BOU 76], [MEK 93]. 10.3. Behavior laws and couplings 10.3.1. Electromagnetic phenomena In the case of ferromagnetic steel, the electrical resistivity varies with the temperature (Figure 10.4) and the material permeability depends on the magnetic field (saturation) and temperature (Figure 10.5). Thus, the electromagnetic equation is nonlinear and the electromagnetic phenomena are coupled with thermal phenomena. 10.3.2. Thermal phenomena The nonlinearity of the PDE is due to the fact that the physical properties (specific heat, thermal conductivity) depend on the temperature (Figures 10.6 and 10.7). These thermal phenomena are coupled with electromagnetism through the heating sources consisting of electromagnetic powers. In the case of heating fluids (plasma), thermal phenomena are coupled with mechanical phenomena through the flow velocity. 10.3.3. Flow phenomena The mechanical phenomenon of plasma flow is coupled with electromagnetism by the magnetic forces. Its coupling with thermal phenomena is due to the fact that the gas properties (viscosity, etc.) depend on temperature [BOU 76], [MEK 93]. 10.4. Resolution methods 10.4.1. Numerical methods The resolution of the electromagnetic PDEs can be carried out by using the FDM, the FVM or the FEM in ferromagnetic mediums (part to be heated, magnetic circuit, etc.) [FEL 91] and the SIC (surface impedance condition) [AYM 97] or BIM [KRA 97] in the linear areas (the inductor, air, etc.). The solution to the thermal equation which describes the temperature field in a nonlinear medium (part to be heated) requires the use of the FDM, FEM or FVM [MET 96].
410
The Finite Element Method for Electromagnetic Modeling
Knowing the field distribution of magnetic vector potential A, the magnetothermal coupling terms can be determined, i.e. induced current Ji and power densities p. Ji p
[10.7]
jZVA 1 2
2
VZ AA
[10.8]
In addition, it is possible to determine the electromechanical coupling terms by calculating the average value of the force density due to the interaction between the induced currents in the gas (plasma) and magnetic induction B. Fx
1 2
Re( Ji . B y )
Fy
1 2
Re( Ji . B x )
[10.9]
The use of numerical methods thus leads to the field distribution (electromagnetic: vector potential, etc., thermics: temperature, mechanics: flow velocity) [BOU 76], [MEK 93]. 10.4.2. Semi-analytical methods The integral method makes it possible to provide a PDE solution using a Biot and Savart law expression [LED 84], [MAO 97]. When applied to the study of induction systems with axisymmetric structure, this method (CCM: coupled circuits method) allows the impedance of the system to be calculated, which is an important feature regarding the adaptation of impedance with the supply generator. In this case, we associate with the integral form of the solution a subdivision of the solenoidal inductor in basic turns. By applying Kirchoff laws to these basic circuits, we obtain an algebraic system whose solution leads to the distribution of the current densities [MAO 97]. In the case of significant skin effect, only the surface of the turn is discretized in basic circuits [LED 84]. The CCM, which discretizes only the active parts (inductor, non-magnetic load, etc.), can be advantageously coupled with a numerical method representing nonlinear mediums (magnetic load). Moreover, the CCM allows easy simulation of the systems with moving parts [MAO 98].
Magneto-thermal Coupling
411
10.4.3. Analytical-numerical methods The main analytical resolution method for linear PDE of a Laplacian equation is the method of separation of variables (MSV). It applies to simple geometries. This method is useful when it is coupled with numerical methods (finite elements). This coupling of analytical (MSV, etc.) and numerical (MEF, etc.) solutions relates to two main cases. 10.4.3.1. Systems with moving parts Here we consider the induction heating of a short work piece in displacement under the inductor [MOH 98]. In this case, the mesh of the zone situated between the inductor and the load becomes deformed during the movement. In order to avoid meshing the zone for each load position again with respect to the inductor, coupling techniques of analytical resolutions and finite elements can be used. We can thus calculate an analytical solution in the “movement zone” coupled with a finite element resolution of the inductor and load zones. The coupling is implemented by imposing the tangential component continuity of the magnetic field between the zones [MOH 98]. 10.4.3.2. Systems with significant skin effect This is for example the case of the cold crucible of plasma torches [LOU 96]. In this case, we use the surface impedance technique and its coupling with the finite elements. In the case of complex geometries, it is preferred to use modified surface impedance (superposition of two 1D solutions) [AYM 97]. In the nonlinear case (magnetic saturation, thermal effect), variations of permeability should be taken into account. This requires the definition of nonlinear surface impedance when possibly taking the temperature into account [AZZ 01]. 10.4.4. Magneto-thermal coupling models Three main modes of resolving of the coupled problems can be used: the model of alternating coupling (MAC) [MAS 85], [PAN 89], [FEL 92], [OUL 99], [PAN 00], the model of direct coupling (MDC) [FEL 91] and the model of parameterized coupling (MPC) [MIE 97]. In the first case (Figure 10.1a), the PDEs are solved separately and the coupling is carried out by data transfer from one problem to another. This calculation requires the intervention of the user or an executive routine for data transfer by tabulation of files. However, this can generate numerical errors if we do not take the precaution of using the same mesh structure for the two problems and avoiding the interpolation of the data. A “prediction-correction” method automatically allows the
412
The Finite Element Method for Electromagnetic Modeling
time step of the transient problems to be determined. It should be noted that this procedure can be burdensome and can have convergence problems [MAS 85]. Initialization Initialization
Initialization Initialization Initialization Initialization
Solving the electro magnetic Solving the electromagnetic
Solution of electro magnetic
equation equation
equation
Calculate power density Calculate power density
for a range of de tempera tempera tures ture s
Solving thermal Solving thermal equation equation
Simultaneous solution of Computing power Computing powerdensity densityfunction function electro magnetic and thermal
(adp) (adp)
Updating pro theperties pro prieties P 7 V 7
equations
Solving thermal Solving thermal
equation equation
no Converg ence ? (b ) (c ) End yes
End
(a ) End
Figure 10.1. Coupling algorithms (a): MAC, (b): MDC, (c): MPC
The second mode of coupling, which uses the MDC (Figure 10.1b), consists of solving the equations simultaneously, this enables us to carry out accurate simulations that are simple to implement. The MDC can advantageously be used in the case of strongly coupled problems. However, the number of iterations is more significant than in the case of using MAC. In addition, the matrix of the algebraic system has a relatively large size and is not well conditioned, and thus its inversion requires the use of a direct method (Gaussian method), which is expensive in calculation times [FEL 91]. A third possibility (Figure 10.1c) consists of applying a coupling by parameterization. This coupling allows us to avoid the alternate resolution of the coupled equations (electromagnetic, thermal, etc.) and thus prevents the transfer of the data from one problem to the other. This mode of coupling is based on the determination of an average density of power (ADP) localized in the skin depth of
Magneto-thermal Coupling
413
the work piece. This function is calculated through the resolution by finite elements of the electromagnetic equation for a given range of temperatures and for a fixed power supply (current or voltage). Then the ADP function will be used as a source term for the thermal equation. Thus, a modification of the thermal properties (calorific capacity, thermal conductivity, etc.) relates only to the thermal problem and does not require a new electromagnetic calculation. However, this mode of coupling is available only in the case of weak magnetic saturation [MIE 97]. 10.5. Heating of a moving work piece This concerns heating of a magnetic steel tube moving at the speed of 10 mm/s along the axial direction (z>0) (Figure 10.2) under a solenoidal inductor.
Figure 10.2. Study domain
Figure 10.3 presents an enlargement of a portion of the tube and its mesh structure.
Figure 10.3. Finite element mesh (4 layers of elements in the skin thickness)
414
The Finite Element Method for Electromagnetic Modeling
The electric and thermal characteristics of the load are given in Figures 10.410.7 [MIE 97]. The winding has 20 turns with a flowing current having an rms value of 410 amps and a frequency of 4,500 Hz.
Temperature
Figure 10.4. Electric resistivity
Temperature (qC)
Figure 10.5. Saturation magnetization
Magneto-thermal Coupling
415
Temperature (qC)
Figure 10.6. Specific heat
Temperature (qC)
Figure 10.7. Thermal conductivity
The results, in terms of potential and density of power, are provided in Figures 10.8 and 10.9 respectively.
416
The Finite Element Method for Electromagnetic Modeling
The temperature distribution in the work piece is given in Figure 10.10. Calculation was carried out in steady state conditions and the maximum temperature reached is: Tmax = 615 K (342°C). The distribution of the power density along a path A-B (Figure 10.9) shows a high concentration of this density in the area situated in front of the inductor. In addition, it can be noted that the area of the part located at the bottom of the inductor practically does not heat whereas that situated in front of the inductor is subject to a strong temperature variation (Figure 10.10). In the exit area of the inductor, the tube preserves a relatively high temperature (thermal conduction).
Figure 10.8. Equipotentials A
Figure 10.9. Distribution of the power density along path A-B
Magneto-thermal Coupling
417
Figure 10.10. Distribution of the temperature along path A-B
10.6. Induction plasma 10.6.1. Introduction Since its earliest investigation by Wilhelm Hittorf in 1884 [HIT 1884], and through more recent work [THO 29], [LAN 28], [BAB 47], our understanding of the inductive discharge in gases has not stopped evolving. It has moved from an interesting laboratory tool, to a source of plasma used on an industrial scale. Plasma is an ionized gas composed of molecules, ions, electrons and photons. The ionization level in plasma is dependent on the energy brought to the gas. This energy is provided either by the capacitive effects (arc plasma) or by the inductive effects (electrode-less plasma). Inductive thermal plasmas, which have only recently penetrated the industrial market in comparison with arc plasmas, are very attractive for several industrial applications, especially in the material processing. Their advantages are particularly due to the absence of electrodes, thus offering [REE 61]: – easy operation in a wide range of conditions with inert, oxidizing or reducing gases with atmospheric pressure or low pressure; – a high temperature medium and very high purity; – a relatively large residence time for the reactive elements.
418
The Finite Element Method for Electromagnetic Modeling
10.6.2. Inductive plasma installation An inductive plasma installation is made up of a high frequency triode generator, an impedance adaptation system, an inductor and an applicator into which a gas is injected (see section 10.11). The applicator can be a quartz tube or a cooled metal cage. Load
Triode
Plasma
generator
Inductor
C
Cool cage Figure 10.11. High frequency inductive plasma installation (4 to 5 MHz)
Modeling of the whole system from the generator to the plasma is the subject of several research studies [PLO 96], [PLO 97]. In this section, we study the modeling of the set composing of the inductor and the applicator. We replace the generator by a source of constant voltage and calculate the impedance of this whole set with respect to the generator. 10.6.3. Mathematical models Plasma can be assimilated to a conductive cylinder. The difference lies in the fact that the electric conductivity of plasma is a function of the temperature (see Figure 10.12). It thus varies from one point to another. However, for the calculation of the impedance of plasma, it is necessary to know the conductivity of plasma at any point in the field of study. The calculation of the temperature distribution is thus essential for the calculation of this impedance. In addition, temperature is a function of the electromagnetic power injected into plasma and gas flow velocity.
Electric conductivity (S/m)
Magneto-thermal Coupling
419
1E+5 1E+4 1E+3 1E+2 1E+1 1E+0 1E-1 1E-2 1E-3 1E-4 1E-5 1E-6 1E-7 1E-8 1E-9 1E-10 1E-11 1E-12 1E-13 1E-14 1E-15 1E-16 1E-17 1E-18 1E-19 1E-20 1E-21 1E-22 1E-23 0
5000
10000
15000
20000
25000
Temperature (°K)
Figure 10.12. Variation of the electric conductivity of plasma as a function of the temperature
As the particles are charged and viscous, speed is a function of the magnetic field and the temperature. Thus, the electromagnetic, heat transfer and flow equations are nonlinear and coupled between them. The discretization of the resolution field and the approximation of unknown variables through numerical methods lead to a nonlinear set of equations. In the case of an axisymmetric system the variables to be calculated in plasma are: – the temperature T; – the potential vector A (real and imaginary parts); – the axial speed vz, the radial speed vr, and the pressure p. If the source term is a voltage, it is necessary to add the inductor current to these quantities which, in this case, is also unknown. The nonlinearity of the characteristics and the significant number of unknown physical variables make solving the equation system very difficult. Simplifying the system is thus essential to reduce the calculation time and the memory requirement. 10.6.3.1. Simplification Among the six unknown physical variables, three are associated with the flow equation (vr, vz and p). Knowledge of the field speed is of paramount importance when we want to study the trajectory of the particles injected into plasma.
420
The Finite Element Method for Electromagnetic Modeling
In our case, for the calculation of impedance, an approximation of the field speed is sufficient. Solving the flow equation shows that the axial speed component is prevalent [MOS 86], [MEK 93]. This leads us to assume a laminar flow of gas following in direction z. Thus, from all the flow equations, we use only the continuity equation which can be written: w U vz wz
0 thus: U v z
cte
U v z 0
[10.10]
where U is the density of the gas. Since the flow rate and the gas density at the input of the torch are known, we can evaluate speed vz, everywhere in the plasma. This assumption has been validated by the work by A. Chentouf [FOU 93], [CHE 94], [CHE 95]. 10.6.3.2. Solving the equations 10.6.3.2.1. Electromagnetic equation The system being axisymmetric, the magnetic vector potential, in cylindrical coordinates, is given by: w §1 w · w § wA · (rA) ¸ ¨ ¸ ¨ wr © r wr ¹ wz © wz ¹
jPVZA
[10.11]
Equation [10.11] is solved by the Coupled Circuit Method (CCM) in the inductor and at the plasma border and by the FEM inside the plasma. a) The inductor and the plasma border (CCM) The potential vector created in a point of space is the sum contribution of the potentials created by the elements of the load and those of the inductor: A=A:+Aind
[10.12]
A: and Aind are obtained by the integration of the Biot-Savart’s law on the elements of the load and those of the inductor. For that purpose, the inductor and load are cut out on Ni and Nc basic turns respectively (see Figure 10.13).
Magneto-thermal Coupling
421
The potential vector created by an element K of Sk section, at the middle of an element i is thus º P0 rik ª§ K2 · ¸ J K J K » dS ³³S Jk r,z «¨1 1 2 » k 2S k ri «¨ 2 ¸ ¹ ¬© ¼ 1 rik ª r ri z zi º 2 ¬ ¼ 1 r ri 2 K 2 rik Ai,k
[10.13]
J1(K) and J2(K) are Legendre functions of 1st and 2nd species and r and z are cylindrical coordinates. The general equation for a circuit element can be written: – inside the inductor: Nt
2S ri (Ui Ji jZ ¦ Ai,k )
ui
i 1.........N t
[10.14]
k 1
– at the plasma border: Nt
2S ri (Ui Ji jZ ¦ Ai,k )
0
i 1.........N t
k 1
ui: voltage at the element terminals of a turn and Nt = Ni + Nc.
Figure 10.13. Cutout of the inductor and the applicator in basic turns
[10.15]
422
The Finite Element Method for Electromagnetic Modeling
Therefore, by using the circuit equations, the following matrix form is obtained:
> M11
M12
ª J ind º « » M13 @ « J * » «¬ J : »¼
> U@
[10.16]
where Jind, J:, J* represent current density vectors in the inductor, inside plasma and at the plasma border respectively. In the same manner, for the plasma border, we obtain:
> M 21
M 22
ª Jind º « » M 23 @ « J * » «¬ J : »¼
> 0@
[10.17]
b) Inside plasma If the current densities at the plasma border are known, we can solve equation [10.11] using the FEM. Thus, it follows:
> M32
ªJ* º M 33 @ « » ¬J: ¼
[10.18]
0
If the electric conductivity of plasma is known, equations [10.16], [10.17] and [10.18] give the current distribution and the electromagnetic power injected into plasma. This power provides the source term for solving the heat equation. 10.6.3.2.2. Heat equation The thermal equation to be solved is: wT wT ) U Cp (v z vr wz wr
1w wT w wT (rk ) (k ) S r wr wr wz wz
[10.19]
Magneto-thermal Coupling
423
This equation should be simultaneously solved with the electromagnetic equation to obtain source term S. By neglecting vr and replacing (U vz) with (U vz)0, we can write: (U v z ) 0 C p
wT wz
wT 1 w w wT (rk ) (k ) S wr r wr wz wz
[10.20]
The boundary conditions associated with the equation are: – at the entrance of the torch: – on the symmetry axis:
T
Te
wT wr
0
– at the internal layer of the torch: k
wT wr
h T Ta with h global convective
exchange coefficients; – at the output of the torch: it is possible to neglect the conduction term in favor of the transport term, which leads to:
wT wz
0
Equation [10.20] is solved by using the finite volume method (FVM) based on the mesh structure in Figure 10.13. The FVM uses one analytical solution corresponding to a 1D problem in each mesh. The equation is considered without a source term:
d § dT · = ¨k ȡ v C dT ¸ dz dz © dz ¹ z
[10.21]
p
with the boundary conditions:
T
Tj
to
z
0
T
T j 1
to
z
L
where Tj is the temperature in the center of element j, Tj+1 is the temperature in the center of element j+1 and L is the distance between the two elements.
424
The Finite Element Method for Electromagnetic Modeling
The solution of this equation is given by: T Tj Tj 1 Tj
with Pe
exp(Pe z / L) 1 exp(Pe ) 1
U vz Cp L
[10.22]
Cte in the volume element.
k
The thermal conductivity of plasma is strongly nonlinear (see Figure 10.14). The temperature variation in plasma is very significant going from 1,000 K at the outer layer to 10,000 K in the center of the torch on a radius of 2 cm. Let us consider the interface between two nodes P and E (see Figure 10.14): 5
.
4
3
2
1
0 0
5000
10000
15000
20000
25000
Figure 10.14. Thermal conductivity of argon as a function of the temperature
dr dr-
P
dr+ e
E
Figure 10.15. Interface between two elements
Magneto-thermal Coupling
425
A 1D analysis, without a source term, leads to the expression: ke
with E e
E 1 E e 1 ( e ) kP kE dr dr
[10.23]
.
The source term in equation [10.20] can be written: S = P – Q where P and Q represent respectively the heat power density created by the induced currents and the power density lost through radiation. P comes from solving the electromagnetic equation and can be written:
P
0.5 V T Z2 A 2 .
[10.24]
Q comes from experimental measurements or theoretical calculations. The source term contains V(T) and Q(T) which are strongly nonlinear. We can linearize the source term by processing the Taylor development of S in the vicinity of a T0 value: dS S S TT ( ) 0 0 dT T
with S c
dS S T ( ) 0 0 dT T
T 0 T 0
S S T c p and S p
dS ( ) dT
[10.25]
The introduction of the local solutions [10.22], [10.23] and [10.25] in equation [10.20] and the discretization of this equation on the mesh of Figure 10.13 leads to a nonlinear system of the form:
> A @ > T @ > B@
[10.26]
where A is a sparse nonlinear matrix. 10.6.3.3. Coupling algorithm The complete matrix system for the calculation of currents and temperature in plasma is composed of equations [10.16], [10.17], [10.18] and [10.26]. This system has simultaneously full submatrices as well as linear and nonlinear parts. Solving the whole set of these equations simultaneously poses numerical problems. In order
426
The Finite Element Method for Electromagnetic Modeling
to overcome these problems, we can use the hierarchical algorithm of Figure 10.16. This algorithm allows decoupling solutions for the various equations from the system.
o convergence yes
no
convergence
Figure 10.16. Coupling algorithm
10.6.4. Results
Figure 10.17 shows the temperature field inside plasma and Figure 10.18 gives the comparison between the temperatures calculated by the model and those measured by an enthalpic probe. The calculated results are very close to those measured.
Magneto-thermal Coupling
427
10.6.5. Conclusion
The modeling of a coupled and complex system requires the use of a set of modeling techniques adapted to each part of the system and to each physical phenomenon. In a magneto-thermal coupled system, without a speed term, the FEM is applicable for electromagnetic as well as thermal problems. On the other hand if the thermal load moves at high speeds, the use of the finite volume method for solving the thermal equation becomes essential.
Figure 10.17. Temperature distribution
Figure 10.18. Comparison between the calculated and measured temperatures
428
The Finite Element Method for Electromagnetic Modeling
10.7. References [AYM 97] AYMARD N., FELIACHI M., PAYA B., “An improved modified surface impedance for transverse electric problems”, IEEE Trans. Mag., vol. 33, no. 2, March 1997. [AZZ 01] AZZOUZ F., FELIACHI M., “Non-linear surface impedance taking account of thermal effect”, IEEE Trans. Mag., September 2001. [BAB 47] BABAT G.I., “Electrodeless discharges and some allied problems,” J. Inst. Elec. Engrs., vol. 94, pp. 27-37, 1947. [BOU 76] BOULOS M.I., “Temperature and flow field in the fire ball of an inductively coupled plasma”, IEEE Trans. Plasma Sci., PS-4, 1976, p. 28. [CHE 94] CHENTOUF, Contribution à la modélisation électrique, magnétique et thermique d’un applicateur de plasma inductif haute fréquence, Thesis, University of Nantes, 1994. [CHE 95] CHENTOUF A., FOULADGAR J., DEVELEY G. “A simplified method for the calculation of the impedance of an induction plasma installation”, IEEE, Trans. On Mag., May 1995. [DEL 84] DELAGE D., ERNST R., “Prédiction de la répartition des courants dans un inducteur à symétrie de révolution destiné au chauffage par induction MF et HF”, RGE4/84, 1984, pp. 225-230. [DUT 84] DU TERRAIL Y., SABONNADIERE J.C., MASSE P., COULOMB J.L., “Nonlinear complex finite element analysis of electromagnetic field in steady state AC devices”, IEEE Trans. Mag., vol. 20, no. 4, 1984. [FEL 91] FELIACHI M., DEVELEY G., “Magnetothermal behavior finite element analysis for ferromagnetic materials in induction heating devices”, IEEE Trans. Mag., vol. 27, no. 6, November 1991, pp. 5235-5237. [FEL 92] FELIACHI M., PERRONNET A., DEVELEY G., “Modélisation par éléments finis des phénomènes couplés électromagnéto-thermiques caractérisant le frittage par Induction”, J. Phys. III, November 1992, pp. 2005-2013. [FEL 95] FELIACHI M., BENZERGA D., A 2D Finite Element Analysis for the Modeling of RF Plasma Devices FED by Voltage Driven Coil, Elsevier Science B.V., 1995. [FEL 96] FELIACHI M., BENZERGA D., MIMOUNE S.M., FOULADGAR J., “On the finite element analysis of high frequency induction heating systems fed by voltage driven inductor”, Studies in Applied Electromagnetics Series, The IOS Press, Cardiff, UK, 1996 pp. 851-854. [FOU 91] FOULADGAR J., FELIACHI M., DEVELEY G., “Distribution de l’énergie dans un cylindre chauffé par induction”, RGE, no. 2/91, February 1991, pp. 1-4. [FOU 93] FOULADGAR J., CHENTOUF A., “The calculation of the impedance of an induction plasma by a hybrid finite-element boundary element method”, IEEE Trans on Mag., vol. 29, no. 6, November 1993.
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[HIT 1884] HITTORF W., “Ueber die Elektrizitaetsleitung der Gase” (“About the Conduction of Electricity through Gases”), Wiedemanns Ann. d. Physik., vol. 21, pp. 90-139, 1884. [KRA 97] KRAHENBUHL L., FABREGUE O., WANSER S., DE SUSA DIAS M., NICOLAS A., “Surface impedance, BIEM and FEM coupled with 1D non-linear solutions to solver 3D high frequency eddy current problems”, IEEE Trans. on Mag., vol. 33, no. 2, March 1997, pp. 1167-1172. [LAG 28] LANGMUIR I., “Oscillations in Ionised Gases,” Proc. Nat. Soc. Science, vol. 14, 1928. [LAV 93] LAVERS J.D., “Electromagnetic field computation in power engineering”, IEEE Trans. on Mag., vol. 29, no. 6, November 1993, pp. 2347-2352. [LOU 96] LOUAI F.Z., BENZERGA D., FELIACHI M., BOUILLAULT F., “A 3D finite element analysis coupled to the impedance boundary condition for the magnetodynamic problem in radio frequency plasma devices”, IEEE Trans. on Mag., vol. 32, no. 3, May 1996. [MAO 97] MAOUCHE B., FELIACHI M., “Analyse de l’effet des courants induits sur l’impédance d’un système électromagnétique alimenté en tension BF ou HF. Utilisation de la méthode des circuits couples”, Journal de Physique III, no. 10, October 1997. [MAO 98] MAOUCHE B., FELIACHI M., “A discretized integral method for eddy current computation in moving objects with the coexistence of the velocity and time terms”, IEEE Trans. on Mag., September 1998. [MAS 85] MASSE P., MOREL B., BREVILLE T., “A finite element prediction correction scheme for Magnetothermal coupled problem during Curie transition”, IEEE Trans. on Mag., vol. 21, no. 5, 1985. [MEK 93] MEKIDECHE M.R., FELIACHI M., “An axially symmetric finite element model for the electromagnetic behaviour in an RF plasma device”, IEEE Trans. on Mag., vol. 29, no. 6, 1993, pp. 2476-2478. [MET 96] METAXAS A.C., Foundation of Electroheat, John Wiley & Sons, 1996. [MIE 97] MIEGEVILLE L., FELIACHI M., PAYA B., “Couplage magnéto-thermique par la paramétrisation”, NUMELEC’97, Lyon, March 1997. [MOH 98] MOHELLEBI H., LATRECHE M.E., FELIACHI M., “Coupled axisymmetric analytical and finite element analysis for induction devices having moving parts”, IEEE Trans. on Mag., September 1998. [MOS 86] MOSTAGHIMI J., PROULX P., BOULOS M.I., “A two temperature model of the inductively coupled plasma”, J. Appl. Phys., vol. 61, no. 5, pp. 1753-60, March 1986. [OUL 99] OULED AMOR Y., FELIACHI M., “Considération de l’hystérésis magnétique et de son comportement thermique dans un calcul de champ par éléments finis”, 6th International Conf., ELECTRIMACS’99, 14-16 September 1999.
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[PAN 89] PAN Q., KLADAS A., RAZEK A., “Modélisation du phénomène couplé thermique magnétique dans un système électromagnétique”, Rev. Gén. Therm. Fr., no. 334, October 1989, pp. 575-582. [PAN 00] PANTELYAT M.G., FELIACHI M., “Magneto-thermo-elastc-plastic behaviour of metal workpieces in induction heating devices”, IEEE-CEFC’2000, June 2000. [PLO 96] PLOTEAU J.P., FOULADGAR J., AUGER F., CHENTOUF A., DEVELEY G. “Numerical modeling of an induction plasma installation including the generator state space model”, IEEE Trans. on Mag., October 1996. [PLO 97] PLOTEAU J.P., Modèle numérique d’un ensemble générateur et applicateur de plasma inductif. Validation du modèle par des mesures locales, Thesis, University of Nantes, 1997. [REE 61] REED T.B., “Induction-coupled plasma torch”, Journal of Applied Physics, vol. 32, pp. 821-824, May 1961. [THO 27] THOMSON J.J., “The electrodeless discharge through gases”, Philos. Mag., vol. 4, pp. 1128-1160, 1927.
Chapter 11
Magneto-mechanical Modeling
11.1. Introduction The primary function of electrical machines is to ensure energy conversion. Indeed, the required function of these machines is to transform electrical energy into mechanical energy, and conversely, to transform mechanical energy into electrical energy. It concerns both power generators and all motors and actuators. This transformation is carried out in two successive stages: for one, by transforming electrical energy (U,I) into magnetic energy (B,H); for another, by transforming this magnetic energy into mechanical energy. Even the static equipment (transformers, inductors, converters, cables, etc.), where by definition no part is in movement and no displacement is required, is subject to magneto-mechanical problems, that is, for example, the vibrations generated or the mechanical resistance in the event of a short circuit or an electric shock. Efficient magneto-mechanical modeling is thus a key design challenge, though numerical modeling of coupled magneto-mechanical phenomena has only recently begun to be seriously addressed. This is due to the many specificities of this modeling. Indeed, the scope and diversity of the corresponding problems explain the slow pace of developments. In order to deal with this complex modeling, numerical models for electromagnetic fields are, of course necessary. They must be combined with numerical models for mechanical structures. Depending on the physical nature of the coupling, this can be performed in different ways. It can be performed using methods for calculating magnetic forces or methods taking into account the coupling between magnetic field and elastodynamic equations. Chapter written by Yvan LEFEVRE and Gilbert REYNE.
432
The Finite Element Method for Electromagnetic Modeling
The improvement of numerical models for magneto-mechanical phenomena answers the demand and requirements of industry. For instance, the development of appropriate methods to take into account global magnetic force in magnetic software has improved electromechanical modeling, leading to more efficient design of conventional electrical actuators in terms of energy density and of available torque per mass or volume. The need to further improve the performances of these actuators is of main concern. It addresses the reduction of the magnetic force ripples or the minimization of the vibrations and noises of electromagnetic origin. A more recent trend in industry consists of diversifying the principles of electromechanical conversion by the use of new electro-active components such as piezoelectric or magnetostrictive rods. This requires the development of specific numerical models to take into account the strong electromechanical coupling phenomena inside these “smart” materials. These parallel evolutions of numerical models and their industrial applications have allowed significant advances in the development of both basic electrical actuators or of new ones based on electro-active materials. Thus, industries, where significant research efforts have been made, such as home automation, road, rail, air and sea transportation and energy production and conversion, have developed high quality and diversified new products. In such highly competitive markets we can guess that this basic trend will be further enhanced both by new materials, new knowledge and the development of improved numerical models and software for magneto-mechanical modeling of interactions between fields, solids and fluids (fluid damping noises or magnetohydrodynamics for instance). A brief review of the general principles of coupled magneto-mechanical models will start this chapter. Then, presented by increasing complexity, examples of industrial problems will give shape to these principles. Discussions will follow on the expressions “weak coupling” and “strong coupling” in the context of magnetomechanical problems. It will be shown that the same expressions have different meanings and apply differently depending on the scientific community. Indeed, physicists and specialists in numerical modeling actually use them differently. Finally, examples of modeling problems accounting for either weak or strong magneto-mechanical couplings will be presented. 11.2. Modeling of coupled magneto-mechanical phenomena The coupled phenomena to be modeled result from the interaction between the magnetic field in an electrical actuator and its mechanical structure. The most appropriate methods for electromagnetic numerical modeling have been developed in the previous chapters.
Magneto-mechanical Modeling
433
Basic numerical modeling of a mechanical structure is now presented. This presentation allows the variational principle in mechanics to be briefly introduced. This principle is very general in physics and allows us to understand most of the physical phenomena and their couplings. Then, the modeling of a magneto-elastic system is explained. It enables the simultaneous calculation of magnetic quantities and mechanical displacements in an electromechanical structure submitted to both external forces and magnetic field sources. 11.2.1. Modeling of mechanical structure 11.2.1.1. Hamilton principle (also called the variational principle) The movement of a mechanical system is completely determined by Hamilton’s variational principle. The term “action” indicates the time-dependent integral S [IMB 84]: t1
S
³ Ldt
[11.1]
t2
where L is the Lagrangian of the studied mechanical system evolving between two instants t1 and t2. The Lagrangian is composed of two terms: L=T-V
[11.2]
where T refers to the kinetic energy of the studied system and V is the system’s total potential energy. The system’s real movement is that which makes the action S stationary (principle of least action). 11.2.1.2. Different energy definitions The kinetic energy of the mechanical system is the integral: T
1 2
³ Uvvdv
[11.3]
vol
where vol is the volume of the entire mechanical system and v the velocity vector of a volume element dV of mass density U. The total potential energy V of the systems is equal to the difference between the elastic strain energy U and the potential energy W of the applied force: V=U-W
[11.4]
434
The Finite Element Method for Electromagnetic Modeling
The movement of the structure results from diverse forces applied on the mechanical structure: some volume forces fv, surface forces fs and localized forces fk exerted on precise points. The total potential energy of the applied forces is given by the sum of each force system’s potential energy: Np
W
³
vol
f v d dv
³
6
f s d ds
¦f
kdk
[11.5]
k 1
d represents the displacement vector related to the velocity vector v by:
v
w d wt
[11.6]
These forces cause a structural deformation characterized by the strain tensor e and the stress tensor V. The resulting elastic strain energy is given by: U
1 2
t
³ ı edv
[11.7]
vol
The strain tensor e and constraint tensor V are related to the displacement vector d through the elasticity relations: e Dd ı Ce
[11.8]
where D is the differential elasticity operator and C is the matrix of the constitutive properties of the material used characterized by their Young’s magnitude E and Poisson’s coefficient Q (Hooke’s law). Thus: U
1 2
t
³ e Cedv
[11.9]
vol
The application of the principle of least action allows us to find the displacement field of a mechanical system that undergoes volume forces fv, surface forces fs and localized forces fk. However, in practical terms, a numerical model is required to solve the problem and to provide a good approximation of the solution. 11.2.1.3. Numerical model In order to calculate an approximate solution for the elastodynamic problem, a method of separation of variables is used. Each component dk (k = 1.2 or 3) of the
Magneto-mechanical Modeling
435
displacement vector is approximated by a linear combination of space functions Pki(x,y,z): noe
uk
¦P
ki ( x,
y, z )q ki (t )
[11.10]
i 1
where qki(t) are time-dependent coefficients which are called generalized coordinates of the displacement field vectors in the base Pki(x,y,z). In the nodal finite element method, a nodal approximation is used. The generalized coordinates qki(t) are the values of the displacement vector components at each node. In fact, the principle is to discretize the domain and work on a finite dimensional vector space of functions. The dimension of this vector space is equal to Nl = nl*noe where nl is the number of degrees of freedom at each node (2 or 3) and noe is the number of nodes in the discretized domain. Then, the generalized displacement vector q is composed by the values of the displacement vector components at each node. The same operation is achieved for the generalized velocity vector at the nodes. The following relationship is directly obtained: x
q
d q dt
[11.11]
11.2.1.4. Energy expression in the numerical model The different energies of the mechanical system are expressed in a very simple way according to the displacement and generalized velocity vectors. Thus, the kinetic energy is expressed as a quadratic form of the generalized velocity vector components: T
1 2
xt
x
q Mq
[11.12]
where M is the generalized mass matrix. Its coefficients are dependent on the domain mesh and the mass densities of the used material. Similarly, the elastic strain energy is a quadratic form of the components of the generalized displacement vector: U
1 2
q t Kq
[11.13]
where K is the global rigidity matrix whose coefficients depend on Young’s modulus and on Poisson coefficients for the materials used.
436
The Finite Element Method for Electromagnetic Modeling
The potential energy of a force is expressed as a linear form of the generalized displacement vector: W=ft q
[11.14]
where f is the generalized force vector applied to the nodes. It represents the volume, surface or localized forces applied to the system. The construction of the symmetric matrices M and K as well as the generalized force vector f is carried out by means of the calculation and assembly techniques similar to those used in the numerical models for electromagnetic fields. These techniques take into account the imposed boundary conditions. 11.2.1.5. Lagrange’s movement equations in the numerical model The Lagrangian L, L=T-U+W is thus, for a given meshing, a function of time t, the generalized displacement and velocity vectors, thus: Nl
x
§ ¨fq ¨ i i 1©
¦
L(t , q, q)
i
Nl
§
¦ ¨¨© M j 1
i, j
x x ·· qi q j K i , j q i q j ¸¸ ¸ ¹ ¸¹
[11.15]
By applying the principle of least action, on each degree of freedom, the Lagrange equations of the movement of the system are deduced: w wL wL wt x wqi w qi
[11.16]
0
We thus obtain Nl differential equations for i, varying from 1 to Nl: Nl
¦M
xx
i, j
q j K i, j q j
fi
[11.17]
j 1
xx
This equation shows q j t is the acceleration of node j according to time. In matrix form, the following system of differential equations is obtained: xx
M q Kq
f
[11.18]
Magneto-mechanical Modeling
437
The generalized acceleration vector is related to the generalized displacement vector through the usual definition: xx
q
d2
[11.19]
q
dt 2
For a magneto-mechanical problem, the generalized force vector f represents the magnetic force exerted on the mechanical structure. Knowing the vector f as a function of time, various numerical methods are available to calculate the displacement vector q as a function of time. 11.2.2. Coupled magneto-mechanical modeling The study of the interaction between a mechanical structure and a magnetic field can be very complex. Most of the time, there are no other computation methods except numerical ones. As shown below, this complexity appears even on a simple condition such as the theoretical static interaction. Indeed, let us consider an electromechanical system with coils supplied by DC currents and subject to nonmagnetic, external and constant volume forces. To simplify further, there are no moving parts and mechanical as well as magnetic properties are considered to be linear. Consequently, the mechanical structure will only deform due to external volume forces and magnetic forces created by currents. The least action principle, presented earlier, is applied to calculate the magnetic field and the resulting mechanical deformation field. 11.2.2.1. Total potential energy of a magneto-elastic system For a static system, the Lagrangian is reduced to the system’s total potential energy V. As just seen above, for a static mechanical system, the expression of this energy, considering that the system is only subjected to volume forces, is given by: V mec
1 2
t
³ e Ce dv ³ f
Vol
vd
dv
[11.20]
Vol
For a magnetostatic system, the total potential energy Vmag can be defined. It is equal to the difference between the magnetic energy Umag contained in the system and the potential energy of sources Wmag. Field sources are considered to be only winding currents and the magnetic circuit is linear. Consequently, the expression of Vmag is: V mag
1 2
1
³ P bbdv ³ JAdv
Vol
Vol
[11.21]
438
The Finite Element Method for Electromagnetic Modeling
where P is the magnetic permeability. For a mechanical system, the displacement field resulting from the application of external volume forces corresponds to the extremum of Vmec. For a magnetic system, the vector potential field resulting from the current densities J corresponds to the extremum of Vmag. Similarly, for a magneto-mechanical system on which external forces fv and current densities J are imposed, the mechanical displacement field d and the resulting magnetic vector potential field A correspond to the extremum of the system’s total potential energy Vtot. It results that:
Vtot
[11.22]
V mag V mec
To simplify the calculations, and in particular to understand the origin of the coupling terms, the associated numerical model is considered. 11.2.2.2. Numerical model In the numerical model, obtained after meshing the domain, the displacement field d corresponds to the generalized displacement vector q at each node. Similarly, the vector potential field corresponds to the generalized vector a, the components of which are the components of the vector potential at each node. The total potential energies can be expressed in the following quadratic form:
V mec
1 2
q t Kq f t q
V mag
1 2
a t Ra s t a
[11.23]
where s is the equivalent current density vector applied on each node. R is a symmetric matrix which depends on the magnetic permeability of the materials. The total energy of the system can be expressed as: Nl
Vtot
§ ¨ f q s a i i ¨ i i 1©
¦ i
Nl
¦ K j 1
i, j qi q j
· Ri , j a i a j ¸ ¸ ¹
[11.24]
It can be seen that the number of degrees of freedom is twice that of a simple system, Nl mechanical variables qi and Nl magnetic variables ak. The extremum condition of the total potential energy Vtot is written for each degree of freedom xi: wVtot wxi
0
where xi is put for qi or ai.
[11.25]
Magneto-mechanical Modeling
439
Thus: wVtot wq i
Nl
fi
¦ K
i, j q j
Ri*, j a j
[11.26]
j 1
where R*i,j is a coefficient of a matrix that takes into account the influence of the displacement field on the total potential energy of the magnetic field: Ri*, j
Nl
¦ k 1
wRk , j wq i
[11.27]
ak
In matrix form, the first sub-system of the magneto-elastic problem is thus: Kq R * a
[11.28]
f
The second term of the first member is homogenous to a force. This force is of magnetic origin. It is caused by the change in magnetic energy due to the displacement field. Similarly: wVtot wa i
Nl
si
¦ K
*
i, j q j
Ri , j a j
[11.29]
j 1
where K*i,j is a coefficient of a matrix term that takes into account the influence of the magnetic field on the elastic strain energy: K i*, j
Nl
¦ k 1
wK k , j wa i
qk
[11.30]
In matrix form, the second sub-system of the magneto-elastic problem is thus: K * q Ra
s
[11.31]
The first term of the first member is homogenous to the vector s which is the nodal current density vector. It represents a source of field induced by the mechanical deformations.
440
The Finite Element Method for Electromagnetic Modeling
The final system of algebra differential equations of the discretized magnetoelastic problem is thus: Kq R * a
f
*
K q Ra
[11.32]
s
where the unknowns are the generalized displacement vector components and the potential vector components at each node. It should also be noted that this system is nonlinear, even with linear magnetic and mechanical characteristics for the materials. To solve this issue, it is necessary to know how the elastic strain energy varies depending on the magnetic field and how the magnetic energy varies depending on the mechanical displacement field. To be honest, this approach is only valid for very simplifying assumptions. However, it clearly demonstrates that, even with such assumptions, associated magneto-mechanical coupling is still tricky to model. 11.2.2.3. Physical origin of coupling terms The first magneto-mechanical coupling term highlighted above is the force of magnetic origin due to the variation in magnetic energy caused by the displacement field. To briefly illustrate the physical mechanisms at the root of this force, let us write magnetic energy in the form: U mag
³u
[11.33]
mag dv
vol
The energy contained in a volume element dv is: w mag
[11.34]
u mag dv
The differential of this energy compared to a variation Gq of the displacement field can be written:
Gwmag
Gu mag
Gq
Gq
dv u mag
Gdv Gq
[11.35]
Thus:
Gu mag Gq
G
1 2
K GH GP G G HH PH Gq Gq
[11.36]
Magneto-mechanical Modeling
441
Then:
Gwmag Gq
G
1 2
K GH GP G G Gdv HHdv PH dv u mag Gq Gq Gq
[11.37]
The first term of the second member indicates that variations in magnetic energy density are partly explained by variations in the material’s magnetic permeability induced by the displacement field. Indeed, such stress-dependent variations of magnetic permeability can be experimentally measured for different materials [HIR 95] [TRE 93]. This is called magnetostriction. However, the physical laws that govern these phenomena are, in general, too complex to be modeled [DAN 04]. The second and third terms of the second member account for energy variations even in the absence of stress-induced permeability variations. By using a finite element numerical model, the local Jacobian derivative method provides the resultant of these two terms in each node in the meshing [COU 83] [REN 94] [SAN 06]. The second term of magneto-mechanical coupling identified in the preceding section is a magnetic field source due to the variations in mechanical energy caused by magnetic field. Let us write the elastic strain energy in the form: U mec
³u
mec dv
[11.38]
Vol
The energy contained in a volume element is: wmec= umec dv
[11.39]
The energy differential with respect to a variation GA of the magnetic vector potential field is:
Gwmec GA
Gu mec dv GA
[11.40]
Given the expression of the strain energy, we have:
Gu mec GA
1 2
et
GC e GA
[11.41]
442
The Finite Element Method for Electromagnetic Modeling
It is to be remembered that e represents the strain tensor field and C is a matrix which depends on the mechanical properties of the material. This relationship shows that the variations in local strain energy density are induced by the change in the material’s mechanical properties due to the magnetic field. Changes in the mechanical and magnetic properties of magnetic materials have been studied intensively in research on magnetostriction phenomena [TRE 93]. The applications of these phenomena are numerous. In the last part of this chapter, some examples of actuators based on these phenomena will be addressed.
11.2.3. Conclusion The general principles for modeling a magneto-elastic system have been presented. The complexity of the numerical model obtained was highlighted on an elastic electromechanical system. The algebraic system can be extremely complex even if the mechanical and magnetic properties of the materials used are linear. This model has also allowed the physical origins of the coupling terms between the magnetic field and the deformation field to be heuristically demonstrated. Magnetomechanical modeling is often confronted with the difficulty of correctly modeling these coupling terms. The actual modeling of these couplings usually requires simplifying assumptions to be made. This will be clearly demonstrated on the remaining parts of this chapter. Indeed, several examples illustrating diverse applications of the principles just described will be presented in order of increasing complexity. For each application presented, specifically adapted assumptions will be considered. Thus, by just simplifying the model and applying correct assumptions, many problems can be modeled with existing numerical calculation codes.
11.3. Numerical modeling of electromechanical conversion in conventional actuators The most commonly used approach in electromechanical modeling is demonstrated in this first example. It answers the basic requirements for most conventional electric motors and actuators. Indeed, global displacements, speed, torque, acceleration, etc., of the mobile part are the most important data [SAD 92b] [SAL 95]. The system is basically characterized by the interaction between the electrical supply circuit, the magnetic field and the movement of non-deformable parts of the mechanical structure. It is assumed that magneto-mechanical interaction can be
Magneto-mechanical Modeling
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decomposed in two successive steps. Firstly, the magnetic field and magnetic forces are calculated, and then the mechanical quantities are deduced using dynamic motion equations. As will be explained in more detail in the last part, this method consists of applying a weak numerical coupling to solve the physical problem, wherever it actually is a weak or a strong magneto-mechanical physical coupling.
11.3.1. General simulation procedure In order to accurately describe the dynamic operation of these actuators in the most general way possible, it is necessary to model the interaction between the electrical circuit supplying the actuator, the magnetic field in the actuator and the movement of non-deformable parts. Given that the mechanical constants of the movement are typically very large (low damping and high inertia) compared to the time constants of the electrical circuit coupled to the magnetic field, it is possible to decompose the problem in three steps [BAS 03]: – first, for a given position of the mobile part, the electric variables such as currents in the coils and the magnetic field values (scalar or vector) are calculated; – then, from the magnetic field map, magnetic forces on the moving part are determined; – finally, the velocity and the new position of the mobile part are known by solving the equations of motion [VAS 91]. This requires, in terms of finite element-based magnetic field computation software: – the introduction of coupled “field-circuit” models; – the models to calculate the magnetic forces applied on the mobile part; – the method to solve the mechanical equations of motion of mobile parts considered as rigid bodies; – the methods to take into account the movement of rigid parts in finite element models. The reader should refer to specific chapters regarding: – the coupling methods for field equations and electric circuits, which allow us to determine the variables of the electrical circuit and of the magnetic field at each time step; – the methods to take movement into account. The most commonly used methods in magneto-mechanical software are presented hereafter.
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The Finite Element Method for Electromagnetic Modeling
11.3.2. Global magnetic force calculation method 11.3.2.1. Remote forces and induction lines In order to calculate the global magnetic forces acting on a rigid magnetic body, several methods exist. The diversity of possible methods is based on a remarkable electromagnetic property which is actually derived directly from the fact that they are remote forces. This property is that all information relating to global forces and torques are contained in the “free space” or, to express it another way, in the magnetic field lines that surround the considered object on which the magnetic force applies. This is true regardless of the nature of the magnetic field material inside the object. Whether it actually is currents or magnets or iron or superconducting material does not matter. This is a well known surprising property of electromagnetism. Thus, for two interacting magnetic objects, the magnetic interaction force corresponds to a precise pattern of the magnetic field lines around these objects. This pattern is a deformation of the magnetic field lines and results in a change in the gradients of magnetic quantities in the surrounding space. Thus, interaction forces may be “read” on the magnetic field line patterns in the usually non-magnetic space (air-vacuum) that surrounds the objects. More precisely, all the qualitative and quantitative information on the magnetic force on a magnetic object can be deduced from the values of field lines that surround it. If this was just a remarkable property of electromagnetism only a few years ago, the consequences on the use of digital software tools for the treatment of magnetomechanical coupling is essential at two levels: – first for the calculation tools themselves, as will be explained later; – then for the engineers who use these software tools. Finite element software provides magnetic field or magnetic flux density maps. Just a glance at the magnetic field line pattern in the air surrounding the object will provide the advised engineer with both qualitative and quantitative information on the magnetic forces in the system. 11.3.2.2. The principle of rubber bands (elastics) A characteristic of finite element software is their ability to easily provide the user with a “field map” more correctly called “map of magnetic field lines”. We can thus visualize an invisible physical quantity: the magnetic field that previously only iron filings would materialize. Moreover, this new information is vital. It is generally the best way to quickly check that the problem has been correctly treated without major errors. Furthermore, magnetic field lines provide information on high magnetic flux densities areas, leakage and even forces (qualitatively anyway).
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For forces or torques, it is sufficient to imagine that magnetic lines are stretched rubber bands (elastics) attached on the objects. This immediately makes sense and both forces and torques can be “felt” in most cases. Physically, it is linked to the fact that the existence of a magnetic flux density in a given volume corresponds to a magnetic energy stored in this medium. The expression of magnetic energy density is:
Wvol
b.h 2P 0P r
[11.42]
The relative permeability of air is usually very low compared to the soft ferromagnetic materials used. Thus, for most applications, the essential part of magnetic energy is stored in the air. The principle of energy minimization consequently leads any system to try to suppress air-gaps (air-gap forces, contactors, magnet sticking forces, etc.). Another consequence is that the magnetic field lines, especially in the air, seek the shortest route, as if they were tensed rubber bands. 11.3.2.3. Principle of calculating the global magnetic force All the information needed to calculate the magnetic interaction force between two objects is contained in the space surrounding the objects. It follows that all the techniques that replace magnetic objects, whatever they are, by global magnetic equivalents – that respect the created field – give the right global magnetic interaction force. In particular we can list the equivalent magnetic current method and the equivalent magnetic charge method. A volume calculation on the object, which globally replaces the real object can be carried out. Two other methods calculate directly on a surface surrounding the object. They are based on the information contained on the external field map. In a way, the calculation of the force on a 3D real object or its magnetic equivalents has been replaced by a 2D calculation on a closed surface, in the space surrounding the object. Note that, in theory, we can carry out this calculation in the same way whether it is directly on the surface of the real object or a certain distance away. These are the so-called “Maxwell tensor” and energy-based methods applied to finite elements (as proposed by Jean-Louis Coulomb). 11.3.2.4. Maxwell’s tensor method Traditionally, the Maxwell’s tensor method was the most known and widely used in software tools for calculating a torque or a global force. Here is the global force calculation using Maxwell’s tensor [BRO 62] [DUR 68] [STR 61]: F
³ b.n h 6
1 2
bh n ds
[11.43]
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The Finite Element Method for Electromagnetic Modeling
The integration surface 6 is a surface that completely surrounds the magnetic body where the force is to be calculated. The accuracy of the calculation depends on the formulation chosen to calculate the magnetic field on the mesh of the domain and on the choice of surface 6. This method is very general and applicable in all circumstances, as long as the magnetic field maps in the domain are calculated. 11.3.2.5. Energy methods These methods are based on magnetic energy or co-energy derivatives. Therefore, these methods are definitely the most well documented as it is the direct application of the physical principle of minimum energy. These methods are increasingly being used in finite element calculation tools. Among these methods, there is the energy or co-energy difference method calculated for two successive positions of the moving part, and the local Jacobian derivative. The first method requires two successive field calculations for two nearby positions of the moving part, while maintaining either the fluxes, or the currents in the windings, constant. This makes its use improbable in a simulation of the actuator’s dynamic operation, where the currents or fluxes are also to be determined. However, this method is very useful if we want to know, with a constant current, the force or the torque exerted on the moving parts depending on its relative position with respect to the fixed part. Thus, the calculation of the coenergy is carried out in the entire domain for all possible close positions of the mobile part. If the displacement step is sufficiently small, the difference ratio between the co-energies for two successive positions and the distance provides the force in the corresponding direction. While perfect theoretically, this principle, due to FEM numerical errors leads to very inaccurate or even totally false results. Unless specific methods are used, numerical errors on each of the nearby positions can often exceed the expected energy differential. As for optimization, FEM tools allowing an easy calculation of the derivative from a given solution are missing [SAD 92b]. The second method was developed by Jean-Louis Coulomb to make up for those drawbacks and is well adapted to the finite element calculation. This method allows us to calculate, with a single field calculation, the global force on a magnetic body by deriving magnetic energy or coenergy with respect to a virtual displacement. Therefore, this method can be used in dynamic operation simulations where the currents and the field should be determined simultaneously. The force expressed by the local Jacobian derivative depends on the formulation chosen to calculate the field [COU 83, 84]. If we choose a formulation in magnetic vector potential, the magnetic force is calculated by the magnetic energy derivative. Thus, the force in a given direction x is: Nel
Fx
1
¦³P e 1 Ve
e
b w x b det(J) w(b)w x det J d: e
[11.44]
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where b is the magnetic flux density, Nel the number of elements in the mesh, Ve the volume of element e, J the Jacobian matrix of element e,Pe is the magnetic permeability of element e, and w (b) is the magnetic energy density: b
w(b)
³ hd b
[11.45]
0
This method can be used even on saturated magnetic materials. REN adapted this method to different calculating field formulations, in 3D, using Whitney elements [BOS 92] [REN 92a, 94]. 11.3.3. Conclusion The global force calculation methods most commonly used in existing computing tools are Maxwell’s tensor and energy-based methods such as the local Jacobian derivative. Both methods are valid even in the case of saturated magnetic materials. The second method is very specific to calculation by finite elements and the force expression depends on the formulation used to calculate the magnetic field. 11.4. Numerical modeling of electromagnetic vibrations 11.4.1. Magnetostriction vs. magnetic forces Electromagnetic vibrations (short-cut for “mechanical vibrations of electromagnetic origin”) have two main sources: on the one hand magnetostriction, and on the other hand magnetic forces. By “magnetostriction” we mean the deformation of ferromagnetic materials related to internal changes of matter, during the magnetization process (displacement of Bloch walls between magnetic areas, then rotation of magnetization areas). It is this meaning that researchers give to the word magnetostriction, while less specialized persons, tend to call “magnetostriction” the total deformation of a magnetic body due to magnetization, thus, including both “magnetostriction” and magnetic forces [BES 96] [HIR 95] [REY 87] [TRE 93] [VAN 01, 04]. The two phenomena being linked and appearing simultaneously, the distinction is not straightforward. This is true, both theoretically and experimentally, whether on a steel sheet or within an electrical machine. In addition, for materials typically used in industry, magnetostriction leads to deformations of identical orders of magnitude (from a few ppm to 10-20 ppm) rather than magnetic forces. In light of these figures, the reader may say that these deformations remain very low (10-6 to
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The Finite Element Method for Electromagnetic Modeling
10-5) and are thus negligible. Though, they are the only source, for vibrations of static apparatuses. Moreover, vibrations relate to the dynamic mechanical response. Thus, the resonance phenomena and the corresponding frequencies do play a decisive role with regards to the final amplitude of vibrations. However, many machines, and in particular most static apparatus, are made of oriented grain sheets. Within these materials, magnetization variations are essentially due to wall displacements at 180º. Magnetization then remains in the direction of easy magnetization, with only the direction being changed. Hence, the overall deformation due to magnetostriction is zero (this is no longer true, for example, for transformer angles). For non-oriented grain sheets (most rotating machines), the magnetization also begins with a wall displacement at 180º (without generating global magnetostriction). Further on, rotations away from the easy magnetization directions appear. However, in these sheets, the pattern of surface magnetic domains is complex and there are wall displacements at 180° and 90° simultaneously, starting at the low induction levels. It results that, for many applications, the primary vibration source is not magnetostriction but the magnetic forces. However, this suffers from notable exceptions, in particular for machines where the materials are being pushed to their limits (saturation) beyond easy magnetization directions, but also some structures without air-gap (transformers), where the magnetic forces appear only between the steel sheets. In the following section, vibrations due to electromagnetic forces are the only vibrations considered. This can be theoretically handled with a general model while magnetostriction depends on the materials used and is specific to a given sheet type [MOO 84]. Whether it is necessary to eventually take magnetostriction into account is an extremely complex issue, and depends greatly on many parameters (quality, implementation, clamping of sheets, etc.). The second example is to illustrate problems with a weak interaction between the magnetic field and a deformable mechanical structure. As in the previous one, the calculation is carried out in several steps, though, the global magnetic force exerted on the mechanical structure will no longer be considered. It is now necessary to determine the distribution of local magnetic forces acting on the overall mechanical structure. Then, by using elastodynamic equations, the deformation field of the structure is calculated. This kind of problem appears in the study of electromagnetic vibrations in electric motors or static apparatus [GAB 99] [REY 94] [SAD 96].
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11.4.2. Procedure for simulating vibrations of magnetic origin For most vibrations, the mechanical response may be considered as linear as only low magnetic sheet deformations, are experienced. Under such conditions, the influence of the mechanical structure’s deformation on the magnetic field remains negligible. These considerations allow magnetic vibrations to be calculated in two stages: 1) determine the evolution of all the magnetic forces acting on the structure [BEN 93/3], [IMH 89, 90]; 2) determine the dynamic response of the mechanical structure subjected to these forces [HEN 92] [LEF 89]. The calculation of magnetic forces can be accomplished by considering the nonlinear behavior of a motor’s magnetic circuits. However, as the mechanical equations are linear, we can apply the superposition theorem to calculate the mechanical response. Generally, magnetic forces have a time period. The superposition theorem allows the mechanical response of the structure to be studied in two stages: 1) make a harmonic decomposition of the applied forces; 2) calculate the mechanical response for each harmonic. As the force harmonics are sinusoidal, it is possible to use a complex formulation of elastodynamic equations for calculating the mechanical structure’s response. This significantly reduces the calculation time compared to a step by step simulation. 11.4.3. Magnetic forces density If we neglect magnetostriction phenomena, the volume force in an isotropic material that is incompressible and non-homogenous is given by the expression [WOO 68]:
fv
j b 12 hh gradP
[11.46]
The first term is the Lorentz force density, on current density j. It is used to calculate the force densities on current carrying conductors. The second term represents the force due to magnetization. Its expression is the result of a thermodynamic approach using energy balance laws. However, it is seldom used to calculate the magnetic volume force density in a magnetic body. Indeed, in most practical applications it can be assumed that the magnetic materials have a linear behavior. Thus, it is only used to deduce the surface force density exerted on the external surface of a magnetic, isotropic, homogenous and linear medium.
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The Finite Element Method for Electromagnetic Modeling
Under these assumptions, the magnetic permeability P is uniform and its gradient is zero. If, in addition, the medium is not conducting, the magnetic volume force density is zero everywhere in the medium. Magnetic surface forces exist as permeability is discontinuous at the surface. The magnetic surface force density can then be calculated as follows: first, it is assumed that the external surface has a certain thickness and that the permeability varies continuously through it, then this thickness tends towards zero. This provides a surface force perpendicular to the surface, whose expression is given by [MUL 90] [SAD 92a]: fs
1 2
ª 1· º 2 § 1 ¸¸bn2 » n « P P 0 ht ¨¨ «¬ © P 0 P ¹ »¼
[11.47]
ht and bn being respectively the tangential magnetic field intensity component, and the normal magnetic flux density component, outside the surface. This expression is most commonly used in field calculation software to determine the magnetic surface force density. It can also be deduced from Maxwell’s tensor applied on a box including the surface. Though, to be theoretically valid, the box surface itself, should only be in one linear magnetic medium. When the medium has a nonlinear behavior, in a finite elements calculation code, we can use the local Jacobian derivative which, according to Ren, directly determines the magnetic force exerted on each node [REN 92b, 94]. Physically, materials which have a high permeability essentially undergo a surface force. It arises directly from the spatial distribution of the magnetic energy and from the virtual works principle (force deriving from energy variations generated by a virtual displacement). Yet, as we have seen in section 11.3.2.2, energy is often concentrated in the air and air-gaps. Magnetic energy density in iron, due to its very high permeability (typically 103 to 105), is usually negligible compared to the magnetic energy density in the air. Therefore, most significant energy variations are due to the iron surface displacements. When iron replaces an air zone, it significantly decreases the magnetic energy stored in the system. The consequence of the application of the principle of minimum energy is that most of the forces will be applied on the iron surface and directed towards the air. The virtual works principle directly gives the force expression from energies of both sides. In most cases (all airgaps) the iron energy density is negligible, the surface force density is simplified and directed toward the normal beyond the iron. Its value is: fs
1 2
B air B air
P0
n
[11.48]
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The same energy considerations, based on the increase of stored energy in iron when high levels of saturation are reached, indicate that local volume forces in the same order of magnitude as the surface forces may appear.
Figure 11.1. Elementary contactor and corresponding induction lines [FLUX2D]
Figure 11.2. Magnetic forces on the mobile section of the contactor
Figure 11.3. Magnetic forces on the fixed section
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The Finite Element Method for Electromagnetic Modeling
To sum up, for the vast majority of applications, the two forces to be taken into account are Lorentz force in JxB on current carrying conductors and the surface force on the sheets Bair2/2Po (Figures 11.1, 11.2 and 11.3) [MEL 81] [REY 87, 88] [WOO 68]. Regarding the magnets, they are generally sufficiently small for the vibration problem to be dealt with simply using the global force exerted on them. For the force distribution within the magnets, see [MED 99, 00].
11.4.4. Case of rotating machine teeth In electrical machines, the winding system is carried by a fixed structure, the stator, which contains slots in which conductors are inserted. Electromagnetic forces are usually the main source of vibrations and noises emitted beyond the stator’s mechanical structure. In order to calculate these vibrations, it is necessary to determine the distribution of electromagnetic forces on the stator for each time step [KIM 05]. It is possible to have a good idea of this distribution by calculating the forces on the conductors and the local forces on the stator teeth. The magnetic flux density is usually guided by the teeth. Thus it is very weak in the slots where the conductors are located. These conductors, contrary to common sense, often make a negligible contribution to the total torque. The forces on the conductors are calculated using the Lorentz force expression. The local forces on the teeth can be obtained by integration of the surface force given earlier on the tooth contour. However, it has been shown experimentally, without theoretical proof, that the partial integration of the force density given by Maxwell’s stress tensor on the tooth contour gives a good representation of the force measured on it [LEF 88]. The advantage of this later method is that it can be used even in the case of saturated magnetic medium. The use of global forces rather than magnetic force densities is advantageous from a practical view point, especially if we need to make an harmonic decomposition before calculating the mechanical response. Indeed, it can be brought to a single equivalent force per tooth, which simplifies the problem. This approximation is generally very well adapted to the treatment of vibration problems, given the wavelength of the vibration modes to be taken into account. Figure 11.4 gives example results of force distribution calculations, obtained using a 2D finite element software.
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Figure 11.4. Examples of distribution of the local forces calculated on the teeth of an electrical motor using a 2D finite element software
11.4.5. Mechanical response modeling Once the magnetic force harmonics exerted on the studied structure are known, the analysis of the mechanical response is carried out using a finite element tool. In this case, the study of the mechanical response consists of studying the response to each force harmonic. The following simulation procedure can therefore be applied [LEF 89, 90] [IMH 89, 90]: 1) calculate the magnetic forces over a period of time corresponding to the first excitation force frequency (mechanical rotation frequency, electrical frequency or twice the electrical frequency); 2) decompose each of the local forces calculated by a Fourier analysis; 3) group the magnetic force harmonics of the same frequency, but with different application points and phases; 4) for each group, calculate the mechanical response of the stator. The discretization of the elastodynamic equations by the finite element method leads to a system of differential equations. If all the solicitation forces are sinusoidal and have the same pulsation, this system can be written in the following complex form:
K jZC Z M q 2
f
[11.49]
where K is the stiffness matrix, obtained from Young’s modulus and Poisson’s coefficients of the materials used. M is the generalized mass matrix obtained from the mass density of the material used. C is the damping matrix of the discretized structure.
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The Finite Element Method for Electromagnetic Modeling
f is the vector of the generalized forces applied to the nodes of the mesh and represents the harmonics groups of magnetic forces on the structure at the pulsation Z. q is the vector of generalized displacements at each node. Coefficients of matrices K and M are real numbers. f and q are vectors whose components are complex numbers. The velocity or accelerations on the node vectors are defined by: x
q xx
q
jZ q
[11.50]
2
Z q
Generally, the damping matrix is very difficult to calculate. In practice, the mechanical structures studied present low damping. It is thus possible to neglect this matrix as long as the response of the structure in the vicinity of the resonance frequency is not considered. In this latter case, which is the cause of the most important vibrations phenomena, the response will mainly depend on this damping matrix. The only solution, then, is to provide a response estimate involving experimental measurement of the damping value. Then the measured damping values are to be included in the modeling [CLE 95] [JAV 95]. Figure 11.4 shows the force distributions corresponding to fundamental harmonics at twice the electric and slot frequency in a synchronous motor with permanent magnets and four poles. We recognize the slot harmonic as all the forces are in phase. Figure 11.5 shows the deformations due to these two force distributions. These deformations indicate that the force distribution, corresponding to the fundamental harmonic, excites mainly mode four of the stator’s mechanical structure whereas that at the slot harmonic excites mode zero or “breathing mode”. For slot harmonics, all teeth forces are in phase, and all points of the structure move in phase. That explains why the vibrations at this frequency often have significant levels and are feared.
Figure 11.5. Deformations due to force distributions corresponding to the fundamental harmonic (left) and to the slot harmonic (right) (from a 2D finite element analysis)
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11.4.6. Modal superposition method
Even if the damping matrix C is difficult to calculate, we still know how to measure the modal damping coefficients of a real structure. The method of modal superposition allows these factors to be taken into account for an existing structure. For a structure still at the design stage, this method allows their influence to be studied by parametric variation if we have an idea of their orders of magnitude [IMB 84]. To use this method, it is first necessary to determine the eigenmodes of vibration of the mechanical structure. Since the structures studied generally present low damping, in practice, we calculate the vibration modes of the undamped structure. In a numerical model, this consists of calculating the eigenvalues :2 and the eigenvectors X corresponding to the homogenous equation of the free undamped vibration of the structure: KX
: 2 MX
[11.51]
The eigenvectors X1, X2, etc. define a new basis, called a modal basis, of finite dimensional space functions on which an approximate solution is sought. These vectors are orthogonal, two by two, and by normalizing them an orthonormal basis is obtained. The generalized displacement vectors and generalized force vectors can be expressed on this new basis: Nml
q
¦Q X i
i
i 1 Nml
f
¦F X i
[11.52]
i
i 1
where Nml is the total number of free modes calculated, Qi is the modal coordinates of the displacement vector, Fi is modal coordinates of the vector of generalized force. As the eigenvectors form an orthonormal basis: Qi
qt Xi
Fi
f t Xi
X ti f
[11.53]
Qi and Fi are components of new vectors, Q and F, related to the originals by the matrix relationships: q
X mQ
f
X mF
[11.54]
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The Finite Element Method for Electromagnetic Modeling
The matrix Xm is the basis transformation matrix from the eigenspace to the original space. The columns of Xm correspond to the eigenvectors Xi:
>X1 , X 2 ,...X Nml @
Xm
[11.55]
As eigenvectors form an orthonormal basis: 1 Xm
t Xm
[11.56]
The substitution of these relationships in the complex equation for discretized elastodynamic gives:
X tm K jZC Z 2 M X m Q
F
[11.57]
which can be put in the form:
k jZc Z m Q 2
[11.58]
F
where the matrices k and m are diagonal matrices: k
X tm KX m
m
X tm MX m
[11.59]
X tm CX m
c
The matrix c is generally not diagonal because the eigenmodes calculated are those of the free undamped structure and the modal equations can be coupled by viscous dampers. In practice, it is assumed that the damping matrix C is a linear combination of matrices M and K. In this case, the modal damping matrix c is diagonal. The modal equations are thus decoupled into a series of independent equations with a single degree of freedom:
k
i
jZ c i Z 2 m i Q i
Fi
[11.60]
The coefficients ki, mi and ci are the diagonal terms of k, m and c matrices. The pulsation :i of the eigenmode i is given by: : i2
ki mi
[11.61]
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It is to be remembered that Z is the pulsation of the applied forces. It is interesting to introduce the modal damping coefficients Hk: ck 2: k m k
Hk
[11.62]
We thus obtain the relation: Qk
Fk
m k : k2
j 2H k : k Z Z 2
[11.63]
This shows that, for each mode, we have a mechanical transfer function:
H k Z
1
m k : 2k
j 2H k : k Z Z 2
[11.64]
The structure’s response is given by: Nml
q
¦H
k
Z X k Fk
[11.65]
k 1
Taking this relation into account gives: Nml
q
¦H
k
Z X k X tk f
[11.66]
k 1
This relation can be put in the form: q
G Z f
[11.67]
where G(Z) is a transfer matrix, called a dynamic flexibility matrix. It allows the mechanical structure’s response to a distribution of sinusoidal forces of Z to be calculated directly. This matrix is given by the relation: G Ȧ
Nml
¦H k 1
k
Z X k X tk
[11.68]
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The Finite Element Method for Electromagnetic Modeling
Each coefficient Gi,j of this matrix can be calculated from the components xki of the eigenvectors Xk: Nml
Gi, j
¦H
k
Z x ki x kj
[11.69]
k 1
In practice only the structure’s first and most significant eigenmodes are considered: 1) The first eigenfrequencies as well as associated eigenmodes. 2) If there are experimental data, an estimate of the structure’s modal damping factors is done by interpolation or extrapolation of known structures’ damping factors. Otherwise, a range of damping values for each mode is assigned. 3) The real forces applied on the structure is computed by using magnetic field calculation software, allowing local magnetic forces to be calculated as a function of time. 4) Each force is decomposed into harmonics and harmonic groups of the same pulsation are formed. 5) For each harmonic group, the modal coordinates of each harmonic group Fi are calculated. 6) For each mode, the modal component Qi of the displacement vector is determined by using the transfer function Hi(Z). 7) The structural deformation, is recomposed, i.e. the nodal displacement vector q due to the harmonic groups of the force with pulsation Z. It is not worth reconstructing the temporal displacement vector’s evolution since it is the displacement spectrum, velocity and acceleration that are useful in practice. The advantage of the modal superposition method, in addition to the fact it enables us to take into account the modal damping factors, is that there is no matrix inversion to be performed when calculating the mechanical response. The calculations that take the most time are those of eigenmodes. However, these calculations are carried out once and for all, and can be used for any other solicitations [LEF 97]. 11.4.7. Conclusion
The proposed study of vibrations of magnetic origin shows that a physical and practical analysis of the problem in its context, combined with a good understanding of the numerical models, can greatly facilitate the simulation of those phenomena. This analysis allows well suited simplifying assumptions to be formulated for the
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physical problem of a specific application: neglected magnetostriction phenomena, local teeth forces and the linearity of mechanical equations. The latter assumption, thanks to the superposition principle (study of the mechanical response for each magnetic force harmonic) and to the modal superposition principle, greatly facilitates the simulation procedure. Consequently, the application of such a simulation appears realistic for a real industrial design of an electric motor taking high vibratory discretion into account. 11.5. Modeling strongly coupled phenomena
In this last section, the terms “weak” and “strong” coupling will be discussed. These terms are to be defined more precisely both from a physical view point and from a numerical simulation view point. These precisions explain the way the different corresponding examples are modeled. Then, two examples presenting a higher coupling level will be considered. This will lead to a third type of application. The first example is that of a moving magnetic structure that modifies the magnetic field, and then undergoes a major deformation. This could have been magnetoforming but a structure close to the simple switching of a contactor was chosen: it is the bi-stable deformation of a micro-beam under the action of a magnetic field created by a small electromagnet. The second example is that of an active magnetostrictive material, sometimes called “smart” material. Both bulk material and thin layer applications will be considered. The nonlinear coupling is then intrinsic to the material. 11.5.1. Weak coupling and strong coupling from a physical viewpoint
As has been presented above, physical problems where magneto-mechanical coupling occurs are extremely diverse in nature. There are two main types: – weak coupling: the magnetic quantities are hardly changed by the deformation or the rigid body motion of the studied structure; – strong coupling: the magnetic quantities are significantly changed by the deformation or the rigid body motion of the studied structure. To formalize these ideas, let us consider some examples of weak coupling: – motion of the needle of a compass in Earth’s magnetic field; – electromagnetic forces applied on fixed part of a device. This may be the case of carrying objects using an electromagnet, when it is necessary to ensure that a
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The Finite Element Method for Electromagnetic Modeling
magnetically “glued” structure will actually be kept. This calculation is also required to ensure that a structure withstands a short circuit, a lightning shock or simply the internal forces in normal operations; – small electromagnetic vibrations. The magnitude of the majority of mechanical vibrations is in fact very small compared to air-gaps of corresponding structures and, therefore, the change in magnetic quantities is negligible. Conversely, numerous structures present a strong physical coupling. They may include, for example: – most machines and actuators. They are characterized by a magneto-mechanical conversion. Except for very special cases, the displacements either significantly alter the air-gaps and therefore fields, or create variations in the magnetomotive force. Moreover, the displacement, the start-up or the acceleration due to magnetic forces or torques are already, in themselves, a strong coupling since they change the magnetic field; – vibrations significant enough for their influence on the magnetic quantities to be taken into account; – the magnetostrictive phenomena where the magnetic and mechanical quantities are intrinsically linked within the materials; – all cases of large mechanical deformation under the effect of stress of magnetic origin, for example, the magneto-forming [AZZ 99] [BEN 97]. However, the distinction is unclear and the limit is difficult to specify. Magnetic vibrations, as has been seen, will be ranked as weak or strong coupling on the basis of little or no influence on the magnetic quantities. Similarly, some machines and actuators supplied, for example, at constant current or presenting small displacements do not significantly alter the magnetic quantities. In this case, the coupling is often limited to a simple interaction via the electromagnetic forces and the dynamic motion equation. In this case, the magnetic quantities are not changed. They only generate mechanical forces that produce a simple displacement. This is called weak physical coupling. The terminology can be clarified by saying that weak coupling is used when it is unidirectional and strong coupling when, in contrast, there is a significant reciprocal interaction between magnetism and mechanics. 11.5.2. Weak coupling or strong coupling problem from a numerical modeling analysis
It may be surprising that a distinction must be made, but the terminology of “weak” or “strong” coupling are not used in the same way, for a specialist of numerical modeling or for a physicist.
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This leads specialists of numerical modeling to use the expressions differently. Some researchers, do, as do physicists, prefer to use strong numerical coupling when the physical problem treated also has a strong coupling characteristic, regardless of the numerical technique used for the solution. However, we decided thereafter, to use these expressions, according to the majority of researchers working in finite element modeling. For finite element modeling, the term of strong coupling is used when the magnetic and mechanical equations are solved simultaneously with coupling terms. More precisely it consists of solving an algebro-differential system containing coupling terms between the magnetic unknowns and mechanical unknowns. So, is there any difference? Throughout this chapter it will be demonstrated that a physical problem where the magnetic and mechanical quantities are actually coupled in a strong manner may in fact be solved by a weak numerical coupling. This is usually accomplished by successive solutions alternating a magnetic solution and a mechanical solution in an iterative and converging process. In the same way, a significant displacement can be taken into account using a step by step approach. The calculation of magnetic force at any given time allows the mechanical position at the next step to be determined, through the use, for instance, of a dynamic equation. The corresponding magnetic calculation that takes the displacement into account is now possible. This is often used and clearly demonstrates that many (if not most) strongly coupled physical problems are actually solved with successive or iterative magnetic, and then, mechanical calculations, that is a weak numerical coupling. 11.5.3. Weak coupling and intelligent use of software tools
When coupling is strong and the magnetic and mechanical unknowns at each node are solved simultaneously, no choice is possible: the same software and the same meshing should address the whole problem. However, questions arise for weak numerical couplings which are actually used to solve most applications. It concerns coupled problems that are often complex or nonlinear. Thus, simplifying assumptions are welcomed if not mandatory. Whether it concerns a simple user or a developer, the question arises if modeling must be carried out by the same mesh and the same software for the magnetic and the mechanical computations. There is obviously no systematic response. It will depend on the tools available, on the case to be handled and the context.
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However, it is good to keep in mind a number of matters, including: – for magnetism, the problem can often be handled by the finite element method (FEM) in two dimensions, either a plane model or an axisymmetric model. This is possible as most structures present a magnetic circuit that guides fluxes efficiently and minimizes magnetic leakage flux in the third dimension. This is usually irrelevant in mechanics. Sometimes, simple analytical processing often provides the acceleration, speed and position of an actuator. Then, no mechanical numerical computing is necessary and, conversely, more complex 3D mechanical FEM solving may be required, for instance, for the mechanical response to shocks or vibrations; – the mesh required for a magnetic problem is often a very dense mesh, especially near air-gaps and is a mesh composed of many triangular elements in 2D and tetrahedral elements in 3D. This is due to the necessity of meshing very complex shapes, such as, for instance, the shape of the air region that surrounds an object. In mechanics, meshing the air is useless. The rectangular elements or bricks are more efficient and a much lower mesh density often solves the question. In mechanics, reducing the number of unknowns is even more important as the number of degrees of freedom per node is typically much higher. Furthermore, highly stressed regions are required to be finely meshed while the same regions, magnetically, are not. It may therefore be appropriate or even necessary, to fully differentiate the magnetic and mechanical meshes. They should also be optimized respectively for each of the problems. In this case, techniques must be developed to transfer data issued from one mesh to the other, for example, by interpolating values or passing from distributed magnetic forces density to the equivalent global force for the magnetic problem (see the motor stator’s teeth in section 11.4.4); – beyond meshing, it is the structure itself which may be different from the magnetic problem to the mechanical problem. As was already seen, the air regions do not need to be meshed in mechanics. Other examples are non-magnetic materials such as plastics or woods that do not require to be modeled in magnetism. Similarly, if the deformation problem is of concern, it is enough, in mechanics, to consider only the deformable part on which the original magnetic forces will be transferred. Eventually, mechanics often simplifies matters by considering 3D bodies as hulls or beams. It follows that it can be very advantageous not to consider the same field of study, the same objects or the same geometry at all. It results that it is essential, though difficult, to transfer data from one problem (geometry and mesh) to the other. This leads to the development of interpolation, extraction or data projection techniques [REY 94]. As a conclusion, it is obvious that the natural trend to cope with a weak coupling is not necessarily the optimal solution. Just to find the “best” software, enter the problem, mesh it and successively solve it for magnetic and then mechanical data often provides poor results.
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11.5.4. Displacement and deformation of a magnetic system
A case similar to those addressed in section 11.3 is now under consideration. However, the difference is that the mechanical structure is now deformed. It is no longer a coupling where the determination of magnetic forces is simply injected into the mechanical problem by using only the dynamic motion equation. There is a real mechanical problem to be handled with reciprocal interaction between magnetic and mechanical quantities.
Figure 11.6. Basic principle of optical micro-switch with magneto-mechanical operation
This problem is relatively simple. It is an optical micro-switch with magnetic control. It can be transposed to any type of bi-stable micro-switches. A basic silicon microstructure consisting of a suspended silicon beam (cantilever) supporting a vertical mirror is attracted by an electromagnet thanks to soft material deposition. The magnetic flux density of the system is modified by the cantilever deformation and its displacement from zero position to latching. It is a bi-stable system, with the presence of a magnet that allows permanent latching. The other stable position is due to the beam’s mechanical rigidity. In the “open” position (point A) the magnetic force is weak and counterbalanced by the structure’s rigidity (straight line with negative slope). In the “closed” or latching position (point B) the absence of an air-gap allows the force created by the magnet (central hyperbolic curve) to prevail over the beam’s rigidity. The system is bi-stable when the two curves have two points of intersection. The bi-stability therefore results from a tradeoff between the two forces, the switching being achieved by supplying the electromagnetic coil in order to have either an additive field (closing, higher hyperbolic curve without any intersection) or a subtractive field (opening, lower curve) for the field generated by the magnet.
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Figure 11.7. Magneto-mechanical bi-stability
Figure 11.8. Optimization of the magnetic structure with FLUX (www.cedrat.com)
As shown in Figures 11.6 and 11.8, the magnet must be placed along the magnetic circuit, “in parallel”. This can be explained as, for such a size, a magnet placed into the magnetic circuit would create far too high magnetic flux density which would require too high switching currents. The maximum thickness of the magnet would have to be in the order of several tenths of a micron. This clever solution of a parallel magnet allows most of the magnetic flux to be deliberately lost by leakages and partial saturation of the corresponding part of the magnetic circuit. The small part of the remaining magnet’s magnetic flux density is advantageously used for the latching. The magnetic modeling is nonlinear and must take the position of the bent cantilever into account. Similarly, the mechanical
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deformation is dynamic (inertial effects) and results from a tradeoff between the structural deformation and force momentum. “Momentum” means that the place of application of the resulting force is important and that it is thus necessary to know the local magnetic force distribution.
Figure 11.9. Magnetic flux density and corresponding magnetic forces on the mobile part
Air gap
Cantilever Winding Permanent magnet
Yoke
Figure 11.10. Model and mesh used for the modeling by ANSYS
The magneto-mechanical calculation was carried out with ANSYS software. As the same software and the same mesh are used, we can either solve step by step through strong coupling at each time step, or through successively solving the magnetic and then the mechanical problems. Both types of solution have provided similar results with comparable computing times. 11.5.5. Structural modeling based on magnetostrictive materials
Magnetostriction effects in industrial steel sheets were presented in section 11.4.1. For such materials, it is usually a side-effect phenomenon and an important
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drawback at the source of noises and vibrations. Giant magnetostrictive materials are now considered. They are interesting for several reasons. Based on alloys of iron and rare-earths, they present a magnetostrictive effect two orders of magnitude higher than conventional materials. Their deformation ranges from a few hundred to 2,000 ppm. This pushes magnetostriction from a side-effect drawback to a direct competitor of piezomagnetic phenomena as possible use for actuation. Giant magnetostrictive materials have been developed both in bulk shapes and thin layers. Equivalent in magnetism to piezoelectric materials, they are the only materials where the dominant effect is magneto-mechanical coupling within the matter itself. Numerical modeling of piezoelectric materials is very close. What follows is therefore partly transposable to piezoelectric materials. However, giant magnetostrictive characteristics present very strong nonlinearities, as shown in Figure 11.11. An important hysteresis phenomenon is also present. The magneto-mechanical characteristics are stress dependent. They are globally of a parabolic shape before saturation effects limit the deformation. The transition from parabolic to saturation states allows the characteristic to present a relatively important linear domain. This linear part is, in practice, intensively used with “small” signal deformation around point B. Deformation
B
H
A 'H AC
Figure 11.11. Typical shape of magnetostrictive curves (deformation vs. field)
A frequency doubler can be obtained when using these materials around point A (Figure 11.11). However, in most uses their point of operation moves around point B in order to obtain the benefit of the strongest possible slope. This is the case when using these materials as shakers, in order to excite large structures and obtain their frequency response or create an active vibration compensation. They must therefore
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be magnetically prepolarized (usually with magnets located in the circuit) and a mechanical pre-stress is also required for performances. As mentioned above, it is a strong physical coupling within the matter itself. A strong numerical treatment is used mostly with a finite element method and a matrix characterized by coupled terms between magnetic and mechanical quantities. Tools have been developed to model this coupling in small-signal dynamics (small variations around point B), especially the software ATILA [ATILA]. The material is supposed to behave linearly around point B. Numerical models are similar to the ones used for piezoelectric materials. However, the method, with the use of complex numbers, cannot take into account the nonlinearities. It is therefore restricted to small variations. Furthermore, the values to be assigned to the coupling terms are stress dependent, and this requires the availability of the corresponding experimental data. 1 2
3
8
2
6
9 7
4
10
5
12
12
11
13
Figure 11.12. Magnetostrictive bar: pre-stressed and pre-polarized
A more precise modeling is possible to take into account real nonlinear characteristics. With strong coupling, static or step by step resolution, nonlinearities and hysteresis, it leads to extremely complex modeling. Furthermore, experimental data necessary for the model are often missing. Another option is to solve the problem through successive iterations of magnetic and mechanical processing. As shown in Figure 11.13, the coupling and the nonlinearities are then introduced at
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The Finite Element Method for Electromagnetic Modeling
each step by the experimental data. As it concerns discrete and multivariate data, it is necessary to handle the appropriate tools of interpolation from databases (for example, spline surfaces) [BEN 93/1, 93/2, 94, 95/1, 95/2] [BES 96] [GRO 98].
o
Figure 11.13. Strong physical coupling calculated by iterative magnetic and mechanical independent calculations
It should be noted that large displacements can be obtained by combining elementary displacements and/or associating the magnetostrictive element with clamps. This is the principle of some magnetostrictive rotating motors with very strong torques or inchworm systems as in Figure 11.14. TRANSLATION
ROTATION
ٞl
H=0
H=0
Ø 9 mm
rٞ
Ø 1.5 mm
rٞ H=0
H=0
40 mm
Figure 11.14. Magnetostrictive mini-inchworm system developed at G2E lab, Grenoble
As mentioned above, giant magnetostrictive materials are available in thin layers. They are therefore used according to the bimorph principle in order to make all types of micro-systems. The control command being magnetic, high frequencies can be achieved. The problem of the material’s high price related to the rare-earths thus disappears since very low quantities of material are required. Remarkable
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advances have been made on this issue and relatively thick layers can now be obtained. They are composed of many successive magnetic and magnetostrictive nano-layers so as to obtain a deformation even at very low fields (PoH on the order of 5 to 10 mT). Numerical modeling is very similar to that used for bulk magnetostrictive materials [BOD 96, 97/1, 97/2, 97/3] [VAN 04]. 11.5.6. Electromagnetic induction launchers
This application illustrates the presence of induced currents associated with displacement as it has not yet been considered in this chapter. This corresponds to a few specific applications such as, for instance, induction motors. Taking into account the relative displacement of moving parts is, thus, a crucial point. This example illustrates another important point which is the need to take into account thermal problems, changes in temperature and its influence on the magnetic or even mechanical characteristics of the materials used. Differences in time constants between magnetism, mechanics and heat can often simplify the modeling. By assuming for instance that temperature changes are slow, they can be neglected for a given magneto-mechanical modeling. Thus, the strong thermocoupling disappears. For induction launchers, the time constants are close. Therefore, it is a real, strong coupling between magnetism, thermal energy and mechanics. It will be necessary at each time step to take into account the thermal, magnetic and mechanical and even electric (power) changes and their reciprocal influence. It is a real strong coupled problem with many variables. For those who are not familiar with electric launchers, they simply consist of finding an electromagnetic way to launch an object very quickly. It is a powerful linear actuator whose applications might be military (replacing powder guns) or civil (transportation of material to, or in, the space). We have chosen this example but it is directly transposable to levitation trains based on induction phenomena.
Figure 11.15. Example of an induction launcher structure
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The Finite Element Method for Electromagnetic Modeling
Induction launchers are based on the creation of induced currents and associated propulsion forces on a conducting projectile. Rapid magnetic field variations are generated thanks to high currents switched in successive coils, usually from sequential capacitor discharges. These launchers are mechanically “without contact”, as opposed to rail launchers which inject the current in the projectile that reacts to Lorentz forces. Consequently, the problem is to model high acceleration with capacitor discharges, high induced currents and important displacements. These displacements of magnetized parts create induced voltages in the windings, thus modifying the discharge currents. Skin effects, heating, displacement and possible structural deformations must be taken into account [JAR 93, 94]. The problem is solved by step by step finite elements analysis. Electrical coupling is taken into account as well as the effects due to projectile movement and corresponding induced currents on the supply. At each time step, the current evolution must be determined (prediction-correction system), the influence of local heating increments on the evolution of material properties must be taken into account, and the forces deduced to determine the displacement for the next step. 11.6. Conclusion
Comprehensive study of all magneto-mechanical coupling is impossible. The domain is much too large and diversified as it has been shown. However, terminology, theoretical formula, numerical tools and many examples have been provided. Moreover, clever ways to simplify and efficiently cope with most complex coupled problems have been provided. Thus, magneto-mechanical modeling of electromagnetic forces, magnetostriction, and electromagnetic vibrations has been proposed. In addition to the basic tools provided, many and diverse examples have been developed. They cover most of the field, addressing the simple determination of a force or a torque, the calculation of displacement in objects or motors, vibration problems, large deformations and lastly very strongly coupled problems, including those with giant magnetostriction materials. Major improvements in numerical modeling in the field of mechanics and electromagnetism allow the simulation of many magneto-mechanical problems with commercial software tools. However, these tools can never address all specific applications. Many problems require a specific response.
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Actual engineer analysis of magneto-mechanical phenomena in a given actuator performing a specific function in a known environment, usually leads to major simplifying assumptions. This eventually allows many complex problems to be solved with existing software tools. However, these problems remain complex to solve. One reason among others is the difficulty of representing the real magnetic and mechanical behavior of materials in a numerical model. Research efforts and experimental set ups involving specialists in mechanics, electromagnetism and numerical analysis are often required to obtain appropriate data and characteristics The industrial demand and requirements for magneto-mechanical modeling, pushes for a rapid development of commercial software. They already provide efficient models and simulations for many applications. However, new more complex couplings are still to be modeled and many improvements are still expected. 11.7. References [ATILA] ATILA, 3D CAD software for piezoelectric & magnetostrictive structures, IEMN/Isen, Lille (F), Distr. Cedrat, Meylan (F) email [email protected] and Magsoft, Troy, New York, USA. [AZZ 99] AZZOUZ F., BENDJIMA B., FELIACHI M., LATRECHE M.E., “Application of macro-element and finite element coupling for the behavior analysis of magnetoforming systems”, IEEE Trans. on Magnetics, vol. 35, no. 3, p. 1845-1848, May 1999. [BAS 03] BASTOS J.P.A., SADOWSKI N., Electromagnetic Modeling by Finite Elements, Marcel Dekker, New York, USA, 2003. [BEN 93/1] BENBOUZID M.E.H., REYNE G., MEUNIER G., “Non-linear finite element modelling of giant magnetostriction”, IEEE Trans. on Mag., vol. 29, no. 6 p. 2467-2469, November 1993. [BEN 93/2] BENBOUZID M.E.H., REYNE G., MEUNIER G., BENDAAS M.C., “Variational formulation for nonlinear FEM of Terfenol-D rods using surface splines for material indata”, Elsevier Studies in Applied Electromagnetics in Materials, Magnetoelastic Effects & Ap., vol. 4, p. 185-191, May 1993. [BEN 93/3] BENBOUZID M.E.H., REYNE G., DEROU S., FOGGIA A., “Finite element modeling of a synchronous machine: electromagnetic forces and mode shapes”, IEEE Trans. on Mag., vol. 29, no. 2, p. 2014-2018, March 1993. [BEN 94] BENBOUZID M.E.H., Modélisation de la magnétostriction géante et application aux dispositifs électromagnétiques à base de TERFENOL-D, PhD thesis, Institut National Polytechnique de Grenoble, Grenoble, 1994.
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[BEN 95/1] BENBOUZID M.E.H., BODY C., REYNE G., MEUNIER G., “Finite element modelling of giant magnetostriction in thin films”, IEEE Trans. on Mag., vol. 31, no. 6, p. 3563-3565, November 1995. [BEN 95/2] BENBOUZID M.E.H., REYNE G., MEUNIER G., KVARNJO L., ENGDAHL G., “Dynamic modelling of giant magnetostriction in Terfenol-D rods by the FEM”, IEEE Trans. on Mag., vol. 31, no. 3, p. 1821-1824, May 1995. [BEN 97] BENDJIMA B., SRAIRI K., FELIACHI M., “A coupling model for analysing dynamical behaviours of an electromagnetic forming system”, IEEE Trans. on Magnetics, vol. 33, no. 2, March 1997. [BES 96] BESBES M., REN Z., RAZEK A., “Finite element analysis of magneto-mechanical coupled phenomena in magnetostrictive materials”, IEEE Trans. on Mag., vol. 32, no. 3, p. 1058-1061, May 1996. [BOD 96] BODY C., Modélisation des couches minces magnétostrictives. Application aux microsystèmes, PhD thesis, Institut National Polytechnique de Grenoble, Grenoble, 1996. [BOD 97/1] BODY C., REYNE G., MEUNIER G., “Modelling of magnetostrictive thin films, application to a micromembrane”, Journal de Physique III, vol. 7, p. 67-85, 1997. [BOD 97/2] BODY C., REYNE G., MEUNIER G., “Non linear FEM of magneto-mechanical phenomenon in giant magnetostrictive thin films”, IEEE Trans. on Mag. vol. 33, no. 2, p. 1620-1623, March 1997. [BOD 97/3] BODY C., REYNE G., MEUNIER G., QUANDT E., SEEMAN K., “Application of magnetostrictive thin films for microdevices”, IEEE Trans. on Mag., vol. 33, no. 2, p. 2163-2166, March 1997. [BOS 92] BOSSAVIT A., “Edge-element computation of force field in deformable bodies”, IEEE Trans. on Mag., vol. 28, no. 2, p. 1263-1266, March 1992. [BRO 62] BROWN W.F., Magnetostatic Principles in Ferromagnetism, North-Holland Publishing Co., Amsterdam, 1962. [CAR 59] CARPENTER C.J., “Surface integral methods of calculating forces on magnetized iron parts”, IEE Monograph, no. 342, August 1959. [CLE 95] CLENET S., JAVADI H., LEFEVRE Y., ASTIER S., LAJOIE-MAZENC M., “Theoretical and experimental studies of the effects of the feeding currents on the vibration of magnetic origin of permanent magnet machines”, IEEE Trans. on Mag., vol. 31, no. 3, p. 1837-1842, May 1995. [COU 83] COULOMB J.L., “A methodology for the determination of global electromechanical quantities for finite element analysis and its application to the evolution of magnetic forces, torques and stiffness”, IEEE Trans. on Mag., vol. 19, no. 6, p. 25142519, November 1983. [COU 84] COULOMB J.L., MEUNIER G., “Finite element implementation of virtual work principle for magnetic and electric force and torque computation”, IEEE Trans. on Mag., vol. 20, no. 5, p. 1894-1896, September 1984.
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[DAN 04] DANIEL L., HUBERT O., BILLARDON R., “Homogenisation of magnetoelastic behaviour: from the grain to the macro scale”, Computation and Applied Mathematics, vol. 23, no. 2-3, p. 285-308, 2004. [DEL 00] DELAERE K., HEYLEN W., HAMEYER K., BELMANS R., “Local magnetostriction forces for finite element analysis”, IEEE Trans. on Mag., vol. 36, no. 5, p. 3115-3118, September 2000. [DUR 68] DURAND E., Magnétostatique, Masson, Paris, 1968. [FLUX2D] FLUX2D & FLUX3D: supplied by CEDRAT, 4301 ZIRST, 38943 Meylan, France and MAGSOFT, Peoples Av., Troy, New York, USA. [GAB 99] GABSI M., CAMUS F., BESBES M., “Computation and measurement of magnetically induced vibrations of switched reluctance machine”, IEE Proc. Elec. Power Appl., vol. 146, no. 5, September 1999. [GRO 98] GROS L., REYNE G., BODY C., MEUNIER G., “Strong coupling magneto mechanical methods applied to model heavy magnetostrictive actuators”, IEEE Trans. on Mag., vol. 34, no. 5, p. 3150-3153, September 1998. [HIR 95] HIRSINGER L., BILLARDON R., “Magneto-elastic finite element analysis including magnetic forces and magnetostriction effects”, IEEE Trans. on Mag., vol. 31, no. 3, p. 1877-1880, May 1995. [HEN 92] HENNEBERGER G., SLATTER P.K., HARDRYS W., SHEN D., “Procedure for the numerical computation of mechnical vibrations in electrical machines”, IEEE Trans. on Mag., vol. 28, no. 2, p. 1351-1354, March 1992. [IMB 84] IMBERT J.F., Analyse des structures par éléments finis, (2nd ed.) CEPADUES, 1984. [IMH 89] IMHOFF J.F., MEUNIER G., REYNE G., FOGGIA A., SABONNADIERE J.C., “Spectral analysis of electromagnetic vibrations in DC machines through the F.E.M.”, IEEE Trans. on Mag., vol. 25, no. 5, p. 3590-3592, September 1989. [IMH 90] IMHOFF J.F., REYNE G., FOGGIA A., SABONNADIERE J.C., “Modelisation des phénomènes electromagnétiques et mécaniques couplés: application à l’analyse vibratoire des machines electriques”, Revue de Physique Appliquee, 07/90, p. 627-648, special no.: “Modélisation des champs électromagnétiques”. [JAR 93] JARNIEUX M., GRENIER D., REYNE G., MEUNIER G., “F.E.M. of eddy currents and forces in moving systems. Application to linear induction launcher”, IEEE Trans. on Mag., vol. 29, no. 2, p. 1989-1992, March 1993. [JAR 94] JARNIEUX M., REYNE G., MEUNIER G., “FEM modelling of the magnetic, thermal, electrical and mechanical transient phenomena in linear induction launchers”, IEEE Trans. on Mag., vol. 30, no. 5, p. 3312-3315, September 1994. [JAV 95] JAVADI H., LEFEVRE Y., CLENET S., LAJOIE-MAZENC M., “Electromagneto-mechanical characterizations of the vibrations of magnetic origin of electrical machines”, IEEE Trans. on Mag., vol. 31, no. 3, p. 1892-1895, May 1995.
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[KIM 05] KIM U., LIEU D.K., “Effects of magnetically induced vibration force in brushless permanent-magnet motors”, IEEE Transactions on Magnetics, vol. 41, no. 6, p. 2164-2172, June 2005. [LAN 84] LANDAU L.D., LIFSCHITZ E.M., PITAEVSKI L.P., Electrodynamics of Continuous Media, 2nd ed., Pergamon Press, Oxford, 1984. [LEF 88] LEFEVRE Y., LAJOIE-MAZENC M., DAVAT B., “Force calculation in electromagnetic devices”, in Electromagnetic Fields in Electrical Engineering, Savini A. and Turowski J. (eds.), Plenum Press, New York, USA, 1988. [LEF 89] LEFEVRE Y., DAVAT B., LAJOIE-MAZENC M., “Determination of synchronous motor vibrations due to electromagnetic force harmonics”, IEEE Trans. on Mag., vol. 25, no. 4, p. 2974-2976, July 1989. [LEF 90] LEFEVRE Y., DAVAT B., LAJOIE-MAZENC M., “Vibration of magnetic origin in permanent magnet synchronous motors”, in Electromagnetic Fields in Electrical Engineering, Turowski J. and Zakrzewski K. (eds.), James & James Sciences Publishers Limited, London, UK, 1990. [LEF 97] LEFEVRE Y., De la modélisation des vibrations d’origine magnétique à la conception des machines silencieuses, Journée des vibrations et des bruits acoustique des machines électriques, ENS Cachan, Paris, p. 24-38, April 1997. [MED 99] MEDEIROS L.H.A., REYNE G., MEUNIER G., “About the distribution of forces in permanent magnets”, IEEE Trans. on Mag., vol. 35, no. 3, p. 1215-1218, May 1999. [MED 00] MEDEIROS L.H.A., REYNE G., MEUNIER G., “A unique distribution of forces in permanent magnets using scalar and vector potential formulations”, IEEE Trans. on Mag., vol. 36, no. 5, p. 3345-3348, September 2000. [MEL 81] MELCHER J.R., Continuum Electromechanics, MIT Press, 1981. [MOO 84] MOON F.C., Magneto Solid Mechanics, John Wiley & Sons, 1984. [MUL 90] MULLER W., “Comparison of different methods of force calculation”, IEEE Trans. on Mag., vol. 26, p. 1058, 1061, 1990. [REN 92a] REN Z., RAZEK A., “Local force computation in deformable bodies using edge elements”, IEEE Trans. on. Mag., vol. 28, no. 2, p. 1212-1215, March 1992. [REN 92b] REN Z., BOSSAVIT A., “A new approach to eddy current problems in deformable conductors and some numerical evidence about its validity”, International Journal of Applied Electromagnetics in Materials, vol. 3, p. 39-46, 1992. [REN 94] REN Z., “Comparison of different force calculation methods in 3D finite element modelling”, IEEE Trans. on Mag., vol. 30, no. 5, p. 3471-3474, September 1994. [REY 87] REYNE G., Analyse théorique et expérimentale des phénomènes vibratoires d'origine électromagnétique, PhD thesis, INPG, 1987. [REY 88] REYNE G., SABONNADIERE J.C., IMHOFF J.F., “Finite element modelling of electromagnetic force densities in DC machines”, IEEE Trans on Mag., vol. MAG-24, no. 6, p. 3171-3173, November 1988.
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[REY 94] REYNE G., MAGNIN H., BERLIAT G., CLERC C., “A supervisor for the successive 3D computations of magnetic, mechanical and acoustic quantities in power oil inductors and transformers”, IEEE Trans. on Mag., vol. 30, no. 5, p. 3292-3295, September 1994. [SAD 92a] SADOWSKI N., LEFEVRE Y., LAJOIE-MAZENC M., BASTOS J.P., “Sur le calcul des forces magnétiques”, Journal de Physique III, p. 859-870, May 1992. [SAD 92b] SADOWSKI N., LEFEVRE Y., LAJOIE-MAZENC M., CROS J., “Finite element torque calculation in electrical machine while considering the movement”, IEEE Trans. on Mag., p. 1410-1413, March 1992. [SAD 92c] SADOWSKI N., LEFEVRE Y., LAJOIE-MAZENC M., BASTOS J.P., “Calculation of transient electromagnetic forces in an axisymetrical electromagnet with conductive solid parts”, Compel, vol. 11, no. 1, p. 173-176, March 1992. [SAD 96] SADOWSKI N., LEFEVRE Y., NEVEC G.C., LAJOIE-MAZENC M., “Finite element coupled to electrical circuit equations in the simulation of switched reluctance drives: Attention to mechanical behavior”, IEEE Trans. on Mag., vol. 32, no. 3, p. 10861089, May 1996. [SAL 95] SALON S.J., SLAVIK C.J., DEBORTOLI M.J., REYNE G., “Analysis of magnetic vibrations in rotating electric machines”, Chapter 4 in Finite Elements, Electromagnetics and Design, p. 116-178, Elsevier, 1995. [SAN 06] SANCHEZ-GRANDIA R., VIVES-FOS R., AUCEJO-GALINDO V., “Magnetostatic Maxwell’s tensors in magnetic media applying virtual works method from either energy or co-energy”, Eur. Phys. J. Appl. Phys., vol. 35, p.61-68, 2006. [STR 61] STRATTON J.A., Théorie de l’électromagnétisme, Dunod, Paris, 1961. [TRE 93] TRÉMOLET DE LACHEISSERIE E., Magnetostriction: Theory and Applications, CRC Press, Boca Raton, USA, 1993. [VAN 01] VANDEVELDE L., MELKEBEEK J.A.A., “Magnetic forces and magnetostriction in ferromagnetic material”, International Journal for Computations and Mathematics in Electrical and Electronic Engineering (Compel), vol. 20, no. 1, p. 32-50, 2001. [VAN 04] VANDEVELDE L., GYSELINCK J., A. C. DE WULF M., MELKEBEEK J.A.A., “Finite-element computation of the deformation of ferromagnetic material taking into account magnetic forces and magnetostriction”, IEEE Transactions on Magnetics, vol. 40, no. 2, p. 565-568, March 2004. [VAS 91] VASSENT E., MEUNIER G., FOGGIA A., REYNE G., “Simulation of induction machine operation using a step by step F.E.M. coupled with circuit and mechanical equations”, IEEE Trans. on Mag., vol. 27, no. 6, Part 2, p. 5232-5234, November 1991. [WOO 68] WOODSON H.H., MELCHER J.R., Electromechanical Dynamics, Part II: Fields, Forces and Motion, John Wiley & Sons, 1968.
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Chapter 12
Magnetohydrodynamics: Modeling of a Kinematic Dynamo
12.1. Introduction 12.1.1. Generalities In this chapter, we are interested in the growth of an electromagnetic instability produced by the motion of an electrically conducting fluid. Let us take an initially non-zero magnetic field. Let us consider a stationary flow and assume that we can regulate the intensity of the flow without changing its geometry. We observe that the magnetic field is deformed by the flow. Let us suppose that subsequently the initial source of magnetic field is abruptly removed, two cases are then possible: – either the magnetic intensity disappears over time. This is called magnetic diffusion; – or the magnetic intensity does not decrease over time. This is then a dynamo instability1. Let us consider for example the case of the deformation of a uniform field by a fluid in rotation. In order to simplify the problem, the flow will be taken as a cylinder in rotation around its revolution axis [PAR 66] [WEI 66]. An initial field perpendicular to the rotation axis of the cylinder is chosen. The motion of the Chapter written by Franck PLUNIAN and Philippe MASSÉ. 1 It can happen that the magnetic intensity increases only in the presence of an initial field and that once this is removed the magnetic intensity decreases. This is then a magnetic amplifier and not a dynamo instability [KOL 61].
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The Finite Element Method for Electromagnetic Modeling
cylinder generates an electric current density. This current density induces a magnetic field which is added to the initial field. If the cylinder rotation rate is constant, then the magnetic field reaches a stable state. The final configuration of the field depends on the cylinder rotation rate (Figure 12.1). For a high rotation rate (the 4th image in Figure 12.1), the magnetic field is expelled towards the periphery of the cylinder. If at a given moment the initial magnetic field is removed then the geometry of the deformed field persists, but its intensity dies out over the time. The body rotation is thus not a dynamo.
Figure 12.1. Lines of magnetic field deformed by a cylinder in rotation (anti-clockwise direction). From left to right, rotation is increasing
Now let us consider the preceding motion to which we add a velocity component in the direction of the axis of the cylinder (screw motion). The geometry of the magnetic field is then organized according to a double helix (Figure 12.2). Beyond a velocity threshold, the magnetic intensity grows over time even if removing the initial source of field. The screw motion is thus a dynamo.
Figure 12.2. Distribution of the magnetic energy generated by a screw motion [PLU 96]
Magnetohydrodynamics
479
To know whether a flow is a dynamo or not, there is no general rule, except some anti-dynamo theorems [MOF 78] [HID 79] [PRO 79], which allows us to exclude flows which are too simple (e.g. 2D flows). Each flow must thus be the subject of a particular study. Of course, it is actually impossible to regulate the flow intensity without changing its geometry. Indeed, starting at high velocity, the geometry of the flow cascades to smaller structures. The size of these structures, their distribution and their behavior evolve in a non-deterministic way (turbulent flow). In addition, even assuming that the flow is known and stationary, the magnetic intensity in the case of dynamo instability cannot increase indefinitely (according to the first principle of thermodynamics). Actually, the magnetic growth is accompanied by the feedback action of the magnetic field on the flow via Laplacian forces. Thus, in accordance with Lenz’s law, the growth of the field is opposed to the fluid motion which generated it. The geometry of the flow, in theory, is thus modified by the dynamo instability. However, if we are interested in the appearance of the dynamo instability (zero growth rate) then the assumption of a geometry of flow determined a priori is relevant. In addition, even in the presence of a strongly turbulent fluid, we can in some cases consider only the average flow and show a posteriori that the turbulent fluctuations do not significantly influence the conditions of appearance of the dynamo effect [KRA 80]. Lastly, we have seen that the appearance of the dynamo instability requires a sufficiently vigorous flow (velocity threshold). Actually, a low flow rate can be compensated by the electric conductivity of the fluid, its magnetic permeability or by the characteristic scale of the flow. A dynamo flow can thus be, in theory, laminar or slightly turbulent, though it is not likely to happen in natural objects or experiments. When the geometry of the flow is considered known, the problem is called kinematic, in opposition to the dynamic problem which requires studying not only the evolution of the magnetic field but also that of the flow. In this chapter we limit ourselves to the study of the kinematic problem. The kinematic theory of the dynamo effect [MOF 78] [ROB 92] helps us to understand the physics of magnetic field generation. Classes of dynamo have been identified: slow or fast dynamos, dynamos with scale separation (mean-field theory [KRA 80]) or not. Dynamo mechanisms have been identified: stretching, twisting, folding, shearing [CHI 95], alpha effect, etc. We will see some dynamo examples and their corresponding mechanisms. The study of dynamo instability was born historically from the interest expressed in the origin of the magnetic field in the sun. It was extended to the majority of astrophysical objects (planets, stars, galaxies, etc.) [WEI 94] [RAD 95]. In fact, although the fluid contained in these objects is actuated by a relatively slow motion, the characteristic scales of the flow are so considerable that the dynamo effect is obtained almost systematically. Conversely, in the experiments undertaken in a
480
The Finite Element Method for Electromagnetic Modeling
laboratory we try to compensate the relatively small characteristic dimension of the flow with a significant flow speed. In addition, it is for this reason that the dynamo effect is difficult to reproduce in a laboratory and does not have any practical or industrial application to date. The major part of the solar magnetic field is in fact undetectable because it is hidden in the bottom of the convective zone. However, this part appears due to excursions out of the convective zone. These eruptions form arches on the surface of the sun (visible with the bare eye during an eclipse or with adapted optical instruments). They are at the origin of the sunspots. Additional evidence of solar magnetic activity is the dynamics of sunspots. These sunspots migrate towards the equator with a cycle of 22 years. Solar dynamo models try to reproduce the characteristics of this cycle (butterfly diagram) [PRI 82] [SCH 92]. On a larger time scale, the appearance of the sunspots is no longer cyclic but of a chaotic nature. The dynamo effect is also at the origin of the terrestrial magnetic field. The nonlinear nature of the phenomenon is at the origin of the chaotic inversions of the magnetic dipole which paleomagneticians have highlighted by analyzing cooled lava or submarine sedimentary layers [COX 69] [MER 95] [VAL 93]. The electrically conducting fluid necessary for the geodynamo is the iron contained in the Earth’s core. Its motion is of thermal origin. At the center, the liquid iron solidifies to form a solid inner core. This solidification is accompanied by the release of heat resulting in a natural convection mode of the liquid core. Dynamo instability has also been the subject of theoretical [PLU 96] [PLU 98] [PLU 99] and experimental [ALE 00] studies in the liquid sodium cooling circuits of fast breeder reactors (FBR). In fact, the large volume of liquid sodium (electrically conducting) contained in the primary and secondary circuits, an appropriate flow geometry and a strong motion make FBR potential candidates for dynamo instability. In addition, if the steel used for the realization of the assemblies in which sodium circulates has a strong magnetic permeability (ferromagnetic steel), then dynamo instability is favored [PLU 95] [SOT 99]. Recently several research teams tried to reproduce a dynamo effect in a laboratory. Two experiments (also in liquid sodium) have been a success to date ([GAI 00] [GAI 01] [BUS 96] [RAD 98] [STI 01]). These are types of semidynamics experiments. Indeed, the flow forcing makes it possible to produce an average flow geometry quasi-independent of the forcing intensity on the one hand and of magnetic intensity on the other hand (similar to the kinematic context). Thus, above a certain threshold forcing power, magnetic energy increases exponentially. After a certain time, it is stabilized at a value depending on the intensity of forcing power. This balance results from the feedback action of the magnetic field on the flow intensity. The average flow is thus slowed down but its geometry is not
Magnetohydrodynamics
481
affected (or only slightly) as it would be it in a complete dynamic context. A second generation of experiments, dynamic experiments, is under study. Recently a third experiment has produced dynamo action within a strongly turbulent flow [MON 06]. In the nonlinear regime and for some parameters of the experiment, chaotic polarity reversals have been obtained [BER 07]. In this chapter, we are interested in modeling the kinematic dynamo effect using the finite element method. This enables us to determine the conditions of appearance of the dynamo instability for a given flow and to understand the physical mechanisms leading to this instability. The formulations which we present have been tested and validated. They have the advantage of being the least constraining possible (for example, at the level of electromagnetic property jumps) to allow discontinuities of electromagnetic properties. Since this chapter was written, other formulations have been proposed [LAG 06, GUE 07]. 12.1.2. Maxwell’s equations and Ohm’s law The usual electromagnetic quantities (magnetic field H, magnetic induction B, electric field E, current density J, electric charge q) have a behavior entirely described by: – Maxwell’s equations
u(H) j B 0 ;
w(HE) ; wt
HE q ;
u(E)
w(B) wt
H B
P
[12.1] [12.2] [12.3] [12.4] [12.5]
– Ohm’s law (expressed for a conducting fluid of velocity V)
j V(E V uB)
[12.6]
P, H, V and q being the magnetic permeability and permittivity, the conductivity and the electric charge respectively. In the case of liquid metals, displacement currents w(HE)/wt are negligible with respect to j in [12.1]. Equation [12.1] ҏis thus reduced to:
u(H) j
[12.7]
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The Finite Element Method for Electromagnetic Modeling
In insulating zones (V = 0), such as air, for example, [12.6] implies that there is no electric current j. It is thus deduced from [12.7] that H is derived from a gradient. In insulating zones, magnetic potential M is thus defined by:
H M
[12.8]
The conditions of continuity at two sub-domain interfaces are deduced from Maxwell’s equations and Ohm’s law:
[Eun] 0 ; [V(E V uB)n] 0 ; [Bn] 0 ; [H un] 0
[12.9a;b;c;d]
where n is defined as the normal vector directed towards the outside of the interface (I) or the border of the integration domain (*). 12.1.3. The induction equation By combining equations [12.2], [12.5], [12.6] and [12.7] the induction equation is obtained:
wB u(V uB)u§ 1 u B · ¨V wt P ¸¹ ©
[12.10]
It can be noted that [12.10] implies w (B) 0 . Relation [12.3] is thus an initial
wt
condition in time for B. By performing the scalar product of [12.10] by B/P, and while integrating on all the modeling space, we obtain:
B2 j2 w d : ( E u H ) ndS ( j u B ) Vd : d: 0 ³³ ³³³ ³³³ wt ³³³ 2P V (:) (*) (:) (:)
[12.11]
The flux of the Poynting vector (2nd term) on a surface laid out ad infinitum is zero for the dynamo. Equation [12.11] thus expresses that the magnetic energy production (1st term) is possible provided that the work of the Laplacian forces (3rd term) is higher than dissipation by the Joule effect (4th term). This condition on the Laplacian forces requires a flow velocity of adequate geometry and of sufficiently high intensity. One of the difficulties of experimental reproduction devices is to find
Magnetohydrodynamics
483
a way of generating a flow whose geometry produces a dynamo effect at a reasonable speed. 12.1.4. The dimensionless equation As often in physics, it is interesting to handle dimensionless equations whenever possible. In the case where V and P are constant in the flow, the induction equation is written:
wB 'u(V'uB)Rm1'u 'uB wt'
[12.12]
where Rm is the magnetic Reynolds number, defined by:
Rm VPVL
[12.13]
The ƍ indicate dimensionless quantities, V and L the characteristic values of speed and length of the flow. For example, in the case of the Earth, L is the radius of the liquid core and V the characteristic speed of natural convection in the core. For the sun, L is the thickness of the convective zone. In Table 12.1, some orders of magnitude leading to the calculation of the magnetic Reynolds number (Rm) are presented for various natural objects [ZEL 83], for a fast breeder reactor as well as for the first two dynamo experiments carried out, one in Riga (Latvia), the other in Karlsruhe (Germany). The third experiment carried out in Cadarache (France) could achieve Rm=50 and the dynamo threshold was about 30 using ferromagnetic discs to produce the motion of liquid sodium (dynamo action was not obtained with stainless steel discs). For a stationary flow (time independent), we show that the solution of induction equation [12.10] has the following form:
B(x, y, z,t) Re(B(x, y,z)e pt) where p is a complex number. The real and imaginary parts of p are the growth rate and the pulsation of B respectively. The dynamo effect corresponds to Re(p)t0 . The absence of dynamo is characterized by Re(p)d0 (diffusion). The complex parameter p depends on the geometry of the flow and on Rm.
484
The Finite Element Method for Electromagnetic Modeling T K
L m
V m s-1
Q m2s-1
Re
Rm
ms
35 105 4 10-4
10-6
3
109
5 102
1
1012
106
Qm 2 -1
Earth’s core
4 103
Jupiter’s core
104
5 107 5 10-2 3 10-6
Convective zone of the sun
105
2 108
– interior area
2 106
105
5 105 5 102 1.5 10-4
108
3 1013
– intermediate area
3 105
106
5 103 3 10-2
1011
1010
103
3 10-5
103
7 1015 2 108
Accretion disk around a black hole 4 10-1
Interstellar environment – area HI
102
3 1017 3 103 2 1015 5 1017
5 105 2 103
– area HII
104
3 1018
104
5 1014 3 1016
5 107
106
– tunnel
106
3 1018
104
3 1022 5 1013
1
6 108
104
1019
104
5 1013
1017
2 109
106
Universe in expansion
4 103
4 1020
104
1013
5 103
4 1011 1021
Fast breeder reactor
673
1
5.5
7 10-7
2 10-1
8 106 25-30
13
6 10-7
8 10-2
3 106
20
-7
-2
5
6-7
Gas galactic disk
Laboratory experiments – Riga – Karlsruhe
423 398
0.125 0.1
5
7 10
8 10
7 10
Table 12.1. Typical parameters of natural objects, industrial and experimental facilities leading to a dynamo effect
For a given flow geometry, we can be interested for example in the minimum value of Rm, for which a dynamo effect occurs. This critical magnetic Reynolds number ( Rmc ) corresponds to a zero growth rate (Re(p)=0). For a given dynamo flow geometry, we can also be interested in the limit of Re(p) when Rm tends towards infinity. Two classes of dynamo are thus defined, “slow” or “fast”, depending on whether this limit is zero or not [CHI 95]. This distinction is relevant for many astrophysical objects (Sun, galaxy, etc.) for which Rm is very high (>108) and for which it is suitable to identify generation mechanisms compatible with such a value of Rm. In addition, the majority of astrophysical dynamo predictions is based on the mean field theory [KRA 80] and implicitly assumes the existence of fast dynamo. The discretization according to space coordinates of the induction equation leads to a system of equations of the form pB M B where the matrix M depends on the
Magnetohydrodynamics
485
method of discretization (finite elements, Fourier series, finite differences, etc.) as well as on the discretization refinement (mesh size, number of Fourier modes, etc.). The possible values of p are thus the set of the eigenvalues of M. The significant growth rate p is thus the eigenvalue which has the largest real part. This eigenvalue can also be calculated using an appropriate temporal scheme of evolution of the equations and is physically measured in an experiment. In some cases, it is interesting to follow several eigenvalues according to the parameters of the problem [PLU 99]. 12.2. Modeling the induction equation using finite elements 12.2.1. Potential (A,I) quadric-vector formulation 12.2.1.1. Definition of magnetic and scalar potentials The magnetic vector potential A and electric scalar potential I are defined by:
B u A ;
E I wA/ wt
[12.14] [12.15]
Any transformation of the form
A' Af ;
I' I
wf wt
[12.16] [12.17]
where f is an unspecified scalar function changes neither E nor B. In order to ensure the unicity of the solution in quadri-vector (A,̓I) itis necessary to impose a gauge condition which must be satisfied by (A,̓I). The most current are the Coulomb and Lorenz gauges. 12.2.1.2. Strong form Induction equation [12.10] can then also be formulated in vector and scalar potentials A and I.
u(u A) V( wA I V uu A)) P wt
[12.18]
Equation [12.18] is also valid for V , unlike equation [12.10]. However, there is one more unknown variable than the number of equations to be solved. It is thus necessary to solve an additional equation which, logically, could result from the choice of gauge that potentials A and I must check. However, we impose the Coulomb gauge:
486
The Finite Element Method for Electromagnetic Modeling
A 0
[12.19]
directly in the weak form of equation [12.18], using a least squares formulation [CSE 82]. This process has the advantage of symmetrizing the diffusion matrix but the disadvantage of no longer ensuring the continuity of the normal component of current density j at the interfaces. To overcome this disadvantage, we choose to solve the additional equation
0 ¨¨ V ¨ wA I V uu A ¸ ¸¸ ¹¹ © © wt §
§
··
[12.20]
This equation ensures in particular that jn = 0 on an insulator. The expression of magnetic induction B is obtained thanks to equation [12.14]. Consequently, equation [12.3] is automatically checked. In order to again make the resolution matrices symmetric, the following change of variable is used [BIR 89]
I wW wt
[12.21]
Finally the following system is solved
w(AW) u(u A) V( V u(u A)) P wt w(AW) §¨V( V u(u A)) ·¸ 0 w t © ¹
[12.22]
with gauge [12.19]. In addition to the continuity conditions at interfaces described by equations [12.9], it is necessary to add the continuity of the normal component of A:
[An] 0
[12.23]
12.2.1.3. Weak form The weak form of the formulation in (A,W) is obtained by projecting equations [12.22] on the space of test functions (D, E).
Magnetohydrodynamics
§
487
w(AW) V uA) ·¸d: wt ¹ [12.24a] § A (D n)D (nu u A) ·dS 0 ³³ ¨ P P ¸¹ (I) (*)©
³³³¨© P1 uD u A P1 D AVD ( (:)
w(AW) V uA) ·¸d: wt ¹ (:) w(AW) VE ( V uA)dS 0 ³³ wt (I) (*) §
³³³¨©VE (
[12.24b]
This corresponds to the resolution of a matrix system MX L dX F(X) where X dt is the quadri-vector (A,W) and M and L the matrices defined with the usual notations by:
M11 M 22 M 33 P 1(w xDi w xD j w yDi w yD j w zDi w zD j ) ; M12 P 1(w yDi w xD j w xDi w yD j ) ; M13 P 1(w zDi w xD j w xDi w zD j ) ; M 21 P 1(w xDi w yD j w yDi w xD j ) ; M 23 P 1(w zDi w yD j w yDi w zD j ) ; M 31 P 1(w xDi w zD j w zDi w xD j ) ; M 32 P 1(w yDi w zD j w zDi w yD j ) ;
M14 M 24 M 34 M 41 M 42 M 43 M 44 0 ; L11 L22 L33 VDi D j ; L12 L13 L21 L23 L31 L32 0 ;
L14 VDi w xD j ; L24 VDi w yD j ; L34 VDi w zD j ; L41 Vw xDi D j ; L42 Vw yDi D j ; L43 Vw zDi D j ; L44 V(w xDi w xD j w yDi w yD j w zDi w zD j ) . We can show that this formulation also corresponds to the minimization of equation [12.11] while taking D wA and E W . wt 12.2.1.4. Validity domain of the formulation The principal advantage of the formulation in quadri-vector potential with Coulomb gauge is its domain of validity. In particular: – discontinuities of V and P are authorized;
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The Finite Element Method for Electromagnetic Modeling
– the conductors can be not simply connected; – magnetic permeability P can be nonlinear and anisotropic; – the formulation is compatible with the modeling of the magnets. However, if the integration domain is of constant magnetic permeability, then a more economic formulation in computer power can be used. In the non-conducting part, B being derived from a potential M it is enough to calculate this potential (only one unknown variable). In the conducting part the calculation of B (3 unknown variables) is also enough. The saving in calculation time compared to the quadrivector potential formulation can be significant. It is the object of the formulation in (B, M) which we will not detail in this chapter; see [LAG 06] [GUE 07]. 12.2.2. 2D1/2 quadri-vector potential formulation
12.2.2.1. Strong form In the particular case where V, V and P are independent of one coordinate, for example of z in Cartesian coordinates, we then have an easy way of making it possible to reduce the previous problem to the resolution of a 2D problem [SOT 98] [SOT 99]. Let us consider the decomposition of the quadri-vector potential in Fourier series with respect to z
(A,W)
¦(A ,W )(x, y,t)e k
k
ikz
[12.25]
k
It is shown that the complex Fourier modes (Ak ,Wk ) are independent of each other. We thus solve a 2D problem depending only on x, y and t for each mode k. The strong form of the 2D1/2 complex quadri-vector potential formulation ( Ak ,Ik ) is thus written:
*u Ak w(Ak *Wk ) *u( V u(*u Ak )) ) V( wt P § · w(Ak *Wk ) * ¨¨V( V u(*u Ak )) ¸¸ 0 wt © ¹ where operator
* is defined by (
The Coulomb gauge is written
w , w ,ik ). wx wy
[12.26]
Magnetohydrodynamics
* Ak 0
489
[12.27]
12.2.2.2. Weak form The weak form of the 2D1/2 complex quadri-vector potential formulation ( Ak ,Ik ) is obtained by projecting equations [12.26] on the space of the complex test functions ( D k , E k ).
· §1 w(Ak *Wk ) * uD )(* u A ) 1 (* D )(* A )VD ( ¨ ( V u*u Ak ) ¸¸d:2 k k k k k ³³ ¨ t P P w (: 2 )© ¹ § * Ak *u Ak · ³ ¨¨ P (Dk n)Dk (nu P ) ¸¸¹dl 0 I 1 *1 © [12.28a]
· § w(Ak *Wk ) * E ( * A ) ¸d: 2 ¨ V V u k k ³³ ¨ ¸ wt (: 2 )© ¹ w(Ak *Wk ) VE k ( V u* Ak )dS 0 ³ w t 1 1 I *
[12.28b]
The weak form of [12.27] is solved again by the least squares method and corresponds to the second term of [12.28a]. Weak formulation [12.28] corresponds to the resolution of a matrix system MX k L dX k F(X k ) where Xk is the complex dt
quadri-vector ( Ak ,Ik ), M and L the complex matrices defined with the usual notations by:
M11 M 22 M 33 P 1(w xDi w xD j w yDi w yD j k 2Di D j ) ; M12 P 1(w yDi w xD j w xDi w yD j ) ; M13 ikP 1(Di w xD j w xDi D j ) ; M 21 P 1(w xDi w yD j w yDi w xD j ) ; M 23 ikP 1(Di w yD j w yDi D j ) ; M 31 ikP 1(w xDi D j Di w xD j ) ; M 32 ikP 1(w yDi D j Di w yD j ) ;
M14 M 24 M 34 M 41 M 42 M 43 M 44 0 ; L11 L22 L33 VDi D j ; L12 L13 L21 L23 L31 L32 0 ; L14 VDi w xD j ; L24 VDi w yD j ; L34 ikVDi D j ;
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The Finite Element Method for Electromagnetic Modeling
L41 Vw xDi D j ; L42 Vw yDi D j ; L43 ikVDi D j ; L44 V(w xDi w xD j w yDi w yD j k 2Di D j ) . 12.2.2.3. Validity domain of the formulation The validity domain of the 2D1/2 complex quadri-vector potential formulation is the same as for the 3D formulation, but with the limitation due to the reduction of the dimension of the integration domain: – discontinuities of V and P are allowed only in plane (x,y); – the conductors can be not simply connected in plane (x, y); – magnetic permeability P can be nonlinear and anisotropic in plane (x,y); – the formulation is compatible with the modeling of the magnet formulation. 12.2.2.4. Other 2D1/2 formulations The case where V, V and P are independent of the azimuthal component T in cylindrical coordinates corresponds to another 2D1/2 formulation. The quadri-vector potential decomposition in Fourier series with respect to T is thus considered:
(A,W)
¦(A ,W )(r, z,t)e m
m
imT
[12.29]
m
It is shown that the complex Fourier modes (Am,Wm ) are independent of each other. A 2D problem is thus solved depending only on r, z and t for each mode m. The strong and weak forms of this formulation are stated in the same way as for the formulation in z while operator
* is defined by * r, z i m eˆT . r
In the cases where V, V and P are independent of both z and T cylindrical coordinates, a new formulation can be stated. The quadri-vector potential decomposition in Fourier series with respect to z and T is thus considered:
(A,W)
¦(A
,Wk,m)(r,t)ei(mT kz)
k, m
[12.30]
k, m
It is shown that complex Fourier modes (Ak, m,Wk,m) are independent of each other. A 1D problem is thus solved depending only on r and t for each couple (k, m). The strong and weak forms of this formulation are stated in the same way as for the formulation in z while operator
* is defined by * r i m eˆT ikzˆ . r
Magnetohydrodynamics
491
The previous 2D1/2 formulations in z, 2D1/2 in T as well as the 1D formulation in quadri-vector potential can be also stated for formulation (B,M). For the 1D formulation, the use of appropriate Bessel functions as test functions reduces the resolution domain to the conducting part only, saving computer power [MAR 06] [PEY 07]. 12.3. Some simulation examples 12.3.1. Screw dynamo (Ponomarenko dynamo)
12.3.1.1. Modeling The screw dynamo was initially solved by Ponomarenko [PON 73] and from then on has defined a dynamo benchmark in cylindrical geometry. It has since been studied with the aim of producing an experimental dynamo [GAI 76]. The experimental results obtained are in very good agreement with theoretical predictions [GAI 00] [GAI 01].
VP V z 0 R1 R2
VPV=0 Figure 12.3. Geometry of the integration domain for the screw dynamo
Let us consider an integration domain composed of two parts with symmetry of revolution (Figure 12.3), which are coaxial and of height H: – a cylinder with a radius R1, a conductivity V1, a permeability P1 and a velocity (Vr ,VT ,Vz ) (0,Zr, FZR1 ) ; – an external crown of radius R2>>R1, conductivity V2, permeability P2 and which is at rest (zero velocity). Non-viscous screwing (Z = Z0) is distinguished from viscous screwing (Z is dependent on r and becomes zero for r R1 ). The magnetic Reynolds number is
492
The Finite Element Method for Electromagnetic Modeling
defined on the basis of characteristics of the moving inner-cylinder: Rm V 1 P1 ZR12 1 F 2 . Since the motion is independent of T and z, this problem can be described alternatively using a 3D, 2D1/2ikz, 2D1/2imT or 1Dikz+imT formulation (section 12.2.2.3). It is thus possible to test and compare the various formulations while taking H 2S 2SFR1 and periodic boundary conditions in z=0 and z=H for k the 3D and 2D1/2imT modeling. The boundary conditions at the external domain border (r=R2) can be Dirichlet or Neumann. This border is taken sufficiently far away from the conducting part ( R2 t10R1 ) in order to be able to compare the numerical results with those obtained for ( R2 of ) by other methods. The initial condition must be non-zero and sufficiently complicated to contain the germ of the mode which will be amplified. An initial white noise condition type is sufficient. 12.3.1.2. Main results – The screw motion is a dynamo with Rmc depending on m, k, Fѽ P2/ҏP V2/ҏV 1.
1
and
– The time evolution of the magnetic energy is exponential in accordance with the theoretical predictions for a stationary flow. – In the homogenous case (P2/ҏP 1=V2/ҏV 1=1), the minimum value of Rmc is 17.73. This is obtained for F = 1.3, k=-0.39 and m=1. – In the non-homogenous case, the results obtained for P2/ҏP 1=a and V2/ҏV 1=b are the same as those obtained for P2/ҏP 1=b and V2/ҏV 1=a. – For Rm t Rmc , the larger Rm , the higher the dominant mode (m, k) (the mode which has the maximum growth rate). – For viscous screwing, the maximum growth rate Re(p) O(Rm1/ 3) is obtained for m,k O(Rm1/ 3) . The fact that Re(p)o0 when Rm of confers its “slow” dynamo nature [GIL 88] [CHI 95] to viscous screwing. The maximum of magnetic energy is confined in a layer of a thickness R1 O(Rm1/ 3) . – For non-viscous screwing, and for a given azimuth mode m, the maximum growth rate Re(p) O(Rm1/ 3) is obtained for k O(Rm1/ 2) also suggesting a “slow” dynamo action. However, this maximum growth rate according to k depends on m and reaches a maximum for m O(Rm1/ 2) . This maximum, according to k and m, is Re(p) O(1) for large Rm , conferring the “fast” nature of the non-viscous screwing dynamo [GIL 88] [CHI 95]. The maximum of magnetic energy is confined in a layer of thickness R1 O(Rm1/ 2) . The neutral curves (zero growth rate) according to k and Rm are represented in Figure 12.4 for several ratios of different conductivity and permeability.
Magnetohydrodynamics
493
25 (a)
(c)
20
(b) 15 Rmc 10
(d)
5 0 0
0.1
0.2
0.3
k
0.4
0.5
0.6
0.7
Figure 12.4. Non-viscous Ponomarenko dynamo, m=1. The critical magnetic Reynolds number Rmc is represented versus the vertical wave number k, for various electromagnetic properties. (a)V2/V1=̓P2/P1 =1; (b) V2/V1=10,̓P2/P1 =1; (c)V2/V1=100,̓P2/P1 =1; (d)V2/V1=̓P2/P1 =10. There is a dynamo action for Rm t Rmc
12.3.1.3. Generation mechanism The field generation mechanism (for both viscous and non-viscous screwing) can be divided into two main phases: stretching by the flow shear and diffusion of the magnetic field lines. In order to understand the stretching phase, it is necessary to first consider the case of a perfectly conducting fluid. It is thus shown that by stretching a magnetic flux tube, the magnetic intensity in the tube increases [MOF 78, MOR 90, CHI 95]. It is thus understood that the stretching related to the velocity gradients is a necessary ingredient for the dynamo effect. This ingredient is common to any dynamo flow, even if the fluid is not perfectly conducting. Here the magnetic field stretching results from a double shear: the one of the horizontal flow component (rotation) and the one of the vertical flow component. Due to the cylinder rotation, the magnetic field is stretched and deformed as represented in Figure 12.5. Due to the stretching the magnetic field intensity increases. We also observe that the magnetic field lines are folded by the cylinder rotation. This folding implies that the magnetic field has opposite signs at the cylinder boundary. Therefore for pure rotation, these opposite field lines would cancel by diffusion (implying that pure rotation is not a dynamo flow). Let us consider now the shear between the cylinder vertical motion and the outer domain at rest. This shear has the effect (in addition to a stretching effect) to pull up the magnetic inner field at a height z different from the initial field and then constitutes the double helix of Figure 6.2. This magnetic double helix has also been observed experimentally [ALE 00]. The magnetic field after stretching is then mainly azimuthal and axial, approximately aligned with the
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The Finite Element Method for Electromagnetic Modeling
velocity shear. This constitutes the main part of the dynamo mechanism: the generation of a helical magnetic field from an initial radial field. To close the mechanism, we need this helical field to generate a radial field. This is done owing to diffusion of the azimuthal component of the helical field. Indeed, writing the magnetic diffusion in cylindrical coordinates clearly indicates that the diffusion of the azimuthal component occurs not only along the azimuthal coordinate but also along the radial coordinate (for non-zero m).
Figure 12.5. Deformation of field lines by the cylinder rotation
This heuristic explanation for the azimuth mode m=1 allows the characteristic mechanisms of a “slow” dynamo to be understood. Indeed, the diffusion plays a major part in the final arrangement of the magnetic field lines. It is shown that for Rm of , the growth rate for m=1 tends towards zero. That is also true for any value of m. However, if we consider an infinite spectrum of azimuth modes m, then we show that in the case of non-viscous screwing, the dynamo is “fast” (Figure 12.6).
Figure 12.6. Non-viscous Ponomarenko dynamo. The maximum according to the vertical wave number k of the growth rate Re(p), versus Log (Rm) and for various values of the azimuth wave number m. For given m, Maxk(Re(p)) tends towards zero with large Rm. On the other hand, the maximum according to k and m of Re(p) tends towards a non-zero value with large Rm
Magnetohydrodynamics
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12.3.2. Two-scale dynamo without walls (Roberts dynamo)
12.3.2.1. Modeling Let us consider an integration domain made up of a square with side 2S and boundary conditions periodic in x and y. The flow is defined in dimensionless Cartesian coordinates by:
(Vx ,Vy ,Vz ) (sin xcos y,cos xsin y, 2 F sin xsin y) This flow consists of parallel vortices moving in opposite directions. Each vortex is included in a cell of square section and the horizontal velocity is at maximum on the edges (Figure 12.7). The fluid is homogenous in electric conductivity and magnetic permeability. This problem was solved initially by G.O. Roberts [ROB 72] and from then on has defined a dynamo benchmark in Cartesian geometry. In addition, it has been studied with the aim of producing an experimental dynamo [BUS 96] [RAD 98]. The experimental results obtained [STI 01] are in good agreement with the theoretical predictions.
Figure 12.7. Current lines of the Roberts flow in the xy plane. They coincide with the iso-values of Vz. The signs + (-) correspond to Vz>0 (<0)
Since the motion is independent of z, this problem can be described alternatively using a 3D or 2D1/2ikz formulation (section 12.2.2.3). This thus allows the various formulations to be tested and compared by taking for the 3D modeling a dimensionless height of the integration domain H 2S , periodic boundary k conditions in z=0 and z=H and on edges x rS and y rS . An initial condition of white noise type is sufficient.
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The Finite Element Method for Electromagnetic Modeling
12.3.2.2. Results – The Roberts flow is a dynamo with Rmc depending on k and F. For given values of F and Rm , there is always a sufficiently small value of k for which the Roberts flow is a dynamo (unlike the screw dynamo). This is due to the concept of separation of horizontal scales between the flow (characteristic dimension equal to 2S) and the average magnetic field (characteristic dimension according to x and y, which is infinite). – The time evolution of the magnetic energy is exponential in accordance with the theoretical predictions for a stationary flow. – For F =1, the maximum growth rate Re(p) O(ln(ln Rm)/ ln Rm) is obtained for kR
1/ 2 m
O ln Rm
1/ 2
.
The fact that Re(p)o0 when Rm of confers to the
Roberts flow is nature of “slow” dynamo [SOW 87] [CHI 95]. The maximum magnetic energy is confined in a layer of a thickness O(Rm1/ 2) . The growth rate according to k and
Rm is represented in Figure 12.8 for F=1.
Figure 12.8. Roberts dynamo growth rate Re(p) versus vertical wave number k, for F=1. Each curve corresponds to Rm 2n with n^2;1;0;1;...;6`. For each value of Rm , the maximum growth rate according to k is indicated by a circle. It reaches a maximum for Rm 8 and tends towards zero for Rm of (slow dynamo)
12.3.2.3. Generation mechanism The magnetic field generation of the Roberts dynamo results mechanism of the stretching, folding or shear type.
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497
Let us consider a magnetic field of the form (Bx , By,Bz ) B0 (coskz,0,0) . The stretching and folding of the magnetic field due to the flow are illustrated in Figure 12.9a in the z=0 plane.
Figure 12.9. Roberts dynamo mechanisms: (a) stretching and folding of the magnetic field in the z=0 plane; (b) sections AA’: shearing of the field by vertical velocity. The white arrows represent the direction of the displacement of the field
The field is stretched along y, consequently its intensity increases (see section 12.3.2.3). Component By, which appears, is of opposite direction on both sides of the borders (folding). Let us now consider the shearing according to z between two cells (section AA’). The effect of this shearing is to drag By in the direction of z, but in opposite directions on both sides of the border between two cells (Figure 12.9b). Consequently a field of the form (Bx , By,Bz ) B0 (0,sin kz,0) is generated. It is shown that this will later generate, through the same process, a field of the form (Bx , By, Bz ) B0 (cos kz,0,0) . Consequently, the total field amplified by dynamo effect will be of the form (Bx ,By, Bz ) B0 (coskz,sin kz,0) . A fundamental characteristic of the Roberts dynamo is the scale separation between the flow and the average magnetic field (spatial average in x and y). In fact, it is shown that the flow on a small scale can generate a large scale field, thanks to a mechanism known as the alpha effect [KRA 80]. This mechanism is present in most natural object dynamos. It offers an alternative and compatible explanation with the stretching, folding and shearing mechanism previously seen. Thus, an initial average field of the form (Bx ,By, Bz ) B0 (coskz,0,0) is deformed by the flow by forming field loops (Figure 12.10). These loops induce a current density j opposed to the field. The average magnetic field induced by this current is of the form (Bx ,By,Bz ) B0 (0,sin kz,0) . It is shown that this will later generate, through the same process, an average field of the form (Bx ,By,Bz ) B0 (coskz,0,0) . Consequently, the total average field amplified by dynamo effect will be of the form (Bx ,By, Bz ) B0 (coskz,sin kz,0) .
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The Finite Element Method for Electromagnetic Modeling
Figure 12.10. Illustration of the alpha effect: the deformation of the average magnetic field B by vortices V generates current density j in the direction opposed to field B. This current density induces a field perpendicular to B and phase shifted in z
It is legitimate to also consider an average magnetic field of arbitrary scale in x and y (which is not necessarily infinite). This study was undertaken within the framework of the experiment built in Karlsruhe. A liquid sodium flow goes over 52 Roberts cells. Thus, the average field has a characteristic dimension equal to the size of these 52 cells. The theoretical results obtained are noticeably different from those obtained in the ideal case of an infinite number of cells (Figure 12.11).
Figure 12.11. Roberts dynamo: critical magnetic Reynolds number versus k, for F=1. The curves correspond to an average magnetic field extending on (a) an infinity, (b) 50, (c) 32, (d) 18, (e) 8 Roberts cells
12.3.3. Two-scale dynamo with walls
12.3.3.1. Modeling Let us consider an integration domain made up of a square of side 2S and boundary conditions periodic in x and y. The flow is defined in dimensionless Cartesian coordinates by
Magnetohydrodynamics
499
(Vx ,Vy ,Vz ) (asin y(1cos x),asin x(1cos y),Ka(1cos x)(1cos y)) with K F / 2 and 2a 1 F 2 1 , where F is a screw factor. This flow consists of parallel vortices and moving in the same direction. Each vortex is included in a cell of square section and the velocity is zero on the walls of the cells. The original problem relates to a homogenous fluid in electric conductivity and magnetic permeability [PLU 99]. Walls of a different thickness and electromagnetic properties different from those of the fluid were also considered [SOT 99]. The velocity being independent of z, this problem can be still described here using a 3D or 2D1/2ikz formulation with the same boundary and initial conditions as for the Roberts problem (section 12.3.2.1). 12.3.3.2. Results – For walls with zero thickness, the results are qualitatively the same as those of the Roberts dynamo (section 12.3.2.2). However, there is a difference concerning the maximum growth rate Re(p)max O(Rm1/ 3) which is obtained for k O(Rm1/ 3) . – Two generation modes were identified according to k and Rm: a Ponomarenko mode and a Roberts mode. – For walls of non-zero thickness, Rmc depends on k, on the thickness of the walls and on the ratios of conductivity and permeability between fluid and walls. The growth rate according to k and Rm is represented in Figure 12.12 for F =1 and walls of zero thickness (without wall).
Figure 12.12. Two-scale dynamo with walls of zero thickness: growth rate versus k for various values of Rm. The small (respectively large) values of k correspond to the Roberts (respectively Ponomarenko) mode
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The Finite Element Method for Electromagnetic Modeling
12.3.3.3. Generation mechanisms The generation mechanism of the two-scale dynamo with walls is again of the stretching, folding or shear type. Its originality lies in the existence of two possible dynamo modes: Roberts or Ponomarenko. Indeed, according to values of k and Rm , we observe (Figure 12.13) a magnetic energy confined in a layer either in the vicinity of the walls (Roberts mode) or inside the cells (Ponomarenko mode). In both cases, the thickness of this layer is about O(Rm1/ 3) . With large Rm , the Ponomarenko mode is dominant and the solution obtained corresponds to an azimuthal mode (at the scale of one each cell) increasing with Rm (Figure 12.14).
a)
c)
b)
d)
Figure 12.13. Two-scale dynamo with walls of zero thickness: iso-values of maximum energy for the (a) Roberts, (b) Ponomarenko, m=1, (c) Ponomarenko, m=2, (d) Ponomarenko, m=3 mode. Parameter m is the azimuth mode in a local cylindrical reference frame whose origin is in the center of the cell
Magnetohydrodynamics
501
Figure 12.14. Two-scale dynamo with walls of zero thickness: the two possible Roberts or Ponomarenko modes are represented in the plane (k, Rmc ). For the Ponomarenko mode, the local azimuthal mode m is indicated
12.3.3.4. Influence of walls At small k, the existence of non-zero thickness walls around each cell is unfavorable to the dynamo (Figure 12.15). Indeed, the mode with small k is a Roberts mode and consequently the field generation results from the cell interaction. The existence of walls slows down this interaction and Rmc increases. At large k, the influence of walls around each cell is negligible. Indeed, the mode is then a (viscous) Ponomarenko mode and the generation takes place in the fluid, not requiring the cell interaction. For permeability (or conductivity) walls higher than that of the fluid, the dynamo is favored at small k (Figure 12.16). The field diffusion in the walls is indeed faster and the interaction between cells is thus easier. At large k, the influence of the walls is negligible (viscous Ponomarenko mode). Lastly, as for the screw dynamo, the electric conductivity and magnetic permeability play a symmetric role. 30
(c)
(b)
20 Rmc
10 (a)
0 0
0.2
0.4
0.6
k
0.8
1
1.2
Figure 12.15. Two-scale homogenous dynamo with walls: Rmc versus k for various wall thicknesses. The ratio wall thickness over cell size is (a) zero, (b) 5%, (c) 10%
502
The Finite Element Method for Electromagnetic Modeling
30
20 Rmc
10
(a)
(b) (c)
0 0
0.2
0.4
k
0.6
0.8
1
1.2
Figure 12.16. Two-scale dynamo with walls: Rmc versus k for a ratio wall thickness over cell size equal to 5%. The ratio of wall permeability over the fluid permeability is (a) 1, (b) 10, (c) 100
12.3.4. A dynamo at the industrial scale
In 1989 and 1990, several unexpected stops occurred during the exploitation cycle of the fast breeder reactor Phénix. Each time, a furtive decrease of the power signal of the reactor was observed, engaging the security and leading to stop automatisms. Several scenarios were considered to try to explain the cause of this mysterious phenomenon. One of them was based on the existence of a mechanical instability caused by a dynamo effect in the reactor. The core of the reactor is actually traversed by liquid sodium at high speed with a flow geometry similar to the two-scale dynamo with walls thus favorable to a spontaneous amplification of magnetic field. In addition, the walls of the cells of the core being built with ferromagnetic materials, the efforts exerted by the interaction of the field and the currents in sodium were sufficient to generate the mechanical instability necessary to trigger the security stops. Thus, a dynamo effect was probably observed in an indirect way in the core of a FBR.
Magnetohydrodynamics
503
Figure 12.17. Iso-value Bz in a horizontal section of an FBR core. The clear (bold) color indicates a positive (negative) value of Bz
The result of an EF3D calculation is presented in Figure 12.17 where the isovalues of the vertical component of the magnetic field in a horizontal section of the core of the reactor are represented. For this calculation, the core consists of 61 cells of hexagonal section. The flow traversing each cell is a viscous screw flow. The main interest of this figure is that it shows a double organization of the magnetic field. The field is organized in a double helix at the scale of each cell as in Figure 12.2 and in accordance with the screw dynamo. The field is also organized in a double helix at the scale of the core. This double organization of the field is characteristic of a two-scale dynamo such as that also observed in the Karlsruhe experiment. 12.4. Modeling of the dynamic problem
The dynamic problem (modeling of the induction and Navier-Stokes equation) was formulated and modeled by the finite element method [BEN 99] [BEN 01] in the case where the whole integration domain is homogenous in conductivity and permeability. The tests carried out present a magnetic Reynolds number too low to obtain a dynamo. The formulation uses the primary variables: speed, pressure and magnetic field. The originality of the method relies on the coupling strategy between the resolution of the Navier-Stokes equations and the induction equation. In addition, it should be noted that there is a dynamo benchmark in spherical geometry which was tested and validated by various research teams [CHR 01]. Generally, the space discretization used is of the spherical harmonics type for the latitudinal and longitudinal dependences. The discretization in the direction of the radius is of the spectral or finite differences type. The use of spherical harmonics is
504
The Finite Element Method for Electromagnetic Modeling
naturally suggested by the spherical geometry of the integration domain. In addition, it has the advantage of being able to select a reduced number of harmonics (as far as this selection can give reasonable results). The calculation time can thus be considerably reduced (as for the 2D1/2 and 1D1/2 resolutions described in section 12.2.2). In comparison, the 3D finite element method does not a priori allow an economical resolution to be chosen unless a time adaptive mesh is used (allowing us to have a refined mesh where the scales are similar). The finite element method could also become more advantageous for massively parallel calculations. 12.5. References [ALE 00] ALEMANY A., MARTY P., PLUNIAN F., SOTO J., “Experimental investigations of dynamo action in the secondary pumps of the FBR Superphénix”, J. Fluid Mech., no. 403, pp. 263-276, 2000. [BEN 99] BEN SALAH N., SOULAIMANI A., HABASHI W.G., FORTIN M., “A conservative stabilized finite element method for the magnetohydrodynamic equations”, IJNMF, vol. 29, no. 535, 1999. [BEN 01] BEN SALAH N., SOULAIMANI A., HABASHI W.G., FORTIN M, “A finite element method for the magnetohydrodynamic”, Comp. Meths. Appl. Mech. Engrg., 2001. [BER 07] BERHANU M. et al., “Magnetic field reversals in an experimental turbulent dynamo”, Eur. Phys. Lett. 77 (5), 59001 2007. [BIR 89] BIRO O., KURT P., “On the use of the magnetic vector potential in the finite element analysis of three dimensional eddy currents”, IEEE Trans. Mag., vol. 25, no. 4, 1989. [BUS 96] BUSSE F.H., MULLER U., STIEGLITZ R., TILGNER A., “A two-scale homogenous dynamo: an extended analytical model and an experimental demonstration under development”, Magnetohydrodynamics, vol. 32, pp. 235-248, 1996. [CHI 95] CHILDRESS S., GILBERT A.D., Stretch, Twist, Fold: The Fast Dynamo, Springer, 1995. [CHR 01] CHRISTENSEN U.R., AUBERT J., CARDIN P., et al., “A numerical dynamo benchmark”, Phys. Earth Planet. Inter., vol. 128, 2001. [COX 69] COX A., “Geomagnetic reversals”, Science, vol. 163, pp. 237-245, 1969. [CSE 82] CSENDES Z.J., WEISS J., HOOLE S.R.H., “Alternative vector potential formulation of 3D magnetostatic field problems”, IEEE Trans. Mag., vol. 18, pp. 367372, 1982. [GAI 76] GAILITIS A., FREIBERGS J., “To the theory of a helical MHD-dynamo”, Magnetohydrodynamics, vol. 12, pp. 127-129, 1976.
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[GAI 00] GAILITIS A., LIELAUSIS O., DEMENT’EV S., et al., “Detection of a flow induced magnetic field eigenmode in the Riga dynamo facility”, Phys. Rev. Lett., vol. 84, pp. 4365-4368, 2000. [GAI 01] GAILITIS A., LIELAUSIS O., PLATACIS E., et al., “Magnetic field saturation in the Riga dynamo experiment”, Phys. Rev. Lett., vol. 84, pp. 4365-4368, 2000. [GIL 88] GILBERT A., “Fast dynamo action in the Ponomarenko dynamo”, GAFD, vol. 44, p. 241, 1988. [GUE 07] GUERMOND J.L., LAGUERRE R., LEORAT J., NORE C., “An interior Penalty Galerkin Method for the MHD equations in heterogeneous domains”, J. Comp. Physics 221, 349, 2007. [HID 79] HIDE R., “Dynamo theorems”, GAFD, vol. 12, pp. 171-176, 1979. [KOL 61] KOLM H.H., MAWARDI O.K., “Hydromagnet: a self-generating liquid conductor electromagnet”, J. of Applied Phys., vol. 32, no. 7, p. 1296, 1961. [KRA 80] KRAUSE F., RÄDLER K.-H., Mean Field Magneohydrodynamics and Dynamo Theory, Akademie-Verlag and Pergamon, 1980. [LAG 06] LAGUERRE R., NORE C., LEORAT J., GUERMOND J.L., “Effects of conductivity jumps in the envelope of a kinematic dynamo flow”, CR Mécanique 334, 593, 2006. [LEO 80] LÉORAT J., POUQUET A., FRISCH U., “Turbulence MHD développée et génération de champ magnétique”, J. Phys., vol. 41, pp. 359-369, 1980. [MAR 06] MARIE L., NORMAND C., DAVIAUD F., “Galerkin analysis of kinematic dynamos in the von Karman geometry”, Phys. Fluids 18, 017102, 2006. [MAS 84] MASSÉ P., “Modeling of continuous media methodology and computer-aided design of finite element programs”, IEEE Trans. Mag., vol. 20, no. 5, pp. 1885-1890, 1984. [MER 95] MERRILL R.T., MCFADDEN P.L., “Dynamo theory and paleomagnetism”, J. Geophys. Res., vol. 100, pp. 317-326, 1995. [MOF 78] MOFFATT H.K., Magnetic Field Generation in Electrically Conductive Fluids, Cambridge University Press, New York, 1978. [MOF 89] MOFFATT H.K., “Stretch, twist and fold”, Nature, 341, 1989. [MON 06] MONCHAUX R. et al., “Generation of magnetic field by a turbulent flow of liquid sodium”, Phys. Rev. Lett. 98, 044502, 2006. [MOR 90] MOREAU R., Magnetohydrodynamics, Kluwer Acad. Publ., Dordrecht/Boston/ London, 1990. [PAR 66] PARKER R.L., “Reconnexion of lines of force in rotating spheres and cylinders”, Proc. Roy. Soc., A291, pp. 60-72, 1966. [PEY 07] PEYROT M., PLUNIAN F., NORMAND C., “Parametric instability of the helical dynamo”, Phys. of Fluids, 19, 054109, 2007.
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[PLU 95] PLUNIAN F., ALEMANY A., MARTY PH., “Influence of MHD parameters on electromagnetic self-excitation in the core of a FBR”, Magnetohydrodynamics, vol. 31, 4, 1995. [PLU 96] PLUNIAN F., Etude de l’effet dynamo dans le coeur du réacteur Phénix, Thesis INPG, 1996. [PLU 96] PLUNIAN F., MASSÉ PH., “An optimal preconditionned scheme for finite element modeling of kinematic MHD dynamo effect”, Proc. of Eccomas 96, France, 1996. [PLU 98] PLUNIAN F., ALEMANY A., MARTY PH., MASSÉ PH., “Direct numerical modeling of kinematic dynamo effect. Application to the geometry of the core of the French fast breeder reactor Phénix”, Progress in Astronautics and Aeronautics, AIAA 182, pp. 537-550, 1998. [PLU 99] PLUNIAN F., MARTY PH., ALEMANY A., “Kinematic dynamo action in a network of screw motions. Application to the core of a fast breeder reactor”, J. Fluid Mech., vol. 382, pp. 137-154, 1999. [PLU 02] PLUNIAN F., RÄDLER K.-H., “Harmonic and subharmonic solutions of the Roberts dynamo model. Application to the Karlsruhe experiment”, Magnetohydrodynamics, 2002. [PLU 02] PLUNIAN F., RÄDLER K.-H., “Subharmonic dynamo action in the Roberts flow”, Geophys. Astr. Fluid. Dyn., vol. 96, no. 2, pp. 115-133, 2002. [PON 73] PONOMARENKO Y.B., “On the theory of hydromagnetic dynamo”, (Zh. Prikl. Mekh. Tekhn. Fiz. USSR, 6, 47-51), J. Appl. Mech. Tech. Phys., vol. 14, pp. 775-779, 1973. [PRI 82] PRIEST E.R., “Solar magnetohydrodynamics”, Geophys. and Astr. Monographs, D. Reidel Pub. Comp., 1982. [PRO 79] PROCTOR M.R.E., “Necessary conditions for the MHD Dynamo”, GAFD, vol. 14, pp. 127-145, 1979. [RAD 95] RÄDLER K.-H., “Cosmic Dynamos”, Rev. Mod. Astron., vol. 8, pp. 295-321, 1995. [RAD 98] RÄDLER K.-H., APSTEIN E., RHEINHARDT M., SCHÜLER M., “The Karlsruhe dynamo experiment – a mean field approach”, Studia geoph. et geod., vol. 42, pp. 224-231, 1998. [ROB 72] ROBERTS G.O., “Dynamo action of fluid motions with two-dimensional periodicity”, Phil. Trans. R. Soc., London, A271, 411, 1972. [ROB 92] ROBERTS P., SOWARD A.M., “Dynamo theory”, Ann. Rev. Fluid Mech., Vol. 24, pp. 459-512, 1992. [SCH 92] SCHMITT D., “The solar dynamo”, IAU Symposium, no. 157, Potsdam, 1992. [SOT 98] SOTO J., PLUNIAN F., MASSÉ PH., “A finite element-spectral formulation for the kinematic MHD dynamo problem”, Proc. of Eccomas 98, Greece, 1998.
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[SOT 99] SOTO, Etude cinématique de l’effet dynamo en milieu non homogène. Application aux réacteurs à neutrons rapides, Thesis, INPG, 1999. [SOW 87] SOWARD A.M., “Fast dynamo action in a steady flow”, J. Fluid Mech., vol. 180, pp. 267-295, 1987. [STI 01] STIEGLITZ R., MÜLLER U., “Experimental demonstration of a homogenous twoscale dynamo”, Physics of Fluids, vol. 13, pp. 561-564, 2001. [VAL 93] VALET J.P., MENADIER L., “Geomagnetic field intensity and reversals during the past four million years”, Nature, vol. 366, pp. 234-238, 1993. [WEI 66] WEISS N.O., “The expulsion of magnetic flux by eddies”, Proc. Roy. Soc., A293, pp. 310-328, 1966. [WEI 94] WEISS N.O., “Solar and stellar dynamos”, in Lectures on Solar and Planetary Dynamos, Proctor M.R.E. and Gilbert A.D. (eds.), Camb. Univ. Press, New York, 1994. [ZEL 83] ZELDOVITCH YA. B., RUZMAIKIN A.A., SOKOLOFF D.D., Magnetic Fields in Astrophysics, Gordon and Breach Science Publishers, New York, 1983.
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Chapter 13
Mesh Generation
13.1. Introduction If the finite element method, modeling of electromagnetic problems by the finite elements, and meshing techniques initially had almost independent evolutions, this is no longer the case. Indeed, little by little, the modeling of electromagnetic fields has become more accurate, more rigorous and has been extended to new problems. In order to reach this goal, the finite element method is adapted and generalized. In the same way, mesh algorithms, which constitute a link between the numerical method and the geometric and physical approach, follow this evolution incrementally, and are integrated into the new numerical and mathematical methods, as well as into new growing problems. Thus, it is increasingly difficult to tackle these three topics, numerical methods, physics of magnetism and meshing, in a disconnected manner, because they are more and more closely interrelated. A general analysis of meshing problems within the finite element method to electromagnetic field calculation is proposed. This chapter has several objectives: to display modern meshing techniques and present the current problems dealt with in electromagnetism so as to characterize their influences and their requirements in terms of meshing techniques.
Chapter written by Yves DU TERRAIL COUVAT, François-Xavier ZGAINSKI and Yves MARÉCHAL.
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The Finite Element Method for Electromagnetic Modeling
13.2. General definition In order to implement the finite element method independently of the physics of the phenomena we want to study, it is necessary to define a finished study domain on which we will seek a spatio-temporal distribution of the state variable which satisfies: – partial differential electromagnetism;
equations,
namely
Maxwell’s
equations
for
– boundary conditions on the domain boundary; – initial conditions in the case of temporal or nonlinear problems. This finished study domain is cut into smaller elements, called finite elements, which form a partition. The process of carrying out this cutting is usually called meshing. This process must adapt itself perfectly to the following three domains: – the geometry and topology of the study domain; – the physics of the phenomena; – the specificities of the mathematical method. It must also ensure the transfer of information from one domain to the other. A simple formal definition of a mesh domain : is as follows [DHA 84]: Definition 1. A mesh of a domain : (in one, two or three dimensions) is a set M of elements Ei such that: : = sum (Ei) Elements Ei are the pinpoint elements (a simple point), line elements (portions of rectilinear lines or curves), surface elements in two and three dimensions (triangles or quadrilaterals plane or curved) and voluminal elements (tetrahedrons, hexahedrons, pentahedrons with plane or curved faces). Definition 2. A mesh domain : is in conformity when the intersection of two elements Ei and Ei of M is: either empty, reduced to a point, reduced to an edge or a face (Figure 13.1). From the computing point of view, the following definitions can be stated: – Mesh algorithm: this is a set of computer programs which make it possible to calculate and “fill” the data structures of the grid in a coherent way: the nodes,
Mesh Generation
511
which are tables of space coordinates, and the finite elements, which are tables describing connections between these nodes (Figure 13.2). – Mesh processor: a mesh processor is a set of programs. Its task is to ensure the link between the geometry of the study domain and the meshing algorithms on one hand and between the meshing algorithms and the physical description modules, the resolution of the equation system generated by the finite element method and the analysis of the results on the other hand.
Figure 13.1. Conformity of the elements: a), b), c) not in conformity, d) in conformity
Node References
x
y
z
1
1.0
0.0
0.0
2
1.0
1.0
0.0
3
0.0
1.0
0.0
Table 13.1. Node coordinates
2D element node
Type
Node 1
Node 2
Node 3
1
Triangle
126
253
3
2
Quadrilateral
6
4
2
3
Triangle
1
2
3
Table 13.2. Structures of elementary data to define a mesh
Node 4 23
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The Finite Element Method for Electromagnetic Modeling
13.3. A short history Meshing methods were developed in the late 1970s mainly for applications in mechanics. The mesh processor, which was effectively manual in the early days, has become increasingly automated in order to be able to treat complex geometries that are not readily “cuttable” with manual processes. The initially 2D mesh processors have been quickly extended to the third dimension. The first automatic 3D mesh processors for electromagnetism were developed in French research centers at the end of the 1980s [DUT 85], [ALB 88]. Equally, automatic meshing techniques for CAD have been improved [GEO 88]. Regardless of the electromagnetic domain, recent works, in particular at INRIA, have attempted to make 3D mesh processors more reliable and faster [GEO 90]. Adaptation techniques have also improved [BOR 95], [HEN 93], allowing us to obtain good quality elements considering a fixed criterion before or after a resolution. In parallel, many teams continue to examine the automatic decomposition of a domain into quadrilaterals [BLA 92] and hexahedrons [MON 96]. 13.4. Mesh algorithms 13.4.1. The basic algorithms Meshing programs use a set of basic algorithms working on the mesh topology and geometry. They are founded on correctly specified data structures. 13.4.1.1. Elements direction This very important concept is obtained by imposing numbering rules to the nodes of elements and that must be observed and checked in all mesh algorithms. Some commonly used rules are: – nodes of triangles and quadrilaterals in two dimensions are positioned in the trigonometric direction; – the faces of a volume element are positioned in the trigonometric direction (in this case, the perpendiculars of the faces are always directed towards the outside of the volume). Thus, for a tetrahedron defined by the four points ordered P1, P2, P3, P4 and presented in Figure 13.2 the four correctly directed faces will be: F1= (P1, P3, P2), F2= (P2, P3, P4), F3= (P3, P1, P4), F4= (P4, P1, P2).
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P4 P1
P1
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P2
P3 P2
Figure 13.2. Example of orientation of the triangle or tetrahedron elements
13.4.1.2. Geometric calculations based on the elements In general, meshing algorithms require basic geometric calculations such as the perimeter, surface, volume and orientation of an element, the calculation of the perpendicular, of the tangent to one side (2D, 3D), the center of gravity, etc. For example, considering a tetrahedron defined by the points: ª x1º « » P1 « y1» , P 2 « » ¬ z1¼
ªx2 º « » « y 2 » , P3 « » ¬z2 ¼
ª x3 º « » « y 3» , P 4 « » ¬ z3 ¼
ªx 4º « » « y 4» , « » ¬z 4 ¼
The determinant of the following matrix makes it possible to obtain its volume and its direction according to the obtained sign [HER 82]:
Volume
ª1 1 1 1 º » « 1 « x1 x 2 x3 x 4 » 3! « y1 y 2 y3 y 4 » » « «¬ z1 z 2 z 3 z 4 »¼
[13.1]
In the case of the tetrahedron element shown in Figure 13.2, this volume is positive. For certain automatic algorithms and in particular the Delaunay algorithm [HER 82], the calculation of the following determinant is used to obtain the position of a point P of coordinates (x,y,z) in relation to the sphere circumscribed with the same tetrahedron: ªl 2 l12 l 22 l 32 l 24 º « » « x x1 x 2 x 3 x 4 » Det ( x, y, z ) « » « y y1 y 2 y 3 y 4 » «z z z z z 4 »¼ 2 3 1 ¬
[13.2]
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with l 2
2 2 x 2 y z 2 and l i
2 x i2 y i z i2 .
The same calculations apply in 2D, while considering the three vertices of a triangle. The extension to quadrilaterals, prisms, pentahedrons and hexahedrons can be easily accomplished by breaking the latter up into triangles or elementary tetrahedrons. The calculation of the volume sign is also used to determine the position of a point compared to a tetrahedron (inside/outside). 13.4.1.3. Quality of the elements The improvement of the element shape is obtained by local subdivision, deformation or topological modification of the meshing. The associated algorithms are based on one or more quality criteria. These give an account of a local quality of a geometric or physical type, and can be isotropic, with cases generally treated in electromagnetism or anisotropy. Some commonly used rules are: – for triangles and tetrahedrons: circle radius (sphere) registered/circle radius (sphere) circumscribed; – generally, the control of the minimum and maximum angles (Figure 13.3), of element edge lengths and element face surfaces in two and three dimensions. Two geometric isotropic criteria are as follows [BOR 95], (see equation [13.3] and Figure 13.3): Quality
2 3
Det ( P1P 2, P1P3)
¦
Pj Pk
[13.3]
2
1d j d k d 3, j z k
1
0 S
S
S
Figure 13.3. Example of angular criterion for 2D elements [BOR 95]
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Ranking the elements in agreement with this calculation, for example in histogram form, makes it possible to have general information on the quality of the meshing. The algorithms which allow the improvement of an existing mesh without local subdivision are [BOR 97]: – relaxation. By means of edge permutation, the number of edges connected to one node is minimized; – shift of points, also called regularization or re-centering. A progressive displacement is applied in each interior node in the direction of the surrounding elements center of gravity; – topological optimization. An edge permutation is carried out if the quality of a group of elements is better. Additionally, the following two techniques, which transform triangles into quadrilaterals, are sometimes used to improve the form or to decrease the total number of elements: – fining. This consists of arranging the triangles side by side in order to obtain quadrilaterals. The algorithm uses neighborhood information while considering all the acceptable cases of sticking around the element, and choosing the best, according to a set criterion; – the break. This technique is used to generate a “quadrangulation” from a meshing in triangles. For that, each triangle is divided into 3 quadrilaterals as indicated in Figure 13.4.
Figure 13.4. Breaking a triangle into quadrangles
13.4.1.4. Localization in a set of elements The localization algorithms are based on surface and volume calculations such as those quoted above. The introduction of vicinity information at the data structures level allows the localization algorithms to be strongly accelerated. Hence, in the case of a triangle characterized by three points P1, P2, P3, if we consider the three faces F1= (P1, P2), F2= (P2, P3), F3= (P3, P1), the reference of neighboring elements is associated with them and rests on each of the faces. The addition of supplementary information which indicates the amount of the shared side on the neighboring element can improve the performance of these algorithms. A traditional
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localization technique consists of seeking among the sides of an element, a side directed towards the point we wish to locate. The volume (or the surface) that it forms by connecting this to the point must be positive, taking into account the orientation rules defined previously. In this manner we can, by vicinity information, gradually find the elements which are in the direction of the point. The algorithm ends when none of the element sides are directed towards the point (Figure 13.5). However, in the case of a concave domain or a domain comprising holes, this algorithm must be adapted.
Figure 13.5. Scheme of a localization algorithm
13.4.1.5. Construction of a connected set of elements This algorithm is necessary when using automatic techniques to delimit the list of elements located inside a connected area of the space closed by one or several boundaries which have geometric definition. There are faces of finite elements which lean on the domain boundaries. Useful information in this research is gained by associating each face of an element which leans on a boundary with the nature of the geometric entity to which this boundary is linked. We can thus identify the elements which border the domain boundaries and which are on the same side of those domain boundaries. Then using the vicinity information associated with each element, various sets are filled in [ALB 88]. Generally, this method proceeds by associating adjacent elements until reaching an element adjoining the boundary. This same algorithm allows elements to be identified within a hole and deletes elements in holes.
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13.4.1.6. Observing the boundaries by meshing The use of free meshing algorithms sometimes leads to not observing the boundary by the generated elements. We can then speak about porcupine effect [DUT 85] [ALB88]. This effect is due to an insufficient discretization of the boundaries. Various techniques allow this problem to be solved: the rejection of elements which break the boundary, the creation of additional nodes on this boundary to force the meshing to account for it, and local modification of the meshing, for example by edge permutation. 13.4.1.7. Application of a size criterion distribution in space The size of an element can be controlled on the basis of purely geometric information or, and this is increasingly the case, on physico-mathematical criteria obtained when solving the problem (the gradient of the unknown variable, for example). The elements which do not check the criteria are identified. These are then broken, either by the intersection or division of the element edges, or by the creation of nodes in the element’s gravity centers and/or faces of elements [HER 82]. 13.4.1.8. Gathering of elements by pinpoint, line, surface or volume areas In order to identify the finite elements of physical areas within a study domain [ABA 00], a selection process must be implemented for the set of elements. The selection criteria employed here are based on geometric or topological information. It is a question of for example identifying the external facets of the elements in the volume area “air” to which a Dirichlet or Neuman condition will be applied. A selection is performed in several stages: 1. definition of criteria: – geometric: z=a, z=a and y
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Figure 13.6. Positioning of the second order nodes
These are created on edges, sometimes in the center of gravity of surface (2D and 3D) and volume elements. The nodes generated at the boundary are necessarily based on the geometric boundary definitions (curve and surface equations). The internal element edges of a higher order than 1 (Figure 13.6) are either rectilinear, or curvilinear angles. 13.4.2. General mesh algorithms The meshing methods that have been developed can be arranged into two large families: structured mesh processors, where the user intervenes, and free mesh processors (or automatic), where the user intervenes at the least. 13.4.2.1. Free mesh processors By definition, free mesh processors are “push-button” programs, i.e. the user intervenes very little in the construction of the mesh. The Delaunay triangulation [HER 82], [DUT 85], [GEO 88] or the frontal methods based on triangles, quadrilaterals or hexahedrons [BLA 91], [RAS 95] are the most used. These techniques have to be considered as methods of creating nodes in space and connecting them together. The triangulation algorithms quoted above generate triangles and tetrahedrons, and they are used when the shape of the objects to be meshed is complex, or when we want to mesh the “negative” of the object (in electrical engineering, the air surrounding the electric machine; in submarine acoustics, the water around the transducer).
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13.4.2.1.1. Delaunay technique
Figure 13.7. 3D mesh of an electromechanical contactor using the Delaunay technique
The Delaunay method is certainly the most common of the triangulation techniques, because it is general, robust and simple, both in its principle and implementation. It is possible to generate a triangulation satisfying the Delaunay criteria [HER 82] from a set of predefined points, representing the meshing nodes. This triangulation is made of the most equilateral elements possible for the set of the starting nodes. It represents the convex covering the initial points and adaptations close to the boundaries, allowing its contours to be observed. This method is very efficient in two or three dimensions [DUT 85], [ALB 88], [GEO 92]. 13.4.2.1.2. Frontal technique The frontal technique starts from the boundary of an object broken up into edges, and gradually generates a mesh towards the interior or the exterior of the boundary. Difficulties appear when the faces in progression are superposed. The frontal methods start in general from a discretization of the domain contours. Nodes and/or meshes are then defined according to expert rules [FOR 94], [GOL 89], [RAS 95]. The process is iterative: starting from a given contour, an analysis of the angles formed by two consecutive segments and their respective lengths allows a starting zone to be selected. Paul-Louis George in [GEO 90] gives an example of a frontal algorithm for a 2D mesh in triangles with a domain defined by boundary segments. Depending on the size of the starting angles, three cases are considered (see Figure 13.8):
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The Finite Element Method for Electromagnetic Modeling
Case 1: D < pi/2, the two segments of angle D are retained and will be the two sides of the unique triangle created. Case 2: pi/2 < D < 2pi/3, starting from the two segments of angle D, an internal point is defined and two triangles are generated. Case 3: 2pi/3< D, only one segment is retained, a triangle is built with this segment as a side and an internal point. A new contour is generated and the mesh front is updated. When the contour is reduced to the empty set, the final meshing is obtained. The difficulties related to these methods come from closing the faces where the degenerated elements can easily appear. On the other hand, some of these frontal methods offer the advantage of generating quadrangles.
Figure 13.8. Strategy of 2D meshing frontal triangle
13.4.2.1.3. Quadtree/octree methods The purpose of this type of mesh processor is the construction of a mesh on the considered domain, starting from contour points: – Stage 1: first a rectangle is built which contains all the points of the contour. The method consists of cutting each rectangle (parent) into four rectangles (offspring), in a recursive way starting with the initial rectangle, until obtaining a partition of the initial rectangle in quadrangular elements, such that each one contains at most only one point of the contour. This last property allows the size of each mesh to be controlled. – Stage 2: the partition resulting from the stage above is balanced (see Figure 13.9) by definition of rectangles so that at most one intermediate point exists on an edge of a mesh.
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– Stage 3: an analysis of the meshes thus generated is carried out: - external mesh not containing any point: such a mesh is destroyed, - interior mesh not containing any point: such a mesh produces a quadrangle (which can be cut out in triangles) if these edges do not have intermediate points or is cut out in triangles in the opposite case, - mesh containing a “piece” of the contour: the intersection points of the contour and the edges of the mesh are created, and hence a mesh partition, of which we keep only the part internal to the domain (the final contour of the grid is thus formed at this stage), is defined. – Stage 4: regularization of the internal points is then carried out (the internal points are the tops except a contour of the partition meshes resulting from stage 2 of this process). The method has unquestionable advantages: – the elements generated inside the volume are of good quality; – the meshing refinements can be obtained very easily. However, there are also some disadvantages: – the elements generated close to the boundary are of a lesser quality; – if the meshing of the boundary of the field is very irregular, there is a risk of obtaining very fine meshing uniformly.
Figure 13.9. Quadtree meshing
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The Finite Element Method for Electromagnetic Modeling
13.4.2.2. Assisted meshing Assisted mesh processors are tools which allow the user to intervene in a more significant way. They require a pre-analysis of the problem and allow objects of simple form to be meshed. By cutting a domain judiciously, this can constitute a union of elementary volumes that can be meshed in a regular way. Regular meshing algorithms are based on extrusion techniques or projection transport, and they generally generate hexahedral elements (bricks) or prismatic elements. Regular mesh processors are used mainly in mechanics. The limitation of this family of mesh generators is that we can only treat volumes of simple geometric form. 13.4.2.2.1. Regulated meshing The principle of the method used is schematized by Figure 13.10 [GEO 90].
Figure 13.10. Principle of regulated meshing
Through this approach, we can create a mesh for any simple topology. The domain partition is related to information on the opposite sides of the domain edge. An important characteristic of this mesh processor is thus a boundary discretization of the treated domain, i.e. the contour mesh. The number of points logically connected on both sides must be identical.
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This technique can be split into several phases: 1) Transport of the points of the contour on the sides of a unit element while preserving the relative distance between points. 2) Canonical meshing of the unit element. This mesh is tantamount to connecting the items corresponding two to two on one side with the other of the unit domain. 3) Transport of this reference mesh on the real domain. Let M be a canonical meshing point, of coordinates x and y, for a domain topologically similar to a rectangle. The following transformation is used: M=(1-y).f1(x)+x.f2(y)+y.f3(x)+(1-x).f4(y)-((1-x).(1-y).a1+x.(1y).a2+x.y.a3+(1-x).y.a4)
[13.4]
where fi is the parametrization of the domain sides and the ai its vertices. fi is defined as follows: fi [0,1] belonging to the curve (ai ,ai+1). This transformation allows the image M of the point M (x, y) to be known. These transformations respect the vertices and the sides of the field (other transformations ensuring the same effect are possible for other forms of geometry). We should note that this treatment can be applied to domains topologically similar to a triangle, or another geometrically simple form. We can generalize this approach to the third dimension. 13.4.2.3. Meshing by extrusion Meshing by extrusion on a domain of dimension d is obtained by transforming a mesh of dimension d-1. It is defined by a transformation (translation, rotation, 3D path), a number of layers and a periodic preliminary meshing of the two faces in opposition. These two faces are related to each other by the transformation. Relatively simple to implement, this technique remains limited to a restricted family of devices. In three dimensions, a traditional technique consists of generating a 2D meshing, then extruding it according to a perpendicular direction or according to a curve (for example, circular). It can be effectively adapted for geometries with almost symmetric revolutions (revolving machines, cold crucibles), and for installations where disproportions between objects are not too significant. Figure 13.11 presents a mesh obtained by “global” extrusion [DUT 00] of a 2D mesh. The latter is created from a projection on a plane of the objects of a study domain.
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The Finite Element Method for Electromagnetic Modeling
a)
b) Figure 13.11. Meshing by extrusion, a) 2D reference mesh, b) 3D extruded mesh. Six inductors are laid out helically around a cylindrical load
13.4.2.4. Coupling of mesh algorithms The following table summarizes the characteristics of the main meshing algorithms presented.
Mesh Generation Automatic or free meshing
Regulated, extrusive meshing
Tetrahedrons
Bricks, prisms
Many elements Non-adapted to problems with anisotropy or with scale effect
525
Control of number of elements is simpler strong Adapted to the problems of skin effect or with scale effect
Delaunay algorithm
Transport projection algorithm
Frontal meshing
Meshing by extrusion
“All purpose”
Requires simple geometry
Table 13.3. Advantages and disadvantages of each mesh algorithm
A tool allowing us to mix these two meshing families in order to improve the conditioning of the finite element problem is presented in [ZGA 96]. It makes it possible to benefit from the advantages of each mesh algorithm. In order to use this type of mesh processor, several obstacles must be overcome: the creation of each mesh type, the treatment of non-conformities and the strong connection by a new element, the pyramidal element. This element was proposed to be of the first order [BED 92]. 13.4.2.4.1. The pyramid element A strong connection is necessary to ensure the continuity of the state variable at the interface between the finite elements. Indeed, if we place two triangular finite elements and one quadrangular element in opposition (see Figure 13.12), it is clear that the form function of node N1 on the level of the rectangle is not continuous with the function N1 of the two triangles. The pyramid makes it possible to manage simply and in a strong way the continuity problem between these two types of facet (rectangular and triangular) that we find in many mixed meshes. The first order pyramid element with five nodes, the incomplete second order element with 13 nodes (located in the middle of the edges) and the complete second order element with 14 nodes (an additional node in the center of the base) were studied and validated. The non-traditional form functions are expressed in rational form. For the calculation of integrals by the Gaussian method, a new collection of points and Gaussian weights has been proposed [ZGA 96]. Other uses of the pyramids can be the connection of brick meshes of different densities, or the case of an extruded triangle mesh around an axis (see Figure 13.13).
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The Finite Element Method for Electromagnetic Modeling
N1
Figure 13.12. Connection between two triangular facets and a quadrangular facet
'
Figure 13.13. Construction of a 3D mesh by extrusion from a 2D triangular mesh rotated around an axis ' in the case of a face having a point on the axis
13.5. Mesh regularization After any meshing phase, whatever the mesh processor used, a phase of mesh regularization is often necessary, significantly improving the quality of the elements. It can be a question of edge permutation or node displacement. This last method is often the most direct and most effective. 13.5.1. Regularization by displacement of nodes In a simple and intuitive way, the mesh regularization by node displacement consists of building a better quality mesh by repositioning the nodes in an optimal way. By design, this method preserves the number of nodes, which is one of its advantages. On the other hand, the topology of the edges can be completely changed, modifying the number of elements at the same time. Indeed, if the node displacement is relatively significant, the finite elements resulting from the initial meshing can be very deformed and even turned over. Mesh regularization by node displacement is thus often accompanied by a complete rebuilding of the elements,
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which does not pose any particular problem for the Delaunay algorithm (this algorithm by design ensures the best mesh for this new distribution of nodes). The first meshing regularization algorithms repositioned the nodes at the barycenter of their first neighbor.
Figure 13.14. Principle of barycentering methods
This method is simple and requires only a little computation effort to find out the new position of each node; However, this approach introduces an iterative process since, once a node is moved to a new optimal location, its first neighbors also need to be moved since the optimal position for these nodes (at the barycenter position of its first neighbors) has consequently changed. Thus, the general algorithm consists of iterating the whole set of nodes several times, displacing each node one by one, to its new optimal position, until all the individual movements reach a prescribed limit. It is also possible to avoid the iterative process and calculate the entire set of displacements in one single shot. The main idea is based on a matrix, representing the optimality criterion (each node is located at the barycenter of its neighbors), the positions of the node being obtained by solving this system. However, this approach is generally not used since its requirements in terms of memory and time are not suited to user needs. Anyway, this barycenter criterion is currently considered less efficient compared to methods such as bubble movements [SHI 95], [YOK 99], [HER 00], [LEC 00] that will be developed in the following section. Still, since the bubble method has some similarities with the barycenter criterion, it was useful to provide some information on this former approach.
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The Finite Element Method for Electromagnetic Modeling
13.5.2. Regularization by bubbles The general principle of this method is simple. Nodes are considered as bubbles or particles, respecting the map of the element’s size (the size of each bubble is determined by the average size of the elements resulting from the node).
Figure 13.15. Illustration of the regularization on a simple domain (on top the bubbles, below the meshing, on the left before regularization, on the right after regularization)
When the mesh is not yet regularized, these bubbles overlap or leave vacuums in places. After regularization, the bubbles will be regularly distributed, and in the case where the number of nodes is optimal, they are tangential. The bubbles are attached to each other by forces which can be, depending on the models, either purely repulsive (like the models of forces of perfect gases) or a
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mixture of attractive/zeroed/repulsive, depending on the distance separating the centers of two bubbles.
Figure 13.16. Comparison of the forces models according to the distance between the centers of the bubbles
Regularizing the mesh is ultimately tantamount to finding the bubbles’ point of equilibrium. According to the model of force selected, the results of the regularization will be different: – a purely repulsive model will tend to distribute the bubbles in all the available space; – a mixed force model will lead to the compaction of the mesh, while making the bubbles tangent with each other, ensuring against “holes” appearing in part of the mesh, if the number of nodes is insufficient.
Figure 13.17. Behavior of the bubbles according to the force model (mixed model on the left, purely repulsive model on the right)
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The Finite Element Method for Electromagnetic Modeling
As before, the optimal position of the nodes is deduced in an iterative way by successive displacements. In order to make the movement more stable, a friction force which introduces a damping is generally added to the attraction/repulsion forces. The movement of each bubble is then governed by the solid mechanics equations, by allocating a mass to each bubble. The Runge-Kutta method allows their trajectory in time to be calculated. 13.5.3. Adaptation of nodes population If the bubbles method is relatively straightforward, it must often be accompanied by a control and adaptation of the number of nodes to really produce very regular meshing. Indeed, whatever meshing method is used, it is quite rare that the number of nodes involved in meshing before regularization corresponds exactly to the number of nodes necessary to pave the space in an optimal way, while respecting the wishes of the user in terms of mesh size. To be convinced, it is enough to use a mixed force model on non-optimized meshing; the bubbles then reorganize themselves with respect to each others, but: – either holes appear in places, a sign of a deficit of nodes; – or the bubbles overlap, a sign of an excess of nodes. 13.5.4. Insertion in meshing algorithms In summary, here follows the general meshing regularization algorithm, applicable to any meshing algorithm. While the meshing is not sufficiently regularized; Calculate the map of neighboring connectivity of the nodes between them Initialize the speed of the bubbles to 0 While nodes move Increment time t For each node of the set of nodes Calculate the new position of current node by solving the differential equation of the movement at time t Position the node at its new position Re-calculate the size of the bubble for this new position End for Calculate the kinetic energy at time t of the meshing bubbles End Adapt the number of nodes locally (adding or deleting nodes) End Remeshing starting from new positions of nodes by a Delaunay algorithm.
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In this algorithm, the kinetic energy calculation enables us to find a relative stop criterion capable of determining the mesh stability. Meshing is declared stable when the kinetic energy of the bubbles is weak on the one hand, and the adaptation of the node population does not introduce or remove bubbles on the other hand. 13.5.5. Value of bubble regularization The regularization method by bubbles has many applications. In accordance with its primary purpose it can obviously improve on the mesh obtained with the traditional meshing methods. The figure below shows an example of a considerably improved mesh; the initial meshing having resulted from a Delaunay algorithm.
Figure 13.18. Mesh of a motor (above: before regularization, below: after regularization)
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The Finite Element Method for Electromagnetic Modeling
However, beyond this natural application, the technique can also be used to build new meshing methods. The frontal method, for example, is known for the quality of mesh produced, but also presents a major difficulty when the fronts meet, leaving a complex and thin domain where paving is generally of a much lower quality. Applying regularization by bubbles, followed by a Delaunay mesh, on the node cloud obtained by a frontal method also solves this difficulty. Taking this a step further, we should note that coupling the regularization method with the adaptation of the node population can be regarded as a meshing method itself. Indeed, it makes it possible to gradually pave the space without necessarily requiring an initial meshing; the adaptation of the population being sufficient to produce new nodes. However, the greatest value of the regularization method by bubbles undoubtedly lies in its capacity to produce elastic meshes, able to be adapted to large geometric deformations [LEC 01]. This property is particularly useful when modeling electromagnetic actuators, for example. During the displacement of the moving part, the deformable air zone must obviously be remeshed. Appreciating the difficulty in meshing these domains and the sensitivity of numerical solutions to mesh disturbances, many computing codes adopt a hybrid finite elements/boundary integrals method to circumvent this problem. The disadvantage of this method is that it requires matrix filling, which involves a loss of performance in computing times. The regularization method by bubbles once again proposes an elegant alternative. With each displacement of the moving part, the position of the nodes in the deformable air zone is recalculated, according to the following principle: the bubbles at the interface between the air and the moving part follow, without requiring any particular development, the displacement of the moving part. Then they will gradually “push back” the internal bubbles in the deformable air zone, particularly for the bubbles close to the moving part and much less for the more distant bubbles. Thus, the meshes are largely only deformed, which minimizes the mesh disturbance. Imagining air as a viscous fluid makes it possible to understand the phenomenon which occurs with this regularization method. Additionally, it is on this same principle that certain avalanche models operate, where the snow slides are comparable with variable sized bubbles.
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Figure 13.19. Use of a method of regularization by bubbles for the taking into account of large displacements
13.6. Mesh processor and modeling environment 13.6.1. Some typical criteria – In isotropic mediums, elements must be “well proportioned” [BAK 89], i.e. the ratio of their larger sized elements to smaller elements must be as close to 1 as possible. This ratio is called element distortion ratio. In a surface meshing, the ideal elements are equilateral triangles and squares. In a volume meshing, the ideal meshes are regular tetrahedrons and regular hexahedrons. – In anisotropic mediums, element edges can be to a great extent unbalanced [BOR 97]. A typical example of anisotropic elements is the case of the induction heating in transverse flux of thin sections. The field perpendicular to the plate varies little in the thickness direction and in an important way according to the plane of the plate. We will try to mesh the plate surface correctly, whereas one or two quadrilateral elements in two dimensions, along with hexahedrons in three dimensions, will be enough to mesh its thickness. The rule applied consists of finely meshing in the direction of large gradients. – Meshing should not be unnecessarily fine. It is well known that the finer the meshing, the greater the number of unknown variables is, and consequently the greater the calculation time is. The user must finally make a trade-off between the accuracy of the geometric representation and the calculation time. When he has a rough idea of the end result, he can decide to mesh coarsely in certain areas and more finely in others. Generally an examination of the calculation results leads us to re-start it with an altered meshing. This work may be automated with respect to a predefined criterion of error. This is called “auto-adaptability”.
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In order to satisfy the two preceding criteria as effectively as possible (good element proportions and element amounts) meshing methods were developed. They can be divided into two big families: the structured mesh processors, where the user must intervene, and free (or automatic) mesh processors, where user intervention is minimal. 13.6.2. Electromagnetism and meshing constraints Meshing problems are common in electrical engineering, and are essentially caused by two factors: the geometry of the electrotechnical application, which is often complex, and the strong variations in the distribution of state variables in certain physical cases. 13.6.2.1. The magnetic physical problem and its influence on meshing An exhaustive analysis of the problems and applications of magnetics recently treated in finite elements by research and industrial centers makes it possible to characterize the difficulties encountered, and the researchers’ ingenious means to avoid them or solve them in quite a complete way. It is sometimes difficult to decide if a new formulation, numerical tool or meshing algorithm will be the optimum means to eliminate the problem. Thus, it appears more judicious to evaluate all the problems by topics and to present their solutions, either by meshing techniques, or by numerical or mathematical techniques. The following topics continually appear: – magnetic air-gap, – movement of conducting materials under a magnetic field, – boundary layers, – deformations under the effect of electromagnetic forces, – magnetic saturation, – magnetic field at the infinite. Several of the topics quoted above are present in any industrial application involving electromagnetic systems. For example, when studying the dynamics of electromagnetic systems in movement (electrical motors, electromechanical systems, induction heating in parade devices), we find the following problems: the magnetic air-gap, rotor movement with respect to a stator, the magnetic saturation of the magnetic poles and their heating, the position of the windings, magnetic leakage in the surrounding air and mechanical forces.
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13.6.2.1.1. Magnetic air-gap The magnetic air-gap problem raises the broader problem of dividing spaces that exist between various parts of the device into finite elements. This space can be very small (or very large) compared to the size of the parts. We seek to mesh an air-gap of a few tenths of a millimeter and a device of about 10 centimeters (current transformer). Here we speak of scale effect. The meshing techniques are either traditional, so that the space is meshed whatever its size, or adapted or simplified for some areas [VIN 98]. A particular numerical or mathematical treatment is thus left as the responsibility of the assembly and numerical solving modules: – In the first case, the mesh processor must be able to manage densities of variable elements and to control the form of the elements and their sizes. For small air-gaps, the use of quadrangular elements is always preferred above triangular elements. – In the simplified example, the air-gap is replaced by a layer of elements without thickness (shell elements), line shape in 2D and surface shape in 3D. – In the adapted case, the air-gap is replaced by particular elements, such as the macro-element, to calculate the fields in the air-gap separating the moving parts and the fixed parts of a revolving machine. – In all cases, taking into account the space between the parts of the device, which is generally very complex because of the parts shape and their spatial positioning, generally forces us to use a global meshing method. Required functions: global meshing, scale effect, macro-element, internal boundary elements. 13.6.2.1.2. Conducting material movement under magnetic field Taking into account the moving parts of an electromechanical system is treated, according to the case, as a static problem (heating in transverse flux of a steel blade in a rolling mill, eddy currents brake), or as a dynamic problem (synchronous motor). In the static case, the meshing is fixed. We should also consider the mesh size of the moving part. Indeed when speeds are high, the transport term of the magnetic field can be significantly “higher” than the diffusion term of the field. If no upwind is used during resolution, the mesh process must allow us to observe the Peclet criterion in the mesh of the moving part [KUR 98]. The mesh size in the direction of the displacement should be lower than: size 2 / ( P * V * speed )
[13.5]
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The Finite Element Method for Electromagnetic Modeling
where P is the magnetic permeability and V electric conductivity. In the dynamic case, meshing is mobile. Several alternatives are proposed to simulate these devices: – Systematic remeshing with each time step [TAN 00], [TAN 98]; the mesh processor must automatically remesh, either locally or overall, the study domain. For this purpose, the geometry transmitted to the meshing algorithms must be parameterized. Moreover the set of meshing information such as the discretization of the geometric and physical boundaries and the mesh size distribution in space must be completely controlled. Localization algorithms become necessary in order to interpolate the values of a former time step on the current time step. The simultaneous or sequential management several grids is also necessary. Required functions: parameterized meshing, multi-meshing, localization. – The regular meshing without meshing in sliding conformity [ALB 00], [EMS 98], [KUR 98]; in this configuration, the meshing is defined only once. The moving part is meshed regularly, with a space step between the meshes related to the time step of the study, therefore to the speed. The movement is accounted for by the solving module, by processing for example with a circular shift of the variables and a modification of the boundary conditions during time. Required functions: regular meshing at the boundary. – The non-conformity sling grid [MUR 00], [BUF 00], [GOL 98]; it is difficult to resist temptation to push the finite element method towards domains for which it was not considered. We here authorize a non-conformity of the meshes during time, which seems to be the most natural. The numerical difficulty is to ensure a continuity of the state variable during the crossing of this boundary in nonconformity. Several numerical techniques exist, such as: - the incomplete form functions [BID 98], - the macro-element [TAR 98], - the definition of a transition boundary whose grid is invariant during time and on which the quantities are interpolated [KOM 00], [TOD 98], [BOU 98]. Required functions: meshing in non-conformity, in surface and volume, macroelements. 13.6.2.1.3. Frequency effects The frequency of the device’s supply sources raises several difficulties: taking into account the electromagnetic skin effect and the calculation of the system impedances:
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– electromagnetic skin effect: the penetration depth G of the field inside conductive materials is dependent on the three parameters, electric conductivity V , magnetic permeability P and pulsation Z according to the formula G
2 ZVP .
Two methods are commonly used: – In-depth meshing of the skin depth [ABA 00], [DUT 01]. The knowledge of G is used to generate either an adapted initial meshing or, if the solution allows it, to adapt the meshing during an iterative resolution. The skin/surface meshing is controlled by two considerations: the penetration depth according to the perpendicular to the conductor, and the gradient of the state variable according to the tangent. Hence, a rule commonly used consists of imposing two second order elements according to thickness. We then compare the curve of exponential decay to two parabolas. Once the skin thickness is crossed, a free meshing can be applied. The meshing on the surface is however related to the type of element. Thus, if only triangles (or tetrahedrons) are available, the balanced proportionality of each element will necessarily lead to a very large number of elements. If the quadrilaterals (or hexahedrons) are used a significant disproportion, albeit controlled between the dimensions of the largest sides and those of the small sides, can be introduced without consequence on the result quality. We can then achieve grids of reasonable sizes and carry out calculations in three dimensions. In the particular case of a conducting plasma material the boundary is generally unknown and its determination is the result of a thermo-hydraulic calculation. Electric conductivity strongly varies with the temperature. The temperature map of the plasma area makes it possible to define a distribution of electric conductivity in space and therefore the thickness of the skin. Required functions: skin meshing in boundary, accurate check of the element sizes, quadrangular mesh form anisotropy, automatic meshing based on spatial distribution of the mesh size.
– The fine layer elements. The material is modeled by its boundaries with boundary finite elements on which particular boundary condition equations [ABA 00], [GUE 95], [REN 98] are introduced. The work of the mesh processor thus consists of allowing the identification of these boundaries and the generation of the corresponding line elements (or surface). Required functions: selection of boundary areas.
– Calculation of the system impedances: the calculation of the total impedance of a system requires us to take into account all the conductors (turns and charges) as massive materials. The formulations employed use the electric scalar potential. First, we find the meshing problem of the electromagnetic skin for all the conducting
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The Finite Element Method for Electromagnetic Modeling
materials [SIM 01] including the supply turns. Moreover, in the 3D case, it is necessary to define cut surfaces on which a reference of the scalar potential is imposed [DUL 00]. The mesh processor must thus allow these surfaces to be selected and these line (or surface) elements to be created. Required functionalities: adapted skin meshing, selection of the cut surface elements.
13.6.2.1.4. Deformations under the influence of electromagnetic forces Deformations under the influence of a magnetic field appear in many applications. Levitation in a cold crucible, electromagnetic forming, the deformation of “free” liquid metal surfaces under the influence of electromagnetic pressure (steel continuous casting process). In all the cases, the geometry of one or more areas of space is unknown and is obtained by balancing the equations coupled between various phenomena (magnetic, thermal, mechanical). The geometric modifications introduce nonlinear terms into the algorithmic solution system, whether they are transient or static. Thus with each iteration, new geometries are translated on the level of the mesh, either by a node displacement (small deformations), or by remeshing. In the case of small deformations, a modification of node coordinates, as small as it may be, can have considerable consequences for the whole mesh. In the simple case where only one displacement of boundary nodes is considered, the modification is applied initially to first order nodes and then to higher order nodes if necessary. The problem of “repositioning” higher order nodes on the boundary can be handled by the solver which corrects the coordinates while being based on the element form functions or by using analytical smoothing functions (spline, Bézier curves). In a more complex case, where we consider a general mesh deformation without topological modification, the coordinates of the top nodes of internal elements can be obtained by a traditional algorithm of barycentric re-centering, or, if the solver allows it, by the calculation of a “distribution of the displacements” in each point of the areas concerned. This last technique consists of solving a Laplace equation on each deformed area by considering the displacement vector as a state variable. The displacement vectors at the boundary are used as Dirichlet boundary conditions for Laplace’s equation. Required functionalities: displacement of nodes, smoothing of curves and surfaces, positioning of high order nodes, improving algorithms for forms with constant topology.
In the case of large deformations, in addition to the problems quoted above, the deformation can be such that a remeshing is necessary. This stage has significant consequences on the functions of the mesh processor. The smoothing of the
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modified surface is necessary, the distribution of the nodes on this surface must be controlled automatically, and, if necessary, internal meshing by integrating skin meshing for high frequencies. Moreover, multi-meshing management proves to be necessary in order to interpolate the quantities between the various meshes. 13.6.2.1.5. Magnetic saturation The magnetic saturation problem thus appears, with strong fields in the zones with strong potential gradients. Generally, this phenomenon is solved by using nonlinear algorithms with a positioning control of the operating points on curve B(H) of magnetic materials. A fine meshing is necessary in the nonlinear zones. This makes it possible to approach the local variations of magnetic permeability (or reluctivity) more effectively, expressed according to the induction and favor the convergence of the solving algorithms [JAN 00]. Required functions: refining according to the gradient, adaptability, coupling with the solver.
13.6.2.1.6. Infinite magnetic field The problem of “infinite” meshing remains a contemporary issue when applying the finite element method in electromagnetism. This problem is particularly crucial for modeling electromagnetic waves. This task can be handled by the mesh processor, at the risk of creating a very large number of elements. Element size can progress incrementally between the active parts and the infinite. Alternative techniques have appeared, such as the infinite element method [IMH 89], [BAR 98], absorbing boundary conditions and coupling with boundary integrals method. The treatment of infinite elements requires the mesh processor to locate a zone of the meshed domain on which particular functions of interpolation will be used. A characterization of boundary elements will allow absorbing boundary conditions to be defined. In the same way, in the case of coupling with the boundary integral method, only a selection of boundary element faces common to both methods is necessary. Required functionalities: control of the mesh size in the infinite direction, localization of infinite elements, localization of boundary elements.
13.6.2.2. The geometric problem and its influence on the meshing The geometric description problem arises on several levels. Firstly, independently of the electromagnetic problem, the link with the CAD tools is a cutting edge field of research, the obvious goal being the total control of the design chain, or even of manufacturing. It is then rare that a domain of study fully complies with the reality. It is often simplified for various reasons such as the limitations of geometric models, the prevention of massive calculations, or calculation redundancy
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The Finite Element Method for Electromagnetic Modeling
in certain parts or portions of pieces. Lastly, it is important to note the geometric discretization error resulting from cutting a domain [DHA 84]. The problem of the exact meshing of curved boundaries is important in electromagnetism, for example to correctly model the distribution of the currents in the inductors. These three aspects of the link between geometry and meshing and their consequences in the design and the development of mesh processors (meshers) are detailed below: – Link with the CAD: several ways are possible to establish this link: - the passage of information by standardized files (IGES, SET). Tools for information exchange with these files must be programmed within the mesh processors [MA 00]; - the integration of meshing algorithms in CAD computing codes. The computer implementation of the meshing algorithms, data structures and computer programming language must be in agreement with the CAD software; - the full integration of meshing computing codes within CAD codes. Each tool can then preserve the management of its own information. It is necessary to write programs of information exchanges between the data structures of the various software tools. – Simplifications of the geometry of the domain [VIN 98], [HAS 00]: the simplifications of the geometry of the objects in a domain of study have been initiated following an analysis integrating the evaluation of active zones from the electromagnetic point of view, the desired accuracy level of the computation result and the computer environment (main memory, computing time). To our knowledge, no finite element modeling tool exists with the ability to fully and exactly describe the geometry of a motor with the complexity of its winding, the current supplies and other elements of detail and of calculating the electromagnetic fields and internal currents. In such a case, the wire windings are generally gathered in “massive turn” inside which the current density is either constant, or parameterized according to the section in order to take into account a non-homogenous distribution, or even coupling effects between wires. In general, from the perspective of meshing, we are only interested in the geometry of the section and not in that of the wire which composes it. – Geometric discretization error: this error can be treated either by the mesh processor, by increasing the number of discretization points on a curve boundary, or by the solver through the use of high order elements [DHA 84] or elements known as “exacts” [VIL 00]. If this treatment is handled by the solver, the mesh processor will have to handle the creation of “big” elements, but will have to calculate the intermediate nodes of the high order elements precisely, while leaning on the geometry (see Figure 13.14).
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Spline
Figure 13.20. Geometric discretization error. A Lagrangian element is used to represent a boundary portion defined by: a) a spline function, b) geometry, c) details of the meshing
13.7. Conclusion
If we refer to the recent scientific literature for the application of the finite element method to electromagnetism, papers dealing only with meshing algorithms are very rare. In general, the meshing methods developed in the 1980s and early 1990s are satisfactory and allow many problems to be solved with the help of some specific adaptations. Besides, the integration of the pre-processor and solver modules is increasingly strong and the work of the mesh processor consists of generating a relevant “initial” meshing. In order to deal with modern problems it is necessary that the meshing tool evolves towards an “expert” tool, encompassing several disciplines: computer engineering, CAD, numerical methods, physics of the continuous mediums. It will thus be able, starting from structured and evolutionary knowledge bases, to accommodate new problems and in particular to prepare the couplings between electromagnetism and other physical phenomena. This last task is necessary to obtain realistic and reliable models. 13.8. References [ABA 00] Abakar A., Meunier G., Coulomb J.L., Zgainski F.X., “3D modelling of shielding structures made by conductors and thin plates”, IEEE Trans. on Magnetics, vol. 36, no. 4, pp. 790, 794, July 2000.
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The Finite Element Method for Electromagnetic Modeling
[ALB 88] Albertini J.B., Contribution à la réalisation d’un logiciel de modélisation des phénomènes électromagnetiques en trois dimensions par la méthode des éléments finis, PhD thesis, INPG, 1988. [ALB 00] Albertz D., Henneberger G., “On the use of the new edge based A-A,T formulation for the calculation of time-harmonic, stationary and transient eddy current field problems”, IEEE Trans. on Magnetics, vol. 36, no. 4, pp. 818-822, July 2000. [BAK 89] Baker T.J., “Element quality in tetrahedral meshes”, Proc. 7th Int. Conf. on Finite Element Method in Flow Problems, Huntsville, AL, April 3-7, 1989. [BAR 98] Bardi I., Biro O., Preis K., “Perfectly matched layers in static fields”, IEEE Trans. on Magnetics, vol. 34, no. 5, pp. 2433, 2436, September 1998. [BED 92] Bedrosian G., “Shape functions for 3-D elements”, International Journal for Numerical Methods in Engineering, vol. 35, 1992. [BID 98] Biddlecombe C.S., Simkin J., Jay A.P., Sykulski J.K., Lepaul S., “Transient electromagnetic analysis coupled to electric circuits and motion”, IEEE Trans. on Magnetics, vol. 34, no. 5, pp. 3182, 3185, September 1998. [BLA 91] Blacker T.D., Stephenson M.B., “Paving: a new approach to automated quadrilateral mesh generation”, Int. J. Num. Meth. Eng., 32, 811-847, 1991. [BOR 95] Borouchaki H., George P.L., Hecht F., Laug P., Saltel E., “Mailleur bidimensionnel de Delaunay gouverné par une carte de métriques. Partie I : Algorithmes”, Report INRIA no. 2741, December 1995. [BOR 97] Borouchaki H., George P.L., “Aspects of 2D Delaunay mesh generation” Int. J. Num. Meth. Eng., vol. 40, pp. 1957-1975, 1997. [BUF 00] Buffa M., Maday Y., Rapetti F., “Calculation of eddy currents in moving structures by a sliding mesh finite element method”, IEEE Trans. on Magnetics, vol. 36, no. 4, pp. 1356, 1359, July 2000. [CAV 85] Cavendish J.C., Field D.A., Frey W.H., “An approach to automatic threedimensional finite element mesh generation”, International Journal for Numerical Methods in Engineering, 21, pp. 329-347, 1985. [DHA 84] Dhatt G., Touzot G., “Une présentation de la méthode des éléments finis”, Collection Université de Compiègne, Second Edition, 1984. [DUL 00] Dular P., Kuo-Peng P., Geuzaine C., Sadowski N., Bastos J.P.A., “Dual magnetodynamic formulations and their source fields associated with massive and stranded inductors”, IEEE Trans. on Magnetics, vol. 36, no. 4, pp. 1293, 1299, July 2000. [DUM 00] Dumont B., Gagnoud A., “3D finite element method with impedance boundary condition for the modelling of molten metal shape in electromagnetic casting”, IEEE Trans. on Magnetics, vol. 36, no. 4, pp. 1329, 1332, July 2000. [DUT 85] Du Terrail Couvat Y., Modélisation géométrique et topologique pour l’application de la méthode des éléments finis en électromagnétisme, PhD thesis, INPG, 1985.
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[DUT 00] Du Terrail Couvat Y., Garnier M., Allemann C., “3D Parametered magnetothermal simulations for modular induction heating devices”, Proceedings of EPM Congress, Nagoya, Japan, April 2000. [EMS 98] Emson C.R.I., Riley C.P., Walsh D.A., Ueda K., Kumano T., “Modelling eddy currents induced by rotating systems”, IEEE Trans. on Magnetics, vol. 34, no. 5, pp. 2593, 2596, September 1998. [EUCLID] CD-ROM demonstration, Matra-Datavision, France. [FOR 94] Forsman K., Kettunen L., “Tetrahedral mesh generation in convex primitives by maximising solid angles”, IEEE Trans. on Magnetics, vol. 30, no. 5, September 1994. [GEO 88] George P.L., Hecht F., Saltel E., “Tétraédrisation automatique et respect de frontière”, INRIA, April 1988. [GEO 90] George P.L., Génération automatique de maillage: Applications aux méthodes d'éléments finis, Paris, Masson, 1990. [GEO 92] Georges P.L., Hermeline F., “Delaunay’s mesh of a convex polyhedron in dimension d. Application to arbitrary polyhedra”, International Journal for Numerical Methods in Engineering, vol. 33, 975-995, 1992. [GOL 89] Golgolab A., “Mailleur 3D automatique pour des géométries complexes”, RR INRIA, no. 1004, 1989. [GOL 98] Golovanov C., Coulomb J.L., Maréchal Y., Meunier G., “3D mesh connection techniques applied to movement simulation”, IEEE Trans. on Magnetics, vol. 34, no. 5, pp. 3359, 3362, September 1998. [GUE 95] Guérin C., Tanneau G., Meunier G., Labie P., Ngnegneu T., Sacotte M., “A shell element for computing 3D eddy currents – application to transformers”, IEEE Trans. on Magnetics, vol. 31, no. 3, pp. 1360, 1363, May 1995. [HAS 00] Hashimoto H., Yamada T., Tani K., Honjo S., Sato Y., Ishii H., “Finite element analysis of AC losses in double helix superconducting cables”, IEEE Trans. on Magnetics, vol. 36, no. 4, pp. 1205, 1208, July 2000. [HEN 93] Henot, Briere de l’isle, George P.L., “Optimisation de maillages tridimensionnels”, INRIA, SDRC, Conference Proceedings of STRUCOME, 1993. [HER 82] Hermeline F., “Triangulation automatique d'un polyèdre en dimension N”, RAIRO Analyse Numérique, vol. 16, pp. 211-242, 1982. [HER 00] Hérault C., “Méthodes sans maillage pour la modélisation en électromagnétisme”, PhD thesis, INPG, 2000. [IMH 89] Imhoff J.F., “Modélisation magnétique et mécanique des machines électriques par la méthode des éléments finis”, PhD thesis, INPG, 1989. [JAN 00] Jänicke L., Kost A., “Convergence properties of the Newton Raphson method for non-linear problems”, IEEE Trans. on Magnetics, vol. 34, no. 5, pp. 2505, 2508, September 1998.
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[KOM 00] Kometani H., Sakabe S., Kameari A., “3-D analysis of induction motor with skewed slots using regular coupling mesh”, IEEE Trans. on Magnetics, vol. 36, no. 4, , pp. 1769, 1773, July 2000. [KUR 98] Kurz S., Fetzer J., Lehner G., Rucker W.M., “A novel formulation for 3D eddy current problems with moving bodies using a Lagrangian description and BEM-FEM coupling”, IEEE Trans. on Magnetics, vol. 34, no. 5, pp. 3068, 3073, September 1998. [LEC 00] Leconte V., “Modélisation des pièces en mouvement dans les contacteurs”, PhD thesis, INPG, 2000. [LEC 01] Leconte V., Herault C., Maréchal Y., Meunier G., Mazauric V., “Optimization of a finite element mesh for large air-gap deformation”, The European Physical Journal – Applied Physics, EDP Sciences, Eur. Phys. J. AP., 13, 137-142, 2001. [MA 00] Ma S., Maréchal Y., Coulomb J.L., “Methodology for an implementation of the STEP standard: a Java proptotype”, IEEE Trans. on Magnetics, vol. 36, no. 4, pp. 1664, 1668, July 2000. [MOU 95] Mounoury V., Stab O., “Automatic quadrilateral and hexahedral finite element mesh generation: review of existing methods”, European Journal of Finite Elements, vol. 4, no 1, 75-102, 1995. [NEK 96] Nekhoul B., Zgainski F.X., Labie P., Morillon F., Bourg S., “Calculating the impedance of a grounding system”, IEEE Trans. on Magnetics, vol 32, no. 3, May 1996. [NET 98] Nethe A., Quast R., Stahlmann H., “Boundary condition for high frequency eddy current problems”, IEEE Trans. on Magnetics, vol 34, no. 5, pp. 3331, 3334, September 1998. [POU 95] Poulbot V., Krahenbühl L., Massé P., Blanpain R., “3D interface elements for modelling complex potential drops. Comparison with a boundary elements method”, IEEE Trans. on Magnetics, vol 31, no. 3, pp. 1684, 1689, May 1995. [RAS 95] Rassineux A., “Maillage automatique tridimensionnel par une méthode frontale pour la méthode des éléments finis”, PhD thesis, University of Nancy I, 1995. [REN 98] Ren Z., “Degenerated whitney prism elements – General nodal and edge shell elements for field computation in thin structures”, IEEE Trans. on Magnetics, vol. 34, no. 5, pp. 2547, 2550, September 1998. [SHI 95] Shimada K., Gossard D., “Bubble mesh: automated triangular meshing of nonmanifold geometry by sphere packing”, in 3rd Symposium on Solid Modeling and Applications, pp. 409-419, 1995. [SIM 01] Simon A., Du Terrail Couvat Y., Gagnoud A., “Finite element method adapted to steady state AC problems”, Proceedings of COMPUMAG 2001, Evian, July 2001. [SZU 00] Szucs A., “Macro elements in the finite element ananlysis of multi-conductor eddycurrent problems”, IEEE Trans. on Magnetics, vol. 36, no. 4, pp. 813-817, July 2000. [TAN 98] Tani K., Yamada T., Kawase Y., “A new technique for 3-D dynamic finite element analysis of electromagnetic problems with relative movement”, IEEE Trans. on Magnetics, vol. 34, no. 5, pp. 3371, 3374, September 1998.
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[TAN 00] Tani K., Yamada T., Kawase Y., “Dynamic analysis of linear oscillatory actuator driven by voltage source using FEM with edge elements and 3-D mesh coupling method”, IEEE Trans. on Magnetics, vol. 36, no. 4, pp. 1830, 1836, July 2000. [TAR 98] Tarhasaari T., Koski A., Forsman K., Kettunen L., “Hybrid formulations for eddy current problem with moving objects”, IEEE Trans. on Magnetics, vol. 34, no. 5, pp. 2660, 2663, September 1998. [TOD 98] Todaka T., Enokizono M., “Dynamic finite element analysis of a magnetic hammer with a new composite mesh scheme”, IEEE Trans. on Magnetics, vol. 34, no. 5, pp. 3339, 3342, September 1998. [VIL 00] Villeneuve D., Webb J.P., “Exact treatment of curved boundaries in large finite elements by re-parametrization”, IEEE Trans. on Magnetics, vol. 36, no. 4, pp. 1527, 1530, July 2000. [VIN 98] Vinsard G., Dufour S., Laporte B., “An approximation for the air-slots of complex shaped motors”, IEEE Trans. on Magnetics, vol. 34, no. 5, pp. 2449-2452, September 1998. [YOK 99] Yokoyama T., Cingoski V., Kaneda K., Yamashita H., “3-D automatic mesh generation for FEA using dynamic bubble system”, IEEE Transactions on Magnetics, vol. 35, issue 3, part 1, pp. 1318-1321, May 1999. [ZGA 96] Zgainski F.X., Un préprocesseur pour l'électromagnétisme, l'électro-mécanique et l'électro-acoustique, PhD thesis, INPG, 1996.
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Chapter 14
Optimization
14.1. Introduction 14.1.1. Optimization: who, why, how? During the design of an electromagnetic device, the designer should propose a configuration satisfying the functional needs as far as possible, and at the same time, making it viable from an economic point of view. Very often, the distribution of the electromagnetic field has a considerable impact on the characteristics to be optimized. Here are some design problems in which a good knowledge of the physical behavior is necessary: – minimization of the reluctant torque in a rotating machine; – obtaining a return force with minimization of Joule losses in an electromagnet; – etc. The criterion to be maximized or minimized is a scalar quantity relevant to the objective to be reached. It can concern an intrinsic variable of the studied device (a torque, a force, total losses, etc.) or a combination of several measurements of space or time (sinusoidal induction in the air-gap of a rotating machine, trajectory of closing of an electromagnetic contactor, etc.). Moreover, the criterion not always being unique, it is sometimes necessary to use a multicriterion optimization.
Chapter written by Jean-Louis COULOMB.
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The Finite Element Method for Electromagnetic Modeling
The search for the best performance of a device, in which structural (number of teeth, notch types, etc.), dimensional (dimensions of an air-gap, width of notches, etc.) and physical (type of materials, current density, etc.) parameters intervene, is a difficult problem. This is due, mainly, to the complexity of the phenomena which appear alone or combined: magnetic saturations, eddy currents, three-dimensional effects, relative movements of parts, electric couplings, multi-physical couplings, etc. Respect of the constraints relating to the feasibility (air-gap higher than a minimal value, etc.) or to the characteristics of the device (saturation of sheets lower than a reasonable rate, losses lower than an imposed limit, etc.) also adds to the difficulty. In fact, this optimization is generally out of reach of a traditional parametric study. It thus requires the use of more powerful procedures combining numerical simulation and optimization tools. 14.1.2. Optimization by numerical simulation: is this reasonable? In order to model the device before its production, we can use computer-aided design (CAD) resources, based on the application of numerical methods. In the domain of field calculation, the finite element method (FEM) occupies a place of prime importance in the numerical panoply, because it is able to also process at the same time the geometries and complex phenomena encountered in electromagnetism [SIL 83], [SAB 86] and related disciplines [ZIE 79], [DHA 84]. Other general methods, such as the boundary integral method [DUR 64] or the finite difference method [DUR 64], [LAI 67], [DUR 68] are used. Sometimes, the use of more specific methods, such as the magnetic dipoles method [HOO 89], the equivalent magnetic circuit method [ROT 41] or the partial equivalent electric circuit [CLA 96], is particularly indicated. However, a numerical simulation is very often expensive in terms of calculating time (one hour, one night, one week, etc.). This characteristic becomes a problem when it is a question of optimizing the studied device. Indeed, the application of the optimization methods requires many numerical simulations (at best, tens, when it is possible to use a powerful deterministic optimization algorithm, in the worst case, thousands, when the search for a global optimum is carried out using an evolutionary algorithm) and can thus induce a prohibitive total simulation cost which puts it out of reach of everyday usage.
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14.1.3. Optimization by numerical simulation: difficulties As has been shown previously, the physical phenomena intervening in the electromagnetic devices are complex. The existing simulation tools are obviously powerful but can also be burdensome. To summarize, the problems of optimization that we encounter have the following characteristics: – the function quality can be multi-objective; – the optimization should respect some constraints; – the function quality f(x) is nonlinear; – the law of direct variation of the function quality f(x) is unknown; – most of the time, we do not have any information on the gradient of f(x); – most of the time, the determination of the function quality f(x) is expensive; – the function quality f(x) is disturbed by method errors, which are due in particular to discretizations in space and time. These characteristics come both from the studied phenomena and the tools modeling them. They are behind many difficulties in the implementation of optimization. 14.1.4. Numerical design of experiments (DOE) method: an elegant solution These problems are known from the experimenters who, before the digital era, had to face the deadlines and costs related to the experimental search for an acceptable solution. The solution inspired by the work of Fisher [FIS 35], Box [BOX 78] and Taguchi [TAG 87] has since then been well established in the fields where experimental study plays a dominating part (agronomy, chemistry, etc.). This concerns the DOE method which has transferred very well in the numerical world and which is thus called the numerical DOE method [BRA 94], [SCH 98], [GIL 98]. The design of experiments method is particularly well adapted to the problems of optimization characterized by quite a high number of parameters (a few tens), a high unit cost of the experiment and results subjected to random effects. The main stages of this method are listed below: – initially, the method proposes to carry out a “screening”, i.e. to determine the most influential parameters in the studied device (the goal is to reduce the number of actual parameters from a few tens to a few units);
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The Finite Element Method for Electromagnetic Modeling
– then, the method proposes to develop a “response surface” giving a numerical approximation of the response according to the most influential parameters; – finally, the method proposes to exploit this response surface, either for prediction (interpolation of the response for a new combination of parameters), or for optimization (search for a combination of parameters which optimizes a certain criterion). The first two stages require experiments to be performed with combinations of the entry parameters imposed by the selected experimental design. The third stage does not require any additional experimentation (except a check of the final result) and is thus of low cost compared to the stages which precede it. In its initial concept, the design of experiments method integrates the inherent variability in any experimental approach. In particular, it proposes to repeat the same experiment several times to average the results and discover the standard deviation. Moreover, the experimental points recommended support the maximum excursion in the domain, which, indirectly, minimizes the impact of errors. In its numerical transposition, it is common to neglect the variability of the simulation results. Indeed, for the same set of entry parameters, the outputs of the simulation program are always identical. Thus, their variability is considered zero. Nevertheless, during the variations of parameters, there are method error variations, such as those introduced by re-meshing, and which would deserve detailed attention. We will not consider this aspect in this chapter. Let us note, however, that by favoring a large excursion of the parameters, the effect of the method is to minimize the influence of these variations. 14.1.5. Sensitivity analysis: an “added value” accessible by simulation The numerical design of experiments method process transposes the traditional design of experiments method, i.e. for each combination of entry parameters, the method waits for a response from the device. In fact, digital modeling can bring much more information on the device than this simple response. Thus, at the price of some mathematical processing in addition to numerical processing, numerical simulation can bring the sensitivity of the response with respect to the entry parameters. The powerful deterministic methods of optimization, exploiting the response and its gradient, thus become usable. This process is generalized since the derivatives of a high nature are also reachable. This then led to the Taylor development of the response according to the parameters [SAL 98], which is another way of building a response surface.
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14.1.6. Organization of this chapter In this chapter, we will thus deal with the optimization of structures by numerical simulation. We will briefly recall the deterministic numerical methods of optimization without constraints, the deterministic methods with constraints and the evolutionary methods. Then, we will present the numerical design of experiments method and some usual response surfaces which are well adapted to the optimization. Then, we will present the principles of sensitivity analysis in the context of the numerical simulation. Lastly, we will finish with a concrete example on which we will clarify the whole optimization process developed in this chapter. 14.2. Optimization methods 14.2.1. Optimization problems: some definitions An optimization problem of dimension n can be written generally in form: – minimize f ( x1 , x2 ,..., xn ) with x
f ( x)
^x1, x2 ,..., xn ` n
[14.1]
– while fulfilling Gi x
0
Gi x d 0 x min j
d xj d
x max j
i
1,..., me
i
me 1,..., m j
[14.2]
1,..., n
where: – quantity f(x) is the criterion to be minimized, also called the objective function; – vector x is composed of n variables xj which represent the parameters of the problem; – functions Gi(x) represent the equality and inequality constraints; – values x min and x max indicate the domain constraints. j j
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The Finite Element Method for Electromagnetic Modeling
g1
Figure 14.1. Acceptable domain and prohibited domain
Figure 14.2. Representation of local minima and global minimum of a function
The search space is limited by the constraints of the domain relating directly to the design variables. The equality and inequality constraints separate this space, in an acceptable domain, in which all the constraints are respected, and a prohibited domain, the remainder (Figure 14.1). An objective function can have several local minima. The smallest of these minima is the global minimum. When a function contains only one local minimum, it is known as unimodal. Otherwise it is called multimodal. We have defined the problem of optimization as being the minimization of a function. However, there are situations where we are more interested in finding a point of maximization, i.e., to maximize the objective function. In this case, it is advisable to transform the maximization problem into a minimization problem. In
Optimization
553
fact, the optimization methods which will be presented below are often implemented based on minimization criteria of the objective function. The simplest of the transformations is: ) f ( x)
[14.3]
f ( x)
To be able to compare the design variables, in particular at the level of the stop criteria of the optimization algorithms, they are normalized. It is common to use as a new interval, either [0,+1], or [-1,+1]. The normalized variables x j are obtained thanks to simple transformations of the type: xj
x j , r x min j, r min x max j, r x j, r
or x j
2
x j , r x min j, r min x max j, r x j, r
1
[14.4]
max in which x j , r is a real variable, x min j ,r its lower limit and x j ,r its upper limit.
14.2.2. Optimization problems without constraints An optimization problem is known as unconstrained if it does not contain constrained functions. Its resolution can be carried out by application of various methods which are subdivided into two large families, the deterministic methods (gradient method, quasi-Newton methods, etc.) and non-deterministic or stochastic methods (Monte Carlo methods, evolutionary algorithms, etc.). The deterministic methods have the following properties: – for a given initial context, they always lead to the same final solution; – they require relatively few evaluations of the objective function; – but they can be trapped on a local optimum. On the other hand, the non-deterministic methods have the following characteristics: – for a given initial context, they can lead to different solutions; – they require a large number of evaluations of the objective function; – they have the ability of finding the optimum total. In general, optimization algorithms are iterative and are similar to the trial and error method (Figure 14.3). An algorithm will be characterized by the choices made to answer the following two questions:
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The Finite Element Method for Electromagnetic Modeling
– How do we determine if the performances are acceptable? – How do we choose new values for parameters when the performances are not acceptable?
Figure 14.3. Iterative search for an acceptable combination of parameters
An optimization algorithm also concretizes a choice between the exploration of the space, necessary to the search for the global optimum, and the exploitation of the results obtained to reduce the cost of the local optimum search. For example, Monte Carlo methods allow a good exploration of the space, since any point has an identical probability being reached, but there is no analysis of the results already obtained. With the gradient method, there is less exploration, but the analysis of the previous data, via the gradients, allows a good local search. It is possible to consider that evolutionary algorithms offer a trade off between exploration and exploitation. In order to illustrate these different behaviors, let us take the function [ALO 97] f ( x, y )
0.01 * (( x 0.5) 4 30 * x 2 20 x ( y 0.5) 4 30 * y 2 20 y )
[14.5]
which has three local minima and a global minimum (Figure 14.4). The optimization is carried out, on the one hand, by the deterministic BFGS method and, on the other hand, using a genetic algorithm.
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Figure 14.4. Function having 3 local minima and one global minimum
Iteration
f(x,y)
Step length
Gradient norm
1
0.00125
1
1.17
2
0.00125
3.4641
1.17
3
-1.63
0.866025
3.51
4
-3.45131
0.866025
1.42
5
-3.68237
0.414836
0.393
6
-3.68392
0.0168431
0.00715
7
-3.68392
0.000318346
2.51e-06
Table 14.1. Optimization of function [14.5] by BFGS. Initial point x0 = 0, y0 = 0
The algorithm BFGS has found a local optimum starting from the initial point x0 = 0, y0 = 0, whereas by starting from x0 = -1, y0 = -1, it has found the global optimum. In both cases, the search process has required only 7 calls to the objective function (Tables 14.1 and 14.2).
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The Finite Element Method for Electromagnetic Modeling Iteration
f(x,y)
Step length
Gradient norm
1
-0.19875
1
1.98
2
-0.89875
3.16228
39.1
3
-4.77471
0.402529
9.59
4
-5.21756
0.192832
1.51
5
-5.23273
0.0446687
0.0678
6
-5.23276
0.00221215
0.00016
7
-5.23276
5.24066e-06
8.95e-10
Table 14.2. Optimization of function [14.5] using BFGS. Initial point x0 = -1, y0 = -1
Generation
fmin
1
-3.614688
6
-3.693985
8
-4.340504
12
-4.453080
18
-4.829776
23
-5.230562
41
-5.232669
74
-5.232748
192
-5.232751
200
-5.232751
Table 14.3. Optimization of function [14.5] using genetic algorithm (200 generations of 15 individuals)
In order to set the parameter for the genetic algorithm, we have chosen 200 generations of 15 individuals, which is a rather minimalist choice. Table 14.3 shows the progression of the population towards the global minimum. In Figure 14.4, the points represent the best individuals of the few selected generations. Compared to the preceding deterministic optimization, this search has required a considerable number of evaluations of the objective function, but the global optimum was in fact found.
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14.2.2.1. Deterministic optimization methods Depending on the dimension of the objective function to be optimized, the deterministic methods can be one-dimensional or multidimensional. The one-dimensional deterministic methods are used in the optimization of functions of only one parameter. These methods, also called line search methods, are based on techniques which allow the minimal point of the function to be located using successive reductions of the search interval. Among these methods, let us quote the dichotomy method [CUL 94], the golden number method [CUL 94], [PRE 92] and the Brent method [BRE 93], [PRE 92]. The multidimensional deterministic methods are devoted to the function optimization of one parameter or more. They can be ranked according to information on the function which they use. They are known as order 0 methods, if they use only the value of the function. They are known as order 1 methods, if they require in addition the gradient of the function. The order 0 methods are in general of low accuracy and converge very slowly towards the optimum [KOW 68]. On the other hand, they offer the advantage of avoiding the calculation of the gradient, which can be very interesting when the function is not differentiable or when the calculation of its gradient is complex or represents a high cost. The order 1 methods allow us to speed up the localization of the optimum, because the gradient gives information on the direction of search for the solution. On the other hand, they are applicable only to the problems in which the function is continuously differentiable. The multidimensional methods can be subdivided into two groups, on the one hand, the analytical or descent methods, and on the other hand, the heuristic or geometric methods. The analytical methods are based on the knowledge of a direction of search, often given by the gradient of the function. The majority of these methods is of order 1 and performs successively linear searches by using a one-dimensional method [PRE 92]. The most significant examples of analytical methods are the largest slope method [CUL 94], the conjugate gradient method [CUL 94], [FLE 87], [PRE 92], the Powell method [POW 65] and the quasi-Newton method [CUL 94], [FLE 87], [PRE 92]. The heuristic methods explore the space by successive tests while seeking the most favorable directions. Unlike the analytical methods, the majority of these methods are order 0 methods. The most used methods are the simplex method
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The Finite Element Method for Electromagnetic Modeling
[NEL 65], the Rosenbrock method [RAO 96] and the local variation method by Hooke and Jeeves [CHE 99]. Figure 14.5 shows the multidimensional deterministic methods most usually used, with their respective order of resolution.
Figure 14.5. Some multidimensional deterministic methods
14.2.2.2. Stochastic optimization methods Stochastic optimization methods are based on mechanisms of probabilistic and random transitions. This characteristic indicates that several successive executions of these methods can lead to different results for the same initial configuration from an optimization problem. These methods have a high capability for finding the global optimum of the problem. Unlike the majority of deterministic methods, they do not require a starting point or the knowledge of the gradient of the objective function to reach the optimal solution. Among the stochastic methods most employed, we distinguish the simulated annealing [KIR 83], the tabu search [GLO 89], [GLO 90], [HU 92] and the evolutionary methods. These latter methods gather various algorithms based on the same principle of exploring the search space by using a set of solutions and not only one single solution. As a representative of evolutionary methods, we have genetic algorithms [HOL 75], [GOL 89] [MIC 94], evolutionary strategies [PRE 90], [REC 94], [KAS 95], evolutionary programming [FOG 94] and genetic programming [KOZ 92]. It is of course possible to use a stochastic algorithm to localize the global optimal and to move on a deterministic algorithm to refine the search [MOH 97]. In the case of multimode objective functions, a local optimum can ultimately prove to be more interesting than the global optimum, because it is quite difficult to
Optimization
559
translate all the selection criteria in only one objective function. In these circumstances, the niche algorithms [SAR 98a], allowing us to determine several optima simultaneously, will be quite useful. Figure 14.6 presents the most commonly used stochastic methods.
Figure 14.6. Some stochastic methods
14.2.3. Constrained optimization problems An optimization problem is known as constrained if it contains at least one constraint function in its description. The set of areas of the search space where the design constraints are checked is called the feasible or acceptable domain. On the other hand, the unfeasible or prohibited domain indicates the set of areas of the space where the constraints are violated (Figure 14.1). The existence of constrained functions in an optimization problem requires special attention during the resolution of the problem. The solution to a constrained problem is obtained by the use of methods that we can separate into two groups: transformation methods and direct methods. 14.2.3.1. Transformation methods Transformation methods or indirect methods transform the original problem with constraints into an equivalent problem without constraints. This is achieved by introducing the design constraints into the objective function. Once the equivalent problem is created, a traditional algorithm of unconstrained optimization is applied.
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The Finite Element Method for Electromagnetic Modeling
In fact, such a brutal transformation would lead to an unconstrained problem too stiff thus difficult, even impossible, to solve. In practice, a succession of small transformations is applied. Each intermediate transformation transforms the original constrained problem into an unconstrained sub-problem which tends towards the unconstrained problem equivalent to the initial problem. On each of these subproblems, an algorithm of unconstrained optimization is used to determine a solution on the basis of the previous solution. The final solution is a solution of the problem in question. Among the most used transformation methods, we have the interior penalty method [CAR 61], the external penalty method [FIA 68] the increased Lagrangian method [HES 69], [POW 69], [ROC 73] the mixed variables method and the mobile asymptotes method [MAH 95]. Transformation methods are often used in the optimization of constrained problems, because they are quite simple from a theoretical point of view and of an acceptable effectiveness from a practical point of view, in particular when they are coupled with evolutionary algorithms. 14.2.3.2. Direct methods Direct methods work directly on the original constrained problem [14.1]-[14.2] by solving the associated Kuhn-Tucker equations which represent a necessary condition so that x * is a local minimum of the problem: m
f ( x*) ¦ O*i .Gi x *
0
[14.6]
i 1
O*i Gi
x *
0
i
1,..., m
The resolution of the Kuhn-Tucker equations is at the basis of several direct methods among which we can mention in particular the recursive quadratic programming method [FLE 87] which is of great effectiveness. 14.2.4. Multi-objective optimization The optimization problem, as it is raised by equations [14.1] and [14.2], is often very far from the concrete problems encountered in practice. In fact, it is rare that a single objective function, associated with constraints, can represent in an adequate way the encountered optimization problem. Most of the time, it is a set of objective functions F ( x ) F1 x , F2 x ,...Fq x which we will have to manage.
^
`
Optimization
561
The multi-objective optimization problem is expressed in the following way: – minimize F ( x)
^F1 x , F2 x ,...Fq x ` with x
^x1, x2 ,..., xn ` n
[14.7]
– while fulfilling Gi x
0
i
1,..., me
Gi x d 0
i
me 1,..., m
x min d x j d x max j j j
[14.8]
1,..., n
In this problem, the components of the objective vector enter into competition and their relative importance is not always known in advance. There is thus no single solution to this problem. To characterize the objectives, we must use the concept of Pareto non-inferiority or optimality. A solution is known as non-inferior if an improvement of an objective necessarily involves the degradation of another one. x2
F2 :
/
C P
F1B ! F1 A
A
F2A
F2 B F2 A
B
F2B
D
x1
F1 F1A
F1B
Figure 14.7. Image of the acceptable domain in the space of objective functions
This concept is illustrated in Figure 14.7. The left-hand side represents the acceptable domain for the two design variables x1 and x2 of a constrained multiobjective optimization problem. The right-hand side shows the image of this domain in the space of the objective functions. The interior points, such as P, are not solutions, because decreases, at the same time, of F1 and F2 are possible in their vicinity. On the other hand, all the points of the arc CD, such as A and B, are solutions within the Pareto concept, because any improvement of one of the objectives is performed to the detriment of another.
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The Finite Element Method for Electromagnetic Modeling
A usual method to solve a multi-objective problem consists of transforming it into a one objective problem by weighted summation of q objectives: f ( x)
q
q
k 1
k 1
¦ wk Fk x wT F ( x ) with 0 wk 1 and ¦ wk 1
[14.9]
The latter can thus be solved by a standard optimization method. Other methods enable this type of problem to be solved effectively. Among them we can mention the goal attainment method [GEM 74]. In this method, it is necessary to have a priori an idea of the final values of each objective (goals to be reached) F *
^F , F ,..., F `. However, a certain inaccuracy in these values is * 1
* 2
* q
allowed thanks to the introduction of a degree of relaxation per goal w w1 , w2 ,..., wq . The optimization consists of minimizing scalar quantity J
^
`
which represents the distance to the goal to be reached in direction w, while respecting the initial constraints and the additional constraints Fk ( x) wk J d Fk* k 1,..., q . 14.3. Design of experiments (DOE) method 14.3.1. The direct control of the simulation tool by an optimization algorithm: principle and disadvantages A natural approach to the problem of optimizing electromagnetic structures consists of controlling the simulation tool directly by an optimization algorithm (Figure 14.8). This way of proceeding, simple in its principle, faces difficulties related to the unit cost of the evaluation, the errors introduced by the simulation and the strategy implemented by the algorithm. Since it is very difficult to know in advance the number of evaluations which will be necessary to the optimization algorithm and the unit cost is high, the global cost of this approach is not really controllable a priori. The absence of control of the cost is not the only, or the most important, problem when simulation uses a meshing of domains (finite elements, finite differences, etc.). Within this framework, it is necessary to take into account the effects of changes of the geometric parameters, because the discontinuities of discretization introduce “numerical noises” which strongly disturb the optimization algorithms. This effect is all the more noticeable the smaller the “steps” of change of the geometric parameters
Optimization
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[HOO 91]. However, the characteristic of traditional optimization algorithms is to advance in small steps when they approach the optimum!
Figure 14.8. Control of the simulator by the optimization algorithm
It is thus seen that in some cases, there can be major incompatibility in the simple coupling of the simulation tool and the optimization tool. It is thus necessary, either to improve the simulation tool by neutralizing the numerical noise (elastic meshing [KAD 93], sensitivity analysis [GIT 89], Taylor development [PET 97], [NGU 99]), or to change the optimization strategy. It is this second way which we propose to explore, first of all, by presenting the design of experiments method whose guiding principle is based on the response service method. 14.3.2. The response surface: an approximation enabling indirect optimization A response surface is an approximation of the response (the objective function) of the device. It is obtained from the results calculated for well chosen experiments, then by application of an interpolation algorithm. It brings a global knowledge of the behavior of the device. A basic response surface, although not very expensive, proves to be extremely useful in the determination of the influential parameters, at the screening process. The influential parameters will thus be the only ones to be preserved for the construction of a more elaborated response surface (more expensive) usable for the localization of the optimum or optima (Figure 14.9). It should be noted that the concept of approximation is very often used by the traditional optimization algorithms, in particular by the deterministic algorithms. They simply use it locally. For example, in every iteration, the search for the local minimum in a direction uses a polynomial quadratic or cubic approximation of the function, the quasi-Newton algorithms replace the objective function with an ellipsoid, etc.
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The Finite Element Method for Electromagnetic Modeling
Figure 14.9. Indirect optimization thanks to a response surface
In our situation, where each simulation is expensive, it is the construction phase of this response surface which is expensive. On the other hand, the use of this approximation can be regarded as free. This opens the way to using all types of optimization algorithms, including the greediest in evaluations of the objective function such as genetic algorithms, for the search for the global optimum. In our case, this approach of indirect optimization is advantageous: – if fewer tests are needed to build the response surface than with direct optimization, in order to minimize costs; – if the “distance” between the test points is relatively large, in order to minimize the impact of the method errors. Once the optimum is found on the response surface, it is necessary to check its validity by direct simulation. If the difference between estimation and verification is too large, the approximation can be refined. This approach is the basis of the adaptive construction of a response surface [ALL 97]. In fact, the approach that we have just outlined (construction of a simple response surface for the screening followed by the construction of a more accurate response surface for the optimization) is inspired from the design of experiments method which proved to be reliable in the experimental field and which is developed very well in the digital field. We will describe the main aspects of these methods below.
Optimization
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14.3.3. DOE method: a short history The design of experiments method results from Fisher’s work in agronomic research (1925). It is a field where the experiments are long and expensive and the results are prone to variability. A method was needed making it possible to draw, from a given number of tests, a maximum of credible information relating to the influence of the factors and their hierarchy. Fisher introduced the basis of modern experimentation (Latin squares, blocks, aliasing and variance analysis). Later, the method was enriched (1945-1960) by the statisticians Yates, Box, Hunter, etc. who introduced new experimental designs: two-level fractional plans, centered composite plans and the associated response surface models. This work was used a lot in the chemical industry. Lastly, a decisive advance was achieved in the industrial world thanks to the quality approach developed by Taguchi (1960). He underlined the need for building quality upstream, starting from the design phase, to design powerful standard products with little variation around this standard and to make the performances less sensitive to the conditions of use, manufacturing risks and ageing (robust design). 14.3.4. DOE method: a simple example We will present the design of experiments method in a simple example from the field of electrochemistry, borrowed from P. Ozil [OZI 97] and consisting of studying the efficiency Y of a chemical reaction. This reaction depends on 3 factors X1, X2 and X3: the temperature varying between 100 and 200°C, the pressure varying between 1 and 2 bars and the catalyst used being either product1, or product2. The corresponding normalized variables x1, x2 and x3, all vary between -1 and +1 (see Table 14.4). The study is undertaken in a sequential way in 3 stages, so as to minimize the number of experiments: – stage 1: estimation of the variability of the response (the efficiency) Y; – stage 2: postulate for response Y of a linear model without interaction; – stage 3 (if the previous assumption is not checked): postulate for Y of a linear model with all interactions.
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The Finite Element Method for Electromagnetic Modeling
Factor
Signification
Level – real
Level + real
Level – coded
Level + coded
X1
Temperature (°C)
100
200
x1 = -1
x1 = +1
X2
Pressure (bar)
1
2
x2 = -1
x2 = +1
X3
Catalyst
Product1
Product2
x3 = -1
x3 = +1
Table 14.4. Domain of variation of factors
14.3.4.1. Stage 1: estimation of the variability of response Y Conventionally, this estimate is carried out by repetition of the experiment in the center of the field. However, in our case, the center does not exist because the X3 factor is a discrete factor. 3 experiments are then chosen to be repeated with X1 = 150°C, X2 = 1.5 bars, X3 = Product2: – the 3 responses are Y1 = 40.3%, Y2 = 39.8% and Y3 = 40.9%; 1 >Y1 Y2 Y3 @ = 40.33%; – the average value is Y 3 2 2 2½ 1 – the variance is s ' ® Y1 Y Y2 Y Y3 Y ¾ = 0.55%. 3 1 ¯ ¿
>
@ >
@ >
@
The experimental error is low; thus only one experiment using a combination of factors will be sufficient. It should be noted that in the case of responses resulting from perfectly reproducible experiments, variability is automatically zero. We will thus consider that the numerical design of experiments are exempt from repetitions. 14.3.4.2. Stage 2: postulate for response Y of a linear model without interaction The linear model with 3 normalized variables and without interaction has as an expression Y
a0 a1x1 a2 x2 a3 x3
[14.10]
This model is based on 4 unknown coefficients (the average a0 and the proper coefficients of the normalized factors a1, a2, a3) whose identification requires at least 4 tests. The DOE method suggests choosing the 4 tests whose combinations of factors are given in Table 14.5. This table also shows the responses measured during each one of these experiments.
Optimization
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Test
Factor X1 (°C)
Factor X2 (bar)
Factor X3
Answer Y (%)
1
200
2
Product2
75
2
100
2
Product1
56
3
200
1
Product1
14
4
100
1
Product2
9
Table 14.5. The four tests for the model without interaction
+++ +
+ +
Figure 14.10. Relative positions of the 4 tests of the fractional factorial design
Concerning the terminology, the selected plan is a two-level fractional factorial design. It has two levels because each factor takes only 2 values (here extreme values). It is fractional, because only 4 combinations of the levels, from the 23 = 8 possibilities, are used (Figure 14.10). The 4 tests are known as orthogonal, because, for a factor at a given level, the two levels of the other factors have the same numbers of representatives. The identification gives, for the answer Y, the following expression: Y
38.5 6 x1 27 x2 3.5 x3
[14.11]
In order to test the validity of this model, we carry out a fifth experiment, by taking one of the combinations of extreme levels still not used (Table 14.6).
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The Finite Element Method for Electromagnetic Modeling Test
Factor X1 (°C)
Factor X2 (bar)
Factor X3
Response Y (%)
5
100
1
Product1
10
Table 14.6. Additional test to test the validity of the linear model without interaction
By introducing the values of normalized factors into the model (x1 = -1, x2 = -1, x3 = -1), we obtain Ypredicted = 2% which is very different from Yexp = 10%. This means that the linear model without interaction does not report the response of the reaction at all. It is necessary to change the model. The DOE method thus proposes to consider a linear model with interactions, which will be the subject of the following stage. 14.3.4.3. Stage 3: postulate, for response Y, of a linear model with interactions The linear model with interactions has as an expression Y
a0 a1 x1 a2 x2 a3 x3 a12 x1 x2 a13 x1 x3 a23 x2 x3 a123 x1 x2 x3
[14.12]
In this model, there are 8 unknown coefficients (the average, the proper coefficients and the interaction coefficients). Thus, at least 8 tests are needed. The DOE method suggests considering the complete factorial design of two levels (23 = 8) which supplements the initial fractional factorial design (Figure 14.11). As 5 tests of this plan are already available (4 of the previous model, plus 1 of the test), there remain only 3 additional experiments to carry out (see Table 14.7). Let us note that this experiment saving in the stage sequencing is partly the interest of the method. The identification of the model gives the following expression as the response: Y
38.63 8.63x1 21.88 x2 2.13x3 1.38 x1x2 5.13x1x3 2.63x2 x3 0.13x1 x2 x3
[14.13]
The absolute value of a coefficient measures the impact of the linear effect or the interaction in the model. Interaction X1X2X3 being negligible, it is eliminated. The model which comes from this stage is thus the following: Y
38.63 8.63x1 21.88 x2 2.13x3 1.38 x1x2 5.13x1 x3 2.63x2 x3
[14.14]
Optimization
569
This model must be tested on at least one test. The initial variability test of the study is perfectly advisable, because its central position makes it possible to detect the possible nonlinearity of the answer. Test
Factor X1 (°C)
Factor X2 (bar)
Factor X3
Response Y (%)
6
200
2
Product1
66
7
200
1
Product2
34
8
100
2
Product2
45
Table 14.7. Three additional tests for the linear model with interactions
Figure 14.11. Relative positions of the 8 tests of the complete factorial design
The introduction of the combination x1 = 0, x2 = 0 and x3 = +1, in the previous model, gives Y = 40.76%, which is very close to the average Y = 40.33% found in experiments. We will consider that this model is acceptable and that it can be usable in prediction. Equation 14.14 is thus the response surface which we will retain. Let us carefully note that, in all the previous sections, we have ignored the statistical validation which is an extremely important element of the method, but that we ignore within the framework of this presentation of the numerical DOE method. 14.3.4.4. The process of the DOE method We have just illustrated on a simple example the process suggested by the DOE method. Below the stages characterizing it are summarized: – define the N factors and the experimental area of interest; – define the coded variables associated with the variable factors; – postulate a linear model without interaction:
570
The Finite Element Method for Electromagnetic Modeling
- carry out the experiments, - determine the linear model without interaction, - validate the model (combination of the level +1 and level -1 not used) (if necessary, tests of transformations Yp, Log (Y), arcsin (Y), etc.), - use of the linear model without interaction; – postulate a linear model with interactions (if previous model is not valid): - carry out the experiments, - determine the linear model with interactions, - validate the model (center of the field) (if necessary, tests of transformations Yp, Log (Y), arcsin (Y), etc.), - use of the linear model with interaction; – postulate a 2nd degree model (if previous model is not valid): - etc. Transformations (Yp, Log (Y), arcsin (Y), etc.), possibly used at the time of the validation stages when the model is not valid, perform contractions or dilations of response scales. In fact, before postulating a more complex model, it is judicious to try to apply a simple model to a transformed response. The effect of these transformations is also stabilizing the variance of the response in the case where variability would exist. 14.3.4.5. Fractional factorial designs One of the main aims of the DOE method is to obtain the maximum amount of information starting from a minimal number of tests. From this point of view, fractional factorial designs represent a fundamental tool in the method. In this context, in order to better understand the performances and limitations, we will use a second example. In this new DOE application, the response which is of interest depends on 4 factors. However, here, thanks to a good knowledge of the operation of the device, we know a priori that only interactions X1X3, X2X3 and X3X4 are present. We will thus postulate for the response, a linear model with interactions comprising only 8 coefficients, instead of the 16 of the complete linear model: Y
a0 a1 x1 a2 x2 a3 x3 a4 x4 a13 x1 x3 a23 x2 x3 a34 x3 x4
[14.15]
Optimization
571
If we implement a 24 factorial design, it would need to be carried out 16 experiments, whereas 8 would be enough! Thus, the idea of taking only a subset of the 24 factorial design has come up. However, we must take care; we should not choose any arbitrary subset, because, to draw the best possible precision from the small number of experiments carried out, it is necessary to respect orthogonality between factors. It is thus best to use the tables which give good experimental designs according to the number of factors and the number of experiments desired [BOX 78], [TAG 87]. However, the gain obtained through the choice of a fractional plan, here 8 experiments instead of 16, has a counterpart, because the identification does not give real effects (a1, a13, etc.), but sums of effects (a1+a234, a13+a24, etc.) called contrasts (it is said that a1 is an alias of a234, etc.) which depend on the selected plan. This situation has introduced confusions which depend on the plan and on the order of factors in this plan. The model obtained will thus be valid only if the neglected effects are really negligible. Let us note that in the literature discussing experimental designs, the usage is not to indicate the effects of the variables, the interactions and contrasts by coefficients a1, a13, etc., a1+a234, a13+a24, etc., but rather directly by the symbols of variables X1, X1.X3, etc., X1+X2.X3.X4, X1.X3+X2.X4, etc. In conclusion, we note that by using its know-how, the experimenter could distinguish between the negligible effects (a234 = a24 = … = 0) and the expected effects (a1, a13, etc.) in calculated contrasts. This is concretized by a gain, which is extremely significant in terms of the experiments to be realized. Of course, the validity of the model obtained depends on the validity of the assumption on which it is based. 14.3.4.6. Conclusion on DOE method The DOE method is, for the experimenter, a very effective means to: – determine the influential factors of a system (screening); – predict the responses of a system (response surface); – optimize a system (response surface of at least the 2nd degree). The analysis of the variability of the responses (not presented in this introduction) makes it possible to: – build quality upstream, starting from the design stage; – design powerful standard products and little variation around this standard;
572
The Finite Element Method for Electromagnetic Modeling
– make performances less sensitive to the conditions of use, manufacturing risks and aging (robust design). For the numerical sizing we will retain, by adapting them: – screening method to eliminate the non-influential factors before optimization; – surface response method in replacement of direct calculation to the objective function. 14.4. Response surfaces 14.4.1. Basic principles
In the presentation of the DOE method we have already seen two families of response surface: the polynomials of the 1st degree without interaction and the polynomials of the 1st degree with interactions. The polynomials of the 2nd degree are also usually used. In addition, the polynomial functions are not the only usable functions. Other types of approximations can be found in the literature based, for example, on radial functions [ALO 97] or on diffuse elements [COS 01]. The general form of these approximation functions is as follows: Yapp
M
¦ a jM j ( x1, x2 ,..., xn )
[14.16]
j 1
The independent basic functions M j ( x1 , x 2 ,..., x n ) being chosen, their weightings a j are still to be determined by the least squares approach, i.e. by minimizing e the
sum of the squares of the differences between K experimental responses and the selected model: e a1, a2 ,..., an
K
¦ >wk Yk Yapp k @2
[14.17]
k 1
The weights wk , known a priori, are used to express the relative importance of each experiment. For example, the polynomial approximations of the 2nd degree grant a more significant weight to the central value.
Optimization
573
In order to determine the weighting coefficients, it is necessary to have at least as many evaluations of the response as unknown coefficients. Moreover, some additional evaluations are necessary to test the validity of the response surface. It is convenient to use only basic, relatively smooth, functions whose values vary between -1 and +1. Thus, the amplitude of their weighting coefficient measures their impact in the summation. If the impact is relatively very low, the basic function can be eliminated. If the eliminated basic function is the unique representation of a factor or an interaction between factors, it is the indication that this factor or this interaction probably has only little influence on the response. This observation is the basis of the screening of the DOE method which has the goal of selecting influential parameters. 14.4.2. Polynomial surfaces of degree 1 without interaction: simple but sometimes useful
The polynomial response surfaces of degree 1 without interaction are the simplest. Their form is as follows: f ( x)
a0
n
[14.18]
¦ a i xi
i 1
There are 1+n coefficients to be determined. The DOE method proposes determining them by means of the fractional factorial designs [SCH 98]. This type of approximation, although basic may be useful in a screening phase when the number of factors is very high and the experiments are very expensive. 14.4.3. Polynomial surfaces of degree 1 with interactions: quite useful for screening
The polynomial response surfaces of degree 1 with interactions have the following form: f ( x)
a0
n
n 1
n
¦ ai xi ¦ ¦ aij xi x j
i 1
[14.19]
i 1 j i 1
n.( n 1) coefficients to determine. The DOE method proposes to 2 determine them by means of the two-level fractional factorial designs [SCH 98]. It is
There are 1
574
The Finite Element Method for Electromagnetic Modeling
necessary to note here the interest of the fractional plans which are adjusted on n.( n 1) request at 1 experimental points, compared to the two-level complete 2 factorial designs for which the number of experiments varies in 2n. For example, for a device comprising n = 10 factors, the number of coefficients to be determined is n.( n 1) = 56, an adapted fractional plan will propose to achieve 64 1 2 experiments, whereas the complete plan would require 2n = 210 = 2,048. 14.4.4. Polynomial surfaces of degree 2: a first approach for nonlinearities
The polynomial response surfaces of degree 2 have the following form: f ( x)
a0
n
n 1 n
n
¦ ai xi ¦ ¦ aij xi x j ¦ aii xi2
i 1
i 1 j i 1
[14.20]
i 1
n.( n 3) coefficients to determine, but here, the design of 2 experiments of two levels are not adapted to the quadratic variations. To be convinced, let us simply take a surface with only one factor,
There are 1
f ( x ) a0 a1 x1 a11 x12 , the two tests in x1 = -1 and x1 = +1 are not enough for the determination of the 3 coefficients!
Three-level factorial designs 3n could be a solution, but they are really too redundant (thus too expensive) beyond 3 factors, as is shown in Table 14.8. Number of factors n nd
Number of terms of 2 degree model n
Number of experiments of 3 design
2
3
4
5
6
6
10
15
21
28
9
27
81
243
729
Table 14.8. Numbers of terms of 2nd degree model and 3n design according to the number of factors
When all the factors of a study are continuous, the central composite designs by Box and Wilson are much more economical, as it is shown in Table 14.9.
Optimization
575
Number of factors n
2
3
4
5
6
Number of experiments of the central composite design (for only 1 point in the center)
9
15
25
27
45
= Factorial design 2n or 2n-1 + 2n points on the axes + Nc points in the center
2 n 2n N c
2 n 1 2n N c
Table 14.9. Numbers of experiments of the central composite design according to the number of factors
Figure 14.12. Relative positions of the experiments of a central composite design in a sphere
Figure 14.13. Relative positions of the experiments of a central composite design in a cube
Figure 14.12 presents the relative arrangement of the proposed experiments in a central composite design in a sphere. For each normalized factor, 5 levels are found (-D, -1, 0, +1, +D), which offers a good trade off, for the experiments, between quasi-orthogonality, isovariance by rotation and uniform accuracy. The value of D and Nc depend on the number of factors. For example, for n = 3, we take D = 1.682 and Nc = 6.
576
The Finite Element Method for Electromagnetic Modeling
Figure 14.13 shows the proposed tests by a central composite design in a cube, whose implementation is simpler since it comprises only 3 levels (-1, 0, +1). 14.4.5. Response surfaces of degrees 1 and 2:, interests and limits
Response surfaces of degrees 1 and 2 are interesting because, although simple, they allow the examination of the main effects of the factors and their interactions. This characteristic is very useful for excluding the least significant factors before optimization. However, they have no possibility of correctly accounting for complex responses and, in particular, multimode responses, i.e. having several optima, such as the function represented in Figure 14.4. To go beyond this limitation, it is possible to use polynomial response surfaces of degrees higher than 2 [SCH 98] or new functions, such as response surfaces through a combination of radial functions or approximations by diffuse elements, as we will discover below. 14.4.6. Response surfaces by combination of radial functions
In order to allow the construction of both unimodal and multimode surfaces, Alotto proposes, under the name general response surface, using a combination of radial basic functions [ALO 97]: f ( x)
M
[14.21]
¦ a j h( x p j )
j 1
where the pj are the centers of the radial basic functions which correspond to the points of experimentation. A possible choice for these functions is: h( x p j )
x pj
2
s
[14.22]
where s is an adjustment parameter of the curve of the surface of dimension n. In every M points of experimentation, interpolation [14.21] has to be equal to the result to be interpolated. That results in a full matrix system, whose unknown factors are M coefficients aj, which has a single solution if the points of experimentation are all distinct.
Optimization
577
Unlike a traditional polynomial response surface, coefficients aj of the general response surface do not give information on the relative influence of any factor. On the other hand, this type of construction does not involve a particular position of the points of experimentation. It is thus easy to impose a concentration, either overall on a regular grid, or locally in the zones where the original function seems more disturbed. 14.4.7. Response surfaces using diffuse elements
The diffuse element method belongs to a family of numerical methods called meshless methods of which the goal is to create a discrete approximation of continuous quantities in a field of study [NAY 91], [HER 99], [COS 02]. This approximation is obtained using a discretization of the domain, based on a group of dots called nodes, on which we calculate the values of the state variables called nodal values. Each node represents the center of an element whose form is usually a ball of radius r representing its zone of influence, as is shown in Figure 14.14.
Figure 14.14. Representation of diffuse elements
Unlike the radial basic functions seen previously from which the range extends in the entire domain, the functions used in the diffuse approximation have an influence only in the ball of radius r. Beyond this, the function is uniformly zero, which characterizes this approximation with an adjustable range. The radius of each ball can be arbitrarily selected, provided that the domain is entirely covered and the number of nodes involved by each ball is higher than or equal to the number of coefficients intervening in the approximation (1 for an approximation of order 0,
578
The Finite Element Method for Electromagnetic Modeling
1+n for an approximation of order 1 on n variables, etc.). In the case of elements created on a regular grid, whose internodal step d is the same as in the n directions, the minimal value of the radius ensuring the covering is: rmin
d. n 2
[14.23]
As an illustration, Figure 14.15 compares the one-dimensional function f ( x)
0.01 * (( x 0.5) 4 30 x 2 20 x )
[14.24]
on the interval [-7, +6], with its diffuse approximations of order 0, 1 and 2 defined on the basis of 7 points of evaluations. real function
order 0
order 1
order 2
6,0 5,0 4,0 3,0 2,0
f(x) 1,0 0,0 -7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
-1,0 -2,0 -3,0
x
evaluation point
Figure 14.15. Comparison between various orders of approximation
We note that the approximation of order 2 gives a better result compared to the other orders. However, when the number of parameters n of the function increases, n.(n 3) the minimal number of nodes 1 that the elements must contain can 2 become rather high. In order to guarantee this minimal number of included nodes, the radius of the elements must be sufficiently large, which compromises the local character of the approximation. Orders 0 and 1 are therefore generally preferred.
Optimization
579
14.4.8. Adaptive response surfaces
The number of nodes used in a response surface based on radial functions or diffuse elements is without doubt the most influential factor in the quality of the approximation. The use of a significant discretization with K levels in each domain direction leads to a better quality approximation. However, when the number n of parameters of the function increases, the number of evaluations of the complete factorial design increases according to law Kn, which leads quickly to a prohibitive cost. For example, 7 levels for 5 parameters, require 75 = 16,807 evaluations. We can limit the number of nodes created for the construction of the approximation and ensure its quality at the same time, by using an adaptive algorithm. This algorithm requires the addition of new evaluation points only in some areas of the domain. Such algorithms, which provide an adaptive response surface, can be found in [ALO 97], [HAM 99], [COS 02]. The placement of new points can be inspired by the fractional factorial designs or directly use a D-optimal criterion [VIV 01]. 14.5. Sensitivity analysis
The most effective deterministic optimization methods use not only the knowledge of the value of the objective function for a state of the parameters, but also the value of its gradient. Several approaches are available to obtain the numerical value of these first derivatives. 14.5.1. Finite difference method
The finite difference method is the most popular of the methods allowing an approximation of a derivative to be obtained. If f(p) is dependent on the scalar parameter p, an approximation of its first derivative is obtained by: 'f df | dp 'p
f ( p 'p ) f ( p ) 'p
[14.25]
580
The Finite Element Method for Electromagnetic Modeling
This approach is simple and general. However, it induces a numerical error if step 'p is too small (difference in two close numbers tainted with errors) and a method error if 'p is too large (terms neglected in the Taylor development). A compromise for step 'p is given by [GIL 83]: 'popt
2
H num
[14.26]
f " ( p)
where f"(p) is an approximation of the second derivative of f and Hnum is an evaluation of the error made during a calculation of f. This error has a random part, due to the basic calculation errors, and a systematic method error. When a discretization is involved, which is the case for the finite element method, it is essential not to modify its topology between the two tests. For this purpose, when parameter p influences the form of the domain, the concepts of deformable medium [HOO 91] or elastic meshing [KAD 93] or interpolation [RAM 97] can be used. Let us note that for the evaluation of n partial derivatives of f, this procedure requires n+1 evaluations of f in the previous off-center version, and 2n calculations in a more precise centered version. 14.5.2. Method for local derivation of the Jacobian matrix
The finite differences method seen previously is very easy to implement, since it requires only implementing the calculation of f according to the parameters. However, it requires several evaluations of the function for each gradient calculation and is subject to significant errors in the case where the parameter variation induces variations of numerical topology. A tempting alternative for the calculation of the partial derivatives of f consists of calculating them explicitly. This approach is particularly useful for the parameters of form p which influence a meshing. The principle is based on the local derivative of the Jacobian matrix, initially developed for the calculation of forces, torques and electromagnetic stiffness [COU 83], then extended to the sensitivity calculation [GIT 89], [PAR 94]. We will present the principle of this method. Let us consider a function f(p) resulting from an integration on domain :, cut out in finite elements :e and whose form can depend on p:
f ( p)
³ g x, p dx
:
¦ ³ g x, p dx
elements :e
[14.27]
Optimization
581
In the finite element method [ZIE 79], it is traditional to carry out a change of variables. It replaces the real coordinates in finite element :e, by local coordinates u in reference element 'e:
f ( p)
¦ ³ g x(u ), p det Jdu
[14.28]
elements ' e
where detJ indicates the determinant of Jacobian matrix J for the change of coordinates for each element. If parameter p influences the shape of the element, this matrix depends on it via the geometric nodes on the element. In expression [14.27], derivative with respect to p poses problem when this parameter influences the integration domain. On the other hand, with expression [14.28], each subdomain of space integration is fixed once and for all. The derivative of f with respect to p is thus obtained simply by integration on the reference element of the integral term derivative. The latter includes the derivative of the Jacobian determinant, which is determined from derivatives of the geometric nodes of the elements, and the derivative of function g, which can be an explicit or implicit function of p. Frequently g involves the gradient, rotation or divergence of the state variable. There still, the passage in the reference coordinate space allows the dependency to the parameter of the differential operator form to be made explicit then confined, in the only Jacobian matrix [COU 83], that the finite element is of nodal, edge, facet or volume type. 14.5.3. Sensitivity of state variables: steadiness of state equations
In the context which interests us, objective function f (for example, an electromagnetic torque, induction, etc.) often depends on state variable A (for example, vector A made up of N nodal values of the vector potential) which depends itself on several parameters pi: f
f ^A( p1 ,..., pi ,..., p n ), p1 ,..., pi ,..., p n `
[14.29]
The sensitivities of the objective function are obtained by chain derivative: df dpi
wf wA
T
dA wf , for i = 1, …, n dpi wpi
[14.30]
582
The Finite Element Method for Electromagnetic Modeling
wf
wf , which can be wpi wA calculated by finite differences or direct derivative, and the sensitivities of state dA , whose determination is not immediate. variable dpi
This expression involves partial derivatives
T
and
In fact, the state variable is itself the solution of a system of N equations built by assembly on the finite elements: F ( A, p1 ,..., pi ,..., p n ) Q ( p1 ,..., pi ,..., p n )
0
[14.31]
As these state equations remain stationary whatever the values of pi, we can deduce from them the following new equation systems: wF wA
T
dA
wF dpi dQ wpi
0 , for i = 1, …, n
[14.32]
The sensitivities of state variable A with respect to parameters pi are thus obtained by resolution of the linear systems: wF dA wAT dpi
dQ wF , for i = 1, …, n dpi wpi
[14.33]
dA , is dpi wA the tangent matrix of equation system [14.31] giving A. If this matrix were already built for the resolution of A, the cost of calculation using [14.33], for each of the derivatives, is reduced to the construction of a new second member and a resolution of a system of linear equations.
Let us note that matrix
wF
T
of matrix system [14.33], giving derivative
dQ wF and , can be built, either by finite dpi wpi differences or by explicit derivative. In this last case, if parameter pi is a parameter of shape, it is judicious to use the method for local derivation of the Jacobian matrix presented previously.
In equation [14.33], derivative
Optimization
583
14.5.4. Sensitivity of the objective function: the adjoint state method
The sensitivity of the objective function with respect to the parameters can also be obtained thanks to the adjoint state method [GIT 89]. The process consists of defining and calculating a vector of adjoint state O such that: wF T O wA
wf wA
In [14.30], let us replace df dpi
OT
[14.34]
wf wAT
by the transpose of the first previous member:
wF dA wf , for i = 1, …, n wAT dpi wpi
[14.35]
By introducing [14.33] into [14.35], that led to the expressions of the sensitivities: df dpi
ª dQ wF º wf , for i = 1, …, n » ¬ dpi wpi ¼ wpi
OT «
[14.36]
This method is very interesting because it avoids the determination of the dA and thus requires only one resolution of linear sensitivities of the state variable dpi system [14.34], whatever the number of parameters pi. 14.5.5. Higher order derivative
With formulae [14.30] or [14.36], we can obtain the gradient of a function with respect to the parameters. The same derivative process can be applied to this gradient to give the second derivative simple and crossed. Thus, a recurrence allows all the derivatives to be obtained until the desired order. Let us note that this recurring process requires only solving matrix systems having all the same matrix which is the tangent matrix of state equation [14.31]. Thus, it is much less expensive than what we could a priori imagine. Once these values of successive derivatives are provided, a Taylor or Padé development with respect to the physical or geometric parameters can be built [GUI 94], [PET 97], [NGU 99]. This development is a response surface making it possible to obtain, instantaneously, an evaluation of the developed function, for an
584
The Finite Element Method for Electromagnetic Modeling
unspecified combination of parameters. This response surface is, in particular, usable in an optimization phase [SAL 98]. 14.6. A complete example of optimization 14.6.1. The problem of optimization
The main stages developed in this chapter i.e. screening, construction of a response surface, optimization and checking, will be illustrated here on problem 25 of the series of international test cases [TAK 96], [COS 01]. The objective of this problem is to optimize the shape of a mold matrix used in the production of permanent magnets.
A3
A2 A1
A4
Figure 14.16. The problem to be optimized (problem 25 of TEAM Workshop)
The mold matrix is parameterized by an interior circle of radius R1, by an ellipse of radii L2 and L3 and by a width defined by L4. The problem of optimization consists of determining the values of R1, L2, L3 and L4 to obtain a constant radial magnetic induction of 0.35T at the 10 points defined on the arc of circle ef (Figure 14.16).
Optimization
585
The function to be minimized is given by the following equation:
F
10
^
¦ Bix 0.35 cosT i 2 Biy 0.35 sin T i 2
i 1
`
[14.37]
where parameter Ti indicates the angular position and Bxi Byi the components of the induction at measurement point i. We have added the 4 additional parameters A1, A2, A3 and A4, to the original problem. We know that these additional parameters do not have any significant influence on the value of the objective function. We simply wish to test the DOE method in the identification of the influential parameters. The variation fields of the 8 parameters are given in Table 14.10. Parameter
Minimal value
Maximal value
R1
5
9.4
L2
12.6
18
L3
14
45
L4
4
19
A1
170
190
A2
70
90
A3
86
88
A4
9.5
11
Table 14.10. Variation fields of the parameters
14.6.2. Determination of the influential parameters by the DOE method
The application of a complete factorial design of two levels on our problem of 8 parameters would require 28 = 256 evaluations of the objective function. In order to identify the influential parameters, we will use a fractional plan requiring only 16 experiments (Taguchi counts L16 from [SAD 91]). The values of the parameters used during each experiment are reproduced in Table 14.11, as well as the corresponding values of the objective function obtained by use of a finite element code [FLU 01]. The contributions of significant contrasts, as well as their composition in a linear model with interaction, are presented in Table 14.12. The sum of all other contrasts represents a contribution lower than 7% which authorizes us to assume them all to be negligible.
586
The Finite Element Method for Electromagnetic Modeling
R1
L2
L3
L4
A1
A2
A3
A4
Fobj
5.0
12.6
14.0
4.0
170.0
70.0
86.0
9.5
0.0522
5.0
12.6
14.0
19.0
170.0
90.0
90.0
11.0
0.0710
5.0
12.6
45.0
4.0
190.0
90.0
90.0
9.5
0.2288
5.0
12.6
45.0
19.0
190.0
70.0
86.0
11.0
0.1712
5.0
18.0
14.0
4.0
190.0
90.0
86.0
11.0
0.1273
5.0
18.0
14.0
19.0
190.0
70.0
90.0
9.5
0.1756
5.0
18.0
45.0
4.0
170.0
70.0
90.0
11.0
0.1665
5.0
18.0
45.0
19.0
170.0
90.0
86.0
9.5
0.2646
9.4
12.6
14.0
4.0
190.0
70.0
90.0
11.0
1.2066
9.4
12.6
14.0
19.0
190.0
90.0
86.0
9.5
0.3466
9.4
12.6
45.0
4.0
170.0
90.0
86.0
11.0
1.1023
9.4
12.6
45.0
19.0
170.0
70.0
90.0
9.5
0.5924
9.4
18.0
14.0
4.0
170.0
90.0
90.0
9.5
0.1230
9.4
18.0
14.0
19.0
170.0
70.0
86.0
11.0
0.0040
9.4
18.0
45.0
4.0
190.0
70.0
86.0
9.5
0.0537
9.4
18.0
45.0
19.0
190.0
90.0
90.0
11.0
0.0188
Table 14.11. Values of the parameters and the objective function used for screening
Contrast
Confusions
(*) = interactions of higher order
Contribution (%)
A
R1 + L2.L3.A1 + L3.L4.A3 + (*)
14.91
B
L2 + R1.L3.A1 + L3.L4.A2 + (*)
25.02
D
L4 + L2.L3.A2 + R1.L3.A3 + (*)
6.23
C
R1.L2 + L3.A1 + L4.A4 + A2.A3 + (*)
33.01
D
R1.L4 + L3.A3 + L2.A4 + A1.A2 + (*)
8.27
E
L2.L4 + L3.A2 + R1.A4 + A1.A3 + (*)
6.10
remain
…
6.46
Table 14.12. Contributions obtained by application of the DOE method
Optimization
587
During a screening operation, the interactions between 3 factors (L2.L3.A1, L3.L4.A3, etc.) and more are generally neglected. By following this practice and taking into account the results obtained, we can suppose that only factors R1, L2 and L4 and interactions R1.L2, R1.L4 and L2.L4 have an influence on the value of the objective function. That means that we can retain only R1, L2 and L4 as parameters in the problem of optimization. That represents a considerable reduction of their number. For the other parameters, we choose L3 = 14, A1 = 180, A2 = 80, A3 = 88 and A4 = 9.5 in order to be consistent with the values proposed by the international problem. 14.6.3. Approximation of the objective function by a response surface
Once the 3 parameters, R1, L2 and L4, considered most significant are chosen, we build a response surface of the objective function. In this example, we use the simplest of the sampling techniques. This consists of calculating the values of the function at the nodes of a grid built a priori. We choose 7 equidistant levels in each direction, i.e. a total of 343 experiments. During this sweeping we note that the minimal value of the objective function, F = 0.00148, is obtained for the node R1 = 8.7, L2 = 17.1, L4 = 16.5. This set of results allows a response surface to be built using an adapted method. Here, we adopt the diffuse element method because it provides a smooth surface with few oscillations. This surface can be used as it is, as a parameterized model of behavior of the device or be introduced into an optimization algorithm. 14.6.4. Search for the optimum on the response surface
The optimization is carried out with respect to the selected 3 parameters. The numbers of calls of the objective function no longer being a problem, since we use the response surface instead of simulations, we can choose an algorithm able to determine the global optimum. We select a genetic algorithm with a population of 30 individuals and evolving over 300 generations. The minimum of the objective function F = 0.000411 is obtained for R1 = 7.1675, L2 = 14.0804, L4 = 14.3550. 14.6.5. Verification of the solution by simulation
The previous configuration of the 3 parameters corresponds to the minimum of the response surface. To check the quality of the proposed solution, we carry out a simulation of control which gives for the objective function the value F = 0.000215 which seems to be satisfactory.
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The Finite Element Method for Electromagnetic Modeling
14.7. Conclusion
We have just presented the main concepts related to the optimization of devices in which electromagnetic phenomena are particularly involved. The deterministic and evolutionary methods have available numerical tools to automate these optimizations. When the evaluation of the criterion is carried out by means of expensive calculational tools, we have seen that the numerical DOE method allows the influential parameters to be detected. In addition, the response surface method advantageously replaces the direct computation with this calculational tool. Many references concerning the optimization of electromagnetic structures are available in the scientific literature. In addition, in this chapter we have only quoted a very small number of them. Beyond the traditional optimization applications, these algorithms are also used in the fields of topological design [DYC 96], [DYC 97] and in many inverse problems in which electromagnetic fields are involved (detection of sources, ferromagnetic bodies, cracks, positions, magnetizations, etc.). These problems are often badly stated and require the implementation of a solution regularization method [TIK 76], [HAN 87], [HAN 93], [ALI 95], [TIK 98]. 14.8. References [ALI 95] ALIFANOV O.M., ARTYUKHIN E.A., RUMYANTSEV S.V., Extreme Methods for Solving Ill-posed Problems with Application Heat Transfer Problems, Begell House, 1995. [ALO 97] ALOTTO P., GAGGERO M., MOLINARI G., NERVI M., “A design of experiment and statistical approach to enhance the generalised response surface method in the optimization of multiminima problems”, IEEE Transactions on Magnetics, vol. 33, no. 2, pp. 1896-1899, 1997. [BOX 78] BOX G.E.P, HUNTER W.G., HUNTER J.S., Statistics for Experimenters, Wiley Interscience, 1978. [BRA 94] BRANDISKI K., PAHNER U., BELMANS R., “Optimal design of a segmental PM DC motor using statistical experiment design method in combination with numerical field analysis”, ICEM 1994, Paris, France, vol. 3, pp. 210-215, September 5-8, 1994. [BRE 73] BRENT R.P., Algorithms for Minimization Without Derivatives, Prentice-Hall, 1973. [CAR 61] CAROLL C.W., “The created response surface technique for optimizing nonlinear restraint systems”, Operations Research, vol. 9, no. 2, pp. 169-184, 1961. [CHE 99] CHERRUAULT Y., Optmisation: Méthodes locales et globales, Presses Universitaires de France, 1999.
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List of Authors
Bernard BANDELIER U2R2M Paris-Sud 11 University – CNRS France Frédéric BOUILLAULT Laboratory of Electrical Engineering of Paris Paris-Sud 11 University France Christian BROCHE Faculté Polytechnique de Mons Belgium Jean-Louis COULOMB G2Elab Grenoble Institute of Technology France Patrick DULAR Lab. of Applied and Computational Electromagnetics University of Liège – FNRS Belgium Yves DU TERRAIL COUVAT SIMAP Grenoble Institute of Technology France
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Mouloud FÉLIACHI IREENA University of Nantes France Javad FOULADGAR IREENA University of Nantes France Christophe GUÉRIN CEDRAT Meylan France Afef KEDOUS-LEBOUC G2Elab CNRS – Grenoble Universities France Vincent LECONTE Schneider Electric – Power BU Electropôle Eybens France Yvan LEFEVRE LAPLACE Institut National Polytechnique de Toulouse – CNRS France Yann LE FLOCH CEDRAT Meylan, France Jacques LOBRY Faculté Polytechnique de Mons Belgium Patrick LOMBARD CEDRAT Meylan France
List of Authors
Yves MARÉCHAL G2Elab Grenoble Institute of Technology France Philippe MASSÉ PHELMA Grenoble Institute of Technology France Gérard MEUNIER G2Elab CNRS – Grenoble Universities France Eric NENS Electrabel Linkebeek Belgium Florence OSSART LGEP Pierre and Marie Curie University France Francis PIRIOU L2EP University of Lille I France Franck PLUNIAN LGIT University Joseph Fourier Grenoble France Zhuoxiang REN Mentor Graphics Corporation USA Gilbert REYNE G2Elab CNRS – Grenoble Universities France
597
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The Finite Element Method for Electromagnetic Modeling
Françoise RIOUX-DAMIDAU U2R2M Paris-Sud 11 University – CNRS France François-Xavier ZGAINSKI EDF/DTG/CEM BG Grenoble France
Index
A, B adjoint state 36, 37, 583 air-gap 245, 256, 257, 269, 270, 313, 346, 347, 358-360, 390, 534, 535, 547, 548 anisotropy 178-180, 182, 192-200, 209, 215, 216, 218, 220, 240, 514, 525, 537 anhysteretic 195, 198, 200, 205-207, 210, 212 astrophysical objects 479, 484 augmented Lagrangian 173 Bean model 230-239 behavior law 71, 74, 75, 85, 88, 89, 105, 177-244, 292, 294, 327, 406, 409 BEM (Boundary Equation Method) 350, 394 bubble 357, 527-533
C, D capacitance 74 circuit equation 210, 277-320, 340, 422 circulation 60, 69, 73, 78, 79, 90, 91, 96, 97, 99, 102, 106, 107, 109, 115, 120, 124, 130, 131, 198, 339, 394 coenergy 4, 39, 194, 195-198, 200 coercivity 159, 161, 162, 220 compatibility of approximation spaces 157 complementary energy bounds 129, 133 constitutive relationship 146, 154, 406
constraints 17, 19, 74, 75, 85, 89, 90, 110, 114, 129, 139, 154-156, 385, 395, 398, 534, 548, 549, 551-553, 559, 560, 562 coupling models 411 current source 79, 163, 184, 186, 278, 281, 283, 284, 301, 303, 310 current-voltage relation 286-289, 295, 307, 310, 314, 317 Delaunay 357, 360, 513, 518, 519, 525, 527, 530-532 diffuse element method 348, 572, 576, 577, 579, 587 dynamic behavior 178, 200, 207, 216 hysteresis 209, 210, 215, 226 dynamo instability 477, 479-481 experiments 483
E eddy currents 117, 121-125, 129, 201, 202, 210, 247, 269, 321, 329, 349, 405, 406, 535, 548 edge approximation 249 eigenvalues 455, 485 eigenvectors 455, 456, 458 elastodynamic 431, 434, 448, 449, 456 electric behavior 178, 228, 229, 230, 233, 278, 292
192, 342,
453, 231,
600
The Finite Element Method for Electromagnetic Modeling
charge 2, 3, 8, 12, 15, 27, 30-32, 34, 38, 39, 64, 70, 73, 74, 91, 117, 165, 167, 481 circuit 277-279, 284, 289, 310, 312, 317, 340, 344, 349, 443, 548 coenergy 29, 30, 39 element 278 energy 26, 27, 28, 38 formulation 118, 119, 122, 123, 126, 128, 130-133 vector potential 42, 81, 89, 113, 225, 289, 291, 313, 315, 317, 330 electromagnetic skin 536, 537 vibration 447, 448, 460, 470 electromagnetism 66, 70, 139-175, 350, 355, 369, 378, 384, 403-406, 409, 444, 470, 471, 509-514, 534, 539-541, 548 electrostatics 1, 2, 4, 10, 45, 62, 63, 71, 74, 76, 77, 79, 80, 85, 102, 107, 109, 111, 112, 147 electrokinetics 70, 71, 74, 79, 80, 86, 89, 90, 109, 113, 114, 317 element quality 512, 514 Eulerian 335, 336, 363 extrusion 246, 522-527
F facet finite element 57, 66, 108, 109 fast breeder reactors 480, 483, 484, 502, 503 ferromagnetic material 71, 123, 178, 245, 247, 445, 447, 502 thin sheet 249 fixed point method 45, 183, 187, 188, 192, 207, 208, 263, 264 fluid mechanics 139, 338, 405, 408 frontal 356, 518, 519, 520, 525, 532 function spaces 69, 70, 75, 82, 83, 84
G, H Galerkine method 34, 45, 47, 261, 337, 338 gauge condition 69, 70, 77, 78, 80, 81, 106, 112, 114, 121, 134, 293, 485
giant magnetostriction 466, 468, 470 global quantities 73, 80, 81, 200 group theory 369, 371, 376, 388, 393, 403 Hamilton variational principle 433 high frequency 214, 418 hole 121, 124, 125, 259, 269, 316, 317, 516, 529, 530 hybrid formulation 119, 127, 154, 156 hysteresis 71, 177-179, 182, 192, 195, 200-216, 225, 226, 237, 240, 466, 487
I indefinite matrix 140 identification 30, 48, 204, 205, 207, 537, 566-568, 571, 585 indirect coupling 284, 285 induction equation 482-485, 503 heating 246, 405, 406, 411, 533, 534 machine 213, 214, 294, 297, 298, 372 plasma 406, 417 implicit method 296, 311, 315 inf-sup condition 157, 158, 159, 161, 162 incidence matrix 106, 110, 294, 295, 314, 315 influence coefficient 31-33 integro-differential formulation 284, 285, 295, 296 iron losses 178, 209, 213, 214, 215
J Jacobian matrix 56, 58, 64, 183, 185-187, 191, 253, 447, 580-582 Jiles-Atherton 202-207 Joule losses 132, 133, 202, 215, 239, 262, 265, 268, 269, 547
K, L Kirchhoff’s current law 278 voltage law 279, 295 kinematic dynamo 477-507 Lagrangian 161, 173, 251, 329, 336, 338, 339, 363, 433, 436, 437, 541, 560 loss surface model 210, 211, 212
Index local Jacobian derivative method 42, 63, 64, 441, 446, 447, 450 quantities 34 line region 245-275 linear model 180, 216, 219, 565, 566, 568-570, 585
M magnetic behavior 75, 85, 177, 178, 194, 216, 228, 292 force densities 408, 452 formulations 118, 123, 125, 126, 128, 130, 131, 133, 134, 248 induction 66, 70, 81, 112, 119, 120, 140-142, 154, 158, 162, 228, 230, 287, 288, 291, 292, 326, 405, 410, 481, 486, 584 losses 210, 215 scalar potential 79, 85, 89, 99, 111, 112, 124, 146, 225, 250, 251, 255-260, 265, 269-272, 277, 288, 289, 305, 310, 313, 315, 317, 330 sheets 192, 194, 449 vector potential 81, 85, 90, 120, 146, 188, 248, 257, 259, 287-289, 293-295, 310, 317, 329, 330, 343, 347, 350, 351, 390, 407, 410, 420, 438, 441, 446, 485 macroscopic magnetization 218 magnetodynamic 79, 117-137, 164, 167, 225, 231, 249, 298, 370, 394, 403, 406 magnetoelasticity 433, 437, 439, 440, 442 magnetohydrodynamic 432, 477-507 magneto-mechanical coupling 432, 440, 441, 459, 466, 470 modeling 431-475 transfer function 457 magneto-thermal 405-430 magnetostriction 441, 442, 447-449, 459, 465, 466, 470 massive conductor 283, 286, 288, 293, 294, 295, 297, 299-303, 309, 310, 313-317 material 237, 537 Maxwell stress tensor 452 tensor 37, 40, 42, 445
601
mesh equation 285, 294, 299, 303, 305, 310, 315 generation 509-546 meshless method 346, 348, 577 minimization problems 140, 147-151, 168, 169, 552 mixed finite elements 92, 139-175 formulation 140-151, 154, 157, 158, 160-173 mixed-hybrid methods 157 modes of vibration 452, 455 modified electric vector potential 120, 127, 309
N Newton-Raphson 34, 44, 45, 141, 232, 300, 306, 311, 315, 341, 344 nodal finite element 1-68, 92, 248, 293, 435 potential 34, 278, 280-283, 299, 303, 305, 310, 315 nonlinear behavior 183, 186, 220, 294, 307, 449, 450 non-simply connected 75, 259, 260, 272, 310, 313, 316, 317 numerical design of experiments 549-551, 566, 569, 588 integration 30, 31, 60, 62, 252
O, P, Q Ohm’s law 327-330, 406, 481, 482 optimization 34, 235, 446, 464, 514, 547593 oriented grains 178, 179, 182, 192, 194, 196, 198-200, 208, 209, 211 penalization methods 168-174 permanent magnet 74, 178, 180, 181, 192, 222, 225, 237, 326, 454, 465, 584 Ponomarenko dynamo 491, 493, 494 potential jump 124, 247-257, 265, 271, 272, 317 powder metallurgy 216 power density 408, 412, 416, 425
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The Finite Element Method for Electromagnetic Modeling
Preisach hysteresis model 182, 202, 203, 205 principle of least action 433-437 pyramid 19, 52, 54, 62, 525 quadri-vector potential formulation 488491
R remeshing 355-357, 360-363, 530, 536, 538 regularization 515, 521, 526-533, 588 response surface 550, 551, 563-577, 579, 583, 584, 587 Ritz method 11, 19, 20, 47 Roberts dynamo 495-499
S saddle point 140, 147, 151-156 scalar potential formulation 70, 256, 268, 310, 317 sensitivity analysis 34, 550, 551, 563, 579 shape function 18, 46, 53, 93, 249, 340 shell element 245, 247, 535 sinusoidal condition 245 skin depth 133, 245-247, 257-264, 269, 271, 275, 412, 537 source field 69, 70, 76-80, 85, 89, 90, 106, 108, 109, 112, 113, 124, 141, 142, 145, 163, 260, 262, 267, 268, 317, 369, 395 special element 245-248, 257, 360 stiffness 37, 42, 66, 354, 453, 580 state equation 281-284, 301-303, 317, 581-583 strong coupling 341, 432, 459-461, 465, 467, 469 superconductor 177, 226, 300 surface air-gap 33 impedance 245, 258, 409, 411 symmetric components 369-404
T tetrahedron 52, 60, 62, 70, 91, 93, 95-98, 100, 101, 161-164, 170, 355, 357, 398, 510, 512, 514, 518, 525, 533, 537 thin conducting region 245, 265, 267, 269272 thermal region 272 time integrated nodal potential 280, 299, 310 Tonti diagram 69, 70, 84, 85, 86, 105 torque 37, 298, 325, 340, 432, 442, 444, 452, 460, 468, 547, 580 total scalar potential 248, 250, 251, 254, 262, 264, 268, 312 transformer 192, 198, 200, 210, 227, 240, 245, 264, 270, 312, 369, 431, 448 translation 322, 325, 328, 347, 356, 360, 371, 374, 406, 523
V variational approach 4, 6, 14, 45, 46, 139 principle 147, 149, 433 vector potential formulation 90, 257, 292, 300, 303, 310, 488 velocity term 337, 363, 407, 408 vibration of magnetic origin 449, 460 virtual work 37, 38, 42, 64, 450
W weak coupling 432, 459, 460, 461, 462 formulation 69, 82, 86, 91, 109, 112, 124, 332, 385, 489 Welsh algorithm 282, 284 Whitney elements 119, 120, 124, 128, 130, 134, 140, 161-163, 166, 447 wound conductors 290, 291, 294, 296, 299, 304, 305, 307, 311
