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

117

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

Table of Contents

vii

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

245

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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Table of Contents

6.8. Thin regions in electrostatic problems: “electric harmonic problems” and electric conduction problems . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9. Thin thermal regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

272 272 273

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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342 346 346 346

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348 350 355 362 363

Chapter 9. Symmetric Components and Numerical Modeling . . . . . . . . Jacques LOBRY, Eric NENS and Christian BROCHE

369

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


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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547 547 548 549

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549

Table of Contents

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

599


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


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


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


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.


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


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]


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 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


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 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


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


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


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


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]


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


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


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


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]


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


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]


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


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


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 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]


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


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]


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


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.


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


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


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


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.


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 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.


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


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]


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


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


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


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.


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


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,


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


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]


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


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


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


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

O11 [

1[ 2

O21 [

1[ 2

[1.143]


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 O12 [ 2

[ .[ 1 O22 [ 2

O32 [ 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


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

O11 [ ,\ [

O21 [ ,\ \

O31 [ ,\ ]

[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

O12 [ ,\ [ .2[ 1 O22 [ ,\ \ .2\ 1 O2 [ ,\ ] .2] 1 3

O42 [ ,\ O2 [ ,\

4[\

[ ,\

4][

5 O62

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.


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

O11 [ ,\ O1 [ ,\

1 [ 1 \ / 4 1 [ 1 \ / 4 2 O31 [ ,\ 1 [ 1 \ / 4 O41 [ ,\ 1 [ 1 \ / 4

Figure 1.16. First-order quadrangular element

[1.149]

52


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

O12 [ ,\ O2 [ ,\ 2

1 [ 1 \ 1 [ \ / 4 1 [ 1 \ 1 [ \ / 4

...

O52 [ ,\ O2 [ ,\ 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


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


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 O11 [ ,\ x E O21 [ ,\ xJ O21 [ ,\

y [ ,\

yD O11 [ ,\ y E O21 [ ,\ yJ O21 [ ,\

[1.155]


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 O12 [ ,\ ... xJ O32 [ ,\ xDE O42 [ ,\ ... xJD O62 [ ,\

y

yD O12 [ ,\ ... yJ O32 [ ,\ yDE O42 [ ,\ ... yJD O62 [ ,\

[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


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]


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 O11 [ ,\ v E O21 [ ,\ vJ O31 [ ,\

v [ ,\

[1.161]

then in a second one by v

vD O12 [ ,\ ... vJ O32 [ ,\ vDE O42 [ ,\ ... vJD O62 [ ,\ [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 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]


>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


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.


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


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.


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


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]


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


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.


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


[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


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


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 *, 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

Static Formulations

83

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.

84


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

Static Formulations

85

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

86


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]

Static Formulations

87

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]

88


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.

Static Formulations

89

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]

90


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]

Static Formulations

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.

92


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]

Static Formulations

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

94


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

Static Formulations

95

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

96


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

98


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

100


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

102


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]

104


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.

106


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.

108


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


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


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


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


[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


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].


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


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


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


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.


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


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.


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


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)


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


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]


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 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.


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


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


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.


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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[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.


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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


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.


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

142


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.


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


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]


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

146


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:


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]

148


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]


149

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].

150


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 * .


151

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:

152


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:


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].

154


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 }


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 .

156


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 .


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 –

158


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]


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.

160


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.


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 :) .

162


– 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


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


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]:


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


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


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


§ 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


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


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:


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


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.


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


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.


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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.


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


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]


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


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.


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).

182


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.


183

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]

184


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]


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]

186


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


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]

188


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]


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: GII >7@-1RI . 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].

190


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 In)) = 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]


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


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.


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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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


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 Hsat and EJc 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:

232


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


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]

234


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]


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.

236


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.


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

238


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)


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.

240


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.


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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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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.

246


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


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

248


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

Conducting thin sheet

A

Formulations in B A, AV, A*, E

Jump

H

Ve >> V

Ve 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.


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]

252


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


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

254


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


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:

256


– 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].


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;

258


– 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]


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 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].

266


The only validity condition of this formulation is the one which states that the region must be thin: e 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.

272


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.


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.

274


[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.


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.


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.

278


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:


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]

280


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

>@


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 ¬

S 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

= LT

= S ji

Sij

Sij

Sij

=

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

346


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.

Modeling of Motion

347

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

348


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]

Modeling of Motion

349

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.

350


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.

Modeling of Motion

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

352


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

354


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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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

356


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.

358


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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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.

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Figure 8.20. Simulation results

361

362


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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[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.


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


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.


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 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.


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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.

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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


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.

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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.


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:


\ 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


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]


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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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


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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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 ,


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:

386


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


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

388


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.


389

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:

390


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.


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.

392


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)


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 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


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

396


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


397

398


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.


Figure 9.19. Current density, real part – global solution

Figure 9.20. Current density, real part Re (J(2))

399

400


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


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.

402

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%.


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


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


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


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


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


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


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].


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


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


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 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


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 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


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


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)


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


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).


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


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]


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 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


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


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.


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


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 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.


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 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


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 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]


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


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]


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 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]


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


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


443

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.

444


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).


445

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]

446


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]


447

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

448


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].


449

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.

450


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]


451

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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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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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 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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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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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


465

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

466


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


467

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

468


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


469

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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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.


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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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


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


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]

482


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


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


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


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).


§

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;

488


– 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]


* 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 ;

490


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


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


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.


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

494


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


495

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 (

@ >

@ >

@

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

567

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).

568

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


- 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


– 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: ea1, 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


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


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


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


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


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


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


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.

588


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.

Optimization

589

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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

596


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

598


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


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

602


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

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