Mathematical and Numerical Modelling in Electrical Engineering

Mathematical and Numerical Modelling in Electrical Engineering

MATHEMATICAL MODELLING: Theory and Applications VOLUME 1 This series is aimed at publishing work dealing with the definition, development and application of fundamental theory and methodology, computational and algorithmic implementations and comprehensive empirical studies in mathematical modelling. Work on new mathematics inspired by the construction of mathematical models, combining theory and experiment and furthering the understanding of the systems being modelled are particularly welcomed. Manuscripts to be considered for publication lie within the following, non-exhaustive list of areas: mathematical modelling in engineering, industrial mathematics, control theory, operations research, decision theory, economic modelling, mathematical programming, mathematical system theory, geophysical sciences, climate modelling, environmental processes, mathematical modelling in psychology, political science, sociology and behavioural sciences, mathematical biology, mathematical ecology, image processing, computer vision, artificial intelligence, fuzzy systems, and approximate reasoning, genetic algorithms, neural networks, expert systems, pattern recognition, clustering, chaos and fractals. Original monographs, comprehensive surveys as well as edited collections will be considered for publication. Editor: R. Lowen (Antwerp, Belgium) Editorial Board: G.J. Klir (New York, USA) J.-L. Lions (Paris, France) F. Pfeiffer (Munchen, Germany) H.-J. Zimmerman (Aachen, Germany)

Mathematical and Numerical Modelling in Electrical Engineering Theory and Applications by Michal Kriiek Academy of Sciences, Prague, Czech Republic and Pekka Neittaanmaki University of Jyviiskylii, Jyviiskylii, Finland Springer-Science+Business Media, B.Y.

A C.I.P. Catalogue record for this book is available from the Library of Congress ISBN 978-90-481-4755-7 ISBN 978-94-015-8672-6 (ebook) DOl 10.1 007/978-94-015-8672-6 Printed on acid-free paper All Rights Reserved 1996 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers 1996. Softcover reprint of the hardcover 1 st eiditon 1996 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

"For the mathematician, not calculation, but clear thinking is characteristic: the ability to disregard inessential things." PETER ROZSA Math. Intelligencer 12 (1990), p. 60

Contents Glossary of symbols... xi Foreword.............................................................. xiii 1. Introduction... 1 2. Mathematical modelling of physical phenomena... 4 2.1. General remarks... 4 2.2. Some models described by differential equations... 6 3. Mathematical background... 12 3.1. Basic definitions and theorems from functional analysis... 12 k- 3.2. The spaces LP(n) and C (n)... 15 3.3. Sobolev spaces Hk(n) and W;(n)... 17 3.4. Classical and variational formulation of a 2nd order elliptic problem 27 3.5. On the abstract Neumann problem... 35 3.6. The space H(div)... 40 3.7. Dual variational formulation....41 3.8. The space H(rot)....43 3.9. Least squares formulation... 46 4. Finite elements... 54 4.1. The main idea of the finite element method... 54 4.2. Finite element spaces... 58 4.3. A posteriori error estimates... 61 4.4. Convergence of the finite element method... 66 4.5. Linear interpolation on polyhedra... 77 4.6. Affine curved elements... 89 5. Conjugate gradients... 98 5.1. Preliminary remarks... 98 5.2. Biconjugate gradient method... 100 5.3. Preconditioned biconjugate gradient method... 106 5.4. Rate of convergence of the conjugate gradient method... 111 6. Magnetic potential of transformer window... 121 6.1. Dimensional reduction... ;... 121 6.2. Example... 127 7. Calculation of nonlinear stationary magnetic fields... 130 7.1. Introduction to theory of monotone operators... 130 7.2. An application... 133 7.3. The main theorem on monotone operators... 139 8. Steady-state radiation heat transfer problem... 148 8.1. Gateaux and Frechet differential... 148 8.2. Classical and variational formulation... 150

viii Contents 8.3. Convergence of finite element approximations. '" "... 154 804. Three-dimensional radiation problem... 157 9. Nonlinear anisotropic heat conduction in a transformer magnetic core. 162 9.1. Non-monotonicity and non-potentionality of the problem... 162 9.2. Existence of the weak and discrete solutions... 167 9.3. Uniqueness and nonuniqueness... 170 904. Convergence of finite element approximations... 173 9.5. An application... 176 9.6. Effect of numerical integration... 178 9.7. Variational crimes... 188 10. Stationary semiconductor equations... 197 10.1. Classical formulation... 197 10.2. Existence of a weak solution... 198 10.3. Uniqueness and nonuniqueness... 203 loa. Finite element approximation... 204 11. Nonstationary heat conduction in a stator... 209 11.1. Classical and weak formulations... 209 11.2. Finite element analysis... 211 11.3. Numerical example... 220 12. The time-harmonic Maxwell equations... 223 12.1. Problem with inhomogeneous conductivities... 223 12.2. FE-approximation of the three-dimensional problem... 228 12.3. Magnetic potential of transformer shieldings... 231 13. Approximation of the Maxwell equations in anisotropic inhomogeneous media... 236 13.1. The initial-boundary value problem and its semidiscrete approximation... 236 13.2. FE-approximation of the time-harmonic problem... 240 13.3. Maxwell problem in polygonal domains... 246 14. Methods for optimal shape design of electrical devices... 251 14.1. Background... 251 14.2. Formulation of the problem... 252 14.3. Design sensitivity analysis... 256 1404. Implementation of optimal shape design procedures... 264 14.5. Industrial applications... 264 References... 277 Author index... 291 Subject index... 295

Glossary of symbols C,C,Ci'... Ad (a, b) [a, b) n measdn IT on n c dist VIK n w nl\w v+,vonv Div OjV ~ grad div rot curl Ker span In e deta AT A C AH Re 1m generic constants (different at each occurrence) d-dimensional Euclidean space scalar product in Rd open interval in Al closed interval in RI problem domain (bounded connected open set in Rd) d-dimensional Lebesgue measure of n closure of n boundary of n outward unit normal to on set of all bounded domains with a Lipschitz continuous boundary distance restriction of function v to set K normal component of vector function w on on tangential component of vector function w on on positive and negative parts of v normal derivative of v on on ith generalized derivative of v (i multi-index) first generalized derivative of v (j E {I,..., d}) Laplace operator gradient divergence rotation of vector function for d = 2, 3 rotation of scalar (vector) function for d = 2 (d = 3) kernel linear span natural logarithm Euler number determinant of matrix A transposed matrix to A complex conjugate matrix to A conjugate transposed matrix to A real part imaginary part ix

x List of symbols z, ), k ~ij C :1 V a.a. a.e. Pk(O) QI(O) U(O), p E [1,00) LOO(O) c(n) ck(n) COO(O) co'(o) Hk(O) HJ(O) HI/2(00) W;(O) H(divO) H(rotO) V V* --+ (b,v) v u 11 llv (., )v II. Ilk,f! I. Ik,f! imaginary unit integer indices (subscripts) Kronecker's symbol: ~ij = 1 for i = j, ~ij = 0 otherwise complex plane there exist( s) for all almost all almost everywhere space of polynomials of degree at most k defined on 0 space of d-linear polynomials defined on 0 Lebesgue space of measurable functions v defined on 0 for which If! Iv( x)ip dx is finite Lebesgue space of measurable essentially bounded functions defined on 0 space of continuous functions defined on n space of functions whose classical derivatives up to order k belong to C (n) space of infinitely differentiable functions in 0 space of infinitely differentiable functions with compact support in 0 Sobolev space of functions whose generalized derivatives up to order k belong to L2(0) space of functions from Hl(n) whose traces vanish on an space of traces of all functions from HI (0) Sobolev space of functions whose generalized derivatives up to order k belong to U(O) space of divergence-free functions space of rotation-free functions Banach space of test functions space of linear continuous functionals on V weak convergence strong convergence scalar product or value of functional b E V* at point v E V test function classical or weak (variational, generalized) solution norm in V scalar product in V norm in Hk(O) seminorm in Hk(O)

List of symbols xi ("')k,o II IIk,p,o (., '}o,oo 1 1 11 11 (.,.) a(,.) F( ) J J' K k PK EK (K,PK,EK) dimpk hk h Th Vh 'Trh V suppv i N {v }i=l 0( ) Ell o x E A x rj.a {x E AIP(x)} ACB AnB AuB A\B f:a---+b x 1-+ f(x) o scalar product in Hk(O) norm in W;(O) scalar product in L2(80) absolute value Euclidean norm Euclidean scalar product bilinear or sesquilinear form linear form potential functional Gateaux differential of J element reference element space of shape functions set of degrees of freedom finite element dimension of PK diameter of K (diam K) discretization parameter triangulation (partition, decomposition) finite element space Vh-interpolant of v support of function v basis functions in Vh Landau's symbol: f(o'.) = O(g(O'.)), if If( 0'.)1 ::; Glg( 0'.)1 as 0'. ---+ 0 or 0'. ---+ 00 direct sum empty set element x belongs to set A element x does not belong to set A set of all elements x from A which possess property P( x) A is subset of set B intersection of sets A and B union of sets A and B subtraction of B from A function f from A to B function which assigns value f(x) to x Halmos symbol

Foreword Mathematical modeling plays an essential role in science and engineering. Costly and time consuming experiments (if they can be done at all) are replaced by computational analysis. In industry, commercial codes are widely used. They are flexible and can be adjusted for solving specific problems of interest. Solving large problems with tens or hundreds of thousands unknowns becomes routine. The aim of analysis is to predict the behavior of the engineering and physical reality usually within the constraints of cost and time. Today, human cost and time are more important than computer cost. This trend will continue in the future. Agreement between computational results and reality is related to two factors, namely mathematical formulation of the problems and the accuracy of the numerical solution. The accuracy has to be understood in the context of the aim of the analysis. A small error in an inappropriate norm does not necessarily mean that the computed results are usable for practical purposes. Analysing the same engineering problem by different methods could sometimes lead to results which are significantly different. This could be caused by various reasons, e.g., different models in mathematical formulations were used or the numerical solution does not approximate the data of interest well. This can happen especially when various modern adaptive codes with a posteriori error estimation are used. Obviously, to understand the reasons for such discrepancy is very important. Decisions in engineering are still made by humans and not by computers, although computers are.the main tools. It is necessary to realize that the computer always provides data, color graphs or movies, correct or incorrect. Hence to understand the basic mathematical background of the modelling is of major importance. This book presents in a balanced way, the main features of modelling in electrical engineering. It clearly addresses without many prerequisites, mathematical and computational aspects. It presents mathematical ideas, and tools and approaches which are essential in these problems, illustrating these ideas with examples. Although no book of this type can make the reader an expert with great experience in the modelling of electrical engineering, it is an excellent book to learn the basic ideas and tools. It contributes significantly to a way of creatively assessing whether the numbers provided by computers are reliable as the basis for crucial engineering decisions. Hence it is a very valuable book. Ivo Babuska xiii