Mathematical and Numerical Modelling in Electrical Engineering Theory and Applications

311 Pages·1996·7.354 MB·English

by  Michal Křížek, Pekka Neittaanmäki (auth.)

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-exhaus
tive list of areas: mathematical modelling in engineering, industrial mathematics,
control theory, operations research, decision theory, economic modelling, mathe
matical programming, mathematical system theory, geophysical sciences, climate
modelling, environmental processes, mathematical modelling in psychology,
political science, sociology and behavioural sciences, mathematical biology,
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Original monographs, comprehensive surveys as well as edited collections will be
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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 ofJ yviiskylii, 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 1s t 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 informa
tion 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
3.2. The spaces LP(n) and C k(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' ... generic constants (different at each occurrence)
Ad d-dimensional Euclidean space
scalar product in Rd
(a, b) open interval in Al
[a, b) closed interval in RI
n problem domain (bounded connected open set in Rd)
meaSdn d-dimensional Lebesgue measure of n
IT closure of n
on boundary of n
n outward unit normal to on
c
set of all bounded domains with a Lipschitz continuous
boundary
dist distance
VIK restriction of function v to set K
n·w normal component of vector function w on on
nl\w tangential component of vector function w on on
v+,v- positive and negative parts of v
onv normal derivative of v on on
Div ith generalized derivative of v (i multi-index)
OjV first generalized derivative of v (j E {I, ... , d})
~ Laplace operator
grad gradient
div divergence
rot rotation of vector function for d = 2, 3
curl rotation of scalar (vector) function for d = 2 (d = 3)
Ker kernel
span linear span
In natural logarithm
e Euler number
detA determinant of matrix A
AT
transposed matrix to A
AC complex conjugate matrix to A
AH
conjugate transposed matrix to A
Re real part
1m imaginary part
ix
x List of symbols
imaginary unit
z, ), k integer indices (subscripts)
= = =
~ij Kronecker's symbol: ~ij 1 for i j, ~ij 0 otherwise
C complex plane
:1 there exist( s)
V for all
a.a. almost all
a.e. almost everywhere
Pk(O) space of polynomials of degree at most k defined on 0
QI(O) space of d-linear polynomials defined on 0
U(O), p E [1,00) Lebesgue space of measurable functions v defined on 0
If!
for which Iv( x)IP dx is finite
LOO(O) Lebesgue space of measurable essentially bounded
functions defined on 0
c(n) space of continuous functions defined on n
ck(n) space of functions whose classical derivatives up to order k
belong to C (n)
COO(O) space of infinitely differentiable functions in 0
cO'(O) space of infinitely differentiable functions with compact
support in 0
Hk(O) Sobolev space of functions whose generalized derivatives
up to order k belong to L2(0)
an
HJ(O) space of functions from Hl(n) whose traces vanish on
HI/2(00) space of traces of all functions from HI (0)
W;(O) Sobolev space of functions whose generalized
derivatives up to order k belong to U(O)
H(divO) space of divergence-free functions
H(rotO) space of rotation-free functions
V Banach space of test functions
V* space of linear continuous functionals on V
weak convergence
--+ strong convergence
(b,v) scalar product or value of functional b E V* at point v E V
v test function
u classical or weak (variational, generalized) solution
11·llv norm in V
(., ·)v scalar product in V
II . Ilk,f! norm in Hk(O)
I . Ik,f! seminorm in Hk(O)
List of symbols xi
("')k,O scalar product in Hk(O)
II· IIk,p,o norm in W;(O)
(., '}o,oo scalar product in L2(80)
1·1 absolute value
11·11 Euclidean norm
(. ,.) Euclidean scalar product
a(·, .) bilinear or sesquilinear form
F(·) linear form
J potential functional
J' Gateaux differential of J
K element
k reference element
PK space of shape functions
EK set of degrees of freedom
(K,PK,EK) finite element
dimPK dimension of PK
hK diameter of K (diam K)
h discretization parameter
Th triangulation (partition, decomposition)
Vh finite element space
Vh-interpolant of v
'TrhV
suppv support of function v
{v i }Ni= l basis functions in Vh
=
0(·) Landau's symbol: f(O'.) O(g(O'.)), if
If( 0'.)1 ::; Glg( 0'.)1 as 0'. ---+ 0 or 0'. ---+ 00
Ell direct sum
o
empty set
x E A element x belongs to set A
x rJ.A element x does not belong to set A
{x E AIP(x)} set of all elements x from A which possess property P( x)
ACB A is subset of set B
AnB intersection of sets A and B
AuB union of sets A and B
A\B subtraction of B from A
f:A---+B function f from A to B
x f(x) function which assigns value f(x) to x
1-+
o
Halmos symbol