Preface
The aim of this book is to provide an introduction to computational electromagnetics
(CEM) with a focus on the most popular techniques used in contemporary
research and development projects. The focus is on the solution
of Maxwell’s equations by means of finite difference methods, the finite element
method (FEM), and the method of moments (MoM). The book treats
the solution of both static and dynamic problems, where also quasi-static
problems are discussed to some extent. Both time-harmonic and transient
formulations are employed for the dynamic problems. We feature convergence
characteristics, error analysis (through extrapolation), and stability analysis
of the computational techniques. Some versions of FEM are directly related to
the corresponding finite difference schemes, which provides useful insight into
some of the underlying aspects of the discretization of Maxwell’s equations. A
collection of MATLAB programs is included to demonstrate the implementation
and performance of the numerical methods. The MATLAB programming
language is chosen, since it offers succinct code, easy-to-use linear algebra routines,
and good visualization tools.
The introductory character of the text makes it useful as a textbook in an
undergraduate course on computational electromagnetics. In fact, this book
is based on the material developed for and used in an undergraduate CEM
course at Chalmers University of Technology, G¨oteborg, Sweden. The prerequisites
for a course based on this book are basic electromagnetic field theory,
numerical analysis, and programming typically included in the first year’s
training in, for example, engineering physics and electrical engineering. The
book also works well for an introduction to CEM at an early graduate level.
The use of computer projects can easily be used to adjust the length of a
course that employs this book. An electrical engineer can use this book to
achieve a sufficiently good understanding of CEM to exploit commercial software
for reliable and efficient design of real-world electromagnetic devices.
Since CEM is a multidisciplinary topic, this book may also interest applied
mathematicians, theoretical electromagnetics researchers, and others who are
working in areas related to CEM.
xii Preface
The book starts with a brief introduction to CEM and Maxwell’s equations
in Chapter 1. The concepts of numerical error, resolution, convergence, and
extrapolation are presented in Chapter 2 by means of an electrostatic example.
Chapter 3 introduces some basic finite difference approximations that are
often used in CEM. Again, an electrostatic example is exploited to demonstrate
the use of finite differences, and the convergence of the capacitance for
a problem with sharp corners is studied. This chapter also treats some characteristics
of finite differences applied to complex exponentials, which provides
the foundation for subsequent discussions on numerical dispersion, spurious
modes, and staggered grids. Chapter 4 develops some different views on eigenvalue
problems for Maxwell’s equations. After a short introduction with some
background theory, a one-dimensional eigenvalue problem is studied when
discretized by finite differences. In the corresponding time-domain eigenvalue
calculation that follows, we exploit these results for a stability analysis, and
the findings are also related to numerical dispersion. The theoretical treatment
is supported by MATLAB examples. The output from the time-domain computations
is analyzed by means of its Fourier transform, and a complementing
technique based on the Pad´e approximation is also discussed.
At this point, we are well equipped for the introduction of the finitedifference
time-domain (FDTD) scheme, which is the topic of Chapter 5.
A review of the one-dimensional wave equation discretized by finite differences
demonstrates the effects of numerical dispersion and stability in the
time domain. This is followed by the corresponding FDTD scheme and its
three-dimensional counterpart. The eigenfrequencies of a brick-shaped threedimensional
cavity resonator are computed by an example implementation in
MATLAB. Next, the FDTD scheme is interpreted in terms of the Maxwell’s
equations on an integral form and it is shown that the FDTD scheme preserves
the condition of solenoidal magnetic flux density in the time-stepping
procedure. Also, a dispersion analysis is provided, and based on these results,
necessary resolutions are indicated by rules of thumb. The chapter on
the FDTD scheme ends with brief discussions on common techniques used in
real-world FDTD computations such as the perfectly matched layers (PML)
for open region problems and the near-to-far-field transformation for the computation
of scattering and radiation properties.
In Chapter 6, the FEM is introduced by means of a recipe for Galerkin’s
method. Then, Galerkin’s method is used for the Helmholtz equation in one
and two dimensions, and the treatment includes a discussion on the Dirichlet,
Neumann, and Robin boundary conditions. Next, a detailed description of the
procedures normally used in the implementation of FEM is presented: elementwise
integration of basis functions, assembling procedure, and management
of unstructured meshes in practice. The account is supported by MATLAB
programs, and the capacitance problem from Chapter 3 is revisited by means
of both uniform grid refinement and adaptive techniques. The concept of the
FEM for the first-order system of Maxwell’s equations is developed for onedimensional
problems, and the resulting discrete representation is compared to
Preface xiii
the corresponding finite difference scheme. These discussions naturally lead to
the use of edge elements for the curl-curl equation for the electric field, which
is the next topic in the FEM chapter. First, rectangular edge elements are
discussed, and then, their relations to the finite differences are provided. A
resonator discretized by a 2×2-element grid is used as a detailed example to
illustrate some of the characteristic properties of edge elements, which is followed
by a similar and well-resolved eigenvalue problem. Next, edge elements
on triangles are introduced and supplemented by a discussion on and implementation
for their use in practice. Also, time-dependent problems treated by
FEM are discussed, and the reader is confronted with the Newmark scheme
for unconditionally stable time stepping. At this point, we use the FEM for
the treatment of magnetostatics and eddy current problems in 2D, which is
followed by an outline on techniques for the corresponding 3D problems. The
chapter ends with variational techniques, their relation to Galerkin’s method,
and, finally, a variational method for Maxwell’s equations.
The third and last technique is the MoM, and it is presented in Chapter 7.
The description of the MoM begins with electrostatics: the integral equation
formulation (with the Green’s function) and its solution by means of FEM
techniques. This presentation is complemented with a capacitance problem
in an unbounded two-dimensional region, and a MATLAB implementation
is used for a convergence study by means of uniform and adaptive grids.
Next, the MoM is applied to electromagnetic scattering problems, and again,
the Green’s function and its related electric field integral equation (EFIE)
is derived. The choice of test and basis functions is discussed before a brief
presentation of the magnetic field integral equation (MFIE) and the combined
field integral equations (CFIE). The chapter on MoM is ended by scattering
from thin wires, which is treated by means of the EFIE. Hall´en’s equation
is derived, and the valid approximation of its kernel is discussed before a
MATLAB implementation and some numerical results are presented.
Finally, Chapter 8 provides a summary and overview of the material in
the book. A system of fixed spatial extent is used to study how the computational
requirements scale with the frequency. The FDTD, FEM, MoM, and a
number of similar techniques are compared in this sense. The other differential
equation solvers mentioned are the finite-volume time-domain scheme, the
transmission line method, and the finite integration technique. The additional
aspects of integral equation solvers involves the fast multipole methods, some
other fast methods, and a brief discussion on schemes for time-domain integral
equations. Finally, a short note on hybrid methods ends the chapter. This
chapter also includes a number of references to the literature, which provides
the reader with some additional starting points for further studies of CEM.
Other powerful tools in CEM are included in appendices, where information
on iterative solvers and multigrid methods can be found.
To our knowledge, the open literature is scarce on books that present
contemporary CEM in an introductory manner that is appropriate for use in
undergraduate education. This book is intended to provide such material and
xiv Preface
prepare the reader for the more advanced literature on CEM. Apart from this
text, the book by Davidson [21] and the book by Sadiku [65] are some of the
very few other examples with such an aim. For more experienced readers, there
is a number of good monographs that treat one or a few techniques. However,
these may not be appropriate for classroom use. The books by Taflove et
al. [75, 77, 76] on the FDTD scheme are indeed excellent and good accounts
on the FEM are given by Jin [38, 39] and Silvester and Ferrari [70]. Peterson’s
book [51] on the MoM and FEM (and some FDTD) is also well worth reading.
We would also like to mention the FEM book by Monk [46], which gives a
more mathematical account of the FEM for Maxwell’s equations. Chew et
al. [19] published a book on fast and efficient algorithms in computational
electromagnetics, which deals with a variety of methods in CEM.
The MATLAB implementations listed in this book are available for download
from the URL
http://ct.am.chalmers.se/edu/books/cem/
We would appreciate it if errors found are brought to our attention. These
will be posted on the website above.
Anders Bondeson (deceased) Thomas Rylander
Department of Electromagnetics
Chalmers University of Technology
Ho¨rsalsva¨gen 11
SE-41296 Go¨teborg, Sweden
rylander@chalmers.se
Pa¨r Ingelstro¨m
Department of Electromagnetics
Chalmers University of Technology
Ho¨rsalsva¨gen 11
SE-41296 Go¨teborg, Sweden
pi@chalmers.se
Series Editors
J.E. Marsden
Control and Dynamical Systems, 107–81
California Institute of Technology
Pasadena, CA 91125
USA
marsden@cds.caltech.edu
L. Sirovich
Division of Applied Mathematics
Brown University
Providence, RI 02912
USA
chico@camelot.mssm.edu
S.S. Antman
Department of Mathematics
and
Institute for Physical Science
and Technology
University of Maryland
College Park, MD 20742-4015
USA
ssa@math.umd.edu
Mathematics Subject Classification (2000): 78M05, 78M10, 78M20, 78-01, 65-01
Library of Congress Control Number: 2005926299
ISBN-10: 0-387-26158-3 eISBN 0-387-26160-5 Printed on acid-free paper.
ISBN-13: 978-0387-26158-4
© 2005 Springer Science+Business Media, Inc.
All rights reserved. This work may not be translated or copied in whole or in part without the
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New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly
analysis. Use in connection with any form of information storage and retrieval, electronic adaptation,
computer software, or by similar or dissimilar methodology now known or hereafter
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if they are not identified as such, is not to be taken as an expression of opinion as to whether
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