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Numerical Techniques in Eletromagnetics_2nd_Matthew N.O.Sadiku.pdf
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Numerical Techniques in Electromagnetics
Numerical Techniques in Electromagnetics
Preface
Acknowledgements
A Note to Students
Contents
Fundamental Concepts
Numerical Techniques in Electromagnetics
Contents
Chapter 1
1.1 Introduction
1.2 Review of Electromagnetic Theory
1.2.1 Electrostatic Fields
1.2.2 Magnetostatic Fields
1.2.3 Time-varying Fields
1.2.4 Boundary Conditions
1.2.5 Wave Equations
1.2.6 Time-varying Potentials
1.2.7 Time-harmonic Fields
1.3 Classification of EM Problems
1.3.1 Classification of Solution Regions
1.3.2 Classification of Differential Equations
1.3.3 Classification of Boundary Conditions
1.4 Some Important Theorems
1.4.1 Superposition Principle
1.4.2 Uniqueness Theorem
References
Problems
Analytical Methods
Numerical Techniques in Electromagnetics
Contents
Chapter 2
2.1 Introduction
2.2 Separation of Variables
2.3 Separation of Variables in Rectangular Coordinates
2.3.1 Laplace’s Equations
Example 2.2
2.3.2 Wave Equation
2.4 Separation of Variables in Cylindrical Coordinates
2.4.1 Laplace’s Equation
2.4.2 Wave Equation
2.5 Separation of Variables in Spherical Coordinates
2.5.1 Laplace’s Equation
2.5.2 Wave Equation
2.6 Some Useful Orthogonal Functions
2.7 Series Expansion
2.7.1 Poisson’s Equation in a Cube
2.7.2 Poisson’s Equation in a Cylinder
2.7.3 Strip Transmission Line
2.8 Practical Applications
2.8.1 Scattering by Dielectric Sphere
2.8.2 Scattering Cross Sections
2.9 Attenuation Due to Raindrops
2.10 Concluding Remarks
References
Problems
Finite Difference Methods
Numerical Techniques in Electromagnetics
Contents
Chapter 3
3.1 Introduction
3.2 Finite Difference Schemes
3.3 Finite Differencing of Parabolic PDEs
3.4 Finite Differencing of Hyperbolic PDEs
3.5 Finite Differencing of Elliptic PDEs
3.5.1 Band Matrix Method
3.5.2 Iterative Methods
3.6 Accuracy and Stability of FD Solutions
3.7 Practical Applications I Û Guided Structures
3.7.1 Transmission Lines
3.7.2 Waveguides
3.8 Practical Applications II Û Wave Scattering (FDTD)
3.8.1 YeeÌs Finite Difference Algorithm
3.8.2 Accuracy and Stability
3.8.3 Lattice Truncation Conditions
3.8.4 Initial Fields
3.8.5 Programming Aspects
3.9 Absorbing Boundary Conditions for FDTD
3.10 Finite Differencing for Nonrectangular Systems
3.10.1 Cylindrical Coordinates
3.10.2 Spherical Coordinates
3.11 Numerical Integration
3.11.1 Euler’s Rule
3.11.2 Trapezoidal Rule
3.11.3 Simpson’s Rule
3.11.4 Newton-Cotes Rules
3.11.5 Gaussian Rules
3.11.6 Multiple Integration
3.12 Concluding Remarks
References
Problems
Variational Methods
Numerical Techniques in Electromagnetics
Contents
Chapter 4
4.1 Introduction
4.2 Operators in Linear Spaces
4.3 Calculus of Variations
4.4 Construction of Functionals from PDEs
4.5 Rayleigh-Ritz Method
4.6 Weighted Residual Method
4.6.1 Collocation Method
4.6.2 Subdomain Method
4.6.3 Galerkin Method
4.6.4 Least Squares Method
4.7 Eigenvalue Problems
4.8 Practical Applications
4.9 Concluding Remarks
References
Problems
Moment Methods
Numerical Techniques in Electromagnetics
Contents
Chapter 5
5.1 Introduction
5.2 Integral Equations
5.2.1 Classification of Integral Equations
5.2.2 Connection Between Differential and Integral Equations
5.3 GreenÌs Functions
5.3.1 For Free Space
5.3.2 For Domain with Conducting Boundaries
5.4 Applications I Û Quasi-Static Problems
5.5 Applications II Û Scattering Problems
5.5.1 Scattering by Conducting Cylinder
5.5.2 Scattering by an Arbitrary Array of Parallel Wires
5.6 Applications III Û Radiation Problems
5.6.1 HallenÌs Integral Equation
5.6.2 PocklingtonÌs Integral Equation
5.6.3 Expansion and Weighting Functions
5.7 Applications IV — EM Absorption in the Human Body
5.7.1 Derivation of Integral Equations
5.7.2 Transformation to Matrix Equation (Discretization)
5.7.3 Evaluation of Matrix Elements
5.7.4 Solution of the Matrix Equation
5.8 Concluding Remarks
References
Problems
Finite Element Method
Numerical Techniques in Electromagnetics
Contents
Chapter 6
6.1 Introduction
6.2 Solution of LaplaceÌs Equation
6.2.1 Finite Element Discretization
6.2.2 Element Governing Equations
6.2.3 Assembling of All Elements
6.2.4 Solving the Resulting Equations
6.3 Solution of PoissonÌs Equation
6.3.1 Deriving Element-governing Equations
6.3.2 Solving the Resulting Equations
6.4 Solution of the Wave Equation
6.5 Automatic Mesh Generation I Û Rectangular Domains
6.6 Automatic Mesh Generation II Û Arbitrary Domains
6.6.1 Definition of Blocks
6.6.2 Subdivision of Each Block
6.6.3 Connection of Individual Blocks
6.7 Bandwidth Reduction
6.8 Higher Order Elements
6.8.1 Pascal Triangle
6.8.2 Local Coordinates
6.8.3 Shape Functions
6.8.4 Fundamental Matrices
6.9 Three-Dimensional Elements
6.10 Finite Element Methods for Exterior Problems
6.10.1 Infinite Element Method
6.10.2 Boundary Element Method
6.10.3 Absorbing Boundary Conditions
6.11 Concluding Remarks
References
Problems
Transmission-line-matrix Method
Numerical Techniques in Electromagnetics
Contents
Chapter 7
7.1 Introduction
7.2 Transmission-line Equations
7.3 Solution of Diffusion Equation
7.4 Solution of Wave Equations
7.4.1 Equivalence Between Network and Field Parameters
7.4.2 Dispersion Relation of Propagation Velocity
7.4.3 Scattering Matrix
7.4.4 Boundary Representation
7.4.5 Computation of Fields and Frequency Response
7.4.6 Output Response and Accuracy of Results
7.5 Inhomogeneous and Lossy Media in TLM
7.5.1 General Two-Dimensional Shunt Node
7.5.2 Scattering Matrix
7.5.3 Representation of Lossy Boundaries
7.6 Three-Dimensional TLM Mesh
7.6.1 Series Nodes
7.6.2 Three-Dimensional Node
7.6.3 Boundary Conditions
7.7 Error Sources and Correction
7.7.1 Truncation Error
7.7.2 Coarseness Error
7.7.3 Velocity Error
7.7.4 Misalignment Error
7.8 Absorbing Boundary Conditions
7.9 Concluding Remarks
References
Problems
Monte Carlo Methods
Numerical Techniques in Electromagnetics
Contents
Chapter 8
8.1 Introduction
8.2 Generation of Random Numbers and Variables
8.3 Evaluation of Error
8.4 Numerical Integration
8.4.1 Crude Monte Carlo Integration
8.4.2 Monte Carlo Integration with Antithetic Variates
8.4.3 Improper Integrals
8.5 Solution of Potential Problems
8.5.1 Fixed Random Walk
8.5.2 Floating Random Walk
8.5.3 Exodus Method
8.6 Regional Monte Carlo Methods
8.7 Concluding Remarks
References
Problems
Method of Lines
Numerical Techniques in Electromagnetics
Contents
Chapter 9
9.1 Introduction
9.2 Solution of Laplace’s Equation
9.2.1 Rectangular Coordinates
9.2.2 Cylindrical Coordinates
9.3 Solution of Wave Equation
9.3.1 Planar Microstrip Structures
9.3.2 Cylindrical Microstrip Structures
9.4 Time-Domain Solution
9.5 Concluding Remarks
References
Problems
Vector Relations
Numerical Techniques in Electromagnetics
Contents
Appendix A
A.1 Vector Identities
A.2 Vector Theorems
A.3 Orthogonal Coordinates
Solving Electromagnetic Problems
Numerical Techniques in Electromagnetics
Contents
Appendix B
B.1 Introduction
B.2 A Brief Description of C++
A. What Every New Programmer to a Language Should See First
B. Types and Declarations
C. Input and Output with cin and cout
D. Pointers
E. Arrays
F. Control Constructs
G. Structures and Unions
H. Functions
B.3 Object-Orientation
A. Inheritance
B. Polymorphism
C. Data Abstraction
D. Encapsulation
E. Overloading
B.4 C++ Object-Oriented Language Features
A. Classes
B. Operator Overloading
C. Templates
D. Exception Handling
E. Files and Streams
B.5 A Final Note
References
Numerical Techniques in C++
Numerical Techniques in Electromagnetics
Contents
Appendix C
Numerical Techniques in C
Listing 1: Finite Difference Program:
Listing 2: Finite Element Program:
Listing 3: Transmission-line-matrix:
Listing 4: Fixed-Random Walk Monte Carlo Method:
Solution of Simultaneous Equations
Numerical Techniques in Electromagnetics
Contents
Appendix D
D.1 Elimination Methods
D.1.1 Gauss’s Method
D.1.2 Cholesky’s Method
D.2 Iterative Methods
D.2.1 Jacobi’s Method
D.2.2 Gauss-Seidel Method
D.2.3 Relaxation Method
D.2.4 Gradient Methods
D.3 Matrix Inversion
D.4 Eigenvalue Problems
D.4.1 Iteration (or Power) Method
D.4.2 Jacobi’s Method
References
Answers to Odd-Numbered Problems
Numerical Techniques in Electromagnetics
Contents
Appendix E
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Numerical
Techniques in
Electromagnetics
Second Edition
Numerical
Techniques in
Electromagnetics
Second Edition
Matthew N. O. Sadiku, Ph.D.
Boca Raton London New York Washington, D.C.
CRC Press
Library of Congress Cataloging-in-Publication Data
Sadiku, Matthew N. O.
Numerical techniques in electromagnetics / Matthew N.O. Sadiku.—[2nd ed.].
p. cm.
Includes bibliographical references and index.
ISBN 0-8493-1395-3 (alk. paper)
1. Electromagnetism. 2. Numerical analysis. I. Title.
QC760 .S24 2000
515—dc21
537
.01
′
′
00-026823
CIP
This book contains information obtained from authentic and highly regarded sources. Reprinted material
is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable
efforts have been made to publish reliable data and information, but the author and the publisher cannot
assume responsibility for the validity of all materials or for the consequences of their use.
Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic
or mechanical, including photocopying, microfilming, and recording, or by any information storage or
retrieval system, without prior permission in writing from the publisher.
The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for
creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC
for such copying.
Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431.
Trademark Notice:
used only for identification and explanation, without intent to infringe.
Product or corporate names may be trademarks or registered trademarks, and are
© 2001 by CRC Press LLC
No claim to original U.S. Government works
International Standard Book Number 0-8493-1395-3
Library of Congress Card Number 00-026823
Printed in the United States of America 1 2 3 4 5 6 7 8 9 0
Printed on acid-free paper
Preface
The art of computation of electromagnetic (EM) problems has grown exponentially
for three decades due to the availability of powerful computer resources. In spite
of this, the EM community has suffered without a suitable text on computational
techniques commonly used in solving EM-related problems. Although there have
been monographs on one particular technique or the other, the monographs are written
for the experts rather than students. Only a few texts cover the major techniques and
do that in a manner suitable for classroom use.
It seems experts in this area are
familiar with one or few techniques and not many experts seem to be familiar with
all the common techniques. This text attempts to fill the gap.
The text is intended for seniors or graduate students and may be used for a one-
semester or two-semester course. The main requirements for students taking a course
based on this text are introductory EM courses and a knowledge of a high-level
computer language, preferably FORTRAN or C. Software packages such as Matlab
and Mathcad may be helpful tools. Although familiarity with linear algebra and
numerical analysis is useful, it is not required.
In writing this book, three major objectives were borne in mind. First, the book is
intended to teach students how to pose, numerically analyze, and solve EM problems.
Second, it is designed to give them the ability to expand their problem solving skills
using a variety of available numerical methods. Third, it is meant to prepare graduate
students for research in EM. The aim throughout has been simplicity of presentation
so that the text can be useful for both teaching and self-study.
In striving after
simplicity, however, the reader is referred to the references for more information.
Toward the end of each chapter, the techniques covered in the chapter are applied
to real life problems. Since the application of the technique is as vast as EM and
author’s experience is limited, the choice of application is selective.
Chapter 1 covers some fundamental concepts in EM. Chapter 2 is intended to put
numerical methods in a proper perspective. Analytical methods such as separation
of variables and series expansion are covered. Chapter 3 discusses the finite differ-
ence methods and begins with the derivation of difference equation from a partial
differential equation (PDE) using forward, backward, and central differences. The
finite-difference time-domain (FDTD) technique involving Yee’s algorithm is pre-
v
sented and applied to scattering problems. Numerical integration is covered using
trapezoidal, Simpson’s, Newton-Cotes rules, and Gaussian quadratures.
Chapter 4 on variational methods serves as a preparatory ground for the next two
major topics: moment methods and finite element methods. Basic concepts such
as inner product, self-adjoint operator, functionals, and Euler equation are covered.
Chapter 5 on moment methods focuses on the solution of integral equations. Chap-
ter 6 on finite element method covers the basic steps involved in using the finite
element method. Solutions of Laplace’s, Poisson’s, and wave equations using the
finite element method are covered.
Chapter 7 is devoted to transmission-line matrix or modeling (TLM). The method
is applied to diffusion and scattering problems. Chapter 8 is on Monte Carlo methods,
while Chapter 9 is on the method of lines.
Since the publication of the first edition, there has been an increased awareness and
utilization of numerical techniques. Many graduate curricula now include courses
in numerical analysis of EM problems. However, not much has changed in compu-
tational electromagnetics. A major noticeable change is in the FDTD method. The
method seems to have attracted much attention and many improvements are being
made to the standard algorithm. This edition adds the noticeable change in incorpo-
rating absorbing boundary conditions in FDTD, FEM, and TLM. Chapter 9 is a new
chapter on the method of lines.
Acknowledgements
I am greatly indebted to Temple University for granting me a sabbatical in Fall 1998
during which I was able to do most of the revision. I specifically would like to thank
my dean, Dr. Keya Sadeghipour, and my chairman, Dr. John Helferty, for their support.
Special thanks are due to Raymond Garcia of Georgia Tech for writing Appendices C
and D in C++. I am deeply grateful to Dr. Arthur D. Snider of the University of
South Florida and Mohammad R. Zunoubi of Mississippi State University for taking
the time to send me the list of errors in the first edition. I thank Dr. Reinhold Pregla
for helping in clarifying concepts in Chapter 9 on the method of lines.
I express
my deepest gratitude to my wife, Chris, and our daughters, Ann and Joyce, for their
patience, sacrifices, and prayers.
A Note to Students
Before you embark on writing your own computer program or using the ones in this
text, you should try to understand all relevant theoretical backgrounds. A computer
is no more than a tool used in the analysis of a program. For this reason, you should
be as clear as possible what the machine is really being asked to do before setting it
off on several hours of expensive computations.
It has been well said by A.C. Doyle that “It is a capital mistake to theorize before
you have all the evidence. It biases the judgment.” Therefore, you should never trust
the results of a numerical computation unless they are validated, at least in part. You
validate the results by comparing them with those obtained by previous investigators
or with similar results obtained using a different approach which may be analytical
or numerical. For this reason, it is advisable that you become familiar with as many
numerical techniques as possible.
The references provided at the end of each chapter are by no means exhaustive but
are meant to serve as the starting point for further reading.
Contents
1 Fundamental Concepts
Introduction
1.1
1.2 Review of Electromagnetic Theory
1.2.1 Electrostatic Fields
1.2.2 Magnetostatic Fields
1.2.3 Time-varying Fields
1.2.4 Boundary Conditions
1.2.5 Wave Equations
1.2.6 Time-varying Potentials
1.2.7 Time-harmonic Fields
1.3 Classification of EM Problems
1.3.1 Classification of Solution Regions
1.3.2 Classification of Differential Equations
1.3.3 Classification of Boundary Conditions
1.4 Some Important Theorems
1.4.1 Superposition Principle
1.4.2 Uniqueness Theorem
References
Problems
2 Analytical Methods
Introduction
2.1
2.2 Separation of Variables
2.3 Separation of Variables in Rectangular Coordinates
2.3.1 Laplace’s Equations
2.3.2 Wave Equation
2.4 Separation of Variables in Cylindrical Coordinates
2.4.1 Laplace’s Equation
2.4.2 Wave Equation
2.5 Separation of Variables in Spherical Coordinates
2.5.1 Laplace’s Equation
2.5.2 Wave Equation
2.6 Some Useful Orthogonal Functions
2.7 Series Expansion
2.7.1 Poisson’s Equation in a Cube
2.7.2 Poisson’s Equation in a Cylinder
2.7.3 Strip Transmission Line
2.8 Practical Applications
2.8.1 Scattering by Dielectric Sphere
2.8.2 Scattering Cross Sections
2.9 Attenuation Due to Raindrops
2.10 Concluding Remarks
References
Problems
3 Finite Difference Methods
Introduction
3.1
3.2 Finite Difference Schemes
3.3 Finite Differencing of Parabolic PDEs
3.4 Finite Differencing of Hyperbolic PDEs
3.5 Finite Differencing of Elliptic PDEs
3.5.1 Band Matrix Method
3.5.2
Iterative Methods
3.6 Accuracy and Stability of FD Solutions
3.7 Practical Applications I — Guided Structures
3.7.1 Transmission Lines
3.7.2 Waveguides
3.8 Practical Applications II — Wave Scattering (FDTD)
3.8.1 Yee’s Finite Difference Algorithm
3.8.2 Accuracy and Stability
3.8.3 Lattice Truncation Conditions
3.8.4
3.8.5 Programming Aspects
Initial Fields
3.9 Absorbing Boundary Conditions for FDTD
3.10 Finite Differencing for Nonrectangular Systems
3.10.1 Cylindrical Coordinates
3.10.2 Spherical Coordinates
3.11 Numerical Integration
3.11.1 Euler’s Rule
3.11.2 Trapezoidal Rule
3.11.3 Simpson’s Rule
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