transmission line matrix (tlm) in computational mechanics - d. de cogan, et al., (crc, 2006) ww

Transmission Line Matrix in
Computational Mechanics
Donard de Cogan
William J. O’Connor
Susan Pulko
Transmission Line Matrix in
Computational Mechanics
A CRC title, part of the Taylor & Francis imprint, a member of the
Taylor & Francis Group, the academic division of T&F Informa plc.
Boca Raton London New York
Published in 2006 by
CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2006 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group
No claim to original U.S. Government works
Printed in the United States of America on acid-free paper
10987654321
International Standard Book Number-10: 0-415-32717-2 (Hardcover)
International Standard Book Number-13: 978-0-415-32717-6 (Hardcover)
Library of Congress Card Number 2004062817
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.
No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic,
mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and

recording, or in any information storage or retrieval system, without written permission from the publishers.
For permission to photocopy or use material electronically from this work, please access www.copyright.com
( or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive,
Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration
for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate
system of payment has been arranged.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only
for identification and explanation without intent to infringe.
Library of Congress Cataloging-in-Publication Data
De Cogan, Donard.
Transmission line matrix (TLM) in computational mechanics : (a new perspective in applied
mathematics for computational engineers) / Donard de Cogan, William J. O'Connor, Susan H. Pulko.
p. cm.
Includes bibliographical references and index.
ISBN 0-415-32717-2
1. Microwave transmission lines Mathematical models. I. O'Connor, William, 1951- II. Pulko,
Susan H. III. Title.
TK7876.D43 2005
620.1'001'5118 dc22 2004062817
Visit the Taylor & Francis Web site at

and the CRC Press Web site at

Taylor & Francis Group
is the Academic Division of Informa plc.
TF1745_Discl.fm Page 1 Monday, September 26, 2005 3:54 PM

Acknowledgments

Those who knew Peter Johns

*

speak glowingly of his inspiration and his
enthusiasm. He achieved so much, and we are certain that he could have
achieved much more had he lived. He was already moving into mechanical
applications of TLM and was discussing nonlinear processes such as the
action of a violin bow on a string. Shortly after his first heart attack he
commenced work on a TLM model of electromechanical interactions in heart
muscle. He was a cohesive factor in all areas of development, which in his
absence have tended toward a bimodal partition: TLM applications that are
related to electromagnetics and TLM applications that are not. Within the
latter grouping, the contributions of Peter Enders, Xiang Gui, and the late
Adnan Saleh have been crucial. We also wish to acknowledge the contribu-
tion of the many TLM researchers who have been happy to share their
experiences freely at various workshops and colloquia and by personal
communication. There have also been the behind-the-scenes contributions
of research students and assistants such as Dorian Hindmarsh and Mike
Morton. We have benefited greatly by the many constructive comments from
specialists such as Kevin Edge (Fluid Power Centre, University of Bath),
Petter Krus (Division of Fluid Power Technology, Linköping University), and
Richard Pearson (Power Train Division, Lotus Cars, Hethel, Norfolk, U.K.).
Many thanks to James Flint for some last minute comments on Doppler
modeling. Finally, there are our editors. Without the input of Donald Degen-
hardt this book would never have passed the initial planning stages. Janie
Wardle has overseen the transition between publishers

**

and our progress

toward completion. And finally, Sylvia Wood of Taylor & Francis, who, in
spite of everything, brought it all together. We are most grateful to them for
their encouragement and support.

*

Two of the authors of this work, DdeC and SHP, share this honor.

**

Gordon & Breach became part of Taylor & Francis while this book was being written.

TF1745_C000.fm Page v Wednesday, September 28, 2005 12:50 PM

TF1745_C000.fm Page vi Wednesday, September 28, 2005 12:50 PM

About the Authors

Donard de Cogan

gained a bachelor’s degree in physical chemistry and a
Ph.D. in solid state physics from Trinity College, Dublin. He undertook
research fellowships in solid state chemistry (University of Nijmegen, Neth-
erlands) and microelectronic fabrication (University of Birmingham) before
joining Philips as a senior development engineer in power electronic semi-
conductors. In 1978 he was appointed a lecturer in electrical and electronic
engineering at the University of Nottingham. His initial research was con-
cerned with the overload impulse withstand capability of a range of electrical
and electronic components, and the results confirmed a requirement for
numerical simulation. He was encouraged to use the transmission line matrix

(TLM) technique, which had been invented at Nottingham, and this soon
became his principal line of research. In 1989 he was appointed a senior
lecturer in what is now the Computing Sciences Department at the Univer-
sity of East Anglia at Norwich, where he leads a TLM research team. In 1994
Dr. de Cogan was promoted to Reader. He is the book reviews editor for the

International Journal of Numerical Modeling

and editor of the Gordon and
Breach (now Taylor & Francis)

Electrocomponent Science

monograph series.
His outside interests include music, sailing, and the history of technology.

William O’Connor

obtained his Ph.D. from the University College, Dublin
(UCD) in 1976 on magnetic fields for pole geometries with saturable mate-
rials. He lectures in dynamics, control, and microprocessor applications in
UCD, National University of Ireland, Dublin, in the Department of Mechan-
ical Engineering (UCD is the largest university in Ireland and the Depart-
ment of Mechanical Engineering is also the largest such department in the
country, enjoying a worldwide reputation for teaching and research). In
addition to both analytical and numerical analysis of magnetic fields and
forces, his research interests include novel numerical modeling methods and
applications, especially in acoustics, mechanical-acoustic systems, and fluids;
development of transmission line matrix and impulse propagation numerical
methods; control of flexible mechanical systems including vibration damp-

ing; vibration-based resonant fluid sensors; and acoustic and infrared sen-
sors. Dr. O’Connor is a Fellow of the Institution of Engineers of Ireland.

TF1745_C000.fm Page vii Wednesday, September 28, 2005 12:50 PM

Susan Pulko

graduated from Imperial College, University of London in
1977. She moved to the University of Nottingham and undertook postgrad-
uate work in solid state physics in the Department of Electrical and Electronic
Engineering. Having obtained a Ph.D., she started working on the transmis-
sion line matrix (TLM) technique as a postdoctoral assistant to Professor P.B.
Johns, concentrating largely on the development of the TLM technique for
use in thermal applications. Dr. Pulko later took up a lectureship in the
Department of Electronic Engineering at the University of Hull, where she
established a TLM research group. This group continued the development
of TLM for thermal problems and applied it in a range of industries from
ceramics to food. It was while the group was working with the ceramics
industry that the desirability of modeling deformation processes by TLM
became apparent. The modeling of propagating stress waves took place from
this point and has been applied to the modeling of ultrasound wave prop-
agation in solids; current work in this area is concerned with modeling
magnetostrictive behavior. She is a consultant to Feonic plc.

TF1745_C000.fm Page viii Wednesday, September 28, 2005 12:50 PM
Table of Contents
Chapter 1 Introduction 1
Chapter 2 TLM and the 1-D Wave Equation 9
2.1 Introduction 9
2.2 The Vibrating String 10

2.3 A Simple TLM Model 11
2.4 Boundary and Initial Conditions 13
2.5 Wave Media, Impedance, and Speed 15
2.6 Transmission Line Junctions 18
2.7 Stubs 19
2.8 The Forced Wave Equation 20
2.9 Waves in Moving Media: The Moving Threadline Equation 21
2.10 Gantry Crane Example 21
2.11 Rotating String: Differential Equation and Analytical Solution 22
2.11.1 Rotating String: TLM Model 23
2.11.2 Rotating String: Results 24
2.12 TLM in 2-D (Extension to Higher Dimensions) 24
2.13 Conclusions 25
Chapter 3 The Theory of TLM: An Electromagnetic Viewpoint 27
3.1 Introduction 27
3.2 The Building Blocks: Electrical Components 28
3.2.1 Resistor 28
3.2.2 Capacitor 28
3.2.3 Inductor 30
3.2.4 Transmission Line 31
3.3 Basic Network Theory 32
3.4 Propagation of a Signal in Space (Maxwell’s Equations) 33
3.5 Distributed and Lumped Circuits 36
3.6 Transmission Lines Revisited 37
3.6.1 Time Discretization 37
3.7 Discontinuities 39
3.8 TLM Nodal Configurations 40
3.9 Boundaries 43
3.10 Conclusion 45
Chapter 4 TLM Modeling of Acoustic Propagation 47

4.1 Introduction 47
TF1745_book.fm Page ix Monday, September 12, 2005 11:56 AM
4.2 1-D TLM Algorithm 47
4.3 2-D TLM Algorithm for Acoustic Propagation 52
4.4 Driven Sine-Wave Excitation 56
4.5 The 2-D Propagation of a Gaussian Wave-Form 60
4.6 Moving Sources 63
4.7 Propagation in Inhomogeneous Media 66
4.8 Incorporation of Stub Lines 68
4.9 Boundaries 74
4.10 Surface Conforming Boundaries 74
4.11 Frequency-Dependent Absorbing Boundaries 77
4.12 Open-Boundary Descriptions 80
4.13 Absorption within a PML Region 84
4.14 Conclusion 85
Chapter 5 TLM Modeling of Thermal and Particle Diffusion 87
5.1 Introduction 87
5.2 Spatial Discretizations and Electrical Networks
for Thermal and Particle Diffusion 88
5.3 TLM Algorithm for a 1-D Link-line Nodal Arrangement 90
5.4 1-D Link–Resistor Formulation 91
5.5 Boundaries 92
5.5.1 Insulating Boundary 92
5.5.2 Symmetry Boundary 92
5.5.3 Perfect Heat-Sink Boundary 93
5.5.4 Constant Temperature Boundaries 93
5.6 Temperature/Heat/Matter Excitation of the TLM Mesh 95
5.6.1 Constant T Boundary as an Input 95
5.6.2 Single Shot Injection into Bulk Material 96
5.7 Flux Injection into Bulk Material 100

5.7.1 Single Heat Source 100
5.8 Multiple Flux Sources 101
5.9 The Extension to Two and Three Dimensions 102
5.9.1 Link-Line Formulations 102
5.9.2 Link-Resistor Formulations 104
5.10 Non-Uniformities in Mesh and Material Properties 106
5.11 Stubs and the Avoidance of Internodal Reflections 111
5.12 Time-Step Variation 114
5.13 Some Aspects of the Theory of Lossy TLM 117
5.13.1 TLM and Finite Difference Formulations
for the Telegrapher’s and Diffusion Equations 117
5.13.2 Anomalous “Jumps-To-Zero” In Link-Line TLM 121
5.13.3 TLM Diffusion Models as Binary Scattering Processes 126
5.13.4 Mesh Decimation 128
5.14 The Statistics of TLM Diffusion Models 130
5.15 TLM and Analytical Solutions of the Laplace Equation 132
TF1745_book.fm Page x Monday, September 12, 2005 11:56 AM
5.15.1 Solution of the Diffusion Equation
with Fixed-Value Boundaries 132
5.15.2 Solution of the Telegrapher’s Equation
with Fixed-Value Boundaries 133
Chapter 6 TLM Models of Elastic Solids 137
6.1 The Behavior of Elastic Materials 137
6.2 The Analogy between TLM and State Space Control Theory 140
6.3 Nodal Structure for Modeling Elastic Behavior 143
6.4 Implementation 149
6.5 Boundaries 152
6.6 Force Boundaries 153
6.7 Conclusion 157
Chapter 7 Simple TLM Deformation Models 159

7.1 Introduction 159
7.2 Review of the Behavior of Materials 159
7.3 Trouton’s Descending Fluid and a TLM Treatment
of a Vertically Supported Column 161
7.4 A Model of Viscous Bending 165
7.5 Numerical Issues and Model Convergence 169
7.6 TLM Models of Viscoelastic Deformation 170
7.6.1 The Parallel Viscoelastic Model 170
7.7 Conclusion 173
Chapter 8 TLM Modeling of Hydraulic Systems 177
8.1 Introduction 177
8.2 Symbols, Analogues, and Parameters 178
8.3 Compressional Waves in Fluids 181
8.4 A Transmission Line Analysis of Fluid Flow 181
8.5 Time-Domain Transmission Line Models of Fluid Systems 183
8.6 Transients in Elastic Pipes 193
8.7 Open-Channel Hydraulics 196
8.8 Conclusions 198
Chapter 9 Application of TLM to Computational Fluid Mechanics 203
9.1 Introduction 203
9.2 Viscosity 204
9.3 Viscosity in the TLM Algorithm 205
9.4 Results 206
9.5 Incompressible Fluids and Velocity Fields 207
9.6 Convective Acceleration and the TLM Model 208
9.7 Comments on the Procedure 211
9.8 Implementation Issues 212
TF1745_book.fm Page xi Monday, September 12, 2005 11:56 AM
Chapter 10 State of the Art Examples 213
10.1 Introduction 213

10.2 The Hanging Cable and Gantry Crane Problems 213
10.2.1 Hanging Cable: Analytical Analysis and Results 213
10.2.2 Hanging Cable: TLM Model 214
10.2.3 Gantry Crane: Results 215
10.3 The Modeling of Rigid Bodies Joined by Transmission Line Joints 216
10.4 Klein–Gordon Equation 220
10.5 Acoustic Propagation and Scattering (Two-Dimensions) 223
10.6 Condenser Microphone Model 225
10.7 Propagation in Polar Meshes 226
10.8 Acoustic Propagation in Complex Ducts (A 3-D TLM Model) 227
10.9 A 3-D Symmetrical Condensed TLM Node
for Acoustic Propagation 229
10.10 Waves in Moving Media 233
10.11 Some Recent Developments in TLM Modeling of Doppler Effect 235
10.12 Simulation of a Thermal Environment for Chilled Foods
during Transport: An Example of Three-Dimensional Thermal
Diffusion with Phase-Change 237
10.12.1 Recent Advances in Inverse Thermal Modeling using TLM 239
10.12.2 Inverse scattering 239
10.12.3 Amplification Factor 241
10.12.4 TLM and Spatio-Temporal Patterns — The Present
and the Future 242
10.12.5 TLM and Diffusion Waves 246
10.12.6 The Logistic Equation in the Presence of Diffusion 248
Index 257
TF1745_book.fm Page xii Monday, September 12, 2005 11:56 AM
1
chapter one
Introduction
The simulation of physical phenomena has been much simplified and

extended by the use of numerical methods, which avoid limitations and
simplifying assumptions frequently inherent in analytical solutions of math-
ematical representations. There are many ways in which this can be done.
The equations can be solved by replacing integrals and derivatives by finite
sums and finite differences. An alternative strategy involves the replacement
of the equations by analogue models, which express the same behavior, on
the basis that these may be easier to solve numerically in particular circum-
stances. Perhaps the best-known example is the equivalent electrical net-
work. The use of electrical network models in mechanics is well established.
There are direct analogues between springs, masses, and dampers on one
side and capacitors, inductors, and resistors on the other. The solution to the
mechanical problem can then be obtained using conventional circuit analysis
techniques with results in either the time or frequency domains. As will be
seen, in the case of transmission line matrix (TLM), the equivalent electrical
analogue has the further major advantage that it leads directly to a simple
and natural numerical discretization scheme.
There is a relatively new time-domain modeling technique, called cellu-
lar automaton (CA) modeling. Particles, which may represent, for example,
concentration, amplitude, or population of a species are distributed on a
mesh, which, in two dimensions, may be a Cartesian or hexagonal grid.
These are then subjected to the repeated application of a simple set of rules
and the evolving behavior is monitored. With the right set of rules it may
be possible to define a CA system whose behavior closely parallels that of
the physical problem of interest. In many instances the set of rules may
appear to have no obvious physical basis and, perhaps because of this,
researchers in this area have worked hard at providing a good theoretical
foundation for their subject.
This book is concerned with the application of the TLM numerical mod-
eling method to a range of problems in mechanics. If we take the view point
from which TLM originates, then the approach is as follows: an electrical

network whose behavior closely mimics the physical problem is constructed,
TF1745_book.fm Page 1 Monday, September 12, 2005 11:56 AM
2 Transmission Line Matrix in Computation Mechanics
based mainly on a network (or mesh or matrix) of transmission lines. The
behavior of transmission lines is well understood and fully described in
electromagnetic theory. Their most important property in this context is the
introduction of a time-delay for signals travelling between points in the
electrical network. The distribution of mesh points in the modeled space
provides the spatial discretization of the problem while the time delays in
the transmission lines provides the spatial discretization. Solution of the
network analogue is then achieved by the repeated application of a set of
relatively simple rules. Thus TLM could be considered as a form of CA
modeling, where the transition rules are determined by the laws of electro-
magnetics.
All numerical techniques involve discretization. In most traditional
approaches the physics is first modeled as a differential or integral equation,
with continuous variables, and then this model is again modeled (or solved)
by a numerical scheme. The final numerical solution is therefore twice
removed from the physical problem and approximations are introduced at
both modeling stages. By contrast, an important and powerful feature of
TLM is that all the required discretization is inherent in the initial model,
which is then solved without any further approximation. All the required
discretization happens in the first modeling stage, which is strongly based
on the physics. This ensures that TLM avoids many of the anomalous effects
that can arise in traditional methods, and the physical implications of dis-
cretization and of the model are easier to identify.
This point is worth emphasizing. The existence of two modeling stages
in traditional methods is frequently overlooked. It has the transparency of
the over familiar. For example, in textbooks on numerical methods, generally,
analytical solutions of the corresponding differential equations are taken as

“exact,” forgetting that the differential equation and its solution are in turn
approximations to the physics. There are examples where “perfect” analyt-
ical solutions to differential equations with boundary conditions can suggest
physically impossible behavior. A simple example is the solution of the
diffusion equation with boundary values imposed at some initial instant:
the exact analytical solution suggests infinite diffusion speeds as the diffu-
sion time approaches zero, which clearly cannot happen physically. With the
TLM solution, such anomalies are avoided.
TLM has a clearly defined birth date: 1971, the publication of the pio-
neering paper by Johns and Beurle.
1
But the roots go back long before that.
While working at EEV Ltd., Chelmsford, U.K., Raymond Beurle (later Head
of the Department of Electrical and Electronic Engineering at Nottingham
University) identified a specific need to express electromagnetic phenomena
in the time domain. He had used an early computer to simulate the propa-
gation of activity in neural networks and later had experience using South-
well's relaxation technique
2
to solve electrostatic field problems. These two
apparently unrelated themes coalesced to suggest that propagation in a
matrix of transmission lines might be used to simulate propagation in space,
TF1745_book.fm Page 2 Monday, September 12, 2005 11:56 AM
Chapter one: Introduction 3
in order to enable high frequency field distributions to be calculated in
arbitrarily shaped cavities.
A TLM matrix was deliberately chosen in preference to a network of
finite inductive and capacitative (L and C) elements because it so greatly
simplified the theory regarding the interaction between a short pulse of
voltage (or current) and each node. Another advantage was that a finite

amount of energy introduced at a source in the matrix could not increase,
and the calculation was therefore unconditionally stable, thus avoiding a
problem that has been encountered with some other methods of calculation.
After a small trial confirmed propagation and reflection at a boundary
in TLM, the idea was suggested to a postgraduate student who subsequently
reported confirming this with a computer simulation. Some time later Peter
Johns, a microwave engineer at the Post Office Telecommunications Research
Laboratory at Dollis Hill, London, was appointed as a lecturer at Notting-
ham. He asked Beurle to suggest a research topic, and as no mathematically
minded postgraduate student had come forward to take this topic Beurle
felt (rightly as it transpired) that this would be a good way of launching
TLM. Events proved that this was indeed so. Peter Johns took up the idea
with an enthusiasm that became legendary. The details of the method were
first published in 1971
1
and Beurle was asked to co-author this first paper
as an acknowledgment of the source of the idea.
The approach is based entirely upon establishing an analogue between
a space- and time-dependent physical problem and an electrical network.
This in itself has a long tradition in modeling and simulation. Johns claimed
that he derived inspiration from the work of Kron who first proposed the
use of electrical network analogues for the electromagnetic equations
3,4
in
the mid 1940s. Such concepts have been further developed by Vine
5
and by
Hammond and Sykulski.
6
There were two novel aspects to the approach that

Johns used. As mentioned earlier, the first was the inclusion of lengths of
transmission line, which imposed an inherent time-delay in the propagation
of information.
It is interesting to note that such a concept was being developed else-
where at about the same time. However, as there was a different starting
point, this led to quite a different formulation. Ivor Catt working for Motor-
ola in the United States in the middle 1960s was particularly concerned with
cross-talk between interconnects in high-speed integrated circuits
7
. There
were many problems for which there were no satisfactory answers, but
back-plane technology in computer-boards led him to think in terms of a
particular type of guided electromagnetic wave, termed a TEM wave. A
regular rectangular mesh interconnect looks very much like a two-dimen-
sional shunt TLM mesh.
There was a realization in the mid 1970s that a capacitor was in fact a
transmission line, and Catt's work shows networks comprising lumped
series inductors and shunt transmission lines. Johns, on the other hand,
drawing on concepts from microwaves, conceived the use of an open-circuit,
half-length stub as an approximation of a capacitor. Johns also demonstrated
TF1745_book.fm Page 3 Monday, September 12, 2005 11:56 AM
4 Transmission Line Matrix in Computation Mechanics
that the short-circuit half-length stub represented an inductor. The explicit
use of these stubs to represent reactive components in discrete electrical
networks was first suggested by Johns and O'Brien
8
and has been consider-
ably extended by Hui and Christopoulos
9
.

The method of excitation that Catt used may also explain why the tech-
nique did not advance in the way that TLM has done. Catt, attempting to
bypass what he felt were erroneous interpretations, based everything on
those concepts first proposed by Heaviside. The price that must be paid for
this is computational complexity as the treatment is distributed in space.
Nevertheless, his formulations of propagating TEM waves involve a network
that looks identical to a two-dimensional series TLM mesh.
Johns’ second innovation was the use of Dirac impulse excitation. Such
an entity, sometimes called a delta pulse, occupies zero time, so that as it travels
on a transmission line, it is influenced by nothing except its immediate sur-
roundings. The external observer is unaware of its presence until the precise
moment of arrival at the point of observation, and once it has passed, it
disappears from sight. In the Johns approach a Heaviside excitation is merely
a stream of independent impulses separated by intervals of ∆t. The represen-
tation of a wave-form as a stream of Dirac impulses would not have seemed
so obvious in 1971 as it does now, when digital signal processing has largely
displaced analogue signal processing. The adoption of this concept means that
the information contained within a stream of impulses is localized in space at
any time, so that nonlocal interactions need not be considered.
Johns came from microwave electromagnetics, and even today the tech-
niques of TLM owe much to his legacy. Catt, coming from more conventional
electromagnetics, continues to raise questions,
10,11
which are only beginning
to be addressed as a result of an increased understanding of the processes
that govern electromagnetic compatibility (EMC). His written works reflect
an element of frustration at the lack of an attentive audience. Nevertheless
any student of time-domain electromagnetics would benefit from consulting
his works.
So, why would someone wish to undertake research in TLM? The

response to this depends on where you are standing. When it came on the
electromagnetic scene, it was like nothing that had existed before. Johns had
contracts with many defence research bodies in the U.K. and the effort of
visiting seven U.S. government research establishments during five days was
probably a major factor contributing to his second and final heart attack.
Finite element and other numerical techniques have now entered the niche
market once occupied by TLM, but an inspection of back issues of the
International Journal of Numerical Modeling (published by John Wiley) will
confirm that electromagnetic applications remain a vibrant research area.
Two of the three authors of this book worked with Johns in the appli-
cation of TLM to heat and mass diffusion. Both were fascinated by his
ingenuity and were spurred on by his encouragement. There were areas
where TLM fared better than the equivalent finite difference formulations,
and there were areas where it did not. The investigation of the properties of
TF1745_book.fm Page 4 Monday, September 12, 2005 11:56 AM
Chapter one: Introduction 5
TLM algorithms and the limits of their applications started to drive research.
The fact that TLM provided a method of solving complex problems without
recourse to obfuscating mathematics became an interest in itself, which was
consistent with the original modeling philosophy of Johns: the modeler,
being in control right up to the point of delivery of the result, is in a better
position to judge the effects of assumptions, rounding errors, etc. Unlike
some other approaches, the algorithms are not difficult to understand or
apply and more often than not, researchers develop their own software,
rather than purchase proprietary packages.
There has also been an accelerating convergence with the broader topic
of CA modeling.
12,13
There was a time when purists would have criticized
CA techniques for their lack of rigor. At least the scattering rules of TLM

can claim a firm basis in electromagnetic theory and, in the meantime, we
remain fascinated with what we continue to discover in this productive
research area.
Just as we have attempted to address the question “why TLM research?,”
we might also be asked to respond to the question “why a book on the
application of TLM to computational mechanics?.” These authors currently
work in university departments of computer science, mechanical engineer-
ing, and electronic engineering respectively. All are aware of the cross-dis-
ciplinary nature of the subject and the extent to which their current work is
of relevance to mechanical engineers. They are also aware that existing
introductions to the subject start with the electromagnetic foundations in a
way that assumes much prior knowledge and uses a strange language. There
is therefore a steep learning curve, which is frequently a problem for those
wishing to break into the subject.
Both the name TLM and the usual practice of deriving TLM algorithms
from circuit theory have long inhibited a wider understanding and use of
the method. The underlying process involves the scattering and propagation
of impulses, so that a name like IPS (impulse propagation and scattering) would
be more generic and more descriptive of the technique, and perhaps more
“user-friendly” to people without electrical engineering backgrounds. Nev-
ertheless, for the purposes of this book we will stick with what is established.
It is the authors’ contention that the method should take its place along-
side such generic numerical modeling techniques as finite element, finite
difference, boundary element, and cellular automata approaches. Certain
important features make it merit this honor, and one of the purposes of this
book is to show how the method can be adapted to a very wide range of
important problems in physics. Our guiding philosophy within this text will
be to introduce concepts, bring the reader up to speed in a number of areas,
and provide pointers to references that provide more extensive coverage to
specific topics. Rather in the manner of

14
we have summarized these
in a table that provides some idea of the range. To stay within reasonable
page bounds, we will omit extensive coverage of the topics that are shown
in bold, and concentrate on those shown in italics. Those that are in plain
type remain as challenges for the future.
Kranys
v
TF1745_book.fm Page 5 Monday, September 12, 2005 11:56 AM
6 Transmission Line Matrix in Computation Mechanics
Thus, we will start with a treatment for one-dimensional TLM based
entirely on mechanical engineering concepts (Chapter 2). The pace will be
quite brisk and will by the chapter-end consider some advanced problems.
In Chapter 3 we will revisit much of the same material, but this time from
the point of view of the more conventional electrical engineering approach.
This will start by assuming little or no background knowledge and will
progress somewhat more slowly.
Readers who are familiar with one or other or both concepts may wish
to skip the appropriate sections. Others may find it useful to become accus-
tomed to the electromagnetics-based syntax, which is used elsewhere in the
book. The fourth chapter is concerned with acoustics and acoustic propaga-
tion models, which use a large part of the theory of the previous chapters.
It will also have a tutorial component, at least at the start, when several of
the problems will be demonstrated using computer code based on the com-
mercial modeling language MATLAB
®
. The tone of the chapters then changes
from the application of general principles to the description of the latest
research in a range of areas (modeling of heat and mass transfer is of par-
ticular importance and is discussed in Chapter 6). Chapter 5 covers models

PDE
*
Equation
Wave equation
15
u
tt
– c
2
∇
2
u = 0
Telegrapher’s: damped wave
α
u
tt
+
β
u
t
+
γ
u – c u
xx
= 0
Forced wave equation u
tt
– c
2
u

xx
= f(x, t, u, u
t
, u
x
, …)
Klein–Gordon u
tt
– c
2
u
xx
+ hu = 0
Sine–Gordon u
tt
– u
xx
+ sin u = 0
Heat/diffusion with source
16
u
t
– a
2
u
xx
= f(x, t)
Moving threadline u
tt
+

α
u
xt
+
β
u
xx
= 0
Rotating string u
tt
= c
2
[(l
2
– x
2
) u
x
]
x
Hanging cable u
tt
= g(x u
xx
+ u
t
)
Laplace
16
∇

2
u = 0
Poisson
16
∇
2
u = f
Helmholtz
17
∇
2
u +
λ
u = 0
Schrödinger (time indep.) ∇
2
u +
α[
E – V(x,y,z)]u = 0
Beam (biharmonic wave) ∇
4
u + (1/p
2
) u
tt
= 0
Stretched, stiff string ∇
4
u – ∇
2

u + (1/p
2
) u
tt
= 0
Biharmonic static ∇
4
u = 0
Euler’s fluid mechanics
ρ
(u
t
+ u.∇u) =
ρ
f – ∇p
Navier Stokes (for incompressible fluids)
ρ
(u
t
+ u.∇u) =
ρ
f – ∇p +
µ
∇
2
u
*
PDE = partial differential equation
TF1745_book.fm Page 6 Monday, September 12, 2005 11:56 AM
Chapter one: Introduction 7

of stress-wave propagation in two and three dimensions. This is particularly
interesting because it demonstrates a technique of dealing with what was
perceived as being a major difficulty with TLM modeling of mechanical
systems, namely the lack of cross-terms in the derivatives of the fundamental
equations (e.g., ). Chapter 7 describes work on simple models for
flow and bending and indicates the extent to which shortcomings due to
lack of cross-derivatives can be circumvented. The next two chapters deal
with fluids. Chapter 8 outlines the current state of work on the application
of TLM to hydraulic systems. There is a significant difference in the language
used by different authors, and we attempt to overcome any interpretative
problems by presenting the concepts in a unified format. This is followed
by an outline of the inroads which TLM has made in the area of computa-
tional fluid dynamics, and the work concludes with a chapter outlining some
state-of-the-art examples.
References
1. Johns P. B. and Beurle R. L. Numerical solution of 2-dimensional scattering
problems using a transmission line matrix, Proceedings IEE, 118 (1971)
1203–1208.
2. Southwell R. V., Relaxation Methods in Engineering Science, Oxford University
Press, Oxford, U.K. (1940).
3. Kron G., Equivalent circuits to represent the electromagnetic field equations,
Phys. Rev., 64 (1943) 126–128.
4. Kron G., Equivalent circuits to the field equations of Maxwell, Proceedings
IRE, 32 (1944) 289–298.
5. Vine J., Impedance networks, in Field Analysis; Experimental and Computation,
Vitkovitch, D., Ed., Van Nostrand, London (1966).
6. Hammond P. and Sykulski J., Engineering Electromagnetism; Physical Processes
and Computation, Oxford Science Publications, Oxford (1994).
7. Catt I., Crosstalk (noise) in digital systems, IEEE Trans. Elect. Comp., EC-16
(1967) 743–763.

8. Johns P. B. and O'Brien M., The use of the transmission line matrix method
to solve non-linear lumped networks, The Radio and Electrical Engineer, 50
(1980) 59–70.
9. Hui S. Y. R. and Christopoulos C., The modeling of networks with frequently
changing topology whilst maintaining a constant system matrix, Int. J. Nu-
merical Modelling, 3 (1990) 11–21.
10. Catt I., The Catt Anomaly: Science Beyond the Crossroads, Westfields Press, West-
fields, U.K. (1996).
11. Catt I., Electromagnetism I, Westfields Press, Westfields, U.K. (1994).
12. Enders P. and de Cogan D., TLM for diffusion: the artefact of the standard
initial conditions and its elimination with an abstract TLM suite, Int. J.
Numerical Modelling, 14 (2001) 107–114.
13. Chopard B. and Droz M., Cellular Automata Modelling of Physical Systems,
Cambridge University Press, London, New York (1998).
14. M., Causal theories of evolution and wave propagation in mathe-
matical physics, Appl. Mech. Rev., 42 (1989) 305–322.
∂∂∂
2
φ/ xy
Kranys
v
Kran
y
s
v
TF1745_book.fm Page 7 Monday, September 12, 2005 11:56 AM
8 Transmission Line Matrix in Computation Mechanics
15. Christopoulos C., The Transmission Line Modeling Method, Oxford University
Press/IEEE Press, Oxford, U.K.,(1995).
16. de Cogan D., Transmission Line Matrix (TLM) Techniques for Diffusion Applica-

tions, Gordon and Breach (1998).
17. Clune F., M.Eng.Sc thesis, University College Dublin (Ireland).
TF1745_book.fm Page 8 Monday, September 12, 2005 11:56 AM
9
chapter two
TLM and the 1-D Wave
Equation
2.1 Introduction
This chapter is intended to be introductory for those unfamiliar with TLM,
and expansive for those knowing about TLM only in electromagnetics.
Emphasis will be more on opening up possibilities rather than on full math-
ematical rigor, and a step-by-step approach will be taken. In keeping with
the desire to make the ideas more accessible to nonelectrical engineers,
analogies with circuit theory will be avoided as they are not necessary and
not very helpful to those unfamiliar with electrical engineering (EE)
concepts.
Readers who would prefer the traditional TLM presentation (whether
they are electrical engineers or not), or would like to review it in conjunction
with the approaches presented here, may proceed directly to the next chapter
or should refer to the considerable volume of literature now available in
both journal papers and in textbooks. The present book is intended to fill a
gap not already covered in this literature.
A good place to start in TLM is modeling the one-dimensional wave
equation. In one dimension (1-D), the entire workings of the TLM algorithm
are simple and easy to visualize, yet the model remains powerful, flexible,
and elegant, and applicable to many interesting physical problems. Further-
more, many of the issues that will arise later in two- and three-dimension
(2-D and 3-D) TLM are encountered in the 1-D model in an easily compre-
hensible form.
TF1745_book.fm Page 9 Monday, September 12, 2005 11:56 AM

10 Transmission Line Matrix in Computation Mechanics
2.2 The Vibrating String
Perhaps the simplest 1-D wave equation is that of a vibrating string, namely
(2.1)
where the wave speed, c, is
(2.2)
T is the tension of the string in Newtons, and
ρ
is the linear mass density in
kg/m. In this equation, x is distance along the string, and y is the departure
of the string from the neutral or stationary position, both in meters.
It is easy to show that solutions to Equation (2.1) take the form of
arbitrary disturbances f(x) and g(x), which propagate to the right and left
without changing their shape, at a constant speed, c. Mathematically, this is
expressed as
y(x,t) = f(x – ct) + g(x + ct) (2.3)
To visualize what is happening in Equation (2.3), it is clear that at any
given value of x, say x = 0, the displacement is varying with time. Then, by
imagining time to be frozen, say at t = 0, it is clear that f and g give the
shapes of two “disturbances” in y as a function of the space variable x. Now
imagine time to advance by an amount corresponding to ct. The same shape
of f that was seen at t = 0 will now be seen at some larger value of x at the
point where x – ct takes on its original value (of 0, in this case). In other
words, the f shape is moving rightwards, by an amount x = ct in time t. That
is, the wave speed is c. Similarly, the g shape moves leftwards at the same
speed.
The functions f and g are often assumed to be sinusoidal, but almost any
continuous function, periodic or not, will propagate perfectly. Furthermore,
waves can superpose on each other to form new shapes. A particularly
curious feature is that two arbitrary, counter-propagating waves (in other

words, going in opposite directions) can pass through each other without
affecting each other in the slightest. Even though each wave is “disturbing”
the same section of the same string, each acts as if it had the string completely
to itself, undisturbed by the other.
Now imagine one wave, of shape f(x),
passing by a particular point x.
It will cause the string to have a velocity u = y/t, in the direction normal to
the string’s length. This velocity will depend both on the shape of f, and how
quickly this shape is passing the particular point. In fact
∂
∂
=
∂
∂
2
2
2
2
2
y
t
c
y
x
c=
T
ρ
TF1745_book.fm Page 10 Monday, September 12, 2005 11:56 AM
Chapter two: TLM and the 1-D Wave Equation 11
(2.4)

In other words, if at some point f has a negative slope with respect to x,
as this shape moves to the right at speed c it will produce a positive velocity
in the string. This first order Equation (2.4) can be taken as a more funda-
mental “wave equation” than the second order Equation (2.1) that normally
bears the name. This local velocity u should be clearly distinguished from
the wave speed c. It is the physical velocity of an element of the string in
the direction normal to the string’s length. By contrast no material moves at
the wave speed c, but only the wave shape and associated energy and
momentum.
Waves can be started, maintained, or stopped in various ways. These
possibilities correspond to different “initial conditions” or “forcing func-
tions” in a model. As far as propagation is concerned, real string will be of
finite length, and sooner or later waves will reach an end point. Typical
boundaries are fixed or free. More complex are boundaries that move, either
as a reaction to the arriving wave, or because they are driven externally, or
perhaps due to a combination of these effects. So the model must also be of
finite length (obviously necessary in any case for computational reasons),
and model “boundary conditions” must be established, which simulate the
physical boundary in an appropriate way.
As Equation (2.1) is probably the most commonly derived wave equa-
tion, the derivation will be skipped here. It is however worth making explicit
the assumptions behind it: that the string is continuous, uniform, and per-
fectly flexible; that the tension is constant in space and time; that gravity
effects are negligible; that departures from the equilibrium position are not
large; that the string’s linear density is constant; and that there is neither
internal nor external damping.
Frequently these assumptions are reasonable, but not always. Neverthe-
less, for the moment, their validity will be assumed. They ensure the linear,
nondispersive behavior described above with reference to Equation (2.3).
2.3 A Simple TLM Model

Without even mentioning a “transmission line” or an “electrical circuit,” an
intuitive yet very useful TLM model of the vibrating string can now be set
up. Figure 2.1 shows such a model at two successive time steps. It is divided
into a number of sections, of length
∆
l. Impulses, or narrow pulses, are
imagined to travel along the string, moving a distance
∆
l in each time step
∆
t. These pulses can be considered as samples of the modeled wave in the
string, with the profile of a stream of pulses corresponding to the wave shape
in space. By making
∆l /∆t = c (2.5)
∂
∂
=−
∂
∂
y
t
c
f
t
TF1745_book.fm Page 11 Monday, September 12, 2005 11:56 AM
12 Transmission Line Matrix in Computation Mechanics
the wave advances at the correct wave speed. A counter-propagating wave
can be added, if required, simply by adding a second stream of pulses, which
go leftwards by
∆

l at each time increment
∆
t.
Regarding the choice of the value of
∆
l, in principle it can be set as fine
as one wishes. The price to pay for finer space increments is a greater
computation load, increased memory requirements, and longer run times.
In so far as this may be an issue, the modeler chooses a value for
∆
l that is
sufficiently fine to capture the detail of interest, yet sufficiently coarse to
keep the computational load acceptable.
If the wave shape is changing smoothly in space, not many “sample”
points are needed, whereas a rapidly changing wave clearly requires a
greater density of pulses to capture the details of the shape. If necessary,
Shannon’s sampling theorem can be used to determine exactly how fine the
pulse separation should be to model a particular wave shape. In other words,
more than two sample pulses are required to fall within the shortest wave-
length component of interest in the modeled waveform. This determines
exactly how coarse the model (or how large
∆
l) can be for safety while
minimizing the computation load.
Once
∆
l has been decided, the wave speed c in Equation (2.3) gives the
value of
∆
t from Equation (2.5). Thus the discretization of space and time,

necessary for all numerical modeling techniques, is established. After this,
as the model runs it preserves all the details exactly. There is no dispersion
or other corruption of the waveform with time or over space. For example,
if a waveform is launched at one end of a string, by injecting a stream of
pulses over successive time increments whose envelope is the desired wave-
form, then exactly the same pulse sequence will arrive at the far end, exactly
Figure 2.1 A 1-D model of two arbitrary, counter-propagating waves at successive
time steps, with impulses shown just leaving the nodes on the line.
TF1745_book.fm Page 12 Monday, September 12, 2005 11:56 AM