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

Inaddition to both analytical and numerical analysis of magnetic fields andforces, his research interests include novel numerical modeling methods andapplications, especially in acoustic

Donard de Cogan William J O’Connor Susan Pulko

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

10 9 8 7 6 5 4 3 2 1

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 (http://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

Taylor & Francis Group

is the Academic Division of Informa plc.

Those who knew Peter Johns* speak glowingly of his inspiration and hisenthusiasm He achieved so much, and we are certain that he could haveachieved much more had he lived He was already moving into mechanicalapplications of TLM and was discussing nonlinear processes such as theaction of a violin bow on a string Shortly after his first heart attack hecommenced work on a TLM model of electromechanical interactions in heartmuscle He was a cohesive factor in all areas of development, which in hisabsence have tended toward a bimodal partition: TLM applications that arerelated to electromagnetics and TLM applications that are not Within thelatter grouping, the contributions of Peter Enders, Xiang Gui, and the lateAdnan Saleh have been crucial We also wish to acknowledge the contribu-tion of the many TLM researchers who have been happy to share theirexperiences freely at various workshops and colloquia and by personalcommunication There have also been the behind-the-scenes contributions

of research students and assistants such as Dorian Hindmarsh and MikeMorton We have benefited greatly by the many constructive comments fromspecialists such as Kevin Edge (Fluid Power Centre, University of Bath),Petter Krus (Division of Fluid Power Technology, Linköping University), andRichard Pearson (Power Train Division, Lotus Cars, Hethel, Norfolk, U.K.).Many thanks to James Flint for some last minute comments on Dopplermodeling Finally, there are our editors Without the input of Donald Degen-hardt this book would never have passed the initial planning stages JanieWardle has overseen the transition between publishers** and our progresstoward completion And finally, Sylvia Wood of Taylor & Francis, who, inspite of everything, brought it all together We are most grateful to them fortheir 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.

About the Authors

Donard de Cogan gained a bachelor’s degree in physical chemistry and aPh.D in solid state physics from Trinity College, Dublin He undertookresearch fellowships in solid state chemistry (University of Nijmegen, Neth-erlands) and microelectronic fabrication (University of Birmingham) beforejoining Philips as a senior development engineer in power electronic semi-conductors In 1978 he was appointed a lecturer in electrical and electronicengineering at the University of Nottingham His initial research was con-cerned with the overload impulse withstand capability of a range of electricaland electronic components, and the results confirmed a requirement fornumerical simulation He was encouraged to use the transmission line matrix(TLM) technique, which had been invented at Nottingham, and this soonbecame his principal line of research In 1989 he was appointed a seniorlecturer 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 andBreach (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 inUCD, 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 thecountry, enjoying a worldwide reputation for teaching and research) Inaddition to both analytical and numerical analysis of magnetic fields andforces, his research interests include novel numerical modeling methods andapplications, especially in acoustics, mechanical-acoustic systems, and fluids;development of transmission line matrix and impulse propagation numericalmethods; 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

Susan Pulko graduated from Imperial College, University of London in

1977 She moved to the University of Nottingham and undertook uate work in solid state physics in the Department of Electrical and ElectronicEngineering 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 foruse in thermal applications Dr Pulko later took up a lectureship in theDepartment of Electronic Engineering at the University of Hull, where sheestablished a TLM research group This group continued the development

postgrad-of TLM for thermal problems and applied it in a range postgrad-of industries fromceramics to food It was while the group was working with the ceramicsindustry that the desirability of modeling deformation processes by TLMbecame apparent The modeling of propagating stress waves took place fromthis point and has been applied to the modeling of ultrasound wave prop-agation in solids; current work in this area is concerned with modelingmagnetostrictive behavior She is a consultant to Feonic plc

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

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

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

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

of the equations by analogue models, which express the same behavior, onthe 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 oneside and capacitors, inductors, and resistors on the other The solution to themechanical problem can then be obtained using conventional circuit analysistechniques with results in either the time or frequency domains As will beseen, in the case of transmission line matrix (TLM), the equivalent electricalanalogue has the further major advantage that it leads directly to a simpleand natural numerical discretization scheme.

There is a relatively new time-domain modeling technique, called lar automaton (CA) modeling Particles, which may represent, for example,concentration, amplitude, or population of a species are distributed on amesh, which, in two dimensions, may be a Cartesian or hexagonal grid.These are then subjected to the repeated application of a simple set of rulesand the evolving behavior is monitored With the right set of rules it may

cellu-be possible to define a CA system whose cellu-behavior closely parallels that ofthe physical problem of interest In many instances the set of rules mayappear to have no obvious physical basis and, perhaps because of this,researchers in this area have worked hard at providing a good theoreticalfoundation for their subject

This book is concerned with the application of the TLM numerical eling method to a range of problems in mechanics If we take the view pointfrom which TLM originates, then the approach is as follows: an electricalnetwork whose behavior closely mimics the physical problem is constructed,

mod-based mainly on a network (or mesh or matrix) of transmission lines The

behavior of transmission lines is well understood and fully described inelectromagnetic theory Their most important property in this context is theintroduction of a time-delay for signals travelling between points in theelectrical network The distribution of mesh points in the modeled spaceprovides the spatial discretization of the problem while the time delays inthe transmission lines provides the spatial discretization Solution of thenetwork analogue is then achieved by the repeated application of a set ofrelatively simple rules Thus TLM could be considered as a form of CAmodeling, where the transition rules are determined by the laws of electro-magnetics

All numerical techniques involve discretization In most traditionalapproaches 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 twiceremoved from the physical problem and approximations are introduced atboth modeling stages By contrast, an important and powerful feature ofTLM is that all the required discretization is inherent in the initial model,which is then solved without any further approximation All the requireddiscretization happens in the first modeling stage, which is strongly based

on the physics This ensures that TLM avoids many of the anomalous effectsthat 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 ofthe 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 turnapproximations to the physics There are examples where “perfect” analyt-ical solutions to differential equations with boundary conditions can suggestphysically impossible behavior A simple example is the solution of thediffusion 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 theTLM solution, such anomalies are avoided

TLM has a clearly defined birth date: 1971, the publication of the 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

pio-of the Department pio-of Electrical and Electronic Engineering at NottinghamUniversity) identified a specific need to express electromagnetic phenomena

in the time domain He had used an early computer to simulate the gation of activity in neural networks and later had experience using South-well's relaxation technique2 to solve electrostatic field problems These twoapparently unrelated themes coalesced to suggest that propagation in amatrix of transmission lines might be used to simulate propagation in space,

propa-in order to enable high frequency field distributions to be calculated propa-inarbitrarily shaped cavities.

A TLM matrix was deliberately chosen in preference to a network offinite inductive and capacitative (L and C) elements because it so greatlysimplified the theory regarding the interaction between a short pulse ofvoltage (or current) and each node Another advantage was that a finiteamount of energy introduced at a source in the matrix could not increase,and the calculation was therefore unconditionally stable, thus avoiding aproblem 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 subsequentlyreported confirming this with a computer simulation Some time later PeterJohns, a microwave engineer at the Post Office Telecommunications ResearchLaboratory at Dollis Hill, London, was appointed as a lecturer at Notting-ham He asked Beurle to suggest a research topic, and as no mathematicallyminded postgraduate student had come forward to take this topic Beurlefelt (rightly as it transpired) that this would be a good way of launchingTLM Events proved that this was indeed so Peter Johns took up the ideawith an enthusiasm that became legendary The details of the method werefirst published in 19711 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 claimedthat he derived inspiration from the work of Kron who first proposed theuse of electrical network analogues for the electromagnetic equations3,4 inthe mid 1940s Such concepts have been further developed by Vine5 and byHammond and Sykulski.6 There were two novel aspects to the approach thatJohns used As mentioned earlier, the first was the inclusion of lengths oftransmission line, which imposed an inherent time-delay in the propagation

of information

It is interesting to note that such a concept was being developed where at about the same time However, as there was a different startingpoint, this led to quite a different formulation Ivor Catt working for Motor-ola in the United States in the middle 1960s was particularly concerned withcross-talk between interconnects in high-speed integrated circuits7 Therewere many problems for which there were no satisfactory answers, butback-plane technology in computer-boards led him to think in terms of aparticular type of guided electromagnetic wave, termed a TEM wave Aregular rectangular mesh interconnect looks very much like a two-dimen-sional shunt TLM mesh

else-There was a realization in the mid 1970s that a capacitor was in fact atransmission line, and Catt's work shows networks comprising lumpedseries 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

that the short-circuit half-length stub represented an inductor The explicituse of these stubs to represent reactive components in discrete electricalnetworks was first suggested by Johns and O'Brien8 and has been consider-ably extended by Hui and Christopoulos9.

The method of excitation that Catt used may also explain why the nique did not advance in the way that TLM has done Catt, attempting tobypass what he felt were erroneous interpretations, based everything onthose concepts first proposed by Heaviside The price that must be paid forthis is computational complexity as the treatment is distributed in space.Nevertheless, his formulations of propagating TEM waves involve a network

tech-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 roundings The external observer is unaware of its presence until the precisemoment of arrival at the point of observation, and once it has passed, itdisappears from sight In the Johns approach a Heaviside excitation is merely

sur-a stresur-am of independent impulses sepsur-arsur-ated by intervsur-als of ∆t The tation of a wave-form as a stream of Dirac impulses would not have seemed

represen-so obvious in 1971 as it does now, when digital signal processing has largelydisplaced analogue signal processing The adoption of this concept means thatthe information contained within a stream of impulses is localized in space atany time, so that nonlocal interactions need not be considered

Johns came from microwave electromagnetics, and even today the niques of TLM owe much to his legacy Catt, coming from more conventionalelectromagnetics, continues to raise questions,10,11 which are only beginning

tech-to be addressed as a result of an increased understanding of the processesthat govern electromagnetic compatibility (EMC) His written works reflect

an element of frustration at the lack of an attentive audience Neverthelessany student of time-domain electromagnetics would benefit from consultinghis works

So, why would someone wish to undertake research in TLM? Theresponse to this depends on where you are standing When it came on theelectromagnetic scene, it was like nothing that had existed before Johns hadcontracts with many defence research bodies in the U.K and the effort ofvisiting seven U.S government research establishments during five days wasprobably a major factor contributing to his second and final heart attack.Finite element and other numerical techniques have now entered the nichemarket 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 hisingenuity and were spurred on by his encouragement There were areaswhere TLM fared better than the equivalent finite difference formulations,and there were areas where it did not The investigation of the properties of

TLM algorithms and the limits of their applications started to drive research.The fact that TLM provided a method of solving complex problems withoutrecourse to obfuscating mathematics became an interest in itself, which wasconsistent 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 betterposition to judge the effects of assumptions, rounding errors, etc Unlikesome other approaches, the algorithms are not difficult to understand orapply 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 TLMcan claim a firm basis in electromagnetic theory and, in the meantime, weremain fascinated with what we continue to discover in this productiveresearch 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 theapplication of TLM to computational mechanics?.” These authors currentlywork 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 existingintroductions to the subject start with the electromagnetic foundations in away that assumes much prior knowledge and uses a strange language There

is therefore a steep learning curve, which is frequently a problem for thosewishing to break into the subject

Both the name TLM and the usual practice of deriving TLM algorithmsfrom circuit theory have long inhibited a wider understanding and use ofthe 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 ertheless, for the purposes of this book we will stick with what is established

Nev-It is the authors’ contention that the method should take its place side such generic numerical modeling techniques as finite element, finitedifference, boundary element, and cellular automata approaches Certainimportant features make it merit this honor, and one of the purposes of thisbook is to show how the method can be adapted to a very wide range ofimportant problems in physics Our guiding philosophy within this text will

along-be to introduce concepts, bring the reader up to speed in a numalong-ber of areas,and provide pointers to references that provide more extensive coverage tospecific topics Rather in the manner of 14 we have summarized these

in a table that provides some idea of the range To stay within reasonablepage 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 plaintype remain as challenges for the future

Kranysv

Thus, we will start with a treatment for one-dimensional TLM basedentirely on mechanical engineering concepts (Chapter 2) The pace will bequite 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 fromthe point of view of the more conventional electrical engineering approach.This will start by assuming little or no background knowledge and willprogress 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 tomed to the electromagnetics-based syntax, which is used elsewhere in thebook The fourth chapter is concerned with acoustics and acoustic propaga-tion models, which use a large part of the theory of the previous chapters

accus-It will also have a tutorial component, at least at the start, when several ofthe problems will be demonstrated using computer code based on the com-mercial modeling language MATLAB® The tone of the chapters then changesfrom the application of general principles to the description of the latestresearch 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

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 utt – c2 uxx = f(x, t, u, ut, ux, …)

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∇2u

* PDE = partial differential equation

of stress-wave propagation in two and three dimensions This is particularlyinteresting because it demonstrates a technique of dealing with what wasperceived as being a major difficulty with TLM modeling of mechanicalsystems, namely the lack of cross-terms in the derivatives of the fundamentalequations (e.g., ) Chapter 7 describes work on simple models forflow and bending and indicates the extent to which shortcomings due tolack of cross-derivatives can be circumvented The next two chapters dealwith fluids Chapter 8 outlines the current state of work on the application

of TLM to hydraulic systems There is a significant difference in the languageused by different authors, and we attempt to overcome any interpretativeproblems 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 tional fluid dynamics, and the work concludes with a chapter outlining somestate-of-the-art examples

computa-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.

∂2φ/ x y∂ ∂

Kranysv

Kranysv

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)

Readers who would prefer the traditional TLM presentation (whetherthey are electrical engineers or not), or would like to review it in conjunctionwith the approaches presented here, may proceed directly to the next chapter

or should refer to the considerable volume of literature now available inboth journal papers and in textbooks The present book is intended to fill agap not already covered in this literature

A good place to start in TLM is modeling the one-dimensional waveequation In one dimension (1-D), the entire workings of the TLM algorithmare 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

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

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 particularlycurious feature is that two arbitrary, counter-propagating waves (in otherwords, going in opposite directions) can pass through each other withoutaffecting 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 22

y

y x

c = Tρ

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 mental “wave equation” than the second order Equation (2.1) that normally

funda-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 Thesepossibilities correspond to different “initial conditions” or “forcing func-tions” in a model As far as propagation is concerned, real string will be offinite length, and sooner or later waves will reach an end point Typicalboundaries are fixed or free More complex are boundaries that move, either

as a reaction to the arriving wave, or because they are driven externally, orperhaps due to a combination of these effects So the model must also be offinite length (obviously necessary in any case for computational reasons),and model “boundary conditions” must be established, which simulate thephysical boundary in an appropriate way

As Equation (2.1) is probably the most commonly derived wave tion, the derivation will be skipped here It is however worth making explicitthe assumptions behind it: that the string is continuous, uniform, and per-fectly flexible; that the tension is constant in space and time; that gravityeffects are negligible; that departures from the equilibrium position are notlarge; that the string’s linear density is constant; and that there is neitherinternal nor external damping

equa-Frequently these assumptions are reasonable, but not always less, for the moment, their validity will be assumed They ensure the linear,nondispersive behavior described above with reference to Equation (2.3)

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

the wave advances at the correct wave speed A counter-propagating wavecan 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 greatercomputation 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 tokeep 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 agreater density of pulses to capture the details of the shape If necessary,Shannon’s sampling theorem can be used to determine exactly how fine thepulse 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 determinesexactly 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 ofpulses over successive time increments whose envelope is the desired wave-form, then exactly the same pulse sequence will arrive at the far end, exactly

time steps, with impulses shown just leaving the nodes on the line

reproducing the launched waveform, and arriving at exactly the correct(modeled) time.

Computation is almost trivial Pulse magnitudes are stored as signedreal numbers and pulse positions as integers At each increment, all right-ward-going pulses are moved one position to the right, and all leftward tothe left It is as simple as that The total wave at any point is the sum of theleftward- and rightward-going pulses at the point

2.4 Boundary and Initial Conditions

Fixed or free boundaries are also easily modeled Imagine a “pulse reflector”placed in the middle of a string element, ∆l, so that a pulse leaving a point

at one time is reflected back to become incident on the same point at thenext time step If the boundary is fixed, the pulse reflected back into thestring is inverted (multiplied by –1), so that when it adds to the outgoingpulse stream the sum will be zero In other words the “zero deflection” or

“fixed” condition is fulfilled For the free boundary, the reflected pulse isunchanged, so that, when added to the outgoing wave, there is a doubling

of the displacement

One way of thinking of the action of such boundaries is that there is avirtual wave beyond the boundary that comes in to the real string at theboundary point, superposing on the existing wave in the string approachingthe boundary This virtual wave is a mirror image of the outgoing wave,either inverted or not Superposition at the boundary ensures that the fixed

or free boundary condition is established

Figure 2.2 shows a model of a string vibrating between two fixed points,

as might arise in a musical instrument It shows the string position at cessive time intervals for one half cycle of vibration, obtained by summingthe leftward- and rightward-going pulses at each point, and running asmooth line through the resulting distribution The system was initialized

suc-by inserting a set of pulses in each direction, whose profile was a sine waveover the interval (0, π), each of amplitude of half the total shown At eachend reflecting boundaries with inversion were implemented as describedabove This string model will keep vibrating indefinitely

For a reader new to TLM modeling willing to start programming, this

is a good starting point The choice of computer language is not important.Computer code for this problem for the programming language Matlab isshown below But the reader might prefer to choose another language withwhich he or she is familiar

% ==========================================================

% File Name: halfsine.m

% Stretched string between two fixed points as boundaries

% with an initial displacement corresponding to a half sine wave.

% ==========================================================

nodes = 40; % Number of nodes in the model

ntime = 40; % Number of time iterations

amp(1:nodes) = sin(pi*((1:nodes)-0.5)/nodes);% Set initial string amplitude right_pulse = amp/2; % in terms of component TLM pulses

left_pulse = amp/2; % half going right, half left

exit_left = left_pulse(1);% Store pulse leaving left boundary

exit_right = right_pulse(nodes);% Store pulse leaving right boundary

left_pulse(1:nodes-1) = left_pulse(2:nodes); % Move left pulses one left right_pulse(2:nodes) = right_pulse(1:nodes-1); % Move right pulses one right

left_pulse(nodes) = - exit_right;% Pulse reflected back with inversion

right_pulse(1) = - exit_left;% Pulse reflected back with inversion

amp = right_pulse + left_pulse;% Total amplitude is sum of component waves

%===========================================================

% Now plot string profile at every second time step

%===========================================================

ends and an initial shape of half a sine wave

2.5 Wave Media, Impedance, and Speed

While staying with 1-D, or plane waves, but going beyond the vibratingstring, some further concepts are now considered Generally, in phenomena

to which the wave equation applies it is found that there are two physicalvariables that can be associated with the wave Each of these variables, onits own, obeys the identical wave equation (Equation [2.1] with a common

value of wave speed c) Furthermore the product of these variables has the

dimension of power and the ratio is some kind of “impedance.” Typically,one of the wave variables can be considered as an “effort,” “force,” “pres-sure,” or “across” variable, the other as a “flux,” “flow,” “velocity,” or

“through” variable Waves arise when the temporal derivative of one of thesevariables is proportional to the (negative of the) spatial derivative of thesecond, and vice versa

For example, the natural choice of two variables is the acoustic pressure,

p, and the acoustic velocity, u Then, applying Newton's second law to an

element of fluid, one gets

(2.6)while the continuity relationship is

(2.7)

where κ is the compressibility of the fluid A pair of first order differentialequations similar to these arises in many situations ranging from longitudi-nal and torsional motion of mechanical shafts to the propagation of signals

on an electrical transmission line By differentiating Equations (2.6) and (2.7)and combining them, the second order wave equation (like Equation [2.1])

in either variable can be obtained

The proportionality “constants” in the two first-order equations, such as

ρ and κ above, are another pair of variables that typically arise in wave

phenomena These characterize the wave medium, and determine the wave

∂

∂ = − ∂∂

p x

u t

speed and wave impedance The latter is the ratio of the effort to flowvariables in a freely propagating wave Table 2.1 gives some examples ofwave variable pairs and the corresponding medium variable pair.

Looking at energy, and again taking acoustics as an example, the kineticand potential energies in acoustic waves are respectively

ρu2 and κp2 (2.8)

and these are equal As the wave propagates, energy is continuously ing form, from kinetic to potential and back again With electrical waves, theinterchanging energy types are electric and magnetic

chang-In the case of the vibrating string considered above, following the usualassumption, the primary dependent wave variable was taken to be the string

displacement, y Exactly the same wave equation arises however, if the mal component of the string tension, –T∂y/∂x, is chosen, or indeed the string

nor-velocity multiplied by the linear density, ρ∂y/∂t Furthermore, this pair of

wave variables has the characteristics mentioned above, as well as otheradvantages to be seen later, and so is shown in Table 2.1 as the wave variablepair

Figure 2.3 shows two counter-propagating TLM pulses, f and g, in a link

transmission line between two nodes, corresponding to one link in Figure2.1 It is assumed that the impulses represent samples of an effort variable.The second wave variable (the flow variable) at a point can be obtained fromthe effort impulse by dividing it by a constant, corresponding to the line

“impedance.” While the effort variable represented directly by the impulse

is typically a scalar quantity, such as acoustic pressure, the flow variable istypically a vector quantity, such as acoustic velocity, whose orientation (in2-D and 3-D problems) is the impulse propagation direction along the line.*

* This assumption is equivalent to what is called a “shunt” node in TLM, with voltage as the effort variable, and current as the flow variable The opposite assumption is equally valid, leading to the “series” TLM node But it is probably less confusing to stay with one assumption initially, and so the other case will not be explored here Note that voltage is inherently a scalar quantity (a scalar potential function) and to this extent is suited to modeling a scalar such as acoustic pressure, whereas current is inherently directed, and is similarly suited to modeling a vector, such as acoustic velocity.

1 2

1 2

∆l

The two impulses can pass each other without mutual interference.Where they meet, the total value of the primary or “force” variable is thesum of these two impulses,

p = f + g, (2.9)

whereas the total value of the flow variable, being directed, is

u = (f – g)/Z (2.10)

in the direction in which f is travelling.

As noted above, by assuming or setting values of ∆l/∆t, a pulse speed is

defined One is then free to assign an arbitrary impedance, Z, to the line As

can be seen from Table 2.1, this implicitly specifies the two medium variables

It is more typical, however, to work the other way around The values of themedium variables to be modeled are specified initially and then the values

of pulse speed ∆l/∆t, and the line impedance Z, follow Finally, the value of

the model time increment ∆t is decided from the specified speed ∆l/∆t and

the requirement that ∆l be sufficiently fine to model spatial detail in the

waveforms or model geometry, as previously discussed

“Fixed” boundaries are those where the flow variable is constrained to

be zero Imagine the link line in Figure 2.3 to be at the rightmost extremity

of the system, with the boundary located at the mid-point, such that impulse

f is leaving the system and g is the reflected pulse coming back in If impulse

g is set equal to f, the flow, by Equation (2.7) will be zero This is reflection

of pulses without inversion, and has the desired effect Note that this sponds to a doubling of the effort variable, Equation (2.6) At the oppositeextreme, “free” boundaries are where the effort variable is constrained to bezero, and the flow variable is doubled, achieved by reflection with inversion,

corre-or setting g = –f.

Table 2.1

Wave type

Wave variable pair

Medium variable pair

Wave speed

Wave impedanceAcoustic ac pressure

ac velocity

compressibility κdensity ρ ( κ /ρ) ( κ ρ)

Young’s modulus Edensity ρ (E/ρ) (Eρ)

Torsional

in rod

torqueangular vel

shear modulus Gdensity ρ (G/ρ) (Gρ)

Transmission

line

voltagecurrent

capacitance–1 C–1

inductance L

(1/LC) (L/C)Electro-

magnetic

E H

permittivity–1ε–1

permeability µ (1/µε) (µ/ε)

Intermediate cases arise where either the effort or the flow variable isspecified at the boundary For example, in acoustics or fluid mechanics, amoving surface at the boundary will determine the flow, whereas a pressurereservoir will determine the effort variable In either case, the boundaryconstraint may be constant or varying with time In all cases, the primary

constraint will be either on the effort or on the flow, specifying either p or u

in Equations (2.6) or (2.7), so that one can easily solve for the reflected pulse g for an outgoing pulse f.

The above assumes the boundary behavior is independent of the incidentwave, typically because there is a large impedance mismatch between thewave and the boundary An interesting case arises, however, when there iscoupling between the wave and the boundary, each driving the other, yeteach also obeying its own internal dynamics

In such cases, the boundary dynamics must be modeled separately fromthe wave dynamics At each time increment, the value of the effort variable

of the wave at the boundary acts as an external force on the boundarydynamics The latter can then be updated to give a new boundary velocity,which then becomes the flow boundary condition for the wave The requireddynamic coupling between the two systems is thereby achieved

2.6 Transmission Line Junctions

So far, TLM in only one dimension has been considered, in which the lem has been treated as a string of elementary transmission line elementsjoined in series at “nodes.” In 2-D and 3-D TLM, mesh junctions are inherent

prob-in the solution scheme, as will be seen later The solution is then carried out

on a mesh of transmission lines that meet at mesh nodes where the TLMimpulses “scatter.” But even in 1-D wave problems, such junctions or nodescan arise One example is in the modeling of a hydraulic system with anetwork of interconnected hydraulic lines A second case is when it is desired

to use transmission line “stubs” to control model parameters (see the lowing section)

fol-Impulses arriving at a junction, or node, are “scattered,” that is, partlytransmitted into the other lines meeting at the node and partly reflected backinto the line they arrived on For physically consistent results, two require-ments must be met during the instant of scattering: (1) the force variableshould be common to all lines meeting at the node, and (2) the total flow(into or out of the node) should be conserved

The common force variable is the sum of the incident and reflected pulse

in each line (see Equation [2.9]), whereas the net flow into the node fromeach line is the incident minus the reflected pulses divided by the impedance

of the line (Equation [2.9]) No matter how many lines meet at the node, ifthe line impedances are known, these two conditions lead to sufficient equa-tions to express all the (unknown) scattered pulses in terms of the (known)incident pulses.*

For example, suppose four lines of impedances Z1 to Z4 meet at a node Assume pulses vi1, vi2, vi3, and vi4 arrive together along these lines The problem is to determine the four scattered values, vs1, vs2, vs3, and vs4 The

first condition is that, during scattering, there is a value of the effort variable

at the node that is common to all four lines Call this vnode Then from

(vi1 – vs1)/Z1 + (vi2 – vs2)/Z2 + (vi3 – vs3)/Z3 + (vi4 – vs4)/Z4 = 0 (2.12)

The fifth variable, vnode, can be eliminated, leaving four equations in

the four unknown scattered values It is usual to express the scattering

algorithm in terms of a scattering matrix, [S], relating the vector of unknown

scattered pulses to the vector of given incident pulses

In the simplest 1-D cases considered above there are just two link lines

of equal impedance meeting at each node When the principles above areapplied, it is found that impulses leaving one line element are transmittedentirely into the next line element, with no reflection Thus the simple modeldescribed above and depicted in Figure 2.1 is obtained as a special case ofmore general scattering principles, as expected

2.7 Stubs

It was explained above that once the impulse speed and impedance hadbeen specified, the two medium parameters were also implicitly specified,and vice versa Sometimes one may wish to model a medium with locallyvarying parameters, in other words, with locally varying wave speeds andimpedances One approach would be to give the link lines different imped-ance values and different lengths (differing “∆l” values) as appropriate, to

set the desired speeds and impedances, modifying the scattering matrixaccordingly While this is feasible in 1-D problems, it becomes problematic

in 2- or 3-dimensional meshes, as the varying lengths will distort the mesh

* The scattering matrix in TLM is usually derived using Thévenin equivalent circuits (see Chapter 3) and other results from circuit theory, which guarantee conservation of charge and

of energy The two conditions specified likewise guarantee conservation of the flow variable and of energy.

For example, an originally square or rectangular mesh cannot remain so afterarbitrarily changing link line lengths.

A more elegant solution involves leaving the mesh geometry unchanged,but adding “stubs,” or half-length transmission lines, to the nodes Thesestubs are of two kinds, modifying each of the two medium parameters, andtherefore the energy storage characteristics of the medium (as well as thewave speeds and wave impedances)

2.8 The Forced Wave Equation

Many physical effects can be described by an equation that can be put in theform

utt – c2 uxx = f(x, t, u, ut, ux, …) (2.13)

where f(x, t, u, u t , u x , …) can be considered some kind of “forcing” function

acting on the “unforced” wave equation Depending on the case, f may be

due to internal or external effects in a given system Examples include theKlein–Gordon equation1,2 of quantum mechanics:

utt = c2 uxx – hu (2.14)The Sine–Gordon equation of solid-state electronics:

utt – uxx + sin u = 0 (2.15)The telegrapher’s equation for lossy propagation in transmission lines:

α utt + β ut – cuxx = 0 (2.16)Equation (2.13) also applies, for example, to the forced vibration ofstrings and to the coupling between acoustic and mechanically vibratingsystems

The forcing function, f, may be a known function, f(x,t), of time and

space, as in the case of forced vibrations of a string, or it may be a function

of the state of the system, as in the Klein–Gordon and Sine–Gordon tions Thus, at each time step in the numerical scheme, its value is eitheravailable or can be determined This value is then imposed on the TLMsolution scheme for the standard wave equation as a “perturbation,” positive

equa-or negative, half of which is added to the leftward-going wave and half tothe rightward-going wave

Cross derivative terms can also be dealt with They arise, for example,

in waves in moving media A 1-D example is known as the “moving line” equation The TLM model involves biasing the medium with notionaldiodes at the nodes, which allow pulses to pass in one direction only

thread-(irrespective of sign) into parallel paths whose impedance values are suitablyset This has the desired effect of increasing the wave speed in one directionand reducing it in the opposite direction by the same amount.

2.9 Waves in Moving Media: The Moving Threadline Equation

Systems whose partial differential equations (PDEs) have cross derivativeterms, such as

utt + αuxt + βuxx = 0 (2.17)arise, for example, with waves in moving media, characterized by direc-tion-dependent wave speeds Such problems arise in fluid mechanics andelastic mechanics and are especially significant in acoustics For this 1-D case,Equation (2.7) takes the form

utt + 2Vuxt + (V2– co) uxx = 0 (2.18)

where u is the wave variable, V is the medium speed, and c o is the

corre-sponding wave speed in a stationary medium (when V = 0).

This problem is an example of the extension of TLM based more onphysical intuition than on computational considerations Effectively thewave propagation characteristics are biased by the speed of the medium.This leads to a TLM model in which the propagation and scattering of theimpulses were biased by notional “diodes,” or better, one-directional trans-mission lines that allow pulses (positive or negative) to pass in one directiononly,3,4 as depicted in Figure 2.4 This approach and its implications will bediscussed in detail later in this book

2.10 Gantry Crane Example

Many of the assumptions behind the wave equation, Equation (2.1) above,for the vibrating string lose their validity, for example, in a heavy cable in agantry crane system,5 such as in Figure 2.5 The cable carries a load mass atthe lower end and is attached to a trolley at the top The tension in the cablewill vary with height due to the cable’s own weight, the swings can be large,

gravity adds a new restoring force to the normal component of tension, andthe cable may not be uniform Each of these effects causes a different kind

of departure from Equation (2.1)

The general TLM model for this case, which is presented in Chapter 10,

is developed in three stages First, a model is developed for thesmall-amplitude vibrations of a light, homogenous string, fixed at one endand rotating freely about this fixed end The novelty here is that the tensionvaries along the length of the cable Then a TLM model of a hanging cableunder gravity is considered, in which gravity adds an external restoring forcewhen the cable departs from the neutral position These cases are chosenbecause analytical solutions are available for both, allowing verification ofthe TLM model Finally, the TLM model of the full gantry crane is presented,involving the additional novelty of net translations of the entire system

2.11 Rotating String: Differential Equation and Analytical Solution

This example gives an indication of the approach that is used to analyze aphysical problem and translate it into a meaningful TLM model

For the light, rotating string, if gravity and air-resistance are neglected,the “equilibrium” position of the string will be a straight line rotating withangular velocity ω in a plane passing through the fixed point of rotation It

is assumed that the amplitude of vibrations is small, and that the stringdisplacement from the equilibrium is parallel to the axis of rotation andperpendicular to the plane of rotation Associated with the rotation is atension supplying the centripetal acceleration of the string mass from anypoint to the end of the string This varies along the string, from a maximum

at the center, to a value of zero if the end is free, or to mlω2 if the string,

length l, is terminated by a lumped mass m.

(2.19)

The differential equation describing the displacement u(x,t) of any point

on the string from the equilibrium rotating straight line is

(2.20)

where c 2 =ω2 /2, x is the distance from the fixed point The general solution

is expressible as

(2.21)

where P m is an nth order Legendre polynomial, m = 1, 2, … , and the constants

A m and B m are determined by the initial conditions

2.11.1 Rotating String: TLM Model

The varying tension in the string causes a continuously varying wave speedand wave impedance along the string This can be modeled in the TLMscheme by inductive stubs of varying inductance (impedance) As the induc-tance is inversely proportional to the tension, and stubs increase the line’sinductance, the region of highest tension (fixed point) will have no stubs,with the stub inductance growing towards the regions of lowest tension (endpoint) Under the assumption of low amplitude vibration, the tension dis-tribution, and therefore the stub inductances, can be assumed to be unvary-ing with time

u x t( , )=∑ {A mcos[ 2m m(2 −1) ]ct +B msin[ 2m m( −1) ]ct} P − x

l

m

2 1( )

2.11.2 Rotating String: Results

Figure 2.6 shows a summary superposition of results An initial shape

u(x,0) = x/l – (x/l) 5 is assumed (ct = 0), with zero initial velocity u t (x,0) = 0,

and zero end mass Snapshots of the waveform are shown for successive

values of ct from 0 to 4.5 The analytical and TLM solutions agree.

2.12 TLM in 2-D (Extension to Higher Dimensions)

To model the 2-D wave equation, a 2-D mesh of lines is needed For a typicalCartesian mesh, there are now four lines meeting at each node, and the

scattering algorithm gives, for an incident pulse f in one of the lines, a transmission of f/2 to the other three lines and a reflection of -f/2 in the

incident line Similar results apply to all four lines, so that the total scattering

is the superposition of these four effects

This scattering and propagation of pulses at the micro level models the2-D wave equation at the macro level, provided the wavelength is greaterthan about 10 times ∆l However, new issues arise in 2-D TLM compared

with the simple 1-D case Some of these will be briefly mentioned here Pulsepropagation through the mesh is now dispersive: the wave speed depends

on the wavelength (and/or frequency) of the “macro” wave Furthermore,

at wavelengths shorter than about 10∆l, the wave speed becomes direction

dependent For long wavelengths however, the wave speed is the same

in all directions This common wave speed is 1/ times the pulse speed,

∆l/∆t Also, the wave impedance, defined as the ratio of the effort variable

to flow variable in a freely propagating wave, (or Z(f + g)/(f – g) in Figure

2.1), is now 1/ times the line impedance, Z These results, often intriguing

at first encounter, are simply noted here and will be covered in later chapters.Suffice it to say for the moment, that provided the mesh is fine enoughrelative to the wavelengths of interest, complete time-domain models of 2-D(and 3-D) wave systems can be set up based exclusively on impulse propa-gation and scattering

3.0 2.0

2

2

2.13 Conclusions

Novel techniques have been presented for successfully modifying 1-D TLM

to model important physical effects causing different kinds of departure fromthe wave equation for a vibrating string or cable A combination of varyingimpedance stubs and “force perturbation” is used for the primary physicaleffects, while the net movement of the gantry crane is achieved by integra-tion Known analytical results for special cases were used to test the ideas

References

1 O’Connor W J and Clune F J., TLM based solutions of the Klein-Gordon

equation (Part I), Int J Numerical Modelling, 14 (2002) 439–449.

2 O’Connor W J and Clune F J., TLM based solutions of the Klein-Gordon

equation (Part II), Int J Numerical Modelling, 15 (2002) 215–220.

3 O’Connor W J., Wave Speeds for a TLM model of moving media, Int J.

Numerical Modelling, 14 (2002) 195–203.

4 O’Connor W J., TLM model of waves in moving media, Int J Numerical

Modelling, 14 (2002) 205–214.

5 O’Connor W J., A TLM model of a heavy gantry crane system, Proceedings

of a meeting on the properties, applications and new opportunities for theTLM numerical method, Hotel Tina, Warsaw 1–2 October 2001, de Cogan D.,Ed., School of Information Systems (University of East Anglia, Norwich)(2002) 3.1–3.7

to provide a bridge, so that those who are familiar with mechanical conceptsshould be able to gain a deeper understanding of the standard theory of thesubject It contains much in common with the equivalent chapter in a relatedbook on TLM modeled of diffusion processes.1 However, in this chapter wewill attempt to make fewer assumptions about the level of expertise inelectrical network theory We will cover the basics of both lossless and lossyTLM algorithms The concepts will initially be treated in terms of losslessprocesses, which can be used to describe a variety of wave propagationphenomena We will start by introducing

• A variety of relevant electrical components (resistors, capacitors, ductors, etc.)

in-• Relevant electrical network theory (Thévenin’s theorem)

• A discussion of mechanical analogues (forces, fields, displacements)Armed with these we will introduce Maxwell’s equations of electromag-netics (only in as much as we need them) We will then consider the behavior

of impulses on a transmission line and at that point we should then havesufficient background to tackle concepts in TLM itself