Transmission Line Matrix (TLM) in Computational Mechanics

Transmission Line Matrix in Computational Mechanics Transmission Line Matrix in Computational Mechanics Donard de Co...

Author: Donard de Cogan | William J. O'Connor | Susan Pulko

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Transmission Line Matrix in Computational Mechanics

Transmission Line Matrix in Computational Mechanics

Donard de Cogan William J. O’Connor Susan Pulko

Boca Raton London New York

A CRC title, part of the Taylor & Francis imprint, a member of the Taylor & Francis Group, the academic division of T&F Informa plc.

TF1745_Discl.fm Page 1 Monday, September 26, 2005 3:54 PM

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 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 http://www.taylorandfrancis.com Taylor & Francis Group is the Academic Division of Informa plc.

and the CRC Press Web site at http://www.crcpress.com

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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 contribution 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 Degenhardt 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.

**

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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, Netherlands) and microelectronic fabrication (University of Birmingham) before joining Philips as a senior development engineer in power electronic semiconductors. In 1978 he was appointed a lecturer in electrical and electronic engineering at the University of Nottingham. His initial research was concerned 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 University 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 materials. He lectures in dynamics, control, and microprocessor applications in UCD, National University of Ireland, Dublin, in the Department of Mechanical Engineering (UCD is the largest university in Ireland and the Department 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 damping; vibration-based resonant fluid sensors; and acoustic and infrared sensors. Dr. O’Connor is a Fellow of the Institution of Engineers of Ireland.

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Susan Pulko graduated from Imperial College, University of London in 1977. She moved to the University of Nottingham and undertook postgraduate work in solid state physics in the Department of Electrical and Electronic Engineering. Having obtained a Ph.D., she started working on the transmission 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 propagation in solids; current work in this area is concerned with modeling magnetostrictive behavior. She is a consultant to Feonic plc.

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Table of Contents

Chapter 1

Introduction................................................................................................1

Chapter 2

TLM and the 1-D Wave Equation....................................................... 9 Introduction .................................................................................................... 9 The Vibrating String .................................................................................... 10 A Simple TLM Model.................................................................................. 11 Boundary and Initial Conditions............................................................... 13 Wave Media, Impedance, and Speed........................................................ 15 Transmission Line Junctions ...................................................................... 18 Stubs............................................................................................................... 19 The Forced Wave Equation ........................................................................ 20 Waves in Moving Media: The Moving Threadline Equation................ 21 Gantry Crane Example................................................................................ 21 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 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11

Chapter 3

The Theory of TLM: An Electromagnetic Viewpoint................27 Introduction .................................................................................................. 27 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 3.1 3.2

Chapter 4 4.1

TLM Modeling of Acoustic Propagation........................................47 Introduction .................................................................................................. 47

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4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14

1-D TLM Algorithm..................................................................................... 47 2-D TLM Algorithm for Acoustic Propagation ...................................... 52 Driven Sine-Wave Excitation ..................................................................... 56 The 2-D Propagation of a Gaussian Wave-Form..................................... 60 Moving Sources............................................................................................ 63 Propagation in Inhomogeneous Media .................................................... 66 Incorporation of Stub Lines ........................................................................ 68 Boundaries .................................................................................................... 74 Surface Conforming Boundaries ............................................................... 74 Frequency-Dependent Absorbing Boundaries........................................ 77 Open-Boundary Descriptions .................................................................... 80 Absorption within a PML Region ............................................................. 84 Conclusion .................................................................................................... 85

Chapter 5 5.1 5.2 5.3 5.4 5.5

5.6

5.7 5.8 5.9

5.10 5.11 5.12 5.13

5.14 5.15

TLM Modeling of Thermal and Particle Diffusion....................87 Introduction .................................................................................................. 87 Spatial Discretizations and Electrical Networks for Thermal and Particle Diffusion ........................................................... 88 TLM Algorithm for a 1-D Link-line Nodal Arrangement...................... 90 1-D Link–Resistor Formulation ................................................................. 91 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 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 Flux Injection into Bulk Material ............................................................. 100 5.7.1 Single Heat Source ........................................................................ 100 Multiple Flux Sources................................................................................ 101 The Extension to Two and Three Dimensions....................................... 102 5.9.1 Link-Line Formulations................................................................ 102 5.9.2 Link-Resistor Formulations ......................................................... 104 Non-Uniformities in Mesh and Material Properties............................. 106 Stubs and the Avoidance of Internodal Reflections.............................. 111 Time-Step Variation................................................................................... 114 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 The Statistics of TLM Diffusion Models ................................................. 130 TLM and Analytical Solutions of the Laplace Equation ...................... 132

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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 6.1 6.2 6.3 6.4 6.5 6.6 6.7

TLM Models of Elastic Solids..........................................................137 The Behavior of Elastic Materials ............................................................ 137 The Analogy between TLM and State Space Control Theory ............. 140 Nodal Structure for Modeling Elastic Behavior .................................... 143 Implementation .......................................................................................... 149 Boundaries .................................................................................................. 152 Force Boundaries........................................................................................ 153 Conclusion .................................................................................................. 157

Chapter 7 7.1 7.2 7.3 7.4 7.5 7.6 7.7

Simple TLM Deformation Models ................................................159 Introduction ................................................................................................ 159 Review of the Behavior of Materials ....................................................... 159 Trouton’s Descending Fluid and a TLM Treatment of a Vertically Supported Column .......................................................... 161 A Model of Viscous Bending.................................................................... 165 Numerical Issues and Model Convergence ........................................... 169 TLM Models of Viscoelastic Deformation.............................................. 170 7.6.1 The Parallel Viscoelastic Model................................................... 170 Conclusion .................................................................................................. 173

Chapter 8 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8

TLM Modeling of Hydraulic Systems..........................................177 Introduction ................................................................................................ 177 Symbols, Analogues, and Parameters..................................................... 178 Compressional Waves in Fluids .............................................................. 181 A Transmission Line Analysis of Fluid Flow ........................................ 181 Time-Domain Transmission Line Models of Fluid Systems ............... 183 Transients in Elastic Pipes ........................................................................ 193 Open-Channel Hydraulics........................................................................ 196 Conclusions................................................................................................. 198

Chapter 9 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8

Application of TLM to Computational Fluid Mechanics.......203 Introduction ................................................................................................ 203 Viscosity ...................................................................................................... 204 Viscosity in the TLM Algorithm .............................................................. 205 Results.......................................................................................................... 206 Incompressible Fluids and Velocity Fields ............................................ 207 Convective Acceleration and the TLM Model....................................... 208 Comments on the Procedure.................................................................... 211 Implementation Issues .............................................................................. 212

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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

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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 mathematical 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 circumstances. Perhaps the best-known example is the equivalent electrical network. 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 cellular 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 modeling 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,

1


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 electromagnetics. 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 discretization 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” analytical 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 diffusion 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 pioneering 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 propagation of activity in neural networks and later had experience using Southwell's relaxation technique2 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,


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 Nottingham. 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 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 claimed that he derived inspiration from the work of Kron who first proposed the use of electrical network analogues for the electromagnetic equations3,4 in the mid 1940s. Such concepts have been further developed by Vine5 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 elsewhere at about the same time. However, as there was a different starting point, this led to quite a different formulation. Ivor Catt working for Motorola in the United States in the middle 1960s was particularly concerned with cross-talk between interconnects in high-speed integrated circuits7. 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-dimensional 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


4


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'Brien8 and has been considerably extended by Hui and Christopoulos9. The method of excitation that Catt used may also explain why the technique 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 surroundings. 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 representation 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 techniques 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 application 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


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 engineering, and electronic engineering respectively. All are aware of the cross-disciplinary 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. Nevertheless, 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 alongside 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 Kranys 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. v


6

Transmission Line Matrix in Computation Mechanics PDE* Wave equation

Equation utt – c ∇ u = 0 2

15

Telegrapher’s: damped wave

α utt + β ut + γ u – c uxx = 0

Forced wave equation

utt – c2 uxx = f(x, t, u, ut, ux, …)

Klein–Gordon

utt – c2 uxx + hu = 0

Sine–Gordon Heat/diffusion with source

utt – uxx + sin u = 0 16

ut – a2 uxx = f(x, t)

Moving threadline

utt + α uxt + β uxx = 0

Rotating string

utt = c2 [(l2 – x2) ux]x

Hanging cable

utt = g(x uxx+ ut)

16

∇2 u = 0

Poisson16

∇2 u = f

Helmholtz17

∇2 u + λ u = 0

Schrödinger (time indep.)

∇2u + α[E – V(x,y,z)]u = 0

Beam (biharmonic wave)

∇4u + (1/p2) utt = 0

Stretched, stiff string

∇4u – ∇2u + (1/p2) utt = 0

Biharmonic static

∇4 u = 0

Euler’s fluid mechanics

ρ (ut + u.∇u) = ρ f – ∇p

Navier Stokes (for incompressible fluids)

ρ (ut + u.∇u) = ρ f – ∇p + µ ∇2u

Laplace

*

2

PDE = partial differential equation

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 accustomed to the electromagnetics-based syntax, which is used elsewhere in the book. The fourth chapter is concerned with acoustics and acoustic propagation 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 commercial 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 particular importance and is discussed in Chapter 6). Chapter 5 covers models


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., ∂ 2 φ/ ∂x∂y ). 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 computational 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. Numerical Modelling, 3 (1990) 11–21. 10. Catt I., The Catt Anomaly: Science Beyond the Crossroads, Westfields Press, Westfields, 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. Kranys M., Causal theories of evolution and wave propagation in mathematical physics, Appl. Mech. Rev., 42 (1989) 305–322. vv


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 Applications, Gordon and Breach (1998). 17. Clune F., M.Eng.Sc thesis, University College Dublin (Ireland).


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 mathematical 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. Furthermore, 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 comprehensible form.

9


10


2.2 The Vibrating String Perhaps the simplest 1-D wave equation is that of a vibrating string, namely

∂2 y ∂2 y = c2 2 2 ∂x ∂t

(2.1)

where the wave speed, c, is

c=

T ρ

(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


Chapter two:

TLM and the 1-D Wave Equation

∂y ∂f = −c ∂t ∂t

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 fundamental “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 functions” 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 equation, the derivation will be skipped here. It is however worth making explicit the assumptions behind it: that the string is continuous, uniform, and perfectly 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. Nevertheless, 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)


12


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.

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 wavelength 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 waveform, then exactly the same pulse sequence will arrive at the far end, exactly


Chapter two:


13

reproducing the launched waveform, and arriving at exactly the correct (modeled) time. Computation is almost trivial. Pulse magnitudes are stored as signed real numbers and pulse positions as integers. At each increment, all rightward-going pulses are moved one position to the right, and all leftward to the left. It is as simple as that. The total wave at any point is the sum of the leftward- 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 the next time step. If the boundary is fixed, the pulse reflected back into the string is inverted (multiplied by –1), so that when it adds to the outgoing pulse stream the sum will be zero. In other words the “zero deflection” or “fixed” condition is fulfilled. For the free boundary, the reflected pulse is unchanged, 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 a virtual wave beyond the boundary that comes in to the real string at the boundary point, superposing on the existing wave in the string approaching the 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 successive time intervals for one half cycle of vibration, obtained by summing the leftward- and rightward-going pulses at each point, and running a smooth line through the resulting distribution. The system was initialized by inserting a set of pulses in each direction, whose profile was a sine wave over the interval (0, π), each of amplitude of half the total shown. At each end reflecting boundaries with inversion were implemented as described above. 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 is shown below. But the reader might prefer to choose another language with which 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. % ========================================================== clear


14


1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 5

10

15

20

25

30

35

40

Figure 2.2 TLM model of a vibrating string, with fixed boundary conditions at both ends and an initial shape of half a 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

%=========================================================== % Now propagate TLM pulses, with reflecting boundaries %===========================================================

for t = 1:ntime

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


Chapter two:


15

if mod(t-1,2) == 0 plot(amp) axis([1 nodes -1.1 1.1]) hold on drawnow end

end

2.5 Wave Media, Impedance, and Speed While staying with 1-D, or plane waves, but going beyond the vibrating string, some further concepts are now considered. Generally, in phenomena to which the wave equation applies it is found that there are two physical variables that can be associated with the wave. Each of these variables, on its 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,” “pressure,” or “across” variable, the other as a “flux,” “flow,” “velocity,” or “through” variable. Waves arise when the temporal derivative of one of these variables is proportional to the (negative of the) spatial derivative of the second, 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

∂p ∂u = −ρ ∂x ∂t

(2.6)

while the continuity relationship is

∂u ∂p = −κ ∂x ∂t

(2.7)

where κ is the compressibility of the fluid. A pair of first order differential equations similar to these arises in many situations ranging from longitudinal 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


16


speed and wave impedance. The latter is the ratio of the effort to flow variables in a freely propagating wave. Table 2.1 gives some examples of wave variable pairs and the corresponding medium variable pair. Looking at energy, and again taking acoustics as an example, the kinetic and potential energies in acoustic waves are respectively 1 2

ρu2 and 21 κp2

(2.8)

and these are equal. As the wave propagates, energy is continuously changing form, from kinetic to potential and back again. With electrical waves, the interchanging energy types are electric and magnetic. In the case of the vibrating string considered above, following the usual assumption, the primary dependent wave variable was taken to be the string displacement, y. Exactly the same wave equation arises however, if the normal component of the string tension, –T∂y/∂x, is chosen, or indeed the string velocity multiplied by the linear density, ρ∂y/∂t. Furthermore, this pair of wave variables has the characteristics mentioned above, as well as other advantages to be seen later, and so is shown in Table 2.1 as the wave variable pair. 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 Figure 2.1. 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 from the 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 is typically a vector quantity, such as acoustic velocity, whose orientation (in 2-D and 3-D problems) is the impulse propagation direction along the line.*

g

f ∆l

Figure 2.3 A link line between two nodes in a 1-D “mesh.”

*

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.


Chapter two:


17

Table 2.1 Wave type Acoustic Stretched string Longitudinal in rod Torsional in rod Transmission line Electromagnetic

Wave variable pair ac. pressure ac. velocity –T y/x ρ y/t stress rate of strain torque angular vel voltage current E H

Medium variable pair compressibility κ density ρ tension T linear density ρ Young’s modulus E density ρ shear modulus G density ρ capacitance–1 C–1 inductance L permittivity–1 ε–1 permeability µ

Wave speed ( κ /ρ)

Wave impedance ( κ ρ)

(T/ρ)

(Tρ)

(E/ρ)

(Eρ)

(G/ρ)

(Gρ)

(1/LC)

(L/C)

(1/µε)

(µ/ε)

The two impulses can pass each other without mutual interference. Where they meet, the total value of the primary or “force” variable is the sum 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 the medium 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 corresponds to a doubling of the effort variable, Equation (2.6). At the opposite extreme, “free” boundaries are where the effort variable is constrained to be zero, and the flow variable is doubled, achieved by reflection with inversion, or setting g = –f.


18


Intermediate cases arise where either the effort or the flow variable is specified at the boundary. For example, in acoustics or fluid mechanics, a moving surface at the boundary will determine the flow, whereas a pressure reservoir will determine the effort variable. In either case, the boundary constraint 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 incident wave, typically because there is a large impedance mismatch between the wave and the boundary. An interesting case arises, however, when there is coupling between the wave and the boundary, each driving the other, yet each also obeying its own internal dynamics. In such cases, the boundary dynamics must be modeled separately from the 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 boundary dynamics. The latter can then be updated to give a new boundary velocity, which then becomes the flow boundary condition for the wave. The required dynamic 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 problem has been treated as a string of elementary transmission line elements joined in series at “nodes.” In 2-D and 3-D TLM, mesh junctions are inherent 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 TLM impulses “scatter.” But even in 1-D wave problems, such junctions or nodes can arise. One example is in the modeling of a hydraulic system with a network of interconnected hydraulic lines. A second case is when it is desired to use transmission line “stubs” to control model parameters (see the following section). Impulses arriving at a junction, or node, are “scattered,” that is, partly transmitted into the other lines meeting at the node and partly reflected back into the line they arrived on. For physically consistent results, two requirements must be met during the instant of scattering: (1) the force variable should 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 from each 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, if the line impedances are known, these two conditions lead to sufficient equations to express all the (unknown) scattered pulses in terms of the (known) incident pulses.*


Chapter two:


19

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 Equation (2.9): vnode = vi1 + vs1 = vi2 + vs2 = vi3 + vs3 = vi4 + vs4

(2.11)

The second condition and Equation (2.10) specify that (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 are applied, it is found that impulses leaving one line element are transmitted entirely into the next line element, with no reflection. Thus the simple model described above and depicted in Figure 2.1 is obtained as a special case of more general scattering principles, as expected.

2.7 Stubs It was explained above that once the impulse speed and impedance had been specified, the two medium parameters were also implicitly specified, and vice versa. Sometimes one may wish to model a medium with locally varying parameters, in other words, with locally varying wave speeds and impedances. One approach would be to give the link lines different impedance values and different lengths (differing “∆l” values) as appropriate, to set the desired speeds and impedances, modifying the scattering matrix accordingly. 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.


20


For example, an originally square or rectangular mesh cannot remain so after arbitrarily 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. These stubs are of two kinds, modifying each of the two medium parameters, and therefore the energy storage characteristics of the medium (as well as the wave speeds and wave impedances).

2.8 The Forced Wave Equation Many physical effects can be described by an equation that can be put in the form utt – c2 uxx = f(x, t, u, ut, ux, …)

(2.13)

where f(x, t, u, ut, ux, …) 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 the Klein–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 of strings and to the coupling between acoustic and mechanically vibrating systems. 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 equations. Thus, at each time step in the numerical scheme, its value is either available or can be determined. This value is then imposed on the TLM solution scheme for the standard wave equation as a “perturbation,” positive or negative, half of which is added to the leftward-going wave and half to the 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 threadline” equation. The TLM model involves biasing the medium with notional diodes at the nodes, which allow pulses to pass in one direction only


Chapter two:


21

(irrespective of sign) into parallel paths whose impedance values are suitably set. This has the desired effect of increasing the wave speed in one direction and 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 derivative terms, such as utt + αuxt + βuxx = 0

(2.17)

arise, for example, with waves in moving media, characterized by direction-dependent wave speeds. Such problems arise in fluid mechanics and elastic mechanics and are especially significant in acoustics. For this 1-D case, Equation (2.7) takes the form utt + 2Vuxt + (V2 – co2) uxx = 0

(2.18)

where u is the wave variable, V is the medium speed, and co is the corresponding wave speed in a stationary medium (when V = 0). This problem is an example of the extension of TLM based more on physical intuition than on computational considerations. Effectively the wave propagation characteristics are biased by the speed of the medium. This leads to a TLM model in which the propagation and scattering of the impulses were biased by notional “diodes,” or better, one-directional transmission lines that allow pulses (positive or negative) to pass in one direction only,3,4 as depicted in Figure 2.4. This approach and its implications will be discussed in detail later in this book. b

b m

a

b m

a

m

a

Figure 2.4 A notional network with diodes, which can provide asymmetrical flow.

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 a gantry crane system,5 such as in Figure 2.5. The cable carries a load mass at the lower end and is attached to a trolley at the top. The tension in the cable will vary with height due to the cable’s own weight, the swings can be large,


22


Figure 2.5 The gantry crane configuration, with trolley, cable, and load.

gravity adds a new restoring force to the normal component of tension, and the 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 the small-amplitude vibrations of a light, homogenous string, fixed at one end and rotating freely about this fixed end. The novelty here is that the tension varies along the length of the cable. Then a TLM model of a hanging cable under gravity is considered, in which gravity adds an external restoring force when the cable departs from the neutral position. These cases are chosen because analytical solutions are available for both, allowing verification of the 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 a physical problem and translate it into a meaningful TLM model.


Chapter two:


23

For the light, rotating string, if gravity and air-resistance are neglected, the “equilibrium” position of the string will be a straight line rotating with angular 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 string displacement from the equilibrium is parallel to the axis of rotation and perpendicular to the plane of rotation. Associated with the rotation is a tension supplying the centripetal acceleration of the string mass from any point 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.

T=

l

∫ ρω xdx =ρω (l 2

2

2

− x2 ) / c

(2.19)

x

The differential equation describing the displacement u(x,t) of any point on the string from the equilibrium rotating straight line is

∂ ⎡ 2 ∂2 u 2 ∂u ⎤ = c2 ⎢ l − x ∂x ⎥ 2 ∂ x ∂t ⎣ ⎦

(

)

(2.20)

where c2 =ω 2/2, x is the distance from the fixed point. The general solution is expressible as

u( x , t) =

∑ {A

m

}

cos[ 2 m( 2 m − 1)ct] + Bm sin[ 2 m( m − 1)ct]

P2 m −1 ( x ) l

(2.21)

where Pm is an nth order Legendre polynomial, m = 1, 2, … , and the constants Am and Bm are determined by the initial conditions.

2.11.1 Rotating String: TLM Model The varying tension in the string causes a continuously varying wave speed and wave impedance along the string. This can be modeled in the TLM scheme by inductive stubs of varying inductance (impedance). As the inductance is inversely proportional to the tension, and stubs increase the line’s inductance, the region of highest tension (fixed point) will have no stubs, with the stub inductance growing towards the regions of lowest tension (end point). Under the assumption of low amplitude vibration, the tension distribution, and therefore the stub inductances, can be assumed to be unvarying with time.


24


ct = 0 4.0 0.0

0.2

0.4

0.6

0.8

1.0

4.5

1.0

x /l

3.0 2.0

Figure 2.6 Rotating string shapes over time, starting with shape x/l – (x/l)5.

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 ut(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 typical Cartesian 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 the 2-D wave equation at the macro level, provided the wavelength is greater than 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. Pulse propagation 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/ 2 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/ 2 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 enough relative 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 propagation and scattering.


Chapter two:


25

2.13 Conclusions Novel techniques have been presented for successfully modifying 1-D TLM to model important physical effects causing different kinds of departure from the wave equation for a vibrating string or cable. A combination of varying impedance stubs and “force perturbation” is used for the primary physical effects, while the net movement of the gantry crane is achieved by integration. 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 the TLM 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.



chapter three

The Theory of TLM: An Electromagnetic Viewpoint 3.1 Introduction The previous chapter introduced many of the concepts of TLM from a mechanical viewpoint. However, a large body of literature on the subject approaches TLM from its origins in electromagnetics. This chapter attempts to provide a bridge, so that those who are familiar with mechanical concepts should be able to gain a deeper understanding of the standard theory of the subject. It contains much in common with the equivalent chapter in a related book on TLM modeled of diffusion processes.1 However, in this chapter we will attempt to make fewer assumptions about the level of expertise in electrical network theory. We will cover the basics of both lossless and lossy TLM algorithms. The concepts will initially be treated in terms of lossless processes, which can be used to describe a variety of wave propagation phenomena. We will start by introducing • A variety of relevant electrical components (resistors, capacitors, inductors, etc.) • 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 electromagnetics (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 have sufficient background to tackle concepts in TLM itself.

27


28


3.2 The Building Blocks: Electrical Components 3.2.1

Resistor

This is an energy dissipating device, which is very similar to a narrow pipe or orifice that controls the flow of a liquid. In the mechanical case we can say that the flow (liters/second) is proportional to the applied pressure (or pressure drop across the device). In electricity the current is the rate of flow of charge and is proportional to the applied voltage. The constant of proportionality is called the resistance and the behavior can be expressed by Ohm’s law (V = IR). The magnitude of resistance depends on the resistivity (ρ), a specific property of the conducting material and on its geometry. Thus the resistance of a conductor of length (L) and cross-sectional area (A) is given by R = ρ L/A

(3.1)

The resistivity depends on the physical processes in the conductor, namely the concentration of charge conducting species and their mobility (a measure of the achievable drift in a unit electric field). The power (Joules/second), which is dissipated in a resistor, can be expressed in three possible ways (IV, I2R, or V2/R).

3.2.2 Capacitor A capacitor is an energy storage device. The entity that is stored is electrical charge and for a given geometry there is a relationship between the charge (Q measured in Coulombs) and the voltage (Q = CV) where the constant of proportionality (C) is called the capacitance and has units (Farads). One of the simplest geometries is an arrangement of two parallel plates of area (A), separated by a distance (L) in vacuum. The capacitance is then expressed as C0 = ε0A/L

(3.2)

ε0 is called the permittivity of free-space and has a value 8.854 × 10–12 Fm–1. If the medium between the two plates is not a vacuum, then it will influence the charge storage capacity (in general it will be possible to store more charge) and the extent is expressed in the “relative permittivity” (εr). Thus the capacitance between two plates in some general medium can be written as C = εrC0. As a general guide to magnitude, we can say that two tailoring pins with 2 mm diameter heads, separated by the width of an average human hair (25 µm) have a capacitance of 1.11 pF (1 pF, pico Farad = 10–12 Farad). A capacitor has an additional effect that must be considered when it is used in a circuit where the applied voltage is a function of time. It introduces a phase delay between applied voltage and the flow of charge (current). This can be best demonstrated for an alternating (AC) signal of constant frequency and is shown in Figure 3.1


Chapter three:

The Theory of TLM: An Electromagnetic Viewpoint

29

0.8 voltage 0.6

amplitude

0.4 0.2

current

0 −0.2 −0.4 −0.6 −0.8 0

1

2

3 angle

4

5

6

Figure 3.1 Phase-lag between AC voltage and current in a capacitor.

It is clear that the phase angle is π/2 and this is normally expressed as a function of frequency by representing the AC equivalent of Ohm’s law using the imaginary number, j ( – 1 ): V = IZ where Z = 1/(jωC) is called the “impedance.” Circuits that involve connections of resistors and capacitors can have an impedance, which is represented using complex vectors as Z = A – jB. Impedances in parallel or series are added exactly as if they were resistors (i.e., given Z1 and Z2 in series the sum is Z1 + Z2, in parallel the sum is [1/Z1 + 1/Z2]–1). The capacitor can be viewed as an energy storage device and the magnitude of the energy stored in the electric field between the conducting surfaces of the capacitor is given by CV2/2. Series connections of resistors and capacitors introduce voltage–current phase delays that are different from π/2. If a step-function change in voltage is applied to such a circuit then a voltage transient is observed across the capacitor. This is given by: V(t) = V(1 – exp[–t/RC])

(3.3)

(V is the fully charged voltage, RC is the circuit time-constant). If the ends of the circuit are subsequently connected so that the capacitor discharges through the resistor, then the voltage at any subsequent time is given by V(t) = V exp[-t/RC]

(3.4)


30


RC charge and discharge have very close analogues in mechanical engineering.

3.2.3

Inductor

An inductor is a device where the stored entity is a magnetic field. If an attempt is made to change the current through a coil that is maintaining this field then there will be a voltage established that tries to prevent this change. This is given by Lenz’s law: V = -L(dI/dt)

(3.5)

dI/dt is the rate of change of current, and the constant of proportionality (L) is called the inductance and has units, Henries. If the medium, which is surrounded by a current carrying coil, is ferromagnetic rather than air or vacuum, then the inductor can store a significantly larger magnetic field. The inductance (L) is related to L0 through the relative permeability, µr , which for the case of ferromagnetic materials can be 10,000 or more. As a measure of inductance we can consider a length of overhead high-voltage line, which has an effective inductance of 1 mH. If this is hit by a lightning stroke so that 1000 A enters the line during 1 µs then dI/dt = 109 As–1. The induced voltage given by L(dI/dt) will be 106 V, which exceeds the withstand capability of the line and must be dissipated in a flashover (see Figure 3.2). An inductor also introduces a 90° phase delay between applied voltage and current, which is shown in Figure 3.3.

Figure 3.2 Insulator necklace on a high voltage support (or pylon) showing a flash-over.


Chapter three:


31

0.8 0.6 current

amplitude

0.4 0.2

voltage

0 −0.2 −0.4 −0.6 −0.8 0

1

2

3 angle

4

5

6

Figure 3.3 Phase-lag between AC voltage and current in an inductor.

This relationship is normally expressed by another representation of Ohm’s law: V = IZ

(3.6)

where Z = jωL. Circuits, which involve connections of resistors and inductors, can have an impedance that is represented using complex vectors as Z = A + jB. Connections of inductors and capacitors have a particularly interesting property, they resonate. This is because the capacitor stores electrical energy while the inductor stores magnetic energy, and the system oscillates as energy continually changes from one form to the other. This is exactly analogous to mechanically oscillating systems such as a pendulum or a spring and mass, where the energy alternates between potential and kinetic.

3.2.4 Transmission Line The transmission line is an arrangement of conductors used to guide electromagnetic energy flow. At its simplest it consists of a pair of wires, or even a single wire close to a ground plane. It has inductance and capacitance distributed along its length. It can be represented as a continuous distribution of series inductors and parallel capacitors that act as shunts to ground. We could measure the capacitance of a length of line and dividing by the length we would get a value Cd, the distributed capacitance per meter. We could make a similar measurement of inductance per unit length to arrive at Ld. Although these are continuous properties we will frequently treat them as discrete parameters, which are lumped together within regions of space and separated from each other by ideal conductors.


32


Electromagnetic theory, which is outside the scope of this book, can be used to predict the impedance of a length of lossless transmission line. This is given by

Z=

Ld = Cd

L length length C

(3.7)

The impedance is independent of the length of the transmission line. If a signal is travelling along a lossless transmission line whose impedance is Z0 (called the characteristic impedance) then it will continue undisturbed until it encounters a discontinuity. This may be the end of the line (open-circuit termination), a short-circuit to ground, or some transmission line with another impedance. If we assume that the termination has an impedance ZT then a portion of the signal (depending on the magnitude of ZT) will be reflected back on itself. The reflection coefficient (ρ) is then given by the equation: ρ=

ZT − Z0 ZT + Z0

(3.8)

ZT might be ∞ (an open-circuit) so that ρ = 1. ZT might be a short-circuit so that ρ = –1. We could consider a TV aerial that has a 50Ω coaxial down-feed cable. The naïve user might decide to connect two TV sets to this without any matching circuit. In this case ZT = 25 so that ρ = –1/3. This effectively means that each TV set obtains 4/9 of the total power, which in weak reception areas may not be satisfactory. Transmission lines have one important effect, which is central to the concept of TLM modeling; they introduce a time-delay. The capacitance and inductance contain parameters ε0 and µ0 and the product 1/(ε0 µ0) is equal to the square of the speed of light in a vacuum. The medium of a transmission line through which an electromagnetic signal travels has permittivity εrε0 and permeability µrµ0 so that the velocity of propagation is reduced. Transmission lines can be specially constructed to act as delay-lines and these have a wide range of applications.

3.3 Basic Network Theory We will assume Ohm’s law, which, in any event, has close analogues in hydraulics (flow is proportional to pressure drop). We are particularly concerned with Thévenin’s theorem, which leads to a very useful approach to analyzing the behavior of an electrical circuit. The theorem says that any circuit where a measurement is made at a pair of terminals can be replaced by:


Chapter three:


33

• A voltage source that is equivalent to the voltage that would be observed if there was no external connection to the terminals (open-circuit voltage) • An impedance that is equivalent to all the impedances of the circuit if all of the voltage sources had been replaced by perfect conductors (short-circuit impedance) If we take the transmission line and look at the time history of an impulse travelling along it as observed from one end, we will see how Thévenin’s theorem is applied in TLM. In the first picture in Figure 3.4 we see a pulse travelling on what it believes is an infinite transmission line. The outside observer is aware that it is not infinite but can see no signal. In the second picture the pulse suddenly becomes aware that it is faced with an open circuit termination and according to Equation (3.6) the reflection coefficient, ρ = 1. At that instant the voltage, which is seen by the outside observer, is the sum of the incident and reflected pulses, i.e., 2V. Thereafter the observer sees nothing and once again, the reflected pulse has no knowledge of the finite nature of its environment. In compliance with the second of the Thévenin requirements, we could take the same line in the absence of any pulse and short the other end. An impedance measurement would yield the value, Z. The Thévenin equivalent circuit of a transmission line is shown in Figure 3.5. The Thévenin equivalent for a transmission line and Figure 3.4 indicate some of the significance of TLM. Because the impulse travelling on the line is unaware of anything except its immediate surrounds we are able to treat each impulse independently of any other. Any interaction occurs only when they meet. This effectively time discretizes the problem, and further, the smallest time unit that can be considered is the interval between arrival of impulses at observation points. This value ∆t is normally chosen so that the velocity along a length of line ∆x can be represented as

V=

∆x ∆t

(3.9)

So, once we have constructed the correct analogue, our algorithms become little more than an efficient method to keep track of impulses. We can also approximate more complicated wave forms by means of a train of impulses as shown in Figure 3.6.

3.4 Propagation of a Signal in Space (Maxwell’s Equations) The behavior of these impulses can be related to the propagation of electromagnetic waves and an analysis leads to a wave equation. Different network formulations lead to expressions for different field components. If an


34

Transmission Line Matrix in Computation Mechanics L=∞

L≠∞ Vobs = 0

V

!!* L≠∞ ZT = ∞

! Vobs = 2V

L=∞

L≠∞ Vobs = 0

V

Figure 3.4 Time-history of an impulse traveling on a transmission line from two points of observation.

Z

2V

Figure 3.5 Equivalent circuit of a transmission line at the moment of arrival of a signal.


Chapter three:


35

Figure 3.6 A discrete sinusoid constructed from a time series of impulses.

assembly of transmission lines is connected as shown in Figure 3.7 then we have what is called a “shunt” transmission-line network. If the network had resistive losses then the electric field vector, Ez would be expressed by the lossy wave equation:

∂ 2 Ez ∂ 2 Ez ∂ 2 Ez ∂Ez + = µε + εσ ∂t ∂x 2 ∂y 2 ∂t2

(3.10)

EZ HX

HY

Figure 3.7 Intersecting pair of two-wire transmission lines.

The first term on the right expresses a wave propagation with a velocity µε = 1/c2. The second term, which contains εσ, expresses an attenuation of the wave. If the transmission line is lossless then σ = 0 and the more familiar wave equation is obtained:

∂ 2 Ez ∂ 2 Ez ∂ 2 Ez + = µε ∂x 2 ∂y 2 ∂t2

(3.11)


36


There are similar expressions for all of the other field components and for a large part of this book Equations (3.10) and (3.11) represent as much knowledge of Maxwell’s equations as is required.

3.5 Distributed and Lumped Circuits Up to now the concept of a transmission line has really been that of a distributed system. There was inductance distributed along the conductor. Resistance, if present, was also along the conductor. Capacitance per unit length was distributed between conductors and if there was leakage between conductors then this was also distributed. It is quite difficult to deal with such systems, and we try where possible to replace them by an equivalent with lumped components. In fact, we have already done this in Figure 3.5 where the parameters of a transmission line are replaced by their Thévenin equivalents. Now, everything that has been presented above about Maxwell’s equations can also be expressed for a lumped component electrical network consisting of resistors, capacitors, and inductors. In this case we will use analogues, where potential, V replaces the electric field, E, and current, I replaces the magnetic field, H. We will start by considering a 1-D case (Figure 3.8), which has lumped components whose values are equivalent to what are called distributed parameters. Thus, Rd is the resistance per unit length = R/∆x. Cd and Ld are similarly defined.

I Rd

- Ld dI/dt

Cd V + ∆V

V

∆x Figure 3.8 A simple inductor resistor capacitor (LRC) circuit.

The change in voltage as a function of distance depends on the voltage drop across the resistor (IR) and the voltage that develops across the inductor (-LdI/dt). Thus, lim ∆x→0

V + ∆V − V ∂V ∂I = = -IR d - Ld ∆x ∂x ∂t

(3.12)


Chapter three:


37

At the same time there is a change in current over this distance due to the change in charge being stored on the capacitor.

I = C

∂V ∂V = C d ∆x ∂t ∂t

so that

lim

∆x→0

I + ∆I − I ∂I ∂V = = - Cd ∆x ∂x ∂t

(3.13)

Now, if Equation (3.12) is differentiated with respect to x then we get

∂2V ∂I ∂I 2 = - Rd - Ld 2 ∂x ∂x ∂t ∂x

(3.14)

Equation (3.13) and its derivative can now be substituted into this to give:

∂2V ∂V 2 ∂V = L C + RdCd d d 2 2 ∂t ∂x ∂t

(3.15)

This is called the telegrapher’s equation and is the basis for the TLM method. The extension of this equation to two and three dimensions will be deferred until we consider specific TLM networks. However, before we proceed further it is necessary to revisit some additional basic electromagnetic theory relating to transmission lines.

3.6 Transmission Lines Revisited 3.6.1 Time Discretization The concept of distributed parameters, which was mentioned, is central to our model for a transmission line. Equation (3.7) represented the impedance of a lossless transmission line, regardless of length. The velocity of an impulse on a transmission line can be given by:

v=

1 Ld C d

(3.16)

Much of the analysis of the behavior of electromagnetic fields is presented in the complex domain2 assuming harmonic (sinusoidal) excitation,


38


so that variables of a given frequency are characterized by a magnitude and a phase. A propagation constant γ can then be defined so that a wave can be presented as Asin (γ x). The propagation constant of an impulse on a line is then given by:

γ = jω Ld Ld

(3.17)

The time taken for an electrical signal impressed on the line to traverse it will be determined by the local velocity of light, which depends on µε as mentioned previously. If the resistance on a line is such that Rd is nonnegligible then the situation is much more complicated

Z0 =

γ=

R + jωL jωC

(3.18)

( R + jωL)jωC

(3.19)

For many years the textbook approaches to electromagnetics have explored ways of circumventing this problem. If Rd is negligible then the telegrapher’s equation reduces to a simple wave equation, and this is the basis for lossless TLM modeled. If, on the other hand, Rd is significant then there is an entire branch of TLM modeled that seeks to ignore the wave component in Equation (3.15) so that it can be treated as a diffusion equation3. The velocity of propagation on a uniform, lossless transmission line can be related to its parameters:

v=

∆x = ∆t

1 = Ld C d

∆x 2 = ∆x LC

1 LC

(3.20)

or

1 = ∆t now, using Z = get

1 LC

(3.21)

L we can eliminate either L or C in the Equation (3.21) to C

Z =

∆t C

or

∆t C d ∆x

(3.22)


Chapter three:


39

These relationships between the line impedance, the line parameters, and the spatial and temporal discretizations are the fundamental building blocks of TLM.

3.7 Discontinuities Equation (3.8) expresses the behavior of a pulse/wave on a line when it encounters a change of impedance. The signal, which is transmitted past a discontinuity in a line, depends on whether we are dealing with current or voltage in our analysis. Both the current and voltage expressions for transmission and reflection are essential in the development of lossless and lossy TLM algorithms and these will be treated in turn. If we are dealing with current (charge per second) then the conservation of charge controls what happens:

L ∆t

Z =

i

Ld ∆x ∆t

or

I = r I + tI

(3.23)

(3.24)

In this case the superscripts i, r, and t indicate whether the current is incident, reflected, or transmitted. The reflected current can be defined using the reflection coefficient as: r

I = ρ iI

(3.25)

Therefore, the transmitted current is: t

I = ( 1 − ρ) i I

(3.26)

An analysis of the scattering of voltage pulses (Figure 3.9) starts from the same position but recalls that voltage is a measure of work done when charge is moved against an electric field. Equation (3.24) can be restated using Ohm’s law as: i

i

t

V =ρ V + V Z0 Z0 ZT

(3.27)

However, by virtue of Equation (3.8) we can write this in terms of tV as: t

V = ( 1 − ρ) iV

ZT = ( 1 + ρ) iV Z0

(3.28)


40


i

V

t

V

Z0

ZT

ρ iV

Figure 3.9 Voltage scattering at an impedance discontinuity.

It is interesting to note that Equations (3.26) and (3.28) can be combined to given the transmitted power, P: t

P = ( 1 − ρ) i I ( 1 + ρ) iV = ( 1 − ρ2 ) i P

(3.29)

Since the reflected power is ρ2 i P this confirms the energy conservation law.

3.8 TLM Nodal Configurations Conventional TLM has commonly used what is called the shunt node configuration. This assumes that current is equivalent to magnetic field and voltage is equivalent to electric field. In a 2-D problem Ix and Iy are the analogues of Hx and Hy, and Vz is equivalent to Ez as can be seen in Figure 3.7. There is another approach called the series node configuration which uses Ex, Ey (equivalent to Vx, Vy), and Hz (equivalent to Iz). Details of the implementation of this and the more complicated 3-D lossless representations can be found in references such as Christopoulos4. We will concentrate here on the development of lossless TLM algorithms based on the shunt node in 1- and 2-D. The conversion of these ideas into practical TLM algorithms requires the use of three simple assumptions, one equation from electromagnetics and one theorem from basic electrical theory, all of which have been covered earlier: • The first assumption is that all data (field amplitudes) are represented by impulses of very short duration. Thus an impulse entering a transmission line has no knowledge about the length of the line; indeed it is unaware that the line is finite until it arrives at a discontinuity (see Figure 3.4). • The second assumption is needed for computational purposes and requires that all pulses moving around the spatial mesh are synchronized, in the sense that they all arrive at the next nodal intersection or discontinuity at the same instant. • The third assumption is that the reflection at an impedance discontinuity given by Equation (3.8) applies to the pulses considered here.


Chapter three:


41

We are now in a position to monitor the progress of a single pulse as it starts to scatter around the network of nodes, or mesh, which describes our problem. For our purposes we will consider a 2-D system and since we are dealing with Cartesian coordinates, we can describe the directions of scatter by the compass directions N, S, E, and W. Let us consider iVW, which is incident from the west onto a 2-D TLM node (x,y) as shown on the left in Figure 3.10. As the pulse (from the west) arrives at the end of its line it sees what it believes are three lines of apparently infinite length (representing north, south, and east) connected in parallel and representing a terminating impedance as shown on the right of Figure 3.10.

1 1 1 1 so that ZT = Z/3. = + + ZT Z Z Z The pulse undergoes scattering according to Equation (3.8) and since ZT = Z/3 we have ρ = –1/2. This means that a pulse of magnitude (–0.5)iVW is returned down the incoming transmission line. The remainder of the signal is transmitted into the other arms. Pulses incident from arms N, S, and E are simultaneously incident and undergo scattering. They also contribute to the voltage at the node center, which can be represented by the principle of superposition. This states that the voltage at a node is the sum of the currents divided by the sum of the admittances (where admittance is reciprocal of impedance): Impedances add in parallel as

φ(x,y) =

k

⎡ 2 iVN 2 iVS 2 iVE 2 iVW ⎤ + + + ⎢ ⎥ I Z Z Z Z ⎦ ⎣ = ⎡1 1 1 1⎤ Y ⎢Z + Z + Z + Z ⎥ ⎦ ⎣

∑ ∑

or

⎡ i VN + i VS + i VE + i VW ⎤ ⎣ ⎦ φ(x,y) = k 2

(3.30a)

In a situation where an n-dimensional network has 2n arms with different impedance, then Equation (3.30a) can be expressed in a general form as: n

k

φ(x,y) =

⎡ 2 iV1 2 iV2 2 iV2 ⎤ + + + . . . . . .⎥ ⎢ Z Z Z 1 2 3 ⎦ =⎣ ⎡ 1 ⎤ 1 1 Yj ⎢Z + Z + Z +. . . . . ⎥ 2 3 ⎣ 1 ⎦

∑I j=1 n

∑ j=1

j

(3.30b)


42


Z

N

node center

i VW

+

Z

W

i

E

2 VW

-

total im pedance Z 3

S Figure 3.10 Lossless TLM node and its Thévenin equivalent.

The pulse, which is scattered back in any direction, is the sum of what is reflected and what is transmitted from all other arms. Thus s

VW = ρ i VW + τ ( i VN + i VS + i VE )

(3.31)

There are similar equations for iVN, iVS , iVE, and iVW and the entire scattering process can be expressed in matrix form as:

⎛ iV ⎞ ⎛ sV ⎞ ⎜ N⎟ ⎜ N⎟ ⎜ iV ⎟ ⎜ sV ⎟ ⎜ S ⎟ =S ⎜ S ⎟ ⎜i ⎟ ⎜s ⎟ ⎜ VE ⎟ ⎜ VE ⎟ ⎜i ⎟ ⎜s ⎟ ⎝ VW ⎠ ⎝ VW ⎠ k k

(3.32)

where S is the scattering matrix and is given by

⎛ρ ⎜τ S=⎜ ⎜τ ⎜ ⎝τ

τ ρ τ

τ τ ρ

τ

τ

τ⎞ τ ⎟⎟ τ⎟ ⎟ ρ⎠

(3.33)

In this general form it can be applied to any type of 2-D TLM (lossless or lossy) scattering. The situation where impedances are equal and do not contain any resistance (as shown in Figure 3.10) has ρ = –0.5 and τ = 0.5. The scattering matrix for lossless TLM is then


Chapter three:


⎛ −1 1⎜ 1 S= ⎜ 2⎜ 1 ⎜ ⎝ 1

1 −1 1

1 1 −1

1

1

1⎞ 1⎟⎟ 1⎟ ⎟ −1⎠

43

(3.34)

Each pulse travels the discretization distance ∆x during the discretization time ∆t after which it becomes an incident pulse at an adjacent node. The connections to other nodes as seen at node (x,y) can be expressed in terms of space and time-step, k + 1 as k+1

i

k+1

k+1

i

i

k+1

s

VN(x,y) = kV S(x,y + 1)

(3.35)

s

VS(x,y) = kV N(x,y – 1) s

VE(x,y) = kV W(x + 1,y)

i

s

VW(x,y) = kVE(x – 1,y)

The repeated application of the processes of scatter (Equation [3.32]), connect (Equation [3.35]), and summation (Equation [3.30]) for every time step constitutes the entire TLM process for a 2-D transmission line network. Before proceeding to the next section, which considers what happens when impulses on TL networks interact with boundaries, we should remind ourselves that lossless formulations that use the scattering matrix as shown in Equations (3.34) can be used to model wave propagation problems. The inclusion of resistive losses within such networks yields reflection coefficients that are different from –0.5. In these situations, which can give good approximations to diffusion processes, Equation (3.33) must be used for the scattering matrix.

3.9 Boundaries Any physical problem has boundaries and we next start to consider how these might be treated in TLM. Because the technique has evolved from electromagnetics and particularly from microwave theory it defines boundaries in these terms, but there are equally good mechanical definitions (e.g., in acoustics boundaries with specified acoustic pressures or velocities) • ZT = corresponds to an electrical open-circuit termination. This means that a voltage (or electric field) pulse, which arrives at a boundary, is reflected in-phase, because ρ = 1. The current at the boundary is zero. This is sometimes called a “mirror” boundary. In acoustics


44

Transmission Line Matrix in Computation Mechanics where we frequently use voltage as the analogue for pressure and current as the analogue for velocity, this corresponds to a rigid boundary, where the velocity into the boundary is zero, and the associated pressure is typically a maximum. • ZT = 0 is equivalent to an electric short-circuit termination. ρ = –1 and any pulse incident on a boundary will be reflected in antiphase. The voltage (electric field) at such a boundary is zero, and typically the current is large. In acoustics, the acoustic velocity is a maximum and the acoustic pressure is zero, which accounts for the description “pressure-release” boundary. • The situation where ZT = Z is called a “matched load” boundary condition and ρ = 0. The use of this definition requires caution because ZT is generally a function of frequency. If ρ = 0 at one particular frequency, then it may be different from zero at all other frequencies. The definition of a broad-band perfectly matched load (PML) boundary will be discussed later.

The concept of open-circuit boundaries can also be used to reduce the size of the problem that needs to be computed. Frequently, an axis of symmetry corresponds to a line along which there is zero current, which is equivalent to a perfectly reflecting boundary, with ρ = 1. Consequently, in the case of the rectangular wave guide that was analyzed by Johns and Beurle,5 the entire cross-section of the waveguide could be analyzed by having two short-circuit boundaries (the outer walls) and two open circuit boundaries, the two orthogonal symmetry axes (see Figure 3.11). ρ=1

obs ρ=1 ex

ρ = −1 (a)

(b)

Figure 3.11 (a) A section of discretized electromagnetic waveguide showing the passage of impulses (arrows) across the two orthogonal symmetry axes. (b) Symmetry-reduced model. (“ex” is the point at which excitation signals are inserted into the mesh; “obs” is the observation point where data is collected for subsequent use in Fourier analysis.)


Chapter three:


45

3.10 Conclusion This chapter has demonstrated the basic simplicity of the algorithms of TLM. Much of the perceived difficulty with the technique lies in the range of concepts, which together constitute the underlying physics. There must be some level of understanding of these if TLM is to be anything other than the repeated application of a set of rules in the manner of a CA. We could reasonably say that TLM is an example of a physical interpretation of certain CA rules. Much TLM research is concerned with the analysis of physical problems and their representation as parameters within a lossless or lossy wave equation. The next chapter develops many of the ideas that were introduced here and applies them to problems in acoustic propagation.

References 1. de Cogan D., Transmission Line Matrix (TLM) Techniques for Diffusion Applications, Gordon and Breach (1998). 2. Cheng D. K., Field and Wave Electromagnetics, 2nd ed., Addison Wesley, Reading, MA (1989). 3. de Cogan D., The relationship between parabolic and hyperbolic transmission line matrix models for heat-flow, Microelectronics J., 30 (1999) 1093–1097. 4. Christopoulos C., The Transmission Line Modeling Method, Oxford University Press/IEEE Press, 1995. 5. Johns P. B. and Beurle R. L., Numerical solution of 2-dimensional scattering problems using a transmission line matrix, Proceedings of the IEE, 118 (1971) 1203–1208.



chapter four

TLM Modeling of Acoustic Propagation 4.1 Introduction Having covered several chapters of theory we now introduce practical implementations with a tutorial component. This will be done in the context of acoustic propagation in a variety of media that support longitudinal (as opposed to transverse) waves. We start by relating free-space propagation to a lossless TLM model. This is followed by some simple examples coded in MATLAB. The remainder of the chapter then goes on to consider the nature and implementation of mesh excitation (stationary and moving sources), propagation in inhomogeneous media, realistic boundaries (surface conforming, absorbing), and open boundary descriptions.

4.2 1-D TLM Algorithm We start with a simple TLM algorithm that demonstrates the behavior of forced excitation of a string that is supported at two ends (see also Figure 2.2). By suitable arrangement of the initial excitations we can construct a resultant standing wave on the line, and in the case that is given below, we have used six components: {sin(x) – sin(2x)/2 + sin(3x)/3 – sin(4x)/4 + sin(5x)/5 – sin(6x)/6}, where x is a function of position along the line. These represent the first six Fourier components of a saw-tooth wave form. In the MATLAB code, which is given below, the spatial domain contains 60 nodes, and at the very start two sets of discretized sine waves are impressed, one moving to the left and one moving to the right. The main routine is run for 120 iterations and comprises: “summation” (line 18), “scatter” (lines 19, 20), and “connect” (lines 21, 22).

47


48

Transmission Line Matrix in Computation Mechanics WAVES ON A STRING 1

% spatio-temporal evolution of a sawtooth wave-form using TLM

2

% spatially distributed excitation comprise the first six Fourier components

3 4

% ********************* INPUT PARAMETERS *********************

5

nmax = 60; problem

% the number of spatial nodes in the

6

Kmax=120;

% the number of iterations

7

Z=zeros(Kmax,nmax);% matrix for storing successive results

8

%*********************** TLM ROUTINE ************************

9 10

for j=1:nmax

11

x(j)=(2*pi/(nmax))*((2*j-1)/2); % construction of spatial domain

12

end

13

% excitations shown below

14

vir = sin(x) – (sin(2*x))/2 + (sin(3*x))/3 – (sin(4*x))/ 4 + (sin(5*x))/5 – (sin(6*x))/6;

15

vil = sin(x) – (sin(2*x))/2 + (sin(3*x))/3 – (sin(4*x))/ 4 + (sin(5*x))/5 – (sin(6*x))/6;

16 17

for k=1:Kmax

% start of the iterative process

18

vtotal = vil +vir;% summation of incident pulses

19

vsl=vir;

% scatter to left

20

vsr=vil;

% scatter to right

21

vil = shiftlr(vsr,1);% connect from left

22

vir = shiftlr(vsl,-1);% connect from right

23

vil(1) = -vsl(1);% apply pressure release boundary condition on left

24

vir(nmax)= -vsr(nmax);% apply pressure release boundary condition on right

25 26 27

for j=1:nmax Z(k,j) = vtotal(j);%fill out row of display matrix for one iteration

28 29

end end

30 31

surf(Z) composite wave

32

view(-14,60)

%display space-time evolution of %rearrange view angle


Chapter four:

TLM Modeling of Acoustic Propagation

49

The “connect” step is the 1-D equivalent of Equation (3.35):

V ( x ) = skVR ( x − 1)

i k +1 L

(4.1)

V ( x ) = skVL ( x + 1)

i k +1 R

For simplicity of coding we have used the “shiftlr” routine of Hansleman and Littlefield1. shiftlr (A,b) shifts the contents of matrix, A to the right by b spaces, if b > 0, and by b spaces to the left if b < 0. This is equivalent to multiplying a matrix by the appropriate Toepliz matrix: If A = ⎡⎣0

⎡⎣0

1

2

1

3

2

3

⎡0 ⎢ ⎢0 0 ⎤⎦ * ⎢0 ⎢ ⎢0 ⎢1 ⎣

0 ⎤⎦ then we have:

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0⎤ ⎥ 0⎥ 0 ⎥ = ⎡⎣0 ⎥ 1⎥ 0 ⎥⎦

⎡0 ⎢ ⎢1 0 ⎤⎦ * ⎢0 ⎢ ⎢0 ⎢0 ⎣

0 0 1 0

0 0 0 1

0 0 0 0

0

0

1

0

1

2

3 ⎤⎦

1⎤ ⎥ 0⎥ 0 ⎥ = ⎡⎣1 ⎥ 0⎥ 0 ⎥⎦

2

3

0

(4.2)

and

⎡⎣0

1

2

3

0 ⎤⎦

The connect processes described by Equations (4.1) and/or (4.2) would cause data to “fall off the edge.” Lines 23 and 24 define short-circuit (zero-velocity) boundaries. Lines 26 to 28 fill one row of the display matrix (Z) at each iteration. Ripples are always evident in conventional representations of the finite summation of Fourier components as an approximation to a wave-form. The addition of the “time” (iteration number) evolution as shown in Figure 4.1 provides a new dimension to the visualization. Adaptation of the code will quickly demonstrate how the wave-length and amplitude of the ripples change as extra terms are included in the excitation. TLM operates in the time-domain, but there are many instances where a frequency response may be required. This can be easily achieved by including


50


amplitude

4 2 0 −2

−4 120

80 60

iter

atio

um nn

ber

100

40 20 0 0

10

20

30

40

50

60

spatial position

Figure 4.1 Space-time evolution of a saw-tooth wave form using a spatially distributed excitation comprising the first six components in the appropriate Fourier expansion.

a discrete Fourier transform or fast Fourier transform (DFT or FFT) as appropriate in the code. This is demonstrated by taking a program of the same basic structure and using as excitation two sinusoids with different frequencies. The temporal data at a single inspection point can then be collected and fed into a fast Fourier transform program, so that we can confirm that the wave form does indeed comprise the components that were initially injected. This 1-D model (shown below) comprises 14 nodes with an excitation arranged so that this space supports λ/2. The excitation comprises the fundamental and the first harmonic. The time-domain data is fed into the observation matrix (line 17), which represents the total voltage at the observation point (defined at line 5) at every iteration. The simulation is run for 256 iterations. Line 25 calls the FFT subroutine that has been specifically designed for a sampling rate of 10kHz.


Chapter four:


51

TWO FREQUENCY EXCITATION OF A STRING

1

%lossless TLM to determine the Fourier components of an acoustic wave where

2

%where the sampling frequency is 10kHz

3

% ********************* INPUT PARAMETERS *********************

4

nmax = 14;

5

N=256;

%N is the number of iterations

5

observation= 4;

%observation is the observation point

7

a=zeros(1,N);

% the observed data matrix

7

%*********************** TLM ROUTINE ************************

9

%input

10

%the no of nodes in the problem

for j=1:nmax

11

x(j)=(pi/(nmax))*((2*j-1)/2); %lambda spans the space

12

end

13

vir = sin(x) + sin(2*x);

14

vil = sin(x) + sin(2*x);

15

for k=1:N

16

vtotal = vil +vir; %summation of incident pulses

17

a(k)=vtotal(1,observation);%feed observation matrix

18

vsl=vir;

%scatter to left

19

vsr=vil;

%scatter to right

20

vil = shiftlr(vsr,1);%connect from left

21

vir = shiftlr(vsl,-1);%connect from right

22

vil(1) = -vsl(1);%apply left boundary condition

23

vir(nmax)= -vsr(nmax);%apply right boundary condition

24

end

25

ftplot

The program calls “ftplot” (line 25) and the code for this, which is based on routines by Hansleman and Littlefield,1 is shown below. FFT CODE 1

Ts = (1/10000);%sampling period (10kHz)

2

b=fft(a);

3

fp=b(1:N/2+1)*Ts;% scaled and Nyquist truncated array of frequency components

4

fs=1/Ts;

5

f=fs*(0:N/2)/N;%array of frequency components

6

plot(f,abs(fp))%plot of frequency versus modulus of fp

%array of frequency components

%sampling frequency


52


The results are plotted in Figure 4.2a and clearly demonstrate the existence of two frequency components. From line 11 of the main program we have x = (pi/14)*(7/2) at the observation position (j = 4), so that sin(x[4]) = 0.707 and sin(2*x[4]) = 1. Discretization and sampling errors in the FFT mean that there is generally an error in the ratio of the observed amplitudes. However, the general principle can be demonstrated by repeating the simulation with the observation point at the middle of the space. In this case line 17 becomes: a(k) = 0.5*(vtotal(1,7) + vtotal(1,8))

(4.3)

The results shown in Figure 4.2b demonstrate that the upper frequency has a node at this point. Now that we have derived this much data we will continue the process in reverse. In a normal simulation we would of course operate the entire process in the forward direction. Let us imagine that the sound velocity was 300 m/s. Inspection of the FFT data yields the frequency components as 351.7625 and 703.125 Hz. One wavelength on the 14 node structure is then 0.4267 m at this sound velocity. Accordingly ∆x = 0.0305 m. Since ∆x/∆t = 300 m/s we can calculate the temporal discretization as ∆t = 1.01389 × 10–4 s. The reciprocal (9.836 kHz) is within 1.65% of the sampling frequency used in the FFT.

4.3

2-D TLM Algorithm for Acoustic Propagation

Equations (2.6) and (2.7) can be used to express the acoustic wave equation

∇2 p = kρ

∂2 p ∂t2

(4.4)

where ρ is the density and κ is the compressibility of the medium. This is the same as

∇2 p =

1 ∂2 p v 2 ∂t2

(4.5)

where v, the velocity of propagation of a longitudinal wave is given by

v=

1 kρ

(4.6)

We now develop this in a 2-D lossless TLM simulation and start with what might appear as the simplest case, a single shot excitation at the center.


Chapter four:


53

0.025

0.02

0.015

0.01

0.005

(a) 0

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

4500

5000

0.025

0.02

0.015

0.01

0.005

(b) 0 0

500

1000

1500

2000

2500

3000

3500

4000

Figure 4.2 (a) Fourier components derived from TLM time-domain data collected at observation point (j = 4). (b) Collected data is the average of values at (j = 7) and (j = 8) which represents the node for the sin(2x) excitation component.


54


The code, which is listed below, uses more MATLAB-based sophistications than were used in earlier code. This example might be considered as the equivalent of throwing a stone into the center of a pond and we might expect to see ripples spreading outwards. However, this is not the case, and the example serves to demonstrate some of the problems associated with TLM. However, once these limitations are appreciated, then most difficulties can be circumvented. 2-D TLM CODE %% 2-D TLM routine with single shot excitation clear ITERS = 60; rows = 101;cols = 101;

%% the number of iterations %% dimension of problem

%% set up connection matrices Cn=[[zeros(rows-1,1) eye(rows-1)];zeros(1,rows)];Cs=Cn'; Ce=[[zeros(cols-1,1) eye(cols-1)];zeros(1,cols)];Cw=Ce'; %% set up transmission and reflection coefficients T = 0.5.*ones(rows,cols); R =-0.5.*ones(rows,cols); %% set up transmission and reflection coefficients %% for boundaries BRn = [ones(1,cols);zeros(rows-1,cols)]; BRs = [zeros(rows-1,cols);ones(1,cols)]; BRe = [zeros(rows,cols-1) ones(rows,1)]; BRw = [ones(rows,1) zeros(rows,cols-1)]; %% make room for incident and scattered voltages In=zeros(rows,cols);Is=zeros(rows,cols); Ie=zeros(rows,cols);Iw=zeros(rows,cols); Sn=zeros(rows,cols);Ss=zeros(rows,cols); Se=zeros(rows,cols);Sw=zeros(rows,cols); phi = zeros(rows,cols);

%% make room for 'phi'

%% single excitation at (51,51) which at instant of arrival sums to 1000 In(51,51) = 500; Is(51,51) = 500; Ie(51,51) = 500; Iw(51,51) = 500; for iter = 1:ITERS% start of TLM routine Sn = In.*R + Is.*T + Ie.*T + Iw.*T;%% calculate scatter pulses Ss = In.*T + Is.*R + Ie.*T + Iw.*T; Se = In.*T + Is.*T + Ie.*R + Iw.*T; Sw = In.*T + Is.*T + Ie.*T + Iw.*R; In = Cs*Ss + BRn.*Sn;

%% calculate incident pulses

Is = Cn*Sn + BRs.*Ss; Ie = Sw*Cw + BRe.*Se; Iw = Se*Ce + BRw.*Sw; phi = (In + Is + Ie + Iw)/2;%% superimpose incident pulses end

%% end of TLM routine

surf(phi);colormap(gray);shadinginterp

The results of the “surf” instruction are shown in Figure 4.3. We can make the following observations:


Chapter four:


55

Figure 4.3 View of the apparent pressure profile after 60 iterations following a single pulse input at the center of the 2-D mesh.

• The wave front is not radial; in some directions it seems to go more slowly • The fastest propagation seems to be slower than expected • There is much fine-structure within the circle of propagation The explanation for these lie in the phenomenon of numerical dispersion, the fact that the velocity of propagation on a rectangular mesh depends on frequency and on direction. A factor 0.707 arises because of the way in which an impulse is forced to travel between neighboring points at opposite diagonals. The distance 2 ∆x is traversed in time 2∆t, so that

apparent velocity =

1 ∆x 2 ∆t

(4.7)

Dispersion can be presented as velocity versus frequency but within TLM it is more usual to plot velocity against a frequency equivalent, ∆x/λ, where λ is the wavelength in question and ∆x is the discretization that is used in the model. This is shown in Figure 4.4, and it can be seen that the velocity drops to zero when ∆x/λ = 0.25, which is termed cut-off. This means


56


Propagation velocity 0.7

0.6

0.5 0.1

0.2

0.3

Normallized frequency (∆x/λ) Figure 4.4 Dispersion plotted as normalized velocity versus ∆x/λ. It will be noted that the maximum velocity is 70.7% of the free-space sound-speed.

that it is not possible to propagate a wave if the discretization is equivalent to four nodes per wavelength. It will be noted that, so long as ∆x/λ ≤ 0.1, then there is not much variation in velocity, and this constitutes a basis for limiting the effects of dispersion in TLM models. Any excitation/s whose frequency spectrum does not contain components with less than the equivalent of ten nodes per wavelength should not exhibit significant dispersion. So long as the modeler does not lose sight of these factors, then it is possible to avoid spurious effects due to dispersion.

4.4 Driven Sine-Wave Excitation The 2-D TLM routine can be modified so as to represent a sinusoidal excitation from a single point. In this case, the incident excitations at (51,51) in the 2-D TLM code are replaced by a discretized sinusoid of the form shown in Figure 4.5. Excitation can either be constrained or free. In the first case, using Figure 4.5 as the example, we use 44 individual discrete contributions to launch one complete wavelength onto the mesh. Thus the north, south, east, and west components at the excitation point are given by


Chapter four:


57

Figure 4.5 Amplitude (vertical axis) versus time (horizontal axis) for a discretized sinusoid

i

⎡ 2π ⎤ VN /S/E/E = 0.5 sin ⎢ k ⎥ (where k is the iteration index) ⎣ 44 ⎦

(4.8)

which ensures that the nodal potential at the excitation point at the kth iteration is given by

⎡ 2π ⎤ k⎥ φ(excitation point) = sin ⎢ ⎣ 44 ⎦

(4.9)

If we have a free excitation with ∆x/λ = 0.05 then this can best be expressed in terms of a scattering event as

s

⎡ 2π ⎤ VN /S/E/W = sVN /S/E/W + 0.25 sin ⎢ k⎥ ⎣ 20 ⎦

(4.10)

which can be used to update the scattered data as shown in the code below. When this is run as a MATLAB program a smooth wave form will be observed with an amplitude that falls off as would be expected in a 2-D model. By changing the divisor within the sine expression in Equation (4.10) it is possible to investigate the influence of dispersion. When the divisor is set to four then this corresponds to ∆x/λ = 0.25, i.e., cut-off.


58

Transmission Line Matrix in Computation Mechanics SINUSOIDAL EXCITATION %% 2-D TLM routine with free sinusoidal excitation at (51,51) clear ITERS = 60; rows = 101;cols = 101;

%% the number of iterations %% dimension of problem

%% set up connection matrices Cn=[[zeros(rows-1,1) eye(rows-1)];zeros(1,rows)];Cs=Cn'; Ce=[[zeros(cols-1,1) eye(cols-1)];zeros(1,cols)];Cw=Ce'; %% set up transmission and reflection coefficients T = 0.5.*ones(rows,cols); R =-0.5.*ones(rows,cols); %% set up transmission and reflection coefficients %% for boundaries BRn = [ones(1,cols);zeros(rows-1,cols)]; BRs = [zeros(rows-1,cols);ones(1,cols)]; BRe = [zeros(rows,cols-1) ones(rows,1)]; BRw = [ones(rows,1) zeros(rows,cols-1)]; %% make room for incident and scattered voltages In=zeros(rows,cols);Is=zeros(rows,cols); Ie=zeros(rows,cols);Iw=zeros(rows,cols); Sn=zeros(rows,cols);Ss=zeros(rows,cols); Se=zeros(rows,cols);Sw=zeros(rows,cols); phi = zeros(rows,cols);

%% make room for 'phi'

for k = 1:ITERS% start of TLM routine Sn = In.*R + Is.*T + Ie.*T + Iw.*T;%% calculate scatter pulses Ss = In.*T + Is.*R + Ie.*T + Iw.*T; Se = In.*T + Is.*T + Ie.*R + Iw.*T; Sw = In.*T + Is.*T + Ie.*T + Iw.*R; Sn(51,51) = Sn(51,51) + 0.25*sin(2*pi*k/20);%% addition of driving component Ss(51,51) = Ss(51,51) + 0.25*sin(2*pi*k/20);%% at node (51,51) Se(51,51) = Sn(51,51) + 0.25*sin(2*pi*k/20); Sw(51,51) = Sn(51,51) + 0.25*sin(2*pi*k/20); In = Cs*Ss + BRn.*Sn;

%% calculate incident pulses

Is = Cn*Sn + BRs.*Ss; Ie = Sw*Cw + BRe.*Se; Iw = Se*Ce + BRw.*Sw; phi = (In + Is + Ie + Iw)/2;%% superimpose incident pulses end


surf(phi)

We can also consider the effect of phased sources on beam steering (first implemented in TLM by Saleh and Blanchfield2). This is frequently done in radio communications where we might have two or more radiating monopoles, length λ/4 with respect to a perfectly reflecting ground-plane and set λ/2 apart. In the case of acoustics it is more usual to use a rigid boundary (ρ = 1) as the reflector, so that the monopoles should be of length λ/2 (see Figure 4.6). This presents an interesting TLM problem that has not been encountered up to this moment. In the example that we will consider we will have a wavelength equivalent to 14∆x. Thus we would have two sources, (7∆x apart) adding driving components sin(2πk/ 14 2 ) and -sin(2πk/ 14 2 ) respectively (k is the iteration time index). However, in most formulations that have


Chapter four:


sin(ωt)

59

sin(ωt + π)

Figure 4.6 Two monopoles of length λ/2 set λ/2 above a rigid surface

been presented up to this moment it is usual to set a boundary half way between two nodes so that what leaves a node at time k is reflected and returns at time, k + 1. Using this, we would be forced to place the boundary at either (6∆x + ∆x/2) or (7∆x + ∆x/2). For this reason we need to adopt a different strategy so that a pulse, which leaves a node, travels to a boundary during ∆t and returns (in-phase) after a further ∆t. The code, which is shown below, is a slight variant on what has been used earlier in this chapter, but the basic concepts are identical. Because MATLAB does not like dealing with zero-valued indices, the entire operation is moved in from the outer boundary. So, we see five sources at (9,160), (9,167), (9,174), (9,181), (9,188). We set the boundary at (2,:). The pulses scattered from (3,:) toward the ground are put into a temporary store (tempe) and the pulses that were in a temporary store (tempo) from the previous iteration now become incident pulses at (3,:), mod(k,2) is used to distinguish between odd and even iterations. The results of this simulation are shown in Figure 4.7a and are an accurate representation of an “end-fire” beam array showing pressure doubling at the rigid boundary. Figure 4.7b shows the steering effect where there is a λ/4 phase difference between successive sources. %% 2-D TLM routine with multiple phased sinusoidal sources clear ITERS = 260;

%% the number of iterations

Lx = 350;Ly = 160;%% dimension of problem In=zeros(Ly,Lx);Is=zeros(Ly,Lx); Ie=zeros(Ly,Lx);Iw=zeros(Ly,Lx); Sn=zeros(Ly,Lx);Ss=zeros(Ly,Lx); Se=zeros(Ly,Lx);Sw=zeros(Ly,Lx); phi = zeros(Ly,Lx); tempe = zeros(Ly,Lx); tempo = zeros(Ly,Lx); for k = 1:ITERS

% start of TLM routine

xite = 100*sin(2*pi*k/19.798989); %% calculate scatter pulses Sn= 0.5*(-In + Is + Ie + Iw); Ss= 0.5*(+In – Is + Ie + Iw); Se= 0.5*(+In + Is – Ie + Iw); Sw= 0.5*(+In + Is + Ie – Iw); for jx=2:Lx-1

%connection process

In(:,jx) = Ss(:,jx-1); Is(:,jx) = Sn(:,jx+1);


60

Transmission Line Matrix in Computation Mechanics end for jy=2:Ly-1 Ie(jy,:) = Sw(jy+1,:); Iw(jy,:) = Se(jy-1,:); end %addition of five driving components In(9,160)=In(9,160)-xite; Is(9,160)=Is(9,160)-xite; Ie(9,160)=Ie(9,160)-xite; Iw(9,160)=Iw(9,160)-xite; In(9,167)=In(9,167)-xite; Is(9,167)=Is(9,167)-xite; Ie(9,167)=Ie(9,167)-xite; Iw(9,167)=Iw(9,167)-xite; In(9,174)=In(9,174)-xite; Is(9,174)=Is(9,174)-xite; Ie(9,174)=Ie(9,174)-xite; Iw(9,174)=Iw(9,174)-xite; In(9,181)=In(9,181)-xite; Is(9,181)=Is(9,181)-xite; Ie(9,181)=Ie(9,181)-xite; Iw(9,181)=Iw(9,181)-xite; In(9,188)=In(9,188)-xite; Is(9,188)=Is(9,188)-xite; Ie(9,188)=Ie(9,188)-xite; Iw(9,188)=Iw(9,188)-xite; if mod(k,2) ==0

%rigid ground at lambda/2 below sources

Iw(3,:) = tempe(3,:); tempe(3,:) = Sw(3,:); else Iw(3,:) = tempo(3,:); tempo(3,:) = Sw(3,:); end phi = (In + Is + Ie + Iw)/2;%% superimpose incident pulses surf(phi);shading interp;view(0,90);pause(0.01); end


4.5 The 2-D Propagation of a Gaussian Wave-Form The propagation of a Gaussian wave-form in two dimensions is an example that highlights some of the difficulties that apply both to the TLM and finite difference techniques. In this section we examine some of the problems that are encountered and suggest a possible explanation. Let a Gaussian distribution in the time domain be expressed by

−

Apeak e

(t−t0 )2 σ

2

(σ2 is the variance) (4.11)

If this waveform is subject to a Fourier transform, the variance in the frequency domain is given by 1/σ2. This means that the distribution of frequency components can be influenced by altering σ. A truncated Gaussian profile in the time domain (see Figure 4.8) was used to excite the mid-point of a 2-D space in a TLM model. This was implemented as a free excitation where 500*exp(–(iter–5)^2/25) replaced the sinusoid in Equation (4.10). The result shown in Figure 4.8 can be improved by taking a larger value of variance, but this does not eliminate the problem. One can see effects, which are reminiscent of Figure 4.3. There are also significant components of negative amplitude. In the first instance, one should remember that the


Chapter four:


61

(a)

(b) Figure 4.7 Examples of acoustic beam-steering using phased arrays. (a) An end-fire beam obtained from an array of five sources set λ/2 above a rigid boundary and λ/2 apart with 180° phase difference between adjacent sources. (b) A steered beam resulting from an array of four sources placed λ/2 above a rigid boundary and λ/2 apart. Reading from the right, the nodes are driven at ωt, (ωt + π/4), (ωt + π/2), (ωt + 3π/4).

high frequencies are not eliminated in a Gaussian excitation, so that there will still be components that fall outside the ∆x/λL/0.1 preferred range. The other problem is due to the use of a mesh approach where the velocity is approximately 70.7% of the free-space velocity. One can confirm that the wave form in Figure 4.9 moves as k ⁄ ( 2 ) (where k is the iteration index), but close inspection of this wave-form using the “ceil” command in MATLAB confirms that there are components, albeit very small ones, at x = k∆x, y = k∆y at time k (see Figure 4.10). Now, using Huyghen’s principle, we know that any wave at any time is the result of constructive interference of wavelets ahead of the advancing wave front and destructive interference of wavelets in the wake of the retreating wave front. If the main front is advancing at a velocity of 1 ⁄ ( 2 ) per iteration, while there are components (albeit very small) that are moving at unit velocity then we can hypothesize that the profile observed in Figure 4.9 is the result


62


1

2

3

4

5

6

7

8

9

Figure 4.8 A nine-point discretized Gaussian with time-step along the horizontal axis. The vertical axis is excitation amplitude corresponding to a particular time-step.

Figure 4.9 The propagation within a 101 × 101 computational space 60 iterations after the application of a nine-point (in iteration time) Gaussian excitation at (51,51) with σ = 5.

of improper cancellation of waves. It is left to the reader as an exercise to truncate the wave-form so that it is symmetrical (in terms of population) about its peak. If the truncation points are rinner and router, then it might be


Chapter four:


63

Figure 4.10 The propagation within a 101 × 101 computational space 48 iterations after the application of a nine-point (in iteration time) Gaussian excitation (Amax = 10) at position (51,51) with σ = 5. The MATLAB command “ceil” was used to round all values toward plus infinity.

expected that the sum of amplitudes over the area between the source-point and rinner should be equal to the sum of the intensity (square of amplitudes) between router and the rectangle bounded by (-k,0 0,k k,0 0,-k).

4.6 Moving Sources The modeling of moving sources requires much care. The early work by Pomeroy3 involved the movement of a source by one mesh-point per iteration. Bearing in mind that, even within the minimum dispersive bounds, the velocity of propagation of a wave on a TLM mesh is only 70.7% of the free-space velocity, a one node shift of the source (or in his case, sound reflecting surface) per iteration is equivalent to Mach 1.414 source and the results were quite spectacular. When this work was repeated for a “sinusoidal” source moving at much lower speeds, the results were less satisfactory. The results for a single-point source moving by one node every five iterations are shown in Figure 4.11 where four distinct regions can be seen. There is a compacted region in front of the moving source, an extended wave behind, and there are two “unperturbed” regions normal to the movement. These are separated by lines of change, which, in the wake of the source, resemble a ship’s bow-wave. Some improvements were noted when a scheme for continuous rather than discrete movement (due to one of these authors) was adopted. The O’Connor method for a source moving from


64

Transmission Line Matrix in Computation Mechanics 200 180 160 140 120 100 80 60 40 20 direction of motion 20

40

60

80

100 120 140 160 180 200

Figure 4.11 A sinusoidal source (∆x / λ = 0.1), which is moved step-wise by one nodal distance every five time-steps. The observations were made after 128 iterations.

position (x) to position (x + ∆x) during time n∆t is to divide the intervening space into n subunits and to take the corresponding weighted mean of the amplitudes at (x) and (x + ∆x). However, when the time-sampled amplitudes are analyzed in the frequency domain, further discrepancies are noted. We see the shifted frequencies due to the advancing and retreating source, but we also see a series of harmonics. Additionally, the shifted frequencies do not quite match those predicted by the normal formulae for Doppler shift and only do so when the 1 ⁄ ( 2 ) factor in the velocity is taken into account. Our investigations into the source of this discrepancy revealed several issues that are frequently overlooked, not only in TLM, but in other forms of time-domain modeling.4 The first of these concerns the manner in which sources are generally defined in space and time. This is best illustrated using a finite line source, which excites a 2-D mesh with a sine wave. The results are shown in Figure 4.12a, where the influence of the ends can be seen to introduce a modulation of the propagating waves. A cross-section of the excitation is shown in Figure 4.12b where it can be seen that the excitation is not a single sinusoid, but comprises two sinusoids of identical frequency that are displaced with respect to each other. The conventional 2-D single-point sinusoidal excitation involves injecting pulses 0.5Vmaxsin(2πk/λ) in four directions at every time-step. If the source is moved in the x-direction there is significant interaction between the independent propagating waves. Those injected to north and south do not experience a frequency shift, while those injected in the east and west directions do. All of this is not surprising in view of the fact that TLM is a space-time process, and the excitation, which is frequently used, is distributed only in time. What


Chapter four:


65

200 180 160 140 120 100 80 60 40 20

(a) 20

40

60

80

100

0.8 0.6

0.2 0 −0.2

Source

Amplitude

0.4

−0.4 −0.6 −0.8

(b) Position

Figure 4.12 (a) Profile produced by a finite line-source in 2-D space. (b) Cross-sectional diagram of part of the wave-form emanating from either side of the line-source at its center.

is surprising is the fact that anomalies are not normally observed in the case of a stationary source, but perhaps the south, east, and west components of the wave injected toward the north are masked by major components in those


66


directions. Experiments involving moving observers and a stationary source have confirmed that the spurious harmonics can be significantly reduced if the source is distributed in both time and space. This then leaves one issue still unresolved, the forward and aft bow-wave. The reason for these is quite simple if one considers that the components of the sinusoidal excitation are a series of pulses (see Figure 4.5). When a train of pulses is injected at a point then the dispersion effects as observed in Figure 4.3 are smoothed out. However, the situation is quite different if the source point is allowed to move. In this case we can consider the pulse injected at the origin (point “o” in Figure 4.13) as an independent source of dispersive signal that was injected at t0 and has propagated up to the present instant. The points of maximum signal along the diagonals, as they appear in Figure 4.3 are denoted by the black dots on the t0 circle (note that this is 45° with respect to the origin. The t1 circle is the outer periphery of the dispersive signal resulting from an injection at t1 at an origin, which is displaced to the right with respect to “o.” The 45° maxima with respect to this new source origin are also marked as black dots. This process is continued right up to the source at its position at the present time, and simple geometric considerations indicate that the angles between the source in its current position and all previous dispersive peaks are:

⎡ N ⎤ tan −1 ⎢ ⎥ for the advancing signal ⎣N −2⎦

(4.12a)

⎡ N ⎤ tan −1 ⎢ ⎥ for the retreating signal ⎣N +2⎦

(4.12b)

where N is the fractional Mach number (V/N). These expressions have been confirmed in experimental simulations between V/4 and V/10 and the spurious harmonics can now be interpreted in terms of ineffective cancellation of dispersive effects due to the moving source, something that can be reduced (at least in high frequency content) by spreading the source over several nodes.

4.7 Propagation in Inhomogeneous Media The algorithms, which have been presented up to now, are entirely scalable for a homogeneous medium; the sound speed is all that is required to go from the time-domain to the frequency domain. The situation becomes more complex where there are factors that give rise to different sound speeds (change in medium density or compressibility, or in the case of submarine acoustics, change in salinity).


Chapter four:


t0

t1

t2

t3

67

t

4

Figure 4.13 The propagation from a moving excitation point at discrete time intervals.

∂2 E ∂t2

Maxwell’s equation

∇2 E= µε

Lossless TLM equation

∂2 V ∂2 V ∂2 V ∂V 2 + + = L C d d ∂x 2 ∂y 2 ∂z2 ∂t2

(4.14)

Acoustic wave equation

∂ 2p ∂ 2p ∂ 2p 1 ∂p 2 + 2+ 2= 2 2 2 c ∂t ∂x ∂y ∂z

(4.15)

(4.13)

The analogy between the Maxwell electromagnetic equation (where µ and ε are the medium permeability and permittivity), the lossless TLM equation, and the acoustic wave equation suggests that variations in propagation velocity will be mirrored by changes in the product LdCd. In other words, changes in velocity can be implemented by changing the impedance within the TLM network. We start this analysis by considering a 2-D space comprising two media (Regions A and B) as shown in Figure 4.14. Signals traveling in Region A are in an environment of impedance, ZA and we can use ρ = –1/2, τ = 1/2. The impedance in Region B is ZB and ρ = –1/2, τ = 1/2 can also be used. It is only when we come to consider the exchange of information at the boundary that we encounter problems, and this is because of the impedance mismatch. At the start of an iteration, pulses will leave nodes (x,y) and (x + 1,y) as shown in Figure 4.14. They each travel a distance ∆x/2, after which they encounter a change in impedance, which causes additional scattering. The sign and magnitude depends on the direction of motion.


68


ZA Z B (x,y)

(x+1,y)

Region A

Region B

Figure 4.14 A material discontinuity halfway between two sets of nodes.

ρ A→ B =

ZB − ZA ZB + ZA

(4.16)

The pulse arriving from A at B will experience a complementary reflection coefficient:

ρB→A =

ZA − ZB ZA + ZB

(4.17)

The transmission coefficients are similarly defined so that the connection process across the boundary now becomes:

V ( x , y ) = ρA→ B skVE ( x , y ) + τ B→ A skVW ( x + 1, y )

i k +1 E

V ( x , y ) = ρB→ A skVW ( x , y ) + τ A→ B skVE ( x , y )

i k +1 W

(4.18)

4.8 Incorporation of Stub Lines There is a totally different way of approaching the problem of material discontinuities, which requires a slightly more complicated scattering matrix, but which has no requirement for an intermediate scattering of the form shown in Equation (4.18). This uses the concept of stubs, which has been borrowed from microwave engineering. Equations (4.14 and 4.15) confirm the relationship between the speed of light in the electromagnetic wave


Chapter four:


69

equation and the permittivity and permeability in the Maxwell equation. So any variations in speed could be modeled by changing either ε or µ or both. In this case we alter ε by changing the electrical capacitance. If we take the two regions in Figure 4.14 we can say that the capacitance of the medium is related to the in-vacuo capacitance through the relative permittivity: CA = εAC0 and CB = εBC0

(4.19)

We can also relate the permittivities in the two regions: ε B = εA + εS so that CB = CA + CS

(4.20)

The impedance within Region B is then

ZB =

∆t ∆t = C B C A + CS

(4.21)

This can be written in reciprocal form as:

C C 1 1 1 = A + S = + ∆t ∆t ZB Z A ZS

(4.22)

This means that ZB can be replaced by a parallel combination of ZA and an additional line called a stub, which ensures the match between the lines as shown in Figure 4.15. The main feature of a stub is that it should act as a storage rather than a “leakage” element. This means that data, which is passed to it during the matching process, is returned to the network and not lost. If the stub is to represent an additional capacitance, then it must be terminated in an open-circuit (ρ = 1). By arranging the stub to have length, ∆x/2 we ensure that signals are returned to the network after one time-step. Changes in medium permeability could be implemented by using a similar, half-length stub with a short-circuit termination. This represents an additional inductance in the network. The use of the half-length line introduces a small change in the definition of ZS, the stub impedance, which now becomes:

ZS =

∆t / 2 ∆t = CS 2C S

(4.23)


70


(x,y)

(x+1,y)

Z Z

Region A

Region B

Figure 4.15 Differences in material properties loaded into stubs.

The scattering process must now take into account additional contributions, namely reflection and transmission of signals from link-lines into lines and into the stub, as well as the reflection of signals from the stub into the stub and transmission of signals from the stub into the link-lines. The analysis can be done as before and yields a matrix equation, which is in fact computationally efficient and avoids some of the complexities of the discontinuity approach.

⎛ SVN ⎞ ⎛ iVN ⎞ ⎜ SV ⎟ ⎜ iV ⎟ 1 ⎜ S⎟ ⎜ S⎟ ⎜ SVE ⎟ = 4Z + Z S ⎜ iVE ⎟ S ⎜S ⎟ ⎜i ⎟ ⎜ VW ⎟ ⎜ VW ⎟ ⎜⎝ SVSt ⎟⎠ ⎜⎝ iVSt ⎟⎠

(4.24)

where

⎛ −(Z + 2ZS ) ⎜ 2ZS ⎜ 2ZS S=⎜ ⎜ 2Z ZS ⎜ ⎜⎝ 2ZS

2ZS −(Z + 2ZS ) 2ZS

2ZS 2ZS −(Z + 2ZS )

2ZS 2ZS 2ZS

2ZS 2ZS

2ZS 2ZS

−(Z + 2ZS ) 2ZS

⎞ ⎟ ⎟ ⎟ 2Z ⎟ ⎟ (Z − 4ZS )⎟⎠ 2Z 2Z 2Z


Chapter four:


71

In contrast to the discontinuity method, the TLM connection process for the stub method is as in Equation (2.35) with the addition of a contribution that takes account of the open-circuit termination in the stub:

V ( x , y ) = skVSt ( x , y )

i k + 1 St

(4.25)

The signal, which arrives into the node from the stub at time (k + 1)∆t, is equal to the signal that was scattered from the node into the stub at time k∆t. The nodal potential must also take account of the additional contribution from the stub-line:

⎡ 2i VN + 2i VS + 2i VE + 2i VW 2i VSt ⎤ + ⎢ Z ZS ⎥⎦ ⎣ φ( x) = 1 ⎤ ⎡4 ⎢Z + Z ⎥ S⎦ ⎣

(4.26)

The storage or delaying nature of the stub can be demonstrated by considering the 1-D propagation of a pulse along a lossless line, which has a single half-length open-circuit stub as shown in Figure 4.16. As in Equation (4.26) the equation for the nodal potential is given by the sum of the incoming currents divided by the sum of all of the impedances taken in parallel:

⎡ 2i VL + 2i VR 2i VSt ⎤ + ⎢ Z ZS ⎥⎦ φ= ⎣ 1 ⎤ ⎡2 ⎢Z + Z ⎥ S⎦ ⎣

(4.27)

Z

Z

ZS o/c

Figure 4.16 A 1-D lossless line with a half-length open-circuit stub.

Initially we will consider a single pulse incident from the left (iVL = 1200) with all others set to zero. We will also set Z = 1 and ZS = 1 so that 0φ = 800. The incoming pulse sees ZS and Z in parallel as a load and undergoes a reflection. The reflection coefficient is –1/3 so that –400 is


72


reflected. Thus (2/3)1200 is transmitted into the other arms. Thus at the end s s s of the scattering process we have 0VL = –400, 0VR = 800, 0VS = 800. s s Because there is no other scattering source present 0VL and 0VR are of s no further interest. However 0VS reaches the end of the stub and is reflected i back in-phase. At the end of the first time-step we then have 1VS = 800 i arriving back at the node, which gives rise to nodal voltage 1φ = (2/3) 1VS . The pulse arriving at the node from the stub now sees two impedances in parallel ahead of it and therefore the reflection coefficient is –1/3. The i pulse scattered back into the stub is then -(1/3)800, which becomes 2VS and thus 2φ = (2/3) [ (–1/3)800 ]. The next nodal voltage will then be (2/3) [(–1/3)(–1/3)800 ] and so on. The process continues on for all subsequent time, although as it can be seen in Figure 4.17 the contributions tend to a negligible level within a few iterations. Stubs provide a very useful method for treating variations in propagation velocity. Willison5 developed a TLM model for the distortion of a plane wave as it moves between two media and demonstrated that the results are consistent with Snell’s law of refraction. In spite of the obvious benefits of using stubs, we should be aware that they do change the dispersion characteristics of the TLM model space. This has been considered in detail by Meliani.6 His results are summarized in Figure 4.18, which indicates the relevant velocity of propagation and the ∆x/λ region that ensures dispersion-free propagation. The parameter “s” in the figure is a measure of the size of the stub impedance (where s = 2 corresponds to ZSt = ∞). 800

nodal voltage

600

400

200

0 −200

0

1

2

3 4 iteration number

5

6

Figure 4.17 The variation of nodal voltage with time-step for an initial injection of 1200 (from the left) onto the 1-D mesh with a single stub (ZS/Z = 1) shown in Figure 4.16.


Chapter four:


73

0.8 s=2

0.6

s=3 s=4 s=5

Vmesh 0.4

s = 10

0.2

0.05

0.1

0.15

0.2

0.25

∆x/λ Figure 4.18 Dispersion characteristics of a stub-loaded mesh after Meliani.6 The mesh velocity is normalized with respect to the free-space velocity.

Under certain atmospheric conditions the direction of propagation of sound appears to be bent in that it may not be heard in regions close to the source, but may be heard some distance away, and this effect is attributed to a temperature inversion, where the temperature (which affects the velocity of sound) near the ground is much colder than it is some distance above the ground. The effects of a stratified medium on acoustic propagation can be easily modeled. A model space was constructed with a rigid boundary along the horizontal axis. A sinusoidal point wave-source was placed at the bottom center of the space. The velocity of propagation at ground level was determined by the value of the stub (ZSt = 0.5) but increased in the vertical using the expression: ZSt = 0.5 + 0.0001*h3 (where h represents the nodal height above the ground). The results are shown in Figure 4.19 and it is quite clear that the horizontal component of the wave travels much more slowly than the more vertical components. It can also be seen that the horizontal region next to the source has only little amplitude, but at a greater distance there is a significant level, which seems to coincide with a diffractive effect. The system appears to be attempting to maintain continuity between waves of different wavelength, and this leads to a region of destructive and later constructive interference near the surface. The figure shows samples of the amplitude of the horizontal signal and the vertical signal. The position of acoustic reinforcement above the ground is found to be dependent on frequency.


74


Figure 4.19 The effect of a stratified medium on the propagation from a sinusoidal point-source. The vertical and horizontal line plots represent the magnitudes of the signal one node above the horizontal surface and vertically above the source.

4.9 Boundaries This section starts by revisiting some of the boundaries that were introduced in Chapter 2, before proceeding to consider frequency dependent and surface conforming boundaries and perfectly matched boundaries. TLM normally uses the same boundaries as are used in the mathematical analysis of microwaves. A short-circuit has ZT = 0. The reflection coefficient, ρ = –1. ZT = ∞ describes an open circuit and conventional TLM treats this using ρ = 1. By conventional we mean that the pulses on the transmission lines represent voltage (electric field). We could just as easily have used the pulses to describe current (magnetic field), but in this case the boundary descriptions would be reversed. This is called the “duality” property. There is a similar situation in acoustics, and it has already been incorporated implicitly in several examples in this chapter. The algorithms outlined here use pressure as the analogue of voltage, but in order to provide a valid physical description, the boundaries are reversed. A rigid surface (zero displacement boundary) is equivalent to an electrical open-circuit (ρ = 1), while an nonimpeding opening to air (pressure release boundary) in acoustics is equivalent to an electrical short circuit (ρ = –1). The geometry of parts of a room can have a profound influence on sound propagation, particularly if some alcove/door + anteroom or recess acts as a Helmholtz resonator. The modeling of such resonators using TLM was first demonstrated by Clune.7

4.10 Surface Conforming Boundaries The examples that have been cited until now have involved boundaries that conform precisely with the Cartesian mesh. Nature is not always so obliging


Chapter four:


75

and we have to consider examples where boundaries do not match rectangular coordinates. In the first instance we can consider a short-circuit boundary that is placed at 45° to the mesh. The question posed in Figure 4.20 concerns a staircase representation of the problem. Should such a boundary be placed inside or outside the boundary it is intended to emulate?

or

Figure 4.20 Discretization of a 45° boundary by an inclusive or exclusive staircase.

The Pomeroy3 method uses superposition from two nodes so that a reflective barrier is two nodes thick with the reflection process occurring in parallel with the scattering process. Pulses incident at a boundary encounter a sink for all of the signal. At the same time it is acting as a line source for the emerging signals. Thus the boundaries are effectively node pairs that reflect either in the north–south direction or in the east–west direction. The processes do not interact and are considered separately although an individual node may be a member of more than one pair. Figure 4.21 shows a node pair with a smooth boundary located at an incremental distance, ε = L/∆x from node 1.

i

s

2

L

1

VW VW

∆x Figure 4.21 A node pair with a smooth boundary located at an incremental distance, ε = L/∆x from node 1.

At the first instance the scattered pulse from node 1 is given by: s k

VW =

i k

(

VW * 1 − ε

)

(4.28)


76


The residue of the incident signal passes through to node 2 and is returned one time-step later, so that the signal scattered at the next iteration is given by:

VW =

s k+1

(

)

()

VW * 1 − ε + ki VW * ε

i k+1

(4.29)

The algorithm involves two arrays for storing ε-values, and a sparse approach has been found to be particularly effective. The principles of surface conforming boundaries can be used in a wide variety of applications and can help to reduce the substantial level of computation that is required if the staircase discretization is to be significantly less than the minimum wave-length of any incident sound. It has been used to investigate the scattering of helicopter rotor noise from rough surfaces in air–sea rescue from cliff-face environments.8 The tidal-wave hazard due to a collapse in the Canary Islands can be simulated using a surface-conforming (ρ = 1) model for the land-masses adjacent to the nearby oceans and seas. An intermediate result is shown in Figure 4.22, courtesy of M. Morton.

700

600

500

400

300

200

100

100

200

300

400

500

600

700

Figure 4.22 Propagation of subsea compressional wave following a major land-slip in the Canary Islands. Courtesy of M. Morton.


Chapter four:


77

4.11 Frequency-Dependent Absorbing Boundaries de Cogan9 describes the way in which TLM can be used to describe lumped components that are used in electrical filter circuits. In conventional electrical theory the impedance of a capacitor is given by ZC = 1/jωC. There is a slightly different situation in microwave engineering where we are frequently interested in the apparent impedance, Zobs, as observed from one point on a transmission line due to a termination at some other point. This impedance sometimes appears to represent a capacitance and at other times an inductance, a property that depends on the signal frequency and on the observation point as explained below: If a transmission line of length ∆x/2 is terminated by an open-circuit the apparent impedance is given by:

Z j tan( ω∆t / 2 )

Zobs =

(4.30)

This represents a capacitance for 0 < ω∆t/2 < π/2 where ω = 2πf (f = frequency). Within this range Z C = Z obs . The tan(ω∆t/2) term in Equation (4.30) can be expanded using a Maclaurin series so that the equality can be rewritten as:

jωC =

⎤ j ⎡ ( ω∆t / 2 ) 3 + ... ⎥ ⎢ω∆t / 2 + Z⎣ 3 ⎦

(4.31)

This means that for small values of ω∆t/2 we can approximate:

C=

∆t 2Z

(4.32)

which is consistent with our stub models developed earlier. Subject to the condition of small values of ω∆t/2, an open-circuit terminated transmission line can be used in place of a capacitor in an electronic circuit. Similarly, we can replace inductors by short-circuit terminated transmission lines (L = Z ∆t). The instantaneous current in the low-pass filter circuit shown in Figure 4.23 is given by:

I=

k

VS − 2 kiVC R + ZC

(4.33)


78


R

R + VS

C

R o/c

+ VS

VC

+ VS

i

2 VC ZC

(a)

(b)

(c)

Figure 4.23 (a) A low-pass filter circuit, (b) with its TLM analogue, and (c) lumped equivalent circuit.

This is now used to calculate the voltages i

k

VC = 2 kVC + kI ZC VR = k I R

k

(4.34)

The voltage, which is scattered into the transmission line is s k VC

i

= k V C – k VC

(4.35)

This is reflected at the termination and arrives back at the next iteration as: i k + 1 VC

s

= k VC

(4.36)

and is fed back into Equation (4.34) for the next time-step. Equations (4.33) to (4.36) are all that is required to predict the circuit current, kI or any of the measurable voltages kVC, kVR at each discrete instant in time, k∆t. Applications of these basic ideas to modeling of more complex electric circuits can be found in references such as Reference 10. If we have a surface whose absorbing properties are known from experiment then we can devise an electrical filter circuit that matches the observed properties. Although this filter can then be attached at the precise location of the boundary, the implementation of the algorithm requires a little care in terms of correct timing. A pulse leaves the node nearest the boundary and travels along a link transmission line during time ∆t/2. As it strikes the surface of the boundary, there is a pulse coming inwards, which will arrive at the node at the end of the time-step. The superposition of these two signals at the boundary together constitute the source voltage, Vs(tk+1/2) in the filter (see Figure 4.24). This causes a signal to propagate into the filter transmission line, which


Chapter four:


79

Figure 4.24 (a) Low-pass filter at the boundary of a TLM network. (b) lumped equivalent circuit for formal analysis.

strikes the open-circuit boundary and is returned, so that it contributes to the superposition at time tk+3/2. The objective is to determine the magnitude of the signal that is scattered back from the boundary toward the node and this is best done using a formal analysis. The lumped equivalent, which is used, is shown in Figure 4.23. Just as elsewhere, we can define the superposition of pulses from the node and from the filter as the sum of currents divided by the sum of admittances: n

⎡ 2 sV 2 iVC ⎤ + ⎢ ⎥ Z R + ZC ⎦ =⎣ ⎤ ⎡1 1 Yj ⎢Z + R + Z ⎥ C ⎦ ⎣

∑I j=1 n

∑ j=1

j

(4.37)

This is in fact the source voltage for the filter so that

VS =

2 sV ( R + ZC ) + 2 iVC Z R + Z + ZC

(4.38)

So, the filter-modified pulse, reflected from the surface, which becomes incident at the node at the next iteration is:

V ( node) = k +1/s2V ( surface) = k +1/2VS − skV ( node)

i k +1

(4.39)


80


Thus the steps of the algorithm are as follows: The pulse scattered from the node at time (k – 1) arrives at the surface and becomes the start-up signal for the filter (the pulse reflected from the surface toward the node is zero). Equations (4.33 to 4.36) are used to run the filter during one cycle to obtain iVC. The pulse scattered from the node at time (k) arrives at the surface, and so Equation (4.38) can be used to calculate kVS. kVS is used to calculate the pulse scattered from the surface using Equation (4.39) and provides the source voltage for the next filter cycle.

4.12 Open-Boundary Descriptions The standard lossy TLM node comprising a series connection of transmission lines and resistors introduces attenuation and phase shift. Additional degrees of freedom can be introduced by the inclusion of a shunt impedance, r (see Figure 4.25). We can define a matching condition as follows: an impulse arriving in from the transmission line on the left and seeing the combination of components in front interprets them as having an effective impedance, Z. This is summarized by the equation:

⎡1 1 ⎤ Z = R+⎢ + ⎥ ⎣r R+Z⎦ Z

R

R

−1

(4.40)

Z

r

Figure 4.25 A basic PML network.

Provided this condition applies, then an incoming signal will not be reflected at the end of the left-hand transmission line. It will however be attenuated by the network and the extent of this can be shown to be

Vout Z - R = Vin Z + R

(4.41)


Chapter four:


81

Although any choice of R will provide an output which is attenuated this analysis gives no consideration for the preservation of the character of the impulse. In order to be consistent with the equivalent “infinite” lossless TLM mesh, which this network is supposed to bound, it is necessary that it does not experience distortion. An impulse travelling on a transmission line will not be distorted if the parameters of the line (inductance, L, capacitance, C, resistance, 2R and conductance, G) fulfill the “Heaviside condition.”

L 2R = C G

(4.42)

In any transmission line the impedance is related to L and C through Z = L / C and for the node in Figure 4.23 G = 1/r. Thus Equation (4.42) can be recast as:

Z

2

= 2R r

(4.43)

If we rearrange Equations (4.40) and (4.43) so that we can plot both the matching and distortionless conditions as a function of either R or r, we find that there is no point of convergence. It is not possible to achieve both using the simple network shown in Figure 4.25. However, all is not lost because we can incorporate a stub, a length of transmission line that can be terminated so as to add inductance, L or capacitance, C at a specific location. Figure 4.26 shows a revised network with an open-circuit terminated stub of length ∆x/2. This is a capacitance loading and the revised Heaviside condition is 2

⎡ zSt ⎤ ⎢ z + 1 ⎥ = 2R r ⎦ ⎣ St R

(4.43a)

R Z St r

Figure 4.26 PML network with the inclusion of an open-circuit stub of impedance ZSt.


82


series resistor (R)

2.0 1.6

heaviside

1.2 matching

0.8 0.4 0.2

0.4

0.6

0.8

shunt resistor (r) Figure 4.27 Plots of the matching and distortionless dependence of R as a function of r for a node with a capacitive stub zSt = 2.

zSt is the ratio of the stub impedance to the line impedance. Figure 4.27 demonstrates that it is now possible to obtain a coincidence of the two constraint conditions (i.e., in principle, having a matched attenuating junction without distorting the signal). However, initial experience with this node indicated that there was still some distortion and this was traced to the additional capacitance/inductance in the terminated stub.11 Accordingly, the terminated stub was replaced by an effectively infinite stub (ρSt = 0) and the problem was immediately removed. However, during these experiments it was observed that the level of the reflected signal was critically dependent on the definition of the coincidence. This was iteratively determined by taking the positive root of the quadratic expression for R in terms of r in Equation (4.40) (with normalized impedance) and substituting the expression of r in terms of R derived from Equation (4.43a). The sign and magnitude of the return signal was found to depend on the level of convergence: R and r defined to 12 decimal places yielded an attenuation of –127dB. The analysis, which has been provided above, demonstrates that it is possible to develop an attenuating network that is perfectly matched and does not introduce distortion. Effectively we are saying that corresponding to any stub of magnitude zSt (with respect to the line impedance Z) there is an attenuation factor, F = vout/viin, determined from Equation (4.41), which applies to both line voltage and current. The overall relationship is shown in the graph in Figure 4.28. Thus the entire network can be reduced to a “black-box,” which is located at the mid-points of link-transmission lines.


Chapter four:


83

1.0

attenuation factor (F)

0.8

0.6

0.4

0.2

0.0 −1

0

1 log Z 10 St

2

3

Figure 4.28 Attenuation factor, F against ZSt, where ZSt is the complementary value that guarantees the matched/distortionless conditions for complementary values of R and r.

The effect of the black-box is to attenuate the incident signal by a factor F so that the connection process for M (denoting current or voltage) becomes i k +1

M N ( x , y ) = F ks MS ( x , y + 1)

i k +1

MS ( x , y ) = F ks M N ( x , y − 1)

i k +1

M E ( x , y ) = F ks MW ( x + 1, y )

i k +1

MW ( x , y ) = F ks M E ( x − 1, y )

(where 0 < F /≤/1)

(4.44)

There are those who might have argued that this was perfectly obvious from the start, but the analysis has provided a greater understanding of the underlying processes. For instance, the inclusion of a stub “loads” the network and has the effect of altering the propagation velocity, which may (or may not) be important in a particular implementation. Experience in the use of Berenger boundaries in electromagnetics suggests that significant improvements can be obtained if the transition from the free-space computational environment to the PML occurs over many nodes. This is called a “graded perfectly matched load” (GPML) and in some of the


84


examples, which are presented below, we will use an attenuation factor of the form: z2 ⎛ − ⎞ F( z) = F ⎜ 1 − e B ⎟ ⎟⎠ ⎜⎝

(4.45)

where z is the distance (in terms of number of nodes) into the PML and B is a decay constant that determines the rate of transition.

4.13 Absorption within a PML Region A series of experiments were undertaken to investigate the nature of the absorption within a uniform PML region. It was observed that the attenuation fell off as:

V ( x ) = V0 e − λx

(4.46)

where x is the thickness of the PML (in number of PML nodes), V0 is the signal level outside the PML and λ is a constant which is a function of F. A series of 1-D experiments were undertaken to characterize λ(F) and it was found that λ(F) = -ln(F). So, Equation (4.46) becomes:

V ( x ) = V0 e x ln( F )

(4.47a)

This reduces to V ( x ) = V0 F x which is consistent with the algorithmic implementation. The situation in two dimensions is much more complicated since there are other forms of attenuation. A signal emanating from a point source falls off as (distance)–1 except that in a TLM mesh we must take account of the mesh velocity. In our test, a pure sine wave was excited in space and time and allowed to propagate radially. After moving a distance, d in one direction the wave encountered a flat, uniform PML region and, as it penetrated this, it was progressively absorbed. At any distance x within the PML region we measured the amplitude, A(x). The signal at a distance (d + x) in the radially opposite direction (free-space) was measured and used as A0. We were able to undertake measurements with constant x while varying distance d and repeat this for different values of x. This avoided phase difference problems. Once again, it was found that the attenuation followed Equation (4.46), but an analysis of the F-dependence of the constant yielded a slightly different result from that in the 1-D case

V ( x ) = V0 e

2 x ln( F )

(4.47b)


Chapter four:


85

4.14 Conclusion This chapter has attempted to cover a broad range of TLM techniques that are of relevance in acoustic propagation. It started with some simple examples of Matlab code in the hope that readers who are new to the subject might be able to get started with their own implementations as soon as possible. The latter part of the chapter has addressed specific problems that have been encountered by the authors. Further examples of acoustic applications may be found in Chapter 10 of this work.

References 1. Hansleman D. and Littlefield R., Mastering Matlab 5: A Comprehensive Tutorial and Reference, Prentice-Hall, Upper Saddle River, New Jersey (1998). 2. Saleh A. and Blanchfield P., Analysis of acoustic radiation patterns of array transducers using the TLM method, Int. J. Numerical Modelling, 3 (1990) 39–56. 3. Jaycocks R. and Pomeroy S. C., The precise placement of boundaries within a TLM mesh with applications, in TLM: The Wider Applications, Proceedings of an informal meeting held at the University of East Anglia (UEA), Norwich (27 June 1996), School of Information Systems (University of East Anglia), Norwich, 1996, 2.1–2.5. 4. Aldridge R. V., de Cogan D., Morton M., O’Connor W., and Sant V., Some comments on the TLM modeling of acoustic Doppler effect, in Transmission Line Matrix (TLM) Modelling Proceedings of a meeting on the properties, applications and new opportunities for the TLM numerical method (Hotel Tina, Warsaw 1–2 October 2001), de Cogan D., Ed., School of Information Systems (UEA), 2002. 5. Willison P. A., Transmission line matrix modeling of underwater acoustic propagation, Ph.D. thesis, University of East Anglia, Norwich, U.K., 1992. 6. Meliani H., Mesh generation in TLM, Ph.D. thesis, University of Nottingham, U.K., October 1987. 7. Clune F., M.Sc. thesis, University College, Dublin. 8. de Cogan D., Morton M., Peel D., and Sant V., Hybrid Modeling for echo location and surface characterization, Int. J. Numerical Modeling, 14 (2001) 145–153. 9. de Cogan D. and de Cogan A., Applied Numerical Modelling for Engineers, Oxford University Press, Oxford (1997) 95. 10. Hui S. Y. R. and Christopoulos C., The modeling of networks with frequently changing topology whilst maintaining a constant system matrix, Int. J. Numerical Modelling, 3 (1990) 11–21. 11. de Cogan D. and Chen Z., Toward a TLM description of an open-boundary condition, 13th annual review of progress in applied computational electromagnetics, Naval postgraduate school, Monterey, California (1997) 655–660.



chapter five

TLM Modeling of Thermal and Particle Diffusion 5.1 Introduction Heat and mass transfer are important topics that can be considered under the umbrella of computational mechanics. The application of TLM to problems in these fields has been covered in a recent monograph1 and it could be argued that a reference to that work is sufficient. That said, a book such as this would not be complete without a section on these subjects. It is surely better to include outline source material within one set of covers rather than the inconvenience of having to consult other texts. At the end of this chapter, the reader should be able to achieve the transitions from problem to TLM algorithm and from algorithm to software implementation. Additionally, while the core material remains unchanged, there have been some significant advances in theory and techniques since the monograph was published. The secondary function is therefore to bring the reader up to date with advances in these areas. It is always difficult to decide on a logic of presentation and it is certain that any approach will not satisfy all. It is therefore our intention to make clear at this stage how we propose to progress. In the first instance, we will discuss 1-D (1-D) TLM algorithms. Considerable progress can be achieved without going too deeply into the fundamental theory. Suffice to say that until recently such algorithms were based on an assumption that the diffusion equation could be modeled using the telegraph equation under conditions where space and time discretization allowed the wave component to be neglected (the basis of this will be addressed in the theory section). The next section extends algorithm development to two and three dimensions, but a discussion on the types of excitation will be reserved for the section after, which will include some new material. At this point, we are in a position to discuss some practical implementations of thermal and particle diffusion. Once all of this is in place, we will present some supporting theory. This has been covered in considerable detail elsewhere and for that reason

87


88


this final section will be concerned with recent developments in our understanding of the underlying processes that influence the development and implementation of TLM algorithms. The presentation of some advanced theory and applications will be held over for the final chapter.

5.2 Spatial Discretizations and Electrical Networks for Thermal and Particle Diffusion In lossless TLM formulations there was nothing particularly special about the spatial discretizations. The situation is somewhat different here. In the first instance, we may wish to place our observation points at different locations: in the center of the node (Figure 5.1a) or at the interface between nodes (Figure 5.1b). There may be very good reasons for this, such as a comparison with an experimental observation or with another numerical technique. One or other arrangement may also be more convenient if a TLM routine is to be interfaced with another technique, e.g., the effect of temperature on mechanical stress modeled using a hybrid consisting of TLM for heat-flow and finite element modeling (FEM) for the stress calculations. The discretizations in Figure 5.1 can be modeled using electrical network analogues. In a finite difference analysis of the equivalent circuit it would be possible to use either a T-network or a Π-network approach. The difference is shown in Figure 5.2a, b. Similar differences are also found in TLM implementations and depend on the relative placement of the transmission lines and resistors within a node. If one node is separated from its neighbors by means of lengths of transmission line, this is called a link-line representation and is equivalent to a Π-network (Figure 5.2c). The alternative is to have the resistors at the interface between nodes and to make observations at the center of the transmission line. This is a link-resistor representation (Figure 5.2d) and is equivalent to a T-network. In one dimension, the link-line and link–resistor treatments are completely equivalent, being simply the translation of the observation point and sampling interval. There are, however, significant differences in 2- and 3-D formulations, which will be mentioned later. The potentials (or currents) in a network comprising many repeats of the components shown in Figures 5.2c and 5.2d can be expressed by the lossy wave equation, also known as the telegrapher’s equation

∇2 φ = Ld C d

∂2 φ ∂φ + RdCd 2 ∂t ∂t

(5.1)

Ld, Cd and Rd are the distributed electrical parameters: inductance, capacitance and resistance. Rd and Cd can be interpreted as the thermal resistance and capacitance per unit length.


Chapter five:

TLM Modeling of Thermal and Particle Diffusion

89

(a)

(b) Figure 5.1 Two discretizations of a length of material showing the different positioning of observation points.

x-1

x

x+1

(b)

(a)

R

Z

R

Z

φ

φ

(c)

(d)

R

Figure 5.2 (a) T-network electrical analogue for diffusion, (b) Π-network analogue, (c) link-line TLM node, (d) link-resistor TLM node.


90


The traditional approach is to state that Equation (5.1) can be used to solve heat-flow problems2 on the basis that if the space and time discretizations are suitably chosen then Equation (5.1) reduces to:

∂φ ∇ 2φ R d Cd ∂t

(5.2)

which is equivalent to the diffusion equation with RdCd = 1/D.

5.3 TLM Algorithm for a 1-D Link-line Nodal Arrangement We can approach the development of a 1-D TLM algorithm for diffusion in a range of ways. We will start with the simple treatment for a link-line nodal arrangement. A voltage impulse entering a link-line node will travel along a transmission line during a time ∆t/2. At this point, it encounters a discontinuity, ZT = (R + R + Z). The reflection coefficient ρ = (ZT – Z)/(ZT + Z) is then: ρ=

R R+Z

(5.3)

Z R+Z

(5.4)

and the transmission coefficient is τ=

Let us start by assuming that at time k two incident pulses, kiVL(x) and V (x) are traveling along transmission lines and approaching the resistors k R at the center of node (x) from left and right respectively. The Thévenin equivalent circuit assumes that these pulses have originated from voltage sources 2kiVL(x) and 2kiVR(x), and we can use a simple potential divider formula to calculate the contribution from each to the voltage at the center of the node: i

k φ( x ) =

2 kiVL ( x )( R + Z ) 2 kiVR ( x )( R + Z ) + = kiVL + kiVR (2 R + 2Z) ( 2 R + 2Z )

(5.5)

The scattering of these incident pulses (reflection and transmission) is described by:

VL = ρ kiVL + τ kiVR

s k

VR = τ kiVL + ρ kiVR

s k

(5.6)


Chapter five:


91

Each of the scattered pulses now takes a time ∆t/2 to travel to the boundaries of the node and a further time ∆t/2 before becoming incident pulses at adjacent nodes:

VL ( x ) = skVR ( x − 1)

i k +1

VR ( x ) = kiVL ( x + 1)

i k +1

(5.7)

The repetition of Equations (5.5), (5.6) and (5.7) complete the requirements for a link-line TLM algorithm.

5.4 1-D Link–Resistor Formulation The algorithm for a link–resistor nodal configuration can also be used to give the potentials at the interface between nodes, because it is simply the summation of left- and right-going pulses. At the start of an iteration, six pulses share three positions, (x – 1), (x) and (x + 1) which are situated at the center of transmission lines as in Figure 5.2d (which is the TLM equivalent of Figure 5.2a). These are skVL ( x − 1) , skVR ( x − 1) , skVL ( x ) , skVR ( x ) , skVL ( x + 1) and skVR ( x + 1) . The pulse that is at (x – 1) traveling to the left is no longer relevant to node (x) and is ignored. The same applies to the one that is traveling to the right from (x + 1). The other four pulses travel for time ∆t/ 2 before they are scattered at the resistors. They then become incident on (x) from left and right as:

VL ( x ) = ρ skVL ( x ) + τ skVR ( x − 1)

i k +1

VR ( x ) = ρ skVR ( x ) + τ skVL ( x + 1)

i k +1

(5.8)

The pulses arrive simultaneously at (x) and they sum to give the instantaneous potential k +1

φ( x ) =

VL ( x ) +

i k +1

i k +1

VR ( x )

(5.9)

Once the pulses pass on their way after this incidence it is useful to re-designate them for use at the next iteration:

VL ( x ) =

i k +1

VR ( x ) =

i k +1

s k +1 s k +1

VR ( x )

(5.10)

VL ( x )

A complete algorithm consists of the repetition of Equations (5.8)–(5.10) for k iterations, where k∆t is the total time of the simulation.


92


5.5 Boundaries As in lossless TLM, the standard descriptions of short-circuit and open-circuit terminations can be used to model certain classes of physical boundaries in heat and matter simulations.

5.5.1 Insulating Boundary Any heat approaching an insulating boundary is reflected back into the physical problem (Figure 5.3). This is the open-circuit (ρ = 1) condition, and in both link-line and link-resistor formulations it is customary to place it at the interface between nodes. Thus, a pulse traveling from a node during time ∆t/2, encounters the boundary and arrives back at the node at the end of the time-step.

∆x/2

R

Z

Figure 5.3 Reflection at an insulating boundary

5.5.2 Symmetry Boundary As in lossless propagation, the computation of heat or matter profiles can sometimes be reduced by exploiting any symmetry in the problem, so that only half the problem need be simulated. The interface along the symmetry axis (Figure 5.4) becomes an open-circuit boundary (ρ = 1).

Z

Z

Figure 5.4 The equivalence between a symmetry boundary and an insulating boundary. Two identical pulses crossing this boundary are identical to a reflected pulse.


Chapter five:

5.5.3


93

Perfect Heat-Sink Boundary

This is covered by the definition of the short-circuit boundary of lossless TLM, but some additional care is required. Once again, the boundary is placed at the interface between two nodes. However, there are slight differences between the link-line and link-resistor formulations. In a link-line model, the pulse is halfway along a transmission line when it sees a termination ZT = 0 in front of it. The reflection coefficient is thus ρ = –1. In a link resistor node the normal load impedance that a pulse sees as it reaches the end of the line is R + R + Z. One of these two resistors is associated with the node. The normal description of a short-circuit condition in such cases is that the short is located immediately outside the node. Thus, the line terminating impedance is ZT = R and the reflection coefficient from a short circuit is given by:

ρ=

R−Z R+Z

(5.11)

5.5.4 Constant Temperature Boundaries In a link-line model the transmission line touches the boundary, whose value is held constant (VC, as shown in Figure 5.5).

V

C

i

V

k+1 L

s

V

k L

R

Z

φ(1)

Pseudo source

k

Surface Figure 5.5 The boundary node in a link-line formulation showing the “ghost” source and line together with the incident and scattered pulses.


94


We will assume that there is a “ghost” node outside the boundary, and that this has a source and transmission line. This ensures that the value of the potential (VC) at the surface, which is the summation of the pulse incident on node 1 at each new time-step, and the pulse scattered from node 1 at the previous time-step is always constant.

V ( 1) + skVL ( 1) = VC

i k +1 L

(5.12)

Because skVL ( 1) is known at the present time-step, this equation can be used to calculate k +1iVL ( 1) . The situation with a link-resistor treatment is quite different because it is the resistor, not the transmission line that touches the boundary. There are then two separate considerations: 1. The input from the source, which can now be situated at the boundary (Figure 5.6a) 2. The history of the pulse that is scattered from node 1 and that subsequently approaches the boundary (Figure 5.6b). The source (VC) on the boundary sees a series connection of resistor and impedance, so that the standard potential divider formula gives the signal injected into the line. The pulse scattered toward the boundary sees a short circuit, so that the total, which is incident from the left at a new time-step, is the sum of these contributions.

⎡R−Z⎤ ⎡ Z ⎤ s V ( 1) = VC ⎢ ⎥ + kVL ( 1) ⎢ R + Z ⎥ + R Z ⎦ ⎣ ⎦ ⎣

i k +1 L

(5.13)

s

V

k L

R

VC

R

Z

Z (a)

(b)

Figure 5.6 (a) The network seen by the input at the boundary; (b) the situation seen by the pulse scattered toward the boundary.


Chapter five:


95

5.6 Temperature/Heat/Matter Excitation of the TLM Mesh We have now completed the outline formulation for a bounded TLM mesh for modeling heat and mass transfer. The next section deals with the appropriate methods for excitation, the equivalent of initial/boundary conditions for the equivalent differential equation. The individual sections will of necessity be longer than heretofore in this chapter. We think that the best logical flow is to present relevant interesting observations as they arise, rather than outlining the basics and returning to the subject later. For that reason, we will start with constant temperature boundaries, and it is then only a short step to using TLM to solve the Laplace equation. As will be seen, there are some results that appear to be specific to this technique.

5.6.1 Constant T Boundary as an Input In a TLM algorithm for heat-flow in a 1-D rod, we start with the length and the time for which the simulation is to be run. After a choice of spatial and temporal discretizations, we can write L = M∆x and t = k∆t. Note that, unlike explicit finite difference schemes, there are no stability conditions and the choice of discretizations is a trade-off between accuracy and computational load. The material parameters of the rod that are relevant to this simulation are the density, the specific heat, and the thermal conductivity. From these we can derive the equivalent electrical properties of the node

R=

∆x (kT is the thermal conductivity and a unit “cross sectional area,” A is assumed) kT

C = d C p ∆x (d is density, Cp is the specific heat and A∆x is the “volume”) It can be easily shown that the nodal capacitance for particular time discretization leads to the impedance of the equivalent transmission line via Z = ∆t/C. If we base our construction on a link-line node and if we have a temperature of 100 fixed at x = 0, then we can write i

VL ( 1, k + 1) = 100 − sVL ( 1, k )

(5.14a)

By similar argument, the impulse reflected at the cold boundary (T = 0) is given by i

VR ( M , k + 1) = 0 − sVR ( M , k )

(5.14b)


96


The local temperature at any iteration is then

T ( x , k ) = iVL ( x , k ) + iVR ( x , k )

(5.15)

The initial excitation of the algorithm seems to be important, and this has strong parallels with finite-difference schemes where an initial excitation Thot/2 from the hot boundary is required to avoid strong oscillatory behavior in the results.3 If the TLM scheme described above is run for large values of iteration time, k, then the transients die away and the ultimate solution is a solution of the Laplace equation, d 2T / dx 2 = 0. As time does not appear in this expression, we are at liberty to choose whichever value of ∆t that we like and the natural thing is to choose the one that gives us the fastest convergence. In the limit as t → ∞, the choice of C in Z = ∆t/C is not important. What is important is the choice of Z, but even here we do not have to be explicit, because we can see in Equation (5.3) that it is just one component that determines the value of ρ, which is the critical parameter in the TLM algorithms. Thus, our goal is to choose the value of ρ that provides the fastest convergence. To be able to assess the optimum reflection coefficient, we take a problem space comprising M nodes sandwiched between two boundaries, one at 0ºC and the other at 100ºC. The TLM scheme is then run for as many iterations as are required for the differences between the calculated results and the analytical results to become less than some defined threshold. We define the normalized global error for time-step, k as

ε(k ) =

1 M

x= M

⎛ Tanalytical ( x) − Tcalculated ( x , k ) ⎞ ⎜ ⎟ Tanalytical ( x) ⎝ ⎠ x =1

∑

2

(5.16)

The results, presented as the number of iterations, k to get below normalized global errors (10–7 and 10–4) as a function of ρ for a five node space (M = 5) are shown in Figure 5.7. It is quite clear that there is an optimum value where convergence can be very quick and detailed numerical experiments show that this is very pronounced. For the case considered here a threshold of 10–7 can be reached after 27 iterations. See Table 5.1 for results for different values of M. Possible explanations for these observations will be discussed in the theory section of this chapter.

5.6.2

Single Shot Injection into Bulk Material

This description covers inputs which ultimately lead to Gaussian distributions of matter, heat or temperature profile. It consists of a voltage source


Chapter five:


97

Figure 5.7 The number of iterations required to get below a threshold ε as a function of reflection coefficient ρ for a five-node space (M = 5). Table 5.1 Results for Different Values of M M

ρopt

τ opt / ρopt

k

ε( k )( ×10 7 )

5

0.3600

1.7778

27

0.9916

10

0.2300

3.3478

59

0.8447

15

0.1650

5.0606

85

0.9223

20

0.1200

7.3333

117

0.9868

25

0.1050

8.5238

133

0.9158

(equivalent to temperature input) or a current source (equivalent to heat input) that is switched across one node point during the first iteration in the simulation. If we were to take the viewpoint of the signal coming from the excitation source, it is clear from Figure 5.8 that it sees a junction with equal impedance to left and to right. Thus, the product IEX∆t represents a charge that models the single-shot injection of impurities into a medium. The current divides into equal quantities moving to left and to right. The subsequent diffusion of these current pulses could be monitored using TLM. However, the introductory treatment that has been given in this chapter has been expressed entirely in terms of voltage. This is not a problem, since a current IEX/2 passing through R + Z on the left will give rise to a voltage iVL. Similarly, there will be a voltage iV moving to the right. R


98


R

R

Z

Z IEX

Figure 5.8 The equal distribution of the output from an excitation source.

In terms of simple diffusion we could say that if 1000 particles were injected at the source then the initial conditions would be: k=0iVL = 1000/2 and i k=0 VR = 1000/2. This would be sufficient to initiate a diffusion algorithm. At this point, it is worth considering a curiosity of the link-line algorithm when the propagation of a single shot input is being simulated. At time k = 0, an excitation at (x) gives rise to pulses traveling left and right. They take time ∆t to reach adjacent nodes, so that at k = 1 the values of φ(x – 1) and φ(x + 1) are defined but φ(x) is not. Similarly, the pulses scattered at (x + 1) and (x – 1) scatter so that at k = 2 the values φ(x), φ(x – 2) and φ(x + 2) are defined while φ(x + 1) and φ(x – 1) are not. This will manifest itself as apparent jumps-to-zero of any node at alternate time steps, which is obviously unphysical. Such behavior is observed in other (non-TLM) techniques and, in fact, represents an anomaly. The reasons for this will be discussed later. This can be observed in the simple piece of code to model the diffusion in a 1-D link-line network during the first few iterations following a single input into the center of the space. ************************************************************************ #include <stdio.h> #define refl 0.5 #define trans 0.5 void main(void) { float voltage[22]={0}, vi_left[22]={0}, vi_right[22]={0}, vs_left[22]={0}, vs_right[22]={0}; int i, j, k; /* print heading */ printf(“\n\nLink line:Results over five iterations for nodes 7 to 15”);


Chapter five:


99

printf(“\n=====================================================\n”); /* input */ vi_left[11] = 256; vi_right[11] = 256; /* iteration */ for(i = 1; i 0. We find that the stability of the formulation does not depend on the sign of p2t. The telegrapher’s equation can be approximated by means of a central difference scheme for the two double derivatives with backward difference representation for the single time derivative. This yields an expression that is identical with Equation (5.50) but where the coefficients are

⎛ ∆t ⎞ K⎜ ⎝ ∆x ⎟⎠ px = ∆t + τ

2

⎛ ∆t ⎞ ∆t + 2 τ − 2 K ⎜ ⎝ ∆x ⎟⎠ pt = ∆t + τ

2

and p2 t =

−τ ∆t + τ

(5.54)

Once again (px+ + px– + pt + p2t) = 1, but we note that p2t is unconditionally negative and we are forced to ask what this means, particularly as such formulations can be stable for pt positive. Some clue may be found in the 1-D formulation of the telegrapher’s equation using the TLM technique. The computational space is represented by a discretized network of transmission lines (impedance Z) and series resistors (magnitude 2R). It can be shown that the impedance Z = ∆t/C = L/∆t (where C is the capacitance given by Cd∆x and L is the inductance given by Ld∆x). Similarly, R = Rd∆x. Cd, Ld, and Rd are the


Chapter five:


121

distributed capacitance, inductance and resistance of the transmission line, respectively. The reflection coefficient of a lossy line is given by ρ = R/(R + Z) and the transmission coefficient by Γ = Z/(R + Z). The voltage at node x and time x + 1, is given in terms of local values and previous times as

k +1

φ( x ) = ⎡⎣ k φ( x + 1) + k φ( x − 1) ⎤⎦ Γ +

k −1

(

φ( x ) ρ2 − Γ 2

)

(5.55)

This is a two-step Markov process with a correlation coefficient (ρ2 – Γ2) which, although it can range in value between +1 and –1, is unconditionally stable. Since (ρ + Γ) = 1 we can rewrite this as k +1

φ( x ) = ⎡⎣ k φ( x + 1) + k φ( x − 1) ⎤⎦ p ' x +

k −1

φ( x ) p '2 t

(5.56)

where p’x = Γ and p’2t = (ρ – Γ), so that (p’x+ + p’x– + p’2t) = ρ + Γ = 1. It can be shown that

⎛ ∆t ⎞ K⎜ ⎝ ∆x ⎟⎠ p' x = τ + ∆t

2

and p'2t =

∆t-τ ∆t+τ

(5.57)

We can now return to the finite difference discretizations of the telegrapher’s equation and note the points of similarity with TLM. p’x is identical with px in Equation (5.54). It is similar in form to Equation (5.53). p’t is not defined for TLM. Now, if ∆t >> τ then p’2t → 1, while if ∆t

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