Transmission Line Matrix (TLM) in Computational Mechanics

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.

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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

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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,

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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

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4

Transmission Line Matrix in Computation Mechanics

that the short-circuit half-length stub represented an inductor. The explicit use of these stubs to represent reactive components in discrete electrical networks was first suggested by Johns and O'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

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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

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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

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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

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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).

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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

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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

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∂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)

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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

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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

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

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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

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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.

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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.

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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.*

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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.

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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

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(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,

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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.

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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.

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Transmission Line Matrix in Computation Mechanics

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.

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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.

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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

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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

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Chapter three:

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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)

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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.

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Chapter three:

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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.

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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:

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Chapter three:

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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

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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.

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Chapter three:

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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)

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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)

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Chapter three:

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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,

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38

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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)

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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)

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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.

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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)

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Transmission Line Matrix in Computation Mechanics

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

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Chapter three:

The Theory of TLM: An Electromagnetic Viewpoint

⎛ −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

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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.)

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Chapter three:

The Theory of TLM: An Electromagnetic Viewpoint

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.

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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

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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

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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

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50

Transmission Line Matrix in Computation Mechanics

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.

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Chapter four:

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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

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Transmission Line Matrix in Computation Mechanics

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.

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Chapter four:

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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.

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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:

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Chapter four:

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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

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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

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Chapter four:

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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.

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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

%% end of TLM routine

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

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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);

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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

%% end of TLM routine

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

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(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

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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

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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

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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

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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

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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).

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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.

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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

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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)

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(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

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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

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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.

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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.

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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

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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)

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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.

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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)

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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

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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)

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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)

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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.

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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.

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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

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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)

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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.

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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

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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.

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(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.

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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)

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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.

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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.

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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.

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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.

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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)

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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

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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

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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”);

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printf(“\n=====================================================\n”); /* input */ vi_left[11] = 256; vi_right[11] = 256; /* iteration */ for(i = 1; i <= 5; i++) { /* summation */ for(j = 1; j <= 21; j++) voltage[j] = vi_left[j] + vi_right[j]; /* print results */ printf(“\n”); for(j = 7; j <= 15; j++) printf(“%8.3f”,voltage[j]); /* scattering */ for(j = 1; j <= 21; j++) { vs_left[j] = refl * vi_left[j] + trans * vi_right[j]; vs_right[j] = refl * vi_right[j] + trans * vi_left[j]; } /* connect */ for(j = 2; j <= 21; j++) vi_left[j] = vs_right[j-1]; for(j = 1; j <= 20; j++) vi_right[j] = vs_left[j+1]; } } ************************************************************************

This phenomenon is not observed in a link-resistor algorithm. The point is clearly demonstrated by running the equivalent program for the same 1-D problem. ************************************************************************ #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 resistor:Results over five iterations for nodes 7 to 15”); printf(“\n======================================================\n”); /* input */ vi_left[11] = 256; vi_right[11] = 256;

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100

Transmission Line Matrix in Computation Mechanics /* iteration */ for(i = 1; i <= 5; i++) { /* summation */ for(j = 1; j <= 21; j++) voltage[j] = vi_left[j] + vi_right[j]; /* print results */ printf(“\n”); for(j = 7; j <= 15; j++) printf(“%8.3f”,voltage[j]); /* scattering */ for(j = 1; j <= 21; j++) vs_left[j] = vi_right[j]; for(j = 1; j <= 21; j++) vs_right[j] = vi_left[j]; /* connect */ for(j = 2; j <= 20; j++) { vi_left[j] = refl * vs_left[j] + trans * vs_right[j-1]; vi_right[j] = refl * vs_right[j] + trans * vs_left[j+1]; } } } ************************************************************************

5.7 Flux Injection into Bulk Material 5.7.1

Single Heat Source

A constant rate of heating means that an additional input will be required at every time-step. It is possible to arrange the input at any point in a TLM formulation, but it is often most convenient to add a pulse (IEX) immediately after the incident step at each iteration. The nodal potential at position (x) that also includes a source is:

⎡ 2 k +1iVL ( x ) ⎤ ⎡ 2 k +1iVR ( x ) ⎤ ⎥ + I EX ⎢ ⎥+⎢ R+Z ⎦ ⎣ R+Z ⎦ ⎣ k + 1 φ( x ) = ⎡ 2 ⎤ ⎢R+Z⎥ ⎦ ⎣

(5.17)

The scattered pulses are then:

VL ( x ) = ρ kiVL ( x ) + τ kiVR ( x ) +

s k

VEX 2

V VR ( x ) = τ VL ( x ) + ρ VR ( x ) + EX 2

s k

i k

i k

(5.18)

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where

VEX = I EX ( R + Z ) In a situation where an input is time varying, this will simply involve the adjustment of the amplitude of excitation at every time-step. Figure 5.9 shows an interesting curiosity that is due to an unphysical definition of an input set next to an open-circuit boundary. Let us suppose that we have a single injection of 1000 at k = 0. The 500 that moves to the left immediately encounters a reflecting boundary and is returned just as the other 500 arrives at node 2. This “diffusion by pairs” repeats itself thereafter and can be considered as the fold-back of a Gaussian curve that is displaced by a distance ∆x. 1000

k=1

500

500

k=2

500

250

250

k=3

375

375

125

125

Figure 5.9 “Diffusion by pairs.” The values at the first four nodes during the first three iterations following a single-shot input of 1000 at a node next to a boundary are shown.

5.8 Multiple Flux Sources At first instance this is not much different to the above in that we can repeat the placement of additional sources based on the ideas presented above. However, if we were to run a simulation for many iterations so that the time transient decay, then we would have a TLM implementation of the Poisson equation. 2

2

( d φ ) ⁄ ( dx ) = a (where a is non-zero constant)

(5.19)

The source is normalized to a unit dimension (per length here, per volume in three dimensions). There is a need for caution when implementing this in TLM because of the convention of placing the boundaries halfway between nodes. If these “end-effects” are included, then the results at convergence are indistinguishable from the analytically derived values and

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the effect of ρ on the rate of convergence parallels the observations for the Laplace equation.5

5.9 The Extension to Two and Three Dimensions 5.9.1 Link-Line Formulations Figure 5.10 shows a node with a pulse incident from the west at the instant before it is reflected. The impedance that the pulse sees at the discontinuity is a resistor in series with a parallel arrangement of three impedances (R + Z). The reflection coefficient is therefore ρ=

3R + R + Z − 3Z 2R − Z = 3 R + R + Z + 3 Z 2R + 2Z

(5.20)

The transmitted component down any arm is τ, where 3τ = 1 – ρ. The lumped circuit due to a single incident pulse is also shown in Figure 5.10, but we can assume that in a general formulation there are pulses incident from all directions. Therefore the potential at (x,y) is best expressed as the current from each contributor passing through the point divided by the total admittance (reciprocal of impedance) of the node:

⎡ 2 kiVN ( x ) ⎤ ⎡ 2 kiVS ( x ) ⎤ ⎡ 2 kiVE ( x ) ⎤ ⎡ 2 kiVW ( x ) ⎤ ⎥+⎢ ⎥ ⎢ ⎥+⎢ ⎥+⎢ R+Z ⎦ ⎣ R+Z ⎦ ⎣ R+Z ⎦ ⎣ R+Z ⎦ ⎣ φ( ) = x k ⎡ 4 ⎤ ⎢R+Z⎥ ⎣ ⎦

=

(5.21)

VN ( x ) + kiVS ( x ) + kiVE ( x ) + kiVW ( x ) 2

i k

The pulse, which is scattered from each arm, consists of one reflected and three transmitted components.

⎛ sVN ⎞ ⎛ ρ ⎜s ⎟ ⎜ ⎜ VS ⎟ = ⎜ τ ⎜ sVE ⎟ ⎜ τ ⎜s ⎟ ⎜ ⎝ VW ⎠ ⎝ τ k

τ ρ

τ τ

τ τ

ρ τ

τ⎞ τ ⎟⎟ τ⎟ ⎟ ρ⎠

⎛ iVN ⎞ ⎜ i ⎟ ⎜ VS ⎟ ⎜ iVE ⎟ ⎜i ⎟ ⎝ VW ⎠ k

(5.22)

The connection process is exactly as in lossless TLM. Until now we have used the points of a compass to designate the direction of scattering to and incidence from. This starts to get out of hand in 3-D

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Z

N

node centre i

VW

103

R

R

+

Z

i

E

2 VW -

S Figure 5.10 A two-dimensional link-line lossy node and its lumped equivalent circuit with a pulse. Note the use of the compass points (North, South, East and West) to designate direction.

formulations, although we could use N,S,E,W with U for up and D for down. Electromagnetic modelers tend to use direction numbers, and this is certainly the only way when you need to keep track of 18 or more pulses*. In this presentation we have only six scattered and six incident pulses, but nevertheless, it is beneficial to get used to direction numbers. We can easily develop the nodal voltage from the equivalent in two dimensions:

1 k φ( x , y , z) = 3

6

∑ V ( x , y , z) i k

(5.23)

j

j=1

The scattering process can also be extended from two dimensions. Because the node is a center of symmetry, it is possible to write:

⎛ sV1 ⎞ ⎛ ρ ⎜s ⎟ ⎜ ⎜ V2 ⎟ ⎜ τ ⎜ sV3 ⎟ ⎜ τ ⎜s ⎟ =⎜ ⎜ V4 ⎟ ⎜ τ ⎜ sV ⎟ ⎜ τ ⎜ s 5⎟ ⎜ ⎝ V6 ⎠ ⎝ τ k *

τ ρ

τ τ

τ τ

τ τ

τ τ τ

ρ τ τ

τ ρ τ

τ τ ρ

τ

τ

τ

τ

τ⎞ τ ⎟⎟ τ⎟ ⎟ τ⎟ τ⎟ ⎟ ρ⎠

⎛ iV1 ⎞ ⎜i ⎟ ⎜ V2 ⎟ ⎜ iV3 ⎟ ⎜i ⎟ ⎜ V4 ⎟ ⎜ iV ⎟ ⎜ i 5⎟ ⎝ V6 ⎠ k

(5.24)

A recent development involves trailing subscripts so that Vxn is a pulse going in the -x direction.

We could then write a connection as implies scattering in +x direction.

i

s

Vzn (x, y, z) = kVzp (x, y, z − 1) , where the ‘p’ subscript

k +1

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104

Transmission Line Matrix in Computation Mechanics The reflection coefficient is similarly derived from Equation (5.24) as:

ρ=

3R − 2Z with 5τ = (1 - ρ) 3 R + 3Z

(5.25)

It is only when we come to the connection process that there is a need to be up front about the direction indices. One possible choice is shown in Figure 5.11, and it should be easy to deduce the set of connection equations from this.

Figure 5.11 An arbitrary choice of direction indices (shown in bold) with the connection to neighboring nodes in three dimensions.

5.9.2 Link-Resistor Formulations In a 2-D link-resistor node the transmission lines intersect as in Figure 5.12. This gives rise to scattering at discrete time intervals that are identical to those in lossless TLM, namely:

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Figure 5.12 The two scattering events in a link-resistor cell separated in space by ∆x/2 and in time by ∆t/2.

⎛ sVN ⎞ ⎛ −1 ⎜s ⎟ ⎜ ⎜ VS ⎟ = 1 ⎜ 1 ⎜ sVE ⎟ 2 ⎜ 1 ⎜s ⎟ ⎜ ⎝ 1 ⎝ VW ⎠ k

1 −1 1 1

1 1 −1 1

1⎞ 1⎟⎟ 1⎟ ⎟ −1⎠

⎛ iVN ⎞ ⎜ i ⎟ ⎜ VS ⎟ ⎜ iVE ⎟ ⎜i ⎟ ⎝ VW ⎠ k

(5.26)

However, the presence of linking resistors gives rise to a second set of four scattering events that occur at the half-time intervals. These are incorporated into the connection equations:

V ( x , y ) = ρ' skVN ( x , y ) + τ' skVS ( x , y + 1)

i k +1 N

V ( x , y ) = ρ' skVS ( x , y ) + τ' skVN ( x , y − 1)

i k +1 S

V ( x , y ) = ρ' skVE ( x , y ) + τ' skVW ( x + 1, y )

(5.27)

i k +1 E

V ( x , y ) = ρ' skVW ( x , y ) + τ' skVE ( x − 1, y )

i k +1 W

where ρ’ = R/(R + Z) and τ’ = Z/(R + Z). This is because the pulse reflected from the node and traveling along a line sees its resistor and the resistor and transmission line of the target node as a mis-matching load The nodal potential is then

k φ( x , y ) =

1 2

4

∑ V (x, y) i k

j=1

j

(5.28)

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The extension of the link-resistor formulation to three dimensions can generally be done as it was in the link-line case. The scattering equations are identical to those for a 3-D shunt line node with ρ = –2/3 and τ = 1/3. It is easy to see that as the dimensionality goes up there is significantly more computation in a link-resistor node, and for this reason link-line formulations are the more favored approach.

5.10 Non-Uniformities in Mesh and Material Properties In the treatment of a real problem we might have spatial variations in the value of Z or R. These parameters may be functions of position, time or the magnitude of the nodal potential. Variations as a function of position could reflect differences in adjacent materials, i.e., material inhomogeneities. These can also be due to the manner in which we choose to mesh a particular problem; a cylindrical geometry is best treated by means of a polar mesh, but this does mean that the volume and area of adjacent nodes is quite different. This discretization shown in Figure 5.12 highlights a problem, which in turn raises a point concerning the basis for theory that has been developed up to now in this chapter. If we were looking down at a 3-D wedge, we would say that each discretized section had a different volume. We would know that the equivalent capacitance, C, given by the product of density, specific heat and segment volume would be different for each segment. Similarly, the resistance, given by Rr = ∆r/(ArkT) is different for each segment. If each of these segments surrounds a node point, then in a link-line formulation we will have transmission lines with different impedances (Z = ∆t/C) in contact at the interface between nodes. Let us assume that these are Zr and Zr+∆r. Depending on the direction of travel of a signal we will have a reflection coefficient that is given by →

ρ=

← Z r + ∆r − Z r Z − Z r + ∆r or ρ = r (where the arrows show the direction of travel) Z r + ∆r + Z r Z r + Z r + ∆r

For either of these cases the current transmission coefficient is τI = (1 – ρ). However, the voltage transmission coefficient is τV = (1 + ρ), which ensures the conservation of energy. So, why have we been using (1 – ρ) to represent the voltage transmission coefficient up to this point? The answer lies in an amazing property of lossy TLM meshes. To demonstrate this, we present the picture of the initial scattering events at a 1-D link-line node using a formalism slightly different from elsewhere. For clarity we will consider signals coming in from the left only on the basis that what is happening on the right is (mutatis mutandis) identical (Figure 5.13). t

VL − R τ i I L = ( 1 + ρ) iVL − R( 1 − ρ) i I L

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Figure 5.13 The reflection and transmission of a left incident current pulse in a link-line node.

The voltage seen at the node center due to this incoming pulse depends on the voltage drop across the resistor, R, because i I L = tVL / Z the drop is i VL {1 + ρ − RZ /[Z(R + Z)]} = iVL . The voltage seen at the right transmission line due to the incoming pulse from the left depends on the voltage drop across the other resistor iVL − R t I L = iVL − R(1 − ρ) i I L Since i I L = tVL / Z the drop is given by i VL {1 − RZ /[Z(R + Z)]} = (1 − ρ) iVL . We will return to the polar mesh in more detail once we have looked at some examples of how TLM researchers have tackled the problem of non-uniform meshes in Cartesian coordinates. In 1981, Saguet and Pic6 devised a scheme for the refinement of a rectangular mesh in the form shown in Figure 5.14. This naturally raised the problem of pulse synchronism. de Cogan et al.7 drew inspiration from this and developed a similar technique that was applied to 1-D heat-flow problems. They considered a uniform material where one node was longer than another by a factor b, that is, it had b times the resistance and b times the capacitance (Figure 5.15).

Figure 5.14 Saguet/Pic mesh refinement.

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∆x

b∆x

A

B

Figure 5.15 The connection of nodes of unequal length.

The impedances are given by ZA = ∆t / ∆xCd ZB = ∆t / b∆xCd . Thus, ZB = ZA / b de Cogan and Enders8 subsequently adopted a variant of this process to provide impedance matching across adjacent materials with different thermal properties. These developments have not been limited to one dimension. The Pulko method9 is essentially a transfer of the Saguet/Pic mesh from electromagnetics to diffusion problems. Wong and Wong10 have developed a technique that allows the density of the mesh to be doubled within certain regions of a problem where more detail is required. Figure 5.16 shows a mesh of length ∆x/2 embedded within a mesh of length ∆x. On the right-hand side of the figure, different types of nodes are identified; some are common to both meshes, some are associated with one or another mesh and there is a set of interface nodes. The steps are as follows: 1. Perform a standard TLM routine on the coarse mesh. 2. Determine the interface node voltages by finite difference interpolation.

Vf 0 = Vf 1 =

VC 1 + VC 2 + VC 3 + VC 4 4 VC 2 + Vf 0 + VC 3 + Vf 2

(5.29)

4

3. Determine the boundary voltages in the fine mesh using the interface voltages as boundary conditions. 4. Transfer fine mesh voltages back to the coarse mesh. Since the node voltages on the coarse mesh have been modified, the scattered (or incident) voltages must also be altered. Whereas the previous method has two independent meshes with interconnection, Ait-Sadi et al.11 have developed an alternative approach that maintains synchronism throughout the entire system. The “dangling” lengths of transmission line in the fine mesh of Figure 5.17 are connected to the coarse mesh by means of bus-bars, a concept borrowed from power electrical engineering. These are placed halfway along each line so that the

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Figure 5.16 A mesh of length ∆x/2 embedded within a mesh of length ∆x with the nodes near the interface identified. Note that it is possible to have multiple levels of mesh bisection within a single TLM routine.

Figure 5.17 (a) The connection of a coarse and fine mesh by means of bus-bars; (b) scattering in the region of a bus-bar.

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nodal incidence and scattering can proceed as if each mesh were independent of the other. The difference lies in the method of connection. In each case, we must consider the scattered pulses and what they would see as they arrive at the junction. If we take iV1(x,y), it is traveling along a line of impedance Z1 but sees impedance Z1 and Z2 in parallel as a load. The incident pulses are then ⎡ 2 Z2 ⎤ s ⎡ 2 Z1 ⎤ s ⎡ Z1 ⎤s V (x,y) = ⎢ kV2 (x',y') + ⎢ kV1 (x,y+1) − ⎢ ⎥ ⎥ ⎥ kV1 (x,y+1) ⎣ Z1 + 2 Z2 ⎦ ⎣ Z1 + 2 Z2 ⎦ ⎣ Z1 + 2 Z2 ⎦

i k+1 1

⎤s ⎡ 2 Z1 ⎤ s ⎡ 2 Z2 ⎤ s ⎡ Z1 V (x,y+1) = ⎢ ⎥ kV2 (x',y') + ⎢ Z + 2 Z ⎥ kV1 (x,y) − ⎢ Z + 2 Z ⎥ kV1 (x,y+1) Z + 2 Z 2 ⎦ 2 ⎦ 2 ⎦ ⎣ 1 ⎣ 1 ⎣ 1

i k+1 1

⎡ Z − 2 Z2 ⎤ s ⎡ 2 Z2 V (x',y')= ⎢ 1 ⎥ kV2 (x',y') + ⎢ Z + 2 Z Z Z 2 + 1 2 2 ⎣ ⎦ ⎣ 1

i k+1 2

⎤ s s ⎥ ⎡⎣ kV1 (x,y) + kV1 (x,y+1) ⎤⎦ ⎦

(5.29) Pulko et al.12 have developed a very similar algorithm involving the use of discontinuous lines to substructure a region of space. From this viewpoint it is clear that the approach is not too removed from the 1-D link-resistor formulations. Summation and scattering are as usual in link-line nodes. However, there is a secondary node located at the interface between the two meshes as shown in Figure 5.18. Pulses pass by at the half time-steps and the potential at the intermediate node is then:

Figure 5.18 The intermediate scattering zone in the Pulko spatial substructuring technique.

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φintermediate

⎡ 2 s VA ⎛ 2 s VB 1 + 2 s VB 2 ⎞ ⎤ +⎜ ⎢ ⎟⎥ Z1 ⎠ ⎥⎦ ⎝ ⎢⎣ Z2 = ⎡ 1 2 ⎤ ⎢Z + Z ⎥ 1⎦ ⎣ 2

111

(5.30)

This interface is in fact an additional scattering zone so that the value φ intermediate can be represented by the sum of the incident plus scattered voltages. Thus, intermediate k+1iVA = φintermediate – ksVA = k+1iVB1 = φintermediate – s i s k VB1 = k+1 VB2 = φintermediate – k VB2. The scattered voltages can then be determined by taking the difference φ intermediate minus incident voltage.

5.11 Stubs and the Avoidance of Internodal Reflections We now return to the polar mesh and will continue our consideration of various aspects of the problem. We will then introduce the stub transmission line, a concept unique to TLM, which can be used to overcome many of the difficulties that are encountered in non-uniform meshes. Polar meshes for lossless TLM models with axial symmetry were first developed by Al-Mukhtar and Sitch13 and examined in detail by Naylor.14 de Cogan and John15 extended the technique to diffusion problems. The calculation of the capacitance and the resistance were mentioned above, but the treatment avoided the question of which area to choose Each node in a cylindrical problem has capacitance given by the product of material density, volume and specific heat. The thermal resistance is given by the length divided by the product of area and thermal conductivity. The volume can be calculated as the nodal area (defined by integration) times the nodal height (∆h). There is, however, a question when we come to choose this area that can have a significant effect on the estimate of radial flux. ∆h(θ(r + ∆r)) gives an over-estimate, while ∆hθr is an under-estimate. We use the mean value. The nature of the outer boundary (the right side in Figure 5.19) is determined by the problem, but there is a difficulty with the treatment of the extremity on the left. It is conventional to treat this as a (ρ = 1) boundary and this can be explained in two ways. 1. The area corresponding to the radial center is obviously zero and therefore the resistance of the point must be infinite. ρ = 1 follows from an electromagnetic definition of reflection coefficient. 2. The radial center in a homogeneous problem is also a center of symmetry. Thus, whatever flows from right to left through the point must be matched by an equal flow from left to right.

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To some extent these are intuitive arguments and Cacoveanu et al.16 have adopted a more rigorous approach that places a finite-sized node about the center. This has 2n branches (to ensure continuity of the radial lines) and an open-circuit stub (to ensure pulse synchronism). The uniform discretization in Figure 5.19 has resulted in a situation where there will be an impedance mismatch at the junction between adjacent nodes. Alternatively, we could use a non-uniform meshing so that the capacitance (and consequently the impedance) of each node remains constant and this is shown in Figure 5.20. The differences in properties could be accommodated by the use of stubs, a concept borrowed from microwaves that was mentioned in the last chapter.

Figure 5.19 Nodes of equal length in a polar mesh.

Figure 5.20 A set of nodes of equal surface area in a polar mesh.

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If we look back at Figure 5.19 it can be seen that the innermost node represents the smallest capacitance in terms of a thermal conduction problem. It is therefore used as the base-line value, CB: the capacitances of the successive nodes are then CB + ∆C1, CB + ∆C2, CB + ∆C3 etc. The component CB is used as the link-line in each node and the other part is incorporated as a half-length, open-circuit stub placed at the center of each node. The choice of half-length ensures that a signal from the node into the stub reaches the termination after ∆t/2. It is reflected (ρ = 1) and returns to the node one full time-step after it entered the stub. The value of the stub is given by:

ZSi =

∆t 2 ∆C i

(5.31)

The development of the TLM algorithm for the stub-loaded mesh follows the same lines as in the acoustics chapter but the presence of the resistors increases the complexity of the derivation. In a 1-D formulation we now have pulses from Left, Right and Stub. Inspection of Figure 5.21 shows that pulses from left and right will be reflected with a coefficient:

ρ=

2 RZs + ( R + Z )( R − Z ) = ( 1 − τ) ( R + Z )( R + Z + 2 Zs )

(5.32)

τ is the component of the incoming pulse that is transmitted into the arm on the other side of the node. The incoming component that transfers to the stub is given by:

τ' =

2Z R + Z + 2 Zs

(5.33)

A pulse that travels along the stub toward the node will see two identical arms at the discontinuity and will have a reflection coefficient:

ρS =

Z + R − 2 Zs = (1 − τS ) s Z + R + 2 Zs

(5.34)

The scattering process is then given in matrix form as:

⎛ sVL ⎞ ⎛ ρ ⎜s ⎟ ⎜ ⎜ VR ⎟ = ⎜ τ ⎜⎝ sVS ⎟⎠ ⎜⎝ τ ' k

τ ρ τ'

τS ⎞ τS ⎟⎟ ρS ⎠⎟

⎛ iVL ⎞ ⎜i ⎟ ⎜ VR ⎟ ⎜⎝ iVS ⎟⎠ k

(5.35)

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The normal connection equations are used with the addition of the incidence due to the open-circuit termination in the stub:

V ( x ) = skVS ( x )

i k +1 S

The nodal potential is

⎡ 2 k +1iVL ( x ) ⎤ ⎡ 2 k +1iVR ( x ) ⎤ ⎡ 2 k +1iVS ( x ) ⎤ ⎥+⎢ ⎢ ⎥+⎢ ⎥ ZS ⎣ ( R + Z) ⎦ ⎣ ( R + Z) ⎦ ⎣ ⎦ x = φ( ) k +1 ⎡ 2 ⎤ 1 ⎢ ( R + Z) + Z ⎥ S ⎦ ⎣

(5.36)

Figure 5.21 A 1-D mesh with stubs.

5.12 Time-Step Variation There are many applications where the efficiency of the TLM routine is sacrificed on account of some external factor that dictates the magnitude of the time-step. This frequently occurs when an inhomogeneous problem involves flux between materials of significantly different diffusion parameters. In terms of heat flow this might mean that a poor thermal conductor (e.g., water) is in contact with a good thermal conductor (e.g., metal). If we were to use a value of ∆t that is totally appropriate to water, then we would not accurately model the detailed space-time history of the metal. If the situation were reversed, the very much smaller time-step appropriate to the metal would mean that we were simulating little or no change in the water. Johns (in private conversations with author deCogan) pointed out that within a short period of time the metal would approach thermal equilibrium and its subsequent time-history would be determined by the water. He suggested that it should be possible to start with a short time-step (to model the transients in the metal) and as time progressed this could be altered until the simulation was operating at the much longer time-step. He developed a technique that was published posthumously by Pulko.17 This considered the scattered

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pulses as they pass through the mid-point of the link transmission line. Thus, VA and VB of Figure 5.22 become V’A and V’B after the time-step transformation. The technique is based on two fundamental assumptions: 1. Total voltage is conserved through the transformation VA + VB = V’A + V’B

(5.37)

2. Total current is conserved through the transformation

VA − VB V ' A − V ' B = Z Z'

(5.38)

Figure 5.22 Time-step transformation when the line impedance changes from Z to Z’.

These two equations can then be used to derive values for the transformed pulses:

⎛ ⎛ V 'A ⎞ 1 ⎜ (1 + ⎜⎝ V ' ⎟⎠ = 2 ⎜ B ⎜ (1 − ⎜⎝

Z' ) Z Z' ) Z

Z' ⎞ ) Z ⎟ ⎛ VA ⎞ ⎟ Z ' ⎜⎝ VB ⎟⎠ ( 1 + )⎟⎟ Z ⎠

(1 −

(5.39)

This approach (as it stands) requires that there be no impedance mismatch on the link-lines where the impedance transformation is to take place. If the difference in link-line impedance is solely due to the difference in time-steps, then pulses can be exchanged without modification.

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Webb and Gui18 have proposed an alternative time transformation that takes account of any impedance mismatch that might be present due to extraneous factors such as nonlinearities in the scattering parameters, which are covered in the next section. If we consider the pulse that is incident from the left at (x) in Figure 5.23. After time transformation and impedance mismatch reflection the signal incident on (x) from the left is:

Figure 5.23 Transmission lines linking (x – 1) and (x) in the Webb/Gui approach, s h o w i n g t h e s i t u a t i o n b e f o r e t i m e t r a n s f o r m a t i o n : ρL ( x ) a f f e c t s SVL ( x ) and τ R ( x − 1) affects SVR ( x − 1) .

V ( x ) = ρ ' L ( x ) skVL ( x ) + τ ' R ( x − 1) skVR ( x − 1)

i k +1 L

(5.40)

The reflection and transmission coefficients are defined after the time transformation and have to be determined. We define

α=

∆t ' Z '( x ) Z '( x − 1) = = ∆t Z( x ) Z( x − 1)

We will also define the transmitted and reflected voltages across the mismatch as TV and RV. Voltage conservation at the mismatch gives:

V=

T

V RV − α α

(5.41)

Current conservation gives T R V V V = − Z '( x ) Z '( x − 1) Z '( x )

(5.42)

τ 'L ( x) ρ 'L ( x) − =1 α α

(5.43)

We then get

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and

τ 'L ( x) ρ ' ( x) 1 − L = Z( x − 1) Z( x ) Z( x )

(5.44)

Therefore

τ 'L ( x) =

( 1 + α)Z( x − 1) Z( x − 1) + Z( x )

Z( x − 1) − αZ( x ) ρ 'L ( x) = Z( x − 1) + Z( x )

(5.45)

5.13 Some Aspects of the Theory of Lossy TLM 5.13.1 TLM and Finite Difference Formulations for the Telegrapher’s and Diffusion Equations Johns put forward TLM as a “simple, explicit and unconditionally stable” method for solving the diffusion equation.2 He discussed the relationship between this and various forms of finite-difference formulations in terms of a technique called Propagation Analysis.2 The treatment is virtually inaccessible for all except those well versed in electromagnetic theory, but a simple presentation is now available.19 In addition, there is the assumption that Equation (5.1) can be represented by (5.2) if the space and time discretizations are chosen correctly. But what does this mean? We will start this section by looking at the relationship between finite difference and TLM schemes and in particular we will present ideas based on transition probabilities that help us toward understanding the reasons for the numerical stability of the TLM method. It was mentioned above that a comparison between TLM and finite difference was not as easy as might first appear, as the link-line node represents a Π electrical network, while the latter is a T electrical network. Early attempts at comparison ignored the fact that the two were separated by ∆x/2 in space and ∆t/2 in time. No such problems arose if the comparison was with a link-resistor network. It is known20 that the telegrapher’s equation (5.1) avoids the paradox of infinite velocity of propagation that is inherent in transmission line analogues of the parabolic diffusion equation (5.2). This equation is often presented in the literature as

⎡ ∂2V ⎤ ∂V + τ ⎢ 2 ⎥ = K ∇ 2V ∂t ⎣ ∂t ⎦

(5.46)

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(K ≡ 1/(Rd Cd) is the diffusion constant and τ = Ld/Rd is the relaxation time). A probabilistic interpretation of Equations (5.1) and (5.2) has been presented by Kaplan21 for the case of heat flow. Depending on the exact formulation and the spatial and temporal discretizations, there might be unconditional stability, conditional stability or unconditional instability. Malachowski22 extended the treatment to include TLM, but to avoid any confusion in the use of symbols, he retained τ for relaxation time and used the symbol Γ for the TLM transmission coefficient, and the essentials of that work will be presented here. If in a finite difference of formulation of Equation (5.2) we represent the time derivative by means of a central difference, then, using a TLM-like representation, we can write

V ( x) =

k +1

V ( x ) + kV ( x + 1) p x − + kV ( x − 1) p x + + kV ( x ) pt

k −1

where p x − = p x + =

(5.47)

2 K ∆t 4 K ∆t and pt = − 2 ∆x ∆x 2

The coefficients px– = px+ can be interpreted as the probability that a signal will move its spatial position by one point to either side of x during a single time interval. The coefficient pt is interpreted as the probability that a signal will remain at position x during a single time interval. Because there is no way in which pt, as it appears in Equation (5.47), can be non-negative, we believe that this is consistent with the well-known instability of such formulations. If we now consider a backward difference treatment where the time derivative is given by:

V ( x ) − k −1V ( x ) ∆t

k

We obtain

In this case, p x =

and

V ( x ) = ⎡⎣ kV ( x + 1) + kV ( x − 1) ⎤⎦ p x + kV ( x ) pt

k +1

(5.48)

K ∆t (i.e., half the value of px– and px+ in Equation (3)) ∆x 2 ⎡ 2 K ∆t ⎤ pt = ⎢1 − ⎥ ∆x 2 ⎦ ⎣

The probability of remaining at point, (x) during one time interval is non-negative if

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K ∆t 1 ≤ ∆x 2 2 (i.e., the stability limit of the backward finite difference formulation). We note that the sum of probabilities, (px+ + px– + pt) = 1. The forward difference scheme is unconditionally stable.23

∂V = ∂t

V ( x ) − kV ( x ) ∆t

k +1

which yields

V ( x ) = ⎡⎣ k +1V ( x + 1) +

k +1

V ( x − 1) ⎤⎦ p x + kV ( x ) pt

k +1

(5.49)

⎡ K ∆t ⎤ ⎢ 2⎥ ∆x ⎦ px = ⎣ ⎡ 2 K ∆t ⎤ ⎢1 + ⎥ ∆x 2 ⎦ ⎣

where

is positive and always less than unity and

pt =

1 ⎡ 2 K ∆t ⎤ ⎥ ⎢1 + ∆x 2 ⎦ ⎣

is positive and exists in the range (0 < pt ≤ 1) Because of the implicit nature of the formulation these probabilities should be defined in a slightly different way from previously. In this case, we are looking back from the time, (k + 1 )∆t and interpreting what has happened. Thus, px– is the probability that a signal has arrived at (x) from an adjacent position during the time-step. Similarly, px+ is the probability that a signal has remained at (x) during the time-step. This definition implies that the sum (px+ + px– + pt) = 1, which is true. As px and pt are positive Equation (5.49) is stable for arbitrarily chosen ∆t and ∆x. We can now move to consider the similar representations of the telegrapher’s equation. Equation (5.1) can be analyzed as a central difference approximation in all three derivatives and the results presented in a probabilistic form:

V ( x ) = ⎡⎣ k +1V ( x + 1) +

k +1

V ( x − 1) ⎤⎦ p x + kV ( x ) pt +

k +1

V ( x) p2 t

k −1

(5.50)

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where the coefficients can be interpreted as follows: px is interpreted as the probability that signals will be transferred from nodes (x – 1) and (x + 1) to node (x) during ∆t.

⎛ ∆t ⎞ 2K ⎜ ⎝ ∆x ⎟⎠ px = ∆t + 2 τ

2

(5.51)

pt is the probability that a signal will remain at node, (x) during ∆t

pt =

∆t2 ⎞ 4 ⎛ τ − K ⎟ ∆t + 2 τ ⎜⎝ ∆x 2 ⎠

(5.52)

p2t is the probability that a signal which is at node (x) will still be at node (x) two time-steps later and is given by.

p2 t =

∆t − 2 τ ∆t + 2 τ

(5.53)

We see in this formulation that (px+ + px– + pt + p2t) = 1. It should be noted that there are conditions of discretization and relaxation time when these probabilities could be negative. However, a series of numerical experiments that have been undertaken in parallel with this work has demonstrated that Equation (5.50) is stable so long as pt > 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

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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 << τ then p’2t → –1 and yet the algorithm remains unconditionally stable.

5.13.2 Anomalous “Jumps-To-Zero” In Link-Line TLM Among other things, anomalous “jumps-to-zero” are dependent on the initial conditions in a system. Single-shot excitation is a primary requirement, as the effects of successive inputs in continuous or band-limited excitation may smear the phenomenon. In the case of multiple excitation, a sawtooth effect may be observed if there are an odd number of excitations at a single point or at an odd number of adjacent points. Boundaries are known to play a part; they can either accommodate or inhibit the effect. The observation of jumps-to-zero following a single-shot excitation of a spatial mesh is well known in a range of numerical models including link-line TLM. The simple random walk due to Taitel20 is an example. If the finite difference solution for the Laplace equation is solved iteratively using a Jacobi scheme then jumps-to-zero will also be observed. However, the use of single-shot excitation does not of itself guarantee the effect. If the finite-difference solution for the Laplace equation is solved iteratively using the Gauss-Seidel method, no such anomaly is observed. This is also the case

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for heat-flow and particle diffusion simulations following single-shot injection into a TLM link-resistor formulation. Before proceeding, we need to put some flesh on a concept that has been only mentioned in passing, the relationship between physical properties, space and time discretizations and the TLM reflection coefficient. The diffusion constant was given at Equation (5.46) as D = 1/(RdCd). Note that we will use the more conventional symbol for this parameter. In heat flow we have concepts of thermal resistance and thermal capacitance that we can carry over into our electrical analogues. In particle diffusion, these are not normally defined and the choice for the modeler is arbitrary. In our formulation of the basic TLM node, we have two resistors within the discretized node. This is entirely a matter of convenience because the reflection coefficient is then easier to handle.

R 1 = R+Z ⎡ Z⎤ ⎢1 + R ⎥ ⎦ ⎣

ρ =

Z=

Because

ρ=

Because

Cd =

C ∆x

(5.58a)

∆t C

1 1 = ⎡ ⎤ ⎡ ∆t ⎤ Z ⎢1 + R ⎥ ⎢1 + RC ⎥ ⎣ ⎦ ⎣ ⎦

Rd =

2R ∆x

ρ=

or C = C d ∆x R =

1 ⎡ 2∆t ⎤ ⎢1 + D 2 ⎥ ∆x ⎦ ⎣

(5.58b)

Rd ∆x 2

(5.58c)

We can now demonstrate jumps-to-zero and sawtooth effects using a 1-D link-line TLM model. We start with a sample that has an initial single-unit magnitude injection to left and to right at node 15 in a 30-node space. The choice of reflection coefficient is not critical for this example. We have chosen ρ = τ = 0.5 because it is the closest match to simple random walk models and explicit finite-difference models at the stability limit. The initial input was allowed to propagate for 10 iterations. The output in Figure 5.24 clearly shows jumps-to-zero.

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Figure 5.24 Concentration as a function of position in a 30-node link-line TLM mesh following a single unit magnitude initial excitation at node 15. The results (given as fraction of initial excitation) are shown after 10 iterations.

We could reduce both the time and space discretizations by a factor 2.

∆t → ∆t/2, ∆x → ∆x/2, so that the propagation velocity is unchanged. The space is now represented by 60 nodes and the reference to Equation (5.58) shows that the scattering coefficients become ρ = 1/3 and τ = 2/3. Instead of a single-shot initial injection we must inject inputs of magnitude 1/4 to left and right in each of two nodes that replaced node 15 of the earlier test, and this must be done during the first two time-steps. The results that were obtained after 20 iterations were then taken and placed at the appropriate positions on the original 30-node mesh and are shown in Figure 5.25. It can be seen that there are no jumps-to-zero. Finally in this set of demonstrations, we reduce the space discretization by a factor 3. Thus, the original 30 nodes become 90 nodes. The simulation is run for 30 iterations with ∆t’ = ∆t/3 and ∆x’ = ∆x/3 so that the scattering coefficients are then ρ = 1/4 and τ = 3/4. The three central nodes, which have replaced node 15 of the original problem, are each excited by inputs of 1/9 to left and right during the first three iterations. The results are then superimposed on the original problem space and can be seen as a sawtooth profile in Figure 5.26. This process of mesh refinement can be continued when it will be observed that every even subdivision yields a smooth profile. The sawtooth profile becomes finer as the level of odd subdivision is increased. The influence of stubs (for different material properties) and boundary placement was also investigated and the additional results can be summarized as a set of negatives: even if there is a single-shot initial excitation jumps-to-zero will not be observed if:

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Figure 5.25 Replication of the problem of Figure 5.25 using ∆t’ = ∆t/2, ∆x’ = ∆x/2 with initial injection spread over two nodes and two time-steps. The 60-node network was then run for 20 iterations. The results were then ported back onto the original 30-node space.

Figure 5.26 Replication of the problem of Figure 5.24 using ∆t’ = ∆t/3, ∆x’ = ∆x/3 with initial injection spread over three nodes and three time-steps. The 90-node network was then run for 30 iterations. The results were then ported back onto the original 30-node space.

• Link-resistor nodal formulations are used (lossy TLM only). • The excitation point is located between two link-line nodes (lossy TLM only). • A boundary is placed between two nodes (applies to lossy and lossless TLM). • Stubs are used (applies to lossy and lossless TLM nodes).

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Pulko et al.24 have noted that observations made at ∆t in a link-line lossy mesh of nodal length ∆x represent a significant redundancy. There is no loss of accuracy if the same mesh is sampled at intervals of 2∆t. This is quite distinct from a mesh of length 2∆x sampled at 2∆t. These observations are consistent with the work of Enders and de Cogan25 on 2-D random-walk models where it is noted that the propagation space comprises two separate subsystems with identical equations of motion but different initial conditions (see Figure 5.27). The behavior of scattering processes can also be interpreted in terms of sampling. In the link-line node, the signals traveling a distance ∆x are sampled at time intervals ∆t and are coincidentally scattered at the same times. In a 1-D link-resistor node, the signals are sampled at the center of the node. They then travel a distance ∆x/2 during time ∆t/2, where they undergo scattering at the interface between transmission line and resistors. The transmitted and reflected pulses then arrive at locations (x – 1), x and (x + 1) and, after another time interval, ∆t/2, where sampling again takes place. Thus, the network is sampled at intervals ∆t, with scattering events occurring at distances ∆x/2 from the sample point. This could be interpreted as a network with fundamental mesh distance ∆x/2 sampled at ∆t, which is the equivalent of the Pulko system with redundancy removed. In 2-D link-resistor nodes there is one scattering event at the node center with reflection coefficient ρnode = –1/2 and τnode = 1/2 . There are four separate scattering events at the resistor links with ρlink = R/(R + Z) and τlink = Z/ (R + Z). Sampling occurs at the node center at intervals, ∆t, but scattering events occur at distances ∆x/2. In summary, we can say that while the nodal positions (center of the transmission lines) on a link-resistor mesh are populated at all discrete time intervals, the interfaces between resistors are populated only at times (2n + 1/2)∆t (where n = 0,1,2, …). Thus, one set of points on such a lossy TLM mesh exhibits jumps-to-zero while another set does not. One set can be sampled at n∆t, while the other should be sampled at 2n∆t. If this is done appropriately, there is no mixing between the two independent meshes. It is believed that the principles presented here have wider implications than have been appreciated up to now. A boundary placed at the interface between nodes forces a mixing between the meshes of a lossless or link-line lossy TLM model. If boundaries are to be located between nodes, a link-resistor network should be used in lossy TLM formulations. Otherwise, the boundary should be located at a distance ∆x from the node, so that its response is returned to the mesh with the correct parity. Similarly, the correct positioning of the excitation point in a single-shot injection is important. Conventional TLM stubs of length ∆x/2 will return a signal to the node after a time interval ∆t. This will give rise to mixing in lossless networks. In lossy meshes the positioning of the stub is also significant. Mesh mixing will normally occur if it is placed between the resistors. For this reason, a stub of length ∆x is recommended. On the other hand, it

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Figure 5.27 A rectangular mesh decomposed into two submeshes. Initial injection at (0,0) was in the odd submesh (●), which is populated at k = 1, 3, 5, etc. The even mesh (❍), which is not excited at the first iteration, is populated at k = 2, 4, 6, etc.

is possible to use a conventional half-length stub so long as it is located at the center of a transmission line. By this means, link-resistor formulations can be successfully sampled at intervals ∆t and link-line formulations at intervals ∆t/2.

5.13.3 TLM Diffusion Models as Binary Scattering Processes Some of the ideas that have been presented above can be used to demonstrate that there is a continuum between the parabolic (diffusion equation) and hyperbolic modeling (telegrapher’s equation) of heat flow. Thus, the major assumption used until now by TLM modelers is unnecessary. The presentation of this work will be based on a symbolic analysis of scattering in a 1-D

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Figure 5.28 The initial steps in a reflect/transmit scatter process where a unit signal is injected from the left at position (x) at time, k = 0.

problem and, as a first step, we introduce “binary scatter operators.” These are denoted by the letters “r” and “t” with caret signs on top. The definitions given below are for a unit magnitude impulse initially incident from the left, as shown in Figure 5.28. The action of the reflection operator is to move the pulse back on its tracks by one position and multiply it by ρ. The action of the transmission operator is to move the pulse forward in its current direction of travel by one node and multiply its current amplitude by τ.

^ r ( x) = ρ( x − 1) and ^t ( x) = τ( x + 1) At the next time-step, the reflection and transmission operators act on these, so that

^ r^ r ( x) = ρ2 ( x − 1 + 1) and ^ r ^t ( x) = ρτ( x + 1 − 1)

and d ^t ^ r ( x) = τρ( x − 1 − 1) and ^t ^t ( x) = τ2 ( x + 1 + 1)

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We can see here that this is not a commutative algebra because

^ r ^t ( x) = ρτ( x + 1 − 1) =ρτ( x) while

^t ^ r ( x) = τρ( x − 1 − 1) = ρτ( x − 2) We can then extend this in a general way, so that a sequence of operators, ^ r ^t ^t ^ r ^t ( x) can be read from right to left as

ρ2 τ 3 ( x + 1 − 1 − 1 − 1 + 1) =ρ2 τ 3 ( x − 1) We can now look at scattering following an initial excitation at (x) from the left and predict that after four time-steps we will have 25 contributions, starting at ρρρρρ and ending up at τττττ representing all the possible paths that start at the source and spread along the bar by binary scattering. The situation is like a Pascal triangle with self similarity; every point is the start-point for a new Pascal triangle. We can identify the domain of influence that comprises nodes (x ± 1), (x – 3), (x ± 5) with 10, 5 and 1 contributions respectively. The domain of dependence (see Figure 5.29a) shows all the scatter paths that start at the origin and land at a particular point. In the case of (x + 1) we have 10 contributions that are shown in Figure 5.29b and that reduce to 5ρ2τ3 + 3ρ4τ + ρ3τ2 + ρτ4.

5.13.4 Mesh Decimation We have shown previously how Equation (5.58) can be used to see what happens when we subdivide a mesh. We will now push this further. If ∆t → ∆x/99 and ∆x → ∆x/99 then the reflection coefficient is ρ = 0.01 and τ = 0.99. The original excitation is modeled by having excitations, each of 1.020304x10–4 to left and right into 99 adjacent nodes during 99 iterations. The resulting profile is identical to Figure 5.25. Now, if we were to simulate the response in a very big mesh following a single-shot unit input from the left using ρ = 0.01, we would observe a pulse of magnitude τk after k-iterations located at (x + k) with a small tail extending behind it, something that is characteristic of a solution of a telegrapher’s equation. The tail becomes less prominent, but more independent of position as ρ gets ever smaller. This behavior can be interpreted in terms of scatter diagrams of the type shown in Figure 5.28. For simplicity we will consider the response after three time-steps to a unit input to left and right into an unbounded space. The c o n t r i b u t i o n s a re t h e n ( ρ ρ ρ, ρ ρ τ , ρ τ ρ, ρ τ τ , τ ρ ρ, τ ρ τ , τ τ ρ, τ τ τ ) L a n d (ρρρ,ρρτ,ρτρ,ρττ,τρρ,τρτ,ττρ,τττ)R where the subscripts denote injection to left

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Figure 5.29 (a) The domain of dependence of the point (x + 1) at time k = 5; (b) all possible paths that start at the origin and end at the point in question together with the reflection/transmission operations, which are read from left to right.

and right respectively. The final locations of these contributions are shown in Table 5.2. Table 5.2 The Final Locations of these Contributions x–4 ρρρ

x–2 ρττ ρρτ τρρ ττρ

ρρρ ρτρ τρτ ρρρ ρτρ τρτ

x

x+2 ρρτ τρρ ττρ ρττ

x+4 τττ

If ρ = 0.01 and τ = 0.99 then the values of the above binary sequence of contributors would be as shown in Table 5.3 below. If these values are inserted into Table 5.1 then it will be seen that we have 0.97 at (x±4) and the tail between these extremes is similar to that shown in Figure 5.30. So, as we start from ρ = 0.5, the profile is clearly parabolic, but as we reduce it toward zero we get a leading pulse τk. All terms in ρ2 and above vanish, so we are left with a constant tail, comprising ρτk–1 at every other occupied location. The idea of reproducing a single-shot excitation by

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Table 5.3 Numerical Values for the Components of Table 5.2 ρρρ

ρρτ

ρτρ

ρττ

τρρ

τρτ

ττρ

τττ

10–6

9.9 10–5

9.9 10–5

9.801 10–3

9.9 10–5

9.801 10–3

9.801 10–3

0.97

multiple shots at multiple locations in a decimated mesh with very small ρ leads us to the conclusion that a macro-scale profile can be synthesized by the superposition of micro-scale solutions of the telegrapher’s equation. Three of the many components that would add together to represent a Gaussian profile at the macro-scale are shown in Figure 5.30.

5.14 The Statistics of TLM Diffusion Models An observation of the relationship between the TLM scattering parameters and the variance of the Gaussian profile that is obtained at different stages in the iteration process was made many years ago.26 In general, it was noted that the variance could be expressed as

σ2 ( k) =

τ k + additional terms ρ

(5.59)

The analytical solution of the diffusion equation under Gaussian conditions suggested an expression of the form

σ2 ( k) =

τ k ρ

(5.60)

The additional terms in Equation (5.59) were interpreted as an outcome of TLM solving the telegrapher’s equation rather than the diffusion equation. Moravec27 was the first to develop a combinatorial derivation of P(x,k), the population at position (x) at iteration k following a single injection, and this was subsequently used to derive an expression for the variance as a function of the iteration number and scattering parameters. Much later, these results were reproduced using the theory of runs, which formed the basis for a formal derivation of the variance for a 1-D link-line TLM network:

(

)

τ τ−ρ τ−ρ σ ( k) = k − + ρ 2 ρ2 2 ρ2 2

k +1

(5.61)

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Figure 5.30 Percentage of initial input (1.020304 × 10–4) injected at various intervals during a simulation that is run for 99 iterations with ρ = 0.01.

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In their paper, Chardaire and de Cogan28 include a similar treatment for a link-resistor network and have also extended the treatment for link-line and link-resistor networks up to n-dimensions.29

5.15 TLM and Analytical Solutions of the Laplace Equation Earlier in this chapter we presented results that showed that under certain circumstances a TLM algorithm for the Laplace equation could converge very rapidly. We are now in a better position to understand why. de Cogan et al.30 have shown that it is possible to use a combinatoric approach like that described above to develop symbolic expressions for P(x,k), a polynomial in ρ, representing the sum of scatter contributions at position (x) at iteration k. If a suitable value of ρ were inserted into P(x,k), then we would get a value of the temperature at that point (assuming that this is a heat-flow problem). We can then get a measure of the error ε(x,k) = Ttrue(x) – P(x,k). Such an expression will have k roots, but not all will lie within the range 0 • ρ • 1. If there are n nodes in the problem space, we have n error polynomials. If these are plotted against ρ, it is found that there is one value of ρ where one root of each ε(x,k) coincides, and this is observed to be identical with the optimum value of reflection coefficient, which yields fastest convergence. In the sections below we present some details of the analytical solution of the Laplace problem approached from the diffusion equation and the telegraph equation.

5.15.1 Solution of the Diffusion Equation with Fixed-Value Boundaries The problem of a bar of length L and diffusion constant D, initially at temperature zero throughout, connected to two constant value contacts at T(x = 0,t = 0) = 0ºC and T(x = L,t = 0) = 100ºC can be treated by solving the diffusion equation subject to the initial and boundary conditions, which after algebraic manipulation yields

T ( x , t) = 100

x 200 + π L

∞

∑ n =1

( −1) n

n

⎡ ⎛ n2 π2 ⎞ ⎤ ⎛ nπx ⎞ sin ⎜ exp ⎢ − ⎜ 2 Dt⎟ ⎥ ⎟ ⎝ L ⎠ ⎠ ⎥⎦ ⎢⎣ ⎝ L

(5.62)

We now translate this using the appropriate TLM definitions based on material properties, space and time discretizations and the resulting scattering parameters. The length of the bar and the x-position are discretized as L = M ∆x , x = m ∆x. The definitions of Rd, Cd and Z imply a relationship among the diffusion constant, the TLM parameters and the space and time discretizations

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Dt =

133

∆x 2 τ k 2 ρ

(5.62)

This yields a value of the temperature in terms of nodal position, m and time-step iteration, k

m 200 T ( m , k ) = 100 + M π

∞

∑ n =1

( −1) n

n

⎡ ⎛ nπ ⎞ 2 τ ⎤ ⎛ nπm ⎞ ⎢− ⎜ sin ⎜ ex p k ⎥ (5.64) ⎝ M ⎟⎠ ⎢ ⎝ M ⎟⎠ 2 ρ ⎥ ⎣ ⎦

5.15.2 Solution of the Telegrapher’s Equation with Fixed-Value Boundaries Clearly, nothing in Equation (5.64) could account for the observed convergence properties of TLM simulations as a function of ρ. However, if one treats the problem as a telegrapher’s equation, the situation is quite different and de Cogan et al.30 have shown that the roots can be real or imaginary. Following conversion into TLM terminology, this transition occurs when

For n = 1

τ M = ρ nπ

(5.65a)

τ M = ρ π

(5.65b)

which means that we have effectively inhibited the contribution from the dominant mode in the system. We therefore expect that as k increases, the value of T(m,k) converges to its analytical value without any oscillations. Table 5.4 compares values of τ/ρ derived from Equation (5.65b) with those observed during TLM simulations. The agreement is close, but there is a small and persistent difference whose origin is as yet unclear. This is probably due to fundamental differences in the interpretation of the boundaries in the TLM and Fourier approaches. In the Fourier approach, we have assumed constant temperature boundaries for all times. TLM does similarly except for the first iteration. As in explicit finite difference schemes, there is a need to have an initial input of Vhot/2 to avoid long-duration oscillatory solutions. This situation has not been expressed in the Fourier solutions presented here. Maybe this reveals the lack of discipline that is inherent in both finite-difference and TLM models for diffusion. Given the order of the underlying differential equation, we simply do not provide enough starting condition for what is, after all, a two-step process. In the case of TLM we have

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Table 5.4 A Comparison of Optimum Convergence Conditions Derived from TLM and from Fourier Analysis for Different Sizes of Computational Space

M

⎛ --τ-⎞ ⎝ ρ⎠ TLM

⎛ --τ-⎞ ⎝ ρ⎠ Fourier

⎛ --τ-⎞ ⎛ --τ-⎞ ⎝ ρ⎠ TLM ⁄ ⎝ ρ⎠ Fourier

( τ ⁄ ρ ) Fourier – ( τ ⁄ ρ ) TLM --------------------------------------------------------- × 100% ( τ ⁄ ρ ) TLM

5

1.7778

1.5915

1.1171

10.5%

10

3.3478

3.1830

1.0518

4.9%

15

5.0606

4.7746

1.0599

5.7%

20

7.3333

6.3662

1.1519

13.2%

25

8.5238

7.9577

1.0711

6.6%

k +1

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

k −1

φ( x ) ( ρ2 − τ 2 )

and yet we only define the initial inputs at k = 0. It is fortunate that diffusion is normally so forgiving, allowing initial errors to be smeared out over the problem space. Perhaps it is only when the wave and diffusive components cancel that our lack of rigour is revealed.

References 1. D. de Cogan, Transmission Line Matrix (TLM) Techniques for Diffusion Applications, Gordon & Breach, 1998. 2. P.B. Johns A simple, explicit and unconditionally stable routine for the solution of the diffusion equation, Int. J. Num. Meth. Eng.11 (1977) 1307–1328. 3. G.M. Dusinberre, Numerical Analysis of Heat Flow, McGraw-Hill, New York 1949, 121–125. 4. P.B. Johns, A simple, explicit and unconditionally stable routine for the solution of the diffusion equation, Int. J. Num. Meth. Eng.11 (1977) 1307–1328. 5. D. de Cogan, A. Chakrabarti and R.W. Harvey TLM algorithms for Laplace and Poisson fields in semiconductor transport, Proc. Conf. Solid State Crystals: Materials Science and Applications, Zakopane (Poland) 2–27 October 1994. SPIE Proc. vol. 2373, 198–206. 6. P. Saguet and E. Pic, Le maillage rectangulaire et le changement de maille dans la methode TLM en deux dimensions, Electron. Lett. 17 (1981) 277–279. 7. D. de Cogan, A.K. Shah and M. Henini, Variable Mesh TLM Modeling of Heat Flow in Semiconductors, Proc. 4th Int. Conf. on Numerical Analysis of Semiconductor Devices (NASECODE IV), Dublin 1985, 255–260. 8. D. de Cogan, P. Enders and X. Gui, Impedance transformations and mesh coarsening in TLM heat-flow modeling, Numerical Heat Transfer 21(B) (1992) 327–342. 9. S.H. Pulko, A. Mallik and P.B. Johns, Application of transmission line modeling (TLM) to thermal diffusion of bodies of complex geometry, Int. J. Num. Meth. Eng. 23 (1986) 2303–2313. 10. C.C. Wong and W.S. Wong, Multigrid TLM for diffusion problems, Int. J. Num. Mod. 2 (1989) 103–111.

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11. R. Ait-Sadi, A.J. Lowery and B. Tuck, Combined coarse-fine transmission line modeling method for diffusion problems, Int. J. Num. Mod. 3 (1990) 111–126. 12. S.H. Pulko, J.A. Halleron and C.F. Phizacklea, Substructuring of space and time in TLM diffusion applications, Int. J. Num. Mod. 3 (1990) 207–214. 13. D.A. Al-Mukhtar and J.E. Sitch, Transmission line matrix method with irregularly graded space, IEE Proc. (H) 128 (1981) 299–305. 14. P. Naylor, Variable mesh size and circular TLM, M.Sc. thesis, Nottingham University 1982. 15. D. de Cogan and S.A. John, A two dimensional TLM model for the punch-through diode, J. Phys. D., 18 (1985) 507–515. 16. R. Cacoveanu, P. Saguet and F. Ndagijimana, TLM method: a new approach for the central node in polar meshes, Electron. Lett. 31 (1995) 297–298. 17. S.H. Pulko, A. Mallik, R. Allen and P.B. Johns, Automatic timestepping in TLM routines for the modeling of thermal diffusion processes, Int. J. Num. Mod. 3 (1990) 127–136. 18. P.W. Webb and X. Gui, Time-step changes in TLM diffusion modeling, Int. J. Num. Mod. 5 (1992) 251–257. 19. D. de Cogan, Propagation analysis for thermal modeling, IEEE Trans. Components, Hybrids and Manufacturing Technology- Part A CHMT21 (1998) 418–423. 20. Y. Taitel, On the parabolic, hyperbolic and discrete formulation of the heat conduction equation, Int. J. Heat Mass Transf. 15 (1972) 1369–1371. 21. B. Kaplan, Probabilistic derivation of the stability condition of Richardson’s explicit finite difference equation for the diffusion equation, Am. J. Phys. 52 (1984) 267. 22. M.J. Malachowski Stability of solution of the diffusion equation using various formulation of finite difference method, Proc. meeting on the properties, applications and new opportunities for the TLM numerical method, .Warsaw, 1–2 October 2001 (D. de Cogan, Ed.) 23. G. Liebmann, The solution of transient heat-flow and heat transfer problems by relaxation, J. Appl. Phys. 6 (1955) 129–135. 24. S.H. Pulko, A.J. Wilkinson and M. Gallagher, Redundancy and its implications in TLM diffusion models, Int. J. Num. Mod. 6 (1993) 135–144. 25. P. Enders and D. de Cogan, Discrete modeling of transport processes in two spatial dimensions, Int. J. Num. Mod. 5 (1992) 121–129 26 . P.B. Johns A simple, explicit and unconditionally stable routine for the solution of the diffusion equation, Int. J. Num. Meth. Eng. 11 (1977) 107–109. 27. K.L. Moravec, Transmission line matrix modeling: An exact solution for a single pulse input, B.Sc. thesis (Part II), UEA Norwich, 1994 28. P. Chardaire and D. de Cogan, Distribution in TLM models for diffusion (Part I: 1-D treatment) Int. J. Num. Mod. 15 (2002) 317–327. 29. P. Chardaire and D. de Cogan, Distribution in TLM models for diffusion (Part II: multi-dimensional treatment) Int. J. Num. Mod. 16 (2003) 509–534. 30. D. de Cogan, W.J. O’Connor and X. Gui, Accelerated convergence in TLM algorithms for the Laplace equation, to appear in Int. J. Num. Meth. Eng. 63 (2005) 122–138.

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chapter six

TLM Models of Elastic Solids In this chapter we consider the propagation of mechanical waves in solid elastic materials. These waves can, of course, be acoustic waves, but can also take the form of lower frequency disturbances causing visible deformation of the body.

6.1 The Behavior of Elastic Materials Considering the movement of an elastic material in a control volume in response to some disturbance, the motion can be shown to be described by1

ρ

∂ σ xx ∂ τ yx ∂ 2 ux = ρg x + + 2 ∂x ∂y ∂t

(6.1)

and

ρ

∂ 2 uy ∂t

2

= ρg y +

∂ τ xy ∂y

+

∂ σ yy ∂y

(6.2)

In Equations (6.1) and (6.2) ux and u y are displacements in the x and y directions, σ xx , σ yy , τ yx and τ xy are components of stress (force per unit area) as defined in Figure 6.1, and ρ is the material density. g x and g y are components of body force and are included only for generality. Stress components2 are related to displacements via

⎛ ∂ ux ∂ uy ⎞ = τ xy τ yx = G ⎜ + ∂ x ⎟⎠ ⎝ ∂y

(6.3)

137

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y

τ yx

τ xy

σxx

σxx τ xy

τ yx

σyy

x Figure 6.1 Stresses acting on a differential control volume.

(

σ xx = 2 G + λ

σ yy = λ

) ∂∂ux

x

+λ

∂ uy ∂y

∂ uy ∂ ux + 2G + λ ∂x ∂y

(

)

(6.4)

(6.5)

where G is the shear modulus and λ is Lamé’s constant in the plane stress situation. G and λ are related to fundamental material properties according to

G=

E 2 (1 + ν )

λ=

νE

( 1 + ν ) (1 − ν )

where, E is Young’s modulus and ν is Poisson’s ratio defined as

ν=

lateral strain axial strain

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139

Substituting for σ xx , σ yy , τ yx and τ xy in Equations (6.1) and (6.2), using Equations (6.3) to (6.5), and ignoring body forces gives

∂ 2 uy ⎛ ⎛ λ ⎞ ∂ 2 ux λ⎞ ⎜⎝ 2 + G ⎟⎠ + ∂ y 2 + ∂ x ∂ y ⎜⎝ 1 + G ⎟⎠

ρ ∂ 2 ux ∂ 2 ux = G ∂ t2 ∂ x2

2 2 2 ρ ∂ uy ∂ uy ∂ uy = + G ∂ t2 ∂ x2 ∂ y2

⎛ λ⎞ λ ⎞ ∂ 2 ux ⎛ 1+ ⎟ 2 + + ⎜ ⎜⎝ ⎟ G⎠ G⎠ ∂ y ∂ x ⎝

or, rewriting,

(

)

(

)

(

)

2G + λ ∂ 2 ux G ∂ 2 ux G + λ ∂ 2 uy ∂ 2 ux = + + ∂x∂y ρ ρ ρ ∂ y2 ∂ x2 ∂ t2 ∂ 2 uy ∂t

2

=

(

)

(6.6)

2 2G + λ ∂ 2 uy G + λ ∂ 2 ux G ∂ uy + + ∂y∂x ρ ∂ x2 ρ ρ ∂ y2

In classical theory, two waves are identified as propagating in the bulk of an elastic material, one is a rotational or transverse wave and the other an irrotational or dilatational wave. G/ρ is identified as the velocity of the rotational wave, and ( 2G + λ ) / ρ as the velocity of the dilatational wave. If we use C T to represent the velocity of the transverse wave and C D to represent the velocity of the dilatational wave, Equation (6.6) becomes 2 2 ∂ 2 uy ∂ 2 ux 2 ∂ ux 2 ∂ ux 2 2 C C C C = + + − D T D T ∂x∂y ∂ t2 ∂ x2 ∂ y2

∂ 2 uy ∂t

2

= C T2

∂ 2 uy ∂x

2

+ C D2

∂ 2 uy ∂y

2

(

)

(

) ∂∂y u∂ x

+ C D2 − C T2

(6.7)

2

x

Comparing Equation (6.7) with the telegrapher’s equation, the equation solved by a TLM network, there are some important differences. First, cross derivatives are present in the equation modeling the movement of an elastic material but do not feature in the telegrapher’s equation. Second, the coefficient of the double derivative with respect to x is different from the coefficient of the double derivative with respect to y. It also seems necessary to maintain a distinction between displacements in the x and y directions, and so it would seem reasonable to envisage using two TLM networks to solve Equation (6.7).

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Most TLM nodal configurations used in thermal or wave problems have been derived from a fundamental consideration of the physical processes involved and, frequently, intuition has played a part in defining the node. However, in the case of the behavior of an elastic material, the TLM nodal configuration was derived intuitively and then parameterized using an analogy in Equation (6.3) between TLM and state–space (SS) control theory.

6.2 The Analogy between TLM and State Space Control Theory Discrete state space control theory is a description of a system in which the system’s dynamic behavior is described using a set of simultaneous ordinary difference equations (Equation 6.4 and Equation 6.5). In SS, the system state is a set of variables knowledge of which at some time, together with knowledge of future inputs, is enough to allow complete determination of the system’s behavior. This means that the system state represents the accumulation of the effect of all past inputs to the system. It is often possible to represent a state by a vector of finite dimension. If the dimension of a state vector is n, then the system state, denoted by x(t), is an n vector. If the input to the system is u(t), and the output is y(t) x(t + 1) = f(x(t),u(t),t)

(6.8)

y(t) = g(x(t),u(t),t)

(6.9)

and

To describe the system we then need to do two things. First, we need to identify an appropriate vector of state variables and, second, to determine the functions f and g corresponding to the dynamic behavior of the system. So long as the principle of superposition applies f(x(t),u(t),t) = Ax(t) + Bu(t)

(6.10)

g(x(t),u(t),t) = Cx(t) + Du(t)

(6.11)

and

Hence, the discrete SS representation is x(t + 1) = Ax(t) + Bu(t)

(6.12)

y(t) = Cx(t) + Du(t)

(6.13)

and

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Chapter six:

TLM Models of Elastic Solids

141

We are now in a position to look for an analogy between TLM and SS. If we identify the state of a node as the vector of pulses incident on that node, the TLM algorithm can be expressed in SS form. The “standard” TLM scattering equations are

⎡ Φ=⎢ ⎢⎣

∑( l

2Vli Rl + Zl

)

⎤1 ⎥ ⎥⎦ Y

where, Y =

∑ ( R +1 Z ) l

l

(6.14)

l

and

Vlr =

(

(

Vli Rl − zl Φzl + Rl + zl Rl + zl

)

(

)

)

(6.15)

where l denotes the branch number and the other symbols have their usual TLM meanings. We can rewrite these equations in state space form so that for a 1-D network we have

⎡ k +1Vni+1,1 ⎤ ⎡Z 1 ⎥= ⎢ ⎢ i R + Z ⎣R ⎢⎣ k +1Vn+1,2 ⎥⎦

(

R ⎤ ⎡ kVni+1,1 ⎤ ⎡ b1 ⎤ ⎥ + ⎢ ⎥ k un ⎥⎢ Z ⎦ ⎢⎣ kVni+1,2 ⎥⎦ ⎣b2 ⎦

)

(6.16)

In equation Equation (6.16), b1 and b2 are the coefficients of B the input matrix and k u n is the nodal input. The nodal output is then k y n , which is equivalent to the nodal potential and is given by

k

y n = ⎡⎣c 1

⎡ Vi ⎤ c 2 ⎤⎦ ⎢ k ni ,1 ⎥ ⎢⎣ kVn ,2 ⎥⎦

(6.17)

where c 1 and c 2 are the coefficients of the output matrix C. SS makes use of shift operators for convenient notation. Rewriting Equation (6.16) using forward time and space shift operators gives

⎡QI ⎢ ⎣0

0 ⎤ ⎡ kVni ,1 ⎤ 1 ⎡z = ⎢ i ⎥ −1 ⎥ ⎢ V QI ⎦ ⎢⎣ k n ,2 ⎥⎦ R + z ⎣R

(

)

R ⎤ ⎡ kVni ,1 ⎤ ⎡ b1 ⎤ ⎥ + ⎢ ⎥ k un ⎥⎢ z ⎦ ⎢⎣ kVni ,2 ⎥⎦ ⎣b2 ⎦

(6.18)

where, I is the forward shift operator in 1-D space, I −1 the backward shift operator in space, and Q the forward shift operator in time. We can now relate the nodal output, k y n (potential), to the nodal input, k u n , via a transfer function, G, which is derived by eliminating the

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142

Transmission Line Matrix in Computation Mechanics

state vector, i.e., the vector of incident pulses, from Equations (6.16) and (6.17). Rearranging Equation (6.16) gives

⎡QI ⎢ ⎣0

0 ⎤ ⎡ kVni ,1 ⎤ 1 ⎡z − ⎢ i ⎥ −1 ⎥ ⎢ V QI ⎦ ⎢⎣ k n ,2 ⎥⎦ R + z ⎣ R

(

R ⎤ ⎡ kVni ,1 ⎤ ⎡ b1 ⎤ ⎥ = ⎢ ⎥ k un ⎥⎢ z ⎦ ⎢⎣ kVni ,2 ⎥⎦ ⎣b2 ⎦

)

(6.19)

Substituting for the vectors of incident pulses in Equation (6.19) using Equation (6.17) yields

k

y n = ⎡⎣c 1

⎡ ⎡QI c 2 ⎤⎦ ⎢ ⎢ ⎢⎣ ⎣ 0

−1

R ⎤ ⎤ ⎡ b1 ⎤ ⎥ ⎥ ⎢ ⎥ k un z ⎦ ⎥⎦ ⎣b2 ⎦

0 ⎤ 1 ⎡z − ⎢ −1 ⎥ QI ⎦ R + z ⎣ R

(

)

(6.20)

The transfer function, G, is then given by

G = ⎡⎣c 1

⎡ ⎡QI c 2 ⎤⎦ ⎢ ⎢ ⎣⎢ ⎣ 0

0 ⎤ 1 ⎡z ⎢ ⎥− QI −1 ⎦ R + z ⎣ R

(

)

−1

R ⎤ ⎤ ⎡ b1 ⎤ ⎥⎥ ⎢ ⎥ z ⎦ ⎥⎦ ⎣b2 ⎦ (6.21)

=

⎛⎛ ⎛ Rc 1 ⎛ ⎞ ⎞ z z Q⎞ Rc 2 ⎞ b1 ⎜ ⎜ − + ⎟ c1 + + QI ⎟ c 2 ⎟ + b2 ⎜ + ⎜− ⎟ R + z⎠ ⎠ ⎠ ⎝⎝ R + z I ⎠ ⎝ R+z ⎝ R+z Q2 −

R2

+

z2

−

( R + z) ( R + z) ( 2

2

zQ zQI − R+z I R+z

)

(

)

SS theory then means that the characteristic equation of the process being represented by the TLM network can be derived by setting the denominator of Equation (6.21) equal to zero. Setting the denominator equal to zero and dividing by Q gives us

⎛ 2 ⎞ z z − R2 ⎟ −1 Q − I + I −1 = 0 Q+⎜ R+z ⎜ R+z 2 ⎟ ⎝ ⎠

(

(

)

)

(6.22)

Rearranging Equation (6.22) yields

(Q − 2 + Q ) + R2+RZ (1 − Q ) = R Z+ Z ( I − 2 + I ) −1

−1

−1

(6.23)

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Chapter six:

TLM Models of Elastic Solids

143

We are now in a position to make some observations. Considering Equation (6.23),

(Q − 2 + Q ) is the discrete equivalent off −1

( ∆t)

2

(1 − Q ) −1

∆t

∂2 ∂ t2

is the discrete equivalent of

∂ ∂t

is the discrete equivalent of

∂2 ∂ x2

and

(I − 2 + I ) −1

( ∆x )

2

( )

2

Dividing Equation (6.23) by ∆t and substituting the continuous differentials for the discrete equivalents used so far gives

∂2 2R ∂ Z = + ∂ t2 ∆t R + Z ∂ t R+Z

(

)

(

( ∆x ) ) ( ∆t)

2

2

∂2 ∂ x2

(6.24)

which is a 1-D discrete version of the damped wave equation (or telegrapher’s equation) represented by a resistance loaded TLM network. We now use the TLM-state space analogy to check and parameterize a candidate nodal structure for representing Equation (6.6) and, hence, modeling elastic behavior of materials.

6.3 Nodal Structure for Modeling Elastic Behavior If we compare Equation (6.6) with the telegrapher’s equation it seems reasonable to suppose that we will need two TLM networks to model elastic wave propagation in solids: one network will be needed to represent u x and a second to represent uy. We will also need to incorporate the cross derivations in Equation (6.6) and to account for the fact that the coefficient of the double derivative with respect to x is not the same as that of the double derivative with respect to y.

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144

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However, we will first consider representations of cross derivatives of the nodal variable in a single network, so that we are representing an equation of the form

∂ 2Φ ∂ 2Φ ∂ 2Φ ∂ 2Φ + +h =M 2 2 ∂x∂y ∂x ∂y ∂ t2

(6.25)

Considering the array of nodes illustrated in Figure 6.2, in operator notation

∂ 2Φ I - 2 +I -1 corresponds to 2 2 ∂x ∆x

( )

∂ 2Φ J - 2 +J -1 corresponds to 2 2 ∂y ∆y

(6.26a)

IJ -I -1 J -IJ -1 +I -1 J -1 ∂ 2Φ corresponds to ∂x∂y 4 ∆x ∆y

(6.26b)

( )

and

( )( )

To a l l o w i n p u t t o t h e n o d e a t 11 f ro m t h e s h i f t e d p o s i tions IJ , I −1 J , IJ −1 , I −1 J −1 , it seems that we will need an eight link-line node, similar to that investigated for possible use in modeling electromagnetics. If the link lines connecting nodes I , 11, I −1 and nodes J , 11, J −1 have impedance Z, then it is reasonable to suggest that the diagonal lines would have impedance 4Z/h where, h is the coefficient of the cross derivative and the factor of four gives consistency of spatial discretization. A reasonable nodal structure is illustrated in Figure 6.3. The scattering matrix corresponding to Figure 6.3 is given in Equation (6.27), where Ψ = ( 4 / z) + (1/ zs ) . The characteristic equation is the determinant of the expression ⎡⎣SI - SC ⎤⎦ , where, SC is the scattering matrix of Equation (6.27) and SI is Equation (6.28).4,5

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Chapter six:

TLM Models of Elastic Solids

145

Figure 6.2 Operator notation for nodal positions relative to a central origin.

z

- 4z h

z

4z h

zs 4z h

z

- 4z z

h

Figure 6.3 Nodal structure for the incorporation of cross-derivatives.

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146

Transmission Line Matrix in Computation Mechanics

⎡2 ⎢z − Ψ ⎢ s ⎢ 2 ⎢ ⎢ zs ⎢ 2 ⎢ ⎢ zs ⎢ 2 ⎢ ⎢ zs ⎢ 2 1 ⎢ 4 1 ⎢ zs + Z Zs ⎢ 2 ⎢ ⎢ zs ⎢ 2 ⎢ ⎢ zs ⎢ ⎢ 2 ⎢ zs ⎢ ⎢ 2 ⎢⎣ zs

2 z

2 z

2 z

2 z

h 2z

h 2z

-h 2z

2 z

2 −Ψ z

2 z

2 z

h 2z

h 2z

-h 2z

2 −Ψ z

2 z

2 z

2 z

h 2z

h 2z

-h 2z

2 z

2 z

2 z

2 −Ψ z

h 2z

h 2z

-h 2z

2 z

2 z

2 −Ψ z

2 z

h 2z

h 2z

h 2z

2 z

2 z

2 z

2 z

h 2z

h −Ψ 2z

-h 2z

2 z

2 z

2 z

2 z

h −Ψ 2z

h 2z

-h 2z

2 z

2 z

2 z

2 z

h 2z

h 2z

-h 2z

2 z

2 z

2 z

2 z

h 2z

h 2z

-h −Ψ 2z

⎤ ⎥ ⎥ -h ⎥ ⎥ 2z ⎥ -h ⎥ ⎥ 2z ⎥ -h ⎥ ⎥ 2z ⎥ h ⎥ ⎥ 2z ⎥ -h ⎥ ⎥ 2z ⎥ -h ⎥⎥ 2z ⎥ ⎥ -h − Ψ⎥ ⎥ 2z ⎥ -h ⎥ 2z ⎥⎦ -h 2z

(6.27) ⎡Q ⎢ ⎢0 ⎢0 ⎢ ⎢0 ⎢0 ⎢ ⎢0 ⎢ ⎢0 ⎢0 ⎢ ⎣0

0 QI 0 0 0 0 0 0

0 0 QI −1 0 0 0 0 0

0 0 0 QJ 0 0 0 0

0 0 0 0 QJ −1 0 0 0

0 0 0 0 0 QIJ 0 0

0

0

0

0

0

0 0 0 0 0 0 QI −1 J −1

0 0 0 0 0 0 0

0 0

QIJ −1 0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ QI −1 J ⎦ 0 0 0 0 0 0 0

(6.28)

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Chapter six: ⎡ 2 ⎢ Q- z + Ψ s ⎢ ⎢ −2 ⎢ zs ⎢ ⎢ −2 ⎢ zs ⎢ ⎢ −2 ⎢ zs ⎢ ⎢ −2 ⎢ zs ⎢ ⎢ −2 ⎢ zs ⎢ ⎢ − 2 ⎢ zs ⎢ ⎢ −2 ⎢ ⎢ zs ⎢ −2 ⎢ ⎢⎣ zs

TLM Models of Elastic Solids −2 z QI-

2 z

−2 +Ψ z

147

−2 z

−2 z

−2 z

-h 2z

-h 2z

h 2z

−2 +Ψ z

−2 z

−2 z

-h 2z

-h 2z

h 2z

2 z

−2 z

−2 z

-h 2z

-h 2z

h 2z

−2 +Ψ z

-h 2z

-h 2z

h 2z

2 z

-h 2z

-h 2z

-h 2z

-h +Ψ 2z

h 2z

QI-1 -

−2 z

−2 z

QJ-

−2 z

−2 z

−2 +Ψ z

−2 z

−2 z

−2 z

−2 z

QIJ-

−2 z

−2 z

−2 z

−2 z

-h +Ψ 2z

−2 z

−2 z

−2 z

−2 z

-h 2z

-h 2z

−2 z

−2 z

−2 z

−2 z

-h 2z

-h 2z

2 z

QJ-1 +

h 2z

QI-1 J-1 -

h 2z

h 2z QIJ-1 +

⎤ ⎥ ⎥ ⎥ h ⎥ 2z ⎥ ⎥ h ⎥ 2z ⎥ ⎥ h ⎥ 2z ⎥ ⎥ -h ⎥ 2z ⎥ ⎥ h ⎥ 2z ⎥ ⎥ h ⎥ 2z ⎥ ⎥ h +Ψ ⎥ ⎥ 2z ⎥ h ⎥ QI-1 J+ 2z ⎥⎦ h 2z

-h 2z

h +Ψ 2z

(6.29) In SS terms, the system matrix is then Equation (6.29) Simplifying, the characteristic equation becomes

( I-2+I-1 ) + ( J-2+J-1 ) +

h z+4zs ( Q-1)2 ( IJ-IJ-1 -I-1 J+I-1 J-1 )= 4 2zs Q

(6.30)

Examining Equation (6.30) it is apparent that Equation (6.25) can be represented by the nodal structure of Figure 6.4 and the associated scattering matrix so long as the impedance of the “cross link-lines” is scaled by the factor, h/4. 1

1 rt

ux

y 1 rt

1 h

1 -h

x

ux

1 1 h

1 -h

uy

1 h

1 -h

1 rt 1 h

1 -h

u x node

1

uy 1

1 rt uy node

Figure 6.4 Nodal structure of the investigation of stress transients in two dimensions.

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Transmission Line Matrix in Computation Mechanics

Having established that the coefficient of the cross derivative can be incorporated by an element in the scattering matrix, it is reasonable to suppose that the coefficients of the double derivatives in space can be represented in a similar way. Taking the nonunity value of the coefficient to be rt yields discrete versions of Equations (6.6).

((I-2+I

-1

) rt + ( J-2+J-1 ) u x +

((I-2+I

-1

) + ( J-2+J-1 ) rt u y +

where ,

)

h 1 ( Q-1)2 ( IJ-IJ-1 -I-1 J+I-1 J-1 )u y = 2 ux 4 Q c

)

h 1 ( Q-1)2 ( IJ-IJ-1 -I-1 J+I-1 J-1 )u x = 2 uy 4 Q c

⎛ ⎛ 1 ρ λ⎞ λ⎞ = , rt = ⎜ 2 + ⎟ , h = ⎜ 1 + ⎟ 2 G G G ⎝ ⎠ ⎝ ⎠ c

(6.31) Having now devised a simple method of representing cross derivatives and different coefficients of the spatial double derivatives in a single network, we start to consider the networks necessary to represent Equation (6.6). Figure 6.4 illustrates the proposed networks, one modeling u x and the other u y . By extension of the arguments used when considering a single network, the system matrix corresponding to the TLM algorithm modeling two networks incorporating the cross derivatives of Equation (6.6) between networks is given by Equation (6.32). ⎡ P(Q +1)− 2 zs ⎢ ⎢ −2 zs ⎢ ⎢ −2 zs ⎢ ⎢ −2 zs ⎢ ⎢ −2 zs ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ −2 ⎢ zs ⎢ −2 zs ⎢ ⎢ −2 zs ⎢ ⎢ −2 zs ⎢⎣

−2 rt

−2 rt

−2

−2

−h 2

−h 2

h 2

h 2

0

0

0

0

0

0

0

0

PQI − 2 rt

P −2

−2

−2

−h 2

−h 2

h 2

h 2

0

0

0

0

0

0

0

0

P −2

PQI −1 − 2

−2

−2

−h 2

−h 2

h 2

h 2

0

0

0

0

0

0

0

0

−h 2

h 2

h 2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

−h 2

−2 rt

−2 rt

PQJ − 2

P −2

−h 2

−2 rt

−2 rt

P −2

PQJ −1 − 2

−h 2

−h 2

h 2

h 2

0

0

0

0

PQIJ

0

0

0

−

2 zs

−2

−2

−2 rt

−2 rt

0

0

0

0

0

PQI −1 J −1

0

0

−

2 zs

−2

−2

−2 rt

−2 rt

0

2 − zs

−2

−2

−2 rt

2 zs

−2

−2

−2 rt

−2

−2

PQI − 2

P −2

−1

0

0

0

0

0

0

PQIJ

0

0

0

0

0

0

0

PQI −1 J

−

0

0

0

0

0

0

0

0

P Q +1 −

0

0

0

0

0

0

0

0

−

(

)

2 zs

2 zs

P−

h 2

P−

0 h 2

−h 2

h 2 h 2

−2 rt

−h 2

−h 2

h 2

−2 rt

−h 2

−h 2

P+

−2 rt

−2 rt

−h 2

−h 2

h 2

−2 rt

−2 rt

−h 2

−h 2

h 2

−h 2

h 2

h 2

0

0

0

0

0

0

0

0

2 − zs

−2 rt

−2 rt

−h 2

0

0

0

0

0

0

0

0

−

2 zs

−2

−2

PQJ − 2 rt

P − 2 rt

−h 2

−h 2

h 2

0

0

0

0

0

0

0

0

−

2 zs

−2

−2

P − 2 rt

PQJ −1 − 2 rt

−h 2

−h 2

h 2

−2 rt

−2 rt

−2

−2

−h 2

h 2

h 2

0

0

0

0

0

PQIJ

0

0

−2 rt

−2 rt

−2

−2

−h 2

h 2

h 2

0

0

0

0

0

0

PQI −1 J −1

0

−2 rt

−2 rt

−2

−2

−h 2

−h 2

h 2

P+

0

0

0

0

0

0

0

PQIJ −1

−2 rt

−2 rt

−2

−2

−h 2

−h 2

P+

0

0

0

0

0

0

0

0

P−

h 2

P−

h 2

h 2

h 2

h 2

P −2

PQI

−1

−2

⎤ ⎥ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ h ⎥ 2 ⎥ h ⎥ 2 ⎥ h P+ ⎥ 2 ⎥ h ⎥ 2 ⎥ h 2 ⎥ h ⎥ 2 ⎥ h ⎥ 2 ⎥ h ⎥ 2 ⎥ h ⎥ 2 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ PQI −1 J ⎥ ⎥⎦ 0

0

P

where, P =

(

1 zs

)

+ 2 rt + 2 .

(6.32)

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Chapter six:

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149

It is possible to evaluate the determinant of Equation (6.32) using a symbolic manipulation package. This yields the characteristic equation AB + CA + CB = 0

(6.33)

where,

A=

1 zs

+ 2 rt + 2 2

(Q − 2 + Q ) − rt ( I − 2 + I ) − ( J − 2 + J ) − h4 ( IJ − I −1

−1

−1

−1 −1

J

− IJ −1 + I −1 J

)

−1 −1

− IJ −1 + I −1 J

)

the u x equatio on

B=

1 zs

+ 2 rt + 2 2

(Q − 2 + Q ) − ( I − 2 + I ) − rt ( J − 2 + J ) − h4 ( IJ − I −1

−1

−1

J

the u y equa ation

C=

(

h IJ − I −1 J −1 − IJ −1 + I −1 J 4

)

The expressions for A, B, and C in Equation (6.33) correspond to Equation (6.31) with the relationship between physical and network parameters given by

1 = c2

1 zs

+ 2 rt + 2 2

6.4 Implementation From intuition combined with the theoretical considerations in earlier sections it seems that, to model elastic wave propagation, we need two TLM networks, one having u x as its nodal potential and the other having uy . Each node in the ux network will be connected to its nearest neighbors in the ux network via ordinary link lines but will have an additional four lines connecting it to its nearest neighbors in the uy modeling network. Similarly, each node in the network modeling uy will have four transmission lines linking it to neighbors in its own network, together with four lines connecting it to neighboring nodes in the ux modeling network. All nodes in both networks will also need a stub line for the usual flexibility. In each network, nodal potentials represent the relevant displacement of points, so that displacements in both the x and the y directions are available throughout the modeling area.

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x

y Free Boundary (0,0)

(1,0)

(0,1)

(1,1)

(4,0)

Figure 6.5 The geometry of a semi-infinite plate.

It is sometimes necessary to predict the stresses developed in a material, and these can be extracted conveniently at any time during the modeling period by applying discrete versions of Equations (6.3), (6.4), and (6.5) after nodal potentials have been calculated. To illustrate implementation we model a semi infinite plate loaded by a force over a short length of its free surface, which has been considered by Shibuya et al.7 The situation is illustrated in Figure 6.5. The material properties are

E = 2.10 × 10 4 Kg mm −2 ρ = 7.86 × 10 −6 Kg mm −2 v = 0.3 The coordinates specified in Figure 6.5 refer to points at which TLM predicted stresses are compared with those obtained by Shibuya et al. using finite difference analysis.7 The coordinates refer to positions on a 1-mm grid whose origin is at the center of the free surface. A ramped excitation was applied starting at time = 0 and uniform square mesh with ∆l = 0.1mm was used. The iteration time-step was ∆t = 10 −4 s . Figure 6.6 illustrates the comparison of TLM results with results produced using the finite difference approach for several points within the modeling area. Clearly agreement is very good in terms of both transient and steady state behavior.

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Chapter six:

TLM Models of Elastic Solids

151

1.2

normalized stress

1.0 0.8 0.6 0.4

σx

0.2 0

0

2

4

6

8

10 (a)

12

14

16

18

20

6

8

10 (b)

12

14

16

18

20

1.2

normalized stress

1.0 0.8 σx 0.6 0.4 0.2 0

0

2

4

Figure 6.6 Comparison of TLM results with those obtained using a finite difference analysis at some of the positions in the semi-infinite place (Figure 6.5) (a) 00, (b) 10, (c) 40, (d) 01. The solid lines represent the TLM results and the dotted lines are the finite difference results for the same spatial discretization.

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152

Transmission Line Matrix in Computation Mechanics 0.1 σx normalized stress

0

−0.1 −0.2

−0.3

−0.4

0

2

4

6

8

10

12

14

12

14

16

18

20

(c) 1.2

normalized stress

1.0

σy

0.8 0.6 0.4

σx

0.2 0

0

2

4

6

8

10

16

18

20

(d)

Figure 6.6 (Continued)

6.5 Boundaries The treatment of boundaries was glossed over in the implementation described in the previous section. However, for TLM models of elastic behavior to be useful it is necessary to be able to represent boundaries specified by a range of parameters such as force, displacement, and restraint. As is usual with a TLM model, boundaries are implemented by surrounding the material to be modeled by a layer of boundary nodes. The simplest type of boundary to consider for an elastic model is the displacement boundary because displacement is the nodal variable in the region of bulk propagation. The nodal structure of such boundary nodes is illustrated in Figure 6.7, which shows a boundary parallel to the y-axis with

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Chapter six:

TLM Models of Elastic Solids

153

Figure 6.7 Information exchange for the displacement boundary.

the bulk elastic material lying to the left of the boundary. Considering the u x modeling network, and a boundary node located at coordinate (i,j), this node is connected to its neighbor in the bulk u x modeling network at (i – 1,j) by a transmission line, and is linked by cross lines to the nodes (i – 1,j – 1) and (i – 1,j + 1) in the network modeling u y . Corresponding connections exist for the boundary node at (i,j) in the u y network. The potential at the boundary node is calculated externally to represent the required condition. Dynamic displacement loading makes specification of boundary velocities and accelerations possible; for a given acceleration or velocity, the displacement applied to the node is obtained by integrating the load with respect to time. The boundary condition required is then implemented simply by applying a new nodal displacement to the boundary node at each iteration. However, it is important to realize that this requires the time step to be small compared with dynamics of the loading.

6.6 Force Boundaries Implementing a force boundary condition is rather more complicated than implementing a displacement boundary. For a 2-D situation, such as is shown in Figure 6.8, applied surface forces are related to the stress at the boundary of the elastic material via

X = lσ x + mτ xy Y = lτ xy + mσ y

(6.34)

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Figure 6.8 Nodes in the Φx and Φy meshes involved in modeling force boundaries.

where X and Y are the surface forces and l and m the directional cosines of the normal to the surface as defined by Equation (6.35).

l = cos θ x m = cos θ y n = cos θ z

(6.35)

In Equation (6.35) θ x , θ y , and θ z are the angles between the line of the force and the normals to the surface in the three coordinate directions. Free boundaries have zero applied force so that

X=Y=0

(6.36)

The TLM boundary nodal structure requires nodal potentials, corresponding to displacements, to be specified at the boundary. Hooke’s law relates the stresses in a body to the strains so that the Equation (6.34) may be rewritten in terms of strains as

(

)

(

X = l (2G + λ )ε x + λε y + m Gγ xy

)

(6.37)

and

(

)

(

) )

(

Y = l G γ xy + m λε x + 2 G + λ ε y

(6.38)

Small strain theory also argues that strains and strains are related by

εx =

∂u , ∂x

εy =

∂v , ∂y

εz =

∂w ∂z

(6.39)

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γ xy =

∂v ∂u + , ∂x ∂y

γ yz =

155

∂v ∂w + , ∂z ∂y

γ xz =

∂w ∂u + ∂x ∂z

(6.40)

where εx, εy, εz are the longitudinal or normal strain components, and γxy, γyz, γzx are the shearing strain components. Substituting into Equation (6.37) and (6.38) for the strains gives

⎛ ⎛ ∂v ∂u ⎞ ⎞ ⎛ ∂u ∂v ⎞ X = l ⎜ 2G + λ + λ ⎟ + m ⎜G⎜ + ⎟⎟ ∂x ∂y ⎠ ⎝ ⎝ ⎝ ∂x ∂y ⎠ ⎠

(

)

(6.41)

⎛ ⎛ ∂v ∂u ⎞ ⎞ ⎛ ∂u ∂v ⎞ Y = l⎜G⎜ + ⎟⎠ ⎟ + m ⎜⎝ λ ∂x + 2 G + λ ∂y ⎟⎠ ∂ x ∂ y ⎝ ⎝ ⎠

(

)

In Equation (6.41) the surface forces are related to the displacements on the boundary, but, because of the presence of the spatial derivatives, it is not possible to specify boundary nodal potentials directly from these equations. However, the derivatives may be expressed as difference equations to give a direct relationship between the surface forces and boundary displacements. However, the derivatives in Equation (6.41) have the discrete equivalents

(

)

(

) ≅ ∂u

(

)

(

) ≅ ∂v

Φ x i + 1, j − Φ x i − 1, j 2 ∆l

∂x

Φy i, j + 1 − Φy i, j − 1 2 ∆l

∂y

(6.42)

(

)

(

) ≅ ∂v

(

)

(

) ≅ ∂u

Φ y i + 1, j − Φ y i − 1, j 2 ∆l

∂x

Φx i, j + 1 − Φx i, j − 1 2 ∆l

∂y

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Replacing the derivatives in Equation (6.41) with their discrete equivalents we have

⎛ ⎛ ∂v ∂u ⎞ ⎞ ⎛ ∂u ∂v ⎞ X = l ⎜ 2G + λ + λ ⎟ + m ⎜G⎜ + ⎟⎟ ∂x ∂y ⎠ ⎝ ⎝ ⎝ ∂x ∂y ⎠ ⎠

(

)

(6.43)

⎛ ⎛ ∂v ∂u ⎞ ⎞ ⎛ ∂u ∂v ⎞ Y = l⎜G⎜ + + m⎜λ + 2G + λ ⎟ ⎟ ∂y ⎟⎠ ⎝ ∂x ⎝ ⎝ ∂x ∂y ⎠ ⎠

(

)

and

⎛ ⎛ ∂v ∂u ⎞ ⎞ ⎛ ∂u ∂v ⎞ X = l ⎜ 2G + λ + λ ⎟ + m ⎜G⎜ + ⎟⎟ ∂x ∂y ⎠ ⎝ ⎝ ⎝ ∂x ∂y ⎠ ⎠

(

)

(6.44)

⎛ ⎛ ∂v ∂u ⎞ ⎞ ⎛ ∂u ∂v ⎞ Y = l⎜G⎜ + + m⎜λ + 2G + λ ⎟ ⎟ ∂y ⎟⎠ ⎝ ∂x ⎝ ⎝ ∂x ∂y ⎠ ⎠

(

)

Examining Equations (6.43) and (6.44) it is clear that some of the nodal potentials involved lie within the bulk material so that the displacement is calculated by the TLM routine, but others lie outside so that their potential must be calculated independently of the bulk model. Figure 6.8 illustrates a template of the boundary region. The boundary of the body lies parallel with the x-axis of the reference coordinate frame, so the direction cosines are

( ) m = cos ( 0 ) = 1

l = cos 90 0 = 0

(6.45)

0

Resolving the applied force into coordinate directions gives

X=0 Y = Fy Substituting for l, m and X into Equations. (6.43) we have

(6.46)

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157

⎛ ⎛ ∂v ∂u ⎞ ⎞ ⎛ ∂u ∂v ⎞ X = l ⎜ 2G + λ + λ ⎟ + m ⎜G⎜ + ⎟⎟ ∂x ∂y ⎠ ⎝ ⎝ ⎝ ∂x ∂y ⎠ ⎠

(

)

(6.47)

⎛ ⎛ ∂v ∂u ⎞ ⎞ ⎛ ∂u ∂v ⎞ Y = l⎜G⎜ + + m⎜λ + 2G + λ ⎟ ⎟ ∂y ⎟⎠ ⎝ ∂x ⎝ ⎝ ∂x ∂y ⎠ ⎠

(

)

Similarly, substituting for l, m, and X into Equation (6.44) yields

⎛ ⎛ ∂v ∂u ⎞ ⎞ ⎛ ∂u ∂v ⎞ X = l ⎜ 2G + λ + λ ⎟ + m ⎜G⎜ + ⎟⎟ ∂x ∂y ⎠ ⎝ ⎝ ⎝ ∂x ∂y ⎠ ⎠

(

)

(6.48)

⎛ ⎛ ∂v ∂u ⎞ ⎞ ⎛ ∂u ∂v ⎞ Y = l⎜G⎜ + + m⎜λ + 2G + λ ⎟ ⎟ ∂y ⎟⎠ ⎝ ∂x ⎝ ⎝ ∂x ∂y ⎠ ⎠

(

)

Solving Equations (6.47) and (6.48) to find the nodal potentials, Φx[i,j – 1] and Φy[i,j – 1], which are boundary potentials needed, yields

Φ x (i , j − 1) = Φ x (i , j + 1) + Φ y (i + 1, j ) − Φ y (i − 1, j )

(

⎛ 2∆lFy − λ Φ x (i + 1, j ) − Φ x (i − 1, j ) Φ y (i , j − 1) = Φ y (i , j + 1) − ⎜ ⎜⎝ 2G + λ

) ⎞⎟ ⎟⎠

(6.49)

This method of boundary treatment uses the central difference approximation of the derivative and is identical to that used in some finite difference models of elastic deformation.7 Investigations into the implementation of force boundary conditions using this approximation by researchers of finite difference methods have shown that the method is unstable given certain elastic parameters.8 As a result, modified implementations of the boundary equations have been introduced. These include the one-sided approximation and composed approximation9 and new composed approximation.10 Both of these have been successfully implemented as boundary conditions for TLM models.

6.7 Conclusion The extension of the method described to three dimensions is straightforward and attempts to model the behavior of materials that are elastically anisotropic have met with encouraging results.11 An application of the

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approach has been in attempting to model the propagation of ultrasound in a structure representing cancellous bone,12 although it has to be said that representing the complexity of real bone and achieving sufficient resolution still pose problems. There is a fundamental inelegance associated with using a finite difference boundary condition in conjunction with a TLM model and the development of truly TLM boundary conditions is an attractive area for future work, as is the extension of the TLM approach to representation of viscoelastic materials.

References. 1. Ronalli G., Rheology of the Earth; Deformation and Flow Processes in Geophysics and Geodynamics, London: Allen and Unwin (1987). 2. Hal A. K. and Singh S. J., Deformation of Elastic Solids, Prentice-Hall, New York (1991). 3. Witwit A. R .M., Wilkinson A. J., and Pulko S. H., A method for algebraic analysis of the TLM algorithm, Int. J. Numerical Modelling 8 (1995) 61–71. 4. Franklin G. F., Powell J. D., and Workman M. L., Digital Control of Dynamic Systems, 2nd ed., Addison Wesley, Reading, Massachusetts (1990). 5. Middleton R. H. and Goodwin G. C., Digital Control and Estimation, a Unified Approach, Prentice-Hall, New York (1990). 6. Simons M. R. S., and Sebak A. A., Spatially weighted numerical models for the two dimensional wave equation, fd algorithm and synthesis of the equivalent TLM model, Int. J. Numerical Modelling 6 (1992) 111–129. 7. Shibuya T., Nakahara I., Koizumi T., Kaibara K., Impact stress analysis of a semi-infinite plate by the finite difference method, Bull. JSME, 18 (1975) 649–655. 8. Timoshenco S., Theory of Elasticity, Eng. Soc. Monographs, McGraw-Hill, New York (1934). 9. Ilan A. and Loewenthal D., Instability of finite difference schemes due to boundary schemes in elastic media, Geophysical Prospecting (1976) 24. 10. Ilan A., Stability of finite difference schemes for the problem of elastic wave propagation in a quarter plane, J. Computational Physics (1978) 29. 11. Langley P., Numerical modelling of the deformation of elastic material by the TLM method, Ph.D. thesis, University of Hull, U.K. (1997). 12. Witwit M., Wilkinson A. J., Pulko S. H., and Langton C., TLM model of the propagation of ultrasound in cancellous bone, Pompei D., Ed., Proceedings of the third international workshop on TLM modelling and applications, Nice, Sophia Antipolis (1999).

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chapter seven

Simple TLM Deformation Models 7.1 Introduction In this chapter we consider TLM models that are simple in so far as they represent only pure bending or pure elongation of materials. However, the simplicity of these models has allowed them to be extended, via physical argument, to represent quite complex material characteristics such as viscoelastic behavior.

7.2 Review of the Behavior of Materials The behavior of real materials is complex and a multitude of theories of varying degrees of complexity exist in humanity’s attempts to describe them.1 Viscous materials are those that flow, energy being required to overcome the viscosity of the material. A physical model frequently used to represent a viscous material is the dashpot. Viscous materials tend to be slow in their response to stimuli, gradually increasing their velocity until, under constant stress, a steady state velocity is reached by every point in the fluid. Purely elastic materials do not flow but stretch; their physical model is the spring. Viscoelastic materials exhibit both flow and stretch in their behavior. Two fundamental arrangements of the dashpot-spring combination are possible and are illustrated in Figure 7.1. Materials behaving according to the series model respond to an applied stress initially by stretching and then by flowing, the stretching being characterized by the spring elasticity G and the flow by the dashpot viscosity η . The time-scale of the stretching process is very much less than that of the viscous flow so that a plot of strain versus time for a material behaving according to this Maxwell model is as illustrated in Figure 7.2. Observing Figure 7.2 the initial stretching is responsible for the fact that the graph has an offset on the strain axis and the straight line then represents

159

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G

η

Figure 7.1 A series viscoelastic (Maxwell) material.

e

strain

γ

γv γ

e

time

tr

Figure 7.2 Maxwell viscoelastic behavior.

viscous flow, assuming that inertial effects can be neglected so that the flow reaches its steady state velocity instantaneously. Figure 7.3 illustrates a parallel arrangement of the dashpot and spring known as the Kelvin model. In this model the elastic stretch is impeded in its speed by the need for a viscous flow, or, to take a different viewpoint, the viscous flow associated with the

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161

F

G

η

F Figure 7.3 A parallel (Kelvin) model of a viscoelastic material.

applied stress is impeded by the restoring force provided by the spring. Whichever view one takes it is important to recognize that, in this model, a steady state strain will be reached, characterized by the elasticity of the spring. In addition, at all times during the transient, the strain associated with the spring is the same as that associated with the dashpot and the total stress is the sum of the stress supported by the spring and that associated with the dashpot. Many more physical models of material behavior exist based on combinations of the series and parallel models. We now consider these mechanisms analytically, starting with the viscous flow model, and then devise TLM models to represent the dynamic behavior of materials whose deformation is described by the physical models.

7.3 Trouton’s Descending Fluid and a TLM Treatment of a Vertically Supported Column “Trouton’s column,” as it is sometimes called, is a stream of viscous fluid descending under its own weight.2 Referring to Figure 7.4, at a distance y, from the top of the stream, the tractive force is provided by the weight of

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y A

v

Figure 7.4 Trouton’s column.

the stream below y, and this force is related to material local acceleration as described by

F ∂v =λ A ∂y

(7.1)

where v is the local velocity, F the tractive force, and A the cross-sectional area. λ is a material dependent constant, known as the coefficient of viscous traction, and is related to the fluid viscosity via

λ = 3η where η is the material’s viscosity. A 1-D element of fluid at y has an associated mass of ρAdy , where ρ is the material density, so that the equation of motion is

∂F Dv + ρAg = ρA ∂y Dt

(7.2)

Substituting for F using Equation (7.1) gives

∂2 v ρ ρ Dv + g= 2 λ λ Dt ∂y where

D is the substantial derivative so that Dt

(7.3)

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163

Dv ∂v ∂v = +v Dt ∂t ∂y Examining Equation (7.3) it seems that the characteristic equation is similar to the diffusion equation insofar as it features a double spatial derivative and a single time derivative; however, an additional term is present, representing the driving force provided by gravity. We are now in a position to start to develop a TLM model for viscous elongation of a column of fluid. Consider the column to be supported by a clamp at the top. If the column remains vertically supported in this way for some time then, so long as material continuity is maintained and no part of the fluid “breaks away,” it will narrow just below the support and the shape will become as shown in discretized form in Figure 7.5. The top of the column is supported and stationary; a no-slip condition is provided by the support. For all other elements we consider the velocity of the element to be that of a node situated, as is usual with TLM, at the center of the element. The total velocity at any node will be the sum of two components: the local velocity as the element elongates causing the node at its center to move downwards, and cumulative velocity, which is the result of the downward motion of all the elements, above it. Similarly, the acceleration at any node is the sum of the local acceleration and the cumulative (convective) acceleration provided by the elements above. If we relate the velocity v to the total velocity, then

∂2 v ρ ρ ∂v + g= 2 λ λ ∂t ∂y

Figure 7.5 Narrowing column.

(7.4)

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We can rewrite Equation (7.4) as

∂2 v ρ ∂(gt) ρ ∂v + = λ ∂t λ ∂t ∂y 2

(7.5)

The variable of the time derivative on the left-hand side is different from that on the right hand side, and so an analogous TLM network equation would be

∂2 Φ ∂Φ' ∂Φ = 2R d C d + 2R d C d ∂t ∂t ∂y 2 The term

(7.6)

∂Φ' is unusual. Φ′ is a distributed potential such that, in general, ∂t Φ′ = g ′t

(7.7)

where g ′ is the relevant component of the acceleration due to gravity. The term ∂Φ ′ / ∂t is the driving term, since there are no other inputs to the system. It provides the circuit with a maximum or minimum distributed potential, Φ′ , the value of which increases with time, as suggested by Equation (7.7). The question now arises as to how this potential can be applied to the TLM network. One method of doing it is by modifying the values of incident pulses appropriately. Since we are considering a 1-D situation, increasing the value of each incident pulse by g ′δt /2 at each iteration will result in the nodal potential’s being modified by Φ′ at each iteration. When pulses are modified in this way, they carry full information about potentials in the network and automatically enforce the limiting behavior of the driving term; in the limit when Φ = Φ′ for all nodes, there is no current flow and no relative motion of nodes. By implementing the pulse modification scheme, gravity acts as an internal driving source at each node, and its effect then diffuses through the body. It is intuitively reasonable to model material viscosity by resistance and density by capacitance in the TLM network. The analogy then yields

Φ = velocity = v C = C d × elemental volume = ρ × elemental volume = elemental mass = m (7.8)

R = Rd

elemental length elemental length = 2 × cross − sectional area 2 × λ × cross − sectional area

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165

Examining the analogy further and recalling that current and potential are related by

I=C

∂Φ ∂t

(7.9)

it is clear that the current at a node represents the viscous tractive force and the net current between nodes represents the net force. The development of treatments presented in this chapter is largely due to Newton,3 and she continues the analysis by considering an interesting difference between the loading approach taken here and that commonly used in representing heat generated or absorbed in a body. Having described a TLM network appropriate to modeling the bulk material, it is now necessary to consider boundary conditions. First of all, if we consider the column to be supported vertically by a clamp, then the upper surface is stationary. This zero velocity can be represented by a node whose potential is maintained at zero throughout the simulation. The element at the free end of the column does not experience any elongation. Hence, since nodal potential represents relative velocity of nodes, its velocity is set, throughout the simulation, equal to that of its neighbor inside the fluid. The sides of the column are insulated, representing no loss of material to the surroundings.

7.4 A Model of Viscous Bending The previous section described a situation of pure elongation in which gravity acted wholly along the fluid. However, we were careful to introduce the quantity g ′ for generality. In this section we consider the situation of pure bending. A bar of viscous material, supported horizontally, will gradually bend, the rate of bending being related to the material viscosity. The beam can be considered to be made up of a row of discrete, rigid elements connected by imperfect hinges, which represent the viscosity. Figure 7.6 illustrates two types of horizontal bar, one supported at both ends and the other a cantilever supported at one end only. Each element is allowed (by the hinges) to rotate about its neighbors. It is clear in Figure 7.6a that, when a bar is supported at both ends, each element rotates about both neighboring elements. For the cantilever of Figure 7.6b, on the other hand, each element rotates only about the neighboring element on the side between it and the support. In both cases elements rotate about neighbors, which form a line of connection to the support(s), and this line can be viewed as a neutral axis. Figure 7.7 illustrates a discretized version of the cantilever as it deforms. Initially, gravity acts parallel to the vertical sides of the element as illustrated in Figure 7.7a and gives rise to a shearing force V. This shearing force is assumed to act along the elemental boundary to the right of the element,

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(a)

(b)

Figure 7.6 Discretized bars prior to viscous bending (a) a beam supported at both ends (b) a singly supported cantilever.

and it produces a bending moment δM . The element behaves as a rigid body and rotates as illustrated in Figure 7.7b. The component of gravity g cos θ now acts parallel to the sides of the element, so that the associated shearing force becomes V cos θ , and the new bending moment is given by

δM = V δl cos θ

(7.10)

If the fluid is highly viscous so that no movement takes place before the forces have developed fully, then, from classical theory,2 the shear force is

V = ρ ( L − x ) cross sectional area ( g cos θ )

(7.11)

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167

θ

V (a)

V cos θ

(b)

Figure 7.7 A discretized singly supported cantilever before (a) and during (b) viscoelastic bending.

where L is the length of the bar and x is the distance from the support. The elemental bending moment then becomes

δM = ρ (n-i)δl 2 cross sectional area(g cosθ)

(7.12)

where

nδl = L and iδl = x . It is clear in Figure 7.7b that, as elements rotate, they overlap at the bottom and correspondingly sized gaps develop between them at the top. This is consistent with conservation of volume and with the concept of the line through hinges constituting a neutral axis. However, real fluids maintain continuity of material. In this example the cantilever would experience tension along the upper edge and compression along the lower edge, and would be stressed tractively so that movement was characterized by the coefficient of viscous traction. In the rigid body approach taken here, movement is purely rotational, and no deformation of fluid elements occurs, implying that the flow is dependent on shear viscosity. This means that, if material continuity is to be assumed in the present model, the coefficient of viscous traction must be used to characterize the deformation. In devising a characteristic equation to describe the bending process, Newton3 took an approach similar to that used in the treatment of Trouton’s column. Replacing the linear momentum parameters of Equation (7.4) with those of angular momentum gives

∂ 2 Iz ω ∂M 1 ρ ∂Izω + = ∂x λ λ ∂t ∂x 2

(7.13)

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where I z = bh 3/12 and ω is the angular velocity. In terms of the shearing force, Equation (7.13) may be rewritten

V ∂ 2 Iz ω ρ ∂Izω + = λ λ ∂t ∂x 2

(7.14).

Recalling Equation (7.9) and substituting for V in Equation (7.14) we have

∂ 2 Iz ω ρ(n-i)δl bh g cosθ ρ ∂Izω + = 2 λ λ ∂t ∂x

(7.15)

When considering the vertically supported column we considered the concepts of accumulated velocity and local velocity. Similar concepts are relevant here; each element rotates both as the result of gravity’s action on it, and as the result of the rotation of neighboring elements. For example, in Figure 7.7, if the second element has rotated through an angle, θ, then all elements further from the support must rotate through θ before the effect of gravity on their own rotation is considered. Substituting for Iz in Equation (7.15) we have

h2ω h2ω ∂ 12 + ρ ∂((n-i)δl g t cosθ) = ρ 12 ∂t λ ∂t λ ∂x 2

∂2

(7.16)

Comparing Equation (7.16) with Equation (7.13) gives the analogy outlined below

F=

h2w 12

C = ρ × (elemental volume ) = mass = µ

R=

elemental length 2 l × (cross − sectional area)

Again by analogy with the elongation model,

F ' = ( n − i) d lg t cos θ so that, at each iteration, pulses are augmented by an amount

(7.17)

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169

( n − i) d lg dt cos θ 2 Recalling Equation (7.9), in the TLM analogy for pure bending, the current at a node has the units of force times distance and represents the viscous bending moment. Therefore, the current flow between nodes represents the net bending moment. In terms of output from the TLM model, we frequently need to know the shape of the material at a particular time. The angular velocity of the element may be determined simply from the nodal potential and the angle of inclination, which is found by integration i.e., t

∫

θ = ωdt 0

The discrete version of the integral can, of course, be determined directly from the TLM model as the sum of ωdt over the period of interest. Some of the boundary conditions applicable to bending models are also similar to those of the elongation model. At the support, a no-slip condition is applied so that the nodal potential is maintained at zero. Along the sides of the 1-D model, reflecting boundaries are implemented because the surroundings have no effect, and at the end of the bar the potential is set equal to that of the neighboring node. The treatment of the free end is rather less justifiable than was the case for the elongation model, and this is discussed in more depth by Newton.3 Extension of the approach described above to more complex shapes and to those with more than one support does not involve new concepts. However, if more than one support exists, the fact that the model represents only bending means that the supports will move towards each other during the simulation. Newton overcame this problem by applying controllers based on PIDtheory to the supports.3 It is also a simple matter to represent shapes having initial curvature so long as it is in steady state.

7.5 Numerical Issues and Model Convergence The viscous deformation model described here models a critically damped situation i.e., it is a model of a diffusion process. On this basis we would expect that its convergence properties would be very similar to those of other, for example, thermal diffusion models, these properties having been investigated originally by Johns and Butter.4 The convergence condition established by them is that, for convergence to the diffusion condition as opposed to the wave equation

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δl → 0

δt2 → 0

as

and this also meets the accuracy condition. However, although this condition applies to the viscous deformation model, the dependence of the dynamics of a viscous material on its parameters is different from the dependence of the thermal behavior of a material on its thermal diffusivity. Consider a thermal problem in which a body is exposed on all sides to a constant temperature ambient. Irrespective of the thermal diffusivity of the material, the final temperature of every point in the body will be the same and will equal the ambient temperature; points within a material of low thermal diffusivity will reach that temperature more slowly than those within a material of high thermal diffusivity. However, it is clear from Equation (7.17) that a fluid of high viscosity is modeled using a lower value of resistance than a body of lower viscosity, and hence is associated with a smaller time constant. Such a highly viscous material will have a lower steady state flow rate than a less viscous material, but will exhibit a fast response and therefore a greater rate of change of velocity. Highly viscous fluids therefore can require shorter timesteps than fluids of lower viscosity. However, it is important to realize that a fluid of low viscosity should not be modeled using discretization so coarse that important information is lost.

7.6 TLM Models of Viscoelastic Deformation 7.6.1 The Parallel Viscoelastic Model We first consider modeling bending via the parallel dashpot-spring physical model illustrated in Figure 7.3. In this Kelvin model the elastic and viscous stresses are additive, so that the total shear stress τtotal is given by

τ

total

=τ

elastic

+τ

viscous

(7.18)

In the viscous model of bending described earlier, the stress was incorporated via an applied potential Φ' such that

τ

viscous

=τ

total

=

shear force 1 ∂Φ' ∂Φ' = C = ρ area bhδl ∂t ∂t

(7.19)

where, for a cantilever F ' = g( n − i) dl cos θt and was incorporated by modifying incident pulses according to

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(n-i)δl cosθ g δt 2

Vi = Vi +

(7.20).

Therefore, in a parallel viscoelastic model

τ

total

=τ

so that

elastic

+ρ

∂Φ' ∂t

(

τ -τ ∂Φ' = total elastic ∂t ρ

(7.21)

)

(7.22)

In terms of incident pulses this means that, at each iteration, we need to modify pulses according to

Vi = Vi +

(τ

total

-τ

elastic

2ρ

)δt

(7.23).

The state of zero applied viscous stress, and zero accompanying viscous flow, is achieved in the model when

∂Φ' =0 ∂t In this condition τ total = τelastic so that the TLM routine yields the elastic solution. Classical theory gives steady state deflections as

θ=

1 EIz

s

∫

M ds =

0

ρg (n-i)δl cosθ ρg (n-i)δl cosθEIz

where

Iz = Rearranging Equation (7.24)

bh 3 12

s

∫ M ds 0

(7.24)

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ρg (n-i)δl cosθ =

θ EIz ρg (n-i)δl cosθ

(7.25)

s

∫ M ds 0

so that τ total = τelastic . We can now substitute τ total and τelastic in Equation (7.23) to express the incident pulses for parallel viscoelastic flow in the form

Vi = Vi +

g (n-i)δl δt cosθ θ EIz g (n-i)δl δt cosθ s 2 2 M ds

(7.26)

∫ 0

Since s

θelastic =

∫ M ds 0

(7.27)

EIz

Equation (7.26) can be rewritten

Vi = Vi +

θ(i) g(n − i)δl δt cosθ g (n-i)δl δt cosθ 2 θelastic (i) 2

(7.28)

Equation (7.28) is a very convenient expression since it involves only quantities known to the model and the steady state elastic deflection of each node which can be calculated once outside the model either analytically or by numerical means. This formulation assumes that the parallel viscoelastic behavior is performed upon a beam whose initial angular deformation is zero along its entire length. If the initial angular orientation is denoted by θ0 i then Equation (7.28) becomes

()

Vi = Vi +

g (n-i)δl δt cosθ (θ(i)-θo (i)) g (n-i)δl δt cosθ 2 θelastic (i) 2

(7.29)

The physical series viscoelastic model is illustrated in Figure 7.1; an initial elastic deflection is followed by a much slower viscous flow. This is very simple to implement since the initial steady state deflection can be calculated externally and this deflected shape can then be used as the starting point for the viscous flow.

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Having now devised models for both series and parallel viscoelastic deformation it is possible to model more complex mechanisms consisting of three or more components by “bolting together” the relevant components. Figure 7.8 illustrates a three-component model observed to represent the high temperature behavior of many ceramics.5 Figure 7.9 is a plot of the angular deflection of a cantilever against time, the cantilever being made of material whose behavior is described by the physical model of Figure 7.8. The nonzero intercept on the y-axis is due to the instantaneous nature of the series elastic component, derived in this case analytically, and applied at time equal to zero. This is followed by the slower rise to a steady state deflection modeled by the TLM parallel viscoelastic treatment. When the load is removed, 0.2 seconds after the start of the simulation, similar behavior is observed as the cantilever returns to “a” its horizontal position. It would also be possible to model a three-component physical mechanism in which the series component was a dashpot. In this case two TLM models would run throughout the simulation, one representing the series viscous flow and the other the parallel viscoelastic movement. The total strain at any node is the sum of the strains due to the viscous and viscoelastic components and the models are coupled insofar as the total strain affects the loading term in each model. It is, of course, quite possible to increase the number of components further.

7.7 Conclusion In this chapter TLM models of viscous and viscoelastic bending have been described. They assume that deformation occurs either in one plane only or in a symmetric mode so that torque and in-plane shear effects can be neglected. This is a reasonable assumption for thin walled shapes, which either have only one region of support or are largely symmetric both in their geometry and their support. For situations in which, for example, in-plane shear is significant a fuller treatment may be required. This might involve coupled networks, one for each of the three spatial directions, and the treatment would be based on those used in Langley’s TLM models of purely elastic behavior.6 In such a model it is likely that it would be unnecessary to provide control inputs to retain the position of the supports. The models are simple and elegant in terms of spatial discretization: as the bodies deform the TLM mesh retains its original configuration since only the loading, modeled by changes to pulse values, depends on the deformation so that remeshing during deformation is unnecessary. The models assume that elasticities and viscosity are constant throughout the period of deformation. However, temperature dependence of parameters should not pose particular numerical problems, although variation in elasticity could prove cumbersome if it is necessary to repeatedly obtain new steady state elastic solutions.

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Es

λ

Ep

Figure 7.8 A series-parallel model of a viscoelastic material as used in many ceramic models. 3.5 3 2.5 angular 2 deformation in 1.5 degrees 1 0.5 0 0

0.1

0.2

0.3

0.4

0.5

time in s

Figure 7.9 Angular deflection of a cantilever against time as load is applied and then removed (after 0.2 seconds). The algorithm is based on the three-component model which is shown in Figure 7.8.

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References 1. Rainer M., Deformation, Strain and Flow, H.K. Lewis & Co, London (1960). 2. Massey B. S., Mechanics of Fluids, Chapter 1, 4th ed., Van Nostrand Reinhold Co, New York (1979). 3. Newton H. R., TLM models of bending and their application to thin walled ceramic shapes, Ph.D. thesis, University of Hull, U.K. (1994). 4. Johns P. B. and Butler G., The consistency and accuracy of the TLM model for diffusion and its relations to existing techniques, Int. J. Numer. Methods Eng. 19 (1983) 1549–1554. 5. Airey A. C. and Birtles J. F., Pyroplastic deformation of whiteware bodies, in Science of Whitewares, Chapter 17, Proceedings of Science of Whitewares Conference, 1995. Henkes V., Onada G. Y., and Carty W. M., Eds., American Ceramic Soc. (1996). 6. Langley P., Numerical modelling of the deformation of elastic material by the TLM method, Ph.D. thesis, University of Hull, U.K. (1997).

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chapter eight

TLM Modeling of Hydraulic Systems 8.1 Introduction A fluid flowing in a uniform frictionless pipe experiences no change in its condition over the length of the pipe. If, however, a valve is closed at the end of the pipe, then the fluid in the immediate vicinity of the face plate is brought to a halt. There is a rise in pressure as the moving fluid compresses that which is already stopped. This causes a pressure wave to propagate back along the pipe. The outcome is a transient (water-hammer), which can give rise to localized pressures that may exceed the mechanical strengths of one or more elements within the system. A fluid, having passed one partially closed valve may experience closure further downstream, which causes a back-propagating transient. This transient may in turn be reflected at the upstream valve constriction. If the losses are sufficiently small then resonances could be established within the system with excessive local under or over-pressures at the nodes or antinodes. Oscillations can also occur in a space between two constrictions (e.g., changes in pipe diameter) and the result is “pulsatile” or oscillatory flow at the outlet. In order to avoid catastrophic situations hydraulic engineers have devised a range of protective components: throttle-valves, surge tanks, and accumulators, but there is a need to fully understand the system in order to place such elements for optimum effect. The situation is a complex one due in part to the broad range of compressibility of different fluids. Additionally, pipes may be rigid or may distort and certain fluids can, as a result of under-pressure, vaporize (“column separation”). There may also be gas release if the fluid contains dissolved air or other gases. Wylie and Streeter1 provide an excellent treatment of hydraulic transients and their causes. Their book starts with continuity equations. The authors then develop models for dynamic components in compressible and 177

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incompressible liquids using lumped models of capacitance and inertance, the fluid equivalent of inductance. Their equations for oscillatory flow are almost indistinguishable from the electromagnetic equations that are relevant to transmission lines. At the time of publication (1978) Wylie and Streeter did not have a particularly broad range of numerical techniques available, and quoting from some of their earlier work,2 they say “In 1968 it was suggested that transient fluid flow in multi-dimensional space could be simulated by representing the space with a latticework of one-dimensional elements, in each of which the fluid response obeys the one-dimensional transient flow equations. At that time the elasticity of the line elements was not properly accounted for in the model, a fault which resulted in a delay in the response predicted by the numerical model . . ..” This statement seems quite strange from the perspective of the authors of this book, since the step to a Johns-like TLM formulation of hydraulic problems appears to be so small. Within the TLM community Boucher and Kitsios3 are generally acknowledged to be the first to have taken this step, but as we will see they still had some way to go. In addition to the Wylie and Streeter foundations, they had other precursors. Pearson and Winterbourne,4 using the Boucher approach to model the processes in intake manifolds in internal combustion engines refer to Auslander’s paper of 1968.5 Subsequent developments have been led by groups at the Universities of Bath (U.K.) and Linköping (Sweden)*, but there appear to be variations in nomenclature and formulation that could give rise to confusion. In the treatment, which is outlined below, we will attempt to develop a uniform treatment that is consistent with the models that have been presented in other chapters of this book.

8.2 Symbols, Analogues, and Parameters Given the diversity of symbols that are used to represent the key parameters in hydraulics, there is a requirement for a uniform system that avoids confusion. Wylie and Streeter used H for “head” while most others use P for pressure. This is consistent with the other disciplines and is the equivalent of V in TLM. Several authors have adopted Q to symbolize the flow of fluid. This could be confused with electrical charge, and so we will follow Boucher and Kitsios and use W as the analogue of electrical current, I = ∫Qdt. Some have used I for inertance, but as it is so close to the concept of inductance, we will use L. All seem agreed that C should represent capacitance and R should represent resistance. It should be noted that some authors have used these symbols for per-unit-length parameters. This can give rise to ambiguities and in our treatment they represent the parameters as measured. Resistance, capacitance, and inductance per unit length are given as Rd, Cd, and Ld.

*

Some of the papers from these groups that are not explicitly mentioned within the text are included in a bibliography at the end of the chapter.

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Most authors have adopted Zc as the characteristic impedance (rather than Z0 as in electromagnetics). We will be generally following their example. Wylie and Streeter have interpreted their parameters in terms of fluid height, h. Pressure is given by ρgh (ρ is density), so that the gravitational constant g appears in their definitions. Most others use pressure explicitly, and we will follow this approach. In several cases the definitions are based on a medium of unit density. We would prefer to include density explicitly, so that there is no confusion when inhomogeneous media are being considered. If we start with a rigid pipe section of length ∆x and cross-sectional area, A, then we can use Newton’s second law to define inertance as:

ρ∆x A

L=

(8.1)

as well as Ld = ρ/ A and L0 = ∆x / A for a fluid of unit density. There are several ways of approaching C = A∆x/Ke (where Ke is the bulk modulus). We can start with Va the velocity of a plane wave moving in the fluid which is given by:

Va =

Ke ρ

(where ρ is density) (8.2)

LC =

∆x 2 Va2

(8.3)

A ∆x ρVa2

(8.4)

Since we already have

we can define

C=

A A ∆x and C 0 = for a fluid of unit density. 2 Va2 ρVa The characteristic impedance is then given by:

as well as C d =

ZC =

ρVa A

(8.5)

with an appropriate definition for Z0, the impedance of a fluid of unit density.

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One of the examples considered by Boucher and Kitsios3 is a lossless pipe (3 m long and 0.15 m in diameter) connected to a closed-volume chamber (64 m3). It is not stated, but they appear to be using air at 25°C (Va = 345 ms–1) as the fluid, and have assumed unit density. This last assumption is valid so long as the system is homogeneous. The capacitance of the chamber is 5.34 × 10–4 ms2 and the characteristic impedance of the pipe is 1.958 × 104 m–1s–1 (these units assume a medium of unit density). It should be noted that these authors appear to swap between impedance, Z and its reciprocal, Y (admittance) whenever the formulation favors one or the other definition. The definition of resistance is not as easy as it is in electricity. This is because the relationship between pressure and flow depends on additional parameters such as viscosity and whether the flow is laminar or turbulent. Although Boucher and Kitsios consider problems where resistance is a factor, their paper does not explain how the parameter is derived. However, working through their figures, it is quite clear that they have adopted the definition from Wylie and Streeter,1 although there is a need for caution due to the latter’s use of fluid head, which involves the gravitational constant (through P = ρgh). Poiseuille’s formula for the relationship between flow and head of fluid in a pipe of length, x and or radius, r is:

W=

πr 4 g h 8 ηx

(8.6)

Using definitions for diameter, D and area, A we obtain a definition for R as

R' =

32 ηx gAD2

(8.7)

The formula given by Wylie and Streeter does not include the length of the pipe because their definition is per unit length. Although the Poiseuille formula strictly applies only to flow in long pipes where kinetic effects are small, we will consider an element, ∆x of pipe and define R in terms of pressure as:

R=

8η ∆x πr4

(8.8)

The above equation is only applicable to laminar flow conditions, i.e., where the Reynolds number (Re = WD/η) is less than 2000. Nalluri and Featherstone6 give a general treatment of steady uniform flow in pipes that is based on the Darcy–Weisbach equation and has shown how this can be extended for turbulent flow in pipes with different friction coefficients. They also show how treatments can be modified for pipes of noncircular section. This information could

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be used to deduce a pressure, flow-rate relationship, which may not be linear. In this situation, the instantaneous local resistance may be a function of flow rate and other parameters. Details of handling nonlinear parameters (albeit in heat-flow problems) may be found in reference.7

8.3 Compressional Waves in Fluids Transients in fluids are characterized by pressure and displacement and these are related by two equations that have electromagnetic analogues:

P=L

dW di equivalent to V = − L dt dt

(8.9)

dP dV equivalent to i = C dt dt

(8.10)

W =C

From these we can easily proceed to

∂2 P ∂2 P = Ld C d 2 2 ∂t ∂x

(8.11)

which is identical to the equation for propagation on a transmission line.

8.4 A Transmission Line Analysis of Fluid Flow Figure 8.1 shows a pipe of length, x with characteristic impedance, ZC. The upstream and downstream pressures are PU and PD respectively. The downstream pressure and flow are related to the upstream pressure and flow via the transfer equation

⎡ cosh γ x ⎡ PD ⎤ ⎢ ⎢ ⎥=⎢ 1 ⎣WD ⎦ ⎢ − Z sinh γ x c ⎣

−Zc sinh γ x ⎤ ⎥ ⎡ PU ⎤ ⎢ ⎥ cosh γ x ⎥ ⎣WU ⎦ ⎥⎦

(8.12)

x PU

PD

WU

WD

Figure 8.1 Pipe showing upstream and downstream pressures and flows.

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We can also write

Zc sinh γ x ⎤ ⎥ ⎡ PD ⎤ ⎢ ⎥ cosh γ x ⎥ ⎣WD ⎦ ⎥⎦

⎡ cosh γ x ⎡ PU ⎤ ⎢ ⎢ ⎥=⎢ 1 ⎣WU ⎦ ⎢ Z sinh γ x ⎣ c

(8.13)

In common with the practice in electromagnetics we can express the impedance perceived at the pipe entrance in terms of its length, characteristic impedance, and the downstream (terminating) impedance.

ZU =

ZD + ZC tanh γ x 1 − ( ZU / ZC ) tanh γ x

(8.14)

which is identical to the expression for electromagnetic transmission lines. The propagation constant is given by γ = α + jβ , where α is the attenuation and β is the phase-lag. It is also given by γ 2 = jωC ( R + jωL) , which is identical with the expression, γ 2 = sC ( R + sL) in Wylie and Streeter,1 where s is the complex frequency or Laplace variable. Pollmeier,8 drawing on work by Viersma,9 and Burton,10 also uses the Laplace variable in his expression for the transfer matrix for a cylindrical pipeline (radius, r) with nonelastic walls (Re < 2000) and where tangential flow is neglected:

(

)

⎡ cosh sT N ( s) ⎢ ⎢ 1 ⎢ sinh sT N ( s) ⎢⎣ Zc

(

)

(

)

Zc sinh sT N ( s) ⎤ ⎥ ⎥ cosh sT N ( s) ⎥ ⎥⎦

(

)

(8.15)

N(s) is a frequency dependent friction factor, which is also applicable in the nonplane wave case. It is defined as:

⎛ J0 ⎜ j ⎝ N ( s) = ⎛ J2 ⎜ j ⎝

s ⎞ r⎟ η ⎠ s ⎞ r⎟ η ⎠

(8.16)

There is a large volume of work on the conventional analysis of physical problems that can be described by transmission line analogues (see de Cogan and de Cogan11) and most of these are in the frequency-domain. Wylie and Streeter12 appear to follow this tradition in their analysis of equations for oscillatory flow.

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8.5 Time-Domain Transmission Line Models of Fluid Systems Although the work of Boucher and Kitsios3 tends to be acknowledged as the first true TLM analysis, even here, they appear to restrict themselves to a single node, as some of the examples will demonstrate. Figure 8.2 shows their model of a chamber plus pipe. The pipe is treated as a single transmission line, and, using the acoustic velocity of air, the transit time for sound, 8.67 ms is taken as the discretization time ∆t. Thus the impedance of region 1, the pipe, in pascal seconds per cubic metre is Z1 = 1.958 × 104 Psm–3. The impedance of region 2 is given by Z2 = ∆t/C2 and is thus 16.2 Psm–3. In the model we will distinguish between flow to the left and right WL and WR respectively. Boucher and Kitsios then present the relationships between flows in the different regions as:

⎡ WL 1 ⎤ 1 ⎢ ⎥= Z Z2 + W 1 ⎣ R2 ⎦

⎡ Z2 − Z1 ⎢ ⎢⎣2 Z1Z2

2 Z1Z2 ⎤ ⎡WR1 ⎤ ⎥⎢ ⎥ Z1 − Z2 ⎥⎦ ⎣WL 2 ⎦

(8.17)

or

⎡ WL 1 ⎤ ⎡ −0.99834 ⎢ ⎥=⎢ ⎣WR2 ⎦ ⎣ 0.0574

0.0574 ⎤ ⎡WR1 ⎤ ⎥ ⎥⎢ 0.99834 ⎦ ⎣WL 2 ⎦

3m 0.15m

(8.18)

64m 3 1

2

Figure 8.2 A 64 m3 chamber attached to a 3 m length of 0.15 m diameter pipe.

The next step (Equation 30 in their paper) requires some explanation as they have amalgamated several processes. It is in fact:

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ρ = -1

ρ

ρ

12

1

ρ =1

21

2

2

1

Figure 8.3 Pipe and chamber problem presented as two transmission lines with appropriate reflection coefficients, constant pressure boundary at open end of pipe, discontinuity boundary at the interface between pipe and chamber and rigid boundary at far end of chamber.

⎡ WL 1 ⎤ ⎡ 0 ⎢ ⎥ ⎢ −0.9983 ⎢WR1 ⎥ =⎢ ⎢WL 2 ⎥ ⎢ 0.0575 ⎢ ⎥ ⎢ ⎢⎣WR2 ⎥⎦t+∆t ⎢⎣ 0

−1 0 0 0

0 0 0 1

0 ⎤ ⎡ WL11 ⎤ ⎡0.0101⎤ ⎥ ⎢ ⎥⎢ ⎥ 0.0575 ⎥ ⎢WR1 ⎥ ⎢ 0 ⎥ + P0 ( t) (8.19) 0.9983 ⎥ ⎢WL 2 ⎥ ⎢ 0 ⎥ ⎥ ⎢ ⎥⎢ ⎥ 0 ⎥⎦ ⎢⎣WR2 ⎥⎦t ⎢⎣ 0 ⎥⎦

where the term on the far right takes account of the step-function excitation, which is applied at the pipe opening from time, t = 0. In modern TLM we would break the system into two units and concentrate on the boundaries and this is shown in Figure 8.3. We would also use scattering coefficients ρ12, τ12, ρ21, and τ21, where ρ12 is defined in terms of a pulse, moving from left to right in region 1 as it encounters region 2 and is given by (Z2 – Z1)/(Z2 + Z1). The scattering coefficient in Equation (8.19) is

⎡ 0 ⎢ ⎢ρ12 ⎢ τ12 ⎢ ⎢⎣ 0

ρ1 0 0

0 0 0

0

ρ2

0 ⎤ ⎥ τ21 ⎥ ρ21 ⎥ ⎥ 0 ⎥⎦

(8.20)

In modern TLM we would position ourselves at the center of each section and monitor the signals as they pass. We would also define flows incident from left and right at our current position, iWL and iWR. As there is no local scattering, we allow the pulses to pass through, taking a summation iW + iW as they go. They are instantaneously rebadged: iW → sW and L R R L iW → sW . The pulses move during a time interval, ∆t/2, at which point L R they scatter at the two outer boundaries and at the interface. After a further

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time interval, ∆t/2 the pulses appear as incident at the observation points, when we can write:

WL ( 1) = ρ1 skWL ( 1)

i k +1

WR ( 1) = ρ12 skWR ( 1) + τ21 skWL ( 2 )

i k +1

(8.21)

WL ( 2 ) = τ12 skWR ( 1) + ρ21 skWL ( 2 )

i k +1

WR ( 2 ) = ρ2 skWR ( 2 )

i k +1

If we know how the net flow changes with time at these inspection points, then it is possible to use Equation (8.9) to calculate the corresponding local pressure. It should also be clear that this concept could be applied to a multinode system. If for instance, we had the pipe divided into 10 nodes so that ∆t = 0.867 ms, then we can expand the useful frequency range by a factor of 10. The next example that Boucher and Kitsios considered was a chamber with two pipes attached. In this case, allowance was made for the losses in the pipes and these were lumped at one point in each pipe as shown in Figure 8.4. Because of the difficulty of defining flow in terms of left and right, the original notation has been retained, so that u1 represents flow to the right in pipe 1. The impedances, Z1 and Z2 were defined in the previous problem (1.958 × 10 4 Psm – 3 , 16.2 Psm – 3 respectively). On the same basis, Z3 = 1.76 × 105 Psm–3. Boucher and Kitsios appear to use the Wylie and Streeter1 definition for resistance per unit length, based on Equation (8.7). Since their time-step is based on the time to transit the 3 m length of pipe, this is justified, although the values, R1 = 3.764 and R3 = 304.9 are based on water head rather than pressure. They then iterate the transition matrix:

u v 0.15m

3m

64m 3

3m

1

2

3

u v

0.05m

u v

Figure 8.4 Chamber with two pipes. The pipe resistances are shown as lumped components at the transition points. The Boucher and Kitsios notation for direction of flows has been used here.

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⎡ v1 ⎤ ⎡ u1 ⎤ ⎢ ⎥ ⎢ ⎥ = S ⎢ u2 ⎥ ⎢ v2 ⎥ ⎢ u3 ⎥ ⎢ v3 ⎥ ⎣ ⎦t+∆t ⎣ ⎦t

(8.22)

where the elements of the scattering matrix are defined as:

⎡ ⎤ Kn s nn = 1 + 2 K n Z n ⎢ − 1⎥ (n = 1,2,3) ⎣ K1 + K2 + K 3 ⎦

s nm =

2K n K m Zn Zm (n=1,2,3 m=1,2,3 n ≠ m)) K1 + K2 + K 3 Kn =

1 (n=1,2,3) Rn + Z n

In modern TLM it is possible to represent the problem using either a shunt or series network. The shunt approach (shown in Figure 8.5) is generally easier to handle and will be discussed here. Those wishing to use series networks should consult Christopoulos.13 In this arrangement the capacitance of region 2 is represented by a half-length, open-circuit stub, so that all scattering processes are synchronous. The analysis is quite straightforward as Z1 + R1, Z3 + R3, and Z2 are all in parallel with a common point. If we define iW1, iW2, and iW3 as the incident flows (equivalent to u1, u2, and u3 of Boucher and Kitsios), then the potential of the common point is given by: 1

R1

R3

3

ρ = -1

ρ = -1

1

3

2

ρ =1 2

Figure 8.5 Shunt TLM network for two pipe plus chamber model (Figure 8.4), showing region 2 as an open circuit (rigid-boundary) stub.

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⎛ iW1 ⎞ ⎛ iW2 ⎞ ⎛ iW3 ⎞ ⎜ R +Z ⎟ +⎜ Z ⎟ +⎜ R +Z ⎟ ⎝ 1 ⎝ 2 ⎠ ⎝ 3 1⎠ 3⎠ φ= ⎛ 1 ⎞ ⎛ 1 ⎞ ⎛ 1 ⎞ ⎜⎝ R + Z ⎟⎠ + ⎜⎝ Z ⎟⎠ + ⎜⎝ R + Z ⎟⎠ 1 1 2 3 3

187

(8.23)

The potential at the end of transmission line 2 is simply φ, but in the case of lines 1 and 3, φ represents the potential across each line, (W1 or W2) plus the pressure drop across the appropriate resistor. This can be approached as will be demonstrated for line 1: Flow in line 1 is

I1 =

φ − 2 iW1 R1 + Z1

(8.24)

Pressure drop across line 1 is

W1 = 2 iW1 + I 1Z1

(8.25)

The scattered pressure pulses are then: s

W1 = W1 − iW1

s

W2 = φ − iW2

s

W3 = W3 − iW3

(8.26)

These pulses travel to the end of their appropriate lines where they are scattered, so that they become incident pulses at the next iteration:

W1 = − skW1

i k +1

W2 = skW2

i k +1

(8.27)

W3 = − skW3

i k +1

Although this is a reasonable representation of the problem, there are those who might justifiably claim that the network of Figure 8.5 is not realistic as lines 1 and 3 are connected more intimately than is really the case; depending on the geometry of region 2 there will be a longer or shorter delay for communication between 1 and 3. This objection could be overcome by a minor alteration, namely that region 2 is divided into two parts. The first represents a transmission line with an impedance appropriate to the acoustic delay between 1 and 3. It is labeled 2' in Figure 8.6 and has a length

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∆x, placed on each side of the common point. The remaining capacitance in region 2 is then lumped into a new stub-line, designated 2". This then becomes a five section problem as shown in Figure 8.6. 1

R1

2'

2'

R3

3

ρ = -1

ρ = -1

1

3

2"

ρ =1 2

Figure 8.6 Shunt TLM network for two pipe plus chamber model where the capacitance in region 2 (of Figure 8.4) has been divided into two lines plus stub, so as to create an appropriate time-delay between regions 1 and 3.

This is more like conventional TLM formulations with multiple nodes. The analysis of the junction represented by the 2 lines and the stub is also simpler, although there is a need to distinguish between pulses on line 2 to the left and to the right of the common-point. The common-point potential is:

φCP

⎛ iW2 ' ( L) ⎞ ⎛ iW2 ' ( R ) ⎞ ⎛ iW2 " ⎞ ⎜ Z ⎟ +⎜ Z ⎟ +⎜ Z ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ 2" ⎠ 2' 2' = ⎛ 2 ⎞ ⎛ 1 ⎞ ⎜⎝ Z ⎟⎠ + ⎜⎝ Z ⎟⎠ 2' 2"

(8.28)

There are no resistances within the 2'/2" regions, so that the scattered pulses are s

W2 ' ( L) = φ − iW2 ' ( L)

s

W2 " = φ − iW2 "

s

W2 ' ( R) = φ − iW2 ' ( R)

(8.29)

The pulses that travel to the end of the stub will be reflected in-phase by the rigid boundary. Those that move to left and right will interact with regions 1 and 3, so that in each case, the pulses from 2'(L) and 2'(R) will reflect the effects of scattering at the appropriate discontinuities.

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It is only a small step from here to a treatment that divides the problem into many nodes and thereby extends the frequency range. However, this raises a question. Boucher and Kitsios have chosen to lump the resistance at one place, rather than distribute it along a line. In their case this was understandable, as theirs was a minimum node representation. If the resistive component at each node is large, then we will not be solving the wave equation. In extreme situations, such as the dispersion, C(x,t) of a pollutant in a moving stream, then we are solving the drift-diffusion or “advection” equation:

D

∂ 2C( x , t) ∂C( x , t) ∂(C( x , t)) = −v ∂x 2 ∂t ∂x

(8.30)

where v is the drift velocity and D, the diffusion coefficient is related to the line parameters through the distributed resistance and capacitance (D = 1/RdCd). The reflection coefficient in a 1-D RC line is related to D as follows:

ρ=

R = R+Z

1 1 = ⎛ Z⎞ ⎛ ∆t ⎞ 1+ ⎜ ⎟ 1+ ⎜D 2 ⎟ ⎝ R⎠ ⎝ ∆x ⎠

(8.32)

For the case where ρ is very small (i.e., less than approximately 10–2), then de Cogan14 has shown that the result is effectively a solution of the telegrapher’s equation; an impulse of unit initial magnitude is attenuated to τk after k iterations, and leaves a constant trail behind it of magnitude, ρτk–1 (see Figure 8.7). If we look at the relative magnitudes of the resistances and impedances in the problem as defined by Boucher and Kitsios, then we see that the reflection coefficient, given by ρ = 1/(1 + Z/R) fits the criterion. So, for example, ρ1 = 1.9 × 10–4 and τ1 = 1 – ρ1 = 0.999807 in a situation where region 1 has been broken up into 10 nodes, and we can see that the pulse incident at node 5 at the new time is given by

WL ( 5) = 1.9 × 10 −4 kiWL ( 5) + 0.999807 kiWR ( 4 )

i k +1

WR ( 5) = 1.9 × 10 −4 kiWR ( 5) + 0.999807 kiWL ( 6 )

i k +1

Clearly, it is dominated by contributions from nearest neighbors, although there is a small contribution due to back-scatter as a result of drag. Burton, Edge, and Burrows15 also use a minimum node approach, but it is based on concepts first put forward by Viersma.9 The four-pole transmission-line equation is given by

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Magnitude 1 τk

ρτ

k-1

Time k∆ t Figure 8.7 Propagation of a lossy wave, showing the magnitude of the pulse at time, k∆t and the trailing component due to frictional forces (ρ τk–1). This approximation holds true so long as contributors to the tail of the form ρn τk-n for n > 1 can be ignored.

PL − Z N ( s)WL = ⎡PR + Z N ( s)WR ⎤ e − Ts ⎣ ⎦

N (s)

PR − Z N ( s)WR = ⎡PL + Z N ( s)WL ⎤ e − Ts ⎣ ⎦

N (s)

(8.33)

where PL, WL, PR, and WR represent pressures and flows at left and right ends of the pipe, T is the transit-time for a pulse traveling from end-to-end and N(s) is the viscous friction factor. This can be viewed graphically as shown in Figure 8.8. From this we can derive two transmission line end equations where the transit time, T is replaced by the more general discretization time ∆t.

PL ( t) − ZWL ( t) = φ R ( t − ∆t) PR ( t) − ZWR ( t) = φ L ( t − ∆t)

(8.34)

If the pipe is lossless, then N(s) in Equation (8.33) is unity and the characteristic pressures can be expressed as

φ L ( t − ∆t) = PL ( t − ∆t) + ZWL ( t − ∆t) φ R ( t − ∆t) = PR ( t − ∆t) + ZWR ( t − ∆t)

(8.35)

which, as Burton et al.15 point out, means that the two ends of the line are decoupled.

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t

t

t+T

t+T WR

WL

Z

Z

φL

191

Z

Z

φR

WR

WL + PL -

PR

L

+ -

+ PL -

R

L

(a)

+ -

PR

R (b)

Figure 8.8 Transmission line properties of a pipe (a) fluid leaving left side at time, t arrives at right side at time, t + T, (b) fluid leaving right side at time, t, arrives at left side at time, t + T.

Pollmeier8 gives an example of this approach. Figure 8.9 shows a constant-flow source connected to a fluid-filled transmission line, which is connected to a pressure relief valve. The output of the valve is linked with a constant pressure reservoir. The pressures and flows at the different locations are shown in the figure, with the source flow W1 defined as positive into the transmission line. Equation (8.34) is then used to give:

P1 ( t) = φ2 ( t − T ) + ZW1 ( t)

(8.36)

We also have from Equation (8.35)

φ1 ( t) = P1 ( t) + ZW1 ( t)

(8.37)

W1

W2

W3

P1

P2

P3

1

2

3

Figure 8.9 Simple hydraulic circuit. (From Pollmeier K., TLM: The wider applications, Proceedings of an informal meeting held at the University of East Anglia, Norwich on 27 June 1996, de Cogan D. Ed., School of Information Systems, University of East Anglia, Norwich, 1996. With permission.)

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This characteristic pressure is observed at the inlet of the relief valve at the next time-step. We will assume that the valve dynamics are instantaneous, flow is unidirectional and does not take place until the differential pressure exceeds the cracking pressure, Pc and that a linear flow pressure gradient is assumed once the valve has cracked open. We can now proceed to calculate the pressure at the valve inlet:

P2 ( t) = φ( t − T ) + ZW2 ( t)

(8.38)

P3 ( t) = P3 = const. Flow is then determined as

W2 ( t) = −W3 ( t) = K ⎡⎣P2 ( t) − P3 − Pc ⎤⎦ for (P2 – P3) > Pc

(8.39a)

W2 ( t) = W3 ( t) = 0 for (P2 – P3) Pc

(8.39b)

Flow is obtained by substituting the characteristic pressure equation into the valve equation.

W2 ( t) = −W3 ( t) =

K ⎡⎣φ1 ( t − T ) − P3 − Pc ⎤⎦ 1+ K Z

for (P2 – P3) > Pc

(8.39c)

We can ignore the case of (P2 – P3)Pc as it is not a practical proposition for this circuit; there must be a flow through the relief valve if the pump is running. The valve port pressure can be obtained by substitution of this in Equation (8.34):

P2 ( t) = φ1 ( t − T ) − Z

K ⎡⎣φ1 ( t − T ) − P3 − Pc ⎤⎦ 1+ K Z

=

φ( t − T ) + Z( P3 + Pc ) (8.40) 1+ K Z

The characteristic pressure to be transmitted to the flow source is calculated from Equation (8.34):

φ2 ( t) = P2 ( t) + Z W2 ( t)

(8.41)

In addition to the pressure relief/check valve Burton and Burrows.15 discuss models for a wide range of other components that are used in hydraulic circuits.

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8.6 Transients in Elastic Pipes Burton and Burrows15 also discuss the problem of compressible pipe-lines. They recommend that an approximation can be made by setting the transmission delay to the global integration time-step, ∆t. This, they say, implies a modified line-length and consequently, a compensational change in the cross-sectional area, in order to keep the capacitive term, V/B constant (V is volume and B is fluid bulk modulus). They suggest that ∆t should be small, because otherwise the line dynamics could become distorted due to the inertance term, ρL/A (ρ is density, L is length, and A is pipe area). They also mention that, if unrealistic resonance effects are to be avoided, then low-pass filtering*, such as that developed by Krus et al.16 should be used. We would prefer to take a different approach to compressional waves by looking at a discretized pipe-line. Figure 8.10 shows a pressure impulse traveling along the natural gradient within a pipe-line. The nature of the impulse is such that it creates a localized distortion of the pipe. If we know the local pressure P(x,t) then we should be able to calculate the distortion of the pipe. We will assume that the localized nature of the disturbance has a negligible effect on the length of the pipe. Thus, we are concerned only with the instantaneous area, A(P(x,t)) which is greater than A0. We see from Equation (8.5) that the impedance is inversely proportional to the value of A(P(x,t)). Thus, we are dealing with a problem where the impedance is no longer constant, but changes as a function of the pressure-wave that passes through the pipe, and we will now discuss how this problem might be tackled. Pollmeier8 gives a treatment that is based on an idea first proposed by Johns and published posthumously.17 It seems somewhat strange in that it uses the characteristic pressures, but this may be due to the fact that it is for a whole-line model. In our case, we will consider discretized pipes†. The Pulko/Johns17 approach, which is similar to that used by Pollmeier, considers that the line impedance undergoes a transformation Z → Z' as the pressure pulses from left and right pass each other at the center of the line. The pressures, which were PL and PR, now become P'L and P'R. The total pressure at this mid point must be conserved and so must the total flow:

PL + PR = PL' + PR' (8.41)

PL − PR PL' − PR' = Z Z'

*

Effectively an approximation of the frequency-dependent friction process From here onwards we will use PL(x) and PR(x) (rather than WL and WR) to denote local pressures in a pipe. This should ensure greater consistency with the wider TLM literature.

†

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P

distance

Figure 8.10 Impulsive pressure wave creating a localized distortion in a pipe.

If we define γ = Z'/Z, then this yields the following before and after relationship:

⎡ PL' ⎤ ⎢ ' ⎥= ⎣PR ⎦

⎡( 1 + γ ) ⎢ 2 ⎣( 1 − γ ) 1

( 1 − γ ) ⎤ ⎡ PL ⎤ ⎥⎢ ⎥ ( 1 + γ ) ⎦ ⎣PR ⎦

(8.42)

There is a computational problem with this approach. As the pulse moves, the area (hence impedance) must be recalculated for every position at every time. If there is an impedance mismatch between adjacent lines, then the relevant reflection and transmission coefficients must be calculated. So a pair of pulses start off from the left and right ends of a section ∆x of line. They are designated iPL(x) and iPR(x) to show that they are incident on the inspection point, which is located at (x), the mid-point of the line and they reach this point after time, ∆t/2. The line impedance is changed according to the instantaneous local pressure (iPL + iPR). The pulses are immediately rebadged i P' L (x) and i P' R (x) and are now moving in a medium of impedance, Z'. As they pass the mid-point they are no longer incident and so are relabeled as scattered pulses: i

P'L(x)→ sP'R(x) and iP'R(x)→ sP'L(x).

The values of Z'(x), Z'(x – 1) and Z'(x + 1) are used to calculate the current values of scattering coefficients as seen by the three locations looking left and right. So, the reflection coefficients on either side of (x) are given by:

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ρL ( x ) =

195

Z' ( x − 1) − Z' ( x ) = −ρR ( x − 1) Z' ( x − 1) + Z' ( x ) (8.43)

ρR ( x ) =

Z' ( x + 1) − Z' ( x ) = −ρL ( x + 1) Z' ( x + 1) + Z' ( x )

So, when scattering occurs at these boundaries we obtain values for the pulses, which will be incident at (x) during the next time-step

P ( x ) = ρL ( x ) ks PL' ( x ) + τ R ( x − 1) ks PR' ( x − 1)

i k +1 L

(8.44)

P ( x ) = ρR ( x ) ks PR' ( x ) + τ L ( x − 1) ks PL' ( x − 1)

i k +1 R

The alternative method avoids this process of redesignation of repeated internodal scattering. Instead it uses the concept of stubs (see Chapter 3). Each node in the pipe comprises three lines as shown in Figure 8.11. open circuit termination

i

PSt

Z St i

P

i L

Z

(x)

PR

Z

Figure 8.11 1-D node with half-length stub showing the incident pressures before the stub undergoes a pressure-dependent impedance transformation.

For most of the nodes the stub-lines will have infinite impedance (i.e., they do not contribute to the process of wave propagation). However, where the pressure wave does cause local variations, then we will include this in the stub only. In this way, we do not have to contend with internodal scattering. The potential before impedance transformation is given by:

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⎛ i P ( x ) + i PR ( x ) ⎞ 2 i PSt ( x ) 2⎜ L ⎟+ Z Z ⎝ ⎠ St φ( x ) = 2 1 + Z ZSt

(8.45a)

The value of φ(x) is used to determine the local value of impedance. The line impedances are kept constant and the change is reflected only in the change of stub impedance. So, if the capacitance of the node is given by C = A∆x/Ke (where Ke is the bulk modulus), then any capacitance increase is due to an increase in the area = CSt. The stub impedance is given by ZSt = ∆t/ 2CSt. Let us now imagine that at the instant of arrival, the characteristic pressure, given by Equation (8.45a) causes an adjustment of stub impedance, which now goes to Z'St. The value of φ(x) is then given by:

⎛ i P ( x ) + i PR ( x ) ⎞ 2 i PSt' ( x ) 2⎜ L ⎟+ ' Z ZSt ⎝ ⎠ φ( x ) = 2 1 + ' Z ZSt

(8.45b)

Since the flow in the stub is constant we have a relationship between the parameters before and after transformation:

i

' PSt' = ZSt

i

PSt ZSt

(8.46)

The scattered values are now s

PL ( x ) = φ( x ) − i PL ( x )

s

PR ( x ) = φ( x ) − i PR ( x )

s

PSt' ( x ) = φ( x ) − i PSt' ( x )

(8.47)

The precise method of application of these ideas requires some caution and the sequence will be demonstrated by means of an example in the next section.

8.7 Open-Channel Hydraulics The subject of open-channel hydraulics is very broad and generally makes use of approximations such as the Manning equation, which includes a parameter

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Figure 8.12 A disturbance moving along a fluid held in an open rectangular channel.

that takes account of different types of surface.6 We will trespass into this territory only insofar as it is used to demonstrate the use of the principles of the preceding section to treat the propagation of some simple transients. In this treatment we will consider a lossless channel as shown in Figure 8.12. We will assume that the instantaneous local pressure causes the fluid to rise so that P(x) = ρgh(x). We start with a network of nodes as shown in Figure 8.11. Initially, the impedance values of all stubs are set very large, so that they have no influence on the propagation process. The local pressure, φ(x) is calculated using Equation (8.45a). The scattered pulses are calculated using s

PL ( x ) = φ( x ) − i PL ( x )

s

PR ( x ) = φ( x ) − i PR ( x )

s

PSt' ( x ) = φ( x ) − i PSt ( x )

(8.48)

The modified local stub impedance is calculated using Z'St = 1/( 2φ(x)) and this is used to adjust the pulses scattered in the stub just before it arrives at the open-circuit boundary: s

' PSt' = ZSt

s

PSt ZSt

(8.49)

The pulses incident at node (x) at the next time are then given by:

P ( x ) = ks PR ( x − 1)

i k +1 L

P ( x ) = ks PR ( x + 1)

i k +1 R

(8.50)

P ( x ) = ks PSt' ( x )

i k + 1 St

Equation (8.45a) is then used to calculate the local pressures at this new time and the cycle is repeated. When this algorithm was prepared as a MATLAB program, two different excitations were considered. The first was a conventional Gaussian profile with unit maximum height with water as the medium. Initially the program was run with all stubs at a constant and very high value throughout the

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0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

50

100

150

200 250 300 350 nodal position along channel

400

450

500

Figure 8.13 Propagation of a unit step-wave with a Gaussian front along a 500-node open-top hydraulic channel containing water. Results are shown after 400 iterations. The vertical axis, height, is proportional to local pressure since ρgh(x) = φ(x).

simulation. The profile was seen to move from left to right with no change. When stubs were altered as a function of pressure, Z'St(x) = 981/(2φ(x)), then the profile was attenuated and became asymmetrical as it moved from left to right, leaving a decaying tail behind it. The second case utilized a step-function with a Gaussian leading-edge. The MATLAB program for this is given at the end of the chapter. The result, shown in Figure 8.13, indicates that the disturbance moves from left to right leaving a linear pressure gradient in its wake.

8.8 Conclusions In this chapter we have gone some way towards demonstrating that the wave propagation equations that have been used in applications such as engine manifold design4,18 are identical to those that are well known in electromagnetics. There is no reason why the tools that have been developed for the adjustment and analysis of microwave circuits (stubs and the Smith chart19) cannot be used more widely in computational hydraulics. We have attempted to give a unifying presentation of some aspects of an enormous field of work where different researchers have been using slightly different dialects of what is essentially TLM. Our discussions have centered on the situation where the

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velocity of propagation of a transient is very much faster than the flow associated with the fluid in which it travels. Strictly speaking, there is an advective term that should be included. It plays a dominant role in phenomena such as the “hydraulic jump.” Solving the advection equation has presented an interesting challenge for TLM modelers. In the case of diffusion with drift Chakrabarti20 found that his TLM algorithms had regions of instability, but the restrictions were not as severe as the equivalent formulations using finite differences. O’Connor's alternative approach,21,22 which does not appear to have stability problems, is discussed elsewhere in this book. %%

*** TLM PROGRAMME FOR WAVE PROPAGATION IN OPEN-TOP RECTANGULAR HYDRAULIC CHANNEL *** % USER DEFINITIONS %channel length & number of iterations

nmax = 500; kmax = 400; Z= 1;

% ARRAY INITIALISATIONS

ZSt = 1000000000000000000*ones(1,nmax);% stub(x) has very high impedance ZSt_new = ZSt; % likewise for temporary stub array PiL = zeros(1,nmax);PiR = PiL;PiSt= PiL; PsL = PiLPsR = PiL;PsSt= PiL;PsSt_new= PiL; phi = PiL; for k = 1:kmax

% START OF TLM ALGORITHM

if k<=50 % TIME-DEPENDENT EXCITATION PiL(1) = 1*exp(-((50-k)^2)/400) - PsL(1);% Gaussian front else PiL(1) = 1 - PsL(1);

% unit excitation after 50 iterations

end phi = (2*(PiL + PiR)/Z +2*PiSt./ZSt)./(2/Z + 1./ZSt);% SUMMATION OF PULSES PsL = phi - PiL; PsR = phi - PiR; PsSt= phi - PiSt;% SCATTER OF PULSES for x=1:nmax if phi(x)>0 else

% STUB IMPEDANCE ADJUSTMENT

ZSt_new(x) = 981./(2*phi(x));% redefine stub(x) % ( if phi(x)>0 ) else ZSt_new(x) = ZSt(x);% leave it as it was

end end PsSt_new= ZSt_new.*(PiSt./ZSt);% redefine pulse scattered into stub ZSt = ZSt_new;

% redefine old stub array

for x = 2:nmax PiL(x) = PsR(x-1);

% CONNECTION OF PULSES % connect pulses incident from left

end for x = 1:nmax-1 PiR(x) = PsL(x+1);

% connect pulses incident from right

end for x = 1:nmax PiSt(x) = PsSt_new(x);% connect pulses incident from stub end end plot(phi)

% END OF TLM ALGORITHM

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References 1. Wylie E. B. and Streeter V. L., Fluid Transients, McGraw-Hill International, New York (1978). 2. Wylie E. B. and Streeter V. L., Two and three-dimensional transients, J. Basic Eng., Trans. ASME, vol 90, ser. D, No. 4 (1968) 501–510. 3. Boucher R. F. and Kitsios E. F. Simulation of fluid network dynamics by transmission line modelling, Proc. Inst. Mech. Engrs., 200 C1 (1986) 21–29. 4. Pearson R. J. and Winterbourne D. E., A rapid synthesis technique for intake manifold design, Int. J Vehicle Design, 10 (1989) 659–686. 5. Auslander D. M., Distributed system simulation with bilateral delay-line models, Trans ASME J. Basic Eng. (1968) 195–200. 6. Nalluri G. and Featherstone R. E., Civil Engineering Hydraulics, 4th ed., Chapter 4, Blackwell Science, Abingdon, U.K. (2001). 7. de Cogan D., Transmission Line Matrix (TLM) Techniques for Diffusion Applications, Gordon and Breach (1998) 120–121. 8. Pollmeier K., TLM: The wider applications, Proceedings of an informal meeting held at the University of East Anglia, Norwich on 27 June 1996, de Cogan D. Ed., School of Information Systems, University of East Anglia, Norwich, 1996. 9. Viersma T. J., Analysis, Synthesis and Design of Hydraulic Servo-Systems and Pipelines, Elsevier, Amsterdam, New York (1980). 10. Burton J. D., Parallel simulation of hydraulic systems using Transmission Line Modelling (TLM), Ph.D. thesis, University of Bath, U.K. (1994). 11. de Cogan D. and de Cogan A., Applied Numerical Modelling for Engineers, Chapter 8, Oxford University Press, Oxford (1997). 12. Wylie E. B. and Streeter V. L., Fluid Transients, McGraw-Hill International, New York (1978) 205–235. 13. Christopoulos C., The Transmission-Line Modelling Method: TLM, IEEE/OUP (1995) 71–90. 14. de Cogan D., The relationship between parabolic and hyperbolic transmission line matrix models for heat-flows, Microelectronics J., 30 (1999) 1093–1097. 15. Burton J. D., Edge K. A., and Burrows C. H., Analysis of an electro-hydraulic position control servo-system using Transmission Line Modelling (TLM), 2nd JHPS International Symposium on Fluid Power, Tokyo, Japan, 1993. 16. Krus P., Jansson A., Palmberg J-O., and Weddfelt K., Distributed simulation of hydromechanical systems, 3rd Bath International Fluid Power Workshop, 1990. 17. Pulko S. H., Mallik A., Allen R., and Johns P. B., Automatic timestepping in TLM routines for the modelling of thermal diffusion processes, Int. J. Numerical Modelling, 3 (1990) 127–136. 18. Winterbone D. E. and Pearson R. J., Theory of Engine Manifold Design, Professional Engineering Publishing Ltd, London, and Bury St Edmunds, U.K. (2000). 19. Smith P. A. and Packer G. A., Electronic Applications of the Smith Chart in Waveguide, Circuit and Component Analysis, Rhea R. and Hammond C. Eds., Noble Publishing (1995). 20. Chakrabarti A., Transmission Line Matrix modelling for semiconductor transport, Ph.D. thesis, University of East Anglia, Norwich, U.K. (1996). 21. O’Connor W. J., Wave-speeds for a TLM model of moving media, Int. J Numerical Modelling, 15 (2002) 195–203.

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22. O’Connor W. J., TLM model of waves in moving media, Int. J Numerical Modelling, 15 (2002) 205–214.

Bibliography Papers by the Bath and Linköping hydraulics groups not otherwise cited in this chapter. Burton J. D., Edge K. A., and Burrows C. R., Modelling requirements for the parallel simulation of hydraulic systems, Presented at ASME Winter Annual Meeting, Anaheim, CA, 8–13 November 1992 (document 92-WA/FPST-11). Burton J. D., Edge K. A., and Burrows C. R., Computational load balancing of parallel hydraulic circuit simulations employing variable step Transmission Line Modelling, Proceedings of the Third Scandinavian International Conference on Fluid Power, 1993. Burton J. D., Edge K. A., and Burrows C. R., Partitioned simulation of hydraulic systems using Transmission Line Modelling, Presented at ASME Winter Annual Meeting, New Orleans 28 November–3 December, 1993 (document 93-WA/FPST-4). Krus P., Weddfelt K., and Palmberg J-O., Fast pipeline models for simulation of hydraulic systems, Trans. ASME (J. Dynamic Systems, Measurement and Control), 116 (1994) 132–136. Pollmeier K., Burrows C. R., and Edge K., Mapping of large scale fluid power system simulations on a distributed memory parallel computer using genetic algorithms, ASME Collected Papers in Fluid Power Systems and Technology FPST-3 (1996) 83–91. Pollmeier K., Burrows C. R., and Edge K., Partitioned simulation of fluid power systems—an approach for reduced communication between processors, Proc. Inst. Mech. Engrs., 210 (1996) 221–230. Krus P., An automated approach for creating component and subsystem models for the simulation of distributed systems, 9th Bath International Fluid Power Workshop, 1996.

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chapter nine

Application of TLM to Computational Fluid Mechanics 9.1 Introduction Computational fluid dynamics (CFD) models the mechanics of flow. There is an enormous range of fluid dynamics problems of interest to science and engineering, and CFD is a correspondingly broad subject. The considerations here will be limited mainly to steady flow of real, incompressible, viscous fluids, that is, fluids obeying the Navier-Stokes equations of fluid mechanics. While the focus will be mainly on steady-state flow fields, transient effects will also be briefly considered. Finally only 2-D problems will be discussed, although all the ideas can be extended to 3-D in a straightforward way. The basic CFD variables are pressure and velocity, as in acoustics. But CFD is broader. First, while viscosity is generally ignored in acoustics, it is always present in real fluids and CFD generally needs to take it into account. Its effects can range from barely significant to completely dominating, depending on the nature of the fluid and on the flow regime. In what follows, the effects of viscosity will first be applied to the TLM solution of the acoustic wave equation. By iterating the solution, with appropriate boundary conditions, until a steady state is reached, this technique can be used to determine velocity flow fields for steady flow viscous problems. The second effect that is absent from the acoustic approximation, but of great importance in nonuniform flows of real fluids, is the “convective acceleration” and its associated pressure gradient. When these are incorporated into the equations of motion, one gets the Euler equations of fluids. Finally when both viscosity and convective acceleration effects are accounted for, the resulting mathematical formulation corresponds to the

203

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Navier-Stokes equation, which for incompressible Newtonian fluid can be expressed as1

ρ



Du
= f − grad p + µ∇2 u Dt

(9.1)

The significance of these terms and their modeling will be considered below.

9.2 Viscosity In the standard acoustic wave equation viscous effects are implicitly taken as negligible. It is assumed that a fluid particle is accelerated only by the action of the acoustic pressure gradient. This approximation is physically justifiable for many fluids, especially gases, where the forces due to pressure gradients are typically much larger than the viscous forces. But, where the viscosity is significant, it causes both attenuation and dispersion of pressure waves and has a strong influence on the ultimate steady-flow pattern. Because of the nature of the physical effect, the loss mechanism is not properly modeled by the circuit equivalent of resistive losses, in which there is a potential degradation in the direction of the flow and proportional to the flow (Ohm’s law). When a fluid particle moves at a different velocity to that of the surrounding fluid it experiences a viscous force acting in the direction of the motion. This force can be thought of as a shearing effect, or momentum diffusion effect. As one layer of fluid slides past a neighboring layer traveling faster or slower than itself, it tends to drag the other and to be dragged by it, as the layers mix. Depending on the sign of the relative velocities, the force experienced by any fluid particle can be positive or negative, that is, it can tend to cause acceleration or deceleration. In Newtonian fluids, this viscous (shear) force, τ, on each side of the fluid layer is proportional to the rate of change of velocity perpendicular to the direction of motion. The proportionality constant is called the coefficient of viscosity, µ, which characterizes the fluid’s reluctance to flow. Thus, for a fluid traveling at velocity u in the x-direction,

τ=µ

∂u . ∂y

(9.2)

The viscosity coefficient varies widely between fluid types, from close to zero to almost infinity. In 2-D flow, the net force on a particle in the direction of motion is due to the difference between the viscous forces acting on its two sides, each of which is proportional to the velocity gradient. In the limit therefore the net force is proportional to the second spatial derivative of the velocity (the rate

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of change of the rate of change of velocity) in the direction perpendicular to the motion. One can associate with this force a pressure loss, or “viscous pressure gradient,” so that for the case of flow everywhere parallel to the x-axis,

⎡ ∂P ⎤ ∂2 u ⎢ ∂x ⎥ = µ 2 ∂y ⎣ ⎦µ

(9.3)

When the flow is fully 2-D, the situation is more complex1 and the following equations apply:

⎛ ∂2 u ∂2 u ⎞ ⎡ ∂P ⎤ = µ ⎢ ∂x ⎥ ⎜ 2 + 2⎟ ∂y ⎠ ⎝ ∂x ⎣ ⎦µ

(9.4)

⎛ ∂2 v ∂2 v ⎞ ⎡ ∂P ⎤ ⎢ ∂y ⎥ = µ ⎜ 2 + 2 ⎟ ∂y ⎠ ⎝ ∂x ⎣ ⎦µ

(9.5)

where u and v are the components of the velocity in the x and y directions.

9.3 Viscosity in the TLM Algorithm Viscosity causes energy loss. In general, losses in TLM are modeled by scattering at lumped resistors or by lossy stubs. The approach here is an adaptation of a more “distributed” approach described in Reference 2 for ohmic, series losses, but here modified to model the physical viscous effects in an appropriate way. If the velocity of any link line is higher than in the adjacent lines on each side, then viscosity will reduce the velocity, while conserving flow. As with series resistance effects, there must be a corresponding pressure gradient. The difference however is that the gradient is proportional not to the flow (current) but to µ.∇2u, where u is the fluid velocity. At each time interval, after scattering, the acoustic velocity (TLM current) in each line is determined, and a finite difference approximation to the second spatial derivative evaluated, for example, as

∂2 u u( y + 1) − 2 u( y ) + u( y − 1) = ∂y 2 ∆l2

(9.6)

In 2-D flow, as in Equations (9.4) and (9.5), four such terms are needed. Expressions such as Equation (9.6) are readily evaluated for mesh lines away from boundaries. Near boundaries the required neighboring points will not exist and special treatment based on other considerations is required.

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It is a property of viscous flow that close to the boundary the velocity must approach zero, as otherwise the shear force in Equation (9.2) would become infinite even for arbitrarily small values of µ. Furthermore, the flow close to the boundary must be parallel to the boundary, because there is no component of flow either into or out of the boundary. Thus the shear force, and therefore pressure gradient, can be evaluated using a finite difference approximation to Equation (9.2) with unidirectional flow. The viscous force produces the viscous pressure gradients of Equations (9.4) and (9.5), which modify the acoustic pressure gradient and the acoustic velocities as they propagate through the TLM mesh. The required change in the pressure gradient, for example in the x-direction, ∆px, is proportional to the derivative terms in Equation (9.4). There is also an associated change in the velocity in the same direction. In the algorithm, both of these requirements are met by removing half of the ∆px from the pulse moving in the x-direction, and adding the other half to the pulse traveling in the opposite direction (see Chapter 2). This has the effect of both imposing a viscous pressure gradient and causing an appropriate acceleration (velocity change), yet conserving the total flow (see Reference 2 for further details). Clearly, where flow is spatially uniform, ∆px will be negligible, whereas where there are large spatial gradients in velocity, ∆px will be correspondingly large. This is exactly as it should be.

9.4 Results General, closed form analytical results for viscous wave propagation are difficult to obtain.3 The implementation of the scheme is easiest when the flow is parallel to the TLM mesh lines, as in quasi 1-D flow problems. Figure 9.1 shows perhaps the simplest fluid mechanics illustration of viscosity, namely steady viscous flow in a pipe between two pressure boundaries. With pressure boundaries implemented at each end, at different pressure values, and total reflection of TLM pulses at the pipe walls top and bottom, the algorithm was run until steady state was achieved. The expected parabolic velocity distribution across the pipe quickly emerged, as shown, confirming the correct general behavior of the model. Furthermore, the standard fluid mechanics results for total discharge, pressure drop, fluid-energy dissipation rate, and momentum balance are all verified in the solution obtained. The general 2-D case is more complex, but the basic idea still works well. In Figures 9.2 and 9.3, a relatively crude 15x15 TLM mesh was used to simulate the flow in a short pipe without and with a square obstacle in the flow. The results are exactly as expected, and compare very well with results for the same problem from a conventional CFD package. Characterizing a fluid by a single coefficient of viscosity (that is, a constant) is a simplification that may or may not be borne out by the physical behavior of a real fluid. The relationship between viscous shear force and velocity gradient may not be linear. In any case, viscous effects are always

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Figure 9.1 Flow pattern obtained by iterating the viscous wave equation to steady state. 16 14 12 10 8 6 4 2 0

0

2

4

6

8

10

12

14

16

Figure 9.2 Viscous flow through a 15 × 15 mesh model of a pipe with pressure boundaries at each side.

temperature dependent. Furthermore, shear history can also change the viscous properties of fluids. Such “non-Newtonian” fluid behavior is widely encountered. A further advantage of the TLM technique is that even arbitrarily nonlinear, or space dependent, or time dependent viscosity can easily be built into the solution scheme.

9.5 Incompressible Fluids and Velocity Fields In incompressible fluids, the convergence or divergence of the velocity field is everywhere zero. If current is taken as the analogue of fluid velocity, then fluid incompressibility corresponds to conservation of current, something

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that is assured in a properly derived TLM algorithm. Thus, when TLM is used to model incompressible fluids, the incompressibility condition does not require extra constraints on the model. Furthermore, if a TLM model (that captures the high speed acoustic effects very well) is iterated until steady state is reached, the steady flow pattern (velocity field) that emerges obeys the boundary conditions and is physically consistent with conservation of flow and the acoustic effects modeled by the TLM algorithm (including viscosity, if incorporated). This means that the only flow pattern that can become steady implicitly contains all the required information about the “convective pressure gradient” (see below), which can then be determined from it, if required. Thus, if an appropriate TLM scheme is used to solve a steady state CFD problem, the solution scheme need not concern itself with the static convective pressure gradient as part of the solution scheme, but this in turn can be ascertained a posteriori from the solution. This merits further explanation.

9.6 Convective Acceleration and the TLM Model At first sight, one might expect that the acceleration of a fluid at a point could be defined as the time derivative of the fluid’s velocity. Such a description however proves too simple. Consider the case of steady, nonuniform flow, such as stable flow through a variable section pipe or steady flow around a cylinder. As the fluid velocity at every point in space is unchanging with time, the straightforward time derivative of the velocity as seen by a stationary observer is everywhere zero. If one applied the simple definition of acceleration therefore, the fluid would have no acceleration at any point, with

∂u = 0. ∂t

(9.7)

Despite the above, wherever the flow is not perfectly uniform, that is wherever the flow is speeding up or slowing down or changing direction, the fluid particles passing through such a point clearly are accelerating. Thus, from the perspective of an observer moving with the flow, there is certainly an acceleration at such points, and it may be large. The size depends on how quickly the flow pattern is changing from point to point along the flow (that is, on the spatial derivative of the velocity field in the direction of the flow), multiplied by the velocity (that is, how quickly the particles are moving through the changing flow pattern). Thus, for flow in the x-direction



ac = u.∇ u

( )

(9.8)

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This is the “convective” acceleration, and it is a real, physical acceleration that the fluid particles undergo at each point as they flow through the field. The force that brings it about is a difference in pressure across the fluid particle, and so the existence of the convective acceleration implies a corresponding convective pressure gradient. These gradients in the x- and y-directions are

⎛ ∂u ⎡ ∂P ⎤ ∂u ⎞ ⎢ ∂x ⎥ = ρ ⎜ u ∂x + v ∂y ⎟ ⎝ ⎠ ⎣ ⎦C

(9.9)

⎛ ∂v ⎡ ∂P ⎤ ∂v ⎞ ⎢ ∂y ⎥ = ρ ⎜ u ∂x + v ∂y ⎟ ⎠ ⎝ ⎣ ⎦C

(9.10)

The total acceleration of a particle in the flow is the sum of the two kinds of acceleration described above, and is denoted in fluid mechanics by a capital D, thus





Du ∂u = + u.∇ u Dt ∂t

( )

(9.11)

This notation was used in Equation (9.1). In the “acoustic approximations” that lead to the acoustic wave equation, and therefore in acoustic TLM, only the first term on the right hand side is retained and modeled. The second term, the nonlinear convective acceleration, is ignored. This is a sensible simplification in purely acoustics problems where the time derivative term is typically orders of magnitude more significant than the convective acceleration term, and the magnitude of the fluid velocity, u, is small. In other words, one is more interested in the propagation of rapidly changing fluctuations in pressure and velocity rather than the much slower bulk movements of the fluid. Frequently in fluid mechanics the opposite is the case. It is always the case for steady flows, where pressure gradients (which do not change with time) are all associated with convective accelerations. At the other extreme, even in general fluid mechanics, pressure waves traveling at the speed of sound can become the most important issue wherever there are very sudden (that is, high speed) changes in pressure or velocity, such as, for example, in aerodynamics, in aspects of engine analysis and in water hammer effects. In general, both high speed and bulk movement effects will be present to varying degrees. But for steady flows, by definition, the time dependent term is zero, and the remaining convective pressure gradient is static. Figure 9.4 shows the convective pressure gradient corresponding to the geometry of Figure 9.3, with flow past a square obstacle from left to right. As explained, it was determined from the velocity field after the latter had

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Transmission Line Matrix in Computation Mechanics 16 14 12 10 8 6 4 2 0

0

2

4

6

8

10

12

14

16

Figure 9.3 Same as Figure 9.2, but with a square obstacle in the pipe.

30 20 10 0

0

5

10

15

20

Figure 9.4 The convective pressure distribution for the same problem as Figure 9.3.

been obtained by iteration to steady state. First, the convective pressure gradient was evaluated at every point using a finite difference approximation to Equations (9.9) and (9.10). That is,

⎛ uy + _ − uy − _ ⎞ ⎛ ∂P ⎞ ⎛ ux + _ − ux − _ ⎞ ⎟⎠ + v ⎜ ⎜⎝ ⎟⎠ = u ⎜⎝ ⎟⎠ ∆x ∆y ∂x C ⎝

(9.12)

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⎛ vy + _ − vy − _ ⎞ ⎛ ∂P ⎞ ⎛ vx + _ − vx − _ ⎞ ⎟⎠ ⎟⎠ + v ⎜⎝ ⎜⎝ ∂y ⎟⎠ = u ⎜⎝ y ∆ ∆x C

211

(9.13)

Then this pressure gradient field was integrated to obtain the convective pressure differences across the problem. The resulting pressure field is very much as expected. The pressure rises towards the stagnation point in the center of the obstacle, and falls where the fluid accelerates into the narrower sections around the obstacle. Because of corner effects (discussed further below) the convective pressure gradients, and therefore the resultant convective pressures after integration, are unreliable in the vicinity of the corners.

9.7 Comments on the Procedure It might be objected that, in reaching convergence, the TLM algorithm has ignored the convective pressure gradients and bulk fluid velocity effects, and therefore extracting later from the solution is invalid. One response is that these have no effect on the final steady state solution. But one can go further, because neither do they have much effect on the dynamic situation before convergence. In the first place, the convective pressure gradient is not propagating like a wave. Then, as regards the propagating waves modeled by TLM, the wave properties (wave speed and impedance) depend on the bulk modulus and the density of the fluid, and in an incompressible fluid, neither of these is pressure dependent. So that even if one went to the trouble of adding a nonpropagating background pressure field (related to the convective acceleration) to the TLM acoustic pressure field during the dynamic stage, it would not affect the manner of propagation of the acoustic information. So why bother? The case is less clear with the background bulk fluid velocity, which does indeed affect wave speed and impedance, as mentioned in earlier chapters and will be revisited in the next chapter. But even here, such effects become significant only when the fluid velocity becomes comparable to the acoustic wave speed, which is outside the scope of this model. Turning this objection around, a significant advantage of using TLM to solve the CFD problem is precisely that it models both the effects that propagate at sonic speeds, as well as converging to give the correct steady-state solution. The model therefore successfully spans what in the past have tended to be two separate areas, steady flow and rapidly changing or acoustic fluid problems.
Finally, similar comments apply to modeling the body force term, f , in Equation (9.1). Physically this corresponds to any force acting directly on the fluid particle not due to fluid effects, especially the force of gravity. For steady problems, the gravity force gives rise to a static vertical pressure gradient (the “hydrostatic” pressure gradient). Where the vertical

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dimensions of the problem are small, or the density is low, it can frequently be ignored. If required, however, for the steady state problem it can also be added in after the problem has converged, as its effects on the dynamic part of the problem are not significant.

9.8 Implementation Issues In implementing the scheme, a number of difficulties arise in practice. First, if one begins with an “empty” mesh (all TLM pulses assumed zero) and one applies the pressure boundaries, waves then travel up and down the system from boundary to boundary. Of course, this is quite correct from a physical perspective, and the scheme is modeling what would happen if boundary pressures were suddenly changed in a real system. But if one is interested only in the converged scheme, its very power can become a disadvantage as it delays final convergence. In this case, convergence to steady state can be accelerated by both ramping up the pressures to desired values more slowly, and then artificially “damping” the boundary reflection conditions. This is done by calculating the reflected pulse at the pressure boundary in the usual way, but rather than using this value, using a weighted mix of the new and the previously reflected pulse, with the weighting factors determining the level of numerical damping. A more serious problem arises near sharp corners, particularly internal corners. This difficulty is common to most numerical schemes, particularly those related to finite difference schemes on a uniform mesh. Furthermore, this is almost a difficulty of principle, as near a corner the flow is theoretically constrained to be at once parallel to two surfaces that are sharply angled to each other, which of course is physically impossible. This inherent problem is then considerably intensified when derivatives of the velocity field are required, as in the present case. Most pragmatic numerical solutions to this numerical challenge will work well if the mesh can be made arbitrarily fine. The “problem area” then becomes more and more localized, close to the corner.

References 1. Kay J. M. and Nedderman R. M., Fluid Mechanics and Transfer Processes, Cambridge University Press, London, New York (1985). 2. O’Connor W. J., TLM coefficients with distributed losses, Electronics Lett. 35 (1999) 460–462. 3. Duffy D. J., Solutions of Partial Differential Equations, TAB Books Inc., Blue Ridge Summit, Pennsylvania (1986) 292–298.

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chapter ten

State of the Art Examples 10.1 Introduction This closing chapter attempts to demonstrate the present state of the art of TLM modeling of problems that are relevant to the field of computational mechanics and draws on the work of a wider range of authors. It includes examples that were considered to be too advanced for inclusion in earlier chapters.

10.2 The Hanging Cable and Gantry Crane Problems Section 2.11 considered the physics—and the corresponding TLM model—of a rotating string, in which the rotation caused the tension to vary along the string's length. A heavy hanging cable also has a varying tension along its length, now due to the cable's own weight. In addition, gravity plays a second role. To the tension effects it adds a vertical force, acting on each deflected section of the cable, which also modifies the way waves propagate up and down the hanging cable. Regarding boundary conditions, as with the rotating string, at one end (the top) there is simply a fixed point. At the lower end there are at least three boundary conditions of practical interest: a free end, a fixed end, or an inertial, hanging load free to vibrate (or “swing”) laterally. In the third case, the cable vibrations “tug” at the load mass, causing it to accelerate. The moving load, in turn, then drives the cable motion. If of interest, this coupling of the two dynamic systems must also be modeled in TLM.

10.2.1 Hanging Cable: Analytical Analysis and Results The differential equation governing small amplitude vibration of a vertical flexible cable under gravity is

∂2 u ∂2 u ∂u = g ( x + ) ∂t2 ∂x 2 ∂t

(10.1)

213

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where u is the lateral, horizontal displacement of the cable, and x is now the distance from the lower end. By assuming synchronous motion and therefore a solution separable into time and space components, u = U(x)cosωt, the space part U(x), for the free end case, can be shown to obey a Bessel’s differential equation:

d 2U ( z) 1 dU ( z) + + U ( z) = 0 z dz dz2

(10.2)

where z = 2ω√ (x/g). The solution for one particular frequency (mode of vibration) is U(z) = J0(z) or U(x) = J0(2ω√ (x/g)). Again, the general solution can be made up as an infinite sum of such terms for all frequencies and modes.

10.2.2 Hanging Cable: TLM Model For the basic TLM variable in this case, there are significant advantages in choosing, not the cable lateral displacement, u, but the variable -T(∂u/∂x) corresponding to the normal force component of the tension. The negative sign is because a negative gradient in a wave traveling in the positive x-direction produces a positive force. This formulation greatly facilitates the force perturbation approach described below. The lateral velocity, ∂u/∂ t, at any point in the cable is then given by (f – g)/Z, where Z =√ (ρT), is the mechanical impedance of the cable, with ρ the mass per unit length. The instantaneous position of any point on the cable can be obtained by time integration of the velocity, or by spatial integration of the slope. The extra restoring force at any point on the cable is -ρ.∆l.g.∂u/∂x, which is the normal component of the gravity force shown in Figure 10.1. This force constitutes a perturbation on the wave, the effects of which can be shown to cause two equal variations to the force waves that travel in opposite directions. So half of this gravity force should be added directly to each of the two counter-propagating TLM pulses at every point on the cable. The cable tension at the lower end is the load weight (if present) or zero (if free). The tension rises with distance up the cable, reaching the total weight of the cable + load at the upper, fixed end. As before, this is modeled by adding inductive stubs of varying impedance. The coupling of the moving boundary at a load mass and the cable is as follows. The lateral acceleration of the load is the total lateral tension force, -T∂u/∂ x, at the base of the cable, divided by the mass. This is integrated over time to get the instantaneous velocity, v, of the load. The reflection boundary condition for the lower end of the cable is modified by the addition of an amount vZ to the reflected pulse at each time step. The TLM results are found to agree with analytical results for simple cases.

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T + dT

u

u

θ = d u /dx T

∆l.ρ.g

x

Figure 10.1 The forces (left) on a string element of a hanging cable under gravity, and (right) the first mode of vibration of this system.

10.2.3 Gantry Crane: Results The system to be modeled was shown in Chapter 2 (Figure 2.5). The modeling of the variable tension due to the cable’s own weight, and the effects of gravity on the cable vibration were modeled as described above. From a TLM modeling point of view, a third new issue now arises. The system is not just vibrating (or “swinging”) about a neutral axis, but the entire system can undergo a net movement over an arbitrary distance as the trolley travels. Two ways of dealing with this were investigated. At each time step, the new system configuration was obtained by temporal integration of the velocity of each cable element, (f – g)/Z, from the known initial position. Alternatively, the cable and load configuration can be obtained by spatial integration of the cable slope from the known trolley position. The two methods give mutually consistent results. Figure 10.2 illustrates an application of the model to test a novel control strategy in which the aim is to move the load from rest to a target position by combining position control and active swing control of the load mass.

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Position [m]

3

Velocity [m/s]

Exponential arrival at target

Load position

2.5

Trolley position 2

Target: 3 m Max vel: 1 m/s Launch-x

1.5

1 Trolley vel 0.5 Load vel

Exponential vel growth & decay

0

time 0

1

2

3

4

5

6

7

8

Figure 10.2 Simulation of a gantry crane maneuver, using a full TLM model.

10.3 The Modeling of Rigid Bodies Joined by Transmission Line Joints Krus1 has considered transmission line approaches to modeling systems such as the arms of a mechanical digger. One approach might be to have flexible bodies (modeled using transmission lines) connected by rigid joints. He suggests that although it may appear more realistic, the inherent disadvantage is that the parasitic inductances cannot be interpreted as masses, since they are not uniform in all directions. This would make compensation more difficult. Instead, in his 1999 paper he analyzes the alternative of two rigid bodies connected by a flexible joint (Figure 10.3a). In a somewhat similar approach to Equation (8.34), he starts by considering the equations governing a 1-D transmission line where ends 1 and 2 have pressures and flows, p1, q1 and p2, q2 respectively. The characteristic impedance is Zc and the time-interval separating events at the two ends is T.

p1 ( t) = Zc ⎡⎣ q1 ( t) + q2 ( t − T ) ⎤⎦ + p2 ( t − T ) (1)

(10.3)

p2 ( t) = Zc ⎡⎣ q2 ( t) + q1 ( t − T ) ⎤⎦ + p1 ( t − T ) (2)

(10.4)

Since the flows are defined as positive when entering the rod, the speeds when compressing the rod are positive when they will contribute to compression and this leads us to a definition of forces.

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∆xi

body 2

x1 ,y1

∆yi

θ2 θ1

body 1

x2 ,y2

Figure 10.3 (a) Rigid bodies connected by a two-dimensional flexible joint, (b) model parameters.

F1 ( t) = Zc ⎡⎣ x
1 ( t) + x
2 ( t − T ) ⎤⎦ + F2 ( t − T )

(10.5)

F2 ( t) = Zc ⎡⎣ x
2 ( t) + x
1 ( t − T ) ⎤⎦ + F1 ( t − T )

(10.6)

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The stiffness for a rod is defined as

K=

F dF = ∆x d( x1 + x2 )

(10.7)

He is then able to relate stiffness, transit time, and characteristic impedance as

Zc = K T

(10.8)

Newton’s second law together with the difference in the two forces can be used to include the effect of mass (inertia) or inductance in the transmission line analogue. This leads to a definition:

Zc =

m T

(10.9)

He then says “In a simulation where the transmission line equations are used to represent a flexibility, this mass represents an unwanted parasitic inductance, that is, it adds mass (without weight) to the joint which is not of physical origin. It is purely a function of the stiffness, K and the time-step, T.” For the 2-D joint, shown in Figure 10.3a, there are three variable parameters that must be considered, x, y, and θ. If the joint is to be independent of the coordinate system then Zx = KxT must equal Zy = KyT, and so we shall designate these impedances and stiffnesses as Zs and Ks. We end up with a set of six uncoupled equations of the form of Equations (10.5) and (10.6), one for each of the parameters at the start state and at the end state. This can be written in vector form for the two states, separated by time T as


+Θ
F1,t = ZΘ ⎡⎣ −Θ 1 ,t 2 ,t− T ⎤ ⎦ − F2 ,t−T

(10.10)


+Θ
F2 ,t = ZΘ ⎡⎣ −Θ 2 ,t 1 ,t− T ⎤ ⎦ − F1,t−T

(10.11)

Krus then introduces variables, c1 and c2, which he calls characteristics as they represent the waves traveling in each direction through the transmission line


c 1,t = ZΘ Θ 2 ,t− T − F2 ,t− T

(10.12)


c 2 ,t = ZΘ Θ 1 ,t− T − F1 ,t− T

(10.13)

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So that Equations (10.10) and (10.11) can be written as


F1,t = c 1,t − ZΘ Θ 1 ,t

(10.14)


F2 ,t = c 2 ,t − ZΘ Θ 2 ,t

(10.15)

He then defines a rigid body and develops expressions for mass, M and inertia, J in terms of accelerations in x, y, and θ, but based on prior definitions for impedances and characteristics, which yield



= Z Θ
+C MΘ

(10.16)

where

⎡M ⎢ M=⎢0 ⎢0 ⎣

⎛ x
0 ⎞ ⎛ x

0 ⎞ 0⎤ ⎥

⎜ ⎟
⎜ ⎟ 0 ⎥ Θ = ⎜ y

0 ⎟ Θ = ⎜ y
0 ⎟

⎟⎠ ⎜⎝ θ
0 ⎟⎠ ⎜⎝ θ J ⎥⎦ 0

0 M 0

and

⎡ Z e,xx ⎢ Z = ⎢ Z e,yx ⎢ ⎣ Z e,θx

Z e,xy Z e,yy Z e,θy

⎛ c e ,x ⎞ Z e,xθ ⎤ ⎥ ⎜ ⎟ Z e,yθ ⎥ and C = ⎜ c e ,y ⎟ ⎥ ⎜⎝ c ⎟⎠ Z e,θθ ⎦ e ,θ

are the effective impedance and characteristic matrices as defined by Equations (10.43 to 10.51) and (10.52 to 10.54) respectively in his paper.1 Since M is diagonal it can be easily inverted, which yields



= M −1 Z Θ
+ M −1C Θ

(10.17)

This can be written in state–space form as

y
= A y + u

(10.18)

with

⎛ M −1 Z A=⎜ ⎝ I

0⎞ ⎟ 0⎠

(10.19)

Using this as a basis then all parameters can be determined. Krus points out that the parasitic inductance (assuming an isoelastic joint) can be subtracted from the connected bodies, but that this can only be done to zero if

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instability is to be avoided. As in the cases of single node transmission line (TL) formulations for hydraulics systems, the original implementation of this scheme displayed an undamped resonance. This required the inclusion of damping terms at both ends, which was achieved by means of a modification of the characteristics and the addition of two further characteristic terms


2 ,t −T − F2 ,t −T c1’ ,t = ZΘ Θ

(10.20)


1,t −T − F1,t −T c2’ ,t = ZΘ Θ

(10.21)

c1,t = α c1,t −T + (1 − α )c1’ ,t

(0 < α < 0.5)

(10.22)

c2 ,t = α c2 ,t −T + (1 − α )c2’ ,t

(0 < α < 0.5)

(10.23)

Krus also found that there was an additional error due to the fact that the forces are calculated as the summations of speeds, while the spring force is calculated from an integration of speeds instead of directly from displacements. In order to avoid this he introduced a filtered force error vector, which acted as a feedback with a very low gain. The resulting formulation works well and compares favorably with the real situation of jointed pairs of rigid arms in systems such as mechanical diggers.

10.4 Klein–Gordon Equation Equation (10.24) shows a wave equation with an additional term, which is in proportion to the dependent variable 2 ∂2 u 2 ∂ u = c − hu ∂t2 ∂x 2

(10.24)

When this is applied to scalar mesons in quantum mechanics it is known as the Klein–Gordon equation. In classical mechanics it arises where u represents (for example) the lateral displacement of a vibrating string under tension, but with extra, elastic acceleration acting at every point. A typical case might be a flexible string under tension, which is imbedded in a thin elastic sheet held in a rigid frame. An important effect of the additional term is to make the wave problem dispersive.2,3 The challenge was to modify the conventional TLM wave-propagation algorithm to take account of the additional physical term. In the first publication4 of this work O’Connor described two methods where the TLM pulses represented displacement. However, in one case the elastic force was determined by spatial integration, while temporal integration was used for the other. In the second publication5

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the TLM pulses represented the normal force ±Tδu/δx. Once again, two formulations were presented. In the first of these the elastic force was determined by temporal integration of the velocity. In the second, the elastic force was obtained using a spatial integration of the slope. In both of these cases it was possible to add the components directly to the TLM pulses. The reason for this apparent complexity of schemes is that the naive approach does not work. The obvious thing might be to add at each time interval an extra (negative) displacement to the TLM displacement pulses corresponding to the variation in displacement due to the instantaneous value of the extra elastic force at each point along the string. So the displacement adjustment would be (1/2)f∆t2, where f is the acceleration (f = -ku∆x/ρ∆x, the elastic force divided by the mass of the node), shared equally between the left- and right-moving TLM pulses. Sadly, this does not model the physical system correctly. Component solutions of the wave equation obey the relationship

∂u ∂u = −c ∂t ∂x

(10.25)

This implies that by adjusting the spatial slope of the component waveforms we can ensure that on propagation at the wave-speed c they produce the correct velocity change with time. So, in the first formulation, TLM pulses represent displacement. Elastic force, obtained by spatial integration operates the “connect” process of TLM in the conventional way. The “summation” gives the local value of displacement, u(x). We then calculate the extra elastic acceleration, hu at each point, which in turn gives the amount by which the spatial derivative (gradient) must be changed at every point. The next step is to integrate this gradient with respect to distance along the string, adding half of this integral to the left-going TLM pulses and subtracting half of the same from the right-going TLM pulses. This can be started at any arbitrary point (e.g., at a boundary), and with a correspondingly arbitrary constant of integration (e.g., zero). The TLM pulses are scattered in the normal way and the cycle is repeated. The second method (TLM pulses represent displacement with elastic force obtained by temporal integration) involves modeling the elastic force on the string as if it were acting in parallel with, but almost independent of, the propagating wave motion. Over time, the elastic force causes an “extra” acceleration (and corresponding extra velocity and extra displacement) at each point along the string, all three of which can be “tracked” almost as if the displacement waves were not present. To achieve this independent tracking in TLM, a separate array is used to store and update the extra velocity (due to the elastic force alone). This is updated at every time-step, the local velocity change being calculated from the acceleration caused by the elastic force at each point, so that ∆v = [-ku∆x/ρ∆x]∆t. This extra velocity gives the local extra displacement (∆u = v∆t). This is divided into two halves, one

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being added to the right-going TLM pulse and the other to the left-going pulse. The attraction of this approach is that it is a perfect description of the individual terms in Equation (10.24). The lossless TLM wave equation, as used by electromagnetic modelers is usually given in terms of potential. However, it is equally valid, although less common to express it in terms of current:

∂2 i ∂2 i = Ld C d 2 2 ∂x ∂t

(10.26)

In this particular case the TLM pulses represent the normal force (±Tδu/ δx), which acts on the string. In the first of the two cases using this approach the elastic force is obtained from the temporal integration of the velocity ([right pulses + left pulses]/Z) and added directly to the pulses. An additional array is used and this stores the displacement, which is updated at every time-step by integrating the velocity. An example of Matlab-like code that might be used is shown below: f_left = f_left - 0.5*h*ro*delta_x*displacement; f_right = f_right - 0.5*h*ro*delta_x*displacement; displacement = displacement + (delta_t^2/(ro*delta_x))*(f_left + f_right); %(TLM pulses are then propagated to left and right as normal)

The TLM pulses, f_right and f_left are the normal tension component waves, with the total normal tension T δu/δx (-f_right + f_left). In the final method described here, the TLM pulses represent normal force, and the elastic force, determined by spatial integration of the slope, is added directly to the pulses. The integration of the slope is effectively the cumulative sum of [right pulses + left pulses]/T times the TLM pulses along the string, and this has the advantage over the previous method in that (strictly speaking) no extra array is required to be stored, and so there is some saving on memory, but this is achieved at the cost of more calculations per time-step. The key section in the Matlab-like code is shown below: f_left = f_left - 0.5*h*ro*delta_x*displacement; f_right = f_right - 0.5*h*ro*delta_x*displacement; displacement = u_o + (delta_t^2/(ro*delta_x))*cumsum(f_left - f_right); %(TLM pulses are then propagated to left and right as normal)

The “displacement” array stores the string displacement from

u = u x0 +

∫

x

x0

ux=0 is zero.

( ∂u / ∂x ) dx . Typically, the displacement at the boundary u_0 =

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All of the above methods were verified numerically and were found to be completely consistent with other numerical methods and with the appropriate analytical (Fourier) solution to within an arbitrarily small error as the time and space discretization was made finer.

10.5 Acoustic Propagation and Scattering (Two-Dimensions) At UEA Norwich, Dorian Hindmarsh* has been developing 2-D acoustic propagation models for moving vehicles where the computational space is surrounded by a perfectly matched load boundary. Figure 10.4 shows the propagation of the impulsive noise created by the main rotor tips of a helicopter. Kevin Murphy6 in Dublin has examined the scattering of sound by objects and his polar pressure diagrams are entirely consistent with theory. Mike Morton† at UEA has done a similar exercise for scattering from thin-wall metal cylinders, where the observed signatures can include the effects of internal ringing. Both of these workers had a need to define a plane-wave within the computational space. This is not a straight forward exercise, because if a line of nodes is excited with a sinusoidal source, then the results will be as shown in Figure 4.12. The reasons for this were discussed in Chapter 4. Morton divides up the “real” space into three parts as shown in Figure 10.5. The mid-part is the computational space where we see a train of plane waves arriving from the left. The upper and lower portions are replaced by single rows of 2-D TLM nodes. He calls these “infinity” rows as they reflect the behavior of the upper and lower portions of space as they extend to infinity. If we consider the upper interface then the three TLM steps of incident, scatter, and connect are shown in Figure 10.6. Pulses in the infinity row can be exchanged between nodes to the left and right. However, it is bounded top and bottom by open-circuit (ρ = 1) terminations. However, a copy of the pulse that is reflected at the bottom boundary during the connect step is communicated to the main mesh (shown as a dashed line in the figure). This represents the information that would come down to that top-most node in the computational space if it were part of a plane-wave in an infinite space. Meanwhile, any pulse that is transmitted north from the top-most node of the computational space is lost (ρ = 0) at the incident step. The treatment for the lower “infinity” node is (mutatis mutandis) identical. While this approach works for incident plane-waves, it creates problem for dealing with the nonplane waves that may be scattered from the target. Murphy6 bound all parts of his computational space with a PML absorber. His approach to this problem is summarized in Figure 10.7. The node in the infinity row, x0 has the property sVN(x0) = sVS(x0). We are only concerned with the signal that is scattered north at node (x,y) in the main computational *

Work sponsored by the Engineering arm of Lotus Cars. Unpublished work of relevance to U.K. ordnance and pipelines on the sea bed.

†

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Figure 10.4 Model of impulsive noise generated by the tips in a three-rotor helicopter.

Figure 10.5 Incident train of plane-waves in computational space bounded by “infinity” rows.

space. The first step is to remove from it any component due to the plane-wave, and so we write:

VN' ( x , y ) = skVN ( x , y ) − skVN ( x0 )

s k

(10.27)

This modified pulse is then allowed to propagate into the PML region and any attenuated contribution that it makes will not be seen for two

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infinity row

computational space

incident

scatter

connect

Figure 10.6 The three-TLM steps at a node in the “infinity” row.

PML absorber space infinity row computational space

(x0 )

(x,y)

Figure 10.7 Computational space signal modified by Equation (10.27) entering the absorbing region above node (x,y) with the contemporaneous process in the adjacent “infinity” node.

time-steps. Meanwhile at the incident step, we then have the contribution from node (x,y + 1) inside the absorbing region, and the unmodified contribution from the infinity row is now returned:

V ( x , y ) = skVS ( x , y + 1) + skVN ( x0 )

i k +1 N

(10.28)

10.6 Condenser Microphone Model O’Connor7 developed a TLM model where the amplitude of the driver was considered to be small compared with the internodal spacing. In this case, the boundary condition is the common acoustic and driver velocity on the

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face of the driver. The tube resonances will dominate if the driver mass is small, whereas the driver resonances will dominate if the inertia is large compared with the acoustic mass of the tube. In either case, each subsystem is found to modify the resonant frequencies of the other and the TLM results were in excellent agreement with the analytical results given by Kinsler et al.7 The next stage involved the development of a model for the circular membrane in vacuum, and this was tested for free-vibration (initial displacement and/or velocity) and forced vibration (point driving force and uniformly distributed driving force). This was then coupled with the acoustic TLM model. The local membrane velocity (v = dz/dt) caused the injection of acoustic pressure pulses of magnitude ρcv, where ρ is the density of the medium and c is the wave velocity. At the same time, the local acoustic pressure became a force input to the membrane TLM model. These concepts were applied to a simple condenser microphone model. An incoming plane-wave was assumed to approach the microphone normally and impinge on one side of the membrane, behind which was a perforated electrode and then an acoustic chamber. The attenuation due to viscous damping of the air as it passed through the holes was modeled by a viscous damping term added to the membrane TLM system. The original model,8 which nevertheless gave good qualitative agreement with the analytical results due to Morse,9 had to be done in reduced dimensions. This is because of the fact that while the membrane and air TLM models are each 2-D, they are not in the same plane. In order to fully model the interaction between them one would need a 3-D model on the air side, which interacts with a 2-D membrane model. In this case a 1-D membrane model (mass-spring-damper) was used to drive a 2-D airmass through a viscous damper.

10.7 Propagation in Polar Meshes Before the development of surface-conforming boundaries the description of a circular computational space in Cartesian coordinates could introduce spurious effects because of the step-wise approximation. This problem could be avoided by the use of polar meshes. A conventional scattering algorithm involving the interfacing of transmission lines of different impedances can be used, but it is cumbersome and the use of stubs has been found to be easier. The application of this method is well established in heat-flow10 and in electromagnetics.11 In both cases the question of the central node must be addressed as it appears as a circle of infinitesimal circumference. In heat-flow, it is treated as a resistance of zero cross-sectional area (an infinite resistance), and the approach of treating the center as an open-circuit termination has been found to give excellent results. Electromagnetic modelers have a problem because the center is either an open-circuit (ρ = 1) or a short-circuit (ρ = 1), depending on which mode is being considered. This means that the field at the center of the mesh (and some modes) cannot be computed. This problem has been addressed by Cacoveanu et al.12 The circular geometry is

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divided into 2n segments. There is also a discretization along the radii, so that each node, except the center is connected to neighbors via four transmission lines. The center node has 2n transmission line branches and an open-circuit stub. The former ensure continuity, while the latter provides synchronism. If the impedance of each of the lines connected to the center node is Z, and if the impedance of the stub is Zs, then a pulse traveling along a line will see a discontinuity, so that the reflection coefficient is

ρ=

2 Zs ( n − 1) + Z 2 Zs n + Z

(10.29)

and the transmission coefficient is

τ=

2 Zs 2 Zs n + Z

(10.30)

10.8 Acoustic Propagation in Complex Ducts (A 3-D TLM Model) As in the previous example, Pierre Saguet at Grenoble has led an investigation into the effects of higher order modes in the vocal tract and other complex ducts.13 One of their major contentions is that higher order modes play a significant role and help to account for experimental observations that cannot be explained by 1-D propagation models. In the introduction to their paper they acknowledge the pioneering work of Loaseby14 (1-D vocal tract model using TLM) and Kagawa15 (room acoustics and scattering tomography). Although the technique of TLM modeling in three dimensions by having six inductances (in ±x, ±y, ±z directions) with a capacitor shunted to ground is well known as the “scalar TLM” model, this is the first published investigation of its application in acoustics. Computational* loading has been one of the reasons why many other modelers have shied away. In their paper they describe a parallelepipedic mesh, where the dimensions in the coordinate directions are different. They also describe a form of open-boundary condition based on the arrival of a backward wave coming from a virtual node outside the mesh and computed as a linear combination of the forward and backward waves on the last three nodes (see Figure 10.8). They cite other work by Saguet,16 which suggests that excellent results can be obtained using the Taylor series expansion r

Vy ( N y , t) = 2.5 r Vy ( N y − 1, t − ∆t) − 2 r Vy ( N y − 2 , t − 2 ∆t) + 0.5 r Vy ( N y − 3 , t − 3 ∆t)

(10.31)

If ∆x = 1 cm is used to model the acoustic behavior at 2.5 kHz inside a cube of side 1 m, then one million nodes must be calculated for every time-step and 2500 time-steps are needed for one second of “real” time.

*

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Ny-4

Ny-3

Ny-2

Ny-1

Ny

t-4∆t

t-3∆t

t-2∆t

t-∆t

t

Figure 10.8 The nodes that appear in the Saguet infinite-space boundary condition.

where rVy(Ny,t) is the sound pressure reflected at node Ny at time, t on one transmission line along the y-axis. In order to provide a metric for their analysis they define a transfer function T(f), so that if a duct is considered where vni(f) and vno(f) are the normal components of particle velocity (at frequency, f ) at the input and output respectively (measured along the y direction), then we define

T( f ) =

∑v ∑v

no

(f )

ni

(f )

(10.32)

where the summations are made in the x – z directions (cross-sections) at the input and output. The TLM model for a duct is then driven at a fixed frequency at the input, and the transfer function is then monitored. The propagation of sound waves along the y-axis in a rectangular duct of transverse dimensions Lx, Lz, has multiple solutions (vibrational modes) and a cut-on (as opposed to cut-off in electromagnetics) frequency can be defined as 2

f m ,n =

⎛ n⎞ c ⎛ m⎞ +⎜ ⎟ ⎜ ⎟ 2 ⎝ Lx ⎠ ⎝ L2 ⎠

2

(10.33)

Above fm,n any mode (m,n) will propagate. Otherwise it is evanescent. These workers have found that the radiation from a flanged duct (e.g., the human mouth) is strongly dependent on the mode numbers (m,n). They have investigated this by conducting a series of experiments on the influence of the location of the excitation in the x – z plane on the transfer function. If a monopolar excitation is used at x = Lx/2, z = Lz/2, y = 0, then the first transverse mode is not excited. If the excitation is located in a corner of the

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duct, then the effect on this mode will be maximum. A simple subtraction of two sets of results has been used to isolate the transverse mode and to measure its radiation characteristics in terms of reflection coefficient. The same techniques were then used to measure the resonant modes in a rectangular duct with a 90° sharp bend. The rigid walls represented open-circuit (ρ = 1) boundaries, while the ends were terminated in pressure zero (ρ = –1) boundaries. For the physical dimensions that were chosen, the TLM model yielded peaks that could be identified as longitudinal modes ω1(1,0,0) = 1300 Hz, ω2(2,0,0) = 3000 Hz, ω3(3,0,0) = 4150 Hz, and coupling (combination of longitudinal and transverse) modes ω4(2,1,0) = 4600 Hz, ω5(3,1,0) = 4900 Hz. An equivalent TLM model of a rectangular duct with a 90° bifurcation (T-section) yields a resonance that does not correspond to any 2-D equivalent straight duct mode, which confirms the importance of using 3-D models, where appropriate. As a concluding demonstration in their paper13 they construct a 3-D model of the vocal tract and compute the transfer function of a spoken vowel "a". TLM and plane-wave theory agree with the theoretical modal analysis at low frequencies. Above about 3.8 kHz, plane-wave theory no longer seems applicable, while small quantitative discrepancies between the other two techniques are attributed to the limited number of modes used in the modal analysis. Above all, a zero at about 5 kHz, which is observed in experimental measurements on real subjects, is predicted. It does not appear in plane-wave theory.

10.9 A 3-D Symmetrical Condensed TLM Node for Acoustic Propagation While the “scalar” TLM model based on a 3-D shunt node is obviously very successful in acoustics, it is rarely used in electromagnetics. This is because, in this area, modelers are interested in fields as vector quantities, rather as in computational fluid-dynamics. For reasons that were explained in Chapter 3, the conventional shunt node involves three field parameters, two magnetic and one electric, Hx, Hy, and Ez. The other field parameters, Ex, Ey, and Hz can be obtained using a 2-D “series” TLM mesh as shown in Figure 10.9. The treatment of these nodes differs from the shunt node only in the scattering equation. This is identical to Equation (3.32) except that the scattering matrix is given by

⎛ 1 ⎜ 1 1 S= ⎜ 2⎜ 1 ⎜ ⎝ −1

1 1

1 −1

−1 1

1 1

−1⎞ 1⎟⎟ 1⎟ ⎟ 1⎠

(10.34)

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i 2 V3 +

Z + V3

V2

i

i

2 V2

2 V4

I

V

I

+

4

Z

Z

V

1

i

2 V1

Z (a)

+

(b)

Figure 10.9 A TLM series node (a) and its lumped equivalent circuit (b).

The first attempts to combine these two configurations in TLM resulted in what was called the “expanded node network” and this is shown in Figure 10.10. Porti and Morente17 at the University of Granada in Spain have been the first to develop a symmetrical condensed node (SCN) for use in 3-D acoustic propagation models. They start with the acoustic equations in vector form:

∇.u = −σ

∂p ∂t

(10.35a)

and

∇p = − ρ

∂u ∂t

(10.35b)

Their nomenclature for parallel and series nodes is shown in Figure 10.11. The field information contained in Equation (10.35a) for an element of the medium is expressed by the parallel node (Figure 10.11a), while the series node in Figure 10.11b expresses the field information concerning the x-component of Equation (10.35b). There are equivalent series networks involving (V5, V6, and V9) for the y-component and (V3, V4, and V10) for the z-component. In formulating the scattering matrix we must recognize that line 1 contributes a pressure V1 and a particle velocity I1∆x propagating along the x-direction. Its contribution is Y0∆t/2 to the compressibility factor and Z0∆t/(2∆x2) to the density, where ∆t is the time-step and the admittance, Y0 is 1/Z0. Line 7 of characteristic admittance YY0 only defines a pressure

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Hz Hx Ey

Hy D F Ex Ez

Figure 10.10 An expanded node network that can be used in 3-D TLM models, but it should be noted that each of the six field parameters is defined at a different apex of the spatial cube. y

x

V4 V5 V1

z V2 V7

V8 V1

V2

V6 V3 (a)

(b)

Figure 10.11 The parallel network (a) and the series network (b) for the x-component in the Porti and Morente SCN.

V7 and adds compressibility YY0∆t/2. The SCN has three short-circuit stubs (the x-, y-, and z-components of Figure 10.11b) which have characteristic impedances ZxZ0, ZyZ0, ZzZ0. These only define particle velocity, adding density to the acoustic medium. The scattering processes are then described in a 10 × 10 matrix as:

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1

2

3

4

5

6

7

8

9

10

1

ax

bx

c

c

c

c

f

-hx

2

bx

ax

c

c

c

c

f

hx

3

c

c

ay

by

c

c

f

-hy

4

c

c

by

ay

c

c

f

hy

5

c

c

c

c

az

bz

f

-hz

6

c

c

c

c

bz

az

f

hz

7

d

d

d

d

d

d

g

8

-ex

ex

jx

9

-ey

ey

jy

10

-ez

ez

jz

(10.36) where the various parameters are given by

a=

1 ⎡Z − 2⎤ 1 ⎡Y + 2⎤ − 2 ⎢⎣ Z + 2 ⎥⎦ 2 ⎢⎣ Y + 6 ⎥⎦

c=d=

g=

2 Y+6

Y −6 Y+6

e=

h=

e Z

2Z Z+2

j=

b= −

1 ⎡Z − 2⎤ 1 ⎡Y + 2⎤ − 2 ⎢⎣ Z + 2 ⎥⎦ 2 ⎢⎣ Y + 6 ⎥⎦

f = cY

2−Z 2+Z

In their paper Porti and Morente17 discuss methods of medium definition, excitation, dispersion, and the determination of maximum time-step. They describe their own form of absorbing boundary condition, which is interesting in that it attempts to replace Equations (10.35a and 10.35b) by equivalent expressions with stretched coordinates, something very similar to what is being done in the PML that was described in Chapter 4. The model has been tested on a cavity with dimensions 20 cm × 30 cm × 40 cm using nodes with dimensions 2 cm × 3 cm × 4 cm, so that there were 10 nodes in each direction.. Using c = 331 m/s and ∆t = 3.48852 × 10–5s, the resonant modes lying between 413.8 Hz and 994.5 Hz were found to agree with theory to less than 0.5%. The model was also tested successfully on a Helmholtz resonator. The application to acoustic filters confirmed results that could not have been predicted using plane-wave theory.

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10.10 Waves in Moving Media In Chapter 2 there was mention of the “moving threadline” problem and the use of a TLM network incorporating diodes. This is concerned with wave propagation in a system where the medium is moving and is described by the equation

∂2 φ ∂2 φ ∂2 φ + 2V + (V 2 − c 02 ) 2 = 0 2 ∂x ∂t ∂t ∂x

(10.37)

where φ is the wave variable, V is the medium speed, and co is the corresponding wave speed in a stationary medium (when V = 0). The approach was presented as an example of the extension of TLM, which was based more on physical intuition than on computational considerations. Effectively the wave propagation characteristics are biased by the speed of the medium. To derive the scattering algorithm for this network we should consider the pulses incident at a node just before scattering. This is shown in Figure 10.13 along with the situation immediately after scattering. Note that in Figure 10.13a, of the four “parallel” lines a and b only two (those shown with diodes in Figure 10.12) have incident pulses at this node because of the diode actions at the adjacent nodes. Similarly, the presence of the diodes at the node will ensure that there are no signals scattered from the node into two of the four parallel lines. In order to develop an algorithm we need to determine the values of the scattered pulses sVr, sVl (in the lines without diodes with admittance, m) as well as sVa and sVb in terms of the four incident pulses, iVr, iVl, iVa, and iVb. b

b m

a

b m

a

m

a

Figure 10.12 A network with flow-controlling diodes. The letters m, a, and b refer to the relative admittances of the transmission line arms.

Four independent equations are required. These can be obtained by applying Kirchoff’s laws during scattering. At the instant of scattering the node voltage φ is common to the four connected lines, the remaining two lines being isolated by the diodes. This common voltage must equal the incident plus reflected pulses in the main lines m and also equal the scattered pulses in the other two lines (which have no incident pulses). Thus φ = iVl + sVl = iVr + sVr = sVa = sVb

(10.38)

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b

m

b

b

iV b

iV l i

b

iV r

s m

m

sV

Vb

s

r

m

sV a

V

l

V

a

a

a

a

(a)

a (b)

Figure 10.13 The connections at a node (a) before and (b) after scattering.

The fourth equation can be obtained from the conservation of current during scattering. Assume that the admittances in the respective lines are m/Z0, a/Z0, and b/Z0, where Z0 is the reference impedance. Then we have m(iVr + iVl – sVr – sVl) + a(iVa – sVa) + b(iVb – sVb) = 0

(10.39)

Rearranging Equation (10.39) and the components of Equation (10.38) we get i ⎡ i Va + iVb ⎤ i ⎢ m Vl + m Vr + ⎥ 2 ⎦ φ=2⎣ 2m + a + b

(10.40)

and

Vr = φ − iVr

(10.41)

Vl = φ − iVl

(10.42)

Va = sVa = φ

(10.43)

s

s

s

Wave speeds in TLM are normally derived either from circuit theory19 or from the evaluation of eigenvalues of a matrix representation of the problem using Floquet’s theorem.20,21 In this instance a new time-domain technique was developed22 and yields all necessary parameters. The two wave speeds are

c l ,r =

( a − b) ± ( a − b)2 − 4 m( m + a + b) ∆x 2( m + a + b) ∆t

(10.44)

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The medium speed is

V=

a−b ∆x 2( m + a + b) ∆t

(10.45)

and the two mesh (or wave) impedances for waves to left and right are

Zl , r =

Z0 [ m + a + b][ m + ( a − b) c l ,r ( ∆t / ∆x )

(10.46)

The magnitude of the two velocities can be expressed as

c l ,r = c 0 ± V

(10.47)

where c0 is the speed when the medium is stationary, i.e., when a = b. This scheme has been compared with the “moving threadline” problem as described by Swope and Ames.23 This arises in the transverse vibration of a moving string under tension, e.g., a cotton thread winding onto a bobbin. Swope and Ames used D’Alembert’s method to develop an analytical solution for the problem. The TLM approach gave excellent qualitative agreement with their results (see front cover of this book). The quantitative differences were small for the right-moving wave, but some dispersion/diffusion of the pulse was observed in the wave moving against the flow. The pulse peak is reduced in sharpness at the top, and the lower “skirt” is slightly wider by a corresponding amount. This is attributed to the standard TLM dispersion effect, whereby the mesh acts as a low-pass filter and is more evident in the upstream where the pulse has higher spatial frequency components.

10.11 Some Recent Developments in TLM Modeling of Doppler Effect Just before this manuscript was presented to the publishers in its final form, James Flint of Loughborough University, on hearing of the difficulties encountered with Doppler modeling, suggested an approach that appears to overcome the problems, outlined in Chapter 4 [private communication]. He starts by considering a 1-D mesh of nodes, where the source, designated S in Figure 10.14a is stationary, while the background medium moves at ∆x ms–1 in the –x direction. At the (k + 1)th time-step the locations of the initial pressures must be as shown in Figure 10.14b. Simply calculating the pressure at node (n + 1) prior to TLM scattering and injecting this value into node (n) can be used to model movement in the medium. If a source, S is present within the mesh, then the situation is equivalent to the source moving in the +x direction. However, since the mesh

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(a)

P1

P2

P3

Pn

x

∆x Reference point S P1

(b)

P2

P3

P4

Pn+1

Figure 10.14 (a) Static mesh TLM pressures at each node at time-step k, (b) Nodal pressures for dynamic situation with propagation speed ∆x ms–1 at time-step (k + 1).

is effectively following the source, the movement contributes no additional dispersion. In particular, there is not need to take account of the upper Doppler frequency when discretizing the problem to ensure that ∆x/λmin is less than 0.1. In situations where the velocity of the source is less than the acoustic propagation velocity on the mesh (i.e., v = ∆x/∆t < 1) a remapping is required and this is shown in Figure 10.15. It should be noted that this conserves the energy of the pressure field. Reference point

P0

P1

P2

Pn -1

(1 −v ).P1

(1−v ).P2

(1−v ).P3

(1−v ).Pn

+v.Po

+v.P1

+v.P2

+v.Pn −1

Pn

v (1−v ).Po

Figure 10.15 A slow-moving mesh, showing the nodal pressure resampling that is required before scattering is carried out.

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Figure 10.16 Wave-front comparison with the analytical solution for a receding wave at v = 0.4 and v = 0.8 respectively. The black lines are analytical results for lines of equal phase.

These concepts were extended to sound propagation in two-dimensions with source movement in the x-direction only. Some results are shown in Figure 10.16, which agree with theory, although there is some instability at high Mach numbers, which is the subject of continuing study.

10.12 Simulation of a Thermal Environment for Chilled Foods during Transport: An Example of Three-Dimensional Thermal Diffusion with Phase-Change In recent years several “gel-type” refrigerants have appeared on the market. These take the form of pouches or cells of refrigerant, an array of which make up a full refrigerant sheet. The current market for such refrigerants is currently chiefly domestic, e.g,. use in conjunction with “coolboxes” for picnics. In this context, the use of gel-type refrigerants could offer economic and environmental advantages over the traditional approaches involving, for example, dry ice or refrigerated vehicles. However, if they are to be used to maintain a suitable thermal environment for temperature-sensitive foods during transport, then several questions arise. These include how much refrigerant is necessary and at what temperature, what other components are needed to make up the package, e.g., box type and packing material, and how these components should be arranged for successful delivery. In their paper Stubbs et al.24 describe the use of numerical simulation, based on TLM, to predict the thermal experience of foods in such a situation.

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Using TLM, it is possible to model the thawing process25,26. In the current case, heat corresponding to the latent heat necessary to sustain melting is removed from each spatial element whose temperature is above 0oC until all the latent heat associated with the mass of the element has been removed. Following this, the thermal diffusion process continues within the element. Figure 10.17 illustrates the usual representation of the change of state model. At the point of latent heat, the nodal potential is clamped at the latent heat temperature, which in this case is 0oC. The TLM pulse incident upon any nodes representing ice, Vi, is modified to equal the reflected pulse so that Vimod = Vr. The amount of heat taken from the node by this process, given by (Vi – Vr) / Z, is then added to a latent heat store. This process repeats until the latent heat store is full, upon which the normal TLM algorithm is resumed. This model of latent heat provides a good representation of latent heat provided we are not interested on how it interacts with surrounding materials. In a case where we are interested in the interaction between one material and another, such as between ice and food in our case, we must use a different latent heat model. Figure 10.18 shows a modified latent heat model that can be used where a proper description of the thermal interaction within the latent heat period is required. In this approach, when a material reaches latent heat its temperature is clamped at that temperature, as in the previous model. However, instead of modifying the pulses incident upon the node, the nodal potential, Vnode, is calculated as normal. At each time step the excess heat, I, that flows into the node, is added to a latent heat store, which is given by

I = (Vnode − Vclamp ) Y where Y is the nodal admittance (reciprocal of impedance).

Figure 10.17 Modified latent heat model.

(10.48)

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Figure 10.18 Representation of a change of state (the exchange of latent heat).

The node remains clamped and the process repeats until the latent heat store is full. Because no incident pulses are modified, the correct amount of heat still flows between different materials. The model described above was used to represent the thawing of a gel refrigerant, and good agreement has been observed between predicted and measured temperatures. Based on this validation it was possible to model refrigerant thawing in a complex, 3-dimensional, inhomogeneous problem of predicting temperatures in foodstuffs packed with the refrigerant in expanded polystyrene (EPS) boxes. In doing this, the stability of the TLM algorithm has been exploited. The results of a large number of simulations show that such simulation is useful insofar as some wrapping strategies maintain food at suitably low temperatures for a realistic delivery period, whereas others do not. They highlight the fact that EPS box size is important, as is ambient temperature; the exact nature of the foodstuff also has an influence but, for most foods, this is less important than other factors. Overall, simulations performed to date suggest that the technique might enable valuable information to be provided for food suppliers.

10.12.1 Recent Advances in Inverse Thermal Modeling using TLM Early work in this area was undertaken by de Cogan et al.27,28 and Badger et al.29. Recent developments by Koay et al.30 have highlighted the importance of the amplification factor and the differences in stability between the parabolic and hyperbolic representations of heat flow when used in inverse thermal modeling

10.12.2 Inverse scattering In the forward TLM equation,

V i = SV r

(10.49)

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However, in the inverse TLM scatter, the inverse TLM equation is now

V r = S −1V i

(10.50)

Equation (10.50) means that, given a full knowledge about a current mesh, it is possible to effectively move in timesteps of –∆t and return to a previous mesh state. This is conditional upon the ability of the scattering matrix, S, to invert. Figure 10.19 illustrates the process of inverse scattering at a node in a mesh.The potential at a lossy node such as that in Figure 10.19 is

⎛ φ = ⎜2 ⎜⎝

4

⎞

∑ R V+ Z ⎟⎟⎠ Y1 i

l

l=1

l

(10.51)

l

where the admittance is given by Y = 4 / Rl + Zl Rewriting Equation (10.51) to make the vector of incident pulses the subject yields

⎛ Zl ⎞ Rl + Zl V i = ⎜V r − φ Rl + Zl ⎟⎠ Rl − Zl ⎝

(10.52)

The results obtained from the forward and inverse scatter simulations are almost similar if the resultant of the term R – Z is large. As the value of R – Z creeps closer to 0, the inverse scatter simulation becomes more unpredictable and varies from the referenced forward scatter simulation. This is due to the fact that the invertibility of the matrix, S, is dependent on the values of R and Z. The inversion is only possible when R ≠ Z as the term R + Z R − Z would be impossible to be determined if the values of R and Z were equal.

(

)(

)

Figure 10.19 Inverse scatter process at a node which is presumed to be lossy (resistors and transmission lines have been omitted for clarity.

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10.12.3 Amplification Factor In TLM forward scatter, the initial input signal is scattered to the rest of the mesh. This will cause a signal attenuation of

1

(−R + Z )

k

(10.53)

with k being the number of iterations in the scattering process. Similarly, in the inverse scattering process, the scattered signals are accumulated to form the initial input signal, resulting in a signal amplification that can be derived from Equation (10.53). Deriving the inverse scattering equation from the forward scattering equation in terms of incident pulses gives

(10.54) Rewriting Equation (10.54) to relate incident pulses in space and time to incident pulses at the same place at a previous iteration in the inverse scattering now gives

(10.55) From the determinant of the inverse matrix in Equation (10.54) multiplied by the common factor of the same matrix in Equation (10.55), the overall amplification factor of the inverse scatter for this is

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( (

)( )(

) )

⎡ Z + 4Z + R R + Z 3 ⎤ s ⎢ ⎥ ⎢ Z + 4Z − R − R + Z 3 ⎥ s ⎣ ⎦

k

(10.56)

with k being the number of iterations in the scattering process. Analyzing Equation (10.56), the determinant of the amplification factor will yield two singularities. The first term, Z + 4 Zs - R will produce different points of singularity in the graph depending on the percentage of stubs inserted in the transmission line. The second term would create another singularity at the point where R = Z. Figure 10.20 illustrates the resultant of varying (R-Z) term on the x-axis, the amplification factor (from Equation (10.56)) on the y-axis and the percentage of capacitance allocated to the link lines on the z-axis. The conclusions drawn from the graphs are • When the percentage of stub in the capacitance is increased, the discontinuity at the R – Z = 0 range grows wider. • The singularity from the (Z + 4Zs – R) term moves closer to R – Z = 0 when more stub is allocated to the capacitance. • The term (R – Z) generates a static singularity when R = Z. • Koay et al.30 discuss the significance of these singularities. The analysis of inverse modeling of heat flow using the telegrapher’s equation was based on State-Space Equation analogues (described in Chapter 6) and the expression for the amplification factor was derived symbolically using Maple. The surface plot of amplification factor as a function of capacitance loaded into the associated stub is shown in Figure 10.21 and it is quite clear that, unlike the parabolic case, there is only one singularity. Hence, this approach is more predictable, stable and flexible because parameters can be varied according to the desired end results.

10.12.4 TLM and Spatio-Temporal Patterns — The Present and the Future Ever since the pioneering work of D’Arcy Thomson31 spatio-temporal processes and particularly the regular patterns that arise as a result of their manifestation have special significance in biology. In a remarkable edited collection of papers32 there is much debate on Turing theory vs. mechanochemical theory. We will not enter the debate here except to note the mechanical aspects of any biological system. Pattern formation/growth generally involves chemical reaction plus diffusion processes, which of themselves are amenable to TLM treatments. A very simplistic form of TLM was used many years ago to model the formation of periodic precipitations (Liesegang rings)33. Dew and Gui34 have developed schemes for grain growth

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Figure 10.20 Plot of Equation (10. 56) showing the regions of instability as a function of the percentage of nodal capacitance contained in a stub.

Figure 10.21 Identical plot as in Figure 10.21, but based on the Telegrapher’s equation.

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in solids, and their methods for node insertion and deletion are of particular importance in the present context. Much of the present work on spatio-temporal pattern formation has been based on morphological approaches and (of more interest to us in the present context) cellular automaton (CA) modeling. A significant volume of work has been done on providing theoretical foundations to the CA modeling of physical systems35 and there are those who believe that its application is ubiquitous36. It should be obvious that TLM is a CA modeling system, but there is a difference. Cellular automata generally utilize a set of transition rules simply because they give interesting results (as in Conway's Game-of-Life37) or credible results (as in Forest-Fire CAs38). By contrast, TLM traces its transition rules to the laws of electromagnetics — if we accept the one, then the other follows. Thus, a large part of this book has been concerned with reinterpreting problems of computational mechanics so that the transition rules in our ‘CA’ (aka TLM) can be interpreted in an unambiguous way*. TLM has been applied to general problems in chemical kinetics39 and to the solution of a range of differential equations which describe limit-cycle systems40. Examples of these can be found in the literature41 and we will reproduce one here, the Brusselator model for an oscillating chemical reaction system42 whose governing differential equations

d[ X ] = [A] + [X[2[Y] – [B] [X] – [X] dt

(10.56)

d[Y ] = [B] [X] – [X]2[Y] dt describe the set of reactions A⎯⎯⎯⎯⎯⎯⎯⎯⎯→X 2X + Y⎯⎯⎯⎯⎯⎯⎯⎯⎯→3X B + X⎯⎯⎯⎯⎯⎯⎯⎯⎯→Y + D X⎯⎯⎯⎯⎯⎯⎯⎯⎯→E equivalent to A + B ⎯⎯⎯⎯⎯⎯⎯⎯⎯→ E + D From Glansdorff and Prigogine43 we know that the system is in equilibrium when [X]0 = [A] and [Y]0 = [B]/[ A] and that instability occurs if [B] > 1 – [A]2. Beyond instability the steady state enters a limit cycle. *

G.D. Smith at the University of East Anglia describes TLM as “a meta-control system for CAs”

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TLM simulations predict the correct limit-cycle behavior, and time-plots of [X(t)], [Y(t)], the concentration of the intermediates in this regime and the results are shown in Figure 10.22. Impressive though these results are, they are nevertheless derived from a treatment that applies only to homogeneous, well-stirred systems. Accordingly, until recently TLM was not able to treat phenomena of the type that might deliver spatio-temporal patterns. In this section we will outline two developments that should indicate the way forward. 4

3 [X] 2 (a)

1

10

20

30

40

50

40

50

Time (sec) 5

4

3 [Y] (b)

2

1

10

20

30 Time (sec)

Figure 10.22 The TLM simulated concentration of intermediate species in the oscillating chemical reaction (Equation 10.56) using initial concentrations [A]0 = 1, [B]0 = 3 with ∆t = 0. 001 sec. Plot (a) is intermediate X and plot (b) is intermediate Y.

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10.12.5 TLM and Diffusion Waves We can avoid the restrictions of a well-mixed system by returning to the concept that was first put forward by de Cogan and Henini44 to model first order kinetics in a morphological way: the amount of species that arrives at a location is the initial amount that was scattered minus the amount that decayed during the transit time, ∆t. It has been found by experience that it is probably best to use a link-resistor formulation because we can separate the scattering from the sampling. We recall that the incident pulses from left and right at (x) at time k + 1 are given as

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

i k +1 R

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

i k +1 L

(10.57)

Let us start with first order decay with a rate constant, κ. If we express the concentration of the species in TLM terms, then it is possible to write

∆φ = −κ φ ∆t

(10.58)

If this represents the total amount of species that is removed during the iteration time, then we can assume that half of this is due to right-moving pulses and half is due to left-moving pulses. When this is inserted into Equation (10.57) we get

⎡ ⎡ κ φ( x ) ∆t ⎤ κ φ( x + 1) ∆t ⎤ V ( x ) = ρ ⎢ skVR ( x ) − k + τ ⎢ skVL ( x + 1) − k ⎥ ⎥ 2 2 ⎣ ⎦ ⎣ ⎦

i k +1 R

(10.57)

⎡ ⎡ κ φ( x ) ∆t ⎤ κ φ( x − 1) ∆t ⎤ + ρ ⎢ skVL ( x ) − k V ( x ) = τ ⎢ skVR ( x − 1) − k ⎥ ⎥ 2 2 ⎣ ⎦ ⎣ ⎦

i k +1 L

A TLM scheme based on the above incident pulses was run for a variety of rate constants and time discretizations. Results were plotted as loge(φ) vs. iteration number for fixed position (x) and in all cases the slope was identical to –κ. It may have been noted that in Equation (10.57) it was possible to use the nodal potentials at one iteration to calculate the incident pulses at the next. Once we move to second order kinetics this is no longer possible on account of the inherently nonlinear process. We will therefore adopt a more rigorous approach to treat the chemical reaction A + B → B.

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Figure 10.23 Sections of two adjacent nodes that are to be considered in the link-resistor TLM scheme for a second order chemical reaction.

Figure 10.23 shows the parts of the nodes that will be considered in the treatment that follows. We see the transmission line of length ∆x/2 to the left of node (x) together with its associated resistor. We see the right-hand arm of node (x – 1) together with its resistor and we see the zone boundary that designates the end of one node and the start of the next. Any signal that leaves a node will take a time ∆t/2 to reach this interface and anything scattered at the resistors will take a similar time to return to the adjacent nodes. We start with pulses of species A and B, which leave node (x) and move to the left. These react en route through the transmission line so that the amount of each species that arrives at the interface is given by s

VLA ( x ) − κ sVLA ( x ) sVLB ( x )

∆t 2

➀

VLB ( x ) + κ sVLA ( x ) sVLB ( x )

∆t 2

➁

s

At the same time, we have pulses of species A and B which leave node (x –- 1) and move to the right. These react en route through the transmission line so that the amount of each species that arrives at the interface is given by s

VRA ( x − 1) − κ sVRA ( x − 1) sVRB ( x − 1)

∆t 2

➂

s

VRB ( x − 1) + κ sVRA ( x − 1) sVRB ( x − 1)

∆t 2

➃

These are represented by encircled numbers so that the subsequent derivation can be kept clear. Note also that the iteration time index, k, has been omitted for clarity. Reflection occurs at the interface between the two nodes as it normally does in a link-resistor formulation, The components that end up in the left arm of node (x) are

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A reflected at interface ρA × ➀ B reflected at interface ρB × ➁ A transmitted at interface τA × ➂ B transmitted at interface τB × ➃

I II III IV

(the Roman numerals are used to avoid algebraic obfuscation) Finally, these pulses travel back toward node (x) and appear as incident pulses after time ∆t/2

V ( x) = ( I + III ) − κ ( I + III ) ( II + IV )

i k +1 LA

∆t 2 (10.58)

V ( x) = ( II + IV ) + κ ( II + IV ) ( I + III )

i k +1 LB

∆t 2

There are a similar set of expressions for pulses incident on (x) from the right from which the nodal concentration can then be calculated. The above scheme was applied to a system comprising unit concentration of species A throughout the 200 node computational space. The initial concentration of species B was set at zero at all locations except at node 100, where s0VLB (100) = 0.001 and s0VRB (100) = 0.00 As the simulation progressed, species A was depleted and species B was created. A snapshot of a typical result is shown in Figure 10.24. The technique has been ported to two dimensions and have given results that are typical of a “diffusion-wave.”

10.12.6 The Logistic Equation in the Presence of Diffusion The “logistic map” is a deceptively simple nonlinear difference equation that predicts the concentration of a species at the next discrete time step in terms of its present concentration45.

Y = r kY ( 1 − kY ) (r is a constant)

k +1

(10.57)

It is frequently used as a basis for “predator–prey” and other population dynamic models. In this section we will outline the basis for this use, investigate some of the features and demonstrate that the inclusion of spatial diffusion using TLM yields some fascinating results. As this is largely unexplored territory, some MATLAB code is included so that readers may try it for themselves. We start with a continuous kinetic system where rate of growth is first order and rate of decay is second order, a combination of processes that might be encountered in a mechanical control system

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Figure 10.24 Concentration of species A and B (vertical axes) vs. position (horizontal axes) for the second order reaction A + B → 2B with diffusion where ρA = 0. 9 and ρB = 0. 5.

dY = r1 Y − r2 Y 2 dt

(10.58)

If Y(t = 0) = Y0 then this has an analytical solution

Y ( t) =

r1 ⎛ ( r1 − Y0 r2 ) e − r1t ⎞ r2 + ⎜ ⎟ Y0 ⎝ ⎠

We can treat this as a finite difference scheme if we consider how ∆Y varies with ∆t

Y − kY = r1 kY − r2 kY 2 ∆t

k +1

(10.59)

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This yields

(

)

Y = r1 ∆t + 1 kY − r2 ∆t kY 2

k +1

(

)

(10.60)

If we restrict ourselves to the situation r1 ∆t + 1 = r2 = r we obtain Equation (10.57) It is known that for r < 1 kY → 0 as k → ∞ (any population becomes extinct). If the equation is operated in the range 1 < r < 3 then the population grows towards a steady state, but tends to show an oscillatory starting transient. Above this point kY, after the initial rise, may switch between two distinct values (a period 2 cycle). When r is increased a little higher we will see a transition to a period 4 cycle. This period doubling continues until the system becomes chaotic, but regions of chaos may be interspersed with regions of apparent order. This is normally plotted as Y vs. r, but it is also very instructive to undertake an FFT on the data for different values of r and plot the frequencies against r. The system as presented above is zero-dimensional but it could be applied to an entire population so long as it was uniform. This could be replaced by a one-dimensional line of discrete nodes with the same starting concentration at every position. Alternatively, we could arrange for the starting concentration to be zero everywhere except at one point. Each of these scenarios would yield the same concentration-time behavior in the regions that had a non-zero starting population. If we now allow some diffusion of the species to take place, then at some time logistic growth-decay starts in regions that were previously unoccupied. The resulting behavior appears to be extremely complicated but one significant observation is that the point of transition from an ordered to a chaotic state can be manipulated by changing the diffusion constant (altering the reflection coefficient, ρ). The resulting system may take a considerable number of iterations to settle down or, alternatively, having been well-behaved for some time, it may spontaneously become chaotic. It may be ordered in one region and disordered in another (try r = 3.90, ρ = 0.5 in the code that is included below). The spatially dispersed populations may oscillate in phase and this may manifest itself as time-varying spatial patterns. Figure 10.25 shows the concentration vs. position for one half of a symmetrical computational space (following an initial input at position 1000) for different operational conditions. There is an enormous amount to be studied in systems such as these and the extension to the second dimension has been particularly promising. An initially populated point at the center of a Cartesian mesh has shown radial patterns, which suggests that the system should really be treated using a polar mesh. We might imagine a myriad of situations in geomechanics, biomechanics, etc., where reactant is supplied at one rate and removed (incorporated into the static environment) at another, so that we are left with growth patterns. Spatial variation of the diffusion constant (different values of ρ) and temporal variation of the rate r could go a long way toward accounting for spatio-temporal patterns. The future is wide open.

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Figure 10.25 Plots of concentration (vertical axis) versus position for a logistic equation with diffusion (a) r = 3. 2, ρ = 0. 95, (b) r = 3. 2, ρ = 0. 96. Note the differences in scale in the horizontal axes.

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References 1. Krus P., Modeling of mechanical systems using rigid bodies and transmission line joints, Trans. ASME (J. Dynamic Systems, Measurement Control), 121 (1999) 606–611. 2. Zauderer E., Partial Differential Equations of Applied Mathematics, Wiley, New York (1983) 112–113. 3. Whitham G. B., Linear and Non-Linear Waves, Wiley, New York (1974). 4. 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. 5. 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. 6. Murphy K. E., Transmission Line Matrix Software for Modelling Acoustic Devices, M.Eng.Sc. thesis, University College, Dublin, 2000. 7. Kinsler L. E., Frey A. R., Coppens A. B., and Sanders J. V., Fundamentals of Acoustics, 3rd ed., Wiley, New York (1980) 210–214. 8. O’Connor W. J., TLM modelling of devices with acoustic/mechanical coupling, in TLM Applications beyond Electromagnetics, Proceedings of an informal meeting held at the University of Hull, 24 June 1997, de Cogan D., Ed., School of Information Systems, Norwich, U.K.: UEA (1997) 2.1–2.8. 9. Morse P. M., Vibration and Sound, 2nd ed., McGraw-Hill, New York (1948) 357–360. 10. de Cogan D., Transmission Line Matrix (TLM) Techniques for Diffusion Applications, Gordon and Breach (1998). 11. Meliani H., Mesh generation in TLM, Ph.D., thesis, University of Nottingham, U.K., October (1987). 12. Cacoveanu R., Saguet P., and Ndagijimana F., TLM method: a new approach for the central node in polar meshes, Electronics Lett., 31 (1995) 297–298. 13. El-Masri S., Pelorson X., Saguet P., and Badin P., Development of the Transmission line matrix method in acoustic applications to higher order modes in the vocal tract and other complex duct, Int. J. Numerical Modelling, 11 (1998) 133–151. 14. Cross T. E., Johns P. B., and Loaseby J. M., Transmission line modelling of the vocal tract and its application to the problem of speech synthesis, Proceedings IEE Conference, Speech Input/Output: Techniques and Applications No. 258, March (1986), 71–76. 15. Kagawa Y., Computational acoustics- theories of numerical analysis in acoustics with emphasis on transmission line matrix modelling, Proceedings International Symposium on Simulation, Visualisation, and Auralisation for Acoustic Research and Education, Tokyo, Japan (1997) 19–26. 16. Saguet P., TLM method for three-dimensional analysis for microwave and mm-structures, International Workshop of German IEEE MTT/Ap Chapter (1991). 17. Porti J. A. and Morente J. A., TLM method and acoustics, Int. J. Numerical Modelling, 14 (2001) 171–183. 18. O’Connor W. J., TLM model of waves in moving media, Int. J. Numerical Modelling, 14 (2002) 205–214. 19. Hoefer W. J. R., The transmission line method—theory and applications, IEEE Transactions on Microwave Theory and Techniques MTT, 33,(1985) 882–893.

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20. Morente J. A., Giménez G., Porti J. A., and Khalladi M., Dispersion analysis for a TLM mesh of symmetrical condensed nodes with stubs, IEEE Transactions on Microwave Theory and Techniques MTT, 43 (1995) 452–456. 21. Trenkic V., Christopoulos C., and Benson T. M., Analytical expansion of the dispersion relation for the TLM condensed node. IEEE Transactions on Microwave Theory and Techniques MTT, 44 (1996) 2223–2230. 22. O’Connor W. J., Wave Speeds for a TLM model of moving media, Int. J. Numerical Modelling, 14 (2002) 195–203. 23. Swope R. D. and Ames W. F., Vibrations of a moving threadline, J. Franklin Inst., 275 (1963) 36–55. 24. D. Stubbs, S. H. Pulko, A. J. Wilkinson, Simulation of a Thermal Environment for Chilled Foods during Transport, Proceedings of a meeting on the properties, applications and new opportunities for the TLM numerical method (Hotel Tina, Warsaw 1–2 October 2001) School of Information Systems (UEA) 2002 (ISBN 1 898290 16 6) Paper 10. 25. A. I. Hurst, S. H. Pulko, TLM treatments of changes of phase, Int. J. Numerical Modelling 7 (1994) 201–207. 26. G. Butler and P. B. Johns, The solution of moving boundary heat problems using the TLM method of numerical analysis, Numerical Methods in Thermal Problems (Ed. R. W. Lewis and K. Morgan), Swansea: Pineridge Press (1979) 189–195. 27. D. de Cogan and A. Soulos, Inverse thermal modelling using TLM, Numerical Heat Transfer: Part B, 29 (1996) 125–135. 28. D. de Cogan, A. Soulos and K. O. Chichlowski, Sub-surface feature location and identification using inverse TLM techniques, Microelec. J. 29 (1998) 215–222. 29. I. Badger, S. H. Pulko and A. J. Wilkinson, Reverse time modelling of thermal problems using the transmission line matrix approach, Numerical Heat Transfer, Part B, 40 (2001) 1–17. 30. A. L. Koay, S. H. Pulko and A. J. Wilkinson, Time Reversal inverse TLM modelling of thermal problems, Proceedings of a meeting on the properties, applications and new opportunities for the TLM numerical method (Hotel Tina, Warsaw 1- 2 October 2001) School of Information Systems (UEA) 2002 (ISBN 1 898290 16 6) Paper 1. 31. D’Arcy Wentworth Thompson, On Growth and Form: Spatio-temporal Pattern Formation in Biology 1917 32. M. A. J. Chaplain, G. D. Singh, J. C. McLachlan, (Eds.), On Growth and Form: Spatio-temporal Pattern Formation in Biology Wiley Series in Mathematical and Computational Biology (1999) ISBN 0 471 98451 5. 33. D. de Cogan and M. Henini, TLM Modelling of the Liesegang phenomenon, J. Chem. Soc. Faraday Trans. 2, 83 (1987) 837–841. 34. S. K. Dew and X. Gui, Use of a dynamic network for the TLM solution of diffusion problems, Int. J. Numerical Modelling 11 (1998) 259–271. 35. B. Chopard and M. Droz, Cellular Automata Modelling of Physical Systems Collection Alea-Saclay: Monographs and Texts in Statistical Physics. Cambridge University Press (1998) ISBN 0 521 46168 5. 36. S. Wolfram, A New Kind of Science, Wolfram Media Inc. (2002) ISBN 1 57955 008 8.

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37. Game of Life: see M. Gardner, Mathematical Games, Scientific American 224 (1971) Feb. p 112, March p 106, April p 114. M. Gardner Mathematical Games, Scientific American 226 (1972) Jan. p 104. 38. I. Karafyllidis and A. Thannailakis, A Model for predicting forest fire spreading using cellular automata, Ecol. Modelling 99 (1997) 87–97. 39. A. H. M. Saleh and D. de Cogan, Numerical solution of inhomogeneous and non-linear differential equations using the TLM multi-compartment model, Int. J. Numerical Modelling, 3 (1990) 215–228. 40. A. H. M. Saleh, D. de Cogan, TLM Models for Limit Cycle and Other Non-Linear Differential Equations, 4th Int. Conf. Non-linear Engineering Computations (NEC-91), Sept. 1991, Pineridge Press, Swansea. 41. D. de Cogan, Transmission Line Matrix (TLM) Techniques for Diffusion Applications, Gordon and Breach 1998 ISBN 90 5699 129 9 pp 189–198 42. J. J. Tyson, Some further studies of non-linear oscillations in chemical systems, J. Chem Phys 58 (1973) 3919–3930. 43. P. Glansdorff and I. Prigogine, Thermodynamic Theory of Structure, Stability and Fluctuations, John Wiley and Sons, New York 1971. 44. D. de Cogan and M. Henini, Transmission line matrix (TLM): A novel technique for modelling reaction kinetics, J. Chem. Soc. Faraday Transactions 2 83 (1987) 843–855. 45. R. M. May, Simple mathematical models with very complicated dynamics, Nature 261 (1976) 459–467.

MATLAB code for a one-dimensional logistic process with the addition of a diffusive component. The input data included ensures that the results go through an initial random period before settling down to a regular pattern. % **************************************************************** % One-dimension logistic reaction + diffusion using link-resistor TLM network % D. de Cogan, CMP, UEA (12/12/04) % **************************************************************** clear maxspace = 2000; maxtime = maxspace;% to avoid any end-effects r = 3. 90;% rate constant ro = 0. 50;tau = 1 – ro;% reflection/transmission coefficients viL=zeros(1,maxspace);% initialise all a pulses incident from left and right viR = viL;vsL = viL;vsR = viL;phia = vsL + vsR; X = linspace(1,maxspace,maxspace); vsL(1,maxspace/2) = 0. 00001;% redefine one value at centre as a logistic equation seed vsR(1,maxspace/2) = 0. 00001; for j = 1:maxtime% ************* start of TLM iterative loop *************** for x = 2:maxspace% define a pulses incident from left viL(x) = ro*(vsL(x) +(r-1)*vsL(x) – r*vsL(x)*vsL(x)) + tau*(vsR(x-1) +(r-1)*vsR(x-1) – r*vsR(x-1)*vsR(x-1)); end

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for x = 1:maxspace-1% define a pulses incident from right viR(x) = ro*(vsR(x) +(r-1)*vsR(x) – r*vsR(x)*vsR(x)) + tau*(vsL(x+1) +(r-1)*vsL(x+1) – r*vsL(x+1)*vsL(x+1)); end

phia = viL + viR;% define the instantaneous superposition for left and right

plot(X(200:1000),phia(200:1000));pause(0. 1);% plot output vsL = viR;% pulses incident from right become 'scattered' to left vsR = viL;% pulses incident from left become 'scattered to right

end % ************* end of TLM iterative loop ***************

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Index A Acoustic pressure 15, 17 Acoustic propagation 223–225 1-D TLM algorithm 47–52 2-D TLM algorithm 52–56 3-D TLM node 229–232 complex ducts 227–229 Gaussian wave-form 60–62 in inhomogenous media 66–68 moving sources 63–66 polar meshes 226–227 sinu-wave excitation 56–60 Acoustic velocity 15, 17 Acoustic wave equation 17, 67 Admittance 41, 102, 180, 240 Advection equation 189 Ait-Sadi, R. 108 Aldridge, R.V. 64 Allen, R. 193 Al-Mukhtar, D.A. 111 Ames, W.F. 235 Amplification factor 241–242 Angular velocity 17, 168–169 Attenuation factor 82–83 Auslander, D.M. 178

B Backward shift operator 141 Badger, I. 239 Badin, P. 227 Beam 6 Beam steering 61 Bending moment 166–167 Benson, T.M. 234 Berenger boundaries 83 Beurle, R.L. 2–3, 44 Biharmonic static equation 6 Biharmonic wave 6 Binary scattering 126–128 Black-box 83 Blanchfield, P. 58 Boucher, R.F. 178–185, 189 Boundaries 43–44 constant temperature 93–96

displacement 152–153 fixed-value 132–133 force 153–157 frequency-dependent absorbing 77–80 insulating 92 matched-load 44 mirror 43 models 13 perfect heat-sink 93 pressure-release 44 surface conforming 74–76 symmetry 92 telegrapher’s equation 133–134 Brusselator model 244 Bulk modulus 179, 196 Burrows, C.H. 189–193 Burton, J.D. 182, 189–193 Bus-bars 108 Butler, G. 169, 238

C Cacoveanu, R. 112, 226 Canary Islands, tidal wave hazard in 76 Cantilever 167 Capacitance 28 distributed 36–37, 88 symbol 178 transmission lines 17 Capacitors 3, 28–30 Catt, I. 3, 4 Cellular automaton (CA) 1, 244 Chakrabarti, A. 102, 199 Characteristic impedance 32, 179, 216 Chardaire, P. 132 Charge 28 Chen, Z. 82 Cheng, D.K. 37 Chichlowski, K.O. 239 Christopoulos, C. 4, 40, 186, 234 Circuit time-constant 29 Circuits 31 Clune, F.J. 20, 74, 220 Coefficient of viscosity 204 Coefficient of viscous traction 162 Compressibility 15, 17, 52

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Compressional waves 181 Computational fluid dynamics (CFD) 203 convective acceleration 208–211 incompressible fluids 207–208 velocity fields 207–208 viscosity 205–206 Condenser microphone 225–226 Conductors 28 Constant temperature boundaries 93–96 Convective acceleration 203, 208–211 Coppens, A.B. 226 Cracking pressure 192 Cross, T.E. 227 Current 17 Current transmission coefficient 106 Cut-off frequency 55 Cut-on frequency 228

D Darcy-Weisbach equation 180 Dashpot viscosity 159 de Cogan, A. 77, 182 de Cogan, D. 64, 76, 82, 102, 107, 108, 111, 114, 125, 132, 133, 182, 189, 191, 239, 242, 246 Delta pulse 4 Density 17 Dew, S.K. 242 Diffusion coefficient 189 Diffusion equation 6 fixed-value boundaries 132–133 logistic 248–251 TLM models 130–132 Diffusion waves 246–248 Dilatational wave 139 Dirac impulse excitation 4 Discontinuities 39–40 Discretization 37–39, 55, 88, 123 Dispersion 55 Displacement boundaries 152–153 Distributed circuits 36–37 Distributed parameters 36 Domain of dependence 128 Domain of influence 128 Doppler effect 64, 235–237 Drift velocity 189 Drift-diffusion equation 189 Duality 74 Duffy, D.J. 206

E Edge, K.A. 189–190 Effective impedance 80 Elastic materials 137–140 boundaries 152–153 force boundaries 153–157 implementation 149–150 nodal structure 143–149 Elastic pipes 193–196 Electrical current 178 Electromagnetic theory 32 Electromagnetic wave 17 El-Masri, S. 227 Enders, P. 108, 125 Euler’s equation 6 Expanded polystyrene (EPS) 239

F Fast Fourier transform (FFT) 50–52 Featherstone, R.E. 180, 197 Ferromagnetic materials 30 Finite element modeling (FEM) 88 Fixed-value boundaries 132–133 Flint, J. 235 Floquet’s theorem 234 Flow 192 Fluid bulk modulus 193 Fluid flow 177–178 compressional waves 181 parameters 179–181 time-domain transmission line models 183–192 transmission line analysis 181–182 Fluid velocity 205 Flux injection 100–101 Force boundaries 153–157 Forced wave equation 6, 20–21 Forward shift operator 141 Free excitation 57 Frey, A.R. 226

G Gallagher, M. 125 Gantry crane 21–22, 215 Gaussian wave-form 60–62 Gauss-Seidel method 121 Gimenez, G. 234 Glansdorff, P. 244 Graded perfectly matched load (GPML) 83

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Gravity 165 Gui, X. 108, 116, 133, 242

H Halleron, J.A. 110 Hammond, P. 3 Hanging cable 6, 213–214 Hansleman, D. 49, 51 Harvey, R.W. 102 Heat-sink boundaries 93 Heaviside condition 4, 81 Helicopter rotor noise 76 Helmholtz equation 6 Helmholz resonator 74 Henini, M. 107, 242, 246 Henries 30 Hindmarsh, D. 223 Hoefer, W.J.R. 234 Hui, S.Y.R. 4 Hurst, A.I. 238 Huyghen’s principle 61 Hydraulic systems 177–178 analogues 178–179 compressional waves 181 elastic pipes 193–196 open-channel 196–198 parameters 179–181 symbols 178–179 time-domain transmission line models 183–192

I Ilan, A. 157 Impedance 15–17 definition of 29 nodes 144 terminating 41 Impulse 38 Impulse propagation and scattering (IPS) 5 Impulses 11, 17 Incident pulses 90 Incompressible fluids 6, 207–208 Inductance 17, 30, 88 Inductors 30–31 Inertance 179 Insulating boundaries 92 Internodal reflections 111–114 Inverse scattering 239–240 Inverse thermal modeling 239 Irrotational wave 139 Iteration index 57

J Jacobi scheme 121 Jansson, A. 193 John, S.A. 111 Johns, P.B. 3–5, 44, 108, 114, 117, 169, 193, 227 Jumps-to-zero 121–126 Junctions 18–19

K Kagawa, Y. 227 Kaibara, K. 150 Kaplan, B. 118 Karafyllidis, I. 244 Kay, J.M. 204 Kelvin viscoelastic material 160–161, 170 Khalladi, M. 234 Kinetic energy 16 Kinsler, L.E. 226 Kirchoff’s laws 233 Kitsios, E.F. 178–185, 189 Klein-Gordon equation 6, 20, 220–223 Koay, A.L. 239 Koizumi, T. 150 Kranys, M. 5 Kron, G. 3 Krus, P. 193, 216–220

L Lamé’s constant 138 Langley, P. 173 Laplace equation 6, 96, 132 Laplace variable 182 Latent heat 238–239 Lateral tension force 214 Lateral velocity 214 Legendre polynomial 23 Lenz’s law 30 Liebmann, G. 119 Liesegang rings 242 Linear density 10, 16, 17 Link-line node 88 1-D TLM algorithm 90–91 formulations 102–104 Link-resistor node 88 1-D TLM algorithm 91 formulations 104–106 Littlefield, B. 49, 51 Loaseby, J.M. 227

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Loewenthal, D. 157 Logistic equation 248–251 Longitudinal wave 17 Lossless TLM equation 67 Lowery, A.J. 108 Low-pass filters 77–79 Lumped circuits 36–37

M Mach number 66 Maclaurin series 77 Malachowski, M.J. 118 Mallik, A. 108, 193 Manning equation 196 Markov process 121 Massey, B.S. 161, 166 Matched load boundaries 44 MATLAB® 6, 47, 197 Maxwell viscoelastic behavior 160 Maxwell’s equations 33–36, 67 May, R.M. 248 Mechanochemical theory 242 Medium variable pairs 17 Meliani, H. 72, 226 Meshes 106–111, 128–130 Mirror boundaries 43 Moravec, K.L. 130 Morente, J.A. 230–232, 234 Morse, P.M. 226 Morton, M. 64, 76, 223 Moving sources 63–66 Moving threadline equation 6, 21, 233–235 Multiple flux sources 101–102 Murphy, K. 223

N Nakahara, I. 150 Nalluri, G. 180, 197 Navier-Stokes equation 6, 204 Naylor, P. 111 Ndagijimana, F. 112, 226 Nedderman, R.M. 204 Newton's second law 15 Newton, H.R. 165, 167, 169 Newton’s second law 179, 218 Newtonian fluids 204 Nodal potential 105, 114, 238 Nodal structure 143–149

O O’Brien, M. 4 O’Connor, W.J. 20, 21, 64, 133, 199, 220, 225 Ohm’s law 28, 29, 31, 39, 204 Open-channel hydraulics 196–198 Open-circuit termination 32, 43, 71, 77 Open-circuit voltage 33 Oscillating chemical reactions 244

P Packer, G.A. 198 Palmberg, J.O. 193 Parallel network 231 Parallel viscoelastic model 170–173 Parallelepipedic mesh 227 Partial differential equations (PDEs) 6, 21 Partial diffusion 88–90 Pascal triangle 128 Pearson, R.J. 178 Peel, D. 76 Pelorson, X. 227 Perfect heat-sink boundaries 93 Perfectly matched load (PML) 84 Permeability 17 Permittivity 17, 28 Perturbation 20 Phased arrays 61 Phizacklea, C.F. 110 Pic, E. 107 Pipes 193–196 Plane-wave theory 229 Poiseuille’s formula 180 Poisson equation 6, 101 Poisson’s ratio 138 Polar meshes 226–227 Pollmeier, K. 182, 191, 193 Pomeroy, S. 63, 75 Porti, J.A. 230–232, 234 Potential energy 16 Power 28 Pressure 179 Pressure valves 191 Pressure-release boundaries 44 Prigogine, I. 244 Propagation analysis 117 Propagation velocity 38, 52–55 Pulko, S.H. 108, 110, 114, 125, 193, 237, 238, 239

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R Rainier, M. 159 Reference impedance 234 Reflected current 39 Reflection coefficient 32, 68, 74, 90, 96–97, 102, 104, 106, 189, 194, 227 Refrigerants 237–239 Relative permeability 30 Relative permittivity 28, 69 Relaxation time 118 Resistance 28, 36, 88, 178 Resistivity 28 Resistors 28 Reynolds number 180 Rigid bodies 216–220 Ronalili, G. 137 Rotating string 6, 23–24 Rotational wave 139

S Saguet, P. 107, 112, 227 Saleh, A.H.M. 58, 244 Sanders, J.V. 226 Sant, V. 64, 76 Scattering coefficient 184 Scattering matrix 18–19, 42, 144–148, 186, 229–232 Schrödinger equation 6 Series network 231 Series node 40 Shah, A.K. 107 Shannon’s sampling theorem 12 Shear force 166, 204 Shear modulus 17, 138 Shibuya, T. 150 Short-circuit impedance 33 Short-circuit termination 44, 77 Shunt impedance 80 Shunt node 16, 40 Sine-Gordon equation 6, 20 Single heat source 100–101 Single-shot injection 96–99 Sinusoidal excitation 56–60 Sitch, J.E. 111 Smith, P.A. 198 Soulos, A. 239 Source flow 191 Southwell, R.V. 2 Spatial discretization 88–90, 123 Specific heat 95 Spring elasticity 159

State space control theory 140–143 State vector 141 Stiffness 218 Strain 17 Streeter, V.L. 177–178, 182, 185 Stress 17, 137 Stretched string 6, 17 String tension 10, 16 Stubbs, D. 237 Stubs 19–20 impedance 69 internodal reflections 111–114 lines 68–73 Surface conforming boundaries 74–76 Swope, R.D. 235 Sykulski, J. 3 Symmetrical condensed node (SCN) 230–232 Symmetry boundaries 92

T Taitel, Y. 121 Taylor series expansion 227 Telegrapher’s equation 6, 117 approximation of 120–121 derivation of 37 expression of electrical networks in 88 fixed-value boundaries 133–134 lossy propagation 20 Television sets 32 Tension 10, 16, 17 Thannailakis, A. 244 Thermal conductivity 95 Thermal diffusion 88–90, 237–239 Thermal resistance 88 Thévenin’s theorem 32–33, 90 Thomson, D. 242 Time discretization 37–39, 123 Time-step transformation 114–117 T-network 88 Toepliz matrix 49 Torque 17 Torsional wave 17 Tractive force 162 Transfer function 141–142 Transmission coefficient 68, 90, 227 Transmission line matrix (TLM) 2–7 2-D wave equation 24 boundaries 43–44 convective acceleration 208–211 diffusion waves 246–248 model 11–13 nodal configurations 40–43

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and state space control theory 140–143 Thévenin’s theorem 32–33 viscosity 205–206 Transmission lines 31–32 discontinuities 39–40 distributed circuits 36–37 fluid flow 181–182 junctions 18–19 lossy propagation 20 lumped circuits 36–37 open-boundary descriptions 80–84 rigid bodies 216–220 signal propagation in space 33–36 stubs 19–20 time discretization 37–39 time-domain 183–192 wave equation 17 Transmitted current 39 Transverse wave 139 Trenkic, V. 234 Trouton’s column 161–165 Tuck, B. 108 Turing 242 Turing theory 242 2-D wave equation 24

V Velocity 163, 214–215 Velocity fields 207–208 Velocity of propagation 38, 52–55 Vibrating string 10–11 Viersma, T.J. 182, 189 Vine, J. 3 Viscoelastic deformation 170–173 Viscoelastic materials 159 Viscosity 204–205

coefficient of 204 equations 162 TLM algorithm 205–206 Viscous bending 165–169 Viscous pressure gradient 205 Vnode 19 Voltage 17 calculation of 78 capacitors 28 change in 36 inductors 30 transmission coefficient 106

W Wave equation 6, 15–18 Wave impedance 15–17, 24 Wave speed 10, 15–17, 234 Wave variable pairs 17 Webb, P.W. 116 Wilkinson, A.J. 125, 237, 239 Willison, P.A. 72 Winterbourne, D.E.A. 178 Wong, C.C. 108 Wong. W.S. 108 Wylie, E.B. 177–178, 182, 185

Y Young’s modulus 17, 138

Z Zero deflection 13 Zero displacement boundary 74