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

Transmission Line Matrix in Computational Mechanics Transmission Line Matrix in Computational Mechanics Donard de Cogan William J O’Connor Susan Pulko Boca Raton London New York A CRC title, part of the Taylor & Francis imprint, a member of the Taylor & Francis Group, the academic division of T&F Informa plc TF1745_Discl.fm Page Monday, September 26, 2005 3:54 PM Published in 2006 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2006 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group No claim to original U.S Government works Printed in the United States of America on acid-free paper 10 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 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 TF1745_C000.fm Page v Wednesday, September 28, 2005 12:50 PM Acknowledgments Those who knew Peter Johns* speak glowingly of his inspiration and his enthusiasm He achieved so much, and we are certain that he could have achieved much more had he lived He was already moving into mechanical applications of TLM and was discussing nonlinear processes such as the action of a violin bow on a string Shortly after his first heart attack he commenced work on a TLM model of electromechanical interactions in heart muscle He was a cohesive factor in all areas of development, which in his absence have tended toward a bimodal partition: TLM applications that are related to electromagnetics and TLM applications that are not Within the latter grouping, the contributions of Peter Enders, Xiang Gui, and the late Adnan Saleh have been crucial We also wish to acknowledge the 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 ** TF1745_C000.fm Page vi Wednesday, September 28, 2005 12:50 PM TF1745_C000.fm Page vii Wednesday, September 28, 2005 12:50 PM About the Authors Donard de Cogan gained a bachelor’s degree in physical chemistry and a Ph.D in solid state physics from Trinity College, Dublin He undertook research fellowships in solid state chemistry (University of Nijmegen, 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 TF1745_C000.fm Page viii Wednesday, September 28, 2005 12:50 PM Susan Pulko graduated from Imperial College, University of London in 1977 She moved to the University of Nottingham and undertook 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 TF1745_book.fm Page ix Monday, September 12, 2005 11:56 AM Table of Contents Chapter Introduction Chapter TLM and the 1-D Wave Equation Introduction 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 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 4.1 TLM Modeling of Acoustic Propagation 47 Introduction 47 TF1745_C010.fm Page 248 Friday, September 23, 2005 9:41 AM 248 Transmission Line Matrix in Computation Mechanics • • • • 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 (10.58) V ( x) = ( II + IV ) + κ ( II + IV ) ( I + III ) i k +1 LB ∆t 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 sVLB (100) = 0.001 and sVRB (100) = 0.00 As the simulation progressed, 0 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 ( − 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 TF1745_C010.fm Page 249 Friday, September 23, 2005 9:41 AM Chapter ten: State of the Art Examples 249 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 = and ρB = dY = r1 Y − r2 Y 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 ∆t k +1 (10.59) TF1745_C010.fm Page 250 Friday, September 23, 2005 9:41 AM 250 Transmission Line Matrix in Computation Mechanics This yields ( ) Y = r1 ∆t + kY − r2 ∆t kY k +1 ( ) (10.60) If we restrict ourselves to the situation r1 ∆t + = r2 = r we obtain Equation (10.57) It is known that for r < kY → as k → ∞ (any population becomes extinct) If the equation is operated in the range < r < 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 cycle) When r is increased a little higher we will see a transition to a period 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 TF1745_C010.fm Page 251 Friday, September 23, 2005 9:41 AM Chapter ten: State of the Art Examples 251 Figure 10.25 Plots of concentration (vertical axis) versus position for a logistic equation with diffusion (a) r = 2, ρ = 95, (b) r = 2, ρ = 96 Note the differences in scale in the horizontal axes TF1745_C010.fm Page 252 Friday, September 23, 2005 9:41 AM 252 Transmission Line Matrix in Computation Mechanics References Krus P., Modeling of mechanical systems using rigid bodies and transmission line joints, Trans ASME (J Dynamic Systems, Measurement Control), 121 (1999) 606–611 Zauderer E., Partial Differential Equations of Applied Mathematics, Wiley, New York (1983) 112–113 Whitham G B., Linear and Non-Linear Waves, Wiley, New York (1974) 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 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 Murphy K E., Transmission Line Matrix Software for Modelling Acoustic Devices, M.Eng.Sc thesis, University College, Dublin, 2000 Kinsler L E., Frey A R., Coppens A B., and Sanders J V., Fundamentals of Acoustics, 3rd ed., Wiley, New York (1980) 210–214 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 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 TF1745_C010.fm Page 253 Friday, September 23, 2005 9:41 AM Chapter ten: State of the Art Examples 253 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 898290 16 6) Paper 10 25 A I Hurst, S H Pulko, TLM treatments of changes of phase, Int J Numerical Modelling (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- October 2001) School of Information Systems (UEA) 2002 (ISBN 898290 16 6) Paper 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 471 98451 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 521 46168 36 S Wolfram, A New Kind of Science, Wolfram Media Inc (2002) ISBN 57955 008 TF1745_C010.fm Page 254 Friday, September 23, 2005 9:41 AM 254 Transmission Line Matrix in Computation Mechanics 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, (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 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 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 = 90;% rate constant ro = 50;tau = – 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) = 00001;% redefine one value at centre as a logistic equation seed vsR(1,maxspace/2) = 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 TF1745_C010.fm Page 255 Friday, September 23, 2005 9:41 AM Chapter ten: State of the Art Examples 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 *************** 255 TF1745_C010.fm Page 256 Friday, September 23, 2005 9:41 AM TF1745_bookIX.fm Page 257 Tuesday, September 13, 2005 11:16 AM 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 Beam steering 61 Bending moment 166–167 Benson, T.M 234 Berenger boundaries 83 Beurle, R.L 2–3, 44 Biharmonic static equation Biharmonic wave 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, 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, 52 257 TF1745_bookIX.fm Page 258 Tuesday, September 13, 2005 11:16 AM 258 Transmission Line Matrix in Computation Mechanics 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 Density 17 Dew, S.K 242 Diffusion coefficient 189 Diffusion equation fixed-value boundaries 132–133 logistic 248–251 TLM models 130–132 Diffusion waves 246–248 Dilatational wave 139 Dirac impulse excitation 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 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 TF1745_bookIX.fm Page 259 Tuesday, September 13, 2005 11:16 AM Index 259 Gravity 165 Gui, X 108, 116, 133, 242 H Halleron, J.A 110 Hammond, P 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 Helmholz resonator 74 Henini, M 107, 242, 246 Henries 30 Hindmarsh, D 223 Hoefer, W.J.R 234 Hui, S.Y.R 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) 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 Kron, G 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 TF1745_bookIX.fm Page 260 Tuesday, September 13, 2005 11:16 AM 260 Transmission Line Matrix in Computation Mechanics 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 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 TF1745_bookIX.fm Page 261 Tuesday, September 13, 2005 11:16 AM Index 261 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 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 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 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 TF1745_bookIX.fm Page 262 Tuesday, September 13, 2005 11:16 AM 262 Transmission Line Matrix in Computation Mechanics 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 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 ... 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. .. 11:56 AM 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... 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