Infinite Elements PDF
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Infinite Elements First Edition i ii Infinite Elements by Peter Bettess Department of Marine Technology University of Newcastle upon Tyne Newcastle upon Tyne England Penshaw Press 1992 first published 1992 by Penshaw Press TheLawns,Undercliff,CleadonLane,Cleadon,SUNDERLAND,SR67UX,U.K. British Library Cataloguing in Publication Data Bettess, P Infinite Elements I. Title 620.00151524 ISBN 0-9518806-0-8 Copyright(cid:13)c 1992byPenshawPress. Allrightsreserved. Nopartofthispublica- tion may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publishers Printed and bound in Great Britain by Bookcraft (Bath) Ltd, Wheeler’s Hill Midsomer Norton Avon BA3 2BX UK ii Preface I have had three aims in writing this book. The first was to pull together all the infinite element concepts and ideas which have been published over the past sixteen years or so, into one complete, self-contained volume. My second aim was to try to increase the popularity of the method, which forms a very useful adjunct to finite elements. Many finite element problems can best be regarded as unbounded. Most of these can be modelled more effi- cientlyusinginfiniteelementsontheboundary, insteadofsimplytruncating the finite element mesh, which is what is often done. Methods for dealing with unbounded domains, in conjunction with finite elements fall broadly into two types: ‘Global’ methods in which a global solution such as boundary integrals is linked to the finite element mesh. These methods tend to be accurate but, since they usually destroy the banded nature of the system matrix, expensive. They also tend to be more complicated to program. ‘Local’ methods, in which a local boundary condition, such as a damper, or truncation is applied to the boundary of the finite element mesh. These methods tend to be cheap but inaccurate. They also tend to be easy to program. Infinite elements keep the best features of the above two methods, retaining the bandedness, being easy to program, and being effectively as accurate as the user requires. Mythirdandmostimportantaimwastointroducethereadertothemethod of infinite element techniques, which will allow him or her to extend the ap- plication of finite element methods to large classes of unbounded domain problems. The idea of infinite elements simply involves extending the do- main of a finite element so that it is unbounded. This requires appropriate shape functions which are defined up to infinity and tend to the infinite value in a suitable way. It also requires a means of integration over the un- bounded domain. For many problems these are both fairly easy to obtain. The structure of the book is as follows. First the nature of unbounded iii problems will be discussed. Next the simplest, static, problems will be con- sidered. Static problems are here taken to be problems which do not change with time. Examples are steady state solutions of Laplace’s equation (gov- erning heat flow, ideal fluid flow, seepage etc.), and steady problems of elasticity and viscous flow. There is no time dependence and usually the so- lution tends monotonically to the value at infinity with increasing distance. For many of these problems the Green’s function is known. The historical development of infinite elements for such problems will be described. Then separate chapters will be devoted to the two main types of static infinite element, decay function infinite elements and mapped infinite elements. As well as truly static problems, such elements can also be used in ‘added mass’ type problems. Next come problems in which only the first time der ivative occurs. The model equation governs transient heat conduction and soil consolidation for example. It turns out that methods for static problems are also applicable here. The last two classes of problem involve governing equations which contain second derivatives of time. This covers many wave problems. These can be subdivided into those which exhibit periodic behaviour and those which are completely transient. Strictly speaking we can talk of two types of periodic behaviour, periodic, in which all values arrive back at their initial value after a period, of time, T, and harmonic, in which the time dependence is of the form cosωt, sinωt or expiωt, where the angular frequency, ω = 2π/T. Such problems include the diffraction of waves by fixed objects. The types of wave include elastic waves, surface and pressure waves in fluids and electromagnetic waves. Again decay function and mapped infinite elements have been used successfully, but in general in the wave problem things are more complicated. Another useful method here is the wave envelope infinite element. Finally we have totally transient problems, involving second time deriva- tives, typified by shock waves. In this case infinite elements do not appear to be directly applicable, although some interesting work has been done. No doubt the book contains many errors and omissions. I would be most grateful if readers would point them out to me. Also if you publish anything on finite elements please send me a reprint. iv Acknowledgements Many people have helped me with infinite elements and before that with the finite element method. My first lecturer on this topic was R. T. Severn, and I benefitted from my master’s degree supervision at Imperial College, under A. C. Cassell and the late J. R. H. Otter. I continued to learn at Durham University where I was supervised by G. M. Parton and G. R. Higginson. At the British Ship Research Association my mentors were G. Ward and P. W. Knaggs, and I worked with G. Caveney. After my arrival at Swansea, I gained greatly from the company of my colleagues, in what I still believe is the best centre for finite element research in the world, and which has never had the national backing which it deserves. My departmental colleagues at Swansea were: R. F. Allen, N. Bi´cani´c, J. D. Davies, H. E. Evans, the late A. Gorecki, E. Hinton, D. W. Kelly, R. W. Lewis, R. L¨ohner, A. R. Lux- moore, J. Middleton, K. Morgan, D. J. Naylor, D. R. J. Owen, J. Peraire, K. G. Stagg, C. Taylor, and R. D. Wood. It was a pleasure and a privilege to work with them, and with colleagues from other departments. I also owe a debt to those of my research students and research associates who workedwithmeoninfiniteelements, K.Bando, C.R.I.Emson, H.Hara, S.- C. Liang and more recently P. J. Clark and Christine Barbier, who helped to generate the mapping functions described in Chapter 4 automatically, using computer algebra. Other collaborators have helped me, including R. J. Astley, T. C. Chiam and S. S. Saini and I have had very useful discus- sions with G. Beer, J. M. M. C. Marques, F. Medina, R. Ohayon, B. Peseux, J.-P. Quevat, P. M. Roberts and R. L. Taylor. I also benefitted from the continuous stream of distinguished visitors to Swansea, too numerous to mention here. I found time to do some of the work on infinite elements on visits to E.N.S.M. at Nantes and to N.T.H., Trondheim. I am grateful to these institutions for invitations to visit them and for the time to think about this topic. The biggest influence on me has obviously been that of O. C. Zienkiewicz, who when I came to him with the idea of infinite elements in 1974, encour- aged me to develop it, and collaborated with me in this process. I hope that his extensive contribution comes over in the text. Over the years he has taught me a great deal, and not only about finite elements. I am proud v to count him not only as a colleague, but as a friend. Finally, and most importantly, my wife, Jackie, has been a great support to me, not only in being a loving companion, and giving me the incalculable benefit of a happy home, but with help in all aspects of the infinite element work from the theory, through the programming, the debugging, the plot- ting of results, to the word processing of our joint papers. Without her, I could not have written this book, and existence itself would have been unendurable. vi ∞∞∞∞∞∞∞∞∞∞ These difficulties are real ... But let us remember that we are dealing with infinities and indivisibles, both of which transcend our finite understanding ... In spite of this men cannot refrain from discussing them. Galileo The infinite exists in the imagination: not the object of know- ing imagination but of imagination that is uncertain about its object, suspends further thinking and calls infinite all that it abandons. Just as sight recognises darkness by the experience of not seeing so imagination recognises the infinite by not understanding it. Proclus Cantor, havingprovedthattheinfinityofpointswithinasquare is equal to the infinity of points on one of its sides, wrote to his friend Dedekind: ‘I see, but I do not believe it.’ By adding continuously to a finite size I will pass any limited size. By subtracting, I will in the same way leave one which is smaller than any other.’ Aristotle Willst du ins Unendliche schreiten, Geh nur im Endlichen nach allen seiten. (If to the infinite you want to stride, Just walk to the Finite to every side.) Goethe ∞∞∞∞∞∞∞∞∞∞ vii viii
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