The Mathematics of Finite Elements and Applications. Proceedings of the Brunel University Conference of the Institute of Mathematics and its Applications Held in April 1972 PDF
Preview The Mathematics of Finite Elements and Applications. Proceedings of the Brunel University Conference of the Institute of Mathematics and its Applications Held in April 1972
THE MATHEMATICS OF FINITE ELEMENTS AND APPLICATIONS Proceedings of the Brunei University conference of the Institute of Mathematics and Its Applications held in April 1972 Edited by J. R. WHITEMAN Department of Mathematics Brunei University Oxbridge, Middlesex, England 1973 @ ACADEMIC PRESS · LONDON AND NEW YORK A subsidiary of Harcourt Brace Jooanovich, publishers ACADEMIC PRESS INC. (LONDON) LTD. 24/28 Oval Road London NW1 United States Edition published by ACADEMIC PRESS INC. Ill Fifth Avenue New York, New York 10003 Copyright © 1973 by The Institute of Mathematics and Its Applications All Rights Reserved No part of this book may be reproduced in any form by photostat, microfilm, or any other means, without written permission from the publishers Library of Congress Catalog Card Number: 72-7713 ISBN: 0-12-747250-9 PRINTED IN GREAT BRITAIN BY ROYSTAN PRINTERS LIMITED Spencer Court, 7 Chalcot Road London NW1 Contributors J. E. AKIN; Department of Engineering Mechanics, University of Tennessee, Knoxville, Tennessee 37916, U.S.A. A. ALUJEVIC; Nuclear Power Section, Department of Mechanical Engine- ering, Imperial College of Science and Technology, Exhibition Road, London, S.W.7, England. A. J. BARNARD; Department of Transport Technology, Loughborough University of Technology, Loughborough, Leicestershire LEU 3 TU, England. R. E. BARNHILL; Department of Mathematics, Universtiy of Utah, Salt Lake City, Utah 84112, U.S.A. W. S. BLACKBURN; International Research and Development Co., Fossway, Newcastle Upon Tyne 6, England. G. B. BOLLAND; Department of Mechanical Engineering, Lanchester Polytechnic, Priory Street, Coventry CV1 5FB, England. J. C. BRUCH; Department of Mechanical Engineering, University of California, Santa Barbara, California 93106. U.S.A. I. G. CAMERON; United Kingdom Atomic Energy Authority, Atomic Weapons Research Establishment, Aldermaston, Reading RG7 4PR, England. W. C. CARPENTER; Department of Engineering Science, University of Durham, Durham, England. P. G. CIARLET; Laboratoire Central des Ponts et Chaussées, 58, Boulevard Lefebvre, 75 Paris 15, France. Z. CSENDES; Department of Electrical Engineering, McGill University, P.O. Box 6070, Montreal 101, Quebec, Canada. P. DALY; Department of Electrical and Electronic Engineering, University of Leeds, Leeds LS2 9JT, England. D. J. EVANS; Department of Mathematics, Loughborough University of Technology, Loughborough, Leicestershire LEU 3 TU, England. C. C. FLEISCHER; Institute of Sound and Vibration Research, University of Southampton, Southampton S09 5NH, England. v VI CONTRIBUTORS P. W. FRANCE; Department of Civil Engineering, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, Wales. M. FREMOND; Laboratoire Central des Ponts et Chaussées, 58, Boulevard Lefebvre, 75 Paris 15, France. I. FRIED; Department of Mathematics, Boston University, Boston, Massachusetts 02215, U.S.A. P. A. T. GILL; Department of Engineering Science, University of Durham, Durham, England. A. GOPINATH; Department of Electronics, University College of North Wales, Bangor, Wales. 3. L. HEAD; Nuclear Power Section, Department of Mechanical Engine- ering, Imperial College of Science and Technology, Exhibition Road, London, S.W.7., England. U.HEISE; Lehrstuhl für Technische Mechanik, Technische Hochschule Aachen, 51 Aachen, Templergraben 55, West Germany. R. D. HENSHELL; Department of Mechanical Engineering, University of Nottingham, Nottingham NG7 2RD, England. V. HOPPE; Burmeister and Wain's Motor-og Maskinfabrik, Torvegade 2, DK 1449 Copenhagen K, Denmark. y A. JEZERNIK; Central Electricity Generating Board, Computing Bureau, 85, Park Street, London, S.E.I., England. A. V. KRISHNA MURTY; Department of Aeronautical Engineering, Indian Institute of Science, Bangalore 12, India. A. LEECH; Central Electricity Generating Board, Computing Bureau, 85, Park Street, London,, S.E.I., England. A. C. LOCK; Department of Civil and Structural Engineering, University College Cardiff, P.O. Box 78, Cardiff CF1 1XL, Wales. A. R. MITCHELL; Department of Mathematics, University of Dundee, Dundee DD1 4HN, Scotland. B. NOBLE; Oxford University Computing Laboratory, 19, Parks Road, Oxford 0X1 3PL, England. J. T. ODEN; Department of Engineering Mechanics, Division of Engine- ering, University of Alabama, P.O. Box 1274, Huntsville, Alabama 35807, U.S.A. C. PATTERSON; Department of Mechanical Engineering, University of Sheffield, Sheffield SI 3JD, England. M. PETYT; Institute of Sound and Vibration Research, University of Southampton, Southampton S09 5NH, England. CONTRIBUTORS VÜ A. B. SABIR; Department of Civil and Structural Engineering, University College Cardiff, P.O. Box 78, Cardiff CF1 1XL, Wales. G. H. SCHMIDT; Mathematisch Instituut, Rijksuniversiteit te Groningen, Postbus 800, Groningen, Nederlands. L. H. SEITELMAN; Pratt and Whitney Aircraft, East Hartford, Connecticut 06108, U.S.A. P. SILVESTER; Department of Electrical Engineering, McGill University, P.O. Box 6070, Montreal 101, Quebec, Canada. C. J. TAYLOR; Department of Civil Engineering, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, Wales. O. TINGLEFF; Institute for Numerical Analysis, Technical University of Denmark, 2800 Lyngby, Denmark. G. VENKATESWARA RAO; Department of Aeronautical Engineering, Indian Institute of Science, Bangalore 12, India, J. R. WHITEMAN; Department of Mathematics, Brunei University, Uxbridge UB8 3PH, Middlesex, England. O. C. ZIENKIEWICZ; Department of Civil Engineering, University of Wales Swansea, Singleton Park, Swansea, SA2 8PP, Wales. M. ZLÄMAL; Computing Centre, Technical University, L.P.S., Obrancu Miru 21, Brno, Czechoslovakia. G. ZYVOLOSKI; Department of Mechanical Engineering, University of California, Santa Barbara, California 93106, U.S.A. Preface This book contains the papers presented at the conference with the same title held at Brunei University for the period 18-20 April, 1972, which I organised in conjunction with the Institute of Mathematics and Its Applica- tions. The meeting was attended by over 240 participants of whom approx- imately half came from outside the U.K. I was fortunate to have with me on the conference committee, whose main task was the selection of the con- tributed papers, J. Crank, L. Fox, E. T. Goodwin, K. L. Stewart, A. L. Yet- tram and O. C. Zienkiewicz. The field of finite elements is expanding at an unprecedented rate. However, the expansion of the mathematical theory on the one hand, and the engine- ering and physical applications on the other, have been taking place in more or less divorced situations with little interchange between disciplines. The tendency has been for the mathematics involved to become more and more abstract and for the theory and applications paths to diverge. Thus my original idea for the conference, which in the event was closely adhered to, was for numerical analysts, applied mathematicians, mathematical physicists, engine- ers and computer scientists to come together for the purpose of useful interchange, for there to be sessions of contributed papers, and for four speakers to be invited to give one hour expository talks. There were thus two such talks, one introductory and one a survey of recent advances, in each of the fields of the mathematical theory and the application of finite element methods. The invited mathematicians were A. R. Mitchell and M. Zlâmal and the engineers O. C. Zienkiewicz and J. T. Oden. The programme fell naturally into three sections: the mathematics of finite elements, applications, algorithms and computational techniques. In the selection of contributed papers the committee tried to ensure that as wide a range of applications as possible was covered. In editing this book one of my aims has been to keep as short as possible the time interval between the conference and the publication of the proceed- ings. I have not thought it my place to check the mathematics or the technical content of the papers, both of which are thus the responsibility of the authors. With authors from so many different countries, I have on occasion found it necessary to make grammatical changes to aid clarity. I should like to thank those authors concerned for the cheerful and charming way in which they IX X PREFACE have accepted my alterations. There are also inevitably variations in notation. These are marked in inner products for which many different definitions are given. I much enjoyed working with the Institute of Mathematics and Its Applications in running this conference, and my thanks are due to the Insti- tute for so ably arranging the financial and domestic details. I would also like to thank the Conference Committee, the Session Chairmen and the Speakers for their friendly cooperation. Finally, my thanks go to J. Crank and R. E. Barnhill for their wise counsel during the months prior to the meet- ing, to Mrs. D. Barnett for her dedicated secretarial help and to my wife and N. Papamichael for compiling the subject index for this book. J. R. WHITEMAN Brunei University January, 1973 FINITE ELEMENTS—THE BACKGROUND STORY O. C. ZlENKIEWICZ University of Wales, Swansea. 1. INTRODUCTION The engineer whose life and livelihood are concerned with quantifying information for his designs is naturally most conscious of the "numbers game". Mere lack of an elegant theory or suitable tools provided by the mathematician must not deter him from solving his problems. It is from such a background that important innovation often springs and subsequent theories develop. As with other areas of human endeavour, so with numerical methods, necessity becomes the mother of invention. Whether an invention can in fact be cast later as a rediscovery of facts already known but not before used is a matter of pedantry. It is in this context that the few historical remarks concerning the present development of the finite element method will be presented. Just as in practice it was Southwell who put the finite-difference method on the map in the early 1940s so in the finite element context the group of engineers working in the mid 1950s were the true originators of the process. In the subsequent sections we shall attempt to trace the stages of develop- ment of the finite element method and to give an indication of its present adaptability to the solution of various problems. It appears convenient to subdivide the progress chronologically into three eras. The first—"Mediaeval" period in which, as in history, "faith moved mountains !" The second—"Renaissance" period in which the classical models are recognised. Inevitably the third period—into which we are now entering is the "Baroque" in which decoration is added but in which also new suitable bases are sought. Perhaps, as in the historic and artistic periods, not only a process of natural evolution superseding the previous achievements is represented. In a sense we see here different ways of viewing the same subject—and some merit is to be recognised in each. 1 2 O. C. ZIENKIEWICZ Elements 77777 FIG. 1. Some discrete engineering problems and their 'elements' FINITE ELEMENTS—THE BACKGROUND STORY 3 THE 'MEDIAEVAL' PERIOD (FAITH MOVES MOUNTAINS) 2. THE DISCRETE PROBLEM Engineering problems often present a difficulty due not to their mathematical complexity, but simply caused by the number of individual components present. In a bar type structure or in an electric circuit (Fig. 1) a very large number of essentially simple elements is present and over the years a standard approach has been developed to deal with such cases. First a close look is taken at the nature of each individual component. "Input" and "output" variables are specified for each "element" at its connection points and a relationship between these established. In a matrix form we have Ψ = kV + F e 0 or (1) F,· = k /u/ + F * f 0i (implying summation for repeated indices) Here, for instance, F can stand for forces or currents in structural and electrical examples respectively, where u may represent displacements or voltages. Equations (1) imply linearity of behaviour—but extension to non-linear situations is evident. In the second stage connection properties are established (a) by identifying one set of variables, such as u, for the assembled system(continuity) U, = U/ (2) (b) equilibrating (summing) the second set of variables at each node and equating to zero Σ F<e = °- (3) Immediately the system equations are obtained as Ku + F = 0 (4) 0 with K = Ky = t V. Fo = F = Σ ΪΌΛ (4a) 0i which may be solved—even though large. The simple assembly properties of (4) are the feature which makes both the understanding and practice con- venient, and which we shall see carries over throughout the finite element process.
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