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The Finite Element Method
An Introduction with Partial Diﬀerential Equations
A. J. DAVIES
Professor of Mathematics University of Hertfordshire
Great Clarendon Street, Oxford ox2 6dp Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With oﬃces in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York c A. J. Davies 2011 The moral rights of the author have been asserted Database right Oxford University Press (maker) First Edition published 1980 Second Edition published 2011 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose the same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Data available Typeset by SPI Publisher Services, Pondicherry, India Printed in Great Britain by Ashford Colour Press Ltd, Gosport, Hampshire ISBN 978–0–19–960913–0 1 3 5 7 9 10 8 6 4 2
In the ﬁrst paragraph of the preface to the ﬁrst edition in 1980 I wrote: It is not easy, for the newcomer to the subject, to get into the current ﬁnite element literature. The purpose of this book is to oﬀer an introductory approach, after which the well-known texts should be easily accessible.
Writing now, in 2010, I feel that this is still largely the case. However, while the 1980 text was probably the only introductory text at that time, it is not the case now. I refer the interested reader to the references. In this second edition, I have maintained the general ethos of the ﬁrst. It is primarily a text for mathematicians, scientists and engineers who have no previous experience of ﬁnite elements. It has been written as an undergraduate text but will also be useful to postgraduates. It is also suitable for anybody already using large ﬁnite element or CAD/CAM packages and who would like to understand a little more of what is going on. The main aim is to provide an introduction to the ﬁnite element solution of problems posed as partial diﬀerential equations. It is self-contained in that it requires no previous knowledge of the subject. Familiarity with the mathematics normally covered by the end of the second year of undergraduate courses in mathematics, physical science or engineering is all that is assumed. In particular, matrix algebra and vector calculus are used extensively throughout; the necessary theorems from vector calculus are collected together in Appendix B. The reader familiar with the ﬁrst edition will notice some signiﬁcant changes. I now present the method as a numerical technique for the solution of partial diﬀerential equations, comparable with the ﬁnite diﬀerence method. This is in contrast to the ﬁrst edition, in which the technique was developed as an extension of the ideas of structural analysis. The only thing that remains of this approach is the terminology, for...