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The Finite Element Method

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The Finite Element Method An Introduction with Partial Differential Equations

Second Edition

A. J. DAVIES Professor of Mathematics University of Hertfordshire

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3 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

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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

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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

Preface

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 under- graduate 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 example ‘stiffness matrix’, since this is still in common parlance. The reader familiar with the first edition will notice a change in notation which reflects the move away from the structural background. There is also a change of order of chapters: the introduction to finite elements is now via weighted residual methods, variational methods being delayed until later. I have taken the opportunity to introduce a completely new chapter on boundary element methods. At the time of the first edition, such methods were in their infancy, but now they have reached such a stage of development that it is natural to include them; this chapter is by no means exhaustive and is very much an introduction. I have also included a brief chapter on computational aspects. This is also an introduction; the topic is far too large to treat in any depth. Again the interested reader can follow up the references. In the first edition, many of

vi The Finite Element Method

the examples and exercises were based on problems in journal papers of around that time. I have kept the original references in this second edition.

In Chapter 1, I have written an updated historical introduction and included many new references. Chapter 2 provides a background in weighted residual and variational methods. Chapter 3 describes the finite element method for Poisson’s equation, concentrating on linear elements. Higher-order elements and the isoparametric concept are introduced in Chapter 4.

Chapter 5 sets the finite element method in a variational context and intro- duces time-dependent and non-linear problems. Chapter 6 is almost identical with Chapter 7 of the first edition, the only change being the notation. Chapter 7 is the new chapter on the boundary element method, and Chapter 8, the final chapter, addresses the computational aspects. I have also expanded the appendices by including, in Appendix A, a brief description of some of the partial differential equation models in the physical sciences which are amenable to solution by the finite element method.

I have not changed the general approach of the first edition. At the end of each chapter is a set of exercises with detailed solutions. They serve two purposes: (i) to give the reader the opportunity to practice the techniques, and (ii) to develop the theory a little further where this does not require any new concepts; for example, the finite element solution of eigenvalue problems is considered in Exercise 3.24. Also, some of the basic theory of Chapter 6 is left to the exercises. Consequently, certain results of importance are to be found in the exercises and their solutions.

An important development over the past thirty years has been the wide availability of computational aids such as spreadsheets and computer algebra packages. In this edition, I have included examples of how a spreadsheet could be used to develop more sophisticated solutions compared with the ‘hand’ calculations in the first edition. Obviously, I would encourage readers to use whichever packages they have on their own personal computers.

Well, this second edition has been a long time coming; I’ve been working on it for quite some time. It has been confined to very concentrated two-week spells over the Easter periods for the past six years, these periods being spent with Margaret and Arthur, Les Meuniers, at their home in the Lot, south-west France. The environment there is ideal for the sort of focused work needed to produce this second edition. I am grateful for their friendship and, of course, their hospitality. The production of this edition would have been impossible without the help of Dr Diane Crann, my wife. I am very grateful for her expertise in turning my sometimes illegible handwritten script into OUP LATEX house style.

A.J.D. Lacombrade Sabadel Latronquière Lot August 2010

Contents

1 Historical introduction 1

2 Weighted residual and variational methods 7

2.1 Classification of differential operators 7 2.2 Self-adjoint positive definite operators 9 2.3 Weighted residual methods 12 2.4 Extremum formulation: homogeneous boundary

conditions 24 2.5 Non-homogeneous boundary conditions 28 2.6 Partial differential equations: natural boundary

conditions 32 2.7 The Rayleigh–Ritz method 35 2.8 The ‘elastic analogy’ for Poisson’s equation 44 2.9 Variational methods for time-dependent problems 48 2.10 Exercises and solutions 50

3 The finite element method for elliptic problems 71

3.1 Difficulties associated with the application of weighted residual methods 71

3.2 Piecewise application of the Galerkin method 72 3.3 Terminology 73 3.4 Finite element idealization 75 3.5 Illustrative problem involving one independent variable 80 3.6 Finite element equations for Poisson’s equation 91 3.7 A rectangular element for Poisson’s equation 102 3.8 A triangular element for Poisson’s equation 107 3.9 Exercises and solutions 114

4 Higher-order elements: the isoparametric concept 141

4.1 A two-point boundary-value problem 141 4.2 Higher-order rectangular elements 144 4.3 Higher-order triangular elements 145 4.4 Two degrees of freedom at each node 147 4.5 Condensation of internal nodal freedoms 151 4.6 Curved boundaries and higher-order elements: isoparametric

elements 153 4.7 Exercises and solutions 160

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5 Further topics in the finite element method 171

5.1 The variational approach 171 5.2 Collocation and least squares methods 177 5.3 Use of Galerkin’s method for time-depe