The Finite Element Method Fifth edition Volume 1: The Basis Professor O.C. Zienkiewicz, CBE, FRS ppt

2.2 Direct formulation of ®nite element characteristics 19 2.4 Displacement approach as a minimization of total potential energy 29 2.7 Displacement functions with discontinuity between

The Finite Element Method

Fifth edition

Volume 1: The Basis

Professor O.C Zienkiewicz, CBE, FRS, FREng is Professor Emeritus and Director

of the Institute for Numerical Methods in Engineering at the University of Wales,Swansea, UK He holds the UNESCO Chair of Numerical Methods in Engineering

at the Technical University of Catalunya, Barcelona, Spain He was the head of theCivil Engineering Department at the University of Wales Swansea between 1961and 1989 He established that department as one of the primary centres of ®niteelement research In 1968 he became the Founder Editor of the International Journalfor Numerical Methods in Engineering which still remains today the major journal

in this ®eld The recipient of 24 honorary degrees and many medals, ProfessorZienkiewicz is also a member of ®ve academies ± an honour he has received for hismany contributions to the fundamental developments of the ®nite element method

In 1978, he became a Fellow of the Royal Society and the Royal Academy ofEngineering This was followed by his election as a foreign member to the U.S.Academy of Engineering (1981), the Polish Academy of Science (1985), the ChineseAcademy of Sciences (1998), and the National Academy of Science, Italy (Academiadei Lincei) (1999) He published the ®rst edition of this book in 1967 and it remainedthe only book on the subject until 1971

Professor R.L Taylor has more than 35 years' experience in the modelling and lation of structures and solid continua including two years in industry In 1991 he waselected to membership in the U.S National Academy of Engineering in recognition ofhis educational and research contributions to the ®eld of computational mechanics

simu-He was appointed as the T.Y and Margaret Lin Professor of Engineering in 1992and, in 1994, received the Berkeley Citation, the highest honour awarded by theUniversity of California, Berkeley In 1997, Professor Taylor was made a Fellow inthe U.S Association for Computational Mechanics and recently he was electedFellow in the International Association of Computational Mechanics, and wasawarded the USACM John von Neumann Medal Professor Taylor has written sev-eral computer programs for ®nite element analysis of structural and non-structuralsystems, one of which, FEAP, is used world-wide in education and research environ-ments FEAP is now incorporated more fully into the book to address non-linear and

®nite deformation problems

Front cover image: A Finite Element Model of the world land speed record (765.035 mph) car THRUST SSC The analysis was done using the ®nite element method by K Morgan, O Hassan and N.P Weatherill

at the Institute for Numerical Methods in Engineering, University of Wales Swansea, UK (see K Morgan,

O Hassan and N.P Weatherill, `Why didn't the supersonic car ¯y?', Mathematics Today, Bulletin of the Institute of Mathematics and Its Applications, Vol 35, No 4, 110±114, Aug 1999).

The Finite Element

Method Fifth edition Volume 1: The Basis

O.C Zienkiewicz, CBE, FRS, FREng UNESCO Professor of Numerical Methods in Engineering

International Centre for Numerical Methods in Engineering, Barcelona Emeritus Professor of Civil Engineering and Director of the Institute for Numerical Methods in Engineering, University of Wales, Swansea

R.L Taylor Professor in the Graduate School Department of Civil and Environmental Engineering

University of California at Berkeley

Berkeley, California

OXFORD AUCKLAND BOSTON JOHANNESBURG MELBOURNE NEW DELHI

Butterworth-HeinemannLinacre House, Jordan Hill, Oxford OX2 8DP

225 Wildwood Avenue, Woburn, MA 01801-2041

A division of Reed Educational and Professional Publishing Ltd

First published in 1967 by McGraw-HillFifth edition published by Butterworth-Heinemann 2000

# O.C Zienkiewicz and R.L Taylor 2000

All rights reserved No part of this publication

may be reproduced in any material form (including

photocopying or storing in any medium by electronic

means and whether or not transiently or incidentally

to some other use of this publication) without the

written permission of the copyright holder except

in accordance with the provisions of the Copyright,

Designs and Patents Act 1988 or under the terms of a

licence issued by the Copyright Licensing Agency Ltd,

90 Tottenham Court Road, London, England W1P 9HE.

Applications for the copyright holder's written permission

to reproduce any part of this publication should

be addressed to the publishers

British Library Cataloguing in Publication Data

A catalogue record for this book is available from the British Library

Library of Congress Cataloguing in Publication Data

A catalogue record for this book is available from the Library of Congress

ISBN 0 7506 5049 4Published with the cooperation of CIMNE,

the International Centre for Numerical Methods in Engineering,

Barcelona, Spain (www.cimne.upc.es)

Typeset by Academic & Technical Typesetting, Bristol

Printed and bound by MPG Books Ltd

This book is dedicated to our wives Helen and Mary Lou and our families for their support and patience during the preparation of this book, and also to all of our students and colleagues who over the years have contributed to our knowledge of the ®nite element method In particular we would like to mention Professor Eugenio OnÄate and his group at CIMNE for their help, encouragement and support during the

preparation process.

2.2 Direct formulation of ®nite element characteristics 19

2.4 Displacement approach as a minimization of total potential energy 29

2.7 Displacement functions with discontinuity between elements 332.8 Bound on strain energy in a displacement formulation 34

3 Generalization of the ®nite element concepts Galerkin-weighted residual

3.2 Integral or `weak' statements equivalent to the di€erential equations 42

3.4 Virtual work as the `weak form' of equilibrium equations for

3.5 Partial discretization 55

3.8 `Natural' variational principles and their relation to governing

3.9 Establishment of natural variational principles for linear,

3.11 Constrained variational principles Lagrange multipliers and

4.5 Special treatment of plane strain with an incompressible material 110

7 Steady-state ®eld problems ± heat conduction, electric and magnetic

viii Contents

8.3 Rectangular elements ± some preliminary considerations 168

8.7 Elimination of internal variables before assembly ± substructures 177

8.15 Two- and three-dimensional, hierarchic, elements of the `rectangle'

8.18 Improvement of conditioning with hierarchic forms 197

9.4 Variation of the unknown function within distorted, curvilinear

9.5 Evaluation of element matrices (transformation in , , 

9.7 Convergence of elements in curvilinear coordinates 213

9.9 Numerical integration ± rectangular (2D) or right prism (3D)

9.10 Numerical integration ± triangular or tetrahedral regions 221

9.12 Generation of ®nite element meshes by mapping Blending functions 226

9.14 Singular elements by mapping for fracture mechanics, etc 234

Contents ix

9.15 A computational advantage of numerically integrated ®nite

9.16 Some practical examples of two-dimensional stress analysis 237

10.4 Generalized patch test (test C) and the single-element test 255

10.7 Application of the patch test to plane elasticity elements with

10.8 Application of the patch test to an incompatible element 26410.9 Generation of incompatible shape functions which satisfy the

10.11 Higher order patch test ± assessment of robustness 271

11.6 An iterative method solution of mixed approximations 298

11.8 Concluding remarks ± mixed formulation or a test of element

12.2 Deviatoric stress and strain, pressure and volume change 307

12.4 Three-®eld nearly incompressible elasticity (u±p±"vform) 31412.5 Reduced and selective integration and its equivalence to penalized

12.6 A simple iterative solution process for mixed problems: Uzawa

x Contents

12.7 Stabilized methods for some mixed elements failing the

13.3 Interface traction link of two or more mixed form subdomains 349

13.5 Linking of boundary (or Tre€tz)-type solution by the `frame' of

13.6 Subdomains with `standard' elements and global functions 360

13.7 Lagrange variables or discontinuous Galerkin methods? 361

14.8 Asymptotic behaviour and robustness of error estimators ± the

16.4 Hierarchical enhancement of moving least square expansions 443

Contents xi

16.6 Galerkin weighting and ®nite volume methods 45116.7 Use of hierarchic and special functions based on standard ®nite

elements satisfying the partition of unity requirement 457

17 The time dimension ± semi-discretization of ®eld and dynamic problems

18.5 Some remarks on general performance of numerical algorithms 530

19.4 Partitioned single-phase systems ± implicit±explicit partitions

20.4 Solution module ± the command programming language 590

xii Contents

20.6 Solution of simultaneous linear algebraic equations 609

20.7 Extension and modi®cation of computer program FEAPpv 618

Appendix B: Tensor-indicial notation in the approximation of elasticity

Appendix G: Integration by parts in two and three dimensions

Contents xiii

Volume 2: Solid and structural mechanics

1 General problems in solid mechanics and non-linearity

2 Solution of non-linear algebraic equations

3 Inelastic materials

4 Plate bending approximation: thin (Kirchho€) plates andC1 continuity ments

require-5 `Thick' Reissner±Mindlin plates ± irreducible and mixed formulations

6 Shells as an assembly of ¯at elements

10 Geometrically non-linear problems ± ®nite deformation

11 Non-linear structural problems ± large displacement and instability

12 Pseudo-rigid and rigid±¯exible bodies

13 Computer procedures for ®nite element analysis

Appendix A: Invariants of second-order tensors

Volume 3: Fluid dynamics

1 Introduction and the equations of ¯uid dynamics

2 Convection dominated problems ± ®nite element approximations

3 A general algorithm for compressible and incompressible ¯ows ± the characteristicbased split (CBS) algorithm

4 Incompressible laminar ¯ow ± newtonian and non-newtonian ¯uids

5 Free surfaces, buoyancy and turbulent incompressible ¯ows

6 Compressible high speed gas ¯ow

7 Shallow-water problems

8 Waves

9 Computer implementation of the CBS algorithm

Appendix A Non-conservative form of Navier±Stokes equations

Appendix B Discontinuous Galerkin methods in the solution of the convection±

di€usion equation

Appendix C Edge-based ®nite element formulation

Appendix D Multi grid methods

Appendix E Boundary layer ± inviscid ¯ow coupling

It is just over thirty years since The Finite Element Method in Structural andContinuum Mechanics was ®rst published This book, which was the ®rst dealingwith the ®nite element method, provided the base from which many further develop-ments occurred The expanding research and ®eld of application of ®nite elements led

to the second edition in 1971, the third in 1977 and the fourth in 1989 and 1991 Thesize of each of these volumes expanded geometrically (from 272 pages in 1967 to thefourth edition of 1455 pages in two volumes) This was necessary to do justice to arapidly expanding ®eld of professional application and research Even so, much ®lter-ing of the contents was necessary to keep these editions within reasonable bounds

It seems that a new edition is necessary every decade as the subject is expanding andmany important developments are continuously occurring The present ®fth edition isindeed motivated by several important developments which have occurred in the 90s.These include such subjects as adaptive error control, meshless and point basedmethods, new approaches to ¯uid dynamics, etc However, we feel it is importantnot to increase further the overall size of the book and we therefore have eliminatedsome redundant material

Further, the reader will notice the present subdivision into three volumes, in which the

®rst volume provides the general basis applicable to linear problems in many ®elds whilstthe second and third volumes are devoted to more advanced topics in solid and ¯uidmechanics, respectively This arrangement will allow a general student to studyVolume 1 whilst a specialist can approach their topics with the help of Volumes 2 and

3 Volumes 2 and 3 are much smaller in size and addressed to more specialized readers

It is hoped that Volume 1 will help to introduce postgraduate students, researchersand practitioners to the modern concepts of ®nite element methods In Volume 1 westress the relationship between the ®nite element method and the more classic ®nitedi€erence and boundary solution methods We show that all methods of numericalapproximation can be cast in the same format and that their individual advantagescan thus be retained

Although Volume 1 is not written as a course text book, it is nevertheless directed atstudents of postgraduate level and we hope these will ®nd it to be of wide use Math-ematical concepts are stressed throughout and precision is maintained, although littleuse is made of modern mathematical symbols to ensure wider understanding amongstengineers and physical scientists

In Volumes 1, 2 and 3 the chapters on computational methods are much reduced bytransferring the computer source programs to a web site.1This has the very substan-tial advantage of not only eliminating errors in copying the programs but also inensuring that the reader has the bene®t of the most recent set of programs available

to him or her at all times as it is our intention from time to time to update and expandthe available programs

The authors are particularly indebted to the International Center of NumericalMethods in Engineering (CIMNE) in Barcelona who have allowed their pre- andpost-processing code (GiD) to be accessed from the publisher's web site Thisallows such dicult tasks as mesh generation and graphic output to be dealt witheciently The authors are also grateful to Dr J.Z Zhu for his careful scrutiny andhelp in drafting Chapters 14 and 15 These deal with error estimation and adaptivity,

a subject to which Dr Zhu has extensively contributed Finally, we thank Peter andJackie Bettess for writing the general subject index

OCZ and RLT

1 Complete source code for all programs in the three volumes may be obtained at no cost from the publisher's web page: http://www.bh.com/companions/fem

xvi Preface

is readily understood, and then rebuilding the original system from such components

to study its behaviour is a natural way in which the engineer, the scientist, or even theeconomist proceeds

In many situations an adequate model is obtained using a ®nite number of de®ned components We shall term such problems discrete In others the subdivision

well-is continued inde®nitely and the problem can only be de®ned using the mathematical

®ction of an in®nitesimal This leads to di€erential equations or equivalent statementswhich imply an in®nite number of elements We shall term such systems continuous.With the advent of digital computers, discrete problems can generally be solvedreadily even if the number of elements is very large As the capacity of all computers

is ®nite, continuous problems can only be solved exactly by mathematical tion Here, the available mathematical techniques usually limit the possibilities tooversimpli®ed situations

manipula-To overcome the intractability of realistic types of continuum problems, variousmethods of discretization have from time to time been proposed both by engineersand mathematicians All involve an approximation which, hopefully, approaches

in the limit the true continuum solution as the number of discrete variablesincreases

The discretization of continuous problems has been approached di€erently bymathematicians and engineers Mathematicians have developed general techniquesapplicable directly to di€erential equations governing the problem, such as ®nite dif-ference approximations,1;2 various weighted residual procedures,3;4 or approximatetechniques for determining the stationarity of properly de®ned `functionals' Theengineer, on the other hand, often approaches the problem more intuitively by creat-ing an analogy between real discrete elements and ®nite portions of a continuumdomain For instance, in the ®eld of solid mechanics McHenry,5 Hreniko€,6Newmark7, and indeed Southwell9in the 1940s, showed that reasonably good solu-tions to an elastic continuum problem can be obtained by replacing small portions

of the continuum by an arrangement of simple elastic bars Later, in the same context,Argyris8and Turner et al.9showed that a more direct, but no less intuitive, substitu-tion of properties can be made much more e€ectively by considering that smallportions or `elements' in a continuum behave in a simpli®ed manner.

It is from the engineering `direct analogy' view that the term `®nite element' wasborn Clough10 appears to be the ®rst to use this term, which implies in it a directuse of a standard methodology applicable to discrete systems Both conceptually andfrom the computational viewpoint, this is of the utmost importance The ®rstallows an improved understanding to be obtained; the second o€ers a uni®edapproach to the variety of problems and the development of standard computationalprocedures

Since the early 1960s much progress has been made, and today the purely matical and `analogy' approaches are fully reconciled It is the object of this text topresent a view of the ®nite element method as a general discretization procedure of con-tinuum problems posed by mathematically de®ned statements

mathe-In the analysis of problems of a discrete nature, a standard methodology has beendeveloped over the years The civil engineer, dealing with structures, ®rst calculatesforce±displacement relationships for each element of the structure and then proceeds

to assemble the whole by following a well-de®ned procedure of establishing localequilibrium at each `node' or connecting point of the structure The resulting equa-tions can be solved for the unknown displacements Similarly, the electrical orhydraulic engineer, dealing with a network of electrical components (resistors, capa-citances, etc.) or hydraulic conduits, ®rst establishes a relationship between currents(¯ows) and potentials for individual elements and then proceeds to assemble thesystem by ensuring continuity of ¯ows

All such analyses follow a standard pattern which is universally adaptable to crete systems It is thus possible to de®ne a standard discrete system, and this chapterwill be primarily concerned with establishing the processes applicable to such systems.Much of what is presented here will be known to engineers, but some reiteration atthis stage is advisable As the treatment of elastic solid structures has been themost developed area of activity this will be introduced ®rst, followed by examplesfrom other ®elds, before attempting a complete generalization

dis-The existence of a uni®ed treatment of `standard discrete problems' leads us to the

®rst de®nition of the ®nite element process as a method of approximation to tinuum problems such that

con-(a) the continuum is divided into a ®nite number of parts (elements), the behaviour ofwhich is speci®ed by a ®nite number of parameters, and

(b) the solution of the complete system as an assembly of its elements follows cisely the same rules as those applicable to standard discrete problems

pre-It will be found that most classical mathematical approximation procedures as well

as the various direct approximations used in engineering fall into this category It isthus dicult to determine the origins of the ®nite element method and the precisemoment of its invention

Table 1.1 shows the process of evolution which led to the present-day concepts of

®nite element analysis Chapter 3 will give, in more detail, the mathematical basiswhich emerged from these classical ideas.11ÿ20

2 Some preliminaries: the standard discrete system

Ritz 1909 12

Weighted residuals Gauss 1795 18

Prager±Synge 1947 14

Zienkiewicz 1964 21

Direct continuum elements Argyris 1955 8

Turner et al 1956 9 ÿÿ"

finite differences Varga 1962 17

PRESENT-DAY FINITE ELEMENT METHOD

1.2 The structural element and the structural system

To introduce the reader to the general concept of discrete systems we shall ®rstconsider a structural engineering example of linear elasticity

Figure 1.1 represents a two-dimensional structure assembled from individualcomponents and interconnected at the nodes numbered 1 to 6 The joints at thenodes, in this case, are pinned so that moments cannot be transmitted

As a starting point it will be assumed that by separate calculation, or for that matterfrom the results of an experiment, the characteristics of each element are preciselyknown Thus, if a typical element labelled (1) and associated with nodes 1, 2, 3 isexamined, the forces acting at the nodes are uniquely de®ned by the displacements

of these nodes, the distributed loading acting on the element (p), and its initialstrain The last may be due to temperature, shrinkage, or simply an initial `lack of

®t' The forces and the corresponding displacements are de®ned by appropriate ponents (U, V and u, v) in a common coordinate system

com-Listing the forces acting on all the nodes (three in the case illustrated) of the element(1) as a matrixy we have

q1 ˆ

q1 1

y A limited knowledge of matrix algebra will be assumed throughout this book This is necessary for reasonable conciseness and forms a convenient book-keeping form For readers not familiar with the subject

a brief appendix (Appendix A) is included in which sucient principles of matrix algebra are given to follow the development intelligently Matrices (and vectors) will be distinguished by bold print throughout.

4 Some preliminaries: the standard discrete system

and for the corresponding nodal displacements

Assuming linear elastic behaviour of the element, the characteristic relationship will

always be of the form

q1ˆ K1a1‡ f1

p‡ f1

in which f1represents the nodal forces required to balance any distributed loads acting

on the element and f1"0 the nodal forces required to balance any initial strains such as

may be caused by temperature change if the nodes are not subject to any displacement

The ®rst of the terms represents the forces induced by displacement of the nodes

Similarly, a preliminary analysis or experiment will permit a unique de®nition of

stresses or internal reactions at any speci®ed point or points of the element in

terms of the nodal displacements De®ning such stresses by a matrix r1a relationship

of the form

is obtained in which the two term gives the stresses due to the initial strains when no

nodal displacement occurs

The matrix Keis known as the element sti€ness matrix and the matrix Qeas the

element stress matrix for an element (e)

Relationships in Eqs (1.3) and (1.4) have been illustrated by an example of an

ele-ment with three nodes and with the interconnection points capable of transmitting

only two components of force Clearly, the same arguments and de®nitions will

apply generally An element (2) of the hypothetical structure will possess only two

points of interconnection; others may have quite a large number of such points

Simi-larly, if the joints were considered as rigid, three components of generalized force and

of generalized displacement would have to be considered, the last of these

correspond-ing to a moment and a rotation respectively For a rigidly jointed, three-dimensional

structure the number of individual nodal components would be six Quite generally,

therefore,

qeˆ

qe 1

qe2

with each qei and aipossessing the same number of components or degrees of freedom

These quantities are conjugate to each other

The sti€ness matrices of the element will clearly always be square and of the form

3

The structural element and the structural system 5

in which Keii, etc., are submatrices which are again square and of the size l  l, where l

is the number of force components to be considered at each node

As an example, the reader can consider a pin-ended bar of uniform section A andmodulus E in a two-dimensional problem shown in Fig 1.2 The bar is subject to auniform lateral load p and a uniform thermal expansion strain

"0 ˆ Twhere is the coecient of linear expansion and T is the temperature change

If the ends of the bar are de®ned by the coordinates xi, yiand xn, ynits length can becalculated as

L ˆq‰…xnÿ xi†2‡ …ynÿ yi†2Šand its inclination from the horizontal as

ˆ tanÿ1xynÿ yi

nÿ xiOnly two components of force and displacement have to be considered at thenodes

The nodal forces due to the lateral load are clearly

ˆ ÿ

ÿsin cos ÿsin cos

and represent the appropriate components of simple reactions, pL=2 Similarly, torestrain the thermal expansion "0 an axial force …ETA† is needed, which gives the

L

n

i y

Fig 1.2 A pin-ended bar

6 Some preliminaries: the standard discrete system

EA=L, gives the axial force whose components can again be found Rearranging

these in the standard form gives

The components of the general equation (1.3) have thus been established for the

elementary case discussed It is again quite simple to ®nd the stresses at any section

of the element in the form of relation (1.4) For instance, if attention is focused on

the mid-section C of the bar the average stress determined from the axial tension

to the element can be shown to be

re  ˆE

L‰ÿcos ; ÿsin ; cos ; sin Šaeÿ ETwhere all the bending e€ects of the lateral load p have been ignored

For more complex elements more sophisticated procedures of analysis are required

but the results are of the same form The engineer will readily recognize that the

so-called `slope±de¯ection' relations used in analysis of rigid frames are only a special

case of the general relations

It may perhaps be remarked, in passing, that the complete sti€ness matrix obtained

for the simple element in tension turns out to be symmetric (as indeed was the case

with some submatrices) This is by no means fortuitous but follows from the principle

of energy conservation and from its corollary, the well-known Maxwell±Betti

reciprocal theorem

ˆEAL

cos2 ...

of the element in the form of relation (1.4) For instance, if attention is focused on

the mid-section C of the bar the average stress determined from the axial tension

to the element. .. discrete elements are interconnected Thesemay be of structural, electrical, or any other linear type In the solution:

The ®rst step is the determination of element properties from the geometric...

the storage of the elements within the upper half of the pro®le is necessary, as

shown in Fig 1.4(c)

The third step is the insertion of prescribed boundary conditions into the