The Finite Element Method: Its Basis and Fundamentals, Sixth Edition

803 Pages·2005·7.19 MB·English

by  O. C. Zienkiewicz

The Finite Element
Method: Its Basis and
Fundamentals
Sixth edition
O.C. Zienkiewicz, CBE, FRS
UNESCOProfessorofNumericalMethodsinEngineering
InternationalCentreforNumericalMethodsinEngineering,Barcelona
PreviouslyDirectoroftheInstituteforNumericalMethodsinEngineering
UniversityofWales,Swansea
R.L. Taylor J.Z. Zhu
ProfessorintheGraduateSchool SeniorScientist
DepartmentofCivilandEnvironmentalEngineering ESIUSR&DInc.
UniversityofCaliforniaatBerkeley 5850WaterlooRoad,Suite140
Berkeley,California Columbia,Maryland
AMSTERDAM•BOSTON•HEIDELBERG•LONDON•NEWYORK•OXFORD
PARIS•SANDIEGO•SANFRANCISCO•SINGAPORE•SYDNEY•TOKYO
ElsevierButterworth-Heinemann
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Firstpublishedin1967byMcGraw-Hill
FiftheditionpublishedbyButterworth-Heinemann2000
Reprinted2002
Sixthedition2005
Copyright(cid:1)c 2000,2005. O.C.Zienkiewicz,R.L.TaylorandJ.Z.Zhu.Allrightsreserved
TherightsofO.C.Zienkiewicz,R.L.TaylorandJ.Z.Zhutobeidentifiedastheauthorsofthiswork
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Dedication
ThisbookisdedicatedtoourwivesHelen,MaryLouand
Songandourfamiliesfortheirsupportandpatienceduring
thepreparationofthisbook,andalsotoallofourstudents
andcolleagueswhoovertheyearshavecontributedtoour
knowledgeofthefiniteelementmethod. Inparticularwe
wouldliketomentionProfessorEugenioOn˜ateandhis
groupatCIMNEfortheirhelp,encouragementandsupport
duringthepreparationprocess.
Preface
Itisthirty-eightyearssincetheTheFiniteElementMethodinStructuralandContinuum
Mechanics was first published. This book, which was the first dealing with the finite
elementmethod,providedthebasisfromwhichmanyfurtherdevelopmentsoccurred. The
expandingresearchandfieldofapplicationoffiniteelementsledtothesecondeditionin
1971,thethirdin1977,thefourthastwovolumesin1989and1991andthefifthasthree
volumes in 2000. The size of each of these editions expanded geometrically (from 272
pages in 1967 to the fifth edition of 1482 pages). This was necessary to do justice to a
rapidlyexpandingfieldofprofessionalapplicationandresearch. Evenso,muchfiltering
ofthecontentswasnecessarytokeeptheseeditionswithinreasonablebounds.
Inthepresenteditionwehavedecidednottopursuethecourseofhavingthreecontiguous
volumesbutratherwetreatthewholeworkasanassemblyofthreeseparateworks,each
onecapableofbeingusedwithouttheothersandeachoneappealingperhapstoadifferent
audience. Thoughnaturallywerecommendtheuseofthewholeensembletopeoplewishing
todevotemuchoftheirtimeandstudytothefiniteelementmethod.
InparticularthefirstvolumewhichwasentitledTheFiniteElementMethod: TheBasis
is now renamed The Finite Element Method: Its Basis and Fundamentals. This volume
has been considerably reorganized from the previous one and is now, we believe, better
suitedforteachingfundamentalsofthefiniteelementmethod. Thesequenceofchapters
hasbeensomewhatalteredandseveralexamplesofworkedproblemshavebeenaddedto
thetext. Asetofproblemstobeworkedoutbystudentshasalsobeenprovided.
Inadditiontoitspreviouscontentthisbookhasbeenconsiderablyenlargedbyincluding
more emphasis on use of higher order shape functions in formulation of problems and a
newchapterdevotedtothesubjectofautomaticmeshgeneration. Abeginnerinthefinite
elementfieldwillfindveryrapidlythatmuchoftheworkofsolvingproblemsconsistsof
preparingasuitablemeshtodealwiththewholeproblemandasthesizeofcomputershas
seemedtoincreasewithoutlimitsthesizeofproblemscapableofbeingdealtwithisalso
increasing. Thus,meshescontainingsometimesmorethanseveralmillionnodeshavetobe
preparedwithdetailsofthematerialinterfaces,boundariesandloadsbeingwellspecified.
Therearemanybooksdevotedexclusivelytothesubjectofmeshgenerationbutwefeel
that the essence of dealing with this difficult problem should be included here for those
whowishtohaveacomplete‘encyclopedic’knowledgeofthesubject.
xiv Preface
The chapter on computational methods is much reduced by transferring the computer
sourceprogramanduserinstructionstoawebsite.†Thishastheverysubstantialadvantage
ofnotonlyeliminatingerrorsinprogramandmanualbutalsoinensuringthatthereaders
havethebenefitofthemostrecentversionoftheprogramavailableatalltimes.
Thetwofurthervolumesformagainseparatebooksandherewefeelthatacompletely
differentaudiencewillusethem. ThefirstoftheseisentitledTheFiniteElementMethod
in Solid and Structural Mechanics and the second is a text entitled The Finite Element
MethodinFluidDynamics. Eachofthesetwovolumesisastandalonetextwhichprovides
thefullknowledgeofthesubjectforthosewhohaveacquiredanintroductiontothefinite
elementmethodthroughothertexts. Ofcoursetheviewpointoftheauthorsintroducedin
thisvolumewillbecontinuedbutitispossibletostartatadifferentpoint.
Weemphasizeherethefactthatallthreebooksstresstheimportanceofconsideringthe
finiteelementmethodasauniqueandwholebasisofapproachandthatitcontainsmanyof
theothernumericalanalysismethodsasspecialcases. Thus,imaginationandknowledge
shouldbecombinedbythereadersintheirendeavours.
TheauthorsareparticularlyindebtedtotheInternationalCenterofNumericalMethodsin
Engineering(CIMNE)inBarcelonawhohaveallowedtheirpre-andpost-processingcode
(GiD)tobeaccessedfromthewebsite. Thisallowssuchdifficulttasksasmeshgeneration
andgraphicoutputtobedealtwithefficiently. TheauthorsarealsogratefultoProfessors
EricKasperandJoseLuisPerez-Apariciofortheircarefulscrutinyoftheentiretextand
DrsJoaquimPeiro´ andC.K.Leefortheirreviewofthenewchapteronmeshgeneration.
Resources to accompany this book
Worked solutions to selected problems in this book are available online for teachers and
lecturerswhoeitheradoptorrecommendthetext. Pleasevisithttp://books.elsevier.com/
manualsandfollowtheregistrationandlogininstructionsonscreen.
OCZ,RLTandJZZ
†CompletesourcecodeandusermanualforprogramFEAPpv maybeobtainedatnocostfromthepublisher’s
webpage:http://books.elsevier.com/companionsorfromtheauthors’webpage:http://www.ce.berkeley.edu/˜rlt
Contents
Preface xiii
1 Thestandarddiscretesystemandoriginsofthefiniteelementmethod 1
1.1 Introduction 1
1.2 Thestructuralelementandthestructuralsystem 3
1.3 Assemblyandanalysisofastructure 5
1.4 Theboundaryconditions 6
1.5 Electricalandfluidnetworks 7
1.6 Thegeneralpattern 9
1.7 Thestandarddiscretesystem 10
1.8 Transformationofcoordinates 11
1.9 Problems 13
2 Adirectphysicalapproachtoproblemsinelasticity: planestress 19
2.1 Introduction 19
2.2 Directformulationoffiniteelementcharacteristics 20
2.3 Generalizationtothewholeregion–internalnodalforceconcept
abandoned 31
2.4 Displacementapproachasaminimizationoftotalpotentialenergy 34
2.5 Convergencecriteria 37
2.6 Discretizationerrorandconvergencerate 38
2.7 Displacementfunctionswithdiscontinuitybetweenelements–
non-conformingelementsandthepatchtest 39
2.8 Finiteelementsolutionprocess 40
2.9 Numericalexamples 40
2.10 Concludingremarks 46
2.11 Problems 47
3 Generalizationofthefiniteelementconcepts. Galerkin-weightedresidualand
variationalapproaches 54
3.1 Introduction 54
3.2 Integralor‘weak’statementsequivalenttothedifferentialequations 57
3.3 Approximationtointegralformulations: theweightedresidual-
Galerkinmethod 60
viii Contents
3.4 Virtualworkasthe‘weakform’ofequilibriumequationsforanalysis
ofsolidsorfluids 69
3.5 Partialdiscretization 71
3.6 Convergence 74
3.7 Whatare‘variationalprinciples’? 76
3.8 ‘Natural’variationalprinciplesandtheirrelationtogoverning
differentialequations 78
3.9 Establishmentofnaturalvariationalprinciplesforlinear,self-adjoint,
differentialequations 81
3.10 Maximum,minimum,orasaddlepoint? 83
3.11 Constrainedvariationalprinciples. Lagrangemultipliers 84
3.12 Constrainedvariationalprinciples. Penaltyfunctionandperturbed
lagrangianmethods 88
3.13 Leastsquaresapproximations 92
3.14 Concludingremarks–finitedifferenceandboundarymethods 95
3.15 Problems 97
4 ‘Standard’and‘hierarchical’elementshapefunctions: somegeneralfamilies
ofC continuity 103
0
4.1 Introduction 103
4.2 Standardandhierarchicalconcepts 104
4.3 Rectangularelements–somepreliminaryconsiderations 107
4.4 Completenessofpolynomials 109
4.5 Rectangularelements–Lagrangefamily 110
4.6 Rectangularelements–‘serendipity’family 112
4.7 Triangularelementfamily 116
4.8 Lineelements 119
4.9 Rectangularprisms–Lagrangefamily 120
4.10 Rectangularprisms–‘serendipity’family 121
4.11 Tetrahedralelements 122
4.12 Othersimplethree-dimensionalelements 125
4.13 Hierarchicpolynomialsinonedimension 125
4.14 Two-andthree-dimensional,hierarchicalelementsofthe‘rectangle’
or‘brick’type 128
4.15 Triangleandtetrahedronfamily 128
4.16 Improvementofconditioningwithhierarchicalforms 130
4.17 Globalandlocalfiniteelementapproximation 131
4.18 Eliminationofinternalparametersbeforeassembly–substructures 132
4.19 Concludingremarks 134
4.20 Problems 134
5 Mappedelementsandnumericalintegration–‘infinite’and
‘singularityelements’ 138
5.1 Introduction 138
5.2 Useof‘shapefunctions’intheestablishmentofcoordinate
transformations 139
5.3 Geometricalconformityofelements 143
5.4 Variationoftheunknownfunctionwithindistorted,curvilinear
elements. Continuityrequirements 143
Contents ix
5.5 Evaluationofelementmatrices. Transformationinξ,η,ζ coordinates 145
5.6 Evaluationofelementmatrices. Transformationinareaandvolume
coordinates 148
5.7 Orderofconvergenceformappedelements 151
5.8 Shapefunctionsbydegeneration 153
5.9 Numericalintegration–onedimensional 160
5.10 Numericalintegration–rectangular(2D)orbrickregions(3D) 162
5.11 Numericalintegration–triangularortetrahedralregions 164
5.12 Requiredorderofnumericalintegration 164
5.13 Generationoffiniteelementmeshesbymapping. Blendingfunctions 169
5.14 Infinitedomainsandinfiniteelements 170
5.15 Singularelementsbymapping–useinfracturemechanics,etc. 176
5.16 Computationaladvantageofnumericallyintegratedfiniteelements 177
5.17 Problems 178
6 Problemsinlinearelasticity 187
6.1 Introduction 187
6.2 Governingequations 188
6.3 Finiteelementapproximation 201
6.4 Reportingofresults: displacements,strainsandstresses 207
6.5 Numericalexamples 209
6.6 Problems 217
7 Fieldproblems–heatconduction,electricandmagneticpotential
andfluidflow 229
7.1 Introduction 229
7.2 Generalquasi-harmonicequation 230
7.3 Finiteelementsolutionprocess 233
7.4 Partialdiscretization–transientproblems 237
7.5 Numericalexamples–anassessmentofaccuracy 239
7.6 Concludingremarks 253
7.7 Problems 253
8 Automaticmeshgeneration 264
8.1 Introduction 264
8.2 Two-dimensionalmeshgeneration–advancingfrontmethod 266
8.3 Surfacemeshgeneration 286
8.4 Three-dimensionalmeshgeneration–Delaunaytriangulation 303
8.5 Concludingremarks 323
8.6 Problems 323
9 Thepatchtest,reducedintegration,andnon-conformingelements 329
9.1 Introduction 329
9.2 Convergencerequirements 330
9.3 Thesimplepatchtest(testsAandB)–anecessaryconditionfor
convergence 332
9.4 Generalizedpatchtest(testC)andthesingle-elementtest 334
9.5 Thegeneralityofanumericalpatchtest 336
9.6 Higherorderpatchtests 336
x Contents
9.7 Applicationofthepatchtesttoplaneelasticityelementswith
‘standard’and‘reduced’quadrature 337
9.8 Applicationofthepatchtesttoanincompatibleelement 343
9.9 Higherorderpatchtest–assessmentofrobustness 347
9.10 Concludingremarks 347
9.11 Problems 350
10 Mixedformulationandconstraints–completefieldmethods 356
10.1 Introduction 356
10.2 Discretizationofmixedforms–somegeneralremarks 358
10.3 Stabilityofmixedapproximation. Thepatchtest 360
10.4 Two-fieldmixedformulationinelasticity 363
10.5 Three-fieldmixedformulationsinelasticity 370
10.6 Complementaryformswithdirectconstraint 375
10.7 Concludingremarks–mixedformulationoratestofelement
‘robustness’ 379
10.8 Problems 379
11 Incompressibleproblems,mixedmethodsandotherproceduresofsolution 383
11.1 Introduction 383
11.2 Deviatoricstressandstrain,pressureandvolumechange 383
11.3 Two-fieldincompressibleelasticity(u–pform) 384
11.4 Three-fieldnearlyincompressibleelasticity(u–p–εv form) 393
11.5 Reducedandselectiveintegrationanditsequivalencetopenalized
mixedproblems 398
11.6 Asimpleiterativesolutionprocessformixedproblems: Uzawa
method 404
11.7 Stabilizedmethodsforsomemixedelementsfailingthe
incompressibilitypatchtest 407
11.8 Concludingremarks 421
11.9 Problems 422
12 Multidomainmixedapproximations–domaindecompositionand‘frame’
methods 429
12.1 Introduction 429
12.2 LinkingoftwoormoresubdomainsbyLagrangemultipliers 430
12.3 Linkingoftwoormoresubdomainsbyperturbedlagrangianand
penaltymethods 436
12.4 Interfacedisplacement‘frame’ 442
12.5 Linkingofboundary(orTrefftz)-typesolutionbythe‘frame’of
specifieddisplacements 445
12.6 Subdomainswith‘standard’elementsandglobalfunctions 451
12.7 Concludingremarks 451
12.8 Problems 451
13 Errors,recoveryprocessesanderrorestimates 456
13.1 Definitionoferrors 456
13.2 Superconvergenceandoptimalsamplingpoints 459
13.3 Recoveryofgradientsandstresses 465