Equilibrium finite element formulations

298 Pages·2017·9.213 MB·English

by  Almeida, José Paulo, Maunder, E. A. W

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EquilibriumFinite
ElementFormulations
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Equilibrium Finite Element Formulations
J.P.MoitinhodeAlmeida
DepartmentofCivilEngineering,ArchitectureandGeoresources,InstitutoSuperior
Técnico,UniversityofLisbon,Portugal
EdwardA.W.Maunder
CollegeofEngineering,MathematicsandPhysicalSciences,UniversityofExeter,
UnitedKingdom
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Thiseditionfirstpublished2017
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LibraryofCongressCataloging-in-PublicationData
Names:Almeida,J.P.Moitinhode(JosePaulo),1958-author.|Maunder,E.
A.W.,author.
Title:Equilibriumfiniteelementformulations/J.P.MoitinhodeAlmeida,
EdwardA.W.Maunder.
Description:Chichester,UK;Hoboken,NJ:JohnWiley&Sons,2016.|
Includesbibliographicalreferencesandindex.
Identifiers:LCCN2016039762(print)|LCCN2016045455(ebook)|ISBN
9781118424155(cloth)|ISBN9781118926215(pdf)|ISBN9781118926208
(epub)
Subjects:LCSH:Finiteelementmethod.|Equilibrium.|Structuralanalysis
(Engineering)–Mathematics.
Classification:LCCTA347.F5A3942016(print)|LCCTA347.F5(ebook)|DDC
518/.25–dc23
LCrecordavailableathttps://lccn.loc.gov/2016039762
AcataloguerecordforthisbookisavailablefromtheBritishLibrary.
CoverDesign:Wiley
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Setin10/12ptWarnockbySPiGlobal,Chennai,India
10987654321
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To
• BruceM.Irons
• BaudouinM.FraeijsdeVeubeke
• JohnC.deC.Henderson
Fortheirinspirationtothinkdifferently.
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vii
Contents
Preface xiii
ListofSymbols xvii
1 Introduction 1
1.1 Prerequisites 1
1.2 WhatIsMeantbyEquilibrium?WeaktoStrongForms 2
1.3 WhatDoWeGainFromStrongFormsofEquilibrium? 3
1.4 WhatPathsHaveBeenFollowedtoAchieveStrongFormsofEquilibrium? 5
1.5 IndustrialPerspectives 6
1.5.1 SimulationGovernance 7
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1.5.2 EquilibriuminStructuralDesignandAssessment 7
1.6 TheStructureoftheBook 8
References 9
2 BasicConceptsIllustratedbySimpleExamples 11
2.1 SymmetricBi-MaterialStrip 12
2.2 KirchhoffPlateWithaLineLoad 16
2.2.1 KinematicallyAdmissibleSolutions 16
2.2.2 StaticallyAdmissibleSolutions 19
2.2.3 AssessmentoftheSolutionsObtained 20
References 21
3 EquilibriuminOtherFiniteElementFormulations 22
3.1 ConformingFormulationsandNodalEquilibrium 22
3.2 Pian’sHybridFormulation 25
3.3 MixedStressFormulations 27
3.4 VariantsoftheDisplacementBasedFormulationsWithStrongerFormsof
Equilibrium 28
3.4.1 FraeijsdeVeubeke’sEquilibratedTriangle 29
3.4.2 TriangularEquilibriumElementsforPlateBending 30
3.4.3 OtherVariants 31
3.5 TrefftzFormulations 32
3.6 FormulationsBasedontheApproximationofaStressPotential 33
3.7 TheSymmetricBi-MaterialStripRevisited 33
References 40
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viii Contents
4 FormulationofHybridEquilibriumElements 43
4.1 ApproximationoftheStresses 43
4.2 ApproximationoftheBoundaryDisplacements 45
4.3 AssemblingtheApproximations 48
4.4 EnforcementofEquilibriumattheBoundariesoftheElements 48
4.5 EnforcementofCompatibility 51
4.6 GoverningSystem 53
4.7 ExistenceandUniquenessoftheSolution 54
4.8 ElementsforSpecificTypesofProblem 57
4.8.1 Continuain2D 57
4.8.1.1 ExemplificationoftheAssemblyProcess 58
4.8.1.2 ASimpleNumericalExample 60
4.8.2 Continuain3D 62
4.8.3 PlateBending 63
4.8.3.1 Reissner–MindlinTheory 64
4.8.3.2 KirchhoffTheory 65
4.8.3.3 Example 66
4.8.4 PotentialProblemsofLowerOrder 66
4.9 TheCaseofGeometriesWithaNon-LinearMapping 68
4.10 CompatibilityDefaults 69
4.11 TheDimensionoftheSystemofEquations 70
References 71
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5 AnalysisoftheKinematicStabilityofHybridEquilibrium
Elements 73
5.1 AlgebraicandDualityConceptsRelatedtoSpuriousKinematicModes 73
5.2 SpuriousKinematicModesinModelsof2DContinua 76
5.2.1 SingleTriangularElements 77
5.2.2 APairofTriangularElementsWithaCommonInterface 80
5.2.3 StarPatchesof2DElements 82
5.2.3.1 OpenStarsofDegree0 84
5.2.3.2 ClosedStarsofDegree0 84
5.2.3.3 OpenStarsofDegree1 84
5.2.3.4 ClosedStarsofDegree1 85
5.2.3.5 OpenStarsofDegree2 85
5.2.3.6 ClosedStarsofDegree2 85
5.2.3.7 ExamplesofUnstableClosedStarPatchesofDegree2 86
5.2.3.8 StarsofDegree3orHigher 87
5.2.4 ObservationsforGeneral2DMeshes 87
5.3 SpuriousKinematicModesinModelsof3DContinua 90
5.3.1 SingleTetrahedralElements 90
5.3.1.1 SpuriousModesAssociatedWithaSingleEdge 92
5.3.1.2 SpuriousModesAssociatedWithaSingleFace 94
5.3.2 APairofTetrahedralElements 94
5.3.2.1 PrimaryInterfaceSpuriousModes 95
5.3.2.2 PairsofTetrahedralElementsWithCoplanarFaces 96
5.3.3 StarPatchesofTetrahedralElements 97
5.3.3.1 Edge-CentredPatches 98
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Contents ix
5.3.3.2 Vertex-CentredPatches 98
5.4 SpuriousKinematicModesinModelsofReissner–MindlinPlates 99
5.4.1 ASingleTriangularReissner–MindlinElement 100
5.4.2 APairofReissner–MindlinElements 102
5.4.3 StarPatchesofReissner–MindlinElements 103
5.4.3.1 OpenStarsofDegree1 103
5.4.3.2 ClosedStarPatchesofDegree1 103
5.4.3.3 OpenStarsofDegree2 103
5.4.3.4 ClosedStarPatchesofDegree2 103
5.4.4 ObservationsforGeneralMeshesofReissner–MindlinElements 104
5.5 TheStabilityofPlatesModelledWithKirchhoffElements 105
5.6 TheStabilityofModelsforPotentialProblems 106
5.7 HowDoWeObtainaStableMeshforGeneralStructuralModels? 108
5.7.1 GeneralProcedures 108
5.7.2 Macro-Elements 108
References 109
6 PracticalAspectsoftheKinematicStabilityofHybridEquilibrium
Elements 111
6.1 IdentificationofRigidBodyandSpuriousKinematicModes 111
6.1.1 SpuriousKinematicandRigidBodyModesofanElement 112
6.1.2 SpuriousKinematicandRigidBodyModesofaMesh 113
(cid:2) 6.2 BlockingtheSpuriousModes 115 (cid:2)
6.3 AnIllustrationoftheProcedurestoRemoveSpuriousModes 116
6.4 HowDoWeRecognizeAdmissibleLoads? 117
6.5 Quasi-SimplicialHybridElementsCreatedbyHierarchicalMesh
Refinement 118
6.6 Non-SimplicialHybridElements 120
6.7 ACautionaryTaleof‘NearMisses’ 120
References 125
7 AVariationalBasisoftheHybridEquilibriumFormulation 126
7.1 PotentialEnergyandComplementaryPotentialEnergy 126
7.1.1 ExistenceandUniquenessofSolutions 129
7.1.2 PropertiesoftheExactSolution 129
7.1.3 TheFormalRelationBetweenBothEnergies 130
7.2 HybridComplementaryPotentialEnergy 131
7.3 PropertiesoftheGeneralizedComplementaryEnergy 132
7.4 TheBabuška–BrezziConditionandHybridEquilibriumElements 133
References 134
8 RecoveryofComplementarySolutions 135
8.1 GeneralFeaturesofPartitionofUnityFunctions 136
8.2 RecoveryofCompatibilityFromanEquilibratedSolution 138
8.2.1 DerivationofũE 140
8.2.2 AnIllustrationoftheTechnique 141
8.3 RecoveryofEquilibriumFromaCompatibleSolution 143
8.3.1 RecoveryFromStarPatches:TheGeneralCase 144
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8.3.2 RecoveryFromStarPatches:TheCaseofLinearDisplacements 146
8.3.3 ElementbyElementRecoveryofEquilibrium 150
8.3.3.1 ResolutionoftheVertexForces 150
8.3.3.2 DerivationofStaticallyEquivalentCodiffusiveTractions 153
8.3.3.3 AdmissibilityoftheDerivedTractions 155
8.3.3.4 DerivationoftheElementStressFields 156
8.4 NumericalExamples 157
8.4.1 RecoveryofCompatibilityFromanEquilibratedSolution 157
8.4.2 RecoveryofEquilibriumFromaCompatibleSolution 160
8.5 ExtensionsoftheRecoveryProcedures 163
8.5.1 Reissner–MindlinTheory 163
8.5.2 KirchhoffTheory 163
8.5.2.1 RecoveryofCompatibility 163
8.5.2.2 RecoveryofEquilibrium 164
8.5.3 Non-SimplicialElements 164
References 164
9 DualAnalysesforErrorEstimation&Adaptivity 166
9.1 GlobalErrorBounds 167
9.1.1 Revisitingthesimpleexample 170
9.2 EstimationoftheErrorDistributionandGlobalMeshAdaptation 177
9.2.2 TheConvergenceoftheSimpleExample 180
(cid:2) 9.3 ObtainingLocalQuantitiesofInterest 184 (cid:2)
9.4 BoundingtheErrorofLocalOutputs 187
9.4.1 Background 187
9.4.2 BoundsoftheErrorofOutputsObtainedFromComplementary
Solutions 187
9.5 LocalOutputsfortheKirchhoffPlateWithaLineLoad 189
9.5.1 TheDisplacementattheCorner 190
9.5.2 TheAverageDisplacementontheLoadedSide 192
9.5.3 TheAverageDisplacementontheFreeSide 193
9.6 EstimationoftheErrorDistributionandMeshAdaptationforLocal
Quantities 194
9.7 AdaptivityforMultipleLoadsandMultipleOutputs 195
References 196
10 DynamicAnalyses 199
10.1 Toupin’sPrincipleforElastodynamics 200
10.2 DerivationoftheEquilibriumFiniteElementEquations 201
10.3 AnalysisintheFrequencyDomain 203
10.4 AnalysisintheTimeDomain 205
10.5 NoDirectBoundsoftheEigenfrequencies? 206
10.6 Example 207
10.6.1 Eigenfrequencies 207
10.6.2 ForcedVibrations 209
References 211
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Contents xi
11 Non-LinearAnalyses 212
11.1 ElasticContact 212
11.2 MaterialNon-Linearity 214
11.2.1 Non-LinearElasticity 214
11.2.2 ElastoplasticConstitutiveRelations 215
11.2.2.1 DirectImplementation 216
11.2.2.2 AStandardReturnMappingImplementation 217
11.2.2.3 AReturnMappingImplementationforPlasticityDefinedintheStrain
Space 218
11.2.2.4 ImposingtheYieldConditioninaWeakForm 219
11.3 LimitAnalysis 220
11.3.1 Introduction 220
11.3.2 GeneralStatementoftheProblemasaMathematicalProgramme 220
11.3.2.1 Formulation(1) 221
11.3.2.2 Formulation(2) 221
11.3.2.3 YieldConstraints 222
11.3.2.4 ApplicationoftheComplementary(Dual)Programme 222
11.3.3 ImplementationforPlateBendingProblems 222
11.3.4 NumericalExample 223
11.4 GeometricNon-Linearity 224
11.4.1 WeakCompatibilityforLargeDisplacementsWithSmallStrains 225
11.4.2 Equilibrium 227
(cid:2) 11.4.3 TransformationofBoundaryDisplacementParametersandGeneralized (cid:2)
Tractions 228
11.4.4 GoverningSystem 229
11.4.5 DeterminationoftheRigidBodyDisplacements 229
11.4.6 TangentFormoftheGoverningSystem 230
11.4.6.1 VariationoftheRigidBodyDisplacements 230
11.4.6.2 TheEffectofaVariationintheBoundaryDisplacementParametersonthe
AssociatedTransformations 231
11.4.6.3 TangentFormoftheGoverningSystemforanElement 233
11.4.7 LargeDisplacementsandSpuriousKinematicModes 233
11.4.7.1 NumericalExample 234
References 235
A FundamentalEquationsofStructuralMechanics 237
A.1 TheGeneralElastostaticProblem 237
A.1.1 TwoDimensionalElasticity 237
A.1.2 ThreeDimensionalElasticity 238
A.1.3 ShearStressesandWarpingofaBeamSection 240
A.1.4 PlateBending 245
A.1.4.1 Reissner–MindlinTheory 247
A.1.4.2 KirchhoffTheory 248
A.2 CompatibilityofStrains 250
A.2.1 IntegrabilityConditions 250
A.2.2 EnforcementoftheKinematicBoundaryConditions 251
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xii Contents
A.3 GeneralElastodynamicProblem 252
References 252
B ComputerProgramsforEquilibriumFiniteElement
Formulations 254
B.1 AuxiliaryPrograms 255
B.1.1 gmsh 255
B.1.2 ThemcheandmchfClasses 258
B.1.3 mtimesx 258
B.2 StructureofthePrograms 259
B.2.1 DefinitionoftheMesh 259
B.2.2 DefinitionoftheMaterialPropertiesandBoundaryConditions 260
B.2.3 DefinitionoftheApproximationFunctions 261
B.2.4 EnforcementofBoundaryConditions 263
B.2.5 ProcessingtheSolutions 265
B.2.6 CodeSnippets 265
B.2.6.1 ComputingtheFlexibilityMatrixofanElement 265
B.2.6.2 TheEquilibriumMatrixofaSideofaPlaneElement 267
References 269
SubjectIndex 271
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