The scaled boundary finite element method: introduction to theory and implementation

490 Pages·2018·15.503 MB·English

by  Song, Chongmin

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TheScaledBoundary
FiniteElementMethod
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The Scaled Boundary
Finite Element Method
IntroductiontoTheoryandImplementation
ChongminSong
UniversityofNewSouthWales
Sydney,Australia
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Thiseditionfirstpublished2018
©2018JohnWiley&SonsLtd
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LibraryofCongressCataloging-in-PublicationData
Names:Song,Chongmin,author.
Title:Thescaledboundaryfiniteelementmethod:introductiontotheoryand
implementation/byChongminSong.
Description:Hoboken,NewJersey:JohnWiley&Sons,2018.|Includes
bibliographicalreferencesandindex.|
Identifiers:LCCN2018003949(print)|LCCN2018010627(ebook)|ISBN
9781119388463(pdf)|ISBN9781119388456(epub)|ISBN9781119388159
(cloth)
Subjects:LCSH:Finiteelementmethod.|Boundaryelementmethods.
Classification:LCCQC20.7.F56(ebook)|LCCQC20.7.F56S662018(print)|
DDC518/.25–dc23
LCrecordavailableathttps://lccn.loc.gov/2018003949
Coverdesign:Wiley
Coverimage:CourtesyofChongminSong
Setin10/12ptWarnockbySPiGlobal,Pondicherry,India
10 9 8 7 6 5 4 3 2 1
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ToFeng,HelenaandSophie
fortheirsupport,patience
andmanysacrifices
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Contents
Preface xv
Acknowledgements xix
AbouttheCompanionWebsite xxi
1 Introduction 1
1.1 NumericalModelling 1
1.2 OverviewoftheScaledBoundaryFiniteElementMethod 6
1.3 FeaturesandExampleApplicationsoftheScaledBoundaryFiniteElement
Method 10
1.3.1 LinearElasticFractureMechanics:CrackTerminatingatMaterial
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Interface 11
1.3.2 AutomaticMeshGenerationBasedonQuadtree/Octree 13
1.3.3 TreatmentofNon-matchingMeshes 14
1.3.4 CrackPropagation 17
1.3.5 AdaptiveAnalysis 17
1.3.6 TransientWaveScatteringinanAlluvialBasin 19
1.3.7 AutomaticImage-basedAnalysis 19
1.3.7.1 Two-dimensionalElastoplasticAnalysisofCastIron 20
1.3.7.2 Three-dimensionalConcreteSpecimen 22
1.3.8 AutomaticAnalysisofSTLModels 24
1.4 Summary 26
PartI BasicConceptsandMATLABImplementationoftheScaled
BoundaryFiniteElementMethodinTwoDimensions 27
2 BasicFormulationsoftheScaledBoundaryFiniteElement
Method 31
2.1 Introduction 31
2.2 ModellingofGeometryinScaledBoundaryCoordinates 31
2.2.1 S-domains:ScalingRequirementonGeometry,ScalingCentreandScaling
ofBoundary 31
2.2.2 S-elements:BoundaryDiscretizationofS-domains 37
2.2.3 ScaledBoundaryTransformation 40
2.2.3.1 ScaledBoundaryCoordinates 40
2.2.3.2 CoordinateTransformationofPartialDerivatives 42
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2.2.3.3 GeometricalPropertiesinScaledBoundaryCoordinates 44
2.3 GoverningEquationsofLinearElasticityinScaledBoundary
Coordinates 50
2.4 Semi-analyticalRepresentationofDisplacementandStrainFields 51
2.5 DerivationoftheScaledBoundaryFiniteElementEquationbytheVirtual
WorkPrinciple 53
2.5.1 VirtualDisplacementandStrainFieldsinScaledBoundaryCoordinates 54
2.5.2 NodalForceFunctions 54
2.5.3 TheScaledBoundaryFiniteElementEquation 55
2.6 ComputerProgramPlatypus:CoefficientMatricesofanS-element 63
2.6.1 ElementCoefficientMatricesofa2-nodeLineElement 63
2.6.2 AssemblyofCoefficientMatricesofanS-element 67
3 SolutionoftheScaledBoundaryFiniteElementEquationby
EigenvalueDecomposition 73
3.1 SolutionProcedurefortheScaledBoundaryFiniteElementEquationsin
Displacement 73
3.2 Pre-conditioningofEigenvalueProblems 77
3.3 ComputerProgramPlatypus:SolutionoftheScaledBoundaryFinite
ElementEquationofaBoundedS-elementbytheEigenvalueMethod 78
3.4 AssemblyofS-elementsandSolutionofGlobalSystemofEquations 84
3.4.1 AssemblyofS-elements 84
(cid:2) 3.4.2 SurfaceTractions 85 (cid:2)
3.4.3 EnforcingDisplacementBoundaryConditions 87
3.5 ComputerProgramPlatypus:AssemblyandSolution 87
3.5.1 AssemblyofGlobalStiffnessMatrix 87
3.5.2 AssemblyofLoadVector 95
3.5.3 SolutionofGlobalSystemofEquations 96
3.5.4 UtilityFunctions 97
3.6 ExamplesofStaticAnalysisUsingPlatypus 102
3.7 EvaluationofInternalDisplacementsandStressesofanS-element 111
3.7.1 IntegrationConstantsandInternalDisplacements 111
3.7.2 Strain/StressModesandStrain/StressFields 112
3.7.3 ShapeFunctionsofPolygonElementsModelledasS-elements 114
3.8 ComputerProgramPlatypus:InternalDisplacementsandStrains 114
3.9 BodyLoads 132
3.10 DynamicsandVibrationAnalysis 135
3.10.1 MassMatrixandEquationofMotion 135
3.10.2 NaturalFrequenciesandModeShapes 140
3.10.3 ResponseHistoryAnalysisUsingtheNewmarkMethod 143
4 AutomaticPolygonMeshGenerationforScaledBoundaryFinite
ElementAnalysis 149
4.1 Introduction 149
4.2 BasicsofGeometricalRepresentationbySignedDistanceFunctions 150
4.3 ComputerProgramPlatypus:GenerationofPolygonS-elementMesh 154
4.3.1 MeshDataStructure 157
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4.3.2 CentroidofaPolygon 165
4.3.3 ConvertingaTriangularMeshtoanS-elementMesh 166
4.3.4 UseofPolygonMeshesGeneratedbyPolyMesherinaScaledBoundary
FiniteElementAnalysis 171
4.3.5 DividingEdgesofPolygonsintoMultipleElements 172
4.4 ExamplesofScaledBoundaryFiniteElementAnalysisUsingPlatypus 175
4.4.1 ADeepBeam 178
4.4.1.1 StaticAnalysis 186
4.4.1.2 ModalAnalysis 189
4.4.1.3 ResponseHistoryAnalysis 190
4.4.1.4 PureBendingofaBeam:2LineElementsonanEdgeofPolygons 190
4.4.2 ACircularHoleinanInfinitePlaneUnderRemoteUniaxialTension 193
4.4.3 AnL-shapedPanel 197
4.4.3.1 StaticAnalysis 203
4.4.3.2 ModalAnalysis 204
4.4.3.3 ResponseHistoryAnalysis 207
5 ModellingConsiderationsintheScaledBoundaryFiniteElement
Analysis 209
5.1 EffectofLocationofScalingCentreonAccuracy 209
5.2 MeshTransition 212
5.2.1 LocalMeshRefinement 212
(cid:2) 5.2.2 RapidMeshTransition 214 (cid:2)
5.2.3 EffectofNonuniformityofLineElementLengthontheBoundary
ofS-elements 216
5.3 ConnectingNon-matchingMeshesofMultipleDomains 218
5.3.1 ComputerProgramPlatypus:CombiningTwoNon-matchingMeshes 220
5.3.2 ComputerProgramPlatypus:ModellingofaProblembyMultipleDomains
withNon-matchingMeshes 223
5.3.3 Examples 225
5.4 ModellingofStressSingularities 234
PartII TheoryandApplicationsoftheScaledBoundaryFinite
ElementMethod 237
6 DerivationoftheScaledBoundaryFiniteElementEquationinThree
Dimensions 239
6.1 Introduction 239
6.2 ScalingofBoundary 239
6.3 BoundaryDiscretizationofanS-domain 242
6.3.1 IsoparametricQuadrilateralElements 243
6.3.1.1 Four-nodeQuadrilateralElement 243
6.3.1.2 QuadrilateralElementofVariableNumberofNodes 245
6.3.2 IsoparametricTriangularElements 246
6.3.2.1 Three-nodeTriangularElements 247
6.3.2.2 Six-nodeTriangularElements 248
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6.4 ScaledBoundaryTransformationofGeometry 249
6.5 GeometricalPropertiesinScaledBoundaryCoordinates 253
6.6 GoverningEquationsofElastodynamicswithGeometryinScaledBoundary
Coordinates 257
6.7 DerivationoftheScaledBoundaryFiniteElementEquationbytheGalerkin’s
WeightedResidualTechnique 259
6.7.1 Displacement,StrainFieldsandNodalForceFunctionsinScaledBoundary
Coordinates 259
6.7.2 TheScaledBoundaryFiniteElementEquation 262
6.8 UnifiedFormulationsinTwoandThreeDimensions 267
6.9 FormulationoftheScaledBoundaryFiniteElementEquationasaSystem
ofFirst-orderDifferentialEquations 268
6.10 PropertiesofCoefficientMatrices 269
6.10.1 CoefficientMatrices[E ]and[M ] 270
0 0
6.10.2 CoefficientMatrix[E ] 270
2
6.10.3 Matrix[Z ] 271
p
6.11 LinearCompletenessoftheScaledBoundaryFiniteElementSolution 272
6.11.1 ConstantDisplacementField 272
6.11.2 LinearDisplacementField 273
6.12 ScaledBoundaryFiniteElementEquationinStiffness 278
7 SolutionoftheScaledBoundaryFiniteElementEquationinStatics
(cid:2) bySchurDecomposition 281 (cid:2)
7.1 Introduction 281
7.2 BasicsofMatrixExponentialFunction 283
7.3 SchurDecomposition 287
7.3.1 Introduction 287
7.3.2 TreatmentoftheDiagonalBlockofEigenvaluesof0 288
7.4 SolutionProcedureforaBoundedS-elementbySchurDecomposition 291
7.4.1 TransformationoftheScaledBoundaryFiniteElementEquation 291
7.4.2 EnforcingtheBoundaryConditionattheScalingCentre 292
7.4.3 DeterminingtheSolutionforDisplacementandNodalForceFunctions 294
7.4.4 DeterminingtheStaticStiffnessMatrix 295
7.5 SolutionofDisplacementandStressFieldsofanS-element 295
7.5.1 IntegrationConstants 295
7.5.2 StressModesandStressesontheBoundary 296
7.6 Block-diagonalSchurDecomposition 297
7.7 SolutionProcedurebyBlock-diagonalSchurDecomposition 303
7.7.1 GeneralSolutionoftheScaledBoundaryFiniteElementEquation 303
7.7.1.1 [Z ]HavingNoEigenvaluesofZero 304
p
7.7.1.2 [Z ]HavingEigenvaluesofZero 304
p
7.7.2 SolutionforBoundedS-elements 305
7.7.3 SolutionforUnboundedS-elements 307
7.7.3.1 [Z ]HavingNoEigenvaluesofZero 307
p
7.7.3.2 [Z ]HavingEigenvaluesofZero 308
p
7.8 DisplacementsandStressesofanS-elementbyBlock-diagonalSchur
Decomposition 310
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7.8.1 IntegrationConstantsandDisplacementFields 310
7.8.2 StressModesandStressFields 311
7.8.3 ShapeFunctionsofPolytopeElements 312
7.9 BodyLoads 313
7.10 MassMatrix 315
7.11 Remarks 317
7.12 Examples 319
7.12.1 CircularCavityinFull-plane 319
7.12.2 Bi-materialWedge 322
7.12.3 InterfaceCrackinAnisotropicBi-materialFull-plane 325
7.13 Summary 327
8 High-orderElements 329
8.1 LagrangeInterpolation 330
8.2 One-dimensionalSpectralElements 333
8.2.1 ShapeFunctions 334
8.2.2 NumericalIntegrationofElementCoefficientMatrices 337
8.2.2.1 Gauss-LegendreQuadrature 337
8.2.2.2 Gauss-Lobatto-LegendreQuadrature 338
8.3 Two-dimensionalQuadrilateralSpectralElements 341
8.3.1 ShapeFunctions 341
8.3.2 IntegrationofElementCoefficientMatricesbyGauss-Lobatto-Legendre
(cid:2) Quadrature 342 (cid:2)
8.4 Examples 344
8.4.1 ACantileverBeamSubjecttoEndLoading 345
8.4.2 ACircularHoleinanInfinitePlate 347
8.4.3 AnL-shapedPanel 349
8.4.4 A3DCantileverBeamSubjecttoEnd-shearLoading 351
8.4.5 APressurizedHollowSphere 352
9 Quadtree/OctreeAlgorithmofMeshGenerationforScaled
BoundaryFiniteElementAnalysis 355
9.1 Introduction 355
9.1.1 MeshGeneration 355
9.1.2 TheQuadtree/OctreeAlgorithm 357
9.2 DataStructureofS-elementMeshes 360
9.3 Quadtree/OctreeMeshGenerationofDigitalImages 361
9.3.1 IllustrationofQuadtreeDecompositionofTwo-dimensionalImages
byanExample 361
9.3.2 OctreeDecomposition 366
9.4 SolutionsofS-elementswiththeSamePatternofNodeConfiguration 370
9.4.1 Two-dimensionalS-elements 370
9.4.2 Three-dimensionalS-elements 372
9.5 ExamplesofImage-basedAnalysis 374
9.5.1 A2DConcreteSpecimen 374
9.5.2 A3DConcreteSpecimen 376
9.6 Quadtree/OctreeMeshGenerationforCADModels 378
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9.6.1 Quadtree/OctreeGrid 380
9.6.2 TrimmingofBoundaryCells 381
9.7 ExamplesUsingQuadtree/OctreeMeshesofCADModels 383
9.7.1 SquareBodywithMultipleHoles 384
9.7.2 AnEvolvingVoidinaSquareBody 385
9.7.3 AdaptiveAnalysisofanL-shapedPanel 386
9.7.4 AMechanicalPart 387
9.7.5 STLModels 389
9.8 Remarks 394
10 LinearElasticFractureMechanics 395
10.1 Introduction 395
10.2 BasicsofFractureAnalysis:AsymptoticSolutions,StressIntensityFactors,
andtheT-stress 397
10.2.1 CrackinHomogeneousIsotropicMaterial 397
10.2.2 InterfacialCracksbetweenTwoIsotropicMaterials 401
10.2.3 InterfacialCracksbetweenTwoAnisotropicMaterials 402
10.2.4 Multi-materialWedges 405
10.3 ModellingofSingularStressFieldsbytheScaledBoundaryFiniteElement
Method 406
10.4 StressIntensityFactorsandtheT-stressofaCrackedHomogeneous
Body 407
(cid:2) 10.5 DefinitionandEvaluationofGeneralizedStressIntensityFactors 416 (cid:2)
10.6 ExamplesofHighlyAccurateStressIntensityFactorsandT-stress 432
10.6.1 ASingleEdge-crackedRectangularBodyUnderTension 433
10.6.2 ASingleEdge-crackedRectangularBodyUnderBending 435
10.6.3 ACentre-crackedRectangularBodyUnderTension 437
10.6.4 ADoubleEdge-crackedRectangularBodyUnderTension 438
10.6.5 ASingleEdge-crackedRectangularBodyUnderEndShearing 439
10.7 ModellingofCrackPropagation 440
10.7.1 ModellingofCrackPathsbyPolygonMeshes 442
10.7.2 ModellingofCrackPathsbyQuadtreeMeshes 443
10.7.3 ExamplesofCrackPropagationModelling 444
10.7.3.1 FatigueCrackPropagationUsingPolygonMesh 444
10.7.3.2 CrackPropagationinaBeamwithThreeHoles 447
AppendixA GoverningEquationsofLinearElasticity 449
A.1 Three-dimensionalProblems 449
A.1.1 Strain 449
A.1.2 StressandEquilibriumEquation 450
A.1.3 Stress-strainRelationshipandMaterialElasticityMatrix 451
A.1.4 BoundaryConditions 453
A.2 Two-dimensionalProblems 454
A.2.1 ElasticityMatrixinPlaneStress 455
A.2.2 ElasticityMatrixinPlaneStrain 456
A.3 UnifiedExpressionsofGoverningEquations 457
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