Finite Element Method

16.810(16.682)16.810(16.682)EngineeringDesignandRapidPrototypingEngineeringDesignandRapidPrototypingInstructor(s)FiniteElementMethodJanuary12,[e-mail]@mit.edu16.810(16.682)2PlanforToday„FEMLecture(ca.50min)„FEMfundamentalconcepts,analysisprocedure„Errors,Mistakes,andAccuracy„CosmosIntroduction(ca.30min)„Followalongstep-by-step„ConductFEAofyourpart(ca.90min)„Workinteamsoftwo„FirstconductananalysisofyourCADdesign„Youarefreetomakemodificationstoyouroriginalmodel16.810(16.682)3CourseConcepttoday16.810(16.682)4CourseFlowDiagramCAD/CAM/CAEIntroFEM/SolidMechanicsOverviewManufacturingTrainingStructuralTest“Training”DesignOptimizationHandsketchingCADdesignFEManalysisProducePart1TestProducePart2OptimizationProblemstatementFinalReviewTestLearning/ReviewDeliverablesDesignSketchv1Analysisoutputv1Partv1Experimentdatav1Design/Analysisoutputv2Partv2Experimentdatav2Drawingv1DesignIntroduetodaytodayWednesday16.810(16.682)5NumericalMethodFiniteElementMethodBoundaryElementMethodFiniteDifferenceMethodFiniteVolumeMethodMeshlessMethod16.810(16.682)6WhatistheFEM?Description-FEMcutsastructureintoseveralelements(piecesofthestructure).-Thenreconnectselementsat“nodes”asifnodeswerepinsordropsofgluethatholdelementstogether.-Thisprocessresultsinasetofsimultaneousalgebraicequations.FEM:Methodfornumericalsolutionoffieldproblems.Numberofdegrees-of-freedom(DOF)Continuum:InfiniteFEM:Finite(Thisistheoriginofthename,FiniteElementMethod)16.810(16.682)7FundamentalConcepts(1)ElasticproblemsThermalproblemsFluidflowElectrostaticsetc.Manyengineeringphenomenacanbeexpressedby“governingequations”and“boundaryconditions”GoverningEquation(Differentialequation)()0Lfφ+=BoundaryConditions()0Bgφ+=16.810(16.682)8ElasticdeformationThermalbehavioretc.GoverningEquation:()0Lfφ+=BoundaryConditions:()0Bgφ+=[]{}{}=KuFAsetofsimultaneousalgebraicequationsFEMApproximate!FundamentalConcepts(2)Example:VerticalmachiningcenterGeometryisverycomplex!Youknowalltheequations,butyoucannotsolveitbyhand16.810(16.682)9[]{}{}=KuF1{}[]{}−=uKFPropertyBehaviorActionElasticThermalFluidElectrostaticBehavior{}uProperty[]KAction{}FstiffnessdisplacementforceconductivitytemperatureheatsourceviscosityvelocitybodyforcedialectripermittivityelectricpotentialchargeUnknownFundamentalConcepts(3)16.810(16.682)10ItisverydifficulttomakethealgebraicequationsfortheentiredomainDividethedomainintoanumberofsmall,simpleelementsAdjacentelementssharetheDOFatconnectingnodesFundamentalConcepts(4)Finiteelement:SmallpieceofstructureAfieldquantityisinterpolatedbyapolynomialoveranelement16.810(16.682)11Obtainthealgebraicequationsforeachelement(thisiseasy!)Putalltheelementequationstogether[]{}{}=KuF[]{}{}EEE=KuF[]{}{}EEE=KuF[]{}{}EEE=KuF[]{}{}EEE=KuF[]{}{}EEE=KuF[]{}{}EEE=KuF[]{}{}EEE=KuF[]{}{}EEE=KuF[]{}{}EEE=KuFFundamentalConcepts(5)16.810(16.682)12[]{}{}=KuF1{}[]{}−=uKFSolvetheequations,obtainingunknownvariablessatnodes.FundamentalConcepts(6)16.810(16.682)13Concepts-Summary-FEMusestheconceptofpiecewisepolynomialinterpolation.-Byconnectingelementstogether,thefieldquantitybecomesinterpolatedovertheentirestructureinpiecewisefashion.-Asetofsimultaneousalgebraicequationsatnodes.[]{}{}=KuFPropertyBehaviorActionK:Stiffnessmatrixx:DisplacementF:LoadKxF=KFx16.810(16.682)14BriefHistory-Thetermfiniteelementwasfirstcoinedbycloughin1960.Intheearly1960s,engineersusedthemethodforapproximatesolutionsofproblemsinstressanalysis,fluidflow,heattransfer,andotherareas.-ThefirstbookontheFEMbyZienkiewiczandChungwaspublishedin1967.-Inthelate1960sandearly1970s,theFEMwasappliedtoawidevarietyofengineeringproblems.-MostcommercialFEMsoftwarepackagesoriginatedinthe1970s.(Abaqus,Adina,Ansys,etc.)-Klaus-JurgenBatheinMEatMITReference[2]16.810(16.682)15Canreadilyhandleverycomplexgeometry:-TheheartandpoweroftheFEMCanhandleawidevarietyofengineeringproblems-Solidmechanics-Dynamics-Heatproblems-Fluids-ElectrostaticproblemsCanhandlecomplexrestraints-Indeterminatestructurescanbesolved.Canhandlecomplexloading-Nodalload(pointloads)-Elementload(pressure,thermal,inertialforces)-TimeorfrequencydependentloadingAdvantagesoftheFEM16.810(16.682)16DisadvantagesoftheFEMAgeneralclosed-formsolution,whichwouldpermitonetoexaminesystemresponsetochangesinvariousparameters,isnotproduced.TheFEMobtainsonlyapproximatesolutions.TheFEMhasinherenterrors.Mistakesbyuserscanbefatal.16.810(16.682)17TypicalFEAProcedurebyCommercialSoftwarePreprocessProcessPostprocessUserUserComputerBuildaFEmodelConductnumericalanalysisSeeresults16.810(16.682)18[1]Selectanalysistype-StructuralStaticAnalysis-ModalAnalysis-TransientDynamicAnalysis-BucklingAnalysis-Contact-Steady-stateThermalAnalysis-TransientThermalAnalysis[2]Selectelementtype2-D3-DLinearQuadraticBeamTrussShellSolidPlate[3]Materialproperties,,,,EνραPreprocess(1)16.810(16.682)19Preprocess(2)[4]Makenodes[5]Buildelementsbyassigningconnectivity[6]Applyboundaryconditionsandloads16.810(16.682)20ProcessandPostprocess-Solvetheboundaryvalueproblem[7]Process-Seetheresults[8]PostprocessDisplacementStressSt