Introduction to Finite Element Method

IntroductiontoFiniteElementMethodMathematicModelFiniteElementMethodHistoricalBackgroundAnalyticalProcessofFEMApplicationsofFEMComputerProgramsforFEM1.MathematicalModelPhysicalProblemsMathematicalModelSolutionIdentifycontrolvariablesAssumptions(empiricallaw)(1)Modeling(2)TypesofsolutionExactEq.Approx.Eq.ExactSol.◎◎Approx.Sol.◎◎Eq.Sol.(3)MethodsofSolution(3)MethodofSolutionA.ClassicalmethodsTheyofferahighdegreeofinsight,buttheproblemsaredifficultorimpossibletosolveforanythingbutsimplegeometriesandloadings.B.Numericalmethods(I)Energy:Minimizeanexpressionforthepotentialenergyofthestructureoverthewholedomain.(II)Boundaryelement:Approximatesfunctionssatisfyingthegoverningdifferentialequationsnottheboundaryconditions.(III)Finitedifference:Replacesgoverningdifferentialequationsandboundaryconditionswithalgebraicfinitedifferenceequations.(IV)Finiteelement:Approximatesthebehaviorofanirregular,continuousstructureundergeneralloadingsandconstraintswithanassemblyofdiscreteelements.2.FiniteElementMethod(1)DefinitionFEMisanumericalmethodforsolvingasystemofgoverningequationsoverthedomainofacontinuousphysicalsystem,whichisdiscretizedintosimplegeometricshapescalledfiniteelement.ContinuoussystemTime-independentPDETime-dependentPDEDiscretesystemLinearalgebraiceq.ODE(2)DiscretizationModelingabodybydividingitintoanequivalentsystemoffiniteelementsinterconnectedatafinitenumberofpointsoneachelementcallednodes.3.HistoricalBackgroundChronicleofFiniteElementMethodYearScholarTheory1941HrennikoffPresentedasolutionofelasticityproblemusingone-dimensionalelements.1943McHenrySameasabove.1943CourantIntroducedshapefunctionsovertriangularsubregionstomodelthewholeregion.1947LevyDevelopedtheforce(flexibility)methodforstructureproblem.1953LevyDevelopedthedisplacement(stiffness)methodforstructureproblem.1954Argyris&KelseyDevelopedmatrixstructuralanalysismethodsusingenergyprinciples.1956Turner,Clough,Martin,ToppDerivedstiffnessmatricesfortruss,beamand2Dplanestresselements.Directstiffnessmethod.1960CloughIntroducedthephrasefiniteelement.1960Turneret.alLargedeflectionandthermalanalysis.1961MeloshDevelopedplatebendingelementstiffnessmatrix.1961MartinDevelopedthetetrahedralstiffnessmatrixfor3Dproblems.1962GallagheretalMaterialnonlinearity.ChronicleofFiniteElementMethodYearScholarTheory1963Grafton,StromeDevelopedcurved-shellbendingelementstiffnessmatrix.1963MeloshAppliedvariationalformulationtosolvenonstructuralproblems.1965Cloughet.al3Delementsofaxisymmetricsolids.1967Zienkiewiczet.Publishedthefirstbookonfiniteelement.1968Zienkiewiczet.Visco-elasticityproblems.1969Szabo&LeeAdaptedweightedresidualmethodsinstructuralanalysis.1972OdenBookonnonlinearcontinua.1976BelytschkoLarge-displacementnonlineardynamicbehavior.~1997Newelementdevelopment,convergencestudies,thedevelopmentsofsupercomputers,theavailabilityofpowerfulmicrocomputers,thedevelopmentofuser-friendlygeneral-purposefiniteelementsoftwarepackages.4.AnalyticalProcessesofFiniteElementMethod(1)StructuralstressanalysisproblemA.Conditionsthatsolutionmustsatisfya.Equilibriumb.Compatibilityc.Constitutivelawd.BoundaryconditionsAboveconditionsareusedtogenerateasystemofequationsrepresentingsystembehavior.B.Approacha.Force(flexibility)method:internalforcesasunknowns.b.Displacement(stiffness)method:nodaldisp.Asunknowns.Forcomputationalpurpose,thedisplacementmethodismoredesirablebecauseitsformulationissimple.AvastmajorityofgeneralpurposeFEsoftwareshaveincorporatedthedisplacementmethodforsolvingstructuralproblems.(2)AnalysisproceduresoflinearstaticstructuralanalysisA.Buildupgeometricmodela.1Dproblemlineb.2Dproblemsurfacec.3DproblemsolidB.Constructthefiniteelementmodela.Discretizeandselecttheelementtypes(a)elementtype1Dlineelement2Delement3Dbrickelement(b)totalnumberofelement(mesh)1D:2D:3D:b.Selectashapefunction1Dlineelement:u=ax+bc.Definethecompatibilityandconstitutivelawd.Formtheelementstiffnessmatrixandequations(a)Directequilibriummethod(b)Workorenergymethod(c)MethodofweightResidualse.FormthesystemequationAssembletheelementequationstoobtainglobalsystemequationandintroduceboundaryconditionsC.Solvethesystemequationsa.eliminationmethodGauss’smethod(Nastran)b.iterationmethodGaussSeidel’smethodDisplacementfieldstrainfieldstressfieldD.Interprettheresults(postprocessing)a.deformationplotb.stresscontour5.ApplicationsofFiniteElementMethodStructuralProblemNon-structuralProblemStressAnalysis-truss&frameanalysis-stressconcentratedproblemBucklingproblemVibrationAnalysisImpactProblemHeatTransferFluidMechanicsElectricorMagneticPotential6.ComputerProgramsforFiniteElementMethod結構非線性最佳化複合材料碰撞分析疲勞分析機構分析噪音分析流力分析模流分析熱傳分析ANSYS◎D◎◎D◎◎◎NASTRAN◎D◎◎D◎◎ABAQUS◎◎◎◎◎MARC◎◎◎◎◎LS-DYNA3D◎MSC/DYNA◎ADAMS/DADS◎COSMOS◎D◎◎D◎◎MOLDFLOW◎C-FLOW◎PHOENICS◎◎