有限元课件chpt8-modellingtechniqu.ppt

1 FiniteElementMethod MODELLINGTECHNIQUES CHAPTER8 2 CONTENTS INTRODUCTIONCPUTIMEESTIMATIONGEOMETRYMODELLINGMESHINGMeshdensityElementdistortionMESHCOMPATIBILITYDifferentorderofelementsStraddlingelements 3 CONTENTS USEOFSYMMETRYMirrorsymmetryAxialsymmetryCyclicsymmetryRepetitivesymmetryMODELLINGOFOFFSETSCreationofMPCequationsforoffsetsMODELLINGOFSUPPORTSMODELLINGOFJOINTS 4 CONTENTS OTHERAPPLICATIONSOFMPCEQUATIONSModellingofsymmetricboundaryconditionsEnforcementofmeshcompatibilityModellingofconstraintsbyrigidbodyattachmentIMPLEMENTATIONOFMPCEQUATIONSLagrangemultipliermethodPenaltymethod 5 INTRODUCTION Ensurereliabilityandaccuracyofresults Improveefficiencyandaccuracy 6 INTRODUCTION Considerations ComputationalandmanpowerresourcethatlimitthescaleoftheFEMmodel Requirementonresultsthatdefinesthepurposeandhencethemethodsoftheanalysis Mechanicalcharacteristicsofthegeometryoftheproblemdomainthatdeterminethetypesofelementstouse BoundaryconditionsLoadingandinitialconditions 7 CPUTIMEESTIMATION rangesfrom2 3 Bandwidth b affects minimizebandwidth Aim TocreateaFEMmodelwithminimumDOFsbyusingelementsofaslowdimensionaspossible andTouseascoarseameshaspossible andusefinemeshesonlyforimportantareas 8 GEOMETRYMODELLING Reductionofcomplexgeometrytoamanageableone 3D 2D 1D Combination Using2Dor1Dmakesmeshingmucheasier 9 GEOMETRYMODELLING Detailedmodellingofareaswherecriticalresultsareexpected UseofCADsoftwaretoaidmodelling CanbeimportedtoFEsoftwareformeshing 10 MESHING TominimizethenumberofDOFs havefinemeshatimportantareas InFEpackages meshdensitycanbecontrolledbymeshseeds Meshdensity ImagecourtesyofInstituteofHighPerformanceComputingandSunstarLogistics s PteLtd s 11 Elementdistortion Useofdistortedelementsinirregularandcomplexgeometryiscommonbuttherearesomelimitstothedistortion ThedistortionsaremeasuredagainstthebasicshapeoftheelementSquare QuadrilateralelementsIsoscelestriangle TriangleelementsCube HexahedronelementsIsoscelestetrahedron Tetrahedronelements 12 Elementdistortion Aspectratiodistortion Ruleofthumb 13 Elementdistortion Angulardistortion 14 Elementdistortion Curvaturedistortion 15 Elementdistortion Volumetricdistortion Areaoutsidedistortedelementmapsintoaninternalarea negativevolumeintegration 16 Elementdistortion Volumetricdistortion Cont d 17 Elementdistortion Mid nodepositiondistortion Shiftingofnodesbeyondlimitscanresultinsingularstressfield seecracktipelements 18 MESHCOMPATIBILITY RequirementofHamilton sprinciple admissibledisplacementThedisplacementfieldiscontinuousalongalltheedgesbetweenelements 19 Differentorderofelements Cracklikebehaviour incorrectresults 20 Differentorderofelements Solution UsesametypeofelementsthroughoutUsetransitionelementsUseMPCequations 21 Straddlingelements Avoidstraddlingofelementsinmesh 22 USEOFSYMMETRY Differenttypesofsymmetry Mirrorsymmetry Axialsymmetry Cyclicsymmetry Repetitivesymmetry UseofsymmetryreducesnumberofDOFsandhencecomputationaltime Alsoreducesnumericalerror 23 Mirrorsymmetry Symmetryaboutaparticularplane 24 Mirrorsymmetry Considera2Dsymmetricsolid u1x 0 u2x 0 u3x 0 Singlepointconstraints SPC 25 Mirrorsymmetry Deflection FreeRotation 0 Symmetricloading 26 Mirrorsymmetry Anti symmetricloading Deflection 0Rotation Free 27 Mirrorsymmetry SymmetricNotranslationaldisplacementnormaltosymmetryplaneNorotationalcomponentsw r t axisparalleltosymmetryplane 28 Mirrorsymmetry Anti symmetricNotranslationaldisplacementparalleltosymmetryplaneNorotationalcomponentsw r t axisnormaltosymmetryplane 29 Mirrorsymmetry Anyloadcanbedecomposedtoasymmetricandananti symmetricload 30 Mirrorsymmetry 31 Mirrorsymmetry 32 Mirrorsymmetry Dynamicproblems E g 2halfmodelstogetfullsetofeigenmodesineigenvalueanalysis 33 Axialsymmetry Useof1Dor2DaxisymmetricelementsFormulationsimilarto1Dand2Delementsexcepttheuseofpolarcoordinates Cylindricalshellusing1Daxisymmetricelements 3Dstructureusing2Daxisymmetricelements 34 Cyclicsymmetry uAn uBn uAt uBt Multipointconstraints MPC 35 Repetitivesymmetry uAx uBx 36 MODELLINGOFOFFSETS offsetcanbesafelyignored offsetneedtobemodelled ordinarybeam plate andshellelementsshouldnotbeused Use2 Dor3 Dsolidelements Guidelines 37 MODELLINGOFOFFSETS Threemethods VerystiffelementRigidelementMPCequations 38 CreationofMPCequationsforoffsets Eliminateq1 q2 q3 39 CreationofMPCequationsforoffsets 40 CreationofMPCequationsforofffsets d6 d1 d5ord1 d5 d6 0d7 d2 d4ord2 d4 d7 0d8 d3ord3 d8 0d9 d5ord5 d9 0 41 MODELLINGOFSUPPORTS 42 MODELLINGOFSUPPORTS Propsupportofbeam 43 MODELLINGOFJOINTS Perfectconnectionensuredhere 44 MODELLINGOFJOINTS MismatchbetweenDOFsofbeamsand2Dsolid beamisfreetorotate rotationnottransmittedto2Dsolid Perfectconnectionbyartificiallyextendingbeaminto2Dsolid Additionalmass 45 MODELLINGOFJOINTS UsingMPCequations 46 MODELLINGOFJOINTS Similarforplateconnectedto3Dsolid 47 OTHERAPPLICATIONSOFMPCEQUATIONS Modellingofsymmetricboundaryconditions dn 0 uicos visin 0orui vitan 0fori 1 2 3 48 Enforcementofmeshcompatibility dx 0 5 1 d1 0 5 1 d3 dy 0 5 1 d4 0 5 1 d6 Substitutevalueof atnode3 0 5d1 d2 0 5d3 0 0 5d4 d5 0 5d6 0 Uselowerordershapefunctiontointerpolate 49 Enforcementofmeshcompatibility Useshapefunctionoflongerelementtointerpolate dx 0 5 1 d1 1 1 d3 0 5 1 d5 Substitutingthevaluesof forthetwoadditionalnodes d2 0 25 1 5d1 1 5 0 5d3 0 25 0 5d5 d4 0 25 0 5d1 0 5 1 5d3 0 25 1 5d5 50 Enforcementofmeshcompatibility Inxdirection 0 375d1 d2 0 75d3 0 125d5 0 0 125d1 0 75d3 d4 0 375d5 0 Inydirection 0 375d6 d7 0 75d8 0 125d10 0 0 125d6 0 75d8 d9 0 375d10 0 51 Modellingofconstraintsbyrigidbodyattachment d1 q1d2 q1 q2l1d3 q1 q2l2d4 q1 q2l3 l2 l1 1 d1 l2 l1 d2 d3 0 l3 l1 1 d1 l3 l1 d2 d4 0 Eliminateq1andq2 DOFinxdirectionnotconsidered 52 IMPLEMENTATIONOFMPCEQUATIONS MatrixformofMPCequations Globalsystemequation Constantmatrices 53 Lagrangemultipliermethod Lagrangemultipliers MultipliedtoMPCequations Addedtofunctional Thestationaryconditionrequiresthederivativesof pwithrespecttotheDiand itovanish Matrixequationissolved 54 Lagrangemultipliermethod ConstraintequationsaresatisfiedexactlyTotalnumberofunknownsisincreasedExpandedstiffnessmatrixisnon positivedefiniteduetothepresenceofzerodiagonaltermsEfficiencyofsolvingthesystemequationsislower 55 Penaltymethod Constrainequations 1 2 m isadiagonalmatrixof penaltynumbers Stationaryconditionofthemodifiedfunctionalrequiresthederivativesof pwithrespecttotheDitovanish Penaltymatrix 56 Penaltymethod Zienkiewicz et al 2000 constant 1 h p 1 Characteristicsizeofelement Pistheorderofelementused max diagonalelementsinthestiffnessmatrix or Young smodulus 57 Penaltymethod Thetotalnumberofunknownsisnotchanged Systemequationsgenerallybehavewell Theconstraintequationscanonlybesatisfiedapproximately Rightchoiceof maybeambiguous