第九章-有限元法中相关问题的处理.ppt

1,FiniteElementMethod,有限元建模技术,CHAPTER11:,2,INTRODUCTION,保证有限元计算的结果可靠，稳定提高求解的精度和效率,3,INTRODUCTION,需要考虑的主要因素：计算量和计算规模的大小；明确需求和问题的特点；根据物理性质和几何特征选择合理的单元配置；边界条件的施加；初始条件的加载。,4,CPU时间的估计,(rangesfrom23),Bandwidth,b,affects,-最小化带宽值,Aim:,尽可能控制有限元建模的自由度的数目单元密度的搭配,5,GEOMETRYMODELLING,对模型进行适当的简化3D?2D?1D?或者混合单元形式,(尽可能采用低维数单元),6,MESHING,在重点分析的局部布置较多的单元以增加精度；,单元密度控制,7,Elementdistortion,单元会存在不规则的情况，但是不能逾越有限元法的基本原理.ThedistortionsaremeasuredagainstthebasicshapeoftheelementSquareQuadrilateralelementsIsoscelestriangleTriangleelementsCubeHexahedronelementsIsoscelestetrahedronTetrahedronelements,8,Elementdistortion,单元的横纵比,Ruleofthumb:,9,Elementdistortion,角度的要求,10,Elementdistortion,曲率的要求,11,Elementdistortion,对于面积和体积的要求,不能存在负面积，物理坐标和自然坐标之间的转换,12,Elementdistortion,对于面积和体积的要求,13,Elementdistortion,中部节点位置,可能导致应力场的奇异,14,MESHCOMPATIBILITY,最小势能原理的要求单元边界的协调性,15,不同阶数的单元组合,单元间隙，造成应力场的奇异,16,不同阶数的单元组合,解决方式:UsesametypeofelementsthroughoutUsetransitionelementsUseMPCequations多点约束方程,17,Straddlingelements跨界单元模式,避免跨界单元建模形式,18,USEOFSYMMETRY,不同类型的对称:,Mirrorsymmetry,Axialsymmetry,Cyclicsymmetry,Repetitivesymmetry,UseofsymmetryreducesnumberofDOFsandhencecomputationaltime.Alsoreducesnumericalerror.,19,Mirrorsymmetry,特殊面的对称形式,20,Mirrorsymmetry,考虑二维问题，如何施加约束:,u1x=0,u2x=0,u3x=0,Singlepointconstraints(SPC)单点约束,21,Mirrorsymmetry,Deflection=Free法向偏移无约束Rotation=0转角为0,对称加载,22,Mirrorsymmetry,Anti-symmetricloading反对称加载,Deflection=0偏移为0Rotation=Free转角自由,23,Mirrorsymmetry,Symmetric对称Notranslationaldisplacementnormaltosymmetryplane（垂直于对称面）Norotationalcomponentsw.r.t.axisparalleltosymmetryplane（平行于对称面）,24,Mirrorsymmetry,Anti-symmetric反对称NotranslationaldisplacementparalleltosymmetryplaneNorotationalcomponentsw.r.t.axisnormaltosymmetryplane,25,Mirrorsymmetry,Anyloadcanbedecomposedtoasymmetricandananti-symmetricload任何加载可以分解为对称和反对称的组合,26,Mirrorsymmetry,27,Mirrorsymmetry,28,Mirrorsymmetry,Dynamicproblems(e.g.twohalfmodelstogetfullsetofeigenmodesineigenvalueanalysis)动态问题（模态和特征值分析）,29,Axialsymmetry,采用1D,2D轴对称单元,Cylindricalshellusing1Daxisymmetricelements,3Dstructureusing2Daxisymmetricelements,30,Cyclicsymmetry,uAn=uBn,uAt=uBt,Multipointconstraints(MPC),31,Repetitivesymmetry,uAx=uBx,32,MODELLINGOFOFFSETS,offsetcanbesafelyignored,offsetneedstobemodelled,ordinarybeam,plateandshellelementsshouldnotbeused.Use2Dor3Dsolidelements.,Guidelines:,33,MODELLINGOFOFFSETS,Threemethods:Verystiffelement大刚性单元Rigidelement刚体单元MPCequations多点约束方程,34,CreationofMPCequationsforoffsets多点约束方程,Eliminateq1,q2,q3,35,CreationofMPCequationsforoffsets,36,CreationofMPCequationsforofffsets,d6=d1+d5ord1+d5-d6=0d7=d2-d4ord2-d4-d7=0d8=d3ord3-d8=0d9=d5ord5-d9=0,37,MODELLINGOFSUPPORTS,38,MODELLINGOFSUPPORTS,(Propsupportofbeam),39,MODELLINGOFJOINTS,Perfectconnectionensuredhere,40,MODELLINGOFJOINTS,MismatchbetweenDOFsofbeamsand2Dsolidbeamisfreetorotate(rotationnottransmittedto2Dsolid),Perfectconnectionbyartificiallyextendingbeaminto2Dsolid(Additionalmass),41,MODELLINGOFJOINTS,UsingMPCequations,42,MODELLINGOFJOINTS,Similarforplateconnectedto3Dsolid,43,OTHERAPPLICATIONSOFMPCEQUATIONS,Modellingofsymmetricboundaryconditions,dn=0,uicos+visin=0orui+vitan=0fori=1,2,3,44,Enforcementofmeshcompatibility,dx=0.5(1-)d1+0.5(1+)d3,dy=0.5(1-)d4+0.5(1+)d6,Substitutevalueofatnode3,0.5d1-d2+0.5d3=0,0.5d4-d5+0.5d6=0,Uselowerordershapefunctiontointerpolate,45,Enforcementofmeshcompatibility,Useshapefunctionoflongerelementtointerpolate,dx=-0.5(1-)d1+(1+)(1-)d3+0.5(1+)d5,Substitutingthevaluesofforthetwoadditionalnodes,d2=0.251.5d1+1.50.5d3-0.250.5d5,d4=-0.250.5d1+0.51.5d3+0.251.5d5,46,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=0,47,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),48,IMPLEMENTATIONOFMPCEQUATIONS,(MatrixformofMPCequations),(Globalsystemequation),Constantmatrices,49,Lagrangemultipliermethod,(Lagrangemultipliers),MultipliedtoMPCequations,Addedtofunctional,ThestationaryconditionrequiresthederivativesofpwithrespecttotheDianditovanish.,Matrixequationissolved,50,Lagrangemultipliermethod,ConstraintequationsaresatisfiedexactlyTotalnumberofunknownsisincreasedExpandedstiffnessmatrixisnon-positivedefiniteduetothepresenceofzerodiagonaltermsEfficiencyofsolvingthesystemequationsislower,51,Penaltymethod,(Constrainequations),=12.misadiagonalmatrixofpenaltynumbers,StationaryconditionofthemodifiedfunctionalrequiresthederivativesofpwithrespecttotheDitovanish,Penaltymatrix,52,Penaltymethod,Zienkiewiczetal.,2000:,=constant(1/h)p+1,Characteristicsizeofelement,Pistheorder