有限元程序设计有限元法中相关问题的处理.pptx

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1、1INTRODUCTIONINTRODUCTION保证有限元计算的结果可靠，稳定保证有限元计算的结果可靠，稳定提高求解的精度和效率提高求解的精度和效率第1页/共53页2INTRODUCTIONINTRODUCTION需要考虑的主要因素：需要考虑的主要因素：计算量和计算规模的大小；计算量和计算规模的大小；明确需求和问题的特点；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；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；边界条件的施加；初始条件的加载。初始条件的加载。第2页/共53页3CPUCPU时间的估计时间的估计时间的估计时间的估计(ranges from 2 3)

2、Bandwidth,b,affects-最小化带宽值最小化带宽值Aim:尽可能控制有限元建模的自尽可能控制有限元建模的自由度的数目由度的数目单元密度的搭配单元密度的搭配 第3页/共53页4GEOMETRY MODELLINGGEOMETRY MODELLING对模型进行适当的简化3D?2D?1D?或者混合单元形式(尽可能采用低维尽可能采用低维数单元数单元)第4页/共53页5MESHINGMESHING在重点分析的局部布置在重点分析的局部布置较多的单元以增加精度；较多的单元以增加精度；单元密度控制单元密度控制第5页/共53页6Element distortion单元会存在不规则的情况，但是不能逾

3、越有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限元法的基本原理.The distortions are measured against the basic shape of the elementSquare Quadrilateral elements Isosceles triangle Triangle elements Cube Hexahedron elements Isosceles tetrahedron Tetrahedron elements 第6页/共53页7Element distortion单元的横纵比单元的横纵比Rule of thumb:第7页/共53

4、页8Element distortion角度的要求第8页/共53页9Element distortion曲率的要求第9页/共53页10Element distortion对于面积和体积的要求对于面积和体积的要求不能存在负面积，不能存在负面积，物理坐标和自然坐物理坐标和自然坐标之间的转换标之间的转换第10页/共53页11Element distortion对于面积和体积的要求对于面积和体积的要求第11页/共53页12Element distortion中部节点位置中部节点位置可能导致应力场的奇异可能导致应力场的奇异第12页/共53页13MESH COMPATIBILITYMESH COMPATI

5、BILITY最小势能原理的要求单元边界的协调性第13页/共53页14不同阶数的单元组合不同阶数的单元组合单元间隙，造成应力场的奇异单元间隙，造成应力场的奇异第14页/共53页15不同阶数的单元组合不同阶数的单元组合解决方式:Use same type of elements throughoutUse transition elementsUse MPC equations 多点约束方程第15页/共53页16Straddling elements 跨界单元模式跨界单元模式避免跨界单元建模形式第16页/共53页17USE OF SYMMETRYUSE OF SYMMETRY不同类型的对称:Mir

6、ror symmetryAxial symmetryCyclic symmetryRepetitive symmetryUse of symmetry reduces number of DOFs and hence computational time.Also reduces numerical error.第17页/共53页18Mirror symmetry特殊面的对称形式特殊面的对称形式第18页/共53页19Mirror symmetry考虑二维问题，如何施加约束考虑二维问题，如何施加约束:u1x=0u2x=0u3x=0Single point constraints(SPC)单点约束

7、单点约束第19页/共53页20Mirror symmetryDeflection=Free法向偏移无约束法向偏移无约束Rotation=0转角为转角为0对称加载对称加载第20页/共53页21Mirror symmetryAnti-symmetric loading 反对称加载反对称加载Deflection=0偏移为偏移为0Rotation=Free转角自由转角自由第21页/共53页22Mirror symmetryPlane of symmetryuvwxyzxyFreeFreeFixFixFixFreeyzFixFreeFreeFreeFixFixzxFreeFixFreeFixFreeFi

8、xSymmetric 对称对称No translational displacement normal to symmetry plane（垂直于对称面）（垂直于对称面）No rotational components w.r.t.axis parallel to symmetry plane（平行于（平行于对称面）对称面）第22页/共53页23Mirror symmetryAnti-symmetric 反对称反对称No translational displacement parallel to symmetry planeNo rotational components w.r.t.axi

9、s normal to symmetry planePlane of symmetryuvwxyzxyFixFixFreeFreeFreeFixyzFreeFixFixFixFreeFreezxFixFreeFixFreeFixFree第23页/共53页24Mirror symmetryAny load can be decomposed to a symmetric and an anti-symmetric load 任何加载可以分解为对称和反对称的组合任何加载可以分解为对称和反对称的组合第24页/共53页25Mirror symmetry第25页/共53页26Mirror symmetr

10、y第26页/共53页27Mirror symmetryDynamic problems(e.g.two half models to get full set of eigenmodes in eigenvalue analysis)动态问题（模态和特征值分析）动态问题（模态和特征值分析）第27页/共53页28Axial symmetry采用采用1D,2D轴对称单元轴对称单元Cylindrical shell using 1D axisymmetric elements3D structure using 2D axisymmetric elements第28页/共53页29Cyclic sy

11、mmetryuAn=uBn uAt=uBt Multipoint constraints(MPC)第29页/共53页30Repetitive symmetryuAx=uBx 第30页/共53页31MODELLING OF OFFSETSMODELLING OF OFFSETS,offset can be safely ignored,offset needs to be modelled,ordinary beam,plate and shell elements should not be used.Use 2D or 3D solid elements.Guidelines:第31页/共5

12、3页32MODELLING OF OFFSETSMODELLING OF OFFSETSThree methods:Very stiff element 大刚性单元Rigid element 刚体单元MPC equations 多点约束方程第32页/共53页33Creation of MPC equations for offsets多点约束方程多点约束方程Eliminate q1,q2,q3第33页/共53页34Creation of MPC equations for offsets第34页/共53页35Creation of MPC equations for offfsetsd6=d1

13、+d5 or d1+d5-d6=0d7=d2-d4 or d2-d4-d7=0d8=d3 or d3 -d8=0 d9=d5 or d5-d9=0 第35页/共53页36MODELLING OF SUPPORTSMODELLING OF SUPPORTS第36页/共53页37MODELLING OF SUPPORTSMODELLING OF SUPPORTS(Prop support of beam)第37页/共53页38MODELLING OF JOINTSMODELLING OF JOINTSPerfect connection ensured here第38页/共53页39MODELLI

14、NG OF JOINTSMODELLING OF JOINTSMismatch between DOFs of beams and 2D solid beam is free to rotate(rotation not transmitted to 2D solid)Perfect connection by artificially extending beam into 2D solid(Additional mass)第39页/共53页40MODELLING OF JOINTSMODELLING OF JOINTSUsing MPC equations第40页/共53页41MODELL

15、ING OF JOINTSMODELLING OF JOINTSSimilar for plate connected to 3D solid第41页/共53页42OTHER APPLICATIONS OF MPC EQUATIONSOTHER APPLICATIONS OF MPC EQUATIONSModelling of symmetric boundary conditionsdn=0 ui cos+vi sin=0 or ui+vi tan=0 for i=1,2,3 第42页/共53页43Enforcement of mesh compatibilitydx=0.5(1-)d1+0

16、.5(1+)d3 dy=0.5(1-)d4+0.5(1+)d6 Substitute value of at node 3 0.5 d1-d2+0.5 d3=0 0.5 d4-d5+0.5 d6=0 Use lower order shape function to interpolate第43页/共53页44Enforcement of mesh compatibilityUse shape function of longer element to interpolatedx=-0.5(1-)d1+(1+)(1-)d3+0.5(1+)d5 Substituting the values o

17、f for the two additional nodes d2=0.251.5 d1+1.50.5 d3-0.250.5 d5 d4=-0.250.5 d1+0.51.5 d3+0.251.5 d5 第44页/共53页45Enforcement of mesh compatibilityIn x direction,0.375 d1-d2+0.75 d3-0.125 d5=0-0.125 d1+0.75 d3-d4+0.375 d5=0 In y direction,0.375 d6-d7+0.75 d8-0.125 d10=0-0.125 d6+0.75 d8-d9+0.375 d10=

18、0 第45页/共53页46Modelling of constraints by rigid body attachmentd1=q1 d2=q1+q2 l1 d3=q1+q2 l2 d4=q1+q2 l3(l2/l1-1)d1-(l2/l1)d2+d3=0(l3/l1-1)d1-(l3/l1)d2+d4=0 Eliminate q1 and q2(DOF in x direction not considered)第46页/共53页47IMPLEMENTATION OF MPC EQUATIONSIMPLEMENTATION OF MPC EQUATIONS(Matrix form of M

19、PC equations)(Global system equation)Constant matrices第47页/共53页48Lagrange multiplier method(Lagrange multipliers)Multiplied to MPC equationsAdded to functionalThe stationary condition requires the derivatives of p with respect to the Di and i to vanish.Matrix equation is solved第48页/共53页49Lagrange mu

20、ltiplier methodConstraint equations are satisfied exactly Total number of unknowns is increased Expanded stiffness matrix is non-positive definite due to the presence of zero diagonal termsEfficiency of solving the system equations is lower第49页/共53页50Penalty method(Constrain equations)=1 2 .m is a d

21、iagonal matrix of penalty numbers Stationary condition of the modified functional requires the derivatives of p with respect to the Di to vanish Penalty matrix第50页/共53页51Penalty methodZienkiewicz et al.,2000:=constant(1/h)p+1 Characteristic size of elementP is the order of element usedmax(diagonal elements in the stiffness matrix)orYoungs modulus 第51页/共53页52Penalty methodThe total number of unknowns is not changed.System equations generally behave well.The constraint equations can only be satisfied approximately.Right choice of may be ambiguous.第52页/共53页感谢您的观看。第53页/共53页