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1,Finite Element Method,有限元建模技术,CHAPTER 11:,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,( ranges from 2 3),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,GEOMETRY 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,Element distortion,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. The distortions are measured against the basic shape of the element Square Quadrilateral elements Isosceles triangle Triangle elements Cube Hexahedron elements Isosceles tetrahedron Tetrahedron elements,8,Element distortion,单元的横纵比,Rule of thumb:,9,Element distortion,角度的要求,10,Element distortion,曲率的要求,11,Element distortion,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,Element distortion,对于面积和体积的要求,13,Element distortion,中部节点位置,可能导致应力场的奇异,14,MESH COMPATIBILITY,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: Use same type of elements throughout Use transition elements Use MPC equations 多点约束方程,17,Straddling elements 跨界单元模式,避免跨界单元建模形式,18,USE OF SYMMETRY,不同类型的对称:,Mirror symmetry,Axial symmetry,Cyclic symmetry,Repetitive symmetry,Use of symmetry reduces number of DOFs and hence computational time. Also reduces numerical error.,19,Mirror symmetry,特殊面的对称形式,20,Mirror symmetry,考虑二维问题，如何施加约束:,u1x = 0,u2x = 0,u3x = 0,Single point constraints (SPC) 单点约束,21,Mirror symmetry,Deflection = Free 法向偏移无约束 Rotation = 0 转角为0,对称加载,22,Mirror symmetry,Anti-symmetric loading 反对称加载,Deflection = 0 偏移为0 Rotation = Free 转角自由,23,Mirror symmetry,Symmetric 对称 No translational displacement normal to symmetry plane（垂直于对称面） No rotational components w.r.t. axis parallel to symmetry plane（平行于对称面）,24,Mirror symmetry,Anti-symmetric 反对称 No translational displacement parallel to symmetry plane No rotational components w.r.t. axis normal to symmetry plane,25,Mirror symmetry,Any 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,26,Mirror symmetry,27,Mirror symmetry,28,Mirror symmetry,Dynamic 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）,29,Axial symmetry,采用1D,2D轴对称单元,Cylindrical shell using 1D axisymmetric elements,3D structure using 2D axisymmetric elements,30,Cyclic symmetry,uAn = uBn,uAt = uBt,Multipoint constraints (MPC),31,Repetitive symmetry,uAx = uBx,32,MODELLING 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:,33,MODELLING OF OFFSETS,Three methods: Very stiff element 大刚性单元 Rigid element 刚体单元 MPC equations 多点约束方程,34,Creation of MPC equations for offsets多点约束方程,Eliminate q1, q2, q3,35,Creation of MPC equations for offsets,36,Creation of MPC equations for offfsets,d6 = d1 + d5 or d1 + d5 - d6 = 0 d7 = d2 - d4 or d2 - d4 - d7 = 0 d8 = d3 or d3 - d8 = 0 d9 = d5 or d5 - d9 = 0,37,MODELLING OF SUPPORTS,38,MODELLING OF SUPPORTS,(Prop support of beam),39,MODELLING OF JOINTS,Perfect connection ensured here,40,MODELLING OF JOINTS,Mismatch 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),41,MODELLING OF JOINTS,Using MPC equations,42,MODELLING OF JOINTS,Similar for plate connected to 3D solid,43,OTHER APPLICATIONS OF MPC EQUATIONS,Modelling of symmetric boundary conditions,dn = 0,ui cos + vi sin=0 or ui+vi tan =0 for i=1, 2, 3,44,Enforcement of mesh compatibility,dx = 0.5(1-) d1 + 0.5(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,45,Enforcement of mesh compatibility,Use shape function of longer element to interpolate,dx = -0.5 (1-) d1 + (1+)(1-) d3 + 0.5 (1+) d5,Substituting the values of 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 d5,46,Enforcement of mesh compatibility,In 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 = 0,47,Modelling of constraints by rigid body attachment,d1 = 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),48,IMPLEMENTATION OF MPC EQUATIONS,(Matrix form of MPC equations),(Global system equation),Constant matrices,49,Lagrange multiplier method,(Lagrange multipliers),Multiplied to MPC equations,Added to functional,The stationary condition requires the derivatives of p with respect to the Di and i to vanish.,Matrix equation is solved,50,Lagrange multiplier method,Constraint equations are satisfied exactly Total number of unknowns is increased Expanded stiffness matrix is non-positive definite due to the presence of zero diagonal terms Efficiency of solving the system equations is lower,51,Penalty method,(Constrain equations),=1 2 . m is a diagonal matrix of penalty numbers,Stationary condition of the modified functional requires the derivatives of p with respect to the Di to vanish,Penalty matrix,52,Penalty method,Zienkiewicz et al., 2000 :, = constant (1/h)p+1,Characteristic size of element,P is the order of element used,max (diagonal elements in the stiffness matrix),or,Youngs modulus,53,Penalty method,The 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.,
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