有限元程序设计--第九章 有限元法中相关问题的处理.ppt

* Repetitive symmetry uAx = uBx * 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: * 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 * Creation of MPC equations for offsets多点约束方程 Eliminate q1, q2, q3 * Creation of MPC equations for offsets * 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 * MODELLING OF SUPPORTS * MODELLING OF SUPPORTS (Prop support of beam) * MODELLING OF JOINTS Perfect connection ensured here * 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) * MODELLING OF JOINTS Using MPC equations * MODELLING OF JOINTS Similar for plate connected to 3D solid * 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 * 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 * 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 = 0.25?1.5 d1 + 1.5?0.5 d3 - 0.25?0.5 d5 d4 = -0.25?0.5 d1 + 0.5?1.5 d3 + 0.25?1.5 d5 * 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 * Mo