有限元程序設(shè)計--第九章有限元法中相關(guān)問題的處理-資料下載頁
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【正文】 1 + ? d3 + ? d5 46 Enforcement of mesh patibility d 1 Qu ad Qu ad d 2 d 6 d 8 d 5 d 10 0 1 1 ? d 3 d 4 d 7 d 9 0 .5 0 .5 In x direction, d1 ? d2 + d3 ? d5 = 0 ? d1 + d3 ? d4 + d5 = 0 In y direction, d6? d7+ d8? d10 = 0 ? d6+ d8 ? d9 + d10 = 0 47 Modelling of constraints by rigid body attachment l 1l 3l 2q 2 q 1d 1 d 2 d 3 d 4Ri g id s la bd1 = q1 d2 = q1+q2 l1 d3=q1+q2 l2 d4=q1+q2 l3 (l2 /l11) d1 ( l2 /l1) d2 + d3 = 0 (l3 /l11) d1 ( l3 /l1) d2 + d4 = 0 Eliminate q1 and q2 (DOF in x direction not considered) 48 IMPLEMENTATION OF MPC EQUATIONS ?K D F??C D Q 0(Matrix form of MPC equations) (Global system equation) Constant matrices 49 Lagrange multiplier method ? ?12 Tm? ? ???{ } 0T ??C D Q?p1 {}2T T T? ? ? ?D K D D F C D Q??T?? ? ? ? ??? ? ? ???? ? ? ???DFKCλ QC0(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 nonpositive definite due to the presence of zero diagonal terms ? Efficiency of solving the system equations is lower 51 Penalty method ??t C D Q(Constrain equations) p1122T T T ?? ? ?D K D D F t t??=??1 ?2 ... ?m? is a diagonal matrix of ‘penalty numbers’ [] TT??? ? ?K C C D F C QStationary 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., 2022] : ? = constant (1/h)p+1 Characteristic size of element P is the order of element used 461 . 0 1 0? ? ? ?max (diagonal elements in the stiffness matrix) 581 . 0 1 0? ? ? ?or Young’s 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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