Solution of Equilibrium Equations in Static Analysis: LDLT Solution ...

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−1 −1 −1 n− 1...... 2 1 = LDL T Solution L L L K S 1 . . . 1 Solve for K: where L i = K = L1L2...... Ln− 1S l i+1,i i . l i+2,i . … . l n,i 1 Hence, K = LS 1 l 21 . l 31 l 32 . l with L= L1L2...... L n−1= 41 l 42 . . . 1 . . . . . . . . . l n1 . . . . . . l n,n-1 1 15-Jun-07 Method of Finite Elements I 12
LDL T Solution K = LS Now = % where d =δ s hence = % T S DS and since k =k % ij ij ij , K LDS ij ji , S= L, , so K = LDL T In practice: V = −1 L R ⇐ LV = R T −1 − 1 ( ) T U= L D V ⇐ DL U = V Example: Compute L ‐1 i , i = 1, 2, 3 , L, S, D and V from ⎡ 5 −4 1 0 ⎤ ⎡0⎤ ⎢ 4 6 4 1 ⎥ ⎢ ⎢ − − 1 ⎥ K = ⎥ and R = ⎢ ⎥ ⎢ 1 − 4 6 − 4 ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ 0 1 −4 5 0 ⎦ ⎣ ⎦ 15-Jun-07 Method of Finite Elements I 13
Page 1 and 2: Solution of Equilibrium Equations i
Page 3 and 4: Review ‐ Matrices • Positive‐
Page 5 and 6: Gauss elimination In a Gauss elimin
Page 7 and 8: Physical Interpretation A physical
Page 9 and 10: Physical Interpretation 15-Jun-07 M
Page 11: LDL T Solution The successive matri
Page 15 and 16: LDL T Solution Recall the pivots in
Page 17 and 18: LDL T Solution Algorithm: • Colum
Page 19 and 20: LDL T Solution Practical issues (2)
Page 21 and 22: LDL T Solution Example: K First, ge
Page 23 and 24: LDL T Solution j= 5: g 15 = k 15 =
Page 25 and 26: LDL T Solution V = V − l U r = m
Page 27 and 28: Error Estimate •Now we use t=3 di
Page 29 and 30: Error Estimate Bathe, page 749: Sum

Solution of Equilibrium Equations in Static Analysis: LDLT Solution ...

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