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

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Practical issues (1) LDL T Solution • Simultaneous computation of V and L ‐1 i . • L and V not computed from scratch, but by modifications of K and R. • K is symmetric, banded and dpositive‐definite; ite the two oformer properties permit the compact storage in a 1‐D Array, complemented by a 1‐D address array: K ⎡ 5 −4 1 0 ⎤ ⎢ − − ⎥ 4 6 4 1 = ⎢ ⎥ ⎢ 1 − 4 6 − 4 ⎥ ⎢ ⎥ ⎣ 0 1 −4 5 ⎦ a = [ 5 6 − 4 6 − 4 1 5 − 4 1] non‐zero elements of upper half and diagonal b = [1 2 4 7] array indices of the diagonal elements in a 15-Jun-07 Method of Finite Elements I 16
LDL T Solution Algorithm: • Columnwise calculation l of l ij and d jj for j = 2,...,n, starting ti with d 11 = k 11 : g m , j = km , j g ij j j i ∑ ∑ −1 = kij − lrigrj i = m +1,..., j −1 r= m m j where m = skyline of K, m m = max{m i , m j } now: decomposition of K to the factors D and L (or: L T ) g ij l ij = i = m j +1,..., j −1 dii d jj = k jj − ∑ j−1 r= m j l rj g rj Here: l ij denotes an element of L T 15-Jun-07 Method of Finite Elements I 17
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 and 12: LDL T Solution The successive matri
Page 13 and 14: LDL T Solution K = LS Now = % where
Page 15: LDL T Solution Recall the pivots in
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

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