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

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Algorithm: LDL T Solution T −1 − 1 • compute U via V: = ( ) V i = R i − i ∑ − 1 r= m i l ri V r U L D V now backsubstitution: first, compute Starting with V 1 = R 1 , compute V i for i = 2,…,n V = D −1 V Get (n) U n = V n and then get (successively) for U i‐1 (i = n,...,2) V = V −l U r = m ,...,ii − 1 ( i −1) ( i) Vr Vr lri U i i = ( i−1) U i− 1 V i −1 in the (n ‐ i + 1) th evaluation, i.e. in the evaluation of U i‐1 18 15-Jun-07 Method of Finite Elements I 18
LDL T Solution Practical issues (2) • The sums over the products l rj g rj and l ri g rj do not involve terms outside the skyline (“skyline reduction method”, “column reduction method”, “active column solution”) • Storage: K as compact 1‐D array • l ij replaces g ij • d jj replaces k jj a = [ 5 6 − 4 6 − 4 1 5 − 4 1] b = [1 2 4 7] a = [ d d l d l l d l ] 24 11 22 12 33 23 13 44 34 l • V i replaces R i c = [ R 1 R 2 R 3 R 4] ( j) • V k replaces V k ( c = [ V 1 j) V ( j) 2 V ( j) 3 V ( j) 4 ] 15-Jun-07 Method of Finite Elements I 19
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 and 16: LDL T Solution Recall the pivots in
Page 17: LDL T Solution Algorithm: • Colum
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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