9780203009512.ch11

9780203009512.ch11 - 0749_Frame_C11 Page 153 Wednesday,...

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153 Solution Methods for Linear Problems 11.1 NUMERICAL METHODS IN FEA 11.1.1 S OLVING THE F INITE -E LEMENT E QUATIONS : S TATIC P ROBLEMS Consider the numerical solution of the linear system K γ = f , in which K is the positive-definite and symmetric stiffness matrix. In many problems, it has a large dimension, but is also banded. The matrix can be “triangularized”: K = LL T , in which L is a lower triangular, nonsingular matrix (zeroes in all entries above the diagonal). We can introduce z = L T γ and obtain z by solving Lz = f . Next, γ can be computed by solving L T γ = z . Now Lz = f can be conveniently solved by forw ard substitution . In particular, Lz = f can be expanded as (11.1) Assuming that the diagonal entries are not too small, this equation can be solved, starting from the upper-left entry, using simple arithmetic: z 1 = f 1 / l 11 , z 2 = [ f 2 l 21 z 1 ] / l 22 , z 3 = [ f 3 l 31 z 1 l 32 z 2 ] / l 33 , . Next, the equation L T γ = z can be solved using backw ard substitution . The equation is expanded as (11.2) 11 l ll lll l z z z z f f f f nn n n n n 11 21 22 31 32 33 12 1 2 3 1 2 3 00 0 .. . .. . ... . . . . . . . . . . = . l l l n nn n 11 12 1 22 22 21 2 11 1 1 2 3 0 0 0 0 . . . . . . . . . . ,,, ,, −− γ = f f f f n n n 1 2 1 . . . © 2003 by CRC CRC Press LLC
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154 Finite Element Analysis: Thermomechanics of Solids Starting from the lower-right entry, solution can be achieved using simple arith- metic as γ n = f n / l nn , In both procedures, only one unknown is encountered in each step (row). 11.1.2 M ATRIX T RIANGULARIZATION AND S OLUTION OF L INEAR S YSTEMS We next consider how to triangularize K . Suppose that the upper-left ( j 1) × ( j 1) block K j 1 has been triangularized: In determining whether the j × j block K j can be triangularized, we consider (11.3) in which k j is a ( j 1) × 1 array of the first j 1 entries of the j th column of K j . Simple manipulation suffices to furnish k j and l jj . (11.4) Note that λ j can be conveniently computed using forward substitution. Also, note that l jj = The fact that K j > 0 implies that l jj is real. Obviously, the triangularization process proceeds to the ( j + 1) st block and on to the complete stiffness matrix. As an example, consider (11.5) Clearly, For the second block, (11.6) γγ n n n nnn n n nnn nn n nn fl l l l −− − − =− 11 1 1 2 2 2 2 1 1 2 2 [] / , [ ] / , .
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This note was uploaded on 05/18/2011 for the course MAE 269A taught by Professor Ju during the Spring '11 term at UCLA.

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9780203009512.ch11 - 0749_Frame_C11 Page 153 Wednesday,...

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