Lecture Notes: Introduction to Finite Element Method |
Chapter 4. FE Modeling and Solution Techniques |
III. Equation Solving
Direct Methods (Gauss Elimination):
•Solution time proportional to NB2 (N is the dimension of the matrix, B the bandwidth)
•Suitable for small to medium problems, or slender structures (small bandwidth)
•Easy to handle multiple load cases
Iterative Methods:
•Solution time is unknown beforehand
•Reduced storage requirement
•Suitable for large problems, or bulky structures (large bandwidth, converge faster)
•Need solving again for different load cases
© 1997-2002 Yijun Liu, University of Cincinnati |
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Lecture Notes: Introduction to Finite Element Method |
Chapter 4. FE Modeling and Solution Techniques |
Gauss Elimination - Example:
8 −2 |
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Forward Elimination: |
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(1) + 4 x (2) (2): |
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(2) + 143 (3) (3):
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Back Substitution:
x3 =12 / 2 = 6
x2 = (−2 +12x3 ) /14 = 5 x1 = (2 + 2x2 ) / 8 =1.5
or Ax = b.
1.5 or x = 5 .
6
© 1997-2002 Yijun Liu, University of Cincinnati |
110 |
Lecture Notes: Introduction to Finite Element Method |
Chapter 4. FE Modeling and Solution Techniques |
Iterative Method - Example:
The Gauss-Seidel Method
Ax = b (A is symmetric)
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j =1 |
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Start with an estimate x( 0 ) and then iterate using the following:
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j =1 |
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for i =1, 2,..., N.
In vector form,
x(k +1) = AD −1 [b − AL x(k +1) − ALT x(k ) ],
where
AD = aii is the diagonal matrix of A,
A L is the lower triangular matrix of A,
such that A = AD + AL + ALT .
Iterations continue until solution x converges, i.e.
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≤ ε , |
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where ε is the tolerance for convergence control.
© 1997-2002 Yijun Liu, University of Cincinnati |
111 |