Solving Large Linearized Systems in Mechanics
Abstract
The solution of symmetric linearized systems of equations arising from applications of the finite element method to nonlinear analysis of structures is considered. Two related schemes that are based on Krylov sequences (e.g., Lanczos and conjugate gradient procedures) are examined. The conjugate gradient procedure is derived directly from the Lanczos process. In this derivation it is demonstrated that the conjugate gradient implicitly computes the triangular factors of the reduced tridiagonal system generated by the Lanczos process. The stability of the conjugate gradient procedure depends on that for the triangular factorization that can be guaranteed only for positive definite systems. Loss of orthogonality among the Lanczos vectors is also addressed and the method of partial reorthogonalization is used to maintain orthogonality. This approach improves the robustness of the algorithm but can be expensive for ill-conditioned systems. For such problems, preconditioning can help reduce the number of iterations required for convergence. Three different preconditioning schemes are considered; diagonal, element-by-element (E x E), and substructure-by-substructure (S x S). The above methods were applied to a number of different example problem. The S x S method is more effective than either the diagonal or the E x E methods at reducing both the computation cost and the number of iterations. For ill-conditioned systems the computation cost for E x E was less than that for diagonal preconditioners but as the condition number improved, the cost for the two methods were about the same.
Document Details
- Document Type
- Technical Report
- Publication Date
- Mar 01, 1992
- Accession Number
- ADA252723
Entities
People
- B. Nour-omid