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PROCEEDINGS OF THE ELEVENTH INTERNATIONAL CONFERENCE ON CIVIL, STRUCTURAL AND ENVIRONMENTAL ENGINEERING COMPUTING
Edited by: B.H.V. Topping
Efficient Iterative Solution of Stochastic Finite Element Equations
Department of Civil and Environmental Engineering, University of Cyprus, Nicosia, Cyprus
D.C. Charmpis, "Efficient Iterative Solution of Stochastic Finite Element Equations", in B.H.V. Topping, (Editor), "Proceedings of the Eleventh International Conference on Civil, Structural and Environmental Engineering Computing", Civil-Comp Press, Stirlingshire, UK, Paper 40, 2007. doi:10.4203/ccp.86.40
Keywords: stochastic finite element, conjugate gradient method, iterative, solver, preconditioner, incomplete factorization.
The stochastic finite element (SFE) method is a widely appreciated approach to treat structural mechanics applications involving uncertain material and geometric properties with spatial distribution. The research effort devoted during the last decades to the field of stochastic structural mechanics have led to the development of various rather sophisticated SFE formulations, however little progress has been reported on the computational efficiency and feasibility of such formulations when confronted with large-scale real-world problems. As a result, SFE applications usually involve intentionally simple and small-scale structural models and the results obtained are of limited practical importance. The computational burden associated with SFE analyses may be substantially reduced by appropriately handling the most demanding tasks in terms of processing power and storage space needs. The Monte Carlo simulation (MCS) technique, which is the most effective and widely applicable method for handling large-scale SFE problems with complicated structural response, involves expensive computations due to the successive analyses required. More specifically, successive linear systems of equations with multiple left-hand sides have to be processed, since the stiffness matrix changes in every simulation. The standard direct method based on Cholesky factorization remains the most popular scheme for solving such equations, however this solution approach exhibits poor performance for large-scale problems.
The deficiencies of the direct approach can be overcome with the use of a customized version of the preconditioned conjugate gradient (PCG) method, which allows the adaptation of this solution scheme to the special features of nearby problems encountered in finite element reanalyses. PCG can be customized to take into account the relatively small differences between stiffness matrices in successive simulations, avoiding this way the treatment of each simulation's system as a stand-alone problem . PCG-customization is localized at the preconditioning matrix employed to accelerate PCG convergence during the successive finite element solutions. Hence, the reanalysis problems can be effectively solved using the PCG algorithm equipped with a preconditioner following the rationale of incomplete Cholesky preconditionings. According to this rationale, the preconditioning matrix may be taken as the complete factorized stiffness matrix of the initial simulation. With the preconditioning matrix remaining the same during the successive finite element reanalyses, the repeated solutions required for the preconditioning step of the PCG algorithm can be efficiently treated as problems with multiple right-hand sides. Therefore, the stiffness matrix of the initial simulation is retained in memory in a factorized form throughout all simulations. The present work, which is a continuation of , proposes alternative preconditioning schemes, which are based on incomplete factorizations of the stiffness matrix of the initial simulation. Such preconditioners aim in retaining only the essential numerical information during the factorization of the initial stiffness matrix. Therefore, the matrix terms stored during factorization are selected based on their position within the stiffness matrix (incomplete factorization by position) or their magnitude (incomplete factorization by magnitude). Thus, preconditioner storage demands are reduced, while PCG iteration performance is not strongly affected when incomplete factorization yields a sufficiently strong preconditioner.
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