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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 89
PROCEEDINGS OF THE SIXTH INTERNATIONAL CONFERENCE ON ENGINEERING COMPUTATIONAL TECHNOLOGY
Edited by: M. Papadrakakis and B.H.V. Topping
Paper 76

Preconditioning of Iterative Methods Based on an Aggregation Algorithm

J. Kruis1 and P. Mayer2

1Department of Mechanics, 2Department of Mathematics,
Czech Technical University in Prague, Czech Republic

Full Bibliographic Reference for this paper
J. Kruis, P. Mayer, "Preconditioning of Iterative Methods Based on an Aggregation Algorithm", in M. Papadrakakis, B.H.V. Topping, (Editors), "Proceedings of the Sixth International Conference on Engineering Computational Technology", Civil-Comp Press, Stirlingshire, UK, Paper 76, 2008. doi:10.4203/ccp.89.76
Keywords: aggregation algorithm, iterative methods, preconditioning, plate problems.

Summary
Preconditioning of iterative methods is still at the center of attention of the computational community. Really large systems of algebraic equations are solved by an iterative method if only single processor computation is available. Plate or shell analysis in mechanics very often results in an ill-conditioned problem which is characterized by a large ratio of the maximum to the minimum eigenvalue. The ill-conditioned problems cause severe difficulty because the iterative methods need to perform many iterations to attain prescribed error.

A suitable preconditioning can significantly reduce the number of required iterations. This contribution deals with preconditioning based on the BOSS method which was introduced by Brezina [1]. Unknowns are collected to aggregates and a smoothing procedure is applied. The originally nonoverlapping covering is changed to an overlapping covering with controlled overlap. Several auxiliary matrices are defined on aggregates and correction operators are established. The BOSS method deals also with a coarse problem. Each aggregate is assigned by a one new variable. There is also the coarse correction operator. The method can be used for solution of systems of equations as an iterative method.

The BOSS method is used as a preconditioner of the conjugate gradient method. At this time, a single processor implementation is available but the aim is to implement the BOSS method in a parallel environment together with the conjugate gradient method. Attention has to be devoted to proper ordering of aggregate in order to obtain efficient implementation. It will be a suitable alternative to domain decomposition methods [2].

The behaviour of the conjugate gradient method with the BOSS preconditioner is described on a plane stress problem and two plate problems. The plate problems lead to the matrices with different condition numbers which affect the number of iterations. The proposed algorithm is efficient for the problems with very high condition number while for other problems may lead to greater elapsed time.

References
1
M. Brezina, "Robust Iterative Methods on Unstructured Meshes", Ph.D. Thesis, University of Colorado at Denver, 1997.
2
J. Kruis, "Domain Decomposition Methods for Distributed Computing", Saxe-Coburg Publications, Stirling, Scotland, UK, 2006.

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