Computational Reduction Techniques
7.8 Preconditioned iterative solution technique
There are also variations of iterative solutions when the matrix is precondi-tioned to accelerate the convergence. The essence of preconditioning is by premultiplying the problem with a suitable preconditioning matrix as
P−1Ku = P−1F.
The conjugate gradient algorithm allows the implicit application of the pre-conditioner during the iterative process. The prepre-conditioner is commonly presented as an inverse matrix, because of the goal of approximating the in-verse of the system matrix. Clearly with the selection of
P = K the solution is trivial, since
P−1Ku = K−1Ku = Iu = K−1F.
The cost of the computation of the preconditioner is of course in this case the cost of the factorization. The iterative solution itself is a forward-backward substitution.
More practical approaches use various incomplete factorizations of the ma-trix
K≈ CCT = P.
In an incomplete factorization process only those terms of the factor matrix are computed that correspond to nonzero terms of the original matrix. The fill-in terms are omitted. There are several variations of this approach, such as also computing fill-in terms, but only above a certain threshold value. An-other extension is to compute the fill-in terms inside a certain band of the matrix.
It is another commonly used approach to exploit the fact that the matrix is assembled from finite element matrices. In that case the preconditioners are various factorizations of the element matrices as
Ke≈ CeCeT = Pe, assembled into a global preconditioner as
P = Σne=1Pe.
Such approaches are described in [9] and [10]. The preconditioning approach is the most successful however when the preconditioner captures and exploits some physics modeling specific information.
There are cases when the stiffness matrix may consist of two components, one that is easy to solve and one that is difficult. An example is static aero-elastic analysis, where there is a Ksstructural stiffness matrix and a matrix Ka representing the aero-elastic effects. The static aero-elastic solution is de-scribed by the equation of form
(Ks+ Ka)u = F.
Ks is generally sparse and banded and can be solved directly by itself at a reasonable cost. The Ka matrix is developed by a theory where every point of the structure that is touched by air is coupled to every other point touching air. This leads to a dense, largely un-banded, and unsymmetric matrix.
In general, an unsymmetric problem is at least twice as expensive to solve as a symmetric problem, and the other unfortunate characteristics of Ka result in an order of magnitude increase in solution cost. Therefore, preconditioning the static aero-elastic equation with
P = Ks
The inverse operation is done with a factorization followed by a forward-backward substitution with the vector of unknowns from the right hand side.
The structural stiffness matrix acts as a pre-conditioner. u0 is set to zero, re-sulting in u1being the static solution due to the structural effects only. Only one matrix factorization is required for all the iterations. The troublesome Ka is never factorized.
The physics of the problem dictates that the terms in Ksare much larger than in Kabecause the structure is stiffer than the air it intersects. This leads to a rapid convergence of this solution in connection with, for example, the bi-conjugate gradient method. The cost of the coupled aero-elastic solution by this approach is generally no more that twice the cost of a structure only solution.
There are several other classical splitting methods, such as the Gauss-Seidel or the successive over-relaxation methods [13]. The concept in itself is valu-able for engineers when they combine different physical phenomena and they know the coupling and partitioning a priori.
There are also other minimization based methods, such as the generalized minimum residual method [12]. There is also another class of solutions based on an adaptive idea [11]. Both of these are subjects of strong academic in-terest but have not yet been proven as generally useful in the industry as the conjugate gradient method.
References
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[3] George, A. and Liu, J. W. H.; The evolution of the minimum degree ordering algorithm, SIAM Review, Vol. 31, pp. 1-19, 1989
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49, pp. 409-436, 1952
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Reine Angewandte Math., Vol. 30., pp. 51-94, 1846
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of Scientific and Statistical Computing, Vol. 10, pp. 1136-1145, 1988 [7] Karypis, G. and Kumar, V.; ParMETIS:Parallel graph partitioning and
sparse matrix library, University of Minnesota, 1998
[8] Mayer, S.; Distributed parallel solution of very large systems of linear equations in the finite element method, PhD Thesis, Technical Univer-sity of Munich, 1998
[9] Poschmann, P.; and Komzsik, L.; Iterative solution technique for finite element applications, Journal of Finite Element Analysis and Design, Vol. 14, No. 4, pp. 373-381, 1993
[10] Komzsik, L., Sharapov, I., Poschmann, P.; A preconditioning technique for indefinite linear systems, Journal of Finite Element Analysis and Design, Vol. 26, No. 3, pp. 253-258, 1997
[11] R¨ude, U.; Fully adaptive multigrid methods, SIAM Journal of Numerical Analysis, Vol. 30, pp. 230-248, 1993
[12] Saad, Y. and Schultz, M. H.; GMRES: a generalized minimum residual algorithm for solving nonsymmetric linear systems, SIAM, Journal of Scientific and Statistical Computing, Vol. 7, pp. 856-869, 1986
[13] Varga, R. S.; Matrix Iterative Analysis, Prentice-Hall, Englewod Cliffs, New Jersey, 1962
The Kaa matrix is free of constraints and ready for analysis. However, for computational advantages it may be further reduced. Let us partition the remaining degrees of freedom into two groups. One such physical partitioning may be based on considering the boundary degrees of freedom as one, and the interior degrees of freedom as the other partition. This is a single-level, single-component static condensation, the topic of Section 8.1. One of the first publications related to this topic is [2].
It is also possible to first partition the model into components and apply the boundary reduction for each. This is the single-level, multiple-component condensation of Section 8.2. Finally, the two techniques may be recursively applied yielding the multiple-level, multiple-component method of Section 8.3.
All of these methods allow the possibility of parallel processing. Due to the cost of reduction and back-transformation, however, the computational complexity of solving a certain problem may not change. The static conden-sation methods produce computationally exact results when applied to the linear static problem. The following sections detail the static condensation technique.