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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 73
Edited by: B.H.V. Topping
Paper 89

Influence of Domain Decomposition on Solution of Equation Systems

J. Kruis and Z. Bittnar

Faculty of Civil Engineering, Department of Structural Mechanics, Czech Technical University in Prague, Czech Republic

Full Bibliographic Reference for this paper
J. Kruis, Z. Bittnar, "Influence of Domain Decomposition on Solution of Equation Systems", in B.H.V. Topping, (Editor), "Proceedings of the Eighth International Conference on Civil and Structural Engineering Computing", Civil-Comp Press, Stirlingshire, UK, Paper 89, 2001. doi:10.4203/ccp.73.89
Keywords: domain decomposition, condition number, parallel computations.

The influence of domain decomposition on number of iterations leading to acceptable accuracy is studied in this paper. The usual procedure of the domain decomposition technique consists of the direct method in the first phase which is followed by iterative method for solution of the reduced problem. Influence of shapes of subdomains on the iteration methods is in the center of attention. Several useful rules are mentioned and concluded.

Parallel computers are still much more used for scientific computations in all branches. There are clear reasons for their exploitation because larger and more complex problems are solved and modeled. Sequential computers are not able to fulfill such requirements especially thanks to their small memory. Compared to scientists and researchers, engineers in practice do not employ so considerably parallel computers because they are very busy and they have not enough time to study new technologies and algorithms. Hopefully, complicated problems will push them to exchange routine methods for the new ones. Big changes will emerge when the firms connect their computers for CAD and administration by fast network; in fact, they create cluster of computers. Clusters are more popular than the so-called massive parallel computers because they are much cheaper.

The change of computer platform brings, of course, changes to programs and algorithms. The main difference consists in data distribution because each processor contains only part of all data. It is clear that processors can effectively work with the data collected in their memory but other data are available through communication among nodes of parallel computer. The number of communications and the size of sent data have crucial influence to the efficiency of the program. Many routines used in connection with Finite Element Method can be used without changes, for example assembling of stiffness, mass, conductivity and other matrices.

Coupling among individual subdomains is well preserved in the final system of linear algebraic equations which contains data from all subdomains. This leads to difficulties because communication must be performed and communication is the bottleneck of parallel computation. The solvers of the equation systems must be replaced by the new ones. Domain decomposition methods are probably the most popular technique for solution of equation systems. They are based on the computation of the Schur's complements and on the iterative solution of the resulting system. The engineering community knows the computation of Schur's complements as static condensation or substructuring method which are clear and popular. The resulting system of equations can seldom be solved by a direct method. The main reason for rear usage of such method is nearly dense matrix and lack of memory. The time of solution by the finite method is almost always greater than the time consumed by an iterative preconditioned method.

This article deals with the influence of subdomain shapes on the solution of equation systems. The aim is to build general rules for appropriate decomposition of the original domain into subdomains. The square domains have been used for testing because there are large number of possible decompositions. Irrespective of any type of renumbering, the square domain possesses very broad bandwidth in sequential computation which evokes the obstacles with the memory. The original domain has been decomposed into the smaller square subdomains and into different rectangular subdomains even with aspect ratio . Only relatively small changes in the number of iterations have been observed The problem has been solved with homogeneous and heterogeneous materials. It is clear that the convergence is slower for heterogeneous material and this fact is documented by set of eigenvalues. There is obvious correspondence between the growth of the condition numbers and growth of the number of iterations.

Shape of the subdomains does not only influence the eigenvalues but also affects the memory requirements and the number of resulting unknowns. These two aspects are also very important especially for solution of extremely large problems on limited number of processors. Suitable renumbering in connection with appropriate decomposition can enlarge the size of solved problem. Requirements on equal number of nodes or elements on the subdomains are not enough because skyline profile can extremely differ and causes big differences in the consumed time.

The eigenvalues of the matrix of the resulting system have been computed for each decomposition. All eigenvalues, needed for better insight, have been computed only for selected problems and decompositions. In other cases only the minimal and maximal eigenvalues have been calculated. Distribution of the eigenvalues in the spectrum is not optimal and any preconditioner is feasible.

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