Keywords: Cholesky decomposition, semidefinite matrices, generalized inverse, total FETI, fixing nodes, Perron vector.

With the use of finite element tearing and interconecting (FETI) methods the problem of domain decomposition arises, which is essential for the whole process. There are various software packages for domain decomposition, e.g. Chaco, Jostle, Metis, Scotch. Unfortunately we do not know of any package that fits all our requirements. In this paper, we define our ideal decomposition and we describe the algorithm used to "repair" the decomposition obtained using the Metis software [1]. After obtaining such a domain decomposition, we can compute the partial solution on each subdomain using the generalised inverse scheme.

A typical example where we can exploit a generalized inverse is a system of consistent linear equations with symmetric positive semidefinite (SPS) matrix arising in the stress analysis of a "floating" static structure whose essential boundary conditions are not sufficient to prevent its rigid body motions [2,3,4]. This system can be solved by standard direct methods for the solution of systems with positive definite matrices, such as the Cholesky decomposition, adapted for the solution of systems with only a positive semidefinite matrix. Due to the rounding errors, the main difficulty in implementation of the FETI method is effective elimination of the displacements, in particular the evaluation of the action of the generalized inverse of the SPS stiffness matrices of "floating" subdomains. To alleviate this problem, Farhat and Géradin [2] proposed to combine the Cholesky decomposition with the singular value decomposition (SVD) of a relatively small matrix.

Our modification based on the active choice of the SVD partly uses the fixing nodes strategy to make the system as stiff as possible and has been implemented in our solver [5]. This solver uses the Lagrange multipliers not only for gluing the subdomains along auxiliary interfaces, but also for the implementation of the essential boundary conditions [6].

We have defined the fixing nodes as set of nodes, that stabilize the solution well. For finding fixing nodes we use some results of graph theory, especially from spectral graph theory. As a result, we present the technical problem and its solution and also the parallel scalability of a given problem.

1

G. Karypis, V. Kumar, "METIS manual Version 4.0", http://glaros.dtc.umn.edu/gkhome/views/metis, University of Minnesota, 1998.

2

C. Farhat, M. Géradin, "On the general solution by a direct method of a large scale singular system of linear equations: application to the analysis of floating structures", International Journal for Numerical Methods in Engineering, 41, 675-696, 1998. doi:10.1002/(SICI)1097-0207(19980228)41:4<675::AID-NME305>3.3.CO;2-#

3

C.A. Felipa, K.C. Park, "The construction of free-free flexibility matrices for multilevel structural analysis", Computer Methods in Applied Mechanics and Engineering, 191, 2111-2140, 2002. doi:10.1016/S0045-7825(01)00379-6

4

M. Papadrakakis, Y. Fragakis, "An integrated geometric-algebraic method for solving semi-definite problems in structural mechanics", Computer Methods in Applied Mechanics and Engineering", 190, 6513-6532, 2001. doi:10.1016/S0045-7825(01)00234-1

5

T. Kozubek, A. Markopoulos, T. Brzobohatý, R. Kucera, V. Vondrák, "MatSol - MATLAB efficient solvers for problems in engineering", http://www.am.vsb.cz/matsol, VŠB-Technical University of Ostrava, 2008.

6

Z. Dostál, D. Horák, R. Kucera, "Total FETI - an easier implementable variant of the FETI method for numerical solution of elliptic PDE", Communications in Numerical Methods in Engineering, 22, 1155-1162, 2006. doi:10.1002/cnm.881

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