Keywords: finite elements, substructuring, iterative methods, parallel computing.

The BDD method is reinterpreted as a method that acts on a condensed system of equations, in which the equations of the internal nodes have been condensed on the interface equations. The condensed system of equations is never computed in the proposed technique. The matrix vector multiplication of the condensed system is implemented as a sequence of sparse matrix vector multiplications. The main results of reinterpreting the BDD technique are: reduction of one communication cycle, reduction of the data which need to be communicated to the interface nodes only, reduction of the number of matrices that need to be stored.

An object oriented implementation of the BDD method is briefly described. Both the matrix vector multiplication as a sequence of subdomain matrix vector multiplications and the computation of the correction as the sum of coarse and fine mesh contributions are implemented as a matrix vector multiplication. This allows the use of the BDD correction as a preconditioner of an abstract conjugate gradient method available in our code. The key message is that the intricacies of the BDD method are hidden behind a simple user interface. The method was integrated with our object oriented finite element code [2] and tested on two and three dimensional scalar partial differential equations and systems of partial differential equations.

The efficiency of the implementation was tested when applied to the solid model of an eight story building modeled using GID [3] software. The target machine is a dual quad core Macintosh Pro computer with 12 Gb of core memory. The building was partitioned in a variable number of subdomains using the Metis 4.0 library [4]. Perfect speedup was obtained for all configurations when compared to the scalar times. The total execution time for all configurations was balanced. In the case of eight subdomains most time was spent in inverting the subdomain matrices. In the case of large numbers of subdomains (up to 200) most time was spent in the iterative process.

1

C.R. Dohrmann, "A Preconditioner for Substructuring Based on Constrained Energy Minimization", SIAM J. Sci Comput., 25(1), 246-258, 2003. doi:10.1137/S1064827502412887

2

P.R.B. Devloo, "PZ - Object Oriented Finite Element Software", Technical report, 2005.

3

R. Ribó, M.P. Riera, "GiD 10 - The Personal Pre and post processor (version 10.0)", International Center For Numerical Methods In Engineering (CIMNE), 2010.

4

G. Karypis, "METIS - A Software Package for Partitioning Unstructured Graphs, Partitioning Meshes, and Computing Fill-Reducing Orderings of Sparse Matrices", University of Minnesota - Karypis Lab, 1998.

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