Keywords: parallel direct solver, domain decomposition, nested domain decomposition, h-adaptive refinement, hierarchical data structure, finite element analysis.

Ródenas et al. [1] showed the hierarchical relationships between geometrically similar finite elements. Under this similarity condition, the terms involved in the evaluation of element stiffness matrices, corresponding to parent and child elements, are related by a constant which is a function of the ratio of the element sizes (scaling factor). These and other parent-child relations where then used to develop a basic implementation of a hierarchical h-adaptive refinement program (based on element subdivision, and the use of multi-point constraints to satisfy the C⁰ continuity) in order to create a framework that could help us to explore the possibilities of the use of hierarchical relations in h-adaptivity to improve the performance of adaptive analyses. The program uses a data structure that accounts for hierarchical relations to reduce to a minimum the number of element stiffness matrices that need to be evaluated, simplifies the mesh generation process, reduces the computational cost associated to the evaluation of volume integrals, etc.

The data structure was used in [2,3] to create a nested domain decomposition (NDD) that generates stiffness matrices with a nested arrowhead structure. This allowed for the development of efficient domain decomposition sequential direct solvers. These references already mentioned that the algorithms were suitable for parallelization

This paper shows the following benefits provided by the hierarchical data structure associated to the resolution of the system of equations. On the one hand, the NDD reordering, directly provided by the data structure at no additional cost, has shown to outperform the behaviour of the rest of reordering techniques analyzed in the paper. On the other hand, the hierarchical data structure has shown its usefulness in the definition of parallel domain decomposition solver based on MatlabR [4]. The results obtained are satisfactory but suggest the implementation of the algorithm in a more efficient programming language such as C++ to improve the performance of the code. Further improvements of the parallel solver could also be obtained by using the NDD solver presented in [3].

1

J.J. Ródenas, J.E. Tarancón, J. Albelda, A. Roda, F.J. Fuenmayor, "Hierarquical properties in elements obtained by subdivision: a hierarquical h-adaptivity program", Adaptive Modeling and Simulation 2005, 420-435, 2005.

2

J.J. Ródenas, J. Albelda, C. Corral, J. Mas, "Efficient implementation of domain decomposition methods using a hierarchical h-adaptive Finite Element program", III European Conference on Computational Mechanics. Solids, Structures and Coupled Problems in Engineering, Book of Abstracs, 2006.

3

J.J. Ródenas, C. Corral, J. Albelda, J. Mas, C. Adam, "Solver directo de división recursiva en subdominios basado en un programa de refinamiento h-adaptable de estructura jerárquica", Métodos Numéricos e Computacionais em Engenharia, CMNE CILAMCE, 2007.

4

Matlab® 7.9.0.529 (R2009b), The MathWorks Inc, Natick, MA, 2009.

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