A Parallel Sparse Direct Solver to Improve Scalability of FETI Methods

In structural mechanics, numerical simulations with finite element codes often lead to very large sparse linear systems. Solving these linear systems is really very expensive in terms of computational time and memory requirements. Domain decomposition methods are a natural way to reduce these computational cost by parallelizing these codes. One of the most commonly used method is FETI 1 [1, 2, 3]. It is a non-overlapping domain decomposition method based on a "dual approach" which consists to introduce continuity condition at the interfaces between subdomains. The FETI method has the advantage of being robust and well suited to the problems studied in structural mechanics. However, some studies carried out on large numerical tests (tens of millions of degrees of freedom) showed that the effectiveness of this method is decreasing beyond a number of subdomains (a few hundred). If we want to split largescale models in a reasonable number of subdomains, we will be faced with very large local systems. Moreover, with the evolution of microprocessor technology in terms of power, we are witnessing the birth of new massively multi-core architectures. The essential interest and strength of multi-core solutions is to enable the simultaneous execution of a task by core. Some parallel algorithms and software can take full advantage of these massively multi-core systems. The main object of this paper is to implement a direct solver for sparse linear systems which can be symmetric or non-symmetric, real or complex, with single or multiple right-rand sides. The implementation, based on a nested dissection technique [4], is completed by a useful point in many domain decomposition methods (building a preconditioner or formulation of the FETI operator): handling of zero-energy modes of singular systems. We parallelize the solver through a model of shared memory parallelism (multi-threading) [5] to take advantage of new multi-core processors. We integrate this multithreads version in FETI to solve local problems in parallel. We carry out some parallel tests in ZeBuLon FEA code to evaluate the scalability of FETI method. The results of this paper highlight the usefulness of the work and interest to use as local solver in FETI methods a parallel direct solver which is robust and efficient. This can give access to new ranges of complex problems in structural mechanics.