Comparison of multi-level approaches in domain decomposition for structural analysis
Résumé
Domain decomposition methods are well suited to multiprocessor computers with either shared or distributed memory, which are now the most powerful computers known. When increasing the number of subdomains, the maximum efficiency expected from domain decomposition methods is not attained. Use of a global mechanism to propagate information among all of the substructures can overcome this drawback. Such a mechanism has now been implemented on several domain decomposition-like algorithms, such as the FETI method or the “Balancing Domain Decomposition method”. The approach used in this work is based upon the LArge Time INcrement (LATIN) method, coupled with a substructuring technique, in order to solve the problem concurrently. The initial structure is decomposed into substructures and interfaces. Since each possesses its own behaviour and equations, the unknowns in the problem are both the displacement and the efforts on the interfaces. The resulting approach then is a “mixed” domain decomposition approach. In initial extension to the LATIN approach, which takes into account the two scales arising from the substructuring, has been applied in order to improve performance. A more complete extension pertains to the homogenisation techniques between a macro level and a micro level, which are also derived from the substructuring. This approach is described for the case of linear elasticity and perfect interfaces. The macro level is chosen to be the “homogenised” structure (in order to represent the large-scale effects), to propagate the information globally. The choice of this macro-level problem then allows us to express the operators that transfer information between the micro and macro levels throughout the iterations of the algorithm. The displacement within each substructure, as well as the displacement field on the interfaces, are additively split according to the definition of the two levels. The stress field and effort on the interfaces are then split by duality when separating the energy on the two levels. As a first step, we focus herein on linear elasticity and perfect interfaces; however, the resulting multi-level approach naturally takes place within the LATIN framework, which has formerly been designed for non-linear evolution problems. Last, a comparison of the mixed domain decomposition methods is proposed.
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