Abstract : Although the current numerical tools evolve quickly, the computation of mechanical problems faces many difficulties (complex geometries, complicated behavior, increasingly large sizes of structures ...). To avoid these constraints, many methods had been developped. Two approaches are particularly interesting : multi-scale approaches and parallel computations.
The work presented includes three parts : homogenization with The Extended Finite Element Method (X-FEM), a multi-scale analysis strategy and the development of X-FEM for parallel computations.
Regarding the first topic, the work concerns the field of the periodic homogenization theory which was developed at the time of the appearance of composite materials. This method proposes to define generalized mechanical characteristics of a structure including two or more phases with different properties. The structure is split up into repetitive volumes denoted R.V.E. (representative volume element). The computation of the microscopic problem, i.e. on the R.V.E., enables us to define the homogenized properties. The use of X-FEM, allowing discontinuous strains inside elements, associated to the Level Set technique, bringing an alternative to the representation of complex or random geometries, is employed for this purpose.
The aim of the second part is to solve mechanical problems on large structure containing details using a two-scale analysis. The structural scale is solved using a mesh which does not take into account the detail. The detail is taken into account by an ``extended'' homogenization. The homogenization is obtained from a local analysis of the detail on a sub-mesh of the structural one. On this local mesh, the X-FEM approach and the Level Set technique are used to carry out the analysis.
The last part deals with problems solved on several domains on a parallel machine. The purpose of this application is to increase the capacity of computation for large data problems. More precisely, we developed a strategy able to manage the enrichment of X-FEM on several meshes dispatched on a cluster of computers