International audienceIn multiscale analysis of components, there is usually a need to solve microstructures with complex geometries. In this paper, we use the extended finite element method (X-FEM) to solve scales involving complex geometries. The X-FEM allows one to use meshes not necessarily matching the physical surface of the problem while retaining the accuracy of the classical finite element approach. For material interfaces, this is achieved by introducing a new enrichment strategy. Although the mesh does not need to conform to the physical surfaces, it needs to be fine enough to capture the geometry of these surfaces. A simple algorithm is described to adaptively refine the mesh to meet this geometrical requirement. Numerical experiments on the periodic homogenization of two-phase complex cells demonstrate the accuracy and simplicity of the X-FEM

La méthode des éléments finis étendue X-FEM permet d'enrichir les fonctions de forme éléments finis, et donc de prendre en compte des discontinuités au sein d'un élément. Elle est appliquée ici à la résolution de problèmes de micromécanique, pour faciliter le maillage de la cellule, qui peut ainsi ne pas respecter l'interface matériau. Une nouvelle fonction d'enrichissement est proposée, et une solution numérique de même qualité qu'une approche éléments finis classique est obtenue. Plusieurs exemples numériques sont présentés, sur des applications matériau, et sur une structure câble.ABSTRACT. The eXtended Finite Element Method (X-FEM) allows one to enrich finite element approximation space, thus accounting for discontinuities inside an element. This method is applied in this paper to the solution of micromechanical problem, in order to simplify the mesh generation, since it does not need to conform to the material interfaces. A new enrichment function is proposed, which turns out to have the same accuracy as the classical finite element method. Numerical experiments are presented, for material applications, and for a strandedrope structure. MOTS-CLÉS : homogénéisation, X-FEM, partition de l'unité, level set.