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/. Mpi, , vol.11

, // number of processes in parallel 14 int iproc = mpiRank(comm)

, 2+sin(2*pi*y/eps))/(2+1.8*sin(2*pi*x/eps))

/. Pde,

, // macro to simplify variational formulation 44 macro Aeps(u,v) (Grad(u)^T*Grad(v)*aeps) // 45 macro Grad(u)

. Chik,

, // RHS for the coarse P1 problem 68 69 matrix A=vA(P1Tri,P1Tri)

, // Stiffness matrix for the coarse P1 pb -> this matrix will be used in P1 region and changed for MSFEM region 70 real

, // RHS for the coarse P1 pb 71 real

, // Coefficient for the coupled formulation P1-MSFEM 56 matrix Pglobal

, 57 ifstream Pmat

&. Pmat and . Number,

, 59 real

, 60 int

&. Pmat and . Vala,

[. Pglobal= and J. A. Ia,

. &lt;&lt;, MsFEM solution computing, << endl

, cout << "defining fine problem, << endl

, 71 varf vBf(unused,v)= int2d(Thf)(f*v )

, Résumé Le travail de cette thèse a porté sur la simulation numérique des matériaux multi-échelles

, ) varient à une échelle petite par rapport à la taille du matériau. La thèse s'articule en deux parties qui correspondent à deux aspects di érents des problèmes multi-échelles, On considère des matériaux hétérogènes dont les propriétés physiques ou mécaniques

, Sur le plan théorique, nous avons considéré un matériau faiblement aléatoire (microstructure périodique avec ajout d'une perturbation aléatoire petite). Nous avons montré qu'en utilisant le correcteur standard issu de la théorie de l'homogénéisation aléatoire, nous sommes capables de calculer un tenseur Q qui gouverne complètement les uctuations de la réponse. Ce tenseur, dé ni par une formule explicite, permet d'estimer la uctuation de la réponse sans résoudre le problème n pour de nombreuses réalisations, Dans la première partie, on se place dans le cadre de l'homogénéisation aléatoire et on s'intéresse à une question plus ne que la caractérisation d'un comportement moyen : on cherche à étudier les uctuations de la réponse

.. Dans-la-deuxième-partie-de-la-thèse-;-périodicité, Dans cette deuxième partie, plusieurs taches ont été réalisées. Tout d'abord, une implémentation de plusieurs variantes MsFEM a été e ectuée sous forme de templates dans le logiciel de calcul Éléments Finis FreeFem++. Par ailleurs, plusieurs variantes des MsFEM pâtissent d'une erreur dite de résonance : lorsque la taille des hétérogénéités est proche de la taille du maillage grossier, la méthode devient très imprécise. Pour pallier ce problème, une méthode MsFEM enrichie a été développée : à la base MsFEM classique on rajoute des solutions de problèmes locaux ayant pour conditions aux limites des polynômes de haut degré, on considère un matériau hétérogène déterministe xé où les hypothèses classiques d'homogénéisation