Routes for Efficient Computational Homogenization of Nonlinear Materials Using the Proper Generalized Decompositions

Hajer Lamari (1) , Amine Ammar (2) , Patrice Cartraud (1) , Grégory Legrain (1) , Francisco Chinesta (1) , Frédéric Jacquemin (1)

1 GeM - Institut de Recherche en Génie Civil et Mécanique
2 LAMPA - Laboratoire Angevin de Mécanique, Procédés et InnovAtion

Computational homogenization is nowadays one of the most active research topics in computational mechanics. Different strategies have been proposed, the main challenge being the computing cost induced by complex microstructures exhibiting nonlinear behaviors. Two quite tricky scenarios lie in (i) the necessity of applying the homogenization procedure for many microstructures (e.g. material microstructure evolving at the macroscopic level or stochastic microstructure); the second situation concerns the homogenization of nonlinear behaviors implying the necessity of solving microscopic problems for each macroscopic state (history independent nonlinear models) or for each macroscopic history (history dependant non-linear models). In this paper we present some preliminary results concerning the application of Proper Generalized Decompositions for addressing the efficient solution of homogenization problems. This numerical technique could allow to compute the homogenized properties for any microstructure or for any macroscopic loading history by solving a single but highly multidimensional model. The PGD allows circumventing the so called curse of dimensionality that mesh based representations suffer. Even if this work only describes the firs steps in a very ambitious objective , many original ideas are launched that could be at the origin of impressive progresses.

Domaines: Mécanique [physics.med-ph]

Hajer Lamari, Amine Ammar, Patrice Cartraud, Grégory Legrain, Francisco Chinesta, et al.. Routes for Efficient Computational Homogenization of Nonlinear Materials Using the Proper Generalized Decompositions. Archives of Computational Methods in Engineering, 2010, 17 (4), pp.373-391. ⟨10.1007/s11831-010-9051-4⟩. ⟨hal-01528268⟩