FE2 multiscale in linear elasticity based on parametrized microscale models using proper generalized decomposition

In recent years, a tremendous growth of activity in multiscale modelling has been produced to get the mechanical response of highly heterogenous materials, where the complexity of solving numerically all microscale details is not feasible. In this paper we present a multiscale framework based on the decomposition of the displacement field into coarse (macro) and fine (micro) scales, earlier proposed in the Variational Multiscale approach or the hp-d method. The novelty of this work lies in solving the microscale step, where a multidimensional parametrized model of a generic RVE is solved, by means of the multidimensional model reduction technique, named as proper generalized decomposition (PGD). As result of this previous and off-line step, the displacement field over the RVE is obtained by simple algebraic operations for any combination of parameters (boundary condition, material properties, loads, etc.), allowing a significant reduction in computational cost when solving the macro scale problem. For problems where the detailed structure of the localized displacement, strain and stress fields are of interest, recovery of the fine scale components can be performed immediately. The basic concepts and several 1-D and 2-D linear elastic problems are presented to demonstrate the robustness of the formulation and computer time saving.