A PGD-based multiscale formulation for non-linear solid mechanics under small deformations

Model reduction techniques have became an attractive and a promising field to be applied in multiscale methods. The main objective of this work is to formulate a multiscale procedure for non-linear problems based on parametrized microscale models. The novelty of this work relies in the implementation of the model reduction technique known as Proper Generalized Decomposition for solving the high dimensional parametrized problem resulting from the microscale model. The multiscale framework here proposed is formulated to non-linear problems, specifically to material non-linearities, where material response is governed by a strain dependent evolution law. Two strategies to deal with this kind of problem under small deformations are detailed in this work. Both strategies based on parametrized microscale models solved by PGD have been applied to a problem with a rate-dependent isotropic damage model. First, a procedure where the problem is solved by uncoupling the equilibrium equation to the state variable expression has been explored. In order, to alleviate the parametrized microscale problem, a second strategy for problems with material non-linearity has been proposed, incorporating a parametrized microscale problem at each macroscale increment (FE-PGD). The basis of those procedures are described and compared, highlighting the solution accuracy and computer time consumption in comparison to a traditional finite element analysis.