Non-intrusive proper generalized decomposition involving space and parameters: application to the mechanical modeling of 3D woven fabrics

In our former works we proposed different Model Order Reduction strategies for alleviating the complexity of computational simulations. In fact we proved that separated representations are specially appealing for addressing many issues, in particular, the treatment of 3D models defined in degenerated domains (those involving very different characteristic dimensions, like beams, plate and shells) as well as the solution of parametrized models for calculating their parametric solutions. However it was proved that the efficiency of solvers based on the construction of such separated representations strongly depends on the affine decompositions (separability) of operators, parameters and geometry. Even if our works proved that different techniques exists for performing such beneficial separation prior of applying the separated representation constructor, the complexity of the solver increases in certain circumstances too much, as the one involving the space separation of complex microstructures concerned by 3D woven fabrics. In this paper we explore an alternative route that allows circumventing the just referred difficulties. Thus, instead of following the standard procedure that consists of introducing the separated representation of the unknown field prior to discretize the models, the strategy here proposed consists of proceeding inversely: first the model is discretized and then the separated representation of the discrete unknown field is enforced. Such a procedure enables the consideration of very complex and non separable features, like complex domains, boundary conditions and microstructures as the ones concerned by homogenized models of complex and rich 3D woven fabrics. It will be proved that such a procedure can be also easily coupled with a non-intrusive treatment of the parametric dimensions by using a sparse hierarchical collocation technique.