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

Abstract

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.

Proper Generalized Decomposition for Parametric Study and Material Distribution Design of Multi-Directional Functionally Graded Plates Based on 3D Elasticity Solution

“…For some pioneering works in the context of the application of the PGD technique in plate-like structures, refer to [2][3][4][5]. There have also been some new publications in this field [6][7][8][9][10]. Although PGD is quite efficient in parametric and high-dimensional problems and the parametric analysis/optimization of FGM materials is highly demanding, there are no published works (to the best of the authors' knowledge) regarding the application of the PGD technique in FGM materials.…”

Artificial Intelligence Based Space Reduction of Structural Models

“…Thus, the unknown field (e.g., 3D displacement, temperature, …) is assumed expressed as a finite sum of products of functions, one depending on the in-plane coordinates (x,y) and the other depending on the through-the-thickness coordinate (z). This separated representation was applied in our former works in a diversity of physics: (i) elastic and inelastic problems defined in plate, shell and beam-like geometries, including dynamics, contact, plasticity, damage, fracture, … [1][2][3][4][5]; (ii) thermal models defined in plates and laminates [6][7]; (iii) flows of Newtonian and non-Newtonian fluids in thin flat and rough gaps [8][9][10][11][12]; (iv) electromagnetism in stratified composites [13]; … These formulations, in principle intrusive with respect to their use in existing commercial software, were successfully formulated for reducing intrusiveness [14][15][16][17]. The present paper constitutes a step forward.…”

Spurious-free interpolations for non-intrusive PGD-based parametric solutions: Application to composites forming processes

“…Recent efforts have been undertaken to formulate the PGD in a non-intrusive manner as much as possible [4,5,27]. Some non-intrusive versions of the PGD rely on external flow processing codes that controls validated open-source finite element (FE) codes or commercial software [16,20,26]. The external, homemade, code manages the assigned FE problems to the commercial software.…”