An overview of the proper generalized decomposition with applications in computational rheology

Abstract

is an open access repository that collects the work of Arts et Métiers ParisTech researchers and makes it freely available over the web where possible. b s t r a c tWe review the foundations and applications of the proper generalized decomposition (PGD), a powerful model reduction technique that computes a priori by means of successive enrichment a separated representation of the unknown field. The computational complexity of the PGD scales linearly with the dimension of the space wherein the model is defined, which is in marked contrast with the exponential scaling of standard grid-based methods. First introduced in the context of computational rheology by Ammar et al. [3,4], the PGD has since been further developed and applied in a variety of applications ranging from the solution of the Schrödinger equation of quantum mechanics to the analysis of laminate composites. In this paper, we illustrate the use of the PGD in four problem categories related to computational rheology: (i) the direct solution of the Fokker-Planck equation for complex fluids in configuration spaces of high dimension, (ii) the development of very efficient non-incremental algorithms for transient problems, (iii) the fully three-dimensional solution of problems defined in degenerate plate or shelllike domains often encountered in polymer processing or composites manufacturing, and finally (iv) the solution of multidimensional parametric models obtained by introducing various sources of problem variability as additional coordinates.

Models defined in stratified-plate domains. A review on efficient 3D PGD-based solution procedures

“…However, in the case of richer behaviors and/or geometries these simplified models fail to describe the more complex behaviors of the associated solutions. Thus for example we proved in Chinesta, Ammar et al (2011) that when averaging permeability in composite laminates the computed solution differs drastically from the fully 3D solution. The same conclusions apply in the case of thermal models that are formally similar to the ones related to the flow in porous media.…”

“…We now illustrate the application of separated representations to the modeling of resin transfer molding processes deeply addressed in Chinesta, Ammar et al (2011). We consider the flow within a porous medium in a plate domain Ξ = Ω × I with Ω ⊂ R 2 and I = [0, H] ⊂ R. The governing equation is obtained by combining Darcy's law, which relates the fluid velocity to the pressure gradient…”

“…The interested reader can refer to the recent reviews (Chinesta et al, 2010;Chinesta, Ammar et al, 2011;Chinesta, Ladeveze and Cueto, 2011) and the references therein.…”

“…Thus an in-plane-out-of-plane decomposition was proposed for solving flow problems in laminates (Chinesta, Ammar et al, 2011;Ghnatios et al, 2013), thermal problems in extruded geometries and laminates , elasticity problems in plates (Bognet et al, 2012) and shells geometries . In general the 3D solution was obtained from the solution of a sequence of 2D problems (the ones involving the in-plane coordinates) and 1D problems (the ones involving the coordinate related to the plate thickness).…”

A separated representation of an error indicator for the mesh refinement process under the proper generalized decomposition framework

“…This technique is able to evaluate a solution for all the parametric space circumventing the so called curse of dimensionality, typical of mesh-based techniques. The PGD technique has been successfully applied to a variety of different problems [3][4][5] showing a considerable improvement in computational speed and making possible to perform in real time and in light devices, such as tables or smartphones, complex calculations which, so far, were only possible in expensive computers.…”