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

Article dans une revue avec comité de lecture
Author
CHINESTA, Francisco
10921 Institut de Recherche en Génie Civil et Mécanique [GeM]
AMMAR, Amine
211916 Laboratoire Angevin de Mécanique, Procédés et InnovAtion [LAMPA]
LEYGUE, Adrien
10921 Institut de Recherche en Génie Civil et Mécanique [GeM]
KEUNINGS, Roland
92863 Université Catholique de Louvain = Catholic University of Louvain [UCL]

URI
http://hdl.handle.net/10985/8473
DOI
10.1016/j.jnnfm.2010.12.012
Date
2011
Journal
Journal of Non-Newtonian Fluid Mechanics

Abstract

We 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] and [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 shell-like 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.

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