On the Simulation of Kinetic Theory Models of Complex Fluids Using the Fokker-Planck Approach

330

310

International audienceThis paper revisits a powerful discretization technique, the Proper Generalized Decomposition-PGD, illustrating its ability for solving highly multidimensional models. This technique operates by constructing a separated representation of the solution, such that the solution complexity scales linearly with the dimension of the space in which the model is defined, instead the exponentially-growing complexity characteristic of mesh based discretization strategies. The PGD makes possible the efficient solution of models defined in multidimensional spaces, as the ones encountered in quantum chemistry, kinetic theory description of complex fluids, genetics (chemical master equation), financial mathematics, aEuro broken vertical bar but also those, classically defined in the standard space and time, to which we can add new extra-coordinates (parametric models, aEuro broken vertical bar) opening numerous possibilities (optimization, inverse identification, real time simulations, aEuro broken vertical bar)

Section: Remarkmentioning

confidence: 99%

“…In [34] we considered the simplest one, a fixed point strategy for solving some models associated with entangled polymeric systems.…”

Section: Solving the Fokker-planck Equationmentioning

confidence: 99%

Recent Advances and New Challenges in the Use of the Proper Generalized Decomposition for Solving Multidimensional Models

Chinesta

Ammar

Cueto

2010

330

310

“…If one proceeds to the solution of a model defined in a space of dimension N by using a standard mesh based discretization technique, where M nodes are used for discretizing each space coordinate, the resulting number of nodes reaches the astronomical value of M N . With M = 1000 (a very coarse description in practice) and N = 30 (a very simple model) the numerical complexity results 10 90 . It is important to recall that 10 80 is the presumed number of elementary particles in the universe!…”

Section: Introductionmentioning

confidence: 99%

A Short Review on Model Order Reduction Based on Proper Generalized Decomposition

Chinesta

Ladevèze²,

Cueto³

2011

550

437

This paper revisits a new model reduction methodology based on the use of separated representations, the so called Proper Generalized Decomposition-PGD. Space and time separated representations generalize Proper Orthogonal Decompositions-POD-avoiding any a priori knowledge on the solution in contrast to the vast majority of POD based model reduction technologies as well as reduced bases approaches. Moreover, PGD allows to treat efficiently models defined in degenerated domains as well as the multidimensional models arising from multidimensional physics (quantum chemistry, kinetic theory descriptions, . . .) or from the standard ones when some sources of variability are introduced in the model as extra-coordinates.

“…This technique was successfully applied for solving multidimensional models encountered in the kinetic theory description of complex fluids, involving steady state and transient linear and non-linear models [2][3][4]32]. Time-space separated representations were originally introduced many years ago by P. Ladeveze for addressing complex time-dependent nonlinear models within the LATIN framework (see [26] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

Routes for Efficient Computational Homogenization of Nonlinear Materials Using the Proper Generalized Decompositions

Lamari

Ammar

Cartraud

et al. 2010

Self Cite

Computational homogenization is nowadays one of the most active research topics in computational mechanics. Different strategies have been proposed, the main challenge being the computing cost induced by complex microstructures exhibiting nonlinear behaviors. Two quite tricky scenarios lie in (i) the necessity of applying the homogenization procedure for many microstructures (e.g. material microstructure evolving at the macroscopic level or stochastic microstructure); the second situation concerns the homogenization of nonlinear behaviors implying the necessity of solving microscopic problems for each macroscopic state (history independent nonlinear models) or for each macroscopic history (history dependant non-H. Lamari ( ) · P. Cartraud · G. Legrain linear models). In this paper we present some preliminary results concerning the application of Proper Generalized Decompositions-PGD-for addressing the efficient solution of homogenization problems. This numerical technique could allow to compute the homogenized properties for any microstructure or for any macroscopic loading history by solving a single but highly multidimensional model. The PGD allows circumventing the so called curse of dimensionality that mesh based representations suffer. Even if this work only describes the first steps in a very ambitious objective, many original ideas are launched that could be at the origin of impressive progresses.