1,120 research outputs found
An overview of the proper generalized decomposition with applications in computational rheology
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
Gaussian approximation for finitely extensible bead-spring chains with hydrodynamic interaction
The Gaussian Approximation, proposed originally by Ottinger [J. Chem. Phys.,
90 (1) : 463-473, 1989] to account for the influence of fluctuations in
hydrodynamic interactions in Rouse chains, is adapted here to derive a new
mean-field approximation for the FENE spring force. This "FENE-PG" force law
approximately accounts for spring-force fluctuations, which are neglected in
the widely used FENE-P approximation. The Gaussian Approximation for
hydrodynamic interactions is combined with the FENE-P and FENE-PG spring force
approximations to obtain approximate models for finitely-extensible bead-spring
chains with hydrodynamic interactions. The closed set of ODE's governing the
evolution of the second-moments of the configurational probability distribution
in the approximate models are used to generate predictions of rheological
properties in steady and unsteady shear and uniaxial extensional flows, which
are found to be in good agreement with the exact results obtained with Brownian
dynamics simulations. In particular, predictions of coil-stretch hysteresis are
in quantitative agreement with simulations' results. Additional simplifying
diagonalization-of-normal-modes assumptions are found to lead to considerable
savings in computation time, without significant loss in accuracy.Comment: 26 pages, 17 figures, 2 tables, 75 numbered equations, 1 appendix
with 10 numbered equations Submitted to J. Chem. Phys. on 6 February 200
On the deterministic solution of multidimensional parametric models using the Proper Generalized Decomposition
This paper focuses on the efficient solution of models defined in high dimensional spaces. Those models involve numerous numerical challenges because of their associated curse of dimensionality. It is well known that in mesh-based discrete models the complexity (degrees of freedom) scales exponentially with the dimension of the space. Many models encountered in computational science and engineering involve numerous dimensions called configurational coordinates. Some examples are the models encoun- tered in biology making use of the chemical master equation, quantum chemistry involving the solution of the Schrödinger or Dirac equations, kinetic theory descriptions of complex systems based on the solution of the so-called Fokker–Planck equation, stochastic models in which the random variables are included as new coordinates, financial mathematics, etc. This paper revisits the curse of dimensionality and proposes an efficient strategy for circumventing such challenging issue. This strategy, based on the use of a Proper Generalized Decomposition, is specially well suited to treat the multidimensional parametric equations
Corrections to Einstein's relation for Brownian motion in a tilted periodic potential
In this paper we revisit the problem of Brownian motion in a tilted periodic
potential. We use homogenization theory to derive general formulas for the
effective velocity and the effective diffusion tensor that are valid for
arbitrary tilts. Furthermore, we obtain power series expansions for the
velocity and the diffusion coefficient as functions of the external forcing.
Thus, we provide systematic corrections to Einstein's formula and to linear
response theory. Our theoretical results are supported by extensive numerical
simulations. For our numerical experiments we use a novel spectral numerical
method that leads to a very efficient and accurate calculation of the effective
velocity and the effective diffusion tensor.Comment: 29 pages, 7 figures, submitted to the Journal of Statistical Physic
High order resolution of the Maxwell-Fokker-Planck-Landau model intended for ICF applications
A high order, deterministic direct numerical method is proposed for the
nonrelativistic Vlasov-Maxwell system, coupled
with Fokker-Planck-Landau type operators. Such a system is devoted to the
modelling of electronic transport and energy deposition in the general frame of
Inertial Confinement Fusion applications. It describes the kinetics of plasma
physics in the nonlocal thermodynamic equilibrium regime. Strong numerical
constraints lead us to develop specific methods and approaches for validation,
that might be used in other fields where couplings between equations,
multiscale physics, and high dimensionality are involved. Parallelisation (MPI
communication standard) and fast algorithms such as the multigrid method are
employed, that make this direct approach be computationally affordable for
simulations of hundreds of picoseconds, when dealing with configurations that
present five dimensions in phase space
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