Abstract
Stochastic simulation for finitely extensible non-linear elastic (FENE) dumbbells has been successfully applied (see the review paper of Keunings [R. Keunings, Micro–macro methods for the multiscale simulation viscoelastic flow using molecular models of kinetic theory, in: D.M. Binding, K. Walters (Eds.), Rheology Reviews, British Society of Rheology, 2004, pp. 67–98] and the references therein). The main difficulty in these simulations is related to the high number of realizations required for describing accurately the microstructural state due to Brownian effects. The discretisation of the Fokker–Planck equation with a mesh support (finite elements, finite differences, finite volumes, spectral techniques, … ) allows to go beyond the difficulty related to Brownian effects. However, kinetic theory models involve physical and conformation spaces. Thus, the molecular distribution depends on time, space as well as on the molecular orientation and extension (conformation coordinates). In this form the resulting Fokker–Planck equation is defined in a space of dimension 7. In the reduction technique proposed in this paper, a reduced approximation basis is constructed. The new shape functions are defined in the whole domain in an appropriate manner. Thus, the number of degrees of freedom involved in the solution of the Fokker–Planck equation is significantly reduced. The construction of those new approximation functions is done with an ‘a priori’ approach, which combines a basis reduction (using the Karhunen–Loève decomposition) with a basis enrichment based on the use of some Krylov subspaces. This numerical technique is applied for solving the FENE model of viscoelastic flows.
Highlights
In the last two decades, great progress has been attained in the numerical simulation of complex fluid flows such as polymer flow for example
The number of degrees of freedom involved in the solution of the Fokker–Planck equation is significantly reduced
The construction of those new approximation functions is done with an ‘a priori’ approach, which combines a basis reduction with a basis enrichment based on the use of some Krylov subspaces [13,14]
Summary
In the last two decades, great progress has been attained in the numerical simulation of complex fluid flows such as polymer flow for example. The purpose of this work is to present a robust and simplified model to carry out micro-simulations of molecular behavior within the kinetic theory framework This framework is useful when it allows to obtain constitutive equation for polymer. The purpose of this work is to propose an efficient and accurate discretisation technique able to solve the multidimensional Fokker–Planck equations with a substantial reduction in the number of degrees of freedom involved. This technique operates by extracting automatically, and in a way completely transparent for the user, the most relevant information of the unknown solution to construct the functional approximation from the information just extracted.
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