990 research outputs found
The fully-implicit log-conformation formulation and its application to three-dimensional flows
The stable and efficient numerical simulation of viscoelastic flows has been
a constant struggle due to the High Weissenberg Number Problem. While the
stability for macroscopic descriptions could be greatly enhanced by the
log-conformation method as proposed by Fattal and Kupferman, the application of
the efficient Newton-Raphson algorithm to the full monolithic system of
governing equations, consisting of the log-conformation equations and the
Navier-Stokes equations, has always posed a problem. In particular, it is the
formulation of the constitutive equations by means of the spectral
decomposition that hinders the application of further analytical tools.
Therefore, up to now, a fully monolithic approach could only be achieved in two
dimensions, as, e.g., recently shown in [P. Knechtges, M. Behr, S. Elgeti,
Fully-implicit log-conformation formulation of constitutive laws, J.
Non-Newtonian Fluid Mech. 214 (2014) 78-87].
The aim of this paper is to find a generalization of the previously made
considerations to three dimensions, such that a monolithic Newton-Raphson
solver based on the log-conformation formulation can be implemented also in
this case. The underlying idea is analogous to the two-dimensional case, to
replace the eigenvalue decomposition in the constitutive equation by an
analytically more "well-behaved" term and to rely on the eigenvalue
decomposition only for the actual computation. Furthermore, in order to
demonstrate the practicality of the proposed method, numerical results of the
newly derived formulation are presented in the case of the sedimenting sphere
and ellipsoid benchmarks for the Oldroyd-B and Giesekus models. It is found
that the expected quadratic convergence of Newton's method can be achieved.Comment: 21 pages, 9 figure
Fully-implicit log-conformation formulation of constitutive laws
Subject of this paper is the derivation of a new constitutive law in terms of
the logarithm of the conformation tensor that can be used as a full substitute
for the 2D governing equations of the Oldroyd-B, Giesekus and other models. One
of the key features of these new equations is that - in contrast to the
original log-conf equations given by Fattal and Kupferman (2004) - these
constitutive equations combined with the Navier-Stokes equations constitute a
self-contained, non-iterative system of partial differential equations. In
addition to its potential as a fruitful source for understanding the
mathematical subtleties of the models from a new perspective, this analytical
description also allows us to fully utilize the Newton-Raphson algorithm in
numerical simulations, which by design should lead to reduced computational
effort. By means of the confined cylinder benchmark we will show that a finite
element discretization of these new equations delivers results of comparable
accuracy to known methods.Comment: 21 pages, 5 figure
The finite-volume method in computational rheology
The finite volume method (FVM) is widely used in traditional computational fluid dynamics (CFD), and many commercial CFD codes are based on this technique which is typically less demanding in computational resources than finite element methods (FEM). However, for historical reasons, a large number of Computational Rheology codes are based on FEM. There is no clear reason why the FVM should not be as successful as finite element based techniques in Computational Rheology and its applications, such as polymer processing or, more recently, microfluidic systems using complex fluids. This chapter describes the major advances on this topic since its inception in the early 1990’s, and is organized as follows. In the next section, a review of the major contributions to computational rheology using finite volume techniques is carried out, followed by a detailed explanation of the methodology developed by the authors. This section includes recent developments and methodologies related to the description of the viscoelastic constitutive equations used to alleviate the high-Weissenberg number problem, such as the log-conformation formulation and the recent kernel-conformation technique. At the end, results of numerical calculations are presented for the well-known benchmark flow in a 4:1 planar contraction to ascertain the quality of the predictions by this method
Free-energy-dissipative schemes for the Oldroyd-B model
In this article, we analyze the stability of various numerical schemes for
differential models of viscoelastic fluids. More precisely, we consider the
prototypical Oldroyd-B model, for which a free energy dissipation holds, and we
show under which assumptions such a dissipation is also satisfied for the
numerical scheme. Among the numerical schemes we analyze, we consider some
discretizations based on the log-formulation of the Oldroyd-B system proposed
by Fattal and Kupferman, which have been reported to be numerically more stable
than discretizations of the usual formulation in some benchmark problems. Our
analysis gives some tracks to understand these numerical observations
Symmetric factorization of the conformation tensor in viscoelastic fluid models
The positive definite symmetric polymer conformation tensor possesses a
unique symmetric square root that satisfies a closed evolution equation in the
Oldroyd-B and FENE-P models of viscoelastic fluid flow. When expressed in terms
of the velocity field and the symmetric square root of the conformation tensor,
these models' equations of motion formally constitute an evolution in a Hilbert
space with a total energy functional that defines a norm. Moreover, this
formulation is easily implemented in direct numerical simulations resulting in
significant practical advantages in terms of both accuracy and stability.Comment: 7 pages, 5 figure
Automatic implementation of material laws: Jacobian calculation in a finite element code with TAPENADE
In an effort to increase the versatility of finite element codes, we explore
the possibility of automatically creating the Jacobian matrix necessary for the
gradient-based solution of nonlinear systems of equations. Particularly, we aim
to assess the feasibility of employing the automatic differentiation tool
TAPENADE for this purpose on a large Fortran codebase that is the result of
many years of continuous development. As a starting point we will describe the
special structure of finite element codes and the implications that this code
design carries for an efficient calculation of the Jacobian matrix. We will
also propose a first approach towards improving the efficiency of such a
method. Finally, we will present a functioning method for the automatic
implementation of the Jacobian calculation in a finite element software, but
will also point out important shortcomings that will have to be addressed in
the future.Comment: 17 pages, 9 figure
- …