Finite element approximation of the viscoelastic flow problem: a non-residual based stabilized formulation

In this paper, a three-field finite element stabilized formulation for the incompressible viscoelastic fluid flow problem is tested numerically. Starting from a residual based formulation, a non-residual based one is designed, the benefits of which are highlighted in this work. Both formulations allow one to deal with the convective nature of the problem and to use equal interpolation for the problem unknowns View the MathML sources-u-p (deviatoric stress, velocity and pressure). Additionally, some results from the numerical analysis of the formulation are stated. Numerical examples are presented to show the robustness of the method, which include the classical 4: 1 planar contraction problem and the flow over a confined cylinder case, as well as a two-fluid formulation for the planar jet buckling problem.Peer ReviewedPostprint (author's final draft

Stabilized finite element formulations for the three-field viscoelastic fluid flow problem

The Finite Element Method (FEM) is a powerful numerical tool, that permits the resolution of problems defined by partial differential equations, very often employed to deal with the numerical simulation of multiphysics problems. In this work, we use it to approximate numerically the viscoelastic fluid flow problem, which involves the resolution of the standard Navier-Stokes equations for velocity and pressure, and another tensorial reactive-convective constitutive equation for the elastic part of the stress, that describes the viscoelastic nature of the fluid. The three-field (velocity-pressure-stress) mixed formulation of the incompressible Navier-Stokes problem, either in the elastic and in the non-elastic case, can lead to two different types of numerical instabilities. The first is associated with the incompressibility and loss of stability of the stress field, and the second with the dominant convection. The first type of instabilities can be overcome by choosing an interpolation for the unknowns that satisfies the two inf-sup conditions that restrict the mixed problem, whereas the dominant convection requires a stabilized formulation in any case. In this work, different stabilized schemes of the Sub-Grid-Scale (SGS) type are proposed to solve the three-field problem, first for quasi Newtonian fluids and then for solving the viscoelastic case. The proposed methods allow one to use equal interpolation for the problem unknowns and to stabilize dominant convective terms both in the momentum and in the constitutive equation. Starting from a residual based formulation used in the quasi-Newtonian case, a non-residual based formulation is proposed in the viscoelastic case which is shown to have superior behavior when there are numerical or geometrical singularities. The stabilized finite element formulations presented in the work yield a global stable solution, however, if the solution presents very high gradients, local oscillations may still remain. In order to alleviate these local instabilities, a general discontinuity-capturing technique for the elastic stress is also proposed. The monolithic resolution of the three-field viscoelastic problem could be extremely expensive computationally, particularly, in the threedimensional case with ten degrees of freedom per node. A fractional step approach motivated in the classical pressure segregation algorithms used in the two-field Navier-Stokes problem is presented in the work.The algorithms designed allow one the resolution of the system of equations that define the problem in a fully decoupled manner, reducing in this way the CPU time and memory requirements with respect to the monolithic case. The numerical simulation of moving interfaces involved in two-fluid flow problems is an important topic in many industrial processes and physical situations. If we solve the problem using a fixed mesh approach, when the interface between both fluids cuts an element, the discontinuity in the material properties leads to discontinuities in the gradients of the unknowns which cannot be captured using a standard finite element interpolation. The method presented in this work features a local enrichment for the pressure unknowns which allows one to capture pressure gradient discontinuities in fluids presenting different density values. The stability and convergence of the non-residual formulation used for viscoelastic fluids is analyzed in the last part of the work, for a linearized stationary case of the Oseen type and for the semi-discrete time dependent non-linear case. In both cases, it is shown that the formulation is stable and optimally convergent under suitable regularity assumptions.El Método de los Elementos Finitos (MEF) es una herramienta numérica de gran alcance, que permite la resolución de problemas definidos por ecuaciones diferenciales parciales, comúnmente utilizado para llevar a cabo simulaciones numéricas de problemas de multifísica. En este trabajo, se utiliza para aproximar numéricamente el problema del flujo de fluidos viscoelásticos, el cual requiere la resolución de las ecuaciones básicas de Navier-Stokes y otra ecuación adicional constitutiva tensorial de tipo reactiva-convectiva, que describe la naturaleza viscoelástica del fluido. La formulación mixta de tres campos (velocidad-presión-tensión) del problema de Navier-Stokes, tanto en el caso elástico como en el no-elástico, puede conducir a dos tipos de inestabilidades numéricas. El primer grupo, se asocia con la incompresibilidad del fluido y la pérdida de estabilidad del campo de tensiones, y el segundo con la convección dominante. El primer tipo de inestabilidades, se puede solucionar eligiendo un tipo de interpolación entre las incógnitas que satisfaga las dos condiciones inf-sup que restringen el problema mixto, mientras que la convección dominante requiere del uso de formulaciones estabilizadas en cualquier caso. En el trabajo, se proponen diferentes esquemas estabilizados del tipo SGS (Sub-Grid-Scales) para resolver el problema de tres campos, primero para fluidos del tipo cuasi-newtonianos y luego para resolver el caso viscoelástico. Los métodos estabilizados propuestos permiten el uso de igual interpolación entre las incógnitas del problema y estabilizan la convección dominante, tanto en la ecuación de momento como en la ecuación constitutiva. Comenzando desde una formulación de tipo residual usada en el caso cuasi-newtoniano, se propone una formulación no-residual para el caso viscoelástico que muestra un comportamiento superior en presencia de singularidades numéricas y geométricas. En general, una formulación estabilizada produce una solución estable global, sin embargo, si la solución presenta gradientes elevados, oscilaciones locales se pueden mantener. Con el objetivo de aliviar este tipo de inestabilidades locales, se propone adicionalmente una técnica general de captura de discontinuidades para la tensión elástica. La resolución monolítica del problema de tres campos viscoelástico puede llegar a ser extremadamente costosa computacionalmente, sobre todo, en el caso tridimensional donde se tienen diez grados de libertad por nodo. Un enfoque de paso fraccionado motivado en los algorítmos clásicos de segregación de la presión usados en el caso del problema de dos campos de Navier-Stokes, se presenta en el trabajo, el cual permite la resolución del sistema de ecuaciones que definen el problema de una manera completamente desacoplada, lo que reduce los tiempos de cálculo y los requerimientos de memoria, respecto al caso monolítico. La simulación numérica de interfaces móviles que envuelve los problemas de dos fluidos, es un tópico importante en un gran número de procesos industriales y situaciones físicas. Si se resuelve el problema utilizando un enfoque de mallas fijas, cuando la interfaz que separa los dos fluidos corta un elemento, la discontinuidad en las propiedades materiales da lugar a discontinuidades en los gradientes de las incógnitas que no pueden ser capturados utilizando una formulación estándar de interpolación. Un enriquecimiento local para la presión se presenta en el trabajo, el cual permite la captura de gradientes discontinuos en la presión, asociados a fluidos de diferentes densidades. La estabilidad y la convergencia de la formulación no-residual utilizada para fluidos viscoelásticos es analizada en la última parte del trabajo, para un caso linealizado estacionario del tipo Oseen y para un problema transitorio no-lineal semi-discreto. En ambos casos, se logra mostrar que la formulación es estable y de convergencia óptima bajo supuestos de regularidad adecuados.Postprint (published version

Numerical analysis of a stabilized finite element approximation for the three-field linearized viscoelastic fluid problem using arbitrary interpolations

The original publication is available at www.esaimm2an.org.In this paper we present the numerical analysis of a three-field stabilized finite element formulation recently proposed to approximate viscoelastic flows. The three-field viscoelastic fluid flow problem may suffer from two types of numerical instabilities: on the one hand we have the two inf-sup conditions related to the mixed nature problem and, on the other, the convective nature of the momentum and constitutive equations may produce global and local oscillations in the numerical approximation. Both can be overcome by resorting from the standard Galerkin method to a stabilized formulation. The one presented here is based on the subgrid scale concept, in which unresolvable scales of the continuous solution are approximately accounted for. In particular, the approach developed herein is based on the decomposition into their finite element component and a subscale, which is approximated properly to yield a stable formulation. The analyzed problem corresponds to a linearized version of the Navier-Stokes/Oldroyd-B case where the advection velocity of the momentum equation and the non-linear terms in the constitutive equation are treated using a fixed point strategy for the velocity and the velocity gradient. The proposed method permits the resolution of the problem using arbitrary interpolations for all the unknowns. We describe some important ingredients related to the design of the formulation and present the results of its numerical analysis. It is shown that the formulation is stable and optimally convergent for small Weissenberg numbers, independently of the interpolation used.Peer ReviewedPostprint (author's final draft

Backward Euler method for the Equations of Motion Arising in Oldroyd Fluids of Order One with Nonsmooth Initial Data

In this paper, a backward Euler method is discussed for the equations of motion arising in the 2D Oldroyd model of viscoelastic fluids of order one with the forcing term independent of time or in $L^{\infty}$ in time. It is shown that the estimates of the discrete solution in Dirichlet norm is bounded uniformly in time. Optimal a priori error estimate in L2-norm is derived for the discrete problem with non-smooth initial data. This estimate is shown to be uniform in time, under the assumption of uniqueness condition

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

Energy-stable linear schemes for polymer-solvent phase field models

We present new linear energy-stable numerical schemes for numerical simulation of complex polymer-solvent mixtures. The mathematical model proposed by Zhou, Zhang and E (Physical Review E 73, 2006) consists of the Cahn-Hilliard equation which describes dynamics of the interface that separates polymer and solvent and the Oldroyd-B equations for the hydrodynamics of polymeric mixtures. The model is thermodynamically consistent and dissipates free energy. Our main goal in this paper is to derive numerical schemes for the polymer-solvent mixture model that are energy dissipative and efficient in time. To this end we will propose several problem-suited time discretizations yielding linear schemes and discuss their properties