On a modified non-singular log-conformation formulation for Johnson-Segalman viscoelastic fluids

International audienceA modified log-conformation formulation of viscoelastic fluid flows is presented in this paper. This new formulation is non-singular for vanishing Weissenberg numbers and allows a direct steady numerical resolution by a Newton method. Moreover, an exact computation of all the terms of the linearized problem is provided. The use of an exact divergence-free finite element method for velocity-pressure approximation and a discontinuous Galerkin upwinding treatment for stresses leads to a robust discretization. A demonstration is provided by the computation of steady solutions at high Weissenberg numbers for the difficult benchmark case of the lid driven cavity flow. Numerical results are in good agreement, qualitatively with experiment measurements on real viscoelastic flows, and quantitatively with computations performed by others authors. The numerical algorithm is both robust and very efficient, as it requires a low mesh-invariant number of linear systems resolution to obtain solutions at high Weissenberg number. An adaptive mesh procedure is also presented: it alows representing accurately both boundary layers and the main and secondary vorties

A damped Newton algorithm for computing viscoplastic fluid flows

International audienceFor the first time, a Newton method is proposed for the unregularized viscoplastic fluid flow problem. It leads to a superlinear convergence for Herschel-Bulkley fluids when 0<n<1, where n is the power law index. Performances are enhanced by using the inexact variant of the Newton method and, for solving the Jacobian system, by using an efficient preconditioner based on the regularized problem. A demonstration is provided by computing a viscoplastic flow in a pipe with a square cross section. Comparisons with the augmented Lagrangian algorithm show a dramatic reduction of the required computing time while this new algorithm provides an equivalent accuracy for the prediction of the yield surfaces

A new operator splitting algorithm for elastoviscoplastic flow problems

International audienceThis paper presents an efficient time-dependent decoupled approach for the numerical resolution of the highly nonlinear set of coupled partial differential equations appearing in elastoviscoplastic fluid flow problems. The two main nonlinear difficulties, the viscoplasticity and the viscoelasticity, are then solved separately. Numerical simulations suggest an optimal convergence rate with respect to the space discretization. Finally, numerical results compare well with experimental measurements on liquid foams in a complex geometry. Future works will explore flows of liquid foams for tridimensional geometries where experimental data are available and also compare to flows of others soft glassy materials such as carbopol solutions

Discontinuous Galerkin finite element method applied to the coupled Navier-Stokes/Cahn-Hilliard equations

International audienceTwo-phase flows driven by the interfacial dynamics is studied with a phase-field model to tract implicitly interfaces. The phase field obeys the Cahn-Hilliard equation. The fluid dynamics is described with the Stokes equations with an additional source term in the momentum equation taking into account the capillary forces. A discontinuous Galerkin finite element method is used to solve the coupled Stokes/Cahn-Hilliard equations. The Cahn-Hilliard equation is treated as a system of two coupled equations corresponding to the advection-diffusion equation for the phase field and a non-linear elliptic equation for the chemical potential. First, the variational formulation of the Cahn-Hilliard equation is presented. A numerical test is achieved showing the optimal-order in error bounds. Second, the variational formulation in discontinuous Galerkin finite element approach of the Stokes equations is recalled in which the same space of approximation is used for the velocity and the pressure with an adequate stabilization technique. Finally, numerical simulations describing the capillary rising in a tube is presented

Efficient C++ finite element computing with Rheolef : volume 2: discontinuous Galerkin methods

DEARheolef is a programming environment for finite element method computing. This second volume is dedicated to discontinuous Galerkin methods. This Book presents in details how some simple and more complex problems from solid and fluid mechanics can be solved, most of them in less than 20 lines of code. The concision and readability of codes written with Rheolef is certainly a major keypoint of this environment. Data structures fit the variational formulation concept of partial differential equations: fields, bilinear forms and functional spaces are C++ types for variables. They can be combined in expressions, as you write it on the paper. As a Lego game, these bricks allows the user to solve most complex nonlinear problems. Algorithms refer to the most up-to-date ones: preconditioned sparse solvers for linear systems, incompressible elasticity, Stokes and Navier-Stokes flows, characteristic method for convection dominated heat problems, etc. Also nonlinear generic algorithms such as fixed point and damped Newton methods. Software home page is: http://www-ljk.imag.fr/membres/Pierre.Saramito/rheole

Tensorial rheological model for concentrated non-colloidal suspensions: normal-stress differences

Most existing rheological models for non-colloidal suspensions fail to simultaneously capture the two main non-Newtonian trends of these systems, namely finite normal stress differences and transient effects. We address this issue by extending a previously-proposed minimal model accounting for microstructure anisotropy through a conformation tensor, and which was shown to correctly predict transient effects (Ozenda et al. 2018). The new model is compared to a large experimental dataset involving varying volume fractions, from dilute to concentrated cases. Both transient evolution of apparent viscosity during shear reversal, and normal stress differences in steady state, are quantitatively reproduced in the whole range of volume fraction. Furthermore, the model is validated against particle pressure measurements that were not used for parameter identification. Even if the proposed constitutive equation for the Cauchy stress tensor is more difficult to interpret than in the minimal model, this study opens way for the use of conformation tensor rheological models in applications where the effect of the second normal stress difference is prominent, like elongational flows or migration phenomenon

An implicit high order discontinuous Galerkin level set method for two-phase flow problems

International audienceAn implicit high order time (BDF) and polynomial degree discontinuous Galerkin (DG) level set method is presented in this talk. The major advantage of this new approach is an accurate mass conservation during the convection of the level set function, thanks to the implicit method. Numerical experiments are presented for the Zalesak and the Leveque test cases. The convergence rates versus time and space are investigated for both BDF and DG high orders. The capture of the zero level set interface is then improved by using an auto-adaptive mesh procedure. The problem is approximated by using the discontinuous Galerkin method for both the level set function, the velocity and the pressure fields

Numerical modeling of shallow non-Newtonian flows: Part I. The 1D horizontal dam break problem revisited

International audienceThe dam break problem shallow approximation for laminar flows of viscoplastic non-Newtonian fluids is numerically revisited under a time and space second order adaptive method. Theoretical solutions are compared with experimental measurements from the literature and new ones made on silicon. Asymptotic behaviors are solved numerically and from autosimilar solutions. The obtained theoretical results are finally compared with experiments. These comparisons confirm the validity of the shallow approximation equations for non-Newtonian fluids subject to the horizontal dam break problem

Langage C++ et calcul scientifique

Table des matières 1 Introduction à l'algorithmique numérique en C++ 1.1 Quaternions 1.2 Analyse asymptotique des algorithmes 2 Transformée de Fourier et applications 2.1 Transformée de Fourier 2.2 Discrétisation de problèmes aux limites 2.3 Application aux différences finies multi-dimensionnelles 3 Matrices creuses et méthode des éléments finis 3.1 Algorithme du gradient conjugué 3.2 Matrices creuses 3.3 Maillages 3.4 Méthode des éléments finis A Pré- et post-traitements A.1 Ordonnancement et visualisation des matrices creuses A.2 Génération et visualisation de maillages A.3 Visualisation des solutions de type éléments finis B Corrigé des exercicesDEALa simulation numérique est devenue essentielle dans de nombreux domaines tels que la mécanique des fluides et des solides, la météo, l'évolution du climat, la biologie ou les semi-conducteurs. Elle permet de comprendre, de prévoir, d'accéder là où les instruments de mesures s'arrêtent. Ce livre présente des méthodes performantes du calcul scientifique : matrices creuses, résolution efficace des grands systèmes linéaires, ainsi que de nombreuses applications à la résolution par éléments finis et différences finies. Alternant algorithmes et applications, les programmes sont directement présentés en langage C++. Ils sont sous forme concise et claire, et utilisent largement les notions de classe et de généricité du langage C++. Le contenu de ce livre a fait l'objet de cours de troisième année à l'école nationale supérieure d'informatique et de mathématiques appliquées de Grenoble (ENSIMAG) ainsi qu'au mastère de mathématiques appliquées de l'université Joseph Fourier. Des connaissances de base d'algèbre matricielle et de programmation sont recommandées. La maîtrise du contenu de cet ouvrage permet d'appréhender les principaux paradigmes de programmation du calcul scientifique. Il est alors possible d'appliquer ces paradigmes pour aborder des problèmes d'intérêt pratique, tels que la résolution des équations aux dérivées partielles, qui est abordée au cours de ce livre. La diversité des sujets abordés, l'efficacité des algorithmes présentés et leur écriture directe en langage C++ font de cet ouvrage un recueil fort utile dans la vie professionnelle d'un ingénieur. Le premier chapitre présente les bases fondamentales pour la suite : présentation du langage C++ à travers la conception d'une classe de quaternions et outils d'analyse asymptotique du temps de calcul des algorithmes. Le second chapitre aborde l'algorithme de transformée de Fourier rapide et développe deux applications à la discrétisation d'équations aux dérivées partielles par la méthode des différences finies. Le troisième chapitre est dédié aux matrices creuses et à l'algorithme du gradient conjugué. Ces notions sont appliquées à la méthode des éléments finis. En annexe sont groupés des exemples de génération de maillage et de visualisation graphique. S'il est cependant recommandé de maîtriser les notions du premier chapitre pour aborder le reste du livre, les chapitres deux et trois sont complètement indépendants et peuvent être abordés séparément. Ces chapitres sont complétés par des exercices qui en constituent des développements, ainsi que des notes bibliographiques retraçant l'historique des travaux et fournissant des références sur des logiciels et librairies récents implémentant ou étendant les algorithmes présentés. Les codes C++ présentés au long de ce livre ainsi que dans les exercices sont disponibles librement à l'adresse \url{http://www-ljk.imag.fr/membres/Pierre.Saramito/books} sous la licence GNU public licence. Ce livre est publié sous licence GNU FDL avec les références suivantes : P. Saramito, Language C++ et calcul scientifique, College Publications, London, 2013. ISBN 978-1-84890-101-8

A Maxwell-elasto-brittle rheology for sea ice modelling

International audienceA new rheological model is developed that builds on an elasto-brittle (EB) framework used for sea ice and rock mechanics, with the intent of representing both the small elastic deformations associated with fracturing processes and the larger deformations occurring along the faults/leads once the material is highly damaged and fragmented. A viscous-like relaxation term is added to the linear-elastic constitutive relationship together with an effective viscosity that evolves according to the local level of damage of the material, like its elastic modulus. The coupling between the level of damage and both mechanical parameters is such that within an undamaged ice cover the viscosity is infinitely large and deformations are strictly elastic, while along highly damaged zones the elastic modulus vanishes and most of the stress is dissipated through permanent deformations. A healing mechanism is also introduced, counterbalancing the effects of damaging over large time scales. In this new model, named Maxwell-EB after the Maxwell rheology, the irreversible and reversible deformations are solved for simultaneously, hence drift velocities are defined naturally. First idealized simulations without advection show that the model reproduces the main characteristics of sea ice mechanics and deformation: strain localization, anisotropy, intermittency and associated scaling laws