Existence results for non-smooth second order differential inclusions, Convergence result for a numerical scheme and applications for modelling inelastic collisions

We are interested in existence results for second order differential inclusions, involving finite number of unilateral constraints in an abstract framework. These constraints are described by a set-valued operator, more precisely a proximal normal cone to a time-dependent set. Moreover we extend a numerical scheme, introduced in [8] and proved a convergence result. We propose applications in modelling inelastic collisions between rigid particles too.Comment: 22 pages, 1 figur

Micro-Macro Modelling of an Array of Spheres Interacting Through Lubrication Forces

We consider here a discrete system of spheres interacting through a lubrication force. This force is dissipative, and singular near contact: it behaves like the reciprocal of interparticle distance. We propose a macroscopic constitutive equation which is built as the natural continuous counterpart of this microscopic lubrication model. This model, which is of the newtonian type, relies on an elongational viscosity, which is proportional to the reciprocal of the local fluid fraction. We then establish the convergence in a weak sense of solutions to the discrete problem towards the solution to the partial differential equation which we identified as the macroscopic constitutive equation

Numerical simulation of rigid particles in Stokes flow: lubrication correction for any (regular) shape of particles

We address the problem of numerical simulation of suspensions of rigid particles in a Stokes flow. We focus on the inclusion of the singular short range interaction effects (lubrication effects) in the simulations when the particles come close one to another. The problem is solved without introducing new hypothesis nor model. As in [Lefebvre-Lepot, Merlet, Nguyen, JFM, 2015], the key idea is to decompose the velocity and pressure flows in a sum of a singular and a regular part. In this article, the singular part is computed using an explicit asymptotic expansion of the solution when the distance goes to zero. This expansion is similar to the asymptotic expansion proposed in [Hillairet, Kelai, Asymptotic Analysis, 2015] but is more appropriate for numerical simulations of suspensions. It can be computed for any shape of particles. Using [Hillairet, Kelai, Asymptotic Analysis, 2015] as an intermediate result, we prove that the remaining part is regular in the sense that it is bounded independently of the distance. As a consequence, only a small number of degrees of freedom are necessary to obtain accurate results. The method is tested in dimension 2 for clusters of two or three aligned particles with general rigid velocities. We show that, as expected, the convergence is independent on the distance

Problèmes de contact pour des particules en écoulement cisaillé

National audienceSee http://hal.archives-ouvertes.fr/docs/00/59/28/28/ANNEX/r_X8K316T9.pd

The Sparse Cardinal Sine Decomposition applied to Stokes integral equations

International audienceNumerical simulations of two-phase flows driven by viscosity (e.g. for bubble motions in glass melting process) rely on the ability to efficiently compute the solutions to discretized Stokes equations. When using boundary element methods to track fluid interfaces, one usually faces the problem of solving linear systems with a dense matrix with a size proportional to the system number of degrees of freedom. Acceleration techniques, based on the compression of the underlying matrix and efficient matrix vector products are known (Fast Multipole Method, H-matrices, etc.) but are usually rather cumbersome to develop. More recently, a new method was proposed, called the " Sparse Cardinal Sine Decomposition " , in the context of acoustic problems to tackle this kind of problem in some generality (in particular with respect to the Green kernel of the problem). The proposed contribution aims at showing the potential applicability of the method in the context of viscous flows governed by Stokes equations

Ecoulement dense autour d'une sphère traversant un nuage de grains

Une simulation bidimensionnelle d'une sphère se déplaçant à vitesse constante à l'intérieur d'un nuage de petits grains est présentée avec une méthode de type ?'Non-Smooth Contact Dynamic'' (sans effet de la gravité). Une zone granulaire dense, appelée ?'cluster'', à fraction volumique constante se construit progressivement autour de la sphère jusqu'à ce qu'un régime stationnaire apparaisse caractérisée par une taille constante du cluster en amont de la sphère qui augmente avec la fraction volumique initiale \phi_0 du nuage. Une analyse détaillée du champ de taux de déformation et du champ de contrainte à l'intérieur du cluster révèle que, malgré les variations spatiales de ces champs, le coefficient local de friction \mu et la fraction volumique \phi dépendent uniquement du nombre d'inertie I, ce qui signifie que la rhéologie du milieu granulaire est bien locale dans cet écoulement non parallèle. Les variations spatiales de I à l'intérieur même du cluster ne dépendent pas de la vitesse de déplacement de la sphère et explore une faible gamme allant de 0.01 à 0.1. L'influence des parois latérales sur l'écoulement et les forces est ensuite étudiée

Dense flow around a sphere moving into a cloud of grains

A bidimensional simulation of a sphere moving at constant velocity into a cloud of smaller spherical grains without gravity is presented with a non-smooth contact dynamics method. A dense granular “cluster” zone of about constant solid fraction builds progressively around the moving sphere until a stationary regime appears with a constant upstream cluster size that increases with the initial solid fraction ϕ0 of the cloud. A detailed analysis of the local strain rate and local stress fields inside the cluster reveals that, despite different spatial variations of strain and stresses, the local friction coeffcient μ appears to depend only on the local inertial number I as well as the local solid fraction ϕ, which means that a local rheology does exist in the present non parallel flow. The key point is that the spatial variations of I inside the cluster does not depend on the sphere velocity and explore only a small range between about 10−2 and 10−1. The influence of sidewalls is then investigated on the flow and the forces

Modélisation numérique d'écoulements fluide/particules

This PhD thesis is made of three parts. In the first one, we present a method to simulate fluid/particle flows. We show that the penalty method, combined to a time discretization performed using the method of characteristics leads to a generalized Stokes variational formulation. Numerical tests are implemented with FreeFem++ to study the convergence. We also present three examples using this method. In the second part we propose a model to take into account lubrication forces in direct simulations of fluid/particle flows. We first present a "viscous contact" model in the plane/particle case, obtained as the vanishing viscosity limit of the lubrication model. Then, we describe an algorithm based on a projection of the velocities, at each time step, over a set of admissible velocities. Next, we prove the convergence of the scheme and generalize the algorithm to the multi-particle case. We also present an example of object oriented programming of it. In the last part, we consider a discrete system of spheres (chain in 1D) interacting through a lubrication force. The microscopic model relies on the development of that force at small distance. We propose a macroscopic constitutive equation, of Newtonian type, which relies on an elongational viscosity which is proportional to the reciprocal of the local fluid fraction. We establish the convergence of the microscopic model towards the solution of the proposed macroscopic model.Cette thèse comporte trois parties. Dans la première, nous présentons une méthode de simulation d'écoulements fluide/particules. Nous montrons que la pénalisation du tenseur des contraintes, associée à une discrétisation en temps par la méthode des caractéristiques, conduit à une formulation variationnelle de type Stokes généralisée. Des tests numériques sont effectués sous FreeFem++ afin d'étudier la convergence. Nous en présentons également trois exemples d'utilisation. Dans la seconde partie nous proposons un modèle permettant de prendre en compte les forces de lubrification dans les simulations directes d'écoulements fluide/particules. Nous présentons d'abord un modèle de contact visqueux dans le cas particule/plan, obtenu comme limite, à viscosité nulle, du modèle de lubrification. Nous décrivons ensuite un algorithme reposant sur une étape de projection des vitesses, à chaque instant, sur un espace dit de vitesses admissibles. On montre alors la convergence du schéma et on généralise l'algorithme au cas multi-particules. Nous en présentons également un exemple de programmation orientée objet. Dans la dernière partie, nous considérons un système discret de sphères (boulier en 1D) qui interagissent à travers la force de lubrification. Le modèle microscopique repose sur le développement de cette force à courte distance. Nous proposons une équation constitutive marcoscopique, de type Newtonien, reposant sur une viscosité linéique proportionnelle à l'inverse de la fraction locale de fluide. Nous établissons la convergence du modèle microscopique vers le modèle macroscopique proposé

Numerical simulation of suspensions: lubrication correction, including fluid correction

International audienceWhen simulating systems of particles embedded in a Stokes flow, it is necessary to question the treatment of close interacting particles. Indeed, when two solids come close one to another, it becomes difficult to approximate the velocity and pressure fields which become singular. However, taking the corresponding singu-larity is essential, both from a numerical and physical point of view. Moreover, experimentalists now need more and more precise results, taking into account the effect of these interactions on the whole flow. The method we propose is based on a decomposition of the fluid/particle problem into two subproblems: a singular problem (when the distance between the particles goes to zero) and a regular problem. The singular field is supposed to be known and the resolution of the problem comes back to solving the regular problem. A first approach have been proposed in [6], where the singular field is tabulated. Here, we propose a new method, based on an asymptotic expansion of the singular field. This method allows to catch the effect of the lubrication on the whole velocity and pressure flows and can deal with any forms of particles. We focus on a toy problem in two dimensions to present the method. Numerical results are given, using a finite element discretization