A local adaptive refinement method with multigrid preconditionning illustrated by multiphase flows simulations.

Abstract

Abstract. The aim of this paper is to describe some numerical aspects linked to incompressible three-phase flow simulations, thanks to Cahn-Hilliard type model. The numerical capture of transfer phenomenon in the neighborhood of the interface require a mesh thickness which become crippling in the case where it is applied to the whole computational domain. This suggests the use of a local refinement method which allows to dynamically focus on problematic areas. The notion of refinement pattern, introduced for Lagrange finite elements, allows to build a conceptual hierarchy of nested conformal approximation spaces which is then used to implement the so-called CHARMS local refinement methods. Properties of these methods are proved ensuring in particular the conformity of approximation spaces at every time of simulations. Furthermore, the multilevel structure obtained by this method, is used to construct multigrid preconditioners. Finally, after a validation on a model problem, the performance of the whole method is illustrated on an example of a liquid lens spreading between two stratified fluids.Résumé. L'objectif de l'article est de décrire certains aspects numériques liésà la simulation d'écou-lements incompressiblesà trois phases non miscibles,à l'aide de modèlesà interfaces diffuses de type Cahn-Hilliard. La capture numérique des phénomènes de transfert au voisinage des interfaces requiert une finesse de maillage qui devient rédhibitoire si elle est appliquéeà l'ensemble du domaine de calcul. Ceci suggère l'utilisation de méthodes de raffinement local adaptatif qui permettent de se focaliser dynamiquement sur les zones sensibles. La notion de motif de raffinement, introduite pour deséléments finis de Lagrange, permet de construire une hiérarchie conceptuelle d'espaces d'approximation conformes emboîtés qui est alors utilisée pour mettre en oeuvre les techniques de raffinement local dites CHARMS. Les propriétés de la méthode sont prouvées assurant en particulier la conformité de l'espace d'approximationà tout instant des simulations. En outre, la structure multiniveaux obtenue par cette méthode est exploitée pour construire des préconditionneurs multigrilles. Enfin, après une validation sur un problème modèle, les performances de l'ensemble sont illustrées sur un exemple d'étalement d'une lentille piégée entre deux phases stratifiées.

The Cahn-Hilliard Equation with Logarithmic Potentials

“…We refer the reader to, e.g., [8], [9], [10], [11], [12], [13], [14], [30], [31], [32], [33], [34], [50], [60], [77], [78], [79], [86], [87], [88], [89], [94], [111], [122], [124],…”

An adaptive pressure correction method without spurious velocities for diffuse-interface models of incompressible flows

“…The space discretization is performed using multilevel finite element approximation spaces based on a Q 1 approximation for the order parameter c, the chemical potential µ and the pressure p and on a Q 2 approximation for the velocity u. The reader can refer to [16] for a precise description of the construction of these approximation spaces. The important point is to note that here approximation spaces are not modified during the time marching and that the resolution is accurate in the neighborhood of interface: there is about 10 meshes in the interface.…”

“…The adaptation procedures are based on conforming multilevel finite element approximation spaces which are built by refinement or unrefinement of the finite element basis functions instead of cells. All the details about this method and also various examples (in particular, simulations using the Cahn-Hilliard model considered in this article) are described in [16]. The refinement criterion used in those (un-)refinement procedures imposes the value of the smaller diameter h min of a cell and ensures that refined areas are located in the neighborhood of the interfaces (i.e.…”

An unconditionally stable uncoupled scheme for a triphasic Cahn–Hilliard/Navier–Stokes model

“…The adaptation procedures are based on conforming multilevel finite element approximation spaces which are built by refinement or unrefinement of the finite element basis functions instead of cells. All the details about this method and also various examples (in particular, simulations using the Cahn-Hilliard model considered in this article) are described in [5]. The refinement criterion used in those (un-)refinement procedures imposes the value of the smaller diameter h min of a cell and ensures that refined areas are located in the neighborhood of the interfaces (i.e.…”