Philippe Angot

Fictitious domain methods to solve convection-diffusion problems with general boundary conditions

Since a few years, fictitious domain methods have been arising for Computational Fluid Dynamics. The main idea of these methods consists in immersing the original physical domain in a geometrically bigger and simply-shaped other one called fictitious domain. As the spatial discretization is then performed in the fictitious domain, simple structured meshes can be used. The aim of this paper is to solve convection-diffusion problems with fictitious domain methods which can easily simulate free-boundary with possibly deformations of the boundary without increasing the computational cost. Two fictitious domain approaches performing either a spread interface or a thin interface are introduced. These two approaches require neither the modification of the numerical scheme near the immersed interface nor the use of Lagrange multipliers. Several ways to impose general embedded boundary conditions (Dirichlet, Robin or Neumann) are presented. The spread interface approach is computed using a finite element method as a finite volume method is used for the thin interface approach. The numerical schemes conserve the first- order accuracy with respect to the discretization step as observed in the numerical results reported here. The spread interface approach is then combined with a local adaptive mesh refinement algorithm in order to increase the precision in the vicinity of the immersed boundary. The results obtained are full of promise, more especially as convection-diffusion equations are the core of the resolution of Navier-Stokes equations.

On the well-posed coupling between free fluid and porous viscous flows

We present a well-posed model for the Stokes/Brinkman problem with {\em jump embedded boundary conditions (J.E.B.C.)} on an immersed interface. It is issued from a general framework recently proposed for fictitious domain problems. Our model is based on algebraic transmission conditions combining the stress and velocity jumps on the interface

Une méthode de pénalité-projection pour les écoulements dilatables

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A Penalty-Projection Method Using Staggered Grids for Incompressible Flows

separating the fluid and porous domains. These conditions are well chosen to get the coercivity of the operator. Then, the general framework allows to prove the global solvability of some models with physically relevant stress or velocity jump boundary conditions for the momentum transport at a fluid-porous interface. The Stokes/Brinkman problem with {\em Ochoa-Tapia \& Whitaker (1995)} interface conditions and the Stokes/Darcy problem with {\em Beavers \& Joseph (1967)} conditions are both proved to be well-posed by an asymptotic analysis. Up to now, only the Stokes/Darcy problem with {\em Saffman (1971)} approximate interface conditions was known to be well-posed.

Well-posed Stokes/Brinkman and Stokes/Darcy problems for coupled fluid-porous viscous flows

Nous présentons dans cet article une nouvelle méthode de correction de pression pour les écoulements dilatables. Nommée « méthode de pénalité-projection », cette technique diffère des schémas de projection usuels par l’ajout dans l’étape de prédiction d’un terme de pénalisation, construit pour contraindre la vitesse à satisfaire le bilan de masse. Ce terme est multiplié par un coefficient r, dit paramètre de pénalisation. Nous montrons par des expériences numériques que ce schéma est bien plus précis que la méthode usuelle. L’erreur de fractionnement, dominante à fort pas de temps, est réduite à volonté en augmentant r ; à noter, toutefois, que l’usage d’une valeur trop importante dégrade le conditionnement de l’opérateur associé à l’étape de prédiction. Par ailleurs, les pertes de convergence de la méthode de projection usuelle en cas de conditions aux limites ouvertes sont corrigées, dès que r est non nul.

Penalty Methods for the Hyperbolic System Modelling the Wall-Plasma Interaction in a Tokamak

We deal with the time-dependent Navier-Stokes equations with Dirichlet boundary conditions on all the domain or, on a part of the domain and open boundary conditions on the other part. It is shown numerically that a staggered mesh with penalty-projection method yields reasonable good results for solving the above mentioned problem. Similarly to the results obtained recently by other scientists using finite element method (FEM) [1] and [2] (with the rotational pressure-correction method for the latter), we confirm that the penalty-projection scheme with spatial discretization of the Marker And Cell method (MAC) [3] is compatible with our problem.

A fictitious domain approach with spread interface for elliptic problems with general boundary conditions

We present a well-posed model for the Stokes/Brinkman problem with a family of jump embedded boundary conditions (J.E.B.C.) on an immersed interface with weak regularity assumptions. It is issued from a general framework recently proposed for fictitious domain problems. Our model is based on algebraic transmission conditions combining the stress and velocity jumps on the interface

A general fictitious domain method with immersed jumps and multilevel nested structured meshes

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Error analysis of the Penalty-Projection method for the time dependent Stokes equations

separating the fluid and porous domains. These conditions, well chosen to get the coercivity of the operator, are sufficiently general to get the usual immersed boundary conditions on

A fictitious domain model for the Stokes/Brinkman problem with jump embedded boundary conditions

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