Mourad Ismaïl Associate Professor in Applied Mathematics at Grenoble Alpes University Ph.: +33 4 76 51 48 94 F.: +33 4 76 63 54 95 Mourad.Ismail univ-grenoble-alpes.fr

Dynamique des Fluides Complexes et Morphogénèse Laboratoire Interdisciplinaire de Physique Université Grenoble Alpes CS 4070 38058 Grenoble Cedex 9

Ph.D. Thesis

Download the Dissertation (in french) :

thesis.pdf (8.2M) or thesis.ps (42M). It's also available at the HAL ARCHIVES. Just follow this link.

The aim of my thesis (supervised by Pr. Bertrand Maury) was to develop a new method (The Fat Boundary Method, aka FBM) dedicated to the numerical resolution of elliptic problems in complex geometries in order to be able to simulate fluid-particles flows (3D Direct simulations).

Thesis title:

"The Fat Boundary Method for the Numerical Resolution of Elliptic Problems in Perforated Domains. Application to 3D Fluid Flows."

Examining Board :

• Christine Bernardi (Université Pierre et Marie Curie, France)
• Silvia Bertoluzza (Istituto di Matematica Applicata e Tecnologie Informatiche del C.N.R., Italy)
• Michel Crouzeix (Université de Rennes I, France) "Rapporteur"
• Lassaad El Asmi (École Polytechnique de Tunisie, Tunisia)
• Taïeb Hadhri (Université du 7 novembre à Carthage, Tunisia)
• Yvon Maday (Université Pierre et Marie Curie, France)
• Bertrand Maury (Université de Paris Sud, France)
• Adélia Sequeira (Instituto Superior Técnico, Technical University of Lisbon, Portugal) "Rapporteur"

Mention :

"Trés Honorable" (Highest honor)

Key Words :

Fat Boundary Method, Fictitious Domains, Elliptic Problems, Projection Methods, Characteristics Method, Navier-Stokes equations, Incompressible Fluid Flows.

Abstract :

The aim of this thesis is the mathematical analysis of the Fat Boundary Method (F.B.M.) and its adaptation to the numerical simulation of the 3D incompressible fluid flows in complex geometries (perforated domains).
First, we focus on a simple case of model elliptic problems (Poisson or Helmholtz-like problems) set in a perforated domain (typically a box containing spherical obstacles). Using the F.B.M., the initial problem is replaced by a new one defined in the entire box, making it possible to use a cartesian grid, thus offering a suitable framework for the use of fast solvers. We thus carry out the mathematical analysis of the F.B.M., in particular the convergence and the errors estimate. The obtained theoretical results are also illustrated numerically.
The second part is dedicated to the application of these tools for the Numerical Simulation of three-dimensional incompressible fluid flows. The strategy consists in discretizing the Navier-Stokes equations by combining the F.B.M. (for the space discretization), a Projection Scheme (for the time discretization) and the Characteristics Method (for the treatment of the convection). Finally, we present several three-dimensional Numerical Simulations corresponding to fluid flows in presence of fixed and mobile obstacles (imposed motion).