Research interests
My research focus on the use and the development of models and numerical methods directed towards fluid mechanics. Examples of projects are presented below on this page.
Partial differential equations
- Newtonian fluids (Navier-Stokes, Euler models);
- Complex fluids models, such as viscoelastic ones (Oldroyd-B, FENE-P);
- Special solutions for complex fluids (Poiseuille, Couette flows...);
- Boundary conditions study.
Numerical methods
- Computational Fluid Dynamics (CFD);
- Finite elements and finite differences;
- Staggered finite difference schemes (MAC scheme);
- Optimization methods for solving PDE.
Finite Element resolution of viscoelastic models for living tissues
This is a summary of my internship for the M2 year (second semester, 2023-2024), directed by Pierre Saramito, and Ibrahim Cheddadi, in collaboration with Hélène Delanoë-Ayari and François Graner. No code is available yet as its development is pursued during my PhD, but the report is available here.
Cellular tissues represent a significant challenge for modeling, at the intersection between fluids, solids and biomechanics. This internship addressed this modeling challenge through the use of viscoelastic models (PDEs) : Oldroyd-B and FENE-P. Those are non-linear systems having three unknowns (conformation, tensor; velocity, vector; pressure, scalar). To add complexity, some of the terms contained in the system are not differentiable. The complexity of the system lead to using Galerkin discontinuous schemes for transport terms, finite elements for the solving, and some optimization methods. The weak formulation of each system is expressed as a scalar function on which a damped Newton method is applied to find a steady-state solution. This is where sub-differential tricks have been used.
(left/top) Velocity profile. (right/bottom) Conformation profile.
Some analytical Poiseuille flow have been computed for FENE-P model (see above) in order to have boundary conditions for the 2D domain in the code. This was a challenging task, in addition with the implementation of the variational problem. This was programmed using the Rheolef librairy (C++) from Pierre Saramito. Stick boundary conditions helped check the code before going on slip boundary conditions for a comparison with experiments (see below).
(left/top) Experiment. (right/bottom) Simulation.
The summarized conclusion of this project is that the Finite Elements framework and method is validated, as well as the use of FENE-P model which helps improve numerical comparison. However, data provided by the biomechanics researchers suggest a huge Weissenberg number should be used (We > 100). But, this would mean increasing the difficulty of the code to solve the problem, and it may induce a bad fit for conformation tensor. This thus calls for plasticity, in addition to the viscoelastic model (and there comes my PhD !).
Absorbing boundary conditions for Schrödinger equation
This is a M2 project; it has a GitLab page, and a report is available here. Note that four persons contributed to its development : Linnear Hallin, Éloi Navet, Nicolas Roblet and me.
This project relies on an article from X. Antoine, E. Lorin and Q. Tang (2017). It provides an overview of absorbing boundary conditions (ABCs) for numerical resolution of Quantum Wave equations in finite domains with Finite Differences Methods. Due to artificial boundaries imposed by the transition from paper to computers, borders have to be added on the computational domain, which create reflections of the quantum wave.
(left/top) Absorbing boundary condition at x=10. (right/bottom) Exact solution.
The aim was to understand and implement some of the methods to cope with this issue, on classical quantum wave equations. For instance, the 1D Schrödinger equation has been extensively used, after some tests on the classical wave equation. Two main axis were used : ABCs, which change the nature of the numerical boundary, adding a boundary condition ; PMLs, for perfectly matched layers, which extend the numerical domain on which a damping function for the wave is introduced to avoid reflections.