École Nationale Supérieure de
Techniques Avancées

Abstract

Over the past twenty years, great improvements have been achieved regarding the modeling and the simulation of fluid dynamics. The Navier-Stokes equations are now well mastered, and many softwares have been designed to compute fluid flows. However, dark areas still remain, among which we can quote the simulation of turbulent flows or multifluid mechanics. On this last subject, the stake is crucial, especially regarding biomedical sciences, with more and more physicians claiming for new simulation means in cardiology or neurology. This internship has been supervised between both Jacques-Louis Lions laboratory attached to the University of Paris VI and INRIA Rocquencourt, the organism which funds the project. My work at Paris VI was mostly focused on finite element techniques and level-set methods, whereas my involvement at INRIA was mainly related to mesh adaptation, with a great part of the job consisting in mastering new softwares and computational techniques, all this in a very stimulating and friendly environnement. In the first part of this report, I will describe the model we chose and show to what extent it fits to the biomedical phenomenons we aim at representing. The most important theorical results and numerical analysis concepts will be briefly reminded to the reader. Then, I will give a relatively detailed description of the techniques used to implement the Stokes solver, and show its efficiency on some test cases. The third part deals with adaptive meshing, applied to our Stokes problem, and show in which manner these techniques can improve the quality of the solutions we obtain. To finish with, I will present some recent techniques employed to handle multifluid interfaces dynamics (among them the level-set methods) and I will give some leads to build a new kind of algorithm which could enable to efficiently couple adaptive remeshing and level-set methods.