Discrétisations polyhédriques d’ordre élevé pour les fluides non-newtoniens
Advisors: Alexandre Ern (CERMICS), Xavier Chateau (NAVIER), Jérémie Bleyer (NAVIER)
Started October 2015
The goal of this thesis is to develop, analyze and apply Hybrid High Order (HHO) numerical methods to the simulation of non-Newtonian fluid flow (i.e., yield stress fluid). The ultimate goal is the numerical simulation of problems related to practical applications such as the deformation and the burst of a bubble sheared by a flowing non-Newtonian fluid.
HHO methods have been recently developed by A. Ern and D. Di Pietro (University Montpellier) for linear problems (diffusion, advection-diffusion, elasticity, Incompressible Stokes flows). The principle of HHO methods is to use face-based discrete unknowns for the global problem and a differential operator to reconstruct the solution inside the mesh elements. The discrete unknowns are thus piecewise polynomials on the mesh faces that compose the mesh skeleton. Since no continuity is enforced from one face to a neighboring one, such methods belong to the class of so-called Discontinuous Skeletal methods. HHO methods offer several assets, such as a dimension-independent construction, a locally-balanced formulation expressing fundamental conservation laws at the discrete level, the possibility to employ arbitrary-order polynomial approximation, and, perhaps most importantly, the fact that general grids composed of polyhedral cells with hanging nodes are supported. This is indeed a key feature since it allows one to use locally refined meshes without propagating mesh refinement to ensure mesh conformity.
 Cascavita K., Bleyer J., Château X., and Ern A. (2018) Hybrid discretization methods with adaptive yield surface detection for Bingham pipe flows. J. Sci. Comput., accepted. DOI 10.1007/s10915-018-0745-3, https://hal.archives-ouvertes.fr/hal-01698983