## Modélisation des interfaces en mécanique des fluides

# Participants:

Abdelkader Baggag *(Génie civil)*

André Fortin *(Mathématiques et statistique)*

Lakhdar Remaki (*Basque Center for Applied Mathematics*)

Donald Ziegler (*Alcoa*)

# But:

The intellectual merit of the proposed work lies in the development of a unified framework, suited for parallel computers, for the direct numerical simulation of a class of problems with sharp interfaces, namely the two-phase flow and solidification problems in the aluminum smelting process. Interfaces, between two different materials and which separate the two phases, arise in a wide range of practical applications, are often unknown and must be determined together with the state variables. The level-set method, which is designed to represent evolving features with evolving functions, is chosen to represent the motion of the interface. And the quadrature-free Runge-Kutta discontinuous Galerkin (DG) method is used for the temporal and spatial discretization of the Hamilton-Jacobi equation, written in a conservative form for the two-phase flow, and in an advective form for the solidification. Both approaches require careful implementation and will be numerically analyzed for convergence to the “viscosity” solution. DG finite element methods are highly accurate, compact, robust, parallelizable, and can easily handle complex geometries. The physical parameters of the phases (viscosity and density) are discontinuous across the interface, and hence the extended finite element method (XFEM) is used for the discretization of the discontinuous state variables (pressure, velocity, temperature). The enriched pressure space is used to avoid spurious pressure modes. The perspective of XFEM, when coupled with level-set methods, greatly simplifies the treatment of problems with significant changes in topology, and easily represents such problems without the need of re-meshing, i.e., the mesh is completely independent of the shape and position of the interface, and need not be aligned to it. For the two-phase flow problem, this non-alignment causes severe difficulties for the discretization of the localized surface tension. In this case, a variational improved Laplace-Beltrami operator is used. The key numerical issues are

i) the accurate resolution of the unknown interface;

ii) the treatment of the surface tension force, which acts only at the surface;

iii) the coupling of the flow and the interface dynamics; and

iv) the efficiency and robustness of the iterative solvers.

Because of the high-complexity, parallelization is needed, and is based on the MPI message passing standard for full portability. This requires knowledge of mathematics, numerical methods, high performance computing (HPC), and large-scale computer software development.

# Informations détaillées:

Méthode XFEM.# Résultats

À venir.