Êtes-vous un étudiant de l'EPFL à la recherche d'un projet de semestre?
Travaillez avec nous sur des projets en science des données et en visualisation, et déployez votre projet sous forme d'application sur GraphSearch.
Publication
Êtes-vous un étudiant de l'EPFL à la recherche d'un projet de semestre?
Travaillez avec nous sur des projets en science des données et en visualisation, et déployez votre projet sous forme d'application sur GraphSearch.
A numerical method based on an adaptive octree space discretization for the simulation of displacement of free surfaces is proposed and applied to 3D free surface flow problems. A VOF approach is combined with a mass-conserving semi-Lagrangian time-stepping scheme. An interface prediction algorithm is used to refine the octree at the predicted location of the interface in order to ensure detail preservation. Subsequently, the fluid is advected and a coarsening algorithm adapts the mesh to avoid excess refinement in non-interfacial regions. SLIC and decompression algorithms are used for post-processing to limit numerical diffusion and correct numerical compression of the VOF function. The scheme is unconditionally stable with respect to the CFL number and does not require solving of a linear system. The octree scheme allows anisotropy and refinement of interfacial cells to an arbitrary level. It does not require a 2:1 cell size ratio condition between neighbouring cells. Numerical validation is done on benchmark test cases and results are compared with the structured analog. The scheme is coupled with a Stokes solver on a tetrahedral grid for solving of time-dependent Navier-Stokes equations and numerical results are compared with experimental water wave profile measurements. (C) 2016 The Authors. Published by Elsevier Ltd.
Chargement
Chargement
Chargement
Chargement
Chargement
how?' but also why?', where?' and what for?'.
The motivation for developing structure-preserving algorithms for special classes of problems originates independently in such diverse areas of research as astronomy, molecular dynamics, mechanics, control theory, theoretical physics and numerical analysis, with important contributions from other areas of both applied and pure mathematics. Moreover, it turns out that preservation of geometric properties of the flow not only produces an improved qualitative behaviour, but also allows for a significantly more accurate long-time integration than with general-purpose methods.
In addition to the construction of geometric integrators, an important aspect of geometric integration is the light it sheds on the relationship between geometric properties of a numerical method and favourable error propagation in long-time integration. A very successful organising principle is backward error analysis, whereby the numerical one-step map is interpreted as (almost) the flow of a modified differential equation. In this way, geometric properties of the numerical integrator translate seamlessly into structure preservation on the level of the modified equation. Much insight and rigourous error estimates over long time intervals can then be obtained by combining backward error analysis with the KAM theory and related perturbation theories for Hamiltonian and reversible systems. While this approach has been very successful for ordinary differential equations, much less is currently known about highly oscillatory systems and geometric integrators for partial differential equations.
Geometric numerical integration has been an active interdisciplinary research area since the last decade. Although the subject is in a lively phase of intensive development, the results so far are substantive and they impact on a wide range of application areas and on our understanding of core issues in computational mathematics. This is evidenced by the monographs \cite{HLW:GNI2002,LR:SMH2004}.