Optimal Anisotropic Finite Element Meshes for Multiphase Flow Dynamics

  • Coupez, Thierry (CEMEF/Mines Paris - PSL)

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A wider use of numerical simulation is depending on meshing and adaptive meshing capabilities when complex geometry, multi-domain, moving interface and multiphase flow are involved. Anisotropic meshes can help to solve a lot of numerical modelling problems by providing the optimal numerical approximation, that is, the best possible solution for a given complexity. The success of anisotropic mesh adaptive technologies relies on different ingredients. First, unstructured meshing capabilities based on the flexibility of simplex elements allow to derive iteratively from mesh to mesh just by local modifications, while minimizing the cost of rebuilding a mesh and allowing massively parallel meshing and computing [1]. Moreover, the remeshing process must be driven by a metric field calculated from an error estimate related to the complexity (i.e. the computational cost). Finally, the CFD solver must still work accurately with highly stretched elements as long as they are correctly aligned to the solution. Under these conditions we are in position to address the optimality of the mesh and to adapt dynamically to obtain or maintain the optimal one. In this work, the CFD solver is based on stabilised finite elements, a Petrov-Galerkin formulation in which the stabilisation parameter is shown to be controlled optimally by the trace of a unit continuous metric field and the dimensional physical parameters. The metric calculation relies on the tensorial approach [2], taking into account the length and orientation of the elements and slightly different from the differential approach. The approximation error estimate including both the velocity, pressure and the interface/boundary locations [3] is obtained from the same tensorial approach and provides a unique mesh metric (avoiding metric intersections). This methodology will be described with several examples and industrial applications, in which recurring problems can be solved simply, such as the stress over-continuity, the added-mass problem, the surface tension calculations (without reconstruction of normals and even including he three-phase problem).