Section: New Results

High order mesh generation and mesh adaptation

• Participants: Luca Arpaia , Cécile Dobrzynski, Marco Lorini, Mario Ricchiuto

• Corresponding member: Cécile Dobrzynski

This year several new algorithmic improvements have been obtained which will allow to enhance our meshing tools:

• We have enhanced our work on $r$-adaptation techniques for time dependent equations. These techniques are based on mesh deformations obtained by solving continuous differential equations for the local displacements. These equations are controlled by an error monitor. Several improvements have been made. We have proposed a new mixed model to compute the mesh deformations. This model is based on one hand on a Laplacian model and on the other hand on an Elasticity model. It takes advantages of the two approaches: a refined mesh where the solution varies a lot and a smooth gradation of the edges size elsewhere. We have applied this technic to 2d unsteady compressible simulations and we have preliminary results in three dimensions

• Additional work on $r$-adaptation has also involved a simple extension to spherical coordinates, allowing an efficient treatment of inundation caused by large scale tusnami waves [33], [32]

• A novel strategy to solve the finite volume discretization of the unsteady Euler equations within the ALE framework over tetrahedral adaptive grids have been proposed [11]. The volume changes due to local mesh adaptation are treated as continuous deformations of the finite volumes and they are taken into account by adding fictitious numerical fluxes to the governing equation. This peculiar interpretation enables to avoid any explicit interpolation of the solution between different grids and to compute grid velocities so that the GCL is automatically fulfilled also for connectivity changes. The solution on the new grid is obtained through standard ALE techniques, thus preserving the underlying scheme properties, such as conservativeness, stability and monotonicity. The adaptation procedure includes node insertion, node deletion, edge swapping and points relocation and it is exploited both to enhance grid quality after the boundary movement and to modify the grid spacing to increase solution accuracy. We have demonstrated the ability of the method on three-dimensional simulations of steady and unsteady flow fields.

• We extended our technique for generating high order curved meshes to immersed boundary problem. Based on a level-set function, we curved the mesh according to the 0-level-set. Preliminary results in 2d have been performed for compressible simulations

• Initial work on the use of fitting techniques to exactly compute moving shocks has been performed. The benefit of this approach in completely removing all numerical artefacts related to the capturing of the discontinuity, and in recovering the full order of accuracy have been shown for both straight and mildly curved discontinuities [42]