PhD Position at Inria Saclay-Ile de France Title: Methods for generating quad-dominant and hex-dominant adaptive meshes for CFD applications

Subject: The aim of mesh adaptation is to generate the best mesh to perform a specific numerical simulation [1]. It results in a powerful methodology that reduces significantly the size of the mesh required to reach the desired accuracy. Thus, it impacts favorably the simulation CPU time and memory requirement. Moreover, as the generated adapted mesh is in agreement with the physics of the flow, for some applications, this is the only way to obtain an accurate prediction of the flow [3]. Nowadays, mesh adaptation is a mature tool which is well-posed mathematically [8, 9] (it is based on the concept of metric and metric field (i.e., Riemannian metric space)). And, as it is fully automatic, it has started to be used in industrial R&D departments. Indeed, it has already proved, throughout many publications and applications, its superiority with respect to fixed mesh. It has been applied successfully to inviscid flows [3], viscous turbulent flows [5], unsteady flows [2], and unsteady flows with moving geometries [4]. On the other hand, there is still a strong demand for pure hexahedra meshes as many numerical schemes have proven to be more accurate on quad meshes than triangles meshes, and as many numerical schemes favor structured elements in the boundary layer instead of tetrahedra to simulate viscous turbulent flows. But, the automatic generation of hex meshes with the alignement constraint requested by the user1 is a chimera that has not been solved since forty years. To relax the constraint on the meshing method, we propose to consider methods to generate quad-dominant or hex-dominant meshes. Quite recently, it has been proposed to use frame field to generate quad or hex meshes [6, 7], the difficulty here is to build an adequate frame field. We remark that the notion of metric field used in mesh adaptation is a generalization of the frame field as it is a frame field with a size prescription, and this field is naturally provided by anisotropic error estimates. To generate quasi-structured simplicial (triangles or tetrahedra) meshes metric-aligned and metric-orthogonal adaptive mesh generation methods have been developed [10], see Figure 1. The goal of this thesis is to pursue the development of the metric-aligned and metric-orthogonal approaches in 2D and 3D and to add a new stage in the mesh adaptation loop to generate quad-dominant or hex-dominant meshes.