Alignment and orthogonality in anisotropic metric-based mesh adaptation

Cartesian or structured grids are usually preferred in the numerical simulation community as they have many interesting properties: natural high-order solutions, fast numerical solvers and high robustness. However, when complex geometries are used, robust Cartesian grids are less straightforward to implement, many research exist on the subject in order to intersect a complex geometry with a Cartesian grid.1 As far as mesh adaptation is concerned, reaching a high level of anisotropy remains a challenge as the directions are always aligned with the natural axes. On a di↵erent perspective, unstructured mesh generation is now a mature field of research where many methods are able to generate a 3D volume mesh with respect to a prescribed surface mesh. The most common methods are based on the Delaunay-kernel with boundary recovery,6,12,14 on a frontal approach24,35 or on a combination of both with local reconnections.33 All these techniques have been extended in the framework of anisotropic unstructured mesh adaptation. For robustness issues, most of the strategies rely on local mesh modification operators.8,10,11,20,22,37,40 If these strategies produce highly anisotropic tetrahedra, they fail to naturally produce elements that are aligned with the eigenvectors of the provided metric field. Unfortunately, element shape and alignment with the metric field also impacts accuracy and e ciency of the solution process. One well-known CFD example is the mesh generation of the boundary layer (BL) regions where physical variable gradients vary by orders of magnitude in di↵erent directions. In BL regions the characteristics of a mesh generated using anisotropic unstructured mesh adaptation are not always ideal. Viscous BL regions near a surface often have very stringent numerical requirements as they involve high-gradient and non-linear physics that usually includes turbulence. These regions are known a priori and ideally suited to a pseudostructured approach that generates elements with an advancing layer/normal type process.19,23,30,33,38 The resulting mesh is highly aligned, precisely spaced and very structured in at least the normal direction. The characteristics of a pseudo-structured type mesh are ideal for the BL regions. However, such methods are not compatible with adaptive approaches eliminating the adaptation of the boundary layer region and they usually lead to poor quality mesh near geometry singularities or in the transition between the BL region and the overall field, even if strategies have been proposed to solve this issues.4,5, 30,34 Consequently, the quest of ⇤Researcher, adrien.loseille@inria.fr †Professor, marcum@cavs.msstate.edu ‡Senior Researcher, frederic.alauzet@inria.fr