Entropy solution at concave corners and ridges
Romain Aubry, B. Kaan Karamete, Eric L. Mestreau, Saikat Kumar Dey · 54th AIAA Aerospace Sciences Meeting · 2016
Boundary layer volume mesh generation applied to generic non smooth surfaces gives rise to various challenges. The mesh should present a strong anisotropy normal to the surface, should smoothly transition to the isotropic region, and should take into account complex ridges and corners. A theoretical tool to attack this problem consists in relying on the Eikonal equation, which is a non linear hyperbolic equation capable of generating shocks at concave regions and expansion waves close to convex regions. Typically, expansion waves are discretized with multiple normals. In this work, emphasis is given to the the reversible phenomenon, where shocks appear. This represents the extension to the three dimensional space of. Only few strategies have been advocated for concave situations. In, negative elements are removed, therefore stopping the front progression close to these locations. However, since typically a small jump between layers is enforced, stopping the front prematurely will spread from this location outwards. In, the normal direction is limited. However, this may violate the prescribed size, or generate extremely small cells close to concave corners. Another approach consists in using smoothed normals as extruded directions, or even a blend of both, trading normality for postponed front abortion. As mentioned previously, Athanasiadis et al. is one of the few references to consider special procedures for concave situations, where characteristics coalesce. It is mentioned that a quad surface mesh is expected to be able to collapse the prisms along concave ridges in a structured manner. However, this represents only a particular configuration of a more generic approach. The Voronoi diagram is the key building block of this construction. Considering the boundary layer mesh as a subpart of the three dimensional generalized Voronoi diagram, where triangle faces, edges and vertices are taken into account, the Voronoi bisectors are important spatial locations that ideally would be discretized in the mesh. This would require however the computation of the full three dimensional generalized Voronoi diagram. For triangle surface meshes, concavity and convexity are two important notions because they implicitly give informations on the distance field generated by the triangles in space. Convexity means that, as seen before, the distance field will present expansion waves, while concavity brings shock waves. Regarding the Voronoi diagram, convexity is associated with multiple normals, while concavity implies Voronoi bisectors. Therefore, concave edges will provide the birth of Voronoi faces while concave corners will generate Voronoi edges in three dimensions.