What is the Ideal Mesh?
There are many options to select from when choosing the appropriate meshing technique (which may precipitate decisions regarding preprocessing solutions) to discretize your domain for computer-aided engineering (CAE) applications. Some of these options include:
- Multi-block Structured
- Advancing Front
- …and many others
Choosing from among these options quickly exposes both the advantages and drawbacks of a particular meshing method, especially when comparing one against another. Users typically tailor their meshing habits based on the types of problems they encounter regularly. For example, meshes created using a multi-block structured technique benefit from having flow-aligned hexahedral elements which help provide improved solution accuracy with few cells. However, creating a topology to connect each of the structured block elements can be a challenging and time-consuming task–one which often results in either having localized regions that are insufficiently or overly resolved. This issue sometimes arises when highly resolved regions of the domain are propagated to regions that don’t necessitate resolution nearly as fine, but are maintained due to the point-matching nature of the multi-block structured approach.
Unstructured meshing techniques often provide added flexibility and time savings when creating a mesh, especially meshes involving complex geometries, but do so at the expense of including a profuse number of cells, reducing flow alignment with cell normals, and limiting control for localized mesh refinement. The state-of-the-art in meshing continues to evolve, but to the dismay of many engineers, there isn’t yet an “easy button” to generating a mesh which will provide results that accurately represent the physics involved with a minimum number of cells.
At Pointwise we have been interested in an evolution of the unstructured advancing layer technique, for which Pointwise is known, that may provide flow-aligned cell normals and reduced cell count, combined with the flexibility of generating unstructured meshes using our anisotropic advancing front technology (T-Rex). Pointwise already provides users the ability to combine prisms extruded via T-Rex into a mix of cells including hexahedra, pyramids, and tetrahedra to reduce the number of cells in a given mesh. However, another similar approach involves extruding cells directly from a quad-dominant surface mesh to provide a 3D unstructured hex-dominant grid. Starting with a quad-dominant surface mesh may potentially provide flow-aligned cell normals, particularly near wall boundaries, to help further improve solution accuracy in a similar manner as in a multi-block structured mesh. In this way, one may be able to combine the advantages from both multi-block structured and unstructured advancing front techniques.
To better understand the influence of these mesh characteristics on convergence behavior and solution accuracy, we generated three meshes around an ONERA M6 wing and compared their corresponding CFD solutions to empirical data gathered from a 1979 AGARD report published by NASA. The types of meshes we evaluated were:
- Multi-block Structured
- Unstructured with a mix of tetrahedra and prisms (triangulated surface mesh)
- Unstructured with a mix of hexahedra, pyramids, and tetrahedra (quad-dominant surface mesh)
The model used for this comparison was an ONERA M6 wing section with the following freestream flow conditions:
- M0 = 0.8395
- T0 = 518.688˚R
- Re = 11.72e+06
- α = 3.06˚
The surface meshes for each case are shown in Fig. 1., and a table comparing several characteristics of each mesh is shown in Table 1.
In all of the mesh configurations, the initial cell height from the surface was set to 6.4e-05 inches, which corresponds to a y+ value of 1.0 based on a mean aerodynamic chord of 2.1196 feet. The total number of points used in the multi-block structured mesh here is somewhat larger than the other two meshes due to propagating points used to resolve the wing’s surface to the farfield based on the multi-block topology chosen. The mixed-cell mesh provides some advantage over the unstructured prism mesh for users of cell-centered solvers with a substantial reduction in the number of cells in the mesh.
Figure 3 compares the relative cell volumes at a constant streamwise cut through the mesh for each of the three grids. Where the unstructured prism mesh includes layers of prism elements in the vicinity of the wing surface, the unstructured mixed-cell mesh has primarily hexahedra.
Steady-state RANS solutions were generated using NASA’s FUN3D flow solver (v. 12.7) with their 2003 Menter SST Two-Equation Model implementation. Because FUN3D is a node-based solver, we attempted to keep the number of points resolving the wing’s surface somewhat equitable among the three mesh configurations. A comparison of y+ contours between the three configurations is shown in Fig. 2, and the the number of points used to resolve the wing’s surface in each of the meshes is shown in Table 1.
Table 2 shows solution runtimes as well as integrated lift and drag coefficient results of the ONERA M6 wing for each of the three cases. Unfortunately, there weren’t similar empirical values in the published data to use for comparison. However, Fig. 4 shows a comparison of surface pressures plotted at various spanwise locations along the wing for the three mesh configurations. The numerical results from solutions obtained using each mesh is plotted with empirical data gathered from NASA’s ONERA M6 Wing Validation Archive. These numerical results show pretty good agreement overall with the available empirical data. Towards the wing’s tip, the structured mesh does appear to more closely match the empirical results near the shock boundary on the upper wing surface. However this is likely due to the additional mesh refinement near the wing’s tip that appears in the structured mesh. Similarly, there is an outboard region of the wing between Y/b = 0.80 and 0.95 where the prism and mixed-cell meshes track the shock boundary more closely to experimental results.
A comparison of the convergence behavior from each of the runs is shown in Fig. 5. Convergence between the two unstructured meshes is comparable. Both converged in roughly 100 iterations and reached machine zero in approximately 5,000 iterations. The only substantial difference between the two appears to be the far smoother residual convergence behavior displayed in the unstructured mixed-cell case. The multi-block structured mesh took about 800 iterations to converge and was only about halfway to machine zero in the same number of iterations. This is likely due to the increased number of points combined with smaller cells in regions near the wing’s tip which are present in the multi-block structured mesh.
In the end, all three meshes are suitable, but the smoother residual convergence criteria and reduced wall clock time for the unstructured mixed-cell case didn’t go unnoticed. Looking back on our initial quad-dominant surface mesh, there was certainly some room for improvement when considering our initial desire for flow-aligned cell normals to help compare more directly with the multi-block structured case. After some iterative improvement of the underlying unstructured quad-dominant meshing algorithm, our developers at Pointwise have made considerable progress in pursuit of the ideal mesh–something we think our users are really going to enjoy.