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With anisotropic re nement, it is more di cult to ensure a smooth variation in the mesh. In addition, one has to be careful that at least one of the cells adjacent to the face has the correct picture of it. For example, consider an x-face. If the cell on the left is re ned in y and the cell on the right is 6 Marsha Berger and Michael Aftosmis re ned in z, then neither cell has a correct picture of the face. To prevent such anomalies, each cell shares" its re nement tag with its neighbor. Since this could propagate, invalidating previously acceptable faces, several sweeps over the face list may be necessary. Although a diabolical case may need a large number of sweeps, in practice one or two passes su ces.
The re nement strategy described above for uniform isotropic re nement is actually implemented by sweeping over the 3 coordinate directions individually. Each re nement bisects the cell in one direction only. Each new face that bisects a cell creates one new cell. Both the new and old cells adjust their locations encoded into unique cell names" Aftosmis, et al. 98 . The faces that point to these cells must also be adjusted. Since we do not keep cell-to-face pointers, this is accomplished by keeping a single pointer from old cells to their new sibling. A sweep over faces easily adjusts the a ected cells.
Directional re nement is easily implemented within this framework.
As a model geometry for anisotropic re nement consider a cylinder with its axis aligned with the z-axis. Intuitively, one expects re nement in the x and y directions, since the cylinder has a circular crosssection in the x-y plane. However, there is no curvature in the z direction. Table 1 compares the number of cells in the isotropically re ned mesh with the corresponding anisotropic mesh with the same number of levels as a function of the maximum allowed aspect ratio. The initial coarse mesh is 24 24 24 cells, the re nement criteria uses a variation in the surface normals of 5 degrees, and the bu er zones are three cells wide. Re nement occurs along the length of the cylinder anisotropically, but the criteria triggers isotropic re nement at the corners. Figure 4 shows an aspect ratio 8 mesh around the cylinder. The growth rate in the number of cells is substantially reduced in the higher aspect ratio case, with the total number of cells a factor of 2.2 less with aspect ratio 4, and 2.7 with aspect ratio 8.
Ref. Level Cells AR 1 Cells AR 2 Cells AR 4 Cells AR 8
1. Comparison of number of cells using isotropic re nement versus Table anisotropic re nement, as a function of the maximum allowable aspect ratios.
For a more realistic test case, we compare results for an ONERA M6 wing with anisotropic re nement. The table below shows the total number of cells in the isotropic mesh, and aspect ratio 2, 4 and 8 meshes. Five bu er cells were used. A cell was tagged for re nement if the variation in the components of the normals exceeded 10 degrees, and the directional angle re nement criteria Aspects and Aspect Ratios of Cartesian Mesh Methods 7 was 30 degrees. For the results in the table below, the geometry was tagged for isotropic re nement for 6 levels. After that level of resolution, the directional re nement could begin. Figure 5 shows the Cartesian cut cells that intersect the geometry, color coded by aspect ratio. The leading and trailing edges are both re ned to the maximum aspect ratio allowed. Even for this case, the savings in total number of cells between the isotropic mesh and the aspect ratio 8 mesh is over a factor of 2.
Ref. Level Cells AR 1 Cells AR 2 Cells AR 4 Cells AR 8 Comparison of number of cells using anisotropic re nement for the OnTable 2.
3 Computational Results We use an example of a mesh around a business jet to illustrate the runtime performance of the grid generator. There are 92850 triangles in the initial surface description. The nal mesh has 9 levels of re nement, with
2.29 million hex cells and 7.3M faces. Only 310K of the cells 13 of the mesh intersected the geometry. There were 17827 split polyhedral cells in the mesh, despite the number of levels of re nement.
The total number of cells in the mesh after each level of re nement is shown in Table 3. As the mesh re nes, the algorithm monitors the trend of the cell growth factor. A curve t is used to predict the number of cells in the nal mesh. This information makes it possible to minimize the number of memory re-allocations and copies required during the mesh generation process. For example, this growth factor for the last 3 re nements, as shown in Table 1, is 2.20, 2.18, and 2.14 A bu er zone of 3 cells was used in this mesh. The re nement criteria was 10 degrees. Six levels of mesh were re ned around the body before the algorithm started estimating where to re ne.
The complete grid generation procedure, including the calculation of the face and cell centroids, took 3 minutes 51 seconds on an SGI R10000 workstation. A maximum of 308 Mb was used to generate the 9 level mesh. Figure 6 shows 10 cutting planes through the mesh, which provide a good indication of the re nement criteria's behavior. For comparison, the anisotropic 9 level mesh, with a maximum aspect ratio of 8, had 1225022 cells.
8 Marsha Berger and Michael Aftosmis Ref. Level Total Cells Cut Cells Total Faces Growth of number of cells with re nement level for business jet example.
4 Future Research Two projects currently underway would greatly extend the capabilities of Cartesian mesh methods. The rst computes inviscid ow with moving geometry, including the case with geometry components in relative motion. The di culty here is in developing numerical methods to handle the case where Cartesian cells are newly uncovered during a time step, or become completely covered by the geometry. Some two dimensional work in this direction is in Bayyuk, et al. 96 ; a new approach is being developed by Forrer and Berger 98.
The second extension is to try to use Cartesian meshes to help generate viscous meshes with boundary layer zoning in an automatic way. As contrasted with previous e orts by Coirier 94, the new approach Delanaye et al. 99 uses a prismatic body- tted grid, with a background Cartesian grid.
However, to reduce the dependence of the body- tted grid on the quality of the surface triangulation, a new triangulation based on the intersection of the original surface with the Cartesian grid is used to spawn the new grid.
Since this new approach automatically generates the surface discretization used by the ow solver, it maintains the decoupling between the geometry description and surface mesh seen by the ow solver. This decoupling is one of the biggest advantages of Cartesian meshes for inviscid ows, and retaining it in the viscous case is an important rst step.
Acknowledgements The authors thank Eric Charlton for the use of his business jet surface triangulation. Marsha Berger was supported in part by DOE Grant DEFG02ER25139 and by AFOSR Grant F49620-97-1-0322. Some of this work was performed while at RIACS, whose support is gratefully acknowledged.
Aspects and Aspect Ratios of Cartesian Mesh Methods 9 References Aftosmis, M., Upwind Method for Simulation of Viscous ow on Adaptively Rened Meshes, AIAA J. 32:2, Feb., 1994.
Aftosmis, M., Berger, M. and Melton, J., Robust and E cient Cartesian Mesh Generation for Component-Based Geometry, AIAA J. 36:6, June, 1998.
Aftosmis, M., Melton, J. and Berger, M., Adaptation and Surface Modeling for Cartesian Mesh Methods, AIAA paper 95-1725, June, 1995.
Bayyuk, S., Powell K. and van Leer, B., An Algorithm for Simulation of Flows with Moving Boundaries and Fluid-Structure Interactions, Proc. 1st AFOSR Conf. on Dynamic Motion CFD, June, 1996.
Berger, M. and Melton, J., An Accuracy Test of a Cartesian Grid Method for Steady Flow in Complex Geometries, Proc. 5th Intl. Conf. Hyp. Problems., Stonybrook, NY, June, 1994.
Bonet, J. and Peraire, J., An Alternating Digital Tree ADT Algorithm for Geometric Searching and Intersection Problems, Intl. J. Num. Meth. Engg. 31:1Charlton, E. and Powell, K., An Octree Solution to Conservation Laws Over Arbitrary Regions OSCAR, AIAA paper 97-0198, Jan., 1997.
Coirier, W., An Adaptively Re ned, Cartesian, Cell-Based Scheme for the Euler and Navier-Stokes Equations. PhD Thesis, University of Michigan, 1994.
Coirier, W. and Powell, K., An Accuracy Assessment of Cartesian-Mesh Approaches for the Euler Equations, AIAA paper 93-3335, July, 1993.
Delanaye, M., Aftosmis, M., Berger, M., Liu, Y. and Pulliam, T., Automatic Hybrid-Cartesian Grid Generation for High-Reynolds Number Flows Around Complex Geometries. Submitted to AIAA, Reno, Jan, 1999.
Forrer, H. and Berger, M., Flow Simulations on Cartesian Grids involving Complex Moving Geometries, Proc. 7th Intl. Conf. Hyp. Problems, Zurich, Switz., Feb. 1998.
Johansen, H. and Colella, P., A Cartesian Grid Embedded Boundary Method for Poisson's Equation on Irregular Domains. To appear, J. Comp. Phys., 1999.
Karman Jr., S., SPLITFLOW: A 3D Unstructured Cartesian Prismatic Grid CFD Code for Complex Geometries, AIAA paper 95-0343, Jan., 1995.
Landsberg A. and Boris, J., An E cient Method for Solving Flows around Complex Bodies, NRL Review, 1993.
Melton, J., Automated Three-Dimensional Cartesian Grid Generation and Euler Flow Solutions for Arbitrary Geometries. PhD Thesis, U.C. Davis, June, 1996.
O'Rourke, J., Computational Geometry in C. Cambridge Univ. Press, 1994.
Pember, R.B., Bell, J.B., Colella, P., Crutchfield, W.Y. and Welcome, M.W., Adaptive Cartesian Grid Methods for Representing Geometry in Inviscid Compressible Flow. AIAA paper 91-1542, June, 1991.
10 Marsha Berger and Michael Aftosmis
A general split face matching algorithm has to treat faces where the cell Fig. 2.
on one side only is re ned. The high x face of cell a is split into two face polygons which must be matched with the 5 polygons on the low x face of the small cells.
The algorithm stores ve faces for this case: one for ab, two for ac cell c is split, one for ae and one for adAll 4 faces on the re ned side are split into 8 regions; the coarse side is split into two pieces. The algorithm matches fragments of the face polygons which lie on Cartesian edges.
Aspects and Aspect Ratios of Cartesian Mesh Methods 11
The +" signs illustrate the maximum variation in surface normals for a Fig. 3.
two dimensional example. If the magnitude of the variation exceeds the adaptation threshold, the cell needs to be re ned. If the variation is all in one direction within the slivers marked by the dotted lines, the cell can be re ned anisotropically.
Mesh around a cylinder with anisotropic re nement, aspect ratio 8.
12 Marsha Berger and Michael Aftosmis