Unstructured Grid Generation For Aerospace Applications (1999)

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ICASE/LaRC/ARO/NSF Workshop on Computational Aerosciences in the 21st Century

Unstructured Grid Generation For Aerospace Applications David L. Marcum and J. Adam Gaither NSF Engineering Research Center for Computational Field Simulation P.O. Box 9627 Mississippi State University, MS 39759 e–mail: [email protected] 1. Introduction Unstructured grid technology has the potential to significantly reduce the overall user and CPU time required for CFD analysis of realistic configurations. To realize this potential, improvements in automation, anisotropic generation, adaptation, and integration within the solution process are needed. Unstructured grid generation has advanced to the point where generation of a grid for most any configuration requires only a couple of hours of user time. However, prior to grid generation, the CAD geometry must be prepared. This process can take anywhere from hours to weeks. It is the single most labor–intensive task in the overall simulation process. Adherence to standards and alternative procedures for surface grid generation which account for small gaps and overlaps and generate across multiple surfaces can minimize and potentially eliminate much of the geometry preparation. With improvements in the geometry preparation process the overall grid generation task can be more fully automated. Anisotropic grid generation is another area in need of improvement. Current techniques have not advanced to the level of robustness and generality as for isotropic grid generation. Methodologies such as use of multiple normals or truly unstructured placement of anisotropic points need to be developed into more robust procedures. Also, solution–adaptation is a potential advantage of an unstructured grid approach that has not been developed into a feasible technology for high–resolution three–dimensional simulations. Highly anisotropic adaptation is needed to improve feasibility. In addition, the grid generation, and in some cases CAD geometry, should be fully integrated into the solution process for some applications. This is essential for more automated design optimization or aeroelastic coupling applications as well as those with moving bodies, control surface deflections, maneuvering vehicles, and/or unsteady flow. Many of the grid and CAD tools in use today may require significant enhancement to be usable in a fully coupled and automatic mode within an overall simulation environment. In this article, representative examples are presented to demonstrate the current status of unstructured grid generation and describe areas for improvement. Also, the overall grid generation process is reviewed to illustrate user operations that could be automated. 2. Unstructured Grid Generation Current Status Unstructured grid generation for many engineering applications has advanced to the point where it can be used routinely for very complex configurations. For example, isotropic element grids suitable for inviscid CFD simulations can be generated for complete aircraft. In viscous cases, generation of high–aspect–ratio elements is more limited in usability as available procedures are not as robust or consistent as those for isotropic elements. However, viscous simulations of relatively complex

ICASE/LaRC/ARO/NSF Workshop on Computational Aerosciences in the 21st Century

configurations have been successfully performed using unstructured grids [Mavriplis and Pirzadeh, 1999 and Sheng, et al, 1999]. Currently, research and commercial systems are available with unstructured grid generators integrated along with CAD/CAE tools. Many of these are suitable for inviscid CFD applications and some have high–aspect–ratio element capabilities for viscous cases. Methods used are typically based on either an octree [Shepard and Georges, 1991], advancing–front [Lohner and Parikh, 1988, Peraire et al, 1988, and Pirzadeh, 1996], Delaunay [Baker, 1987, George et al, 1990, Holmes and Snyder, 1988, and Weatherill, 1985] or a combined approach [Marcum, 1995a, Mavriplis, 1993, Muller et al, 1993, and Rebay, 1993]. To demonstrate the current status of unstructured grid generation, the advancing–front/local–reconnection (AFLR) procedure [Marcum, 1995a and 1995b] will be used. This procedure is integrated in research systems (SolidMesh, MSU) and commercial systems (HyperMesh, Altair Computing). 2.1. AFLR Unstructured Grid Generation The AFLR triangular/tetrahedral grid generation procedure is a combination of automatic point creation, advancing type ideal point placement, and connectivity optimization schemes. A valid grid is maintained throughout the grid generation process. This provides a framework for implementing efficient local search operations using a simple data structure. It also provides a means for smoothly distributing the desired point spacing in the field using a point distribution function. This function is propagated through the field by interpolation from the boundary point spacing or by specified growth normal to the boundaries. Points are generated using either advancing–front type point placement for isotropic elements or advancing–normal type point placement for high–aspect–ratio elements. The connectivity for new points is initially obtained by direct subdivision of the elements that contain them. Connectivity is then optimized by local–reconnection with a min–max type (minimize the maximum angle) type criterion. The overall procedure is applied repetitively until a complete field grid is obtained. More complete details and results are presented in [Marcum, 1995a and 1995b]. Procedures for both surface and volume grid generation based on AFLR have been integrated with CAD tools in a research system called SolidMesh [Gaither, 1997 and Marcum, 1996]. This system was used for geometry clean–up and preparation and grid generation for the example cases presented later in this section. For grid generation with the present methodology, the grid point distribution is automatically propagated from specified control points to edge grids, from edge to surface grids, and finally from surface grids to the volume grid. Surface patches, edges, and corner points for a fighter geometry definition are shown in Fig. 1. The first step in the grid generation process is to initially set the desired point spacing to a global value at all edge end–points. Point spacings are then set to different values at desired control points on edges in specific regions requiring further resolution. For example, end–points along leading edges and trailing edges would typically be set to a very fine point spacing. Point spacings can be set anywhere along an edge. A point in the middle of a wing section would typically be set to a larger point spacing than at the leading or trailing edges. As control point spacings are set, a discretized edge grid is created for each edge. Specification of desired control point spacings is typically the only user input required in the overall grid generation process. Surface grid generation is an interactive process that requires only seconds for generation of a hundred thousand faces on either a PC or workstation. High quality surface grids can be consistently generated. For a typical surface grid, the maximum angle is 120 deg. or less, the standard deviation is 7 deg. or less, and 99.5% or more of the elements have angles between 30 and 90 deg. 2

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Volume grid generation is driven directly from the surface grid. For a moderate size isotropic grid (500,000 elements) generation requires approximately 2 minutes on a workstation (Sun Ultra 60). A large isotropic grid (3,000,000 elements) requires approximately 20 minutes. Viscous grid cases require considerably less time. For example, 2–3 minutes for 2,000,000 total elements or approximately 30 minutes for 10,000,000 total elements. Generation times given include all I/O and grid quality statistics. A workstation or server is usually used for volume generation due to memory requirements, which are about 100 bytes per isotropic element generated. For grids with high–aspect–ratio elements the memory requirements are less. High quality volume grids can be consistently generated if the surface grid is also of high quality. Typically, for an isotropic grid, the maximum dihedral element angle is 160 deg. or less, the standard deviation is 17 deg. or less, and 99.5% or more of the elements have dihedral angles between 30 and 120 deg. The minimum dihedral angle is usually dictated by the geometry. 2.1. NASA Space Shuttle Orbiter A grid suitable for inviscid CFD analysis was generated for the NASA Space Shuttle Orbiter. This case demonstrates the level of geometric complexity that can be handled routinely using unstructured grid technology. Geometry clean–up and preparation required approximately twenty labor–hours to complete. In this case, the original geometry definition had extra surfaces, missing surfaces, gaps, and overlaps. This is often the case in a research environment when a geometry has been processed through different CAD/CAE systems and passed between researchers. Surface and volume grid generation related work required approximately four labor–hours. This time included modifications for grid quality optimization and resolution changes based upon multiple preliminary CFD solutions. The surface grid on the orbiter surface is shown in Fig. 2. The total surface grid contains 150,206 boundary faces. A tetrahedral field cut is shown in Fig. 3. Element size varies smoothly in the field. The complete volume grid contains 547,741 points and 3,026,562 elements. Grid quality distributions for the surface and volume grids are shown in Figs. 4 and 5, respectively. Element angle distributions, maximum values, and standard deviations verify that the surface and volume grids are of very high quality. Computed density contours from an inviscid solution are shown in Fig. 6. The overall structure of the flow field is captured, especially near the body. However, away from from the body, the resolution is not very accurate. Solution–adaptive grid generation could be used to improve the flow field resolution considerably. A suitable solution–adapted grid for this case would have to utilize anisotropic elements to efficiently resolve the highly directional solution gradients. While techniques for adaptation have been studied for some time, they have not been developed into a feasible technology for high–resolution three–dimensional simulations. Highly anisotropic adaptation is needed to improve feasibility. The anisotropic elements should be aligned in a structured type manner with each other and the flow physics for optimal solution algorithm efficiency. The adaptation process must be capable of resolving many types of features, such as shock waves, contact discontinuities, expansions, compressions, detached viscous shear layers, vortices, etc. There are promising approaches, such as point movement with enrichment, feature decomposition, etc. However, significant research work is required to develop a usable procedure. 2.2. EET High–Lift Configuration A grid suitable for high Reynolds Number viscous CFD analysis was generated for a high–lift wing body configuration of the Energy Efficient Transport (EET). This case demonstrates the level of geometric complexity that can be handled for viscous flow cases using unstructured grid technology. 3

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Geometry clean–up and preparation required approximately seven labor–hours to complete. In this case, the original geometry definition was relatively ”clean” and much of the time was spent on tolerance issues and surface intersections. Surface and volume grid generation related work required approximately two labor–hours. This time included modifications for grid quality optimization and resolution changes based upon a preliminary CFD solution. The surface grid on the upper and lower surface of the wing are shown in Figs. 7a and 7b. The total surface grid contains 273,500 boundary faces. Tetrahedral field cuts are shown in Figs. 8a and 8b. Element size varies smoothly in the field and there is a smooth transition between high–aspect–ratio and isotropic element regions. Also, in areas where there are small distances between surfaces, the merging high–aspect–ratio regions transition (locally) to isotropic generation. If these regions advance too close, without transition, the element quality can be substantially degraded. For this case, increased grid resolution of the leading and trailing edges of the main wing, flap, slat, and vane would improve the grid quality in merging regions. The complete volume grid contains 2,215,470 points and 12,873,429 elements. Most of the tetrahedral elements in the high–aspect–ratio regions can be combined into pentahedral elements for improved solver efficiency. With element combination, the complete volume grid contains 1,813,111 tetrahedrons, 61,366 five–node pentahedrons (pyramids), and 3,645,862 six–node pentahedrons (prisms). Grid quality distributions for the surface and volume grids are shown in Figs. 4 and 5, respectively. Element angle distributions and maximum values verify that the surface and volume grids are of very high quality. The distribution peaks are at the expected values of near 0, 70, and 90 deg. Computed streamlines from a viscous, turbulent, incompressible solution are shown near the upper and lower surfaces of the wing in Figs. 9a and 9b. Comparison to experimental data (not shown) is reasonable overall [Sheng, et al, 1999]. However, additional resolution is needed on the flap, slat, and vane, particularly at the leading edges. The EET configuration illustrates that unstructured grid technology can be used to simulate viscous flow about relatively complex configurations. However, the unstructured grid generation process is not as advanced as for isotropic elements. Improvements are needed in robustness and element quality for cases with complex geometry and multiple components in close proximity. Several techniques are listed below which could be used to improve the unstructured grid generation process for viscous cases. An anisotropic surface grid could be used to efficiently increase grid density along leading and trailing edges of wing components. Automatic surface refinement of close boundaries with merging boundary layers (as is done in 2D) would improve grid quality. Embedded surfaces in the field could be used to improve accuracy in wake regions. Multiple boundary–layer surface normals could be used to enhance grid quality and resolution at points where the boundary surface is discontinuous. A tetrahedral field cut for an example case with multiple normals is shown in Fig. 10. 3. Overall Unstructured Grid Generation Process As demonstrated in the previous examples, user time required to generate an unstructured grid from a properly prepared geometry definition is only a couple of hours. However, the process of preparing the geometry can take anywhere from hours to weeks. It is the single most labor–intensive task in 4

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the overall CFD simulation process. Much of this time is often spent on repair of gaps and overlaps, which can be minimized through standards. Even with a geometry definition which is truly a solid there can be significant CAD work required to prepare the geometry for CFD analysis. Elimination of features or components not relevant to the analysis can require substantial effort. Geometry preparation can also include further work in grouping of multiple surface definitions. Alternative procedures for CAD preparation and surface grid generation are needed which account for small gaps and overlaps, generate across multiple surfaces, and automatically detect and remove features and components not relevant to the analysis. These procedures would minimize and potentially eliminate much of the geometry preparation. With improvements in the geometry preparation process the overall grid generation task can be more fully automated. This can include automatic specification of appropriate element size, at least, for a given class of configurations. Examples of user steps that could be readily automated and use of multiple surface grouping are presented below. With the procedure described in this article, volume element size and distribution is determined from the boundary. A low–quality surface grid will produce low–quality volume elements near the surface. In most cases, a high–quality surface grid will produce a high–quality volume grid. Low–quality surface elements are usually the result of inappropriate edge spacing. With fast surface grid generation and simple point spacing specification, optimizing the surface quality is a quick process. User input could be eliminated by automatically reducing point spacing in low quality regions. An example of a surface mesh with a low quality triangle, that can be corrected by point spacing placement or reduction, is shown in Fig. 11a. In this case, the surface patch has close edges which can not be eliminated. In Fig. 11a, the initial choice of a uniform spacing at the edge end–points produces a single low–quality triangle. Specifying a single point spacing at the middle of the edge near the close edges, eliminates the low–quality element, as shown in Fig. 11b. Alternatively, the spacing near the close edges can be reduced to produce a more “ideal” grid, at the expense of an increased number of elements, as shown in Fig. 11c. Other conditions can affect volume quality even if the surface grid is of high–quality. An example is shown in Fig. 12. In this case, there are two nearby surfaces with large differences in element size. This results in distorted volume elements between the surfaces, as shown in Fig. 12. These elements can be eliminated by increasing the spacing on the surface which has the smaller elements and/or decreasing the spacing on the surfaces which have the larger elements. From a solution algorithm, perspective, the spacings should probably be reduced. The region between the two objects can not be resolved by the solver without additional grid points. Automatic detection and refinement of “close” boundary surfaces could reduce or eliminate user input for these situations. This could be even more beneficial for viscous cases. Volume grid quality can degrade if the boundary–layer regions from opposing boundaries merge with high–aspect–ratio elements. Surface definition can also impact surface grid quality. This type of problem is usually due to a surface patch with a width that is smaller than the desired element size. An example case with a surface containing 11 surface definition patches is shown in Fig. 13a. The detail view shown in Fig 14a of the top center area reveals a very short edge due to the way the surface patches are defined. Generating individual surface grids for each patch can result in an irregular and low–quality overall surface grid, as shown in Fig. 13b. Very high–aspect–ratio elements are generated in the region of the short edge, as shown in Fig. 14a. Combing the patches into one surface patch improves the quality, as shown in Figs. 13c and 14b. This can be accomplished by replacing the multiple patches with a new single definition. However, that process can require considerable user time and it modifies the original geometry. An alternative is to topologically group the surface definition 5

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patches in an intermediate mapped space. This requires minimal user input (selection of patches to be grouped) and preserves the original geometry definition. The surface grid shown in Figs. 13c and 14b was generated using a preliminary version of this procedure. Grouping of multiple surface patches can also be used to remove unnecessary features from a geometry definition. For example, the surface grid shown in Fig. 15a contains a slightly recessed circular region which is well resolved with the point spacings shown. If a larger point spacing is desired, this feature may not be relevant. Grouping the surface patches resolves this region only to the resolution of the selected point spacings, as shown in Fig. 15b. 4. Summary Representative examples were presented which demonstrated that unstructured grid generation has advanced to the point where generation of a grid for most any configuration requires only a couple of hours of user time. However, the overall grid generation process can take anywhere from hours to weeks. Automation and improvements in procedures for preparing the CAD surface definition would substantially reduce the required user time. For viscous grid generation, enhanced methods are needed to improve grid quality and robustness. Also, significant research work is required to develop viable solution–adaptive procedures for high–resolution of complex three–dimensional flow fields. 5. Acknowledgements The authors would like to acknowledge support for this work from the Air Force Office of Scientific Research, Dr. Leonidas Sakell, Program Manager, Ford Motor Company, University Research Program, Dr. Thomas P. Gielda, Technical Monitor, Boeing Space Systems Division, Dan L. Pavish, Technical Monitor, NASA Langley Research Center, Dr. W. Kyle Anderson, Technical Monitor, and the National Science Foundation, ERC Program, Dr. George K. Lea, Program Director. In addition, the author would like to acknowledge Reynaldo Gomez of NASA Johnson Space Center for providing the Space Shuttle Orbiter geometry and Dr. W. Kyle Anderson of NASA Langley Research Center for providing the Energy Efficient Transport geometry and experimental data. References Baker, T. J., 1987, ”Three–Dimensional Mesh Generation by Triangulation of Arbitrary Point Sets,” AIAA Paper 87–1124. Gaither, J. A., 1997. “A Solid Modelling Topology Data Structure for General Grid Generation,” MS Thesis, Mississippi State University. George, P. L., Hecht, F., and Saltel, E., 1990, ”Fully Automatic Mesh Generator for 3D Domains of any Shape,” Impact of Computing in Science and Engineering, 2, 187. Holmes, D. G. and Snyder, D. D., 1988, ”The Generation of Unstructured Meshes Using Delaunay Triangulation,” Proceedings of the Second International Conference on Numerical Grid Generation in Computational Fluid Dynamics, Eds. Sengupta, S., Hauser, J., Eiseman, P. R., and Thompson, J. F., Pineridge Press Ltd. Lohner, R. and Parikh, P., 1988, ”Three–Dimensional Grid Generation by the Advancing–Front Method,” International Journal of Numerical Methods in Fluids, 8, 1135. 6

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Marcum, D. L. and Weatherill, N. P., 1995a, “Unstructured Grid Generation Using Iterative Point Insertion and Local Reconnection,” AIAA Journal, 33, 1619. Marcum, D. L., 1995b, “Generation of Unstructured Grids for Viscous Flow Applications,” AIAA Paper 95–0212. Marcum, D. L., 1996, “Unstructured Grid Generation Components for Complete Systems,” 5th International Conference on Grid Generation in Computational Fluid Simulations, Starkville, MS. Mavriplis, D. J. and Pirzadeh, S., 1999, ”Large–Scale Parallel Unstructured Mesh Computations for 3D High–Lift Analysis,” AIAA Paper 99–0537. Mavriplis, D. J., 1993, ”An Advancing Front Delaunay Triangulation Algorithm Designed for Robustness,” AIAA Paper 93–0671. Muller, J. D., Roe, P. L., and Deconinck, H., 1993, ”A Frontal Approach for Internal Node Generation in Delaunay Triangulations,” International Journal of Numerical Methods in Fluids, 17, 256. Peraire, J., Peiro, J., Formaggia, L., Morgan, K., and Zienkiewicz, O. C., 1988, ”Finite Element Euler Computations in Three–Dimensions,” International Journal of Numerical Methods in Engineering, 26, 2135. Pirzadeh, S., 1996, ”Three–Dimensional Unstructured Viscous Grids by the Advancing–Layers Method,” AIAA Journal, 34, 43. Rebay, S., 1993, ”Efficient Unstructured Mesh Generation by Means of Delaunay Triangulation and Bowyer–Watson Algorithm,” Journal of Computational Physics, 106, 125. Sheng, C. Hyams, D., Sreenivas, K., Gaither, A., Marcum, D., Whitfield, D., and Anderson W., 1999, ”Three–Dimensional Incompressible Navier–Stokes Flow Computations About Complete Configurations Using a Multi–Block Unstructured Grid Approach,” AIAA Paper 99–0778. Shepard, M.S. and Georges, M. K., 1991, ”Automatic Three–Dimensional Mesh Generation by the Finite Octree Technique,” International Journal of Numerical Methods in Engineering, 32, 709. Weatherill, N. P., 1985, ”A Method for Generation of Unstructured Grids Using Dirichlet Tessellations,” Princeton University, MAE Report No. 1715.

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Fig. 1

Surface patches, edges, and corner points for fighter geometry definition.

Fig. 2

NASA space shuttle orbiter surface grid. 8

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Fig. 3

Symmetry plane surface grid and tetrahedral field cut for NASA space shuttle orbiter grid.

Fig. 4

NASA space shuttle orbiter and EET high–lift wing–body surface grid quality. 9

ICASE/LaRC/ARO/NSF Workshop on Computational Aerosciences in the 21st Century

Fig. 5

NASA space shuttle orbiter and EET high–lift wing–body volume grid quality.

Fig. 6

Computed density contours for NASA space shuttle orbiter. 10

ICASE/LaRC/ARO/NSF Workshop on Computational Aerosciences in the 21st Century

a) Top surface of wing.

b) Bottom surface of wing. Fig. 7

EET high–lift wing–body surface grid. 11

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a) Field cut with wing surface grid.

b) Field cut with wing removed. Fig. 8

Tetrahedral field cuts for EET high–lift wing–body grid. 12

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a) Top surface of wing.

b) Bottom surface of wing. Fig. 9

Computed streamlines for EET high–lift wing–body. 13

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Fig. 10 Tetrahedral field cut for high–aspect–ratio element grid with multiple surface normals.

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a) Surface grid patch with distorted surface element.

b) Surface grid patch improved by applying a point spacing near problem edge.

c) Surface grid patch improved by applying a reduced point spacing near problem edge. Fig. 11 Surface grid problem due to close edges. 15

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Fig. 12 Distorted tetrahedral elements between surface grids which are close and have large differences in surface element size.

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a) Original surface definition patches, edges, and corner points.

b) Surface grid with multiple surface definition patches.

c) Surface grid with one topologically combined surface definition patch. Fig. 13 Surface grid problem due to multiple surface definitions. 17

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a) Surface grid with multiple surface definition patches.

b) Surface grid with one topologically combined surface definition patch. Fig. 14 Detail view of surface grid problem due to multiple surface definitions. 18

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a) Surface grid with detail feature resolved.

b) Surface grid at low resolution with feature unresolved. Fig. 15 Use of combined surface definition patches to eliminate features at different resolutions. 19