The Development of Unstructured Grid Methods For Computational Aerodynamics

1. The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis ICASE NASA Langley Research Center Hampton, VA 23681 USA Cornell University, September 17,2002 Ithaca New York, USA

2. Overview • Structured vs. Unstructured meshing approaches • Development of an efficient unstructured grid solver • Discretization • Multigrid solution • Parallelization • Examples of unstructured mesh CFD capabilities • Large scale high-lift case • Typical transonic design study • Areas of current research • Adaptive mesh refinement • Moving and overlapping meshes Cornell University, September 17,2002 Ithaca New York, USA

3. CFD Perspective on Meshing Technology • CFD Initiated in Structured Grid Context • Transfinite Interpolation • Elliptic Grid Generation • Hyperbolic Grid Generation • Smooth, Orthogonal Structured Grids • Relatively Simple Geometries Cornell University, September 17,2002 Ithaca New York, USA

4. CFD Perspective on Meshing Technology • Sophisticated Multiblock Structured Grid Techniques for Complex Geometries Engine Nacelle Multiblock Grid by commercial software TrueGrid.

5. CFD Perspective on Meshing Technology • Sophisticated Overlapping Structured Grid Techniques for Complex Geometries Overlapping grid system on space shuttle (Slotnick, Kandula and Buning 1994)

6. Unstructured Grid Alternative • Connectivity stored explicitly • Single Homogeneous Data Structure Cornell University, September 17,2002 Ithaca New York, USA

7. Characteristics of Both Approaches • Structured Grids • Logically rectangular • Support dimensional splitting algorithms • Banded matrices • Blocked or overlapped for complex geometries • Unstructured grids • Lists of cell connectivity, graphs (edge,vertices) • Alternate discretizations/solution strategies • Sparse Matrices • Complex Geometries, Adaptive Meshing • More Efficient Parallelization Cornell University, September 17,2002 Ithaca New York, USA

8. Discretization • Governing Equations: Reynolds Averaged Navier-Stokes Equations • Conservation of Mass, Momentum and Energy • Single Equation turbulence model (Spalart-Allmaras) • Convection-Difusion – Production • Vertex-Based Discretization • 2nd order upwind finite-volume scheme • 6 variables per grid point • Flow equations fully coupled (5x5) • Turbulence equation uncoupled Cornell University, September 17,2002 Ithaca New York, USA

9. Spatial Discretization • Mixed Element Meshes • Tetrahedra, Prisms, Pyramids, Hexahedra • Control Volume Based on Median Duals • Fluxes based on edges • Single edge-based data-structure represents all element types Cornell University, September 17,2002 Ithaca New York, USA

10. Spatially Discretized Equations • Integrate to Steady-state • Explicit: • Simple, Slow: Local procedure • Implicit • Large Memory Requirements • MatrixFreeImplicit: • Most effective with matrix preconditioner • Multigrid Methods Cornell University, September 17,2002 Ithaca New York, USA

11. Multigrid Methods • High-frequency (local) error rapidly reduced by explicit methods • Low-Frequence (global) error converges slowly • On coarser grid: • Low-frequency viewed as high frequency Cornell University, September 17,2002 Ithaca New York, USA

12. Multigrid Correction Scheme(Linear Problems) Cornell University, September 17,2002 Ithaca New York, USA

13. Multigrid for Unstructured Meshes • Generate fine and coarse meshes • Interpolate between un-nested meshes • Finest grid: 804,000 points, 4.5M tetrahedra • Four level Multigrid sequence

14. Geometric Multigrid • Order of magnitude increase in convergence • Convergence rate equivalent to structured grid schemes • Independent of grid size: O(N) Cornell University, September 17,2002 Ithaca New York, USA

15. Agglomeration vs. Geometric Multigrid • Multigrid methods: • Time step on coarse grids to accelerate solution on fine grid • Geometric multigrid • Coarse grid levels constructed manually • Cumbersome in production environment • Agglomeration Multigrid • Automate coarse level construction • Algebraic nature: summing fine grid equations • Graph based algorithm Cornell University, September 17,2002 Ithaca New York, USA

16. Agglomeration Multigrid • Agglomeration Multigrid solvers for unstructured meshes • Coarse level meshes constructed by agglomerating fine grid cells/equations Cornell University, September 17,2002 Ithaca New York, USA

17. Agglomeration Multigrid • Automated Graph-Based Coarsening Algorithm • Coarse Levels are Graphs • Coarse Level Operator by Galerkin Projection • Grid independent convergence rates (order of magnitude improvement)

18. Agglomeration MG for Euler Equations • Convergence rate similar to geometric MG • Completely automatic Cornell University, September 17,2002 Ithaca New York, USA

19. Anisotropy Induced Stiffness • Convergence rates for RANS (viscous) problems much slower then inviscid flows • Mainly due to grid stretching • Thin boundary and wake regions • Mixed element (prism-tet) grids • Use directional solver to relieve stiffness • Line solver in anisotropic regions Cornell University, September 17,2002 Ithaca New York, USA

20. Directional Solver for Navier-Stokes Problems • Line Solvers for Anisotropic Problems • Lines Constructed in Mesh using weighted graph algorithm • Strong Connections Assigned Large Graph Weight • (Block) Tridiagonal Line Solver similar to structured grids

21. Implementation on Parallel Computers • Intersected edges resolved by ghost vertices • Generates communication between original and ghost vertex • Handled using MPI and/or OpenMP • Portable, Distributed and Shared Memory Architectures • Local reordering within partition for cache-locality

22. Partitioning • Graph partitioning must minimize number of cut edges to minimize communication • Standard graph based partitioners: Metis, Chaco, Jostle • Require only weighted graph description of grid • Edges, vertices and weights taken as unity • Ideal for edge data-structure • Line Solver Inherently sequential • Partition around line using weigted graphs Cornell University, September 17,2002 Ithaca New York, USA

23. Partitioning • Contract graph along implicit lines • Weight edges and vertices • Partition contracted graph • Decontract graph • Guaranteed lines never broken • Possible small increase in imbalance/cut edges Cornell University, September 17,2002 Ithaca New York, USA

24. Partitioning Example • 32-way partition of 30,562 point 2D grid • Unweighted partition: 2.6% edges cut, 2.7% lines cut • Weigted partition: 3.2% edges cut, 0% lines cut

25. Sample Calculations and Validation • Subsonic High-Lift Case • Geometrically Complex • Large Case: 25 million points, 1450 processors • Research environment demonstration case • Transonic Wing Body • Smaller grid sizes • Full matrix of Mach and CL conditions • Typical of production runs indesign environment Cornell University, September 17,2002 Ithaca New York, USA

26. NASA Langley Energy Efficient Transport • Complex geometry • Wing-body, slat, double slotted flaps, cutouts • Experimental data from Langley 14x22ft wind tunnel • Mach = 0.2, Reynolds=1.6 million • Range of incidences: -4 to 24 degrees Cornell University, September 17,2002 Ithaca New York, USA

27. VGRID Tetrahedral Mesh • 3.1 million vertices, 18.2 million tets, 115,489 surface pts • Normal spacing: 1.35E-06 chords, growth factor=1.3

28. Computed Pressure Contours on Coarse Grid • Mach=0.2, Incidence=10 degrees, Re=1.6M

29. Spanwise Stations for Cp Data • Experimental data at 10 degrees incidence Cornell University, September 17,2002 Ithaca New York, USA

30. Comparison of Surface Cp at Middle Station Cornell University, September 17,2002 Ithaca New York, USA

31. Computed Versus Experimental Results • Good drag prediction • Discrepancies near stall

32. Multigrid Convergence History • Mesh independent property of Multigrid

33. Parallel Scalability • Good overall Multigrid scalability • Increased communication due to coarse grid levels • Single grid solution impractical (>100 times slower) • 1 hour soution time on 1450 PEs

34. AIAA Drag Prediction Workshop (2001) • Transonic wing-body configuration • Typical cases required for design study • Matrix of mach and CL values • Grid resolution study • Follow on with engine effects (2003)

35. Cases Run • Baseline grid: 1.6 million points • Full drag Polars for Mach=0.5,0.6,0.7,0.75,0.76,0.77,0.78,0.8 • Total = 72 cases • Medium grid: 3 million points • Full drag polar for each Mach number • Total = 48 cases • Fine grid: 13 million points • Drag polar at mach=0.75 • Total = 7 cases Cornell University, September 17,2002 Ithaca New York, USA

36. Sample Solution (1.65M Pts) • Mach=0.75, CL=0.6, Re=3M • 2.5 hours on 16 Pentium IV 1.7GHz

37. Drag Polar at Mach = 0.75 • Grid resolution study • Good comparison with experimental data

38. Comparison with Experiement • Grid Drag Values • Incidence Offset for Same CL

39. Drag Polars at other Mach Numbers • Grid resolution study • Discrepancies at Higher Mach/CL Conditions

40. Drag Rise Curves • Grid resolution study • Discrepancies at Higher Mach/CL Conditions

41. Cases Run on ICASE Cluster • 120 Cases (excluding finest grid) • About 1 week to compute all cases Cornell University, September 17,2002 Ithaca New York, USA

42. Timings on Various Architectures Cornell University, September 17,2002 Ithaca New York, USA

43. Adaptive Meshing • Potential for large savings trough optimized mesh resolution • Well suited for problems with large range of scales • Possibility of error estimation / control • Requires tight CAD coupling (surface pts) • Mechanics of mesh adaptation • Refinement criteria and error estimation Cornell University, September 17,2002 Ithaca New York, USA

44. Mechanics of Adaptive Meshing • Various well know isotropic mesh methods • Mesh movement • Spring analogy • Linear elasticity • Local Remeshing • Delaunay point insertion/Retriangulation • Edge-face swapping • Element subdivision • Mixed elements (non-simplicial) • Require anisotropic refinement in transition regions Cornell University, September 17,2002 Ithaca New York, USA

45. Subdivision Types for Tetrahedra Cornell University, September 17,2002 Ithaca New York, USA

46. Subdivision Types for Prisms Cornell University, September 17,2002 Ithaca New York, USA

47. Subdivision Types for Pyramids Cornell University, September 17,2002 Ithaca New York, USA

48. Subdivision Types for Hexahedra Cornell University, September 17,2002 Ithaca New York, USA

49. Adaptive Tetrahedral Mesh by Subdivision Cornell University, September 17,2002 Ithaca New York, USA

50. Adaptive Hexahedral Mesh by Subdivision Cornell University, September 17,2002 Ithaca New York, USA