1. Computational Aerodynamics Using Unstructured Meshes
Dimitri J. Mavriplis
National Institute of Aerospace
Hampton, VA 23666
National Institute of Aerospace March 21, 2003

2. National Institute of Aerospace March 21, 2003
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
Higher-order discretizations
National Institute of Aerospace March 21, 2003

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
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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
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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
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8. National Institute of Aerospace March 21, 2003
Discretization
Governing Equations: Reynolds Averaged Navier-Stokes Equations
Conservation of Mass, Momentum and Energy
Single Equation turbulence model (Spalart-Allmaras)
Convection-Diffusion – Production
Vertex-Based Discretization
2nd order upwind finite-volume scheme
6 variables per grid point
Flow equations fully coupled (5x5)
Turbulence equation uncoupled
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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
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10. Spatially Discretized Equations
Integrate to Steady-state
Explicit:
Simple, Slow: Local procedure
Implicit
Large Memory Requirements
Matrix Free Implicit:
Most effective with matrix preconditioner
Multigrid Methods
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11. National Institute of Aerospace March 21, 2003
Multigrid Methods
High-frequency (local) error rapidly reduced by explicit methods
Low-frequency (global) error converges slowly
On coarser grid:
Low-frequency viewed as high frequency
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12. Multigrid Correction Scheme (Linear Problems)
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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. National Institute of Aerospace March 21, 2003
Geometric Multigrid
Order of magnitude increase in convergence
Convergence rate equivalent to structured grid schemes
Independent of grid size: O(N)
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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
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16. Agglomeration Multigrid
Agglomeration Multigrid solvers for unstructured meshes
Coarse level meshes constructed by agglomerating fine grid cells/equations
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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
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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
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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. National Institute of Aerospace March 21, 2003
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 weighted graphs
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23. National Institute of Aerospace March 21, 2003
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
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24. National Institute of Aerospace March 21, 2003
Partitioning Example
32-way partition of 30,562 point 2D grid
Unweighted partition: 2.6% edges cut, 2.7% lines cut
Weighted partition: 3.2% edges cut, 0% lines cut
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25. Multigrid Line-Solver Convergence
DLR-F4 wing-body, Mach=0.75, 1o, Re=3M
Baseline Mesh: 1.65M pts
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26. 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 in design environment
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27. 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
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28. 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

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

30. Spanwise Stations for Cp Data
Experimental data at 10 degrees incidence
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31. Comparison of Surface Cp at Middle Station
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32. Computed Versus Experimental Results
Good drag prediction
Discrepancies near stall

33. Multigrid Convergence History
Mesh independent property of Multigrid

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

35. 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)

36. National Institute of Aerospace March 21, 2003
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
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37. Sample Solution (1.65M Pts)
Mach=0.75, CL=0.6, Re=3M
2.5 hours on 16 Pentium IV 1.7GHz
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38. Drag Polar at Mach = 0.75 Grid resolution study
Good comparison with experimental data

39. Comparison with Experiment
Grid Drag Values
Incidence Offset for Same CL

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

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

42. Cases Run on Coral Cluster
120 Cases (excluding finest grid)
About 1 week to compute all cases
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43. Timings on Various Architectures
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44. National Institute of Aerospace March 21, 2003
Adaptive Meshing
Potential for large savings through 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
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45. 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
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46. Subdivision Types for Tetrahedra
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47. Subdivision Types for Prisms
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48. Subdivision Types for Pyramids
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49. Subdivision Types for Hexahedra
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50. Adaptive Tetrahedral Mesh by Subdivision
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51. Adaptive Hexahedral Mesh by Subdivision
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52. Adaptive Hybrid Mesh by Subdivision
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53. High-Order Accurate Discretizations
Uniform X2 refinement of 3D mesh:
Work increase = factor of 8
2nd order accurate method: accuracy increase = 4
4th order accurate method: accuracy increase = 16
For smooth solutions
Potential for large efficiency gains
Spectral element methods
Discontinuous Galerkin (DG)
Streamwise Upwind Petrov Galerkin (SUPG)
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54. Higher-Order Methods Most effective when high accuracy required
Potential role in aerodynamics (drag prediction)
High accuracy requirements
Large grid sizes required

55. Higher-Order Accurate Discretizations
Transfers burden from grid generation to Discretization
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56. Spectral Element Solution of Maxwell’s Equations
J. Hestahaven and T. Warburton (Brown University)

57. Combined H-P Refinement
Adaptive meshing (h-ref) yields constant factor improvement
After error equidistribution, no further benefit
Order refinement (p-ref) yields asymptotic improvement
Only for smooth functions
Ineffective for inadequate h-resolution of feature
Cannot treat shocks
H-P refinement optimal (exponential convergence)
Requires accurate CAD surface representation
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58. Conclusions
Unstructured mesh technology enabling technology for computational aerodynamics
Complex geometry handling facilitated
Efficient steady-state solvers
Highly effective parallelization
Accurate solutions possible for on-design conditions
Mostly attached flow
Grid resolution always an issue
Orders of Magnitude Improvement Possible in Future
Adaptive meshing
Higher-Order Discretizations
Future work to include more physics
Turbulence, transition, unsteady flows, moving meshes