Computational Aerodynamics Using Unstructured Meshes

1
Computational Aerodynamics Using Unstructured
Meshes

• Dimitri J. Mavriplis
• National Institute of Aerospace
• Hampton, VA 23666

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
• Higher-order discretizations

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

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

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

8
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

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

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

11
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

12
Multigrid Correction Scheme(Linear Problems)
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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)

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

16
Agglomeration Multigrid

• Agglomeration Multigrid solvers for unstructured
  meshes
• Coarse level meshes constructed by agglomerating
  fine grid cells/equations

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

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

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

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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

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

24
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

25
Multigrid Line-Solver Convergence

• DLR-F4 wing-body, Mach0.75, 1o, Re3M
• Baseline Mesh 1.65M pts

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

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, Reynolds1.6 million
• Range of incidences -4 to 24 degrees

28
VGRID Tetrahedral Mesh

• 3.1 million vertices, 18.2 million tets, 115,489
  surface pts
• Normal spacing 1.35E-06 chords, growth factor1.3

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Computed Pressure Contours on Coarse Grid

• Mach0.2, Incidence10 degrees, Re1.6M

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Spanwise Stations for Cp Data

• Experimental data at 10 degrees incidence

31
Comparison of Surface Cp at Middle Station
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 (gt100 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
Cases Run

• Baseline grid 1.6 million points
• Full drag Polars for Mach0.5,0.6,0.7,0.75,0.76,0.
  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 mach0.75
• Total 7 cases

37
Sample Solution (1.65M Pts)

• Mach0.75, CL0.6, Re3M
• 2.5 hours on 16 Pentium IV 1.7GHz

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

43
Timings on Various Architectures
44
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

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

46
Subdivision Types for Tetrahedra
47
Subdivision Types for Prisms
48
Subdivision Types for Pyramids
49
Subdivision Types for Hexahedra
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Adaptive Tetrahedral Mesh by Subdivision
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Adaptive Hexahedral Mesh by Subdivision
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Adaptive Hybrid Mesh by Subdivision
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)

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

56
Spectral Element Solution of Maxwells 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

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