Post on 28-Dec-2015
National Institute of Aerospace March 21, 2003
Computational Aerodynamics Using Unstructured Meshes
Dimitri J. Mavriplis
National Institute of Aerospace
Hampton, VA 23666
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
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
CFD Perspective on Meshing Technology
• Sophisticated Multiblock Structured Grid Techniques for Complex Geometries
Engine Nacelle Multiblock Grid by commercial software TrueGrid.
CFD Perspective on Meshing Technology
• Sophisticated Overlapping Structured Grid Techniques for Complex Geometries
Overlapping grid system on space shuttle (Slotnick, Kandula and Buning 1994)
National Institute of Aerospace March 21, 2003
Unstructured Grid Alternative
• Connectivity stored explicitly• Single Homogeneous Data Structure
National Institute of Aerospace March 21, 2003
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
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
National Institute of Aerospace March 21, 2003
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
National Institute of Aerospace March 21, 2003
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
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
National Institute of Aerospace March 21, 2003
Multigrid Correction Scheme(Linear Problems)
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
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)
National Institute of Aerospace March 21, 2003
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
National Institute of Aerospace March 21, 2003
Agglomeration Multigrid
• Agglomeration Multigrid solvers for unstructured meshes– Coarse level meshes constructed by agglomerating fine grid
cells/equations
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)
National Institute of Aerospace March 21, 2003
Agglomeration MG for Euler Equations
• Convergence rate similar to geometric MG
• Completely automatic
National Institute of Aerospace March 21, 2003
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
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
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
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
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
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
National Institute of Aerospace March 21, 2003
Multigrid Line-Solver Convergence
• DLR-F4 wing-body, Mach=0.75, 1o, Re=3M– Baseline Mesh: 1.65M pts
National Institute of Aerospace March 21, 2003
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
National Institute of Aerospace March 21, 2003
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
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
Computed Pressure Contours on Coarse Grid
• Mach=0.2, Incidence=10 degrees, Re=1.6M
National Institute of Aerospace March 21, 2003
Spanwise Stations for Cp Data
• Experimental data at 10 degrees incidence
National Institute of Aerospace March 21, 2003
Comparison of Surface Cp at Middle Station
Computed Versus Experimental Results
• Good drag prediction• Discrepancies near stall
Multigrid Convergence History
• Mesh independent property of Multigrid
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
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)
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
National Institute of Aerospace March 21, 2003
Sample Solution (1.65M Pts)
• Mach=0.75, CL=0.6, Re=3M• 2.5 hours on 16 Pentium IV 1.7GHz
Drag Polar at Mach = 0.75
• Grid resolution study• Good comparison with experimental data
Comparison with Experiment
• Grid Drag Values• Incidence Offset for Same CL
Drag Polars at other Mach Numbers
• Grid resolution study• Discrepancies at Higher Mach/CL Conditions
Drag Rise Curves
• Grid resolution study• Discrepancies at Higher Mach/CL Conditions
National Institute of Aerospace March 21, 2003
Cases Run on Coral Cluster
• 120 Cases (excluding finest grid)• About 1 week to compute all cases
National Institute of Aerospace March 21, 2003
Timings on Various Architectures
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
National Institute of Aerospace March 21, 2003
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
National Institute of Aerospace March 21, 2003
Subdivision Types for Tetrahedra
National Institute of Aerospace March 21, 2003
Subdivision Types for Prisms
National Institute of Aerospace March 21, 2003
Subdivision Types for Pyramids
National Institute of Aerospace March 21, 2003
Subdivision Types for Hexahedra
National Institute of Aerospace March 21, 2003
Adaptive Tetrahedral Mesh by Subdivision
National Institute of Aerospace March 21, 2003
Adaptive Hexahedral Mesh by Subdivision
National Institute of Aerospace March 21, 2003
Adaptive Hybrid Mesh by Subdivision
National Institute of Aerospace March 21, 2003
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)
Higher-Order Methods
• Most effective when high accuracy required• Potential role in aerodynamics (drag prediction)
– High accuracy requirements
– Large grid sizes required
National Institute of Aerospace March 21, 2003
Higher-Order Accurate Discretizations
• Transfers burden from grid generation to Discretization
Spectral Element Solution of Maxwell’s Equations
J. Hestahaven and T. Warburton (Brown University)
National Institute of Aerospace March 21, 2003
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
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