![Page 1: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/1.jpg)
Cornell University, September 17,2002 Ithaca New York, USA
The Development of Unstructured Grid Methods For Computational
Aerodynamics
Dimitri J. Mavriplis
ICASE
NASA Langley Research Center
Hampton, VA 23681
USA
![Page 2: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/2.jpg)
Cornell University, September 17,2002 Ithaca New York, USA
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
![Page 3: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/3.jpg)
Cornell University, September 17,2002 Ithaca New York, USA
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
![Page 4: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/4.jpg)
CFD Perspective on Meshing Technology
• Sophisticated Multiblock Structured Grid Techniques for Complex Geometries
Engine Nacelle Multiblock Grid by commercial software TrueGrid.
![Page 5: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/5.jpg)
CFD Perspective on Meshing Technology
• Sophisticated Overlapping Structured Grid Techniques for Complex Geometries
Overlapping grid system on space shuttle (Slotnick, Kandula and Buning 1994)
![Page 6: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/6.jpg)
Cornell University, September 17,2002 Ithaca New York, USA
Unstructured Grid Alternative
• Connectivity stored explicitly• Single Homogeneous Data Structure
![Page 7: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/7.jpg)
Cornell University, September 17,2002 Ithaca New York, USA
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
![Page 8: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/8.jpg)
Cornell University, September 17,2002 Ithaca New York, USA
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
![Page 9: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/9.jpg)
Cornell University, September 17,2002 Ithaca New York, USA
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
![Page 10: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/10.jpg)
Cornell University, September 17,2002 Ithaca New York, USA
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
![Page 11: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/11.jpg)
Cornell University, September 17,2002 Ithaca New York, USA
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
![Page 12: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/12.jpg)
Cornell University, September 17,2002 Ithaca New York, USA
Multigrid Correction Scheme(Linear Problems)
![Page 13: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/13.jpg)
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
![Page 14: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/14.jpg)
Cornell University, September 17,2002 Ithaca New York, USA
Geometric Multigrid
• Order of magnitude increase in convergence• Convergence rate equivalent to structured grid
schemes• Independent of grid size: O(N)
![Page 15: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/15.jpg)
Cornell University, September 17,2002 Ithaca New York, USA
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
![Page 16: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/16.jpg)
Cornell University, September 17,2002 Ithaca New York, USA
Agglomeration Multigrid
• Agglomeration Multigrid solvers for unstructured meshes– Coarse level meshes constructed by agglomerating fine grid
cells/equations
![Page 17: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/17.jpg)
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)
![Page 18: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/18.jpg)
Cornell University, September 17,2002 Ithaca New York, USA
Agglomeration MG for Euler Equations
• Convergence rate similar to geometric MG
• Completely automatic
![Page 19: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/19.jpg)
Cornell University, September 17,2002 Ithaca New York, USA
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
![Page 20: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/20.jpg)
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
![Page 21: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/21.jpg)
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
![Page 22: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/22.jpg)
Cornell University, September 17,2002 Ithaca New York, USA
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
![Page 23: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/23.jpg)
Cornell University, September 17,2002 Ithaca New York, USA
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
![Page 24: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/24.jpg)
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
![Page 25: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/25.jpg)
Cornell University, September 17,2002 Ithaca New York, USA
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
![Page 26: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/26.jpg)
Cornell University, September 17,2002 Ithaca New York, USA
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
![Page 27: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/27.jpg)
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
![Page 28: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/28.jpg)
Computed Pressure Contours on Coarse Grid
• Mach=0.2, Incidence=10 degrees, Re=1.6M
![Page 29: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/29.jpg)
Cornell University, September 17,2002 Ithaca New York, USA
Spanwise Stations for Cp Data
• Experimental data at 10 degrees incidence
![Page 30: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/30.jpg)
Cornell University, September 17,2002 Ithaca New York, USA
Comparison of Surface Cp at Middle Station
![Page 31: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/31.jpg)
Computed Versus Experimental Results
• Good drag prediction• Discrepancies near stall
![Page 32: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/32.jpg)
Multigrid Convergence History
• Mesh independent property of Multigrid
![Page 33: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/33.jpg)
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
![Page 34: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/34.jpg)
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)
![Page 35: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/35.jpg)
Cornell University, September 17,2002 Ithaca New York, USA
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
![Page 36: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/36.jpg)
Sample Solution (1.65M Pts)
• Mach=0.75, CL=0.6, Re=3M• 2.5 hours on 16 Pentium IV 1.7GHz
![Page 37: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/37.jpg)
Drag Polar at Mach = 0.75
• Grid resolution study• Good comparison with experimental data
![Page 38: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/38.jpg)
Comparison with Experiement
• Grid Drag Values• Incidence Offset for Same CL
![Page 39: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/39.jpg)
Drag Polars at other Mach Numbers
• Grid resolution study• Discrepancies at Higher Mach/CL Conditions
![Page 40: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/40.jpg)
Drag Rise Curves
• Grid resolution study• Discrepancies at Higher Mach/CL Conditions
![Page 41: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/41.jpg)
Cornell University, September 17,2002 Ithaca New York, USA
Cases Run on ICASE Cluster
• 120 Cases (excluding finest grid)• About 1 week to compute all cases
![Page 42: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/42.jpg)
Cornell University, September 17,2002 Ithaca New York, USA
Timings on Various Architectures
![Page 43: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/43.jpg)
Cornell University, September 17,2002 Ithaca New York, USA
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
![Page 44: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/44.jpg)
Cornell University, September 17,2002 Ithaca New York, USA
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
![Page 45: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/45.jpg)
Cornell University, September 17,2002 Ithaca New York, USA
Subdivision Types for Tetrahedra
![Page 46: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/46.jpg)
Cornell University, September 17,2002 Ithaca New York, USA
Subdivision Types for Prisms
![Page 47: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/47.jpg)
Cornell University, September 17,2002 Ithaca New York, USA
Subdivision Types for Pyramids
![Page 48: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/48.jpg)
Cornell University, September 17,2002 Ithaca New York, USA
Subdivision Types for Hexahedra
![Page 49: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/49.jpg)
Cornell University, September 17,2002 Ithaca New York, USA
Adaptive Tetrahedral Mesh by Subdivision
![Page 50: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/50.jpg)
Cornell University, September 17,2002 Ithaca New York, USA
Adaptive Hexahedral Mesh by Subdivision
![Page 51: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/51.jpg)
Cornell University, September 17,2002 Ithaca New York, USA
Adaptive Hybrid Mesh by Subdivision
![Page 52: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/52.jpg)
Cornell University, September 17,2002 Ithaca New York, USA
Overlapping Unstructured Meshes
• Alternative to Moving Mesh for Large Scale Relative Geometry Motion
• Multiple Overlapping Meshes treated as single data-structure– Dynamic Determination of active/inactive/ghost cells
• Advantages for Parallel Computing– Obviates dynamic load rebalancing required with mesh
motion techniques– Intergrid communication must be dynamically
recomputed and rebalanced• Concept of Rendez-vous grid (Plimpton and Hendrickson)
![Page 53: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/53.jpg)
Cornell University, September 17,2002 Ithaca New York, USA
Overlapping Unstructured Meshes
• Simple 2D transient example
![Page 54: Cornell University, September 17,2002 Ithaca New York, USA The Development of Unstructured Grid Methods For Computational Aerodynamics Dimitri J. Mavriplis](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649e025503460f94aec989/html5/thumbnails/54.jpg)
Cornell University, September 17,2002 Ithaca New York, USA
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
• Adaptive meshing potential not fully exploited– Refinement criteria require more research
• Future work to include more physics– Turbulence, transition, unsteady flows, moving meshes