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

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

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

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)

Cornell University, September 17,2002 Ithaca New York, USA

Unstructured Grid Alternative

• Connectivity stored explicitly• Single Homogeneous Data Structure

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

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

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

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

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

Cornell University, September 17,2002 Ithaca New York, USA

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

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)

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

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

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)

Cornell University, September 17,2002 Ithaca New York, USA

Agglomeration MG for Euler Equations

• Convergence rate similar to geometric MG

• Completely automatic

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

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

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

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

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

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

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

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

Cornell University, September 17,2002 Ithaca New York, USA

Spanwise Stations for Cp Data

• Experimental data at 10 degrees incidence

Cornell University, September 17,2002 Ithaca New York, USA

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 soution 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)

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

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 Experiement

• 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

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

Cornell University, September 17,2002 Ithaca New York, USA

Timings on Various Architectures

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

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

Cornell University, September 17,2002 Ithaca New York, USA

Subdivision Types for Tetrahedra

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Subdivision Types for Prisms

Cornell University, September 17,2002 Ithaca New York, USA

Subdivision Types for Pyramids

Cornell University, September 17,2002 Ithaca New York, USA

Subdivision Types for Hexahedra

Cornell University, September 17,2002 Ithaca New York, USA

Adaptive Tetrahedral Mesh by Subdivision

Cornell University, September 17,2002 Ithaca New York, USA

Adaptive Hexahedral Mesh by Subdivision

Cornell University, September 17,2002 Ithaca New York, USA

Adaptive Hybrid Mesh by Subdivision

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)

Cornell University, September 17,2002 Ithaca New York, USA

Overlapping Unstructured Meshes

• Simple 2D transient example

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