A General Implicit Artificial Boundary Scheme for Chimera...
Transcript of A General Implicit Artificial Boundary Scheme for Chimera...
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GTSL
GTSL
12th Overset Symposium, 9 Oct 2014
A General Implicit Artificial Boundary Scheme
for Chimera Methods
Dr. Marshall Galbraith Massachusetts Institute of Technology
Department of Aeronautics and Astronautics [email protected]
Dr. Robert Knapke University of Cincinnati
School of Aerospace Systems [email protected]
Dr. Paul D. Orkwis
University of Cincinnati School of Aerospace Systems
Dr. John Benek Air Force Research Laboratory
Computational Science Branch Center of Excellence
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Motivation
β’ Chimera Overset Grid Method β Complex Geometries
β βHot swapβ Geometric
Features
β Moving Grids with Relative
Motion
β’ Store Separation
β’ Rotorcraft
β’ Explicit Artificial Boundaries β Solve Decoupled System
β Limits CFL Number with Increasing
Number of Processor
β’ Implicit Artificial Boundaries β Significant Increased Parallel
Performance
β Itβs easier than it sounds
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Outline
β’ Discretization Assumptions
β’ Explicit/Implicit Chimera
β’ Sparse Iterative Solvers β Preconditioners
β’ Distributed Memory Parallelism
β’ Discontinuous Galerkin Method
β’ Inviscid/Viscous Flow Examples
β’ Conclusion and Future Work
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Outline
β’ Discretization Assumptions
β’ Explicit/Implicit Chimera
β’ Sparse Iterative Solvers β Preconditioners
β’ Distributed Memory Parallelism
β’ Discontinuous Galerkin Method
β’ Inviscid/Viscous Flow Examples
β’ Conclusion and Future Work
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Discretization Assumptions
β’ Euler/Navier-Stokes Equations in Conservation Form
β’ Discrete Form β Finite Difference β Finite Volume β Finite Element
β’ Newton's Method
β’ Chimera Interpolation Operator β Linear Operator (donβt think 2nd order accuracy) β Polynomial Basis Functions β Radial Basis Functions β Trigonometric Basis Functions β etc.
0 QF
0QR
QRQ
Q
QR
iih QQI
QRQA
iih
h QQIQQ
QI
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Explicit/Implicit Artificial Boundaries
β’ Unstructured A Matrix β Explicitly add Ci Matrices
β’ Structured Ai Matrix
β Tri-, Penta-, Hepta-diagonal
β’ Sparse Iterative Solver β No Explicit Ci Matrices
π¨π ππ π¨π
βπΈπ
βπΈπ=πΉπ πΈπ, π°π πΈ
π
πΉπ πΈπ, π°π πΈπ
π¨π =ππΉπ
ππΈπ π¨π =
ππΉπ
ππΈπ
π¨πβπΈπ=πΉπ πΈπ, π°π πΈπ
π¨πβπΈπ=πΉπ πΈπ, π°π πΈπ
πͺπ =ππΉπ
ππΈπ πͺπ =
ππΉπ
ππΈπ
Grid 1 Grid 2
π¨π πͺπ
πͺπ π¨πβπΈπ
βπΈπ=πΉπ πΈπ, π°π πΈ
π
πΉπ πΈπ, π°π πΈπ
π¨π«πΈ = πΉ
Newtonβs Method
Explicit Artificial Boundary
Solve Decoupled System
Implicit Artificial Boundary
πΉπ πΈπ, π°π πΈπ =0 πΉπ πΈπ, π°π πΈ
π =0
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Sparse Iterative Solvers
β’ Iterative methods for sparse linear systems β http://www-users.cs.umn.edu/~saad/books.html
β’ Restarted GMRES β Simple Fortran Code Available
β http://people.sc.fsu.edu/~jburkardt/f_src/mgmres/mg
mres.html
β’ Fundamental Operations β Dot products
β Sparse Matrix-Vector Multiplication
β’ Slow without Preconditioner
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Sparse Iterative Solvers
Preconditioners
β’ Incomplete LU
β’ ARC3D Beam-Warming block tridiagonal scheme.
β’ F3D Steger-Warming 2-factor scheme.
β’ ARC3D diagonalized Beam-Warming scalar pentadiagonal scheme.
β’ LU-SGS algorithm.
β’ D3ADI algorithm with Huang subiteration. β’ ARC3D Beam-Warming with Steger-Warming flux
split jacobians.
β’ SSOR algorithm (with subiteration)
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Implicit Artificial Boundaries
πͺπβπΈπ=ππΉπ πΈπ, π°π πΈ
π
ππΈπβπΈπ
πͺπβπΈπ=ππΉπ πΈπ, π°π πΈ
π
ππ°π πΈπ
ππ°π πΈπ
ππΈπβπΈπ
ππ°π πΈπ
ππΈπβπΈπ= π°π βπΈ
π
πͺπβπΈπ=ππΉππ πΈπ³
π, π°π πΈπ
ππΈπΉπ π°π βπΈ
π = πͺπ π°π βπΈπ
Artificial Boundary Linearization
Chain Rule
Linear Interpolation Operator
Matrix-Vector Product
GMRES: Matrix-Vector Multiplication
πͺπ =ππΉπ
ππΈπ πͺπ =
ππΉπ
ππΈπ
π¨π πͺπ
πͺπ π¨πβπΈπ
βπΈπ=πΉπ πΈπ, π°π πΈ
π
πΉπ πΈπ, π°π πΈπ
Implicit Artificial Boundary
Interior Flux Linearization
ππΉπ πΈπ, π°π πΈπ
ππ°π πΈπ =
ππΉππ πΈπ³
π, π°π πΈπ
ππΈπΉπ
ππΉπ πΈπ³π , πΈπΉ
π
ππΈπ³ ππΉπ πΈπ³
π , πΈπΉπ
ππΈπΉ
Interior Jacobian
Array of Matrices
RHS Interpolation Operator
Array of Vectors
Receiver Grid
Interpolation
Mapping
πΉπ πΈπ³π πΈπΉ
π
πΉππ πΈπ³
π π°π πΈπ
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Parallel GMRES Iterative Solver
Processor 1
Matrix-Vector Multiplication
Mask Communication with Local Calculations ππ+π = π¨ππ
Processor 2
Interpolate π°π πππ Interpolate π°π ππ
π
Non-Blocking Send π°π πππ Non-Blocking Send π°π ππ
π
Compute ππ+ππ = π¨πππ
π Compute ππ+ππ = π¨πππ
π
Receive π°π πππ Receive π°π ππ
π
Compute ππ+ππ = ππ+π
π + πͺπ π°π πππ Compute ππ+π
π = ππ+ππ + πͺπ π°π ππ
π
ππ+ππ
ππ+ππ = π¨π πͺπ
πͺπ π¨ππππ
πππ
Dot Products
Compute Local Dot Product All Reduce πΆπ
π+ πππ π
π
π> πππ π
Parallel Efficient If
Preconditioner Omits C Matrices Jacobi as p β n
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Outline
β’ Discretization Assumptions
β’ Explicit/Implicit Chimera
β’ Sparse Iterative Solvers β Preconditioners
β’ Distributed Memory Parallelism
β’ Discontinuous Galerkin Method
β’ Inviscid/Viscous Flow Examples
β’ Conclusion and Future Work
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FV Approximation
Discontinuous Galerkin Chimera
Scheme
β’ Discontinuous Galerkin β Weak Form
β Approximate Riemann Solver by Roe
β BR2 Viscous Scheme
β’ DG-Chimera β Natural Interpolation Operator (Solution is Polynomials)
β Curved Elements
β Reduces to a Zonal Interface (Abutting Meshes)
β No Orphan Points due to Fringe Points
DG Approximation
xj-1/2 xj xj+1/2
0
dF
e
- Legendre
Polynomials
0,,
ee
dQFdnQQFQQR
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Outline
β’ Discretization Assumptions
β’ Explicit/Implicit Chimera
β’ Sparse Iterative Solvers β Preconditioners
β’ Distributed Memory Parallelism
β’ Discontinuous Galerkin Method
β’ Inviscid/Viscous Flow Examples
β’ Conclusion and Future Work
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β’ Inviscid SKF 1.1 Airfoil
β Mβ = 0.4 β Ξ±=2.5Β°
β’ Viscous Subsonic Circular Cylinder β Mβ = 0.25 β Re = 40
β’ Focus on Solution Time β Explicit vs. Implicit Chimera
Inviscid/Viscous Flow Examples
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Time Integration and
Compute Resources
β’ Steady State β Quasi-Newton
β’ GMRES Krylov Solver β ILU1 Preconditioner β Converged to 1e-11 Each Newton Iteration
β’ Intel Core 2 Duo 3.0 GHz processor 8 GB RAM β 10 Compute Nodes β Ethernet Connection
β’ MPI Parallelism
β Timings for 1, 2, 4, and 8 Processors β 1 MPI Process per Node (Maximize Communication)
β’ Shared Memory Multi-Threaded
β 1 Grid Per Thread
π΄
βπ+ππΉ
ππΈπ«πΈ = πΉ πͺππ³π+π = πͺππ³π
πΉπ
πΉπ
πͺππ³πππ = ππππ
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Single
105x30 Ng=3
SKF 1.1 Airfoil (Mβ = 0.4, Ξ±=2.5Β°)
Meshes
105x16 Ng=3
O-Grid Chimera
104x14 Ng=1
R-Grid Chimera
104x104 Ng=1
105x15 Ng=3
Chimera
Interface
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SKF 1.1 Airfoil (Mβ = 0.4 , Ξ±=2.5Β°)
Explicit Chimera Speedup
ChimeraLocalLocal QQRQQA ,
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SKF 1.1 Airfoil (Mβ = 0.4 , Ξ±=2.5Β°)
Explicit Chimera Convergence History
ChimeraLocalLocal QQRQQA ,
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SKF 1.1 Airfoil (Mβ = 0.4 , Ξ±=2.5Β°)
Implicit Chimera Speedup
ChimeraLocalChimeraLocal QQRQQQA ,,
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SKF 1.1 Airfoil (Mβ = 0.4 , Ξ±=2.5Β°)
Implicit Convergence History
ChimeraLocalChimeraLocal QQRQQQA ,,
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10s
1min
1hr
10s
1min
1hr
10s
1min
1hr
SKF 1.1 Airfoil (Mβ = 0.4 , Ξ±=2.5Β°)
Solution Time
ChimeraLocalLocal QQRQQA ,
ChimeraLocalChimeraLocal QQRQQQA ,,
10s
1min
1hr
10s
1min
1hr
10s
1min
1hr
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SKF 1.1 Airfoil (Mβ = 0.4 , Ξ±=2.5Β°)
Implicit/Explicit Solution Speedup
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SKF 1.1 Airfoil (Mβ = 0.4, Ξ±=2.5Β°)
Cp Contour Lines
N=1
2nd-order
N=2
3rd-order
N=3
4th-order
N=0
1st-order
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Single
50x40 Ng=3
50x18 Ng=3
O-Grid Chimera
50x22 Ng=1
Ng = 1
Ng = 3
R-Grid Chimera
100x100 Ng=1
50x18 Ng=1
Circular Cylinder (Re = 40)
Meshes
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Circular Cylinder (Re = 40)
Solution Time
ChimeraLocalLocal QQRQQA ,
ChimeraLocalChimeraLocal QQRQQQA ,,
10s
1min
1hr
1day
10s
1min
1hr
1day
10s
1min
1hr
1day
10 days!
10s
1min
1hr
1day
10s
1min
1hr
1day
10s
1min
1hr
1day
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Circular Cylinder (Re = 40)
Implicit/Explicit Solution Speedup
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Subsonic Circular Cylinder (Mβ = 0.25)
Cp/Entropy Rise Contour Lines N=1
2nd-order N=2
3rd-order
N=3
4th-order
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Conclusion and Future Work
β’ Implicit Artificial Boundaries β Included with GMRES Matrix-Vector Multiplication β Omitted in Preconditioner β Minimal Information Communicated β Significantly Reduces Execution Time
β’ Few Modifications Required to Existing Codes β ~95% of Code already Exists β Spares Matrix-Vector Multiplication β Restarted GMRES Fortran Code
β’ http://people.sc.fsu.edu/~jburkardt/f_src/mgmres/mgmres.html
β’ Demonstrated on Inviscid/Viscous Flows
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Thank you!
Questions?
This work was supported by the
Department of Defense (DoD)
through the National Defense Science &
Engineering Graduate (NDSEG)
Fellowship Program