a general implicit artificial boundary scheme for chimera...
TRANSCRIPT
![Page 1: A General Implicit Artificial Boundary Scheme for Chimera ...2014.oversetgridsymposium.org/assets/presentations/3_2/Galbraith_ogs_2014.pdfGTSL 5 12th Overset Symposium, 9 Oct 2014](https://reader033.vdocuments.mx/reader033/viewer/2022060707/607368047c72023c4d732ba3/html5/thumbnails/1.jpg)
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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GTSL
2 12th Overset Symposium, 9 Oct 2014
Motivation
• Chimera Overset Grid Method – Complex Geometries
– “Hot swap” Geometric
Features
– Moving Grids with Relative
Motion
• Store Separation
• Rotorcraft
• Explicit Artificial Boundaries – Solve Decoupled System
– Limits CFL Number with Increasing
Number of Processor
• Implicit Artificial Boundaries – Significant Increased Parallel
Performance
– It’s easier than it sounds
![Page 3: A General Implicit Artificial Boundary Scheme for Chimera ...2014.oversetgridsymposium.org/assets/presentations/3_2/Galbraith_ogs_2014.pdfGTSL 5 12th Overset Symposium, 9 Oct 2014](https://reader033.vdocuments.mx/reader033/viewer/2022060707/607368047c72023c4d732ba3/html5/thumbnails/3.jpg)
GTSL
3 12th Overset Symposium, 9 Oct 2014
Outline
• Discretization Assumptions
• Explicit/Implicit Chimera
• Sparse Iterative Solvers – Preconditioners
• Distributed Memory Parallelism
• Discontinuous Galerkin Method
• Inviscid/Viscous Flow Examples
• Conclusion and Future Work
![Page 4: A General Implicit Artificial Boundary Scheme for Chimera ...2014.oversetgridsymposium.org/assets/presentations/3_2/Galbraith_ogs_2014.pdfGTSL 5 12th Overset Symposium, 9 Oct 2014](https://reader033.vdocuments.mx/reader033/viewer/2022060707/607368047c72023c4d732ba3/html5/thumbnails/4.jpg)
GTSL
4 12th Overset Symposium, 9 Oct 2014
Outline
• Discretization Assumptions
• Explicit/Implicit Chimera
• Sparse Iterative Solvers – Preconditioners
• Distributed Memory Parallelism
• Discontinuous Galerkin Method
• Inviscid/Viscous Flow Examples
• Conclusion and Future Work
![Page 5: A General Implicit Artificial Boundary Scheme for Chimera ...2014.oversetgridsymposium.org/assets/presentations/3_2/Galbraith_ogs_2014.pdfGTSL 5 12th Overset Symposium, 9 Oct 2014](https://reader033.vdocuments.mx/reader033/viewer/2022060707/607368047c72023c4d732ba3/html5/thumbnails/5.jpg)
GTSL
5 12th Overset Symposium, 9 Oct 2014
Discretization Assumptions
• Euler/Navier-Stokes Equations in Conservation Form
• Discrete Form – Finite Difference – Finite Volume – Finite Element
• Newton's Method
• Chimera Interpolation Operator – Linear Operator (don’t think 2nd order accuracy) – Polynomial Basis Functions – Radial Basis Functions – Trigonometric Basis Functions – etc.
0 QF
0QR
QRQ
Q
QR
iih QQI
QRQA
iih
h QQIQQ
QI
![Page 6: A General Implicit Artificial Boundary Scheme for Chimera ...2014.oversetgridsymposium.org/assets/presentations/3_2/Galbraith_ogs_2014.pdfGTSL 5 12th Overset Symposium, 9 Oct 2014](https://reader033.vdocuments.mx/reader033/viewer/2022060707/607368047c72023c4d732ba3/html5/thumbnails/6.jpg)
GTSL
6 12th Overset Symposium, 9 Oct 2014
Explicit/Implicit Artificial Boundaries
• Unstructured A Matrix – Explicitly add Ci Matrices
• Structured Ai Matrix
– Tri-, Penta-, Hepta-diagonal
• Sparse Iterative Solver – No Explicit Ci Matrices
𝑨𝟏 𝟎𝟎 𝑨𝟐
∆𝑸𝟏
∆𝑸𝟐=𝑹𝟏 𝑸𝟏, 𝑰𝒉 𝑸
𝟐
𝑹𝟐 𝑸𝟐, 𝑰𝒉 𝑸𝟏
𝑨𝟏 =𝝏𝑹𝟏
𝝏𝑸𝟏 𝑨𝟐 =
𝝏𝑹𝟐
𝝏𝑸𝟐
𝑨𝟏∆𝑸𝟏=𝑹𝟏 𝑸𝟏, 𝑰𝒉 𝑸𝟐
𝑨𝟐∆𝑸𝟐=𝑹𝟐 𝑸𝟐, 𝑰𝒉 𝑸𝟏
𝑪𝟏 =𝝏𝑹𝟏
𝝏𝑸𝟐 𝑪𝟐 =
𝝏𝑹𝟐
𝝏𝑸𝟏
Grid 1 Grid 2
𝑨𝟏 𝑪𝟏
𝑪𝟐 𝑨𝟐∆𝑸𝟏
∆𝑸𝟐=𝑹𝟏 𝑸𝟏, 𝑰𝒉 𝑸
𝟐
𝑹𝟐 𝑸𝟐, 𝑰𝒉 𝑸𝟏
𝑨𝚫𝑸 = 𝑹
Newton’s Method
Explicit Artificial Boundary
Solve Decoupled System
Implicit Artificial Boundary
𝑹𝟏 𝑸𝟏, 𝑰𝒉 𝑸𝟐 =0 𝑹𝟐 𝑸𝟐, 𝑰𝒉 𝑸
𝟏 =0
![Page 7: A General Implicit Artificial Boundary Scheme for Chimera ...2014.oversetgridsymposium.org/assets/presentations/3_2/Galbraith_ogs_2014.pdfGTSL 5 12th Overset Symposium, 9 Oct 2014](https://reader033.vdocuments.mx/reader033/viewer/2022060707/607368047c72023c4d732ba3/html5/thumbnails/7.jpg)
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7 12th Overset Symposium, 9 Oct 2014
Sparse Iterative Solvers
• Iterative methods for sparse linear systems – http://www-users.cs.umn.edu/~saad/books.html
• Restarted GMRES – Simple Fortran Code Available
– http://people.sc.fsu.edu/~jburkardt/f_src/mgmres/mg
mres.html
• Fundamental Operations – Dot products
– Sparse Matrix-Vector Multiplication
• Slow without Preconditioner
![Page 8: A General Implicit Artificial Boundary Scheme for Chimera ...2014.oversetgridsymposium.org/assets/presentations/3_2/Galbraith_ogs_2014.pdfGTSL 5 12th Overset Symposium, 9 Oct 2014](https://reader033.vdocuments.mx/reader033/viewer/2022060707/607368047c72023c4d732ba3/html5/thumbnails/8.jpg)
GTSL
8 12th Overset Symposium, 9 Oct 2014
Sparse Iterative Solvers
Preconditioners
• Incomplete LU
• ARC3D Beam-Warming block tridiagonal scheme.
• F3D Steger-Warming 2-factor scheme.
• ARC3D diagonalized Beam-Warming scalar pentadiagonal scheme.
• LU-SGS algorithm.
• D3ADI algorithm with Huang subiteration. • ARC3D Beam-Warming with Steger-Warming flux
split jacobians.
• SSOR algorithm (with subiteration)
![Page 9: A General Implicit Artificial Boundary Scheme for Chimera ...2014.oversetgridsymposium.org/assets/presentations/3_2/Galbraith_ogs_2014.pdfGTSL 5 12th Overset Symposium, 9 Oct 2014](https://reader033.vdocuments.mx/reader033/viewer/2022060707/607368047c72023c4d732ba3/html5/thumbnails/9.jpg)
GTSL
9 12th Overset Symposium, 9 Oct 2014
Implicit Artificial Boundaries
𝑪𝟏∆𝑸𝟐=𝝏𝑹𝟏 𝑸𝟏, 𝑰𝒉 𝑸
𝟐
𝝏𝑸𝟐∆𝑸𝟐
𝑪𝟏∆𝑸𝟐=𝝏𝑹𝟏 𝑸𝟏, 𝑰𝒉 𝑸
𝟐
𝝏𝑰𝒉 𝑸𝟐
𝝏𝑰𝒉 𝑸𝟐
𝝏𝑸𝟐∆𝑸𝟐
𝝏𝑰𝒉 𝑸𝟐
𝝏𝑸𝟐∆𝑸𝟐= 𝑰𝒉 ∆𝑸
𝟐
𝑪𝟏∆𝑸𝟐=𝝏𝑹𝒊𝟏 𝑸𝑳
𝟏, 𝑰𝒉 𝑸𝟐
𝝏𝑸𝑹𝟏 𝑰𝒉 ∆𝑸
𝟐 = 𝑪𝟏 𝑰𝒉 ∆𝑸𝟐
Artificial Boundary Linearization
Chain Rule
Linear Interpolation Operator
Matrix-Vector Product
GMRES: Matrix-Vector Multiplication
𝑪𝟏 =𝝏𝑹𝟏
𝝏𝑸𝟐 𝑪𝟐 =
𝝏𝑹𝟐
𝝏𝑸𝟏
𝑨𝟏 𝑪𝟏
𝑪𝟐 𝑨𝟐∆𝑸𝟏
∆𝑸𝟐=𝑹𝟏 𝑸𝟏, 𝑰𝒉 𝑸
𝟐
𝑹𝟐 𝑸𝟐, 𝑰𝒉 𝑸𝟏
Implicit Artificial Boundary
Interior Flux Linearization
𝝏𝑹𝟏 𝑸𝟏, 𝑰𝒉 𝑸𝟐
𝝏𝑰𝒉 𝑸𝟐 =
𝝏𝑹𝒊𝟏 𝑸𝑳
𝟏, 𝑰𝒉 𝑸𝟐
𝝏𝑸𝑹𝟏
𝝏𝑹𝒊 𝑸𝑳𝒊 , 𝑸𝑹
𝒊
𝝏𝑸𝑳 𝝏𝑹𝒊 𝑸𝑳
𝒊 , 𝑸𝑹𝒊
𝝏𝑸𝑹
Interior Jacobian
Array of Matrices
RHS Interpolation Operator
Array of Vectors
Receiver Grid
Interpolation
Mapping
𝑹𝒊 𝑸𝑳𝒊 𝑸𝑹
𝒊
𝑹𝒊𝟏 𝑸𝑳
𝟏 𝑰𝒉 𝑸𝟐
![Page 10: A General Implicit Artificial Boundary Scheme for Chimera ...2014.oversetgridsymposium.org/assets/presentations/3_2/Galbraith_ogs_2014.pdfGTSL 5 12th Overset Symposium, 9 Oct 2014](https://reader033.vdocuments.mx/reader033/viewer/2022060707/607368047c72023c4d732ba3/html5/thumbnails/10.jpg)
GTSL
10 12th Overset Symposium, 9 Oct 2014
Parallel GMRES Iterative Solver
Processor 1
Matrix-Vector Multiplication
Mask Communication with Local Calculations 𝒘𝒏+𝟏 = 𝑨𝒗𝒏
Processor 2
Interpolate 𝑰𝒉 𝒗𝒏𝟏 Interpolate 𝑰𝒉 𝒗𝒏
𝟐
Non-Blocking Send 𝑰𝒉 𝒗𝒏𝟏 Non-Blocking Send 𝑰𝒉 𝒗𝒏
𝟐
Compute 𝒘𝒏+𝟏𝟏 = 𝑨𝟏𝒗𝒏
𝟏 Compute 𝒘𝒏+𝟏𝟐 = 𝑨𝟐𝒗𝒏
𝟐
Receive 𝑰𝒉 𝒗𝒏𝟐 Receive 𝑰𝒉 𝒗𝒏
𝟏
Compute 𝒘𝒏+𝟏𝟏 = 𝒘𝒏+𝟏
𝟏 + 𝑪𝟏 𝑰𝒉 𝒗𝒏𝟐 Compute 𝒘𝒏+𝟏
𝟐 = 𝒘𝒏+𝟏𝟐 + 𝑪𝟐 𝑰𝒉 𝒗𝒏
𝟏
𝒘𝒏+𝟏𝟏
𝒘𝒏+𝟏𝟐 = 𝑨𝟏 𝑪𝟏
𝑪𝟐 𝑨𝟐𝒗𝒏𝟏
𝒗𝒏𝟐
Dot Products
Compute Local Dot Product All Reduce 𝑶𝒏
𝒑+ 𝒍𝒐𝒈 𝒑
𝒏
𝒑> 𝒍𝒐𝒈 𝒑
Parallel Efficient If
Preconditioner Omits C Matrices Jacobi as p → n
![Page 11: A General Implicit Artificial Boundary Scheme for Chimera ...2014.oversetgridsymposium.org/assets/presentations/3_2/Galbraith_ogs_2014.pdfGTSL 5 12th Overset Symposium, 9 Oct 2014](https://reader033.vdocuments.mx/reader033/viewer/2022060707/607368047c72023c4d732ba3/html5/thumbnails/11.jpg)
GTSL
11 12th Overset Symposium, 9 Oct 2014
Outline
• Discretization Assumptions
• Explicit/Implicit Chimera
• Sparse Iterative Solvers – Preconditioners
• Distributed Memory Parallelism
• Discontinuous Galerkin Method
• Inviscid/Viscous Flow Examples
• Conclusion and Future Work
![Page 12: A General Implicit Artificial Boundary Scheme for Chimera ...2014.oversetgridsymposium.org/assets/presentations/3_2/Galbraith_ogs_2014.pdfGTSL 5 12th Overset Symposium, 9 Oct 2014](https://reader033.vdocuments.mx/reader033/viewer/2022060707/607368047c72023c4d732ba3/html5/thumbnails/12.jpg)
GTSL
12 12th Overset Symposium, 9 Oct 2014
FV Approximation
Discontinuous Galerkin Chimera
Scheme
• Discontinuous Galerkin – Weak Form
– Approximate Riemann Solver by Roe
– BR2 Viscous Scheme
• DG-Chimera – Natural Interpolation Operator (Solution is Polynomials)
– Curved Elements
– Reduces to a Zonal Interface (Abutting Meshes)
– No Orphan Points due to Fringe Points
DG Approximation
xj-1/2 xj xj+1/2
0
dF
e
- Legendre
Polynomials
0,,
ee
dQFdnQQFQQR
![Page 13: A General Implicit Artificial Boundary Scheme for Chimera ...2014.oversetgridsymposium.org/assets/presentations/3_2/Galbraith_ogs_2014.pdfGTSL 5 12th Overset Symposium, 9 Oct 2014](https://reader033.vdocuments.mx/reader033/viewer/2022060707/607368047c72023c4d732ba3/html5/thumbnails/13.jpg)
GTSL
13 12th Overset Symposium, 9 Oct 2014
Outline
• Discretization Assumptions
• Explicit/Implicit Chimera
• Sparse Iterative Solvers – Preconditioners
• Distributed Memory Parallelism
• Discontinuous Galerkin Method
• Inviscid/Viscous Flow Examples
• Conclusion and Future Work
![Page 14: A General Implicit Artificial Boundary Scheme for Chimera ...2014.oversetgridsymposium.org/assets/presentations/3_2/Galbraith_ogs_2014.pdfGTSL 5 12th Overset Symposium, 9 Oct 2014](https://reader033.vdocuments.mx/reader033/viewer/2022060707/607368047c72023c4d732ba3/html5/thumbnails/14.jpg)
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• Inviscid SKF 1.1 Airfoil
– M∞ = 0.4 – α=2.5°
• Viscous Subsonic Circular Cylinder – M∞ = 0.25 – Re = 40
• Focus on Solution Time – Explicit vs. Implicit Chimera
Inviscid/Viscous Flow Examples
![Page 15: A General Implicit Artificial Boundary Scheme for Chimera ...2014.oversetgridsymposium.org/assets/presentations/3_2/Galbraith_ogs_2014.pdfGTSL 5 12th Overset Symposium, 9 Oct 2014](https://reader033.vdocuments.mx/reader033/viewer/2022060707/607368047c72023c4d732ba3/html5/thumbnails/15.jpg)
GTSL
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Time Integration and
Compute Resources
• Steady State – Quasi-Newton
• GMRES Krylov Solver – ILU1 Preconditioner – Converged to 1e-11 Each Newton Iteration
• Intel Core 2 Duo 3.0 GHz processor 8 GB RAM – 10 Compute Nodes – Ethernet Connection
• MPI Parallelism
– Timings for 1, 2, 4, and 8 Processors – 1 MPI Process per Node (Maximize Communication)
• Shared Memory Multi-Threaded
– 1 Grid Per Thread
𝑴
∆𝒕+𝝏𝑹
𝝏𝑸𝚫𝑸 = 𝑹 𝑪𝑭𝑳𝒏+𝟏 = 𝑪𝑭𝑳𝟎
𝑹𝟎
𝑹𝒏
𝑪𝑭𝑳𝒎𝒂𝒙 = 𝟏𝒆𝟑𝟎
![Page 16: A General Implicit Artificial Boundary Scheme for Chimera ...2014.oversetgridsymposium.org/assets/presentations/3_2/Galbraith_ogs_2014.pdfGTSL 5 12th Overset Symposium, 9 Oct 2014](https://reader033.vdocuments.mx/reader033/viewer/2022060707/607368047c72023c4d732ba3/html5/thumbnails/16.jpg)
GTSL
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Single
105x30 Ng=3
SKF 1.1 Airfoil (M∞ = 0.4, α=2.5°)
Meshes
105x16 Ng=3
O-Grid Chimera
104x14 Ng=1
R-Grid Chimera
104x104 Ng=1
105x15 Ng=3
Chimera
Interface
![Page 17: A General Implicit Artificial Boundary Scheme for Chimera ...2014.oversetgridsymposium.org/assets/presentations/3_2/Galbraith_ogs_2014.pdfGTSL 5 12th Overset Symposium, 9 Oct 2014](https://reader033.vdocuments.mx/reader033/viewer/2022060707/607368047c72023c4d732ba3/html5/thumbnails/17.jpg)
GTSL
17 12th Overset Symposium, 9 Oct 2014
SKF 1.1 Airfoil (M∞ = 0.4 , α=2.5°)
Explicit Chimera Speedup
ChimeraLocalLocal QQRQQA ,
![Page 18: A General Implicit Artificial Boundary Scheme for Chimera ...2014.oversetgridsymposium.org/assets/presentations/3_2/Galbraith_ogs_2014.pdfGTSL 5 12th Overset Symposium, 9 Oct 2014](https://reader033.vdocuments.mx/reader033/viewer/2022060707/607368047c72023c4d732ba3/html5/thumbnails/18.jpg)
GTSL
18 12th Overset Symposium, 9 Oct 2014
SKF 1.1 Airfoil (M∞ = 0.4 , α=2.5°)
Explicit Chimera Convergence History
ChimeraLocalLocal QQRQQA ,
![Page 19: A General Implicit Artificial Boundary Scheme for Chimera ...2014.oversetgridsymposium.org/assets/presentations/3_2/Galbraith_ogs_2014.pdfGTSL 5 12th Overset Symposium, 9 Oct 2014](https://reader033.vdocuments.mx/reader033/viewer/2022060707/607368047c72023c4d732ba3/html5/thumbnails/19.jpg)
GTSL
19 12th Overset Symposium, 9 Oct 2014
SKF 1.1 Airfoil (M∞ = 0.4 , α=2.5°)
Implicit Chimera Speedup
ChimeraLocalChimeraLocal QQRQQQA ,,
![Page 20: A General Implicit Artificial Boundary Scheme for Chimera ...2014.oversetgridsymposium.org/assets/presentations/3_2/Galbraith_ogs_2014.pdfGTSL 5 12th Overset Symposium, 9 Oct 2014](https://reader033.vdocuments.mx/reader033/viewer/2022060707/607368047c72023c4d732ba3/html5/thumbnails/20.jpg)
GTSL
20 12th Overset Symposium, 9 Oct 2014
SKF 1.1 Airfoil (M∞ = 0.4 , α=2.5°)
Implicit Convergence History
ChimeraLocalChimeraLocal QQRQQQA ,,
![Page 21: A General Implicit Artificial Boundary Scheme for Chimera ...2014.oversetgridsymposium.org/assets/presentations/3_2/Galbraith_ogs_2014.pdfGTSL 5 12th Overset Symposium, 9 Oct 2014](https://reader033.vdocuments.mx/reader033/viewer/2022060707/607368047c72023c4d732ba3/html5/thumbnails/21.jpg)
GTSL
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10s
1min
1hr
10s
1min
1hr
10s
1min
1hr
SKF 1.1 Airfoil (M∞ = 0.4 , α=2.5°)
Solution Time
ChimeraLocalLocal QQRQQA ,
ChimeraLocalChimeraLocal QQRQQQA ,,
10s
1min
1hr
10s
1min
1hr
10s
1min
1hr
![Page 22: A General Implicit Artificial Boundary Scheme for Chimera ...2014.oversetgridsymposium.org/assets/presentations/3_2/Galbraith_ogs_2014.pdfGTSL 5 12th Overset Symposium, 9 Oct 2014](https://reader033.vdocuments.mx/reader033/viewer/2022060707/607368047c72023c4d732ba3/html5/thumbnails/22.jpg)
GTSL
22 12th Overset Symposium, 9 Oct 2014
SKF 1.1 Airfoil (M∞ = 0.4 , α=2.5°)
Implicit/Explicit Solution Speedup
![Page 23: A General Implicit Artificial Boundary Scheme for Chimera ...2014.oversetgridsymposium.org/assets/presentations/3_2/Galbraith_ogs_2014.pdfGTSL 5 12th Overset Symposium, 9 Oct 2014](https://reader033.vdocuments.mx/reader033/viewer/2022060707/607368047c72023c4d732ba3/html5/thumbnails/23.jpg)
GTSL
23 12th Overset Symposium, 9 Oct 2014
SKF 1.1 Airfoil (M∞ = 0.4, α=2.5°)
Cp Contour Lines
N=1
2nd-order
N=2
3rd-order
N=3
4th-order
N=0
1st-order
![Page 24: A General Implicit Artificial Boundary Scheme for Chimera ...2014.oversetgridsymposium.org/assets/presentations/3_2/Galbraith_ogs_2014.pdfGTSL 5 12th Overset Symposium, 9 Oct 2014](https://reader033.vdocuments.mx/reader033/viewer/2022060707/607368047c72023c4d732ba3/html5/thumbnails/24.jpg)
GTSL
24 12th Overset Symposium, 9 Oct 2014
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
![Page 25: A General Implicit Artificial Boundary Scheme for Chimera ...2014.oversetgridsymposium.org/assets/presentations/3_2/Galbraith_ogs_2014.pdfGTSL 5 12th Overset Symposium, 9 Oct 2014](https://reader033.vdocuments.mx/reader033/viewer/2022060707/607368047c72023c4d732ba3/html5/thumbnails/25.jpg)
GTSL
25 12th Overset Symposium, 9 Oct 2014
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
![Page 26: A General Implicit Artificial Boundary Scheme for Chimera ...2014.oversetgridsymposium.org/assets/presentations/3_2/Galbraith_ogs_2014.pdfGTSL 5 12th Overset Symposium, 9 Oct 2014](https://reader033.vdocuments.mx/reader033/viewer/2022060707/607368047c72023c4d732ba3/html5/thumbnails/26.jpg)
GTSL
26 12th Overset Symposium, 9 Oct 2014
Circular Cylinder (Re = 40)
Implicit/Explicit Solution Speedup
![Page 27: A General Implicit Artificial Boundary Scheme for Chimera ...2014.oversetgridsymposium.org/assets/presentations/3_2/Galbraith_ogs_2014.pdfGTSL 5 12th Overset Symposium, 9 Oct 2014](https://reader033.vdocuments.mx/reader033/viewer/2022060707/607368047c72023c4d732ba3/html5/thumbnails/27.jpg)
GTSL
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Subsonic Circular Cylinder (M∞ = 0.25)
Cp/Entropy Rise Contour Lines N=1
2nd-order N=2
3rd-order
N=3
4th-order
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GTSL
28 12th Overset Symposium, 9 Oct 2014
Conclusion and Future Work
• Implicit Artificial Boundaries – Included with GMRES Matrix-Vector Multiplication – Omitted in Preconditioner – Minimal Information Communicated – Significantly Reduces Execution Time
• Few Modifications Required to Existing Codes – ~95% of Code already Exists – Spares Matrix-Vector Multiplication – Restarted GMRES Fortran Code
• http://people.sc.fsu.edu/~jburkardt/f_src/mgmres/mgmres.html
• Demonstrated on Inviscid/Viscous Flows
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GTSL
GTSL
12th Overset Symposium, 9 Oct 2014
Thank you!
Questions?
This work was supported by the
Department of Defense (DoD)
through the National Defense Science &
Engineering Graduate (NDSEG)
Fellowship Program