GTSL
GTSL
11th Overset Grid Symposium, 16 Oct 2012
Chimera Overset Method with a
Discontinuous Galerkin Discretization
Marshall Galbraith University of Cincinnati
School of Aerospace Systems [email protected]
Paul D. Orkwis University of Cincinnati
School of Advanced Structures [email protected]
John Benek
Air Force Research Laboratory Computational Science Branch Center
of Excellence [email protected]
11th Overset Grid Symposium, 16 Oct 2012
GTSL
2
Motivation
• Chimera Overset Grid Method – Complex Geometries
– “Hot swap” Geometric
Features
– Moving Grids with Relative
Motion
• Store Separation
• Rotorcraft
• High-Order Methods
– DNS-LES
– Transitional Turbulent Flows
– And more…
• WENO, Compact FD – Large Interior Stencils
– Fringe Points
• Maintain Interior Scheme
• Orphan Points
– High-Order Interpolation
Schemes
• Large Stencil
– Complicated Hole Cutting
• Discontinuous Galerkin Method – Natural Higher Order
Extension to Finite Volume
– Compact Stencil
• Communication – No Fringe Points
• Hole Cutting
– Curved Elements
SD7003
Re 60k
AoA 4°
11th Overset Grid Symposium, 16 Oct 2012
GTSL
3
Outline
• Discontinuous Galerkin Method – Mesh Representation
• Inter-Grid Communication
– Finite Volume and Fringe Points
– DG-Chimera
– Inviscid Flow Examples
• Overset Regions on Curved Geometry – Viscous Flow Examples
• Conclusion and Future Work
11th Overset Grid Symposium, 16 Oct 2012
GTSL
4
Outline
• Discontinuous Galerkin Method – Mesh Representation
• Inter-Grid Communication
– Finite Volume and Fringe Points
– DG-Chimera
– Inviscid Flow Examples
• Overset Regions on Curved Geometry – Viscous Flow Examples
• Conclusion and Future Work
11th Overset Grid Symposium, 16 Oct 2012
GTSL
5
FV Approximation
Discontinuous Galerkin Method
xj-1/2 xj xj+1/2
11th Overset Grid Symposium, 16 Oct 2012
GTSL
6
Discontinuous Galerkin Method
DG Approximation
xj-1/2 xj xj+1/2
• Euler/Navier-Stokes Equations in Conservation Form
• Weak Form
• Approximate Riemann Solver by Roe
• BR2 Viscous Scheme
• Newton-Krylov Solver with GMRES and ILU1 Preconditioner
0 F
0
dF
e
- Legendre
Polynomials
0,,
ee
dQFdnQQFQQR
11th Overset Grid Symposium, 16 Oct 2012
GTSL
7
Discontinuous Galerkin Method
Geometric Mapping
g gN
i
N
j
jiijxx
0 0
,
g gN
i
N
j
jiijyy
0 0
,
yxyxJ
1
yxtJJ
n
1
yxtJJ
n
1
Ng – Degree of Cell Polynomial Jyx Jxy
Jyx Jxy
ξ -1 1
η
-1
1
,,, yx
n
n
n
n
Ng=1
11th Overset Grid Symposium, 16 Oct 2012
GTSL
8
Discontinuous Galerkin Method
Geometric Mapping
g gN
i
N
j
jiijxx
0 0
,
g gN
i
N
j
jiijyy
0 0
,
yxyxJ
1
yxtJJ
n
1
yxtJJ
n
1
,,, yx
ξ -1 1
η
-1
1
n
n
n
n
Ng – Degree of Cell Polynomial Jyx Jxy
Jyx Jxy
Ng=2
11th Overset Grid Symposium, 16 Oct 2012
GTSL
9
Outline
• Discontinuous Galerkin Method – Mesh Representation
• Inter-Grid Communication
– Finite Volume and Fringe Points
– DG-Chimera
– Inviscid Flow Examples
• Overset Regions on Curved Geometry – Viscous Flow Examples
• Conclusion and Future Work
11th Overset Grid Symposium, 16 Oct 2012
GTSL
10
High-Order FD/FV Chimera
Fringe Points
• Fringe Points: – Additional Points on the Boundary to Maintain
Interior Scheme
49
1
29
14
3
1
3
1 2211
11
iiii
iii
uuuuuuu
Compact Difference 6th-order
i-2 i-1 i i+1 i+2 1 2 3 4 5
N-2 N-1 N N-3 N-4
-1 0
N+1 N+2
Fringe Points
Direct Injection:
No Interpolation
11th Overset Grid Symposium, 16 Oct 2012
GTSL
11
High-Order FD/FV Chimera
Fringe Points
• Fringe Points: – Additional Points on the Boundary to Maintain
Interior Scheme
49
1
29
14
3
1
3
1 2211
11
iiii
iii
uuuuuuu
Compact Difference 6th-order
i-2 i-1 i i+1 i+2
N-2 N-1 N N-3 N-4 N+1 N+2
Fringe Points
1 2 3 4 5 -1 0
11th Overset Grid Symposium, 16 Oct 2012
GTSL
12
High-Order FD/FV Chimera
Fringe Points
49
1
29
14
3
1
3
1 2211
11
iiii
iii
uuuuuuu
Compact Difference 6th-order
i-2 i-1 i i+1 i+2
N-2 N-1 N N-3 N-4 N+1 N+2
Fringe Points
1 2 3 4 5
N-2 N-1 N N-3 N-4
-1 0
N+1 N+2
Orphan Points:
No Donor Stencil
From Interior Points
1 2 3 4 5 -1 0
11th Overset Grid Symposium, 16 Oct 2012
GTSL
13
Q-
Q+ Q-
Q-
Q-
Q+
Q+
Q+
DG-Chimera
Inter-Grid Communication
0
,,
e
e
dQF
dnQQFQQR
11th Overset Grid Symposium, 16 Oct 2012
GTSL
14
DG-Chimera
Inter-Grid Communication
Q- Q+
N
n
nnqQ Need
0
,,
e
e
dQF
dnQQFQQR
11th Overset Grid Symposium, 16 Oct 2012
GTSL
15
DG-Chimera
Inter-Grid Communication
• KD-Tree – Locate Node Xk In Bounding Box
• Newton’s Method
– Locate Local Cell Coordinate
Q- Q+
kX
kk XX
,
(x0,y0)
(x1,y1)
(x2,y2)
(xr,yr)
gN
k
kkrr xxx0
ˆ,1
gN
k
kkrr yyy
0
ˆ,1
Face Polynomial
krkrk sysxX ,
Cartesian Gauss Quadrature Nodes
N
n
nnqQ Need
d
XXQsw
q
n
k
kkbjnk
n2
,
Q+ Coefficients
(xr(-1,η),yr(-1,η))
kk XX
,
0
,,
e
e
dQF
dnQQFQQR
11th Overset Grid Symposium, 16 Oct 2012
GTSL
16
DG-Chimera
Zonal Interfaces
• No Additional Cells to Facilitate Communication – No Fringe Points – No Orphan Points Associated with Fringe Points
• Reduces to Zonal Interface
Ng = 1
Ng = 1
Ng = 1
Ng = 1
– Linear Boundary – Linear Mappings
– Curved Boundary – Concave Linear Mapping – Convex Quadratic Mapping
Identical to
Interior Scheme
– Curved Boundary
– Coincident Nodes
– Linear Mappings
Ng = 1
Ng = 2
11th Overset Grid Symposium, 16 Oct 2012
GTSL
17
DG-Chimera
Orphan Faces
Ng = 2
Ng = 1
Gap
Ng = 2
Ng = 1
Insufficient Overlap Sufficient Overlap
Ng = 1 Ng = 1
Gap
Ng = 1
Ng = 1
• Orphans Arise with Gaps Between Mesh Boundaries
• Examples: – Curved Boundary – Concave Quadratic Mapping – Convex Linear Mapping
– Curved Boundary – Non-coincident Nodes – Linear Mappings
11th Overset Grid Symposium, 16 Oct 2012
GTSL
18
• Honor of Dr. Benek, Dr. Steger, and Dr. Dougherty
• Subsonic Circular Cylinder (M∞ = 0.25)
• SKF 1.1 Airfoil (M∞ = 0.4)
Inviscid Flow Examples
11th Overset Grid Symposium, 16 Oct 2012
GTSL
19
70x36 Ng=1
70x1 Ng=3
O-Grid Chimera
60x3 Ng=1 Single
70x40 Ng=3
Subsonic Circular Cylinder (M∞ = 0.25)
Meshes
Hole Cut Chimera
105x105 Ng=1
70x16 Ng=1
Chimera
Interface
11th Overset Grid Symposium, 16 Oct 2012
GTSL
20
Subsonic Circular Cylinder (M∞ = 0.25)
Cp Contour Lines
N=1
2nd-order
N=2
3rd-order
N=3
4th-order
Surface Cp
11th Overset Grid Symposium, 16 Oct 2012
GTSL
21
Single
105x30 Ng=3
SKF 1.1 Airfoil (M∞ = 0.4)
Meshes
105x15 Ng=3
O-Grid Chimera
104x15 Ng=1
Hole Cut Chimera
75x75 Ng=1
105x15 Ng=3
11th Overset Grid Symposium, 16 Oct 2012
GTSL
22
SKF 1.1 Airfoil (M∞ = 0.4)
Cp Contour Lines
Surface Cp
N=1
2nd-order
N=2
3rd-order
N=3
4th-order
11th Overset Grid Symposium, 16 Oct 2012
GTSL
23
SKF 1.1 Airfoil (M∞ = 0.4)
Explicit Chimera Convergence
• Explicit Chimera Boundaries
Single Mesh
ChimeraLocalLocal QQRQQA ,
11th Overset Grid Symposium, 16 Oct 2012
GTSL
24
SKF 1.1 Airfoil (M∞ = 0.4)
Implicit Chimera Convergence
• Implicit Chimera Boundaries
Single O-Grid Chimera Hole Cut Chimera
ChimeraLocalChimeraLocal QQRQQQA ,,
11th Overset Grid Symposium, 16 Oct 2012
GTSL
25
Outline
• Discontinuous Galerkin Method – Mesh Representation
• Inter-Grid Communication
– Finite Volume and Fringe Points
– DG-Chimera
– Inviscid Flow Examples
• Overset Regions on Curved Geometry – Viscous Flow Examples
• Conclusion and Future Work
11th Overset Grid Symposium, 16 Oct 2012
GTSL
26
• Secant Line Geometric Representation • Interpolation from Incorrect Interior Cells
• PEGASUS5 & SUGGAR++: Projection • Discontinuous Galerkin Curved Elements
– Elements Follow Surface of Geometry
– (Dr. Noack Finite Volume)
High-Order FD/FV Chimera
Overset Regions on Curved Geometry
11th Overset Grid Symposium, 16 Oct 2012
GTSL
27
Viscous Flow Examples
• Inspired from 3D Overset Mesh
• Nose Section – M∞=0.5, Re=1,000,000
• Circular Cylinder – M∞=0.1, Re=40
Generic Launch Vehicle
11th Overset Grid Symposium, 16 Oct 2012
GTSL
28
Nose Section
Meshes
Finite Volume Discontinuous Galerkin Ng = 3
1,0
5.0
5.1 3
s
ssy
ssx
41x201
8,000 DOF
27x201
14x201
14x201
20x36
11,500 DOF
12x36
6x36
6x36
11th Overset Grid Symposium, 16 Oct 2012
GTSL
29
Nose Section
Surface Velocity
Finite
Volume
DG
With
Projection
No
Projection
11th Overset Grid Symposium, 16 Oct 2012
GTSL
30
Nose Section
Common Boundary Layer
Finite Volume DG
Mesh 1
Mesh 2
Mesh 1
Mesh 2
11th Overset Grid Symposium, 16 Oct 2012
GTSL
31
Circular Cylinder Re 40
Meshes
Finite Volume Discontinuous Galerkin Ng = 3
93x201
18,500 DOF
50x30
24,000 DOF
81x201 25x201 34x30 22x30
11th Overset Grid Symposium, 16 Oct 2012
GTSL
32
Circular Cylinder Re 40
Surface Velocity
Finite
Volume
DG
With
Projection
No
Projection
11th Overset Grid Symposium, 16 Oct 2012
GTSL
33
Circular Cylinder Re 40
Common Boundary Layer
Finite Volume DG
Mesh 1 Mesh 2 Mesh 1 Mesh 2
11th Overset Grid Symposium, 16 Oct 2012
GTSL
34
Circular Cylinder Re 40
Separation Length
Single Grid Chimera Mesh
Finite
Volume 2.246
2.260 Without
2.152 With
DG 2.245 2.246
Literature Separation Length: 1.9 < Ls < 2.3
Finite Volume Discontinuous Galerkin
11th Overset Grid Symposium, 16 Oct 2012
GTSL
35
Conclusion and Future Work
• DG-Chimera – No Orphan Points Due to Fringe Points
– Naturally Reduces to Zonal Interface
– Inherent Proper Interpolation on Curved Geometry
• Curved Elements
– Demonstrated on Inviscid/Viscous Flows
• Future Work – Extend to 3-D in Space (Mostly complete)
– Shock Capturing (Mostly Complete)
– Parallel Execution with MPI