Chimera Overset Method with a Discontinuous...

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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]


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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°


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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


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Outline




– DG-Chimera





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FV Approximation

Discontinuous Galerkin Method

xj-1/2 xj xj+1/2


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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


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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


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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


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Outline




– DG-Chimera





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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


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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


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


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Fringe Points

49

1

29

14

3

1

3

1 2211

11

iiii

iii

uuuuuuu


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


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Q-

Q+ Q-

Q-

Q-

Q+

Q+

Q+

DG-Chimera

Inter-Grid Communication

0

,,

e

e

dQF

dnQQFQQR


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DG-Chimera


Q- Q+

N

n

nnqQ Need

0

,,

e

e

dQF

dnQQFQQR


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DG-Chimera


• 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


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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


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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


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• 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


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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


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Subsonic Circular Cylinder (M∞ = 0.25)

Cp Contour Lines

N=1

2nd-order

N=2

3rd-order

N=3

4th-order

Surface Cp


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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


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Cp Contour Lines

Surface Cp

N=1

2nd-order

N=2

3rd-order

N=3

4th-order


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Explicit Chimera Convergence

• Explicit Chimera Boundaries

Single Mesh

ChimeraLocalLocal QQRQQA ,


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Implicit Chimera Convergence

• Implicit Chimera Boundaries

Single O-Grid Chimera Hole Cut Chimera

ChimeraLocalChimeraLocal QQRQQQA ,,


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Outline




– DG-Chimera





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• 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)


Overset Regions on Curved Geometry


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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


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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


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Nose Section

Surface Velocity

Finite

Volume

DG

With

Projection

No

Projection


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Nose Section

Common Boundary Layer

Finite Volume DG

Mesh 1

Mesh 2

Mesh 1

Mesh 2


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Circular Cylinder Re 40

Meshes

Finite Volume Discontinuous Galerkin Ng = 3

93x201

18,500 DOF

50x30

24,000 DOF

81x201 25x201 34x30 22x30


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Surface Velocity

Finite

Volume

DG

With

Projection

No

Projection


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Common Boundary Layer

Finite Volume DG

Mesh 1 Mesh 2 Mesh 1 Mesh 2


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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


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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

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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

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