on-going - nasa · 2002. 8. 8. · venditti & d armof al number of nodes cl 10 4 10 5 3.60 3.70...

On-Going Research in Drag Prediction

John C. Vassberg

Technical Fellow

The Boeing Company

Long Beach, CA 90807, USA

AIAA CFD Drag Prediction Workshop

19th Applied Aerodynamics Conference

Anaheim, CA

9-10 June, 2001

Vassberg, DPW On-Going Research, Anaheim, CA 1

OUTLINE

� ADJOINT RECOVERY OF

SUPERCONVERGENT FUNCTIONALS

{ Niles Pierce, Caltech

{ Mike Giles, Oxford

� GRID ADAPTATION &

ADJOINT ERROR ESTIMIATION

{ David Venditti, MIT

{ David Darmofal, MIT


PIERCE & GILES

� ADJOINT RECOVERY

{ Solve Both the Primal and Adjoint PDEs

{ Estimate Sensitivity of Grid on Primal Solution

{ Improve Result of a Boundary Integration

� Boundary Integration is Superconvergent

{ Provides an Error Bound

� Currently, Only Proven for Linear PDEs

� Currently, Working on the Non-Linear Problem


PIERCE & GILES

REFERENCES

Pierce, N.A. & Giles, M.B., "Adjoint Recovery of Superconver-

gent Functionals from PDE Approximations," SIAM Review, Vol.

42, No. 2, pp. 247-264, April, 2000.

Mueller, J.D. & Giles, M.B., "Solution Adaptative Mesh Re-

�nement Using Adjoint Error Analysis," AIAA Paper 2001-2550,

CFD Conference, Anahiem, CA, June, 2001.


PIERCE & GILES4 6 8 10 12 14 16

6

8

10

12

14

16

x

y

Primal Solution

4 6 8 10 12 14 16

6

8

10

12

14

16

x

y

Dual Solution2D Poisson Equation on Warped Square


PIERCE & GILES

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6−13

−12

−11

−10

−9

−8

−7

−6

−5

−4Error Convergence

log 10

(Err

or)

log10

(Cells)

Base ErrorRemaining ErrorError Bound

2D Poisson Equation with Bulk Funcitional


PIERCE & GILES−4 −2 0 2 4

−4

−2

0

2

4

x

y

Primal Solution

−4 −2 0 2 4−4

−2

0

2

4

x

y

Dual Solution

2D Laplace Equation on Joukowski Airfoil


PIERCE & GILES

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8−11

−10

−9

−8

−7

−6

−5

−4

−3

−2Error Convergence

log 10

(Err

or)

log10

(Cells)

Base ErrorRemaining Error

2D Laplace Equation with Boundary Functional


VENDITTI & DARMOFAL

� GRID ADAPTATION &

ADJOINT ERROR ESTIMATION

{ Solve Both the Primal and Adjoint PDEs

{ Project Both Solutions to a Finer Grid

{ Estimate Local Error of Primal Solution

{ Adapt Grid until Local Error is Globally Uniform

{ Comparisons with Adhoc Sensors

� 2D Advanced EET High-Lift Con�guration

� 2D Staggard Supersonic Bi-Plane


VENDITTI & DARMOFAL

REFERENCES

Venditti, D. & Darmofal, D., "An Adaptive Unstructured Method-

ology for Functional Outputs of Compressible Flow Simulations,"

AIAA Paper 2001-2659, CFD Conference, Anahiem, CA, June,

2001.

Venditti, D. & Darmofal, D., "A Multilevel Error Estimation and

Grid Adaptation Strategy for Improving the Accuracy of Integral

Outputs," AIAA Paper 99-3292, CFD Conference, Norfolk, VA,

1999.


VENDITTI & DARMOFAL

Number of Nodes

CL

104 105

3.60

3.70

3.80

3.90

4.00

4.10

4.20

Adjoint Adaptation with CorrectionPressure AdaptationUniformly Refined GridsExtrapolated

Lift Convergence - Advanced EETM∞ = 0.26 - α = 8o

Mach Number

Streamlines

Advanced EET High-Lift Airfoil


VENDITTI & DARMOFAL

8844 nodes

Distant viewAdjoint

Adaptation13600 nodes

Distant viewPressure

Adaptation

Advanced EET High-Lift Airfoil


VENDITTI & DARMOFAL

0 1 2

-1

0

Mach number - M∞ = 3.0

2D Staggard Supersonic Bi-Plane


VENDITTI & DARMOFAL

0 1 2

-1

0

Final grid3810 Nodes

Adjoint-based adaptationCD = 0.0766 (lower airfoil)

0 1 2

-1

0

Final grid37352 Nodes

Pressure-based adaptationCD = 0.0767 (lower airfoil)

2D Staggard Supersonic Bi-Plane


on-going - nasa · 2002. 8. 8. · venditti & d armof al number of nodes cl 10 4 10 5 3.60 3.70...

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