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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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
Vassberg, DPW On-Going Research, Anaheim, CA 2
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
Vassberg, DPW On-Going Research, Anaheim, CA 3
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.
Vassberg, DPW On-Going Research, Anaheim, CA 4
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
Vassberg, DPW On-Going Research, Anaheim, CA 5
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
Vassberg, DPW On-Going Research, Anaheim, CA 6
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
Vassberg, DPW On-Going Research, Anaheim, CA 7
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
Vassberg, DPW On-Going Research, Anaheim, CA 8
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
Vassberg, DPW On-Going Research, Anaheim, CA 9
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.
Vassberg, DPW On-Going Research, Anaheim, CA 10
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
Vassberg, DPW On-Going Research, Anaheim, CA 11
VENDITTI & DARMOFAL
8844 nodes
Distant viewAdjoint
Adaptation13600 nodes
Distant viewPressure
Adaptation
Advanced EET High-Lift Airfoil
Vassberg, DPW On-Going Research, Anaheim, CA 12
VENDITTI & DARMOFAL
0 1 2
-1
0
Mach number - M∞ = 3.0
2D Staggard Supersonic Bi-Plane
Vassberg, DPW On-Going Research, Anaheim, CA 13
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
Vassberg, DPW On-Going Research, Anaheim, CA 14
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