Upload
others
View
0
Download
0
Embed Size (px)
Citation preview
NSF Grant DMI-9979711
2002 NSF-SNL Grantees Meeting, 09/24/2002, Albuquerque NM 0
PROTECTION AGAINST MODELING AND SIMULATION UNCERTAINTIES IN DESIGN
OPTIMIZATION
NSF GRANT DMI-9979711
Bernard Grossman, William H. Mason, Layne T. Watson, Serhat Hosder, and Hongman Kim
Virginia Polytechnic Institute and State University Blacksburg, VA
Raphael T. HaftkaUniversity of Florida
Gainesville, FL
Life-Cycle Engineering Program Meeting and Review24-25 September 2002
Albuquerque, NM
NSF Grant DMI-9979711
2002 NSF-SNL Grantees Meeting, 09/24/2002, Albuquerque NM 1
• Detection and Repair of Poorly Converged Optimization Runs
Ø Statistical Modeling of Structural Optimization Errors due to Incomplete Convergence
Ø Computational Fluid Dynamics (CFD) Simulation Uncertainties
Research Performed Under the Project
NSF Grant DMI-9979711
2002 NSF-SNL Grantees Meeting, 09/24/2002, Albuquerque NM 2
• Estimate error level of the optimization procedure
• Identify probabilistic distribution model of the optimization error
• Estimate mean and standard deviation of errors without expensive, accurate runs
Statistical Modeling of Optimization Errors
Objective
NSF Grant DMI-9979711
2002 NSF-SNL Grantees Meeting, 09/24/2002, Albuquerque NM 3
+ x
z Leading edge radius (fixed)
Outboard LE sweep (fixed)
x y
v1
v3
Location of maximum thickness (fixed)
v2
v4
Wing semi span (fixed)
v5: Fuel Weight
• 250 passenger aircraft, 5500 nm range, cruise at Mach 2.4
• Minimize take off gross weight (WTOGW)
• For this study, a simplified 5DV version is used
Test problem: High Speed Civil Transport (HSCT)
NSF Grant DMI-9979711
2002 NSF-SNL Grantees Meeting, 09/24/2002, Albuquerque NM 4
Structural Optimization and Modeling of Error
• Structural optimization performed a priori for many aircraft configurations to obtain optimum Wing Structural Weight (Ws)
• Multiple optimization results to construct response surface
• With multiple optimization results available, statistical techniques can be used to model the convergence error:
Error = Ws – (Ws)t
• Probabilistic modelError fit (expensive approach)
• Weibull Distribution
From high fidelity optimization
NSF Grant DMI-9979711
2002 NSF-SNL Grantees Meeting, 09/24/2002, Albuquerque NM 5
Previous Results
• Weibull Distribution models the convergence error of the optimization runs successfully
• Difference fit used to estimate the mean and standard deviation of errors without expensive, high fidelity runs
Multiple sets of low fidelity optimization results required for Difference Fit
Changing convergence criteria (previous results)
Changing initial design points (current results )
NSF Grant DMI-9979711
2002 NSF-SNL Grantees Meeting, 09/24/2002, Albuquerque NM 6
design
Opt
imum
win
gst
ruct
ura
lwei
ght
(Ws)
5 10 15 2040000
60000
80000
100000
120000
140000
Case 1Case 2
Case 1 Case 2
Average error in Ws
5.51% 5.34%
§ For Case 2, the initial design points perturbed from that of Case 1, by random factors between 0.1 ~ 1.9
§ High fidelity runs used in error calculations
§ In average, Case 2 has the same level of error as Case 1
Change of the Initial Design Point
NSF Grant DMI-9979711
2002 NSF-SNL Grantees Meeting, 09/24/2002, Albuquerque NM 7
Errors of two optimization runs of different initial design points = W1 – W t
t = W2 – W t , (W t : unknown true optimum, expensive to calculate)
The difference of s and t is equal to W1 – W2
x = s – t = (W1 – W t) – (W2 – W t ) = W1 – W2
We can fit distribution to x instead of s or t.
• s and t are independent • Joint distribution of g(s)
and h(t)
g(s; α1)
s, t
h(t; α2)
x=s-t
∫∞
∞−
−= dsxshsgxf )()(),;( 21 αα
• MLE fit for optimization difference x using f(x)
• No need to estimate Wt
Difference fit to estimate statistics of optimization errors
NSF Grant DMI-9979711
2002 NSF-SNL Grantees Meeting, 09/24/2002, Albuquerque NM 8
Cases Case 1 Case 2 Average of Abs(W1-W2) 5941
From data 4458 4321 Error fit
(discrepancy) 4207
(-5.63%) 3952
(-8.54%) Estimate of mean, lb. Difference fit
(discrepancy) 3804
(-14.7%) 3481
(-19.4%)
From data 8383 9799 Error fit
(discrepancy) 7157
(-14.6%) 7505
(-23.4%) Estimate of STD, lb. Difference fit
(discrepancy) 9393
(12.0%) 9868
(0.704%)
p-value of χ2 test 0.5494
Estimated distribution parameters
NSF Grant DMI-9979711
2002 NSF-SNL Grantees Meeting, 09/24/2002, Albuquerque NM 9
Conclusions for Statistical Modeling of Optimization Errors
• Multiple simulation results enable statistical techniques to estimate the uncertainty level of the simulation error
• “Weibull distribution” successfully used to model the convergence error of the optimization runs
• Multiple starting points used to construct two sets of low fidelity optimizations
• “Difference fit” allowed the estimation of average errors without performing high fidelity optimizations
NSF Grant DMI-9979711
2002 NSF-SNL Grantees Meeting, 09/24/2002, Albuquerque NM 10
CFD Uncertainties
Drag polar results from 1st AIAA Drag Prediction Workshop (Hemsch, 2001)
Motivation
NSF Grant DMI-9979711
2002 NSF-SNL Grantees Meeting, 09/24/2002, Albuquerque NM 11
Objective
• Finding the magnitude of CFD simulation uncertainties that a well informed user may encounter and analyzing their sources
• We study 2-D, turbulent, transonic flow in a converging-diverging channel
• complex fluid dynamics problem• affordable for making multiple runs• known as “Sajben Transonic Diffuser”
in CFD validation studies
NSF Grant DMI-9979711
2002 NSF-SNL Grantees Meeting, 09/24/2002, Albuquerque NM 12
x/ht
y/h t
-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.00.0
0.5
1.0
1.5
2.0
2.5
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Transonic Diffuser Problem
0.3
0.4
0.5
0.6
0.7
0.8
0.9
x/ht
y/h t
-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.00.0
0.5
1.0
1.5
2.0
2.5
0.3
0.4
0.5
0.6
0.7
0.8
0.9
P/P0i
P/P0i
Strong shock case (Pe/P0i=0.72)
Weak shock case (Pe/P0i=0.82)
Separation bubble
experiment CFD
Contour variable: velocity magnitude
streamlines
NSF Grant DMI-9979711
2002 NSF-SNL Grantees Meeting, 09/24/2002, Albuquerque NM 13
Uncertainty Sources (following Oberkampf and Blottner)
• Physical Modeling Uncertainty• PDEs describing the flow
• Euler, Thin-Layer N-S, Full N-S, etc.• boundary conditions and initial conditions • geometry representation• auxiliary physical models
• turbulence models, thermodynamic models, etc.• Discretization Error• Iterative Convergence Error• Programming Errors
We show that uncertainties from different sources interact
NSF Grant DMI-9979711
2002 NSF-SNL Grantees Meeting, 09/24/2002, Albuquerque NM 14
Computational Modeling
• General Aerodynamic Simulation Program (GASP)• A commercial, Reynolds-averaged, 3-D, finite
volume Navier-Stokes (N-S) code• Has different solution and modeling options. An
informed CFD user still “uncertain” about which one to choose
• For inviscid fluxes (commonly used options in CFD)• Upwind-biased 3rd order accurate Roe-Flux scheme• Flux-limiters: Min-Mod and Van Albada
• Turbulence models (typical for turbulent flows) • Spalart-Allmaras (Sp-Al)• k-ω (Wilcox, 1998 version) with Sarkar’s
compressibility correction
NSF Grant DMI-9979711
2002 NSF-SNL Grantees Meeting, 09/24/2002, Albuquerque NM 15
Grids Used in the Computations
640 x 4005
320 x 2004
160 x 1003
80 x 502
40 x 251
Mesh Size (number of cells)
Grid level
A single solution on grid 5 requires approximately 1170 hours of total node CPU time on a SGI Origin2000 with six processors (10000 cycles)
y/ht
Grid 2
Grid 2 is the typical grid level used in CFD applications
NSF Grant DMI-9979711
2002 NSF-SNL Grantees Meeting, 09/24/2002, Albuquerque NM 16
Uncertainty in Nozzle Efficiency
+ + + + + ++ + +
+ + + + ++ + + + +
+
+
x x x x x x x x x x x xx
xx
xx
xx
x
x
o
o
+
xo +
+
x
x
Pe/P0i
n eff
0.70 0.72 0.74 0.76 0.78 0.80 0.82 0.840.700
0.725
0.750
0.775
0.800
0.825
0.850
0.875
0.900
grid 1grid 2
grid 4grid 3
grid 5
Sp-Al
NSF Grant DMI-9979711
2002 NSF-SNL Grantees Meeting, 09/24/2002, Albuquerque NM 17
Uncertainty in Nozzle Efficiency
+ + + + + + + + + + + + + ++ + + + +
+
+
x x x x x x x x x x x xx
xx
xx
xx
x
x
++ + + +
+ + + ++ + + + +
+ + ++ + +
++
+
x x x x x x x x x x x x x x x xx
xx
xx
x
x
o
o
o
o
+
xo
+
xo +
+
x
x
Pe/P0i
nef
f
0.70 0.72 0.74 0.76 0.78 0.80 0.82 0.840.700
0.725
0.750
0.775
0.800
0.825
0.850
0.875
0.900
grid 1grid 2
grid 4grid 3
grid 5
Sp-Al k-ω
NSF Grant DMI-9979711
2002 NSF-SNL Grantees Meeting, 09/24/2002, Albuquerque NM 18
Discretization Error by Richardson’s Extrapolation
)( 1+++= ppexactk hOhff α
0.154
0.463
1.452
4.431
0.8291.660.82
(weak shock)
k- ω
0.404
1.123
3.542
8.001
0.8111.580.82
(weak shock)
Sp-Al
1.094
2.723
6.792
14.301
0.7201.320.72
(strong shock)
Sp-Al
Discretization error (%)
Grid level
estimate of (neff)exact
estimate of p (observed order of
accuracy)
Pe/P0iTurbulence model
order of the method
a measure of grid spacing grid level
error coefficient
NSF Grant DMI-9979711
2002 NSF-SNL Grantees Meeting, 09/24/2002, Albuquerque NM 19
x/ht
y/h
t
-1.0 -0.5 0.0 0.5 1.0 1.5 2.00.98
0.99
1.00
1.01
1.02
1.03
1.04
1.05
1.06
1.07
1.08
1.09
modified experimentaldata points
upper wall contour obtainedwith the analytical equation
upper wall contour of the modified-wallgeometry (cubic-spline fit to the modified data points)
upper wall contour of the modified-wallgeometry (cubic-spline fit to the data points)
x/ht
y/h t
-4 .00 -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00
1.00
1.10
1.20
1.30
1.40
1.50 upper wall contour of the modified-wallgeometry (cubic-spline fit to the data points)
upper wall contour obtainedwith the analytical equation
Error in Geometry Representation
NSF Grant DMI-9979711
2002 NSF-SNL Grantees Meeting, 09/24/2002, Albuquerque NM 20
Downstream Boundary Condition
x/hty/
h t
0.0 2.0 4.0 6.0 8.0 10.0 12 .0 14.00.5
1.0
1.5
2.0Extended geometry, Sp-Al, Van Albada, grid 3ext, Pe/P0i=0.7468
x/ht
y/h t
0.0 2.0 4.0 6.0 8.0 10.0 12 .0 14.00.5
1.0
1.5
2.0Extended geometry, Sp-Al, Van Albada, grid 3ext, Pe/P0i=0.72
x/ht
y/h t
0.0 2.0 4.0 6.0 8.0 10.0 12 .0 14.00.5
1.0
1.5
2.0Original geometry, Sp-Al, Van Albada, grid 3, Pe/P0i=0.72
x/ht
y/h
t
2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.51.0
1.1
1.2
1.3
1.4
1.5
Extended geometry, Sp-Al,Van Albada, grid 3ext, Pe/P0i=0.72
Original geometry, Sp-Al, Van Albada, grid 3, Pe/P0i=0.72
Extended geometry, Sp-Al, Van Albada,grid 3ext, Pe/P0i=0.7468
x/ht
y/h
t
2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.51.0
1.1
1.2
1.3
1.4
1.5
x/ht
y/h
t
2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.51.0
1.1
1.2
1.3
1.4
1.5
NSF Grant DMI-9979711
2002 NSF-SNL Grantees Meeting, 09/24/2002, Albuquerque NM 21
2% (grid 2)
9% (grid 4)
the relative uncertainty due to the selection of turbulence model
1.4%(grid 3, k-ω)
0.5%(grid 3, k-ω)
the uncertainty due to the error in geometry representation
1.1%(grid 2, Sp-Al)
0.8%(grid 3, Sp-Al)
the uncertainty due to the change in exit boundary location
3.5% (Sp-Al)
6% (Sp-Al)
the difference between grid level 2 and grid level 4
4%10%the total variation in nozzle
efficiency
Uncertainty Comparison in Nozzle EfficiencyStrong Shock
Weak ShockMaximum value of
NSF Grant DMI-9979711
2002 NSF-SNL Grantees Meeting, 09/24/2002, Albuquerque NM 22
Conclusions for CFD Uncertainties• Based on the results obtained from this study,
• Informed users may get large errors for the cases with strong shocks and substantial separation
• Systematic uncertainty (discretization error and turbulence models) large compared to numerical noise
• Grid convergence not achieved with grid levels that have moderate mesh sizes
• Uncertainties from different sources interact, especially in the simulation of flows with separation
• We should asses the contribution of CFD uncertainties to design problems that include the simulation of complex flows
NSF Grant DMI-9979711
2002 NSF-SNL Grantees Meeting, 09/24/2002, Albuquerque NM 23
Publications 1. Hosder, S., Grossman, B., Haftka, R. T., Mason, W. H., and Watson, L. T., “Observations on CFD
Simulation Uncertainties,” Proceedings of the 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Paper No. AIAA-2002-5531, Atlanta, GA, Sept. 2002.
2. Kim, H., Papila M., Haftka, R. T., Mason, W. H., Watson, L. T., and Grossman, B., “Estimating Optimization Error Statistics via Optimization Runs From Multiple Starting Points,” Proceedings of the 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Paper No. AIAA-2002-5576, Atlanta, GA, Sept. 2002.
3. Kim, H., Mason, W. H., Watson, L. T., Grossman, B., Papila, M., and Haftka, R. T., "Protection Against Modeling and Uncertainties in Design Optimization," in Modeling and Simulation-Based Life Cycle Engineering, eds.: K. Chung, S.Saigal, S. Thynell and H. Morgan, Spon Press, London and New York, 2002, pp. 231-246.
4. Hosder, S., Watson, L. T., Grossman, B., Mason, W. H., Kim, H., Haftka, R. T., and Cox, S. E., “Polynomial Response Surface Approximations for the Multidisciplinary Design Optimization of a High Speed Civil Transport,” Optimization and Engineering, Vol. 2, No. 4, December 2001, pp.431-452.
5. Kim, H., Papila M., Mason, W. H., Haftka, R. T., Watson, L. T., and Grossman, B., “Detection and Repair of Poorly Converged Optimization Runs,” AIAA Journal, Vol. 39, No. 12, 2001, pp. 2242-2249.
6. Kim H., “Statistical Modeling of Simulation Errors and Their Reduction via Response Surface Techniques,” Ph.D Dissertation, Virginia Tech., July 2001.
7. Kim, H., Haftka, R. T., Mason, W. H., Watson, L. T., and Grossman, B., “A Study of the Statistical Description of Errors from Structural Optimization,” Proceedings of the 8th AIAA/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Paper No. AIAA-2000-4840-CP, Long Beach, CA, Sept. 2000.
NSF Grant DMI-9979711
2002 NSF-SNL Grantees Meeting, 09/24/2002, Albuquerque NM 24
Publications (continued)8. Kim, H., Papila M., Mason, W. H., Haftka, R. T., Watson, L. T., and Grossman, B., “Detection and
Correction of Poorly Converged Optimizations by Iteratively Re-weighted Least Squares,” Paper AIAA-2000-1525, 41st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Atlanta, GA, April 3-6, 2000.