AIAA 2002-5531 9 th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 1 AIAA 2002-5531 OBSERVATIONS ON CFD SIMULATION UNCERTAINTIES Serhat Hosder, Bernard Grossman, William H. Mason, and Layne T. Watson Virginia Polytechnic Institute and State University Blacksburg, VA Raphael T. Haftka University of Florida Gainesville, FL 9 th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization 4-6 September 2002 Atlanta, GA
33
Embed
AIAA 2002-5531 OBSERVATIONS ON CFD SIMULATION UNCERTAINTIES
AIAA 2002-5531 OBSERVATIONS ON CFD SIMULATION UNCERTAINTIES. Serhat Hosder, Bernard Grossman, William H. Mason, and Layne T. Watson Virginia Polytechnic Institute and State University Blacksburg, VA Raphael T. Haftka University of Florida Gainesville, FL - PowerPoint PPT Presentation
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
AIAA 2002-5531
9th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 1
AIAA 2002-5531OBSERVATIONS ON CFD SIMULATION
UNCERTAINTIES
Serhat Hosder, Bernard Grossman, William H. Mason, and Layne T. Watson
Virginia Polytechnic Institute and State University Blacksburg, VA
Raphael T. HaftkaUniversity of Florida
Gainesville, FL
9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization
4-6 September 2002Atlanta, GA
AIAA 2002-5531
9th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 2
Introduction
• Computational fluid dynamics (CFD) as an aero/hydrodynamic analysis and design tool
• CFD being used increasingly in multidisciplinary design and optimization (MDO) problems
• CFD results have an associated uncertainty, originating from different sources
• Sources and magnitudes of the uncertainty important to assess the accuracy of the results
AIAA 2002-5531
9th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 3
• Design uncertainties due to computational simulation error
• Optimization is an iterative procedure subject to convergence error.
• Estimating convergence error may require expensive accurate optimization runs
• Many simulation runs are performed in engineering design. (e.g., design of experiments)
• Statistical analysis of error
Motivation
AIAA 2002-5531
9th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 4
• Find error characteristics of a structural optimization of a high speed civil transport
• 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
• Improve response surface approximation against erroneous simulation runs via robust regression
Objectives
AIAA 2002-5531
9th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 5
+ 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
• Take off gross weight (WTOGW) is minimized
• Up to 29 configuration design variables including wing, nacelle and fuselage geometry, fuel weight, and flight altitude
• For this study, a simplified 5DV version is used
High Speed Civil Transport (HSCT)
AIAA 2002-5531
9th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 6
• For each HSCT configuration, wing structural weight (Ws) is minimized by structural optimization (GENESIS) with 40 design variables
• Structural optimization is performed a priori to build a response surface approximation of Ws
Structural optimization of HSCT
AIAA 2002-5531
9th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 7
design
Op
timu
mw
ing
stru
ctu
ralw
eig
ht
(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 point was perturbed from that of Case 1, by factors between 0.1 ~ 1.9
In average, Case 2 has the same level of error as Case 1
The errors were calculated with respect to higher fidelity runs with tightened convergence criteria
Effects of initial design point on optimization error
AIAA 2002-5531
9th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 8
• To get optimization error we need to know true optimum, which is rarely known for practical engineering optimization
• Optimization error = OBJ* - OBJ*true
• To estimate OBJ*true
• Find convergence setting to achieve very accurate optimization
• This approach can be expensive
Estimating error of incomplete optimization
AIAA 2002-5531
9th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 9
x
f(x)
0 1 2 3 4 50.0
1.0
2.0
3.0
= 0.5, = 1 = 1.0, = 1 = 2.0, = 1 = 4.0, = 1
• Widely used in reliability models (e.g., lifetime of devices)• Characterized by a shape parameter and a scale parameter
otherwise
xifx
x
0
0exp1
2
)1
(1
)2
(2
)1
(
22 Variance
Mean
PDF function according to
The Weibull distribution model
AIAA 2002-5531
9th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 10
• Maximum likelihood estimation (MLE) to estimate distribution parameters of the assumed distribution.
• Find to maximize the likelihood function.
2 goodness of fit test
• Comparison of histograms between data and fit
• p-value indicates quality of the fit
nx
xfl
i
n
ii
size of sample for
1
);()(
Estimation of distribution parameters
AIAA 2002-5531
9th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 11
Errors of two optimization runs of different initial design point
s = W1 – Wt
t = W2 – Wt , (Wt is unknown true optimum.)
The difference of s and t is equal to W1 – W2
x = s – t = (W1 – Wt) – (W2 – Wt ) = 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; 1)
PDF
s, t
h(t; 2)
x=s-t
dsxshsgxf )()(),;( 21
• MLE fit for optimization difference x using f(x)
• No need to estimate Wt
Difference fit to estimate statistics of optimization errors
AIAA 2002-5531
9th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 12
Error in Ws (%)
Fre
qu
en
cie
s
0 10 20 30 40 50 60 700
20
40
60
80
DataExpected by error fitExpected by difference fit
• Difference fit is applied to the differences between Cases 1 and 2, assuming the Weibull model for both cases
• Even when only inaccurate optimizations are available, difference fit can give reasonable estimates of uncertainty of the optimization
Error in Ws (%)
Fre
qu
en
cie
s
0 10 20 30 40 50 60 70 800
20
40
60
80
100
DataExpected by error fitExpected by difference fit
Comparison between error fit and difference fit
AIAA 2002-5531
9th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 13
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 2 test 0.5494
• Another set of optimization with a difference initial design point was enough, which is straightforward and no more expensive
• The difference fit gave error statistics as accurate as the error fits involving expensive higher fidelity runs
Estimated distribution parameters
AIAA 2002-5531
9th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 14
• Robust regression techniques• Alternative to the least squares fit that may be
greatly affected by a few very bad data (outliers)• Iteratively reweighted least squares (IRLS)
• Inaccurate response surface approximation is another error source in design process Improve response surface approximation by
repairing outliers
Outliers and robust regression
AIAA 2002-5531
9th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 15
• Weighted least squares: Initially all points have same weight of 1.0.
• At each iteration, the weight is reduced for points that are far from the response surface.
• This gradually moves the response surface away from outliers.
• IRLS approximation leaves out detected outliers
x
y
IRL
Sw
eig
htin
g
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
1.2
1.4
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5DataLSIRLSweighting
Outliers
1 dimensional example
Iteratively Reweighted Least Squares
AIAA 2002-5531
9th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 16
• Regular weighting functions• Symmetrical• Huber’s function gives non-
zero weighting to big outliers
• Biweight rejects big outliers completely
• Nonsymmetric weighting function• make use of one-sidedness
of error by penalizing points of positive residual more severelyResidual, r
We
igh
ting
,w(r
)
-4 -2 0 2 40
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6HuberBiweightNIRLS
Weighting functions of IRLS
Nonsymmetric weighting function
Aggressive search by reducing tuning constant B
AIAA 2002-5531
9th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 17
RMSE(lb.) R2 Before repair
6755 (8.7%)
0.9297
IRLS repair
3257 (4.7%)
0.9769
NIRLS repair
3042 (4.1%)
0.9824
All 117 pts repaired
2578 (3.5%)
0.9879
Design
Ws
(lb
)
5 10 15 200
20000
40000
60000
80000
100000
120000
140000
GENESIS: RepairedRS: Without repairRS: IRLS repairRS: NIRLS repair
Comparison of RS before and after repair
Improvements of RS approximation by outlier repair
AIAA 2002-5531
9th AIAA/ISSMO Symposium on MAO, 09/05/2002, Atlanta, GA 18