Download Reliability-Based Design Optimization with Confidence

6th China-Japan-Korea Joint Symposium on Optimization of Structural and Mechanical Systems
June 22–25, 2010, Kyoto, Japan
Reliability-Based Design Optimization with Confidence Level for Problems with Correlated
Input Distributions
K.K. Choi, Yoojeong Noh, and Ikjin Lee
Department of Mechanical & Industrial Engineering, College of Engineering
The University of Iowa, Iowa City, IA 52242, USA
[email protected], [email protected], [email protected]
Abstract
For obtaining a correct reliability-based optimum design, an input distribution model, which includes marginal and joint
distributions of input random variables with their associated input parameters such as mean, standard deviation, and
correlation coefficient, needs to be accurately estimated. In engineering reliability-based design optimization (RBDO)
applications, most input distributions arise from loading, material properties, and manufacturing geometric variabilities.
However, only limited data on these input random variables are available due to expensive experimental testing or
measuring costs. The input distribution model estimated from the insufficient data could be inaccurate, which will lead to
the incorrect reliability-based optimum design. Thus, RBDO with the confidence level has been addressed to offset the
inaccurate estimation of the input distribution model by using an adjusted upper bound of confidence interval of the
standard deviation, and an adjusted correlation coefficient.
Keywords: Reliability-based design optimization, input distribution model, confidence level, bootstrap method, Adjusted
Parameters
1. Introduction
In many engineering applications, since input random variables such as fatigue material properties are correlated [1-4],
the input statistical uncertainty needs to be modeled by identifying the input joint cumulative distribution functions
(CDFs) as well as marginal CDFs. In addition, the input physical uncertainty needs to be modeled by quantifying the
input parameters, such as mean, standard deviation, and correlation coefficient, of the identified marginal and joint
distributions. In industrial applications, since only limited data on input variables is available due to expensive
experimental testing or measuring costs, it is difficult to accurately model the input distributions, and thus, the design
engineer will not have confidence on the RBDO result obtained using the input distribution model [5] to meet the target
reliability.
In this paper, to offset the possible inexact identification of the input joint and marginal CDFs and inexact
quantification of the parameters, an input distribution model with a target confidence level is developed for RBDO by
appropriately enlarging the estimated t-contour, for the given the target reliability index t, using the confidence intervals
of the estimated parameters. If the input variables have marginal Gaussian distributions, the confidence intervals of mean
and standard deviation can be obtained explicitly, which are accurate [6]. On the other hand, if the input variables have
non-Gaussian marginal distributions, a bootstrap method [7,8] is used to predict the confidence intervals of the input
parameters for input variables with non-Gaussian marginal distributions. Once these confidence intervals are obtained,
instead of directly using the estimated upper or lower bounds, an adjusted upper bound of confidence interval of the
standard deviation and an adjusted correlation coefficient calculated using confidence interval of the correlation
coefficient are proposed.
A mathematical example with Gaussian and non-Gaussian correlated variables are used to illustrate how the
proposed input distribution model with target confidence level provides the RBDO design that meets the target
confidence level.
2. Generation of Input Distribution Model
If the input random variables are independent, the joint distribution can be obtained easily by multiplying the marginal
distributions. However, if the input random variables are correlated, the joint distribution needs to be generated. Since only
limited data is available in practical engineering applications, it is difficult to obtain the joint distribution directly from limited
data. Thus, the copula [9-15], which is a function of marginal distributions and correlation parameters, is used to model the
input joint distribution.
2.1 Identification of Input Distribution Model
Copulas are multivariate distribution functions whose one-dimensional margins are uniform on the interval [0, 1].
According to Sklar’s theorem [11], a joint cumulative distribution function (CDF) FX1 X n  x1 ,, xn  of random variables
Xi can be expressed in terms of marginal CDFs FX i  xi  of Xi for i=1,…,n, and copula function C as

FX1 ,..., X n  x1 ,..., xn   C FX1  x1  ,..., FX n  xn  θ

(1)
where  is the matrix of correlation parameters among X1,…,Xn. The dimension of the correlation matrix  used in
different copulas may or may not depend on the number of correlated random variables [5]. It has been observed that two
input variables are correlated in many cases [7-10], so that only bivariate copulas are considered in this paper. In this case,
 is a scalar quantity.
For a given limited data, the joint distribution can be identified by a one-step procedure, which directly tests all candidate
joint distributions; or by a two-step procedure, which first identifies marginal distributions and then a copula [9,10]. In
identifications of the input distributions, it is shown in Ref. 10 that the two-step approach is more efficient and accurate than the
one-step approach. Regarding the measure of identification, the weight-based Bayesian method [9,10] and Markov chain
Monte Carlo (MCMC)-based Bayesian method [16] can be used. It is shown that the weight-based method is much more
efficient than the MCMC-based method [10]. Thus, in this paper, the two-step weight-based Bayesian method is used to
identify the input distribution model.
2.2 Quantification of Input Distribution Model
With the marginal and joint CDF types are identified, it is necessary to determine their parameters such as means and
standard deviations of the marginal distributions, and the correlation coefficient. The parameters obtained from limited
data are the sample mean
ns
xi
 
(2)
i 1 ns
and the sample variance
1 ns
2
 
(3)
 xi   2
ns  1 i 1
On the other hand, all two-parameter marginal distributions, except the Gaussian distribution, which are used in
many engineering applications, have their own parameters a and b, which can be expressed explicitly in terms of the mean
and standard deviation ( and ) as shown in Table 3 in Ref. 9. Since the mean and standard deviation determine the
location and variability of the distribution, respectively, the standard deviation can be used to enlarge the t-contour of the
distribution. Accordingly, once the sample mean and standard deviation are calculated from given data, the parameters a
and b of the identified input distributions can be calculated using explicit functions.
Since copula functions have their own correlation parameters, it is desirable to use a common correlation coefficient.
The Kendall’s tau, which is a widely used correlation coefficient [9], is used in this paper. The sample version of
Kendall’s tau is expressed as
cd
 ns 
(4)
 c  d  /  
 
cd
2
can be used, where c and d are the numbers of concordant and discordant pairs, respectively, and ns is the number of
samples. Once the sample version of Kendall’s tau is obtained from Eq. (5), the correlation parameter  can be obtained
using explicit formulas   g ( ) presented in Table 1 in Ref. 9.


3. Generation of Input Distribution Models with Target Confidence Level
In this paper, the output confidence level of the RBDO designs is defined as the probability of RBDO designs, which are
obtained using input distribution models, meet the target reliability. In this definition, the input distribution models,
which are generated using limited test data sets, are treated as random events. However, direct assessment of the
confidence level of the RBDO design is a very difficult undertaking. Thus, in this paper, it is proposed to generate the
input distribution model that provides confidence level. If the t-contour of the proposed input distribution model, which
is obtained using the adjusted standard deviation and correlation coefficient, covers the true t-contour, then the RBDO
design will satisfy the target reliability since the Most Probable Point (MPP) of the RBDO design is on the t-contour. To
generate an input distribution model with appropriately enlarged t-contour to cover the true t-contour with a certain
confidence level, information of the confidence intervals of the estimated parameters of marginal distributions are used.
3.1 Dependence of t-contour on Marginal Input Distribution Parameters
To make the estimated t-contour cover the true t-contour, it is necessary to understand how the input parameters affect
the estimated t -contour shapes. First, consider the independent input random variables, where the joint distribution can be
obtained easily by multiplying the marginal distributions. If we can assume the sample mean is accurate, then a larger
standard deviation can be used to enlarge the estimated t-contour. On the other hand, a larger or smaller sample mean is
related to the position of the estimated t-contour. Thus, enlargement of the estimated t-contour also needs to consider
inaccuracy of the sample mean obtained from the data. For this purpose, we need first obtain the confidence intervals of
the sample mean and sample standard deviations, which will be discussed in Sections 3.2.
Second, if the random variables are correlated, then the problem is more complicate since we have to deal with the
correlation coefficient. Like the mean, neither larger nor smaller sample correlation coefficient makes the estimated
t-contour fully covers the true t-contour. Therefore, instead of using the estimated mean and correlation coefficient,
adjusted parameters using the confidence intervals of the mean and correlation coefficient are proposed to ensure the
estimated t-contour with the adjusted parameters covers the true t-contour, which will lead to a desirable confidence
level of the input model.
3.2 Confidence Intervals of Input Parameters for Gaussian and Non-Gaussian Marginal Distributions
Since only limited data are available, the sample mean  in Eq. (2) and sample standard deviation  from Eq. (3) of the
marginal distributions are used to estimate the population mean  and population standard deviation . A range of values
in which they may be located is called the confidence interval.
3.2.1 Confidence Interval of Sample Mean
If the underlying random variable is Gaussian, the two sided confidence interval that the population mean belongs to this
interval with the confidence level of 100(1)% is [5,6]

 

(5)
Pr    t / 2, ns 1
     t / 2, ns 1
  1
ns
ns 

where ns is the number of samples and t / 2,ns 1 are the values of Student’s t-distribution with (ns1) degree of freedom
evaluated at probabilities of (1/2) and /2, respectively. Note that, as the number of samples increases, the lower and
upper bounds of the confidence interval of the sample mean approach the population mean. If input variables follow
Gaussian distribution, Eq. (5) is accurate. However, if not, the estimated confidence interval may not be accurate. To
circumvent this, a bootstrap method, which does not require the normality assumption on the input variable can be used.
In this study, since the confidence interval of the mean is rather accurately calculated even for the non-Gaussian
distribution [17,18], the bootstrap method is used only for confidence interval of the standard deviation of non-Gaussian
distribution.
3.2.2 Confidence Interval of Sample Standard Deviation
Using a similar procedure of calculating the confidence interval of the mean, if the underlying random variable is
Gaussian, the two sided confidence interval that the population variance belongs to this interval with the confidence level
of 100(1) is [5,6]
  ns  1  2
 ns  1  2 
(6)
2 
Pr 
  1
c / 2, ns 1 
 c1 / 2, ns 1
where c / 2,ns 1 and c1 / 2, ns 1 are the values of the chi-square distribution with (ns1) degrees of freedom evaluated at
probabilities of /2 and (1/2), respectively. Thus, the lower and upper bounds of the 100×(1−α)% confidence interval
for the sample standard deviation are respectively calculated as
 1L 
 ns  1  2
c1 / 2, ns 1
and  1U 
 ns  1  2
c / 2, ns 1
(7)
If input variables follow Gaussian distribution, Eq. (7) is accurate. However, if not, the estimated confidence interval
is not accurate. Thus, the bootstrap method, which does not use the Gaussian distribution of input variables to calculate
the confidence interval of the mean and standard deviation, needs to be used. The bootstrap method calculates the
confidence interval of estimated standard deviation  by constructing a distribution of the standard deviation using the
frequency distribution of  * obtained from randomly generated bootstrap samples based on the given data.
For the bootstrap method, the first step is to construct an empirical distribution Fns  x  or a parametric distribution
F  x a, b  from given samples x   x1 , x2 ,, xns  . In the second step, bootstrap samples are generated from the
constructed empirical distribution or parametric distribution. If a random sample of size ns with replacement is drawn
from the empirical distribution Fns  x  , then this is called a non-parametric approach. If the resample is drawn from the
parametric distribution F  x a, b  , this is called a parametric approach. In this paper, the parametric distribution type is
used for the two-step weight-based Bayesian method [10]. The third step is to calculate  from the bootstrap samples,
yielding  bs* . In the fourth step, the second and third steps are repeated B times (e.g., B=1000). Finally, the fifth step is to
construct a probability distribution from 1* ,  2* , ,  B* . This distribution is the bootstrap sampling distribution of  ,
G *  *  , which is used to calculate the confidence interval of  . To obtain the bootstrap sampling distribution of  ,
there are five different methods: the normal approximation, percentile, bias corrected (BC), percentile-t, or bias corrected
accelerated (BCa) methods.
The normal approximation method assumes that the distribution of  is a Gaussian distribution [19]. The
percentile method calculates the confidence interval for the parameter based on the bootstrap sampling distribution
G *  *  approximating the population distribution G   [20]. When the number of samples is small, G *  *  might be
a biased estimator of G   , i.e.,  * is a biased estimator of  . In that case, the percentile method can be inaccurate.
The bias corrected (BC) method corrects the bias term by introducing an adjusted parameter [19]. The BC method
corrects the bias term, but it still requires the parametric assumption that there exist monotonic transformations of  * and
 . The bias corrected and accelerated (BCa) method generalizes the BC method. The BC method only corrects the bias,
whereas the BCa method corrects both the bias and the skewness. The percentile-t method uses the distribution of a
standardized estimator to calculate the confidence interval. The percentile-t interval is expected to be accurate to the
extent that standardizing depends less on the boot sampling estimator,  * , than the percentile method.
These five methods are tested using a lognormal marginal distribution as the true model. The confidence level of the
standard deviation is assessed by calculating the probability that the upper bound of the confidence interval of the
standard deviation is larger than the true standard deviation over M data sets with ns samples generated from the true
model. It is the most desirable if the estimated confidence level is close to the target confidence level. However, even if
the estimated confidence level is close to the target confidence level, an unnecessarily large upper bound for standard
deviation is not desirable because it will yield unnecessarily conservative RBDO design. Therefore, the most desirable
confidence interval should just include the true standard deviation for the target confidence level.
In summary, when the input marginal distribution is not Gaussian, the parametric percentile method has the most
desirable performance. Even though the bootstrap methods do not achieve the target confidence level for a small number
of samples, as the number of samples increases, the obtained confidence levels tend to converge to the target confidence
level while the method using Gaussian distribution of input variable does not. The bootstrap method can be applied to any
types of distribution, and the test results for various types of distributions are presented in Ref. 18.
3.2.3 Confidence Interval of Sample Correlation Coefficient
Genest and Rivest [21] showed that, as ns goes to infinity, the sample correlation coefficient follows a Gaussian
distribution as
 1  dg 1   2 


 ~ N  , 4w
(8)
 
 ns 
d  


1
where   g ( ) is presented in Table 1 in Ref. 9, w 
1 ns
1 ns
1 ns
2
 wi  w i  2w  , wi   Iij , w i   I ji , and

ns j 1
ns j 1
ns i 1
1 ns
 wi . If x1j<x1i or x2j<x2i (x1i or and x2i are ith sample point for X1 and X2), then Iij=1, otherwise, Iij=0. Thus, the
ns i 1
confidence interval for the correlation parameter for 100(1) of the confidence level is obtained as
w
Pr   h      h   1  
where h  z / 2
1
ns
4w
(9)
dg 1  
and z / 2 is the CDF value of Gaussian distribution evaluated at  / 2 .
d
Using the lower and upper bounds of the confidence interval for the correlation parameter  , the upper and lower
bounds of the confidence interval for the correlation coefficient  can be calculated from   g   . The confidence
interval of correlation parameter can be accurately estimated regardless of copula function types; and thus, the bootstrap
method is not required to obtain the confidence interval of the correlation parameter.
4. Generation of Input Distribution Models with Target Confidence Level
To have an input model with a target confidence level, confidence intervals of the input parameters need to be used to
offset the prediction error of the estimated input parameters. However, neither the upper nor lower bounds of the
confidence intervals of mean and correlation yield reliable design. On the other hand, if the sample mean and sample
correlation coefficient are accurate, the upper bound of the confidence interval of the standard deviation fully covers the
t-contour, and thus, it can be readily used to obtain the t-contour covering the true t-contour, which will lead to a
reliable optimum design.
However, the prediction errors in the mean and correlation coefficient exist in RBDO problems when the available
data is insufficient. Therefore, instead of using the estimated mean and correlation coefficient, adjusted parameters using
the confidence intervals of the mean and correlation coefficient are proposed to ensure that the estimated t-contour with
the adjusted parameters covers the true t-contour, which will lead to a desirable confidence level of the input model.
Since the confidence interval of the mean cannot be used to enlarge t -contour, the change  in the sample
standard deviation caused by the change  in the sample mean is added to the confidence interval of the standard
deviation  U to obtain the adjusted standard deviation  A as
 A   U     U 

 

(10)
 
, and    U       L since the sample mean is the middle point of the

 
confidence interval for the Gaussian distribution. In Eq. (10), the COV (  /  ) works as a scale factor such that the effect
where it is assumed that
of the confidence interval of the mean on the adjusted standard deviation is proportional to  /  . Using the estimated
mean and the adjusted standard deviation, parameters a and b of the identified distribution can be estimated using explicit
functions, which are expressed as mean and standard deviation in Ref. 9.
Like the mean, the t-contours for the lower and upper bounds of the correlation coefficient do not yield reliable
design when the sample standard deviation is not large enough to cover the true contour. To resolve this problem, the
adjusted correlation coefficient is used. As stated before, neither the lower nor upper bounds of the confidence interval of
correlation coefficient cover the true t-contour. However, when the lower bound of the correlation coefficient with the
adjusted standard deviation is used, it yields more reliable design than when the upper bound with the adjusted standard
deviation is used. Thus, when the true correlation coefficient is small, the adjusted correlation coefficient needs to be
close to the estimated one. Otherwise, the adjusted correlation coefficient needs to be underestimated. Thus, the adjusted
correlation coefficient is proposed as
(11)
 A      max U   ,    L


such that it can be applied to both small and large correlation coefficients. As the number of samples increases, the
adjusted correlation coefficient converges to the true correlation coefficient.
5. Numerical Example
In this section, a mathematical example with correlated Gaussian and non-Gaussian input variables are used to show how
the proposed methods yield reliable designs to meet the target confidence level of the output. To carry out RBDO, the
MPP-based DRM [22] is used to more accurately calculate the probability of failure than the FORM.
The RBDO problem is formulated to
minimize cost(d)  d1  d 2
subject to P  Gi  X   0   PFTar
( 2.275%), i  1, 2,3
i
d = μ  X  , 0  d1 , d 2  10
G1 ( X)  1   0.9010 X 1  0.4339 X 2  1.5 
2
(12)
  0.4339 X 1  0.9010 X 2  2  / 20
G2 ( X)  1   X 1  X 2  2.8  / 30   X 1  X 2  12  / 120
2
2
G3 ( X)  1  200 / {2.5(0.9010 X 1  0.4339 X 2  3) 2
 8  0.4339 X 1  0.9010 X 2   5}
First, let X1 and X2 have Gaussian distributions, X1 and X2~(3, 0.32), which are correlated with the Frank copula and 
= 0.8. For the comparison study, three types of input models – one model with estimated parameters (without confidence
level), another model with upper bound for standard deviation (only with confidence level for standard deviation) and
sample correlation coefficient, and the other model with adjusted parameters (with confidence levels for all input
variables) – are tested. The output confidence levels are assessed using 100 data sets with ns=30, 100, and 300, which are
randomly generated from the true input model. The marginal distribution, the copula type, and their parameters are
determined from each data set. Using the generated input models from 100 data sets, the RBDO is carried out. The target
probability of failure is given as 2.275% and the target confidence level is 97.5%.
Table 1 shows the minimum, mean, and maximum values of the probabilities of failure PF1 and PF2 for two active
constraints evaluated at the optimum designs using the Monte Carlo simulation (MCS). The output confidence levels are
estimated by calculating the probability that the obtained probability of failure is smaller than the target probability of
failure. As shown in Table 1, when the input model with the estimated parameters is used for ns=30, the mean values of
PF1 and PF2 (3.426% and 2.594%, respectively) are larger than the target probability of failure 2.275%. In particular,
the maximum values of PF1 and PF2 (14.29% and 7.752%, respectively) are much larger than 2.275%. Thus, the
confidence levels of the output performance are significantly lower than the target confidence level 97.5% as shown in
Table 1.
On the other hand, when the input model with the upper bound of the standard deviation and sample correlation is
used for ns=30, the mean values of PF1 and PF2 (1.074% and 1.562%, respectively) are smaller than 2.275%, which
yield more reliable design than those using the estimated parameters. The obtained output confidence levels are improved
over those using estimated parameters. However, they have not reached the 97.55% target confidence level.
When the input model with the adjusted parameters is used, the obtained output confidence levels using the adjusted
parameters become much closer to the target confidence level. Even though it offsets the quantification error of input
parameters, the estimated output confidence levels are still smaller than the target confidence level due to the incorrect
identification of marginal distribution and copula for a small number of samples, ns=30. However, as the number of
samples increases, the minimum, mean, and maximum values of PF1 and PF2 using the input model with the adjusted
parameters approaches the target probability of failure from a more reliable side than those using other input models.
Moreover, the input model with adjusted parameters provides better output confidence levels than other input models.

Table 1. Probabilities of Failure and Output Confidence Levels for X 1 and X 2 ~ N 3, 0.32
Par.
ns
Min
Mean
30
Max
Conf.
Min
Mean
100
Max
Conf.
Min
Mean
300
Max
Conf.
Estimated
PF1
PF2
0.142
3.426
14.29
35
0.640
2.279
5.829
37
0.854
2.394
3.835
43
0.315
2.594
7.752
54
0.516
2.087
5.941
65
1.064
1.924
3.599
77
Upper Std.
PF1
PF2
0.007
1.074
5.918
83
0.231
1.425
3.694
89
0.510
1.615
2.846
94
0.119
1.562
4.719
81
0.393
1.573
4.091
88
0.894
1.624
3.150
90

Adjusted
PF1
PF2
0.004
0.908
7.082
90
0.192
1.282
3.445
93
0.473
1.518
2.732
95
0.026
0.444
2.595
98
0.255
0.947
2.300
99
0.719
1.262
2.285
98
*Optimum Design Using True Input Model: (2.026, 1.163) ; PF = 2.245%, PF =2.070%
1
2
To test the effect of larger standard deviation on the output confidence level, consider a true input model with X 1
2
2
~ N (3, 0.3 ) and X 2 ~ N (3,1.5 ) . As shown in Table 2, the input model with the estimated parameters still yields
large probabilities of failure and small output confidence levels. The input model with the upper bound for standard
deviation and the sample correlation coefficient provides better results than the one with the estimated parameters, but the
output confidence levels are not as good as the one with the adjusted parameters, especially for a small number of
samples. For ns=300, the output confidence level using the adjusted parameters is larger than the target confidence level
of 97.5%, which means it yields somewhat conservative optimum designs. This is because, for large standard deviation
(i.e.,   1.5 ), the confidence intervals of the mean and standard deviations are slowly converged rather than for small
standard deviation. Even though the output confidence levels are high, the minimum, mean, and maximum values of
probabilities of failure tend to approach the target probability of failure 2.275%, as the number of samples increases as
shown in Table 2.
2
2
Table 2. Probabilities of Failure and Output Confidence Levels for X 1 ~ N (3, 0.3 ) and X 2 ~ N (3,1.5 )
ns
Par.
Min
Mean
30
Max
Conf.
Min
Mean
100
Max
Conf.
Min
Mean
300
Max
Conf.
Estimated
PF1
PF2
0.293
2.583
10.92
58
1.035
2.261
8.804
66
1.140
2.029
3.299
78
0.045
2.895
9.732
48
0.871
2.338
5.058
56
1.238
2.185
3.879
65
Upper Std.
PF1
PF2
0.023
0.992
6.961
90
0.507
1.309
6.564
95
0.791
1.514
2.607
98
0.001
1.256
4.723
94
0.296
1.117
2.880
96
0.746
1.468
2.827
98
Adjusted
PF1
PF2
0.001
0.313
4.324
96
0.196
0.715
3.385
99
0.530
1.076
1.984
100
0.001
0.380
4.457
96
0.121
0.667
1.985
100
0.512
1.100
2.330
99
*Optimum Design Using True Input Model: (2.977, 4.280) ; PF =2.075%, PF = 2.213%
1
2
In summary, the input model with the adjusted parameters has the most desirable confidence level of output
performance compared with input models with the estimated parameters and those with the upper bound for standard
deviation and the sample correlation coefficient.
Next, let X1 and X2 have lognormal and Gaussian distributions, X 1 ~ LN  3,1.52  and X 2 ~ N  3, 0.32  , which are
correlated with the Frank copula and   0.7 . Table 3 shows the minimum, mean, and maximum values of the
probabilities of failure PF1 and PF2 for two active constraints evaluated at the optimum designs. As shown in Table 3,
when the input model with the estimated parameters is used for ns=30, the mean value of PF1 (4.313%) is larger than the
target probability of failure, 2.275%. The maximum value of PF1 (20.842%) is not even close to 2.275% due to the
wrong identification and quantification of the input model. Thus, the output confidence level (54%) is significantly
smaller than the target confidence level of 97.5%.
On the other hand, when the input model with the adjusted parameters is used, the mean values of PF1 and PF2 are
smaller than 2.275%. Accordingly, the obtained output confidence levels using the adjusted parameters become much
closer to the target confidence level, 97.5%. When the number of samples is small, e.g., ns=30, the estimated output
confidence levels using the bootstrap method are still smaller than the target confidence level due to the incorrect
identification of marginal distribution and the copula. However, as the number of samples increases, the output
confidence levels using the bootstrap method are closer to the target confidence level.
Table 3. Probabilities of Failure and Output Confidence Levels
Estimated
Adjusted (Bootstrap)
ns
Par.
PF1
PF2
PF1
PF2
Min
0.079 0.317
0.000
0.050
Mean 4.313 2.127
0.600
0.995
30
Max 20.842 9.577
10.633
6.176
Conf.
54
73
94
93
Min
0.190 0.454
0.004
0.394
Mean 2.668 1.725
0.562
1.467
100
Max
16.71 4.833
8.277
3.727
Conf.
48
82
97
93
Min
0.700 0.684
0.067
0.865
Mean 2.393 1.695
0.707
1.675
300
Max
4.647 3.275
2.300
2.960
Conf.
49
91
99
96
*Optimum design using true input model is (3.623, 1.770); PF1 = 2.250%, PF2 =1.922%
6. Conclusions
In many engineering applications, only insufficient test data are available for input variables due to expensive
experimental testing or measuring costs. The input statistical model obtained from the insufficient data could be
inaccurate. Thus, the RBDO with confidence level is proposed to offset the inaccurate estimation of the input model by
using the adjusted standard deviation and correlation coefficient.
If the input variables have Gaussian distribution, the method using the Gaussian distribution of input variables to
calculate the confidence interval of standard deviation, which is exact, is used. If not, it yields an inaccurate estimation of
the confidence interval of standard deviation. Thus, in this paper, the bootstrap method is proposed to be used to calculate
the confidence interval of standard deviation and, thus, the adjusted standard deviation. To offset the inaccurate
estimation of the input model, the adjusted standard deviation and correlation coefficient involving confidence intervals
for all input parameters (mean, standard deviation, and correlation coefficient) are proposed such that they compensate the
inaccurate estimation of the input parameters. Numerical results show that the input models without confidence level and
those only with confidence level for standard deviation do not yield desirable confidence levels for the input model and
output performance on RBDO results. On the other hand, the input models with adjusted parameters yield desirable input
confidence levels, and the obtained RBDO results are considerably reliable, which leads to desirable confidence levels of
the output performances.
7. Acknowledgement
Research is jointly supported by the Automotive Research Center, which is sponsored by the U.S. Army TARDEC and
ARO Project W911NF-09-1-0250. These supports are greatly appreciated.
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