Abstract

For reliability-based design optimization (RBDO), generating an input statistical model with confidence level has been recently proposed to offset inaccurate estimation of the input statistical model with Gaussian distributions. For this, the confidence intervals for the mean and standard deviation are calculated using Gaussian distributions of the input random variables. However, if the input random variables are non-Gaussian, use of Gaussian distributions of the input variables will provide inaccurate confidence intervals, and thus yield an undesirable confidence level of the reliability-based optimum design meeting the target reliability βt. In this paper, an RBDO method using a bootstrap method, which accurately calculates the confidence intervals for the input parameters for non-Gaussian distributions, is proposed to obtain a desirable confidence level of the output performance for non-Gaussian distributions. The proposed method is examined by testing a numerical example and M1A1 Abrams tank roadarm problem.

References

1.
Du
,
L.
,
Choi
,
K. K.
, and
Young
,
B. D.
, 2006, “
An Inverse Possibility Analysis Method for Possibility-Based Design Optimization
,”
AIAA J.
,
44
(
11
), pp.
2682
2690
.
2.
Du
,
L.
, and
Choi
,
K. K.
, 2008, “
An Inverse Analysis Method for Design Optimization With Both Statistical and Fuzzy Uncertainties
,”
Struct. Multidiscip. Optim.
,
37
(
2
), pp.
107
119
.
3.
Rao
,
S. S.
, and
Cao
,
L.
, 2002, “
Optimum Design of Mechanical Systems Involving Interval Parameters
,”
ASME J. Mech. Des.
,
124
(
3
), pp.
465
472
.
4.
Parkinson
,
A.
, 1995, “
Robust Mechanical Design Using Engineering Models
,”
ASME J. Vib. Acoust.
,
117
(
1
), pp.
48
54
.
5.
Penmetsa
,
R. C.
, and
Grandhi
,
R. V.
, 2002 “
Efficient Estimation of Structural Reliability for Problems with Uncertain Intervals
,”
Comput. Struct.
,
80
(
12
), pp.
1103
1112
.
6.
Gunawan
,
S.
, and
Papalambros
,
P. Y.
, 2006, “
A Bayesian Approach to Reliability Based Optimization With Incomplete Information
,”
ASME J. Mech. Des.
,
128
, pp.
909
018
.
7.
Youn
,
B. D.
, and
Wang
,
P.
, 2008, “
Bayesian Reliability-Based Design Optimization Using Eigenvector Dimension Reduction (EDR) Method
,”
Struct. Multidiscip. Optim.
,
36
(
2
), pp.
107
123
.
8.
Noh
,
Y.
,
Choi
,
K. K.
,
Lee
,
I.
,
Gorsich
,
D.
, and
Lamb
,
D.
, 2011, “
Reliability-Based Design Optimization with Confidence Level under Input Model Uncertainty due to Limited Test Data
,”
Struct. Multidiscip. Optim.
,
43
(
4
), pp.
443
458
.
9.
Haldar
,
A.
, and
Mahadevan
,
S.
, 2000,
Probability, Reliability, and Statistical Methods in Engineering Design
,
John Wiley & Sons
,
New York
.
10.
Efron
,
B.
, 1982,
The Jackknife, the Bootstrap, and Other Resampling Plans
,
SIAM
,
Philadelphia
.
11.
Efron
,
B.
, and
Tibshirani
,
R.
, 1993,
An Introduction to the Bootstrap
,
Chapman and Hall
,
London/CRC, Boca Raton
.
12.
Picheny
,
V.
,
Kim
,
N. H.
, and
Haftka
,
R. T.
, 2010, “
Application of Bootstrap Method in Conservative Estimation of Reliability With Limited Samples
,”
Struct. Multidiscip. Optim.
,
41
(
2
), pp.
205
217
.
13.
Bonett
,
D. G.
, 2006, “
Approximate Confidence Interval for Standard Deviation of Nonnormal Distribution
,”
Comput. Stat. Data Anal.
,
50
, pp.
775
782
.
14.
Cojbasic
,
V.
, and
Tomovic
,
A.
, 2007, “
Nonparametric Confidence Intervals for Population Variance of One Sample and the Difference of Variances of Two Samples
,”
51
, pp.
5562
5578
.
15.
Rosenblatt
,
M.
, 1952, “
Remarks on A Multivariate Transformation
,”
Ann. Math. Stat.
,
23
, pp.
470
472
.
16.
Nelsen
,
R. B.
, 1999,
An Introduction to Copulas
,
Springer
,
New York
.
17.
Kendall
,
M.
, 1938, “
A New Measures of Associations
,”
Biometrika
,
30
, pp.
81
89
.
18.
Noh
,
Y.
,
Choi
,
K. K.
, and
Lee
,
I.
, 2010, “
Identification of Marginal and Joint CDFs Using the Bayesian Method for RBDO
,”
Struct. Multidiscip. Optim.
,
40
(
1
), pp.
35
51
.
19.
Socie
,
D. F.
, Seminar notes: “
Probabilistic Aspects of Fatigue
,” 2003, http://www.fatiguecaculator.comhttp://www.fatiguecaculator.com [cited May, 8 2008].
20.
Annis
,
C.
, 2004, “
Probabilistic Life Prediction Isn’t as Easy as It Looks
,”
J. ASTM Int.
,
1
(
2
), pp.
3
14
.
21.
Efstratios
,
N.
,
Ghiocel
,
D.
, and
Singhal
,
S.
, 2004,
Engineering Design Reliability Handbook
,
CRC Press
,
New York
.
22.
Pham
,
H.
, 2006,
Springer Handbook of Engineering Statistics
,
Springer
,
London
.
23.
Huard
,
D.
,
Évin
,
G.
, and
Favre
,
A.-C.
, 2006, “
Bayesian Copula Selection
,”
Comput. Stat. Data Anal., COMSTA3137
,
51
(
2
), pp.
809
822
.
24.
Genest
,
C.
,
Ghoudi
,
K.
, and
Rivest
,
L. P.
, 1995, “
A Semiparametric Estimation Procedure of Dependence Parameters in Multivariate Families of Distribution
,”
Biometrika
,
82
(
3
), pp.
543
552
.
25.
Hürliman
,
W.
, 2004, “
Fitting Bivariate Cumulative Returns With Copulas
,”
Comput. Stat. Data Anal.
,
45
(
2
), pp.
355
372
.
26.
Roch
,
O.
, and
Alegre
,
A.
2006, “
Testing the Bivariate Distribution of Daily Equity Returns Using Copulas. An Application to the Spanish Stockmarket
,”
Comput. Stat. Data Anal.
,
51
(
2
), pp.
1312
1329
.
27.
Noh
,
Y.
,
Choi
,
K. K.
, and
Lee
,
I.
, 2010, “
Comparison Study Between MCMC-based and Weight-Based Bayesian Methods for Identification of Joint Distribution
,”
Struct. Multidiscip. Optim.
,
42
(
6
), pp.
823
833
.
28.
Silva
,
R. S.
, and
Lopes
,
H. F.
, 2008, “
Copula, Marginal Distributions and Model Selection: A Bayesian Note
,”
Stat. Comput.
,
18
(
3
), pp.
313
320
.
29.
Shao
,
J.
, and
Tu
,
D.
, 1995,
The Jackknife and Bootstrap
,
Springer
,
New York
.
30.
Bonate
,
B. L.
, 2005,
Pharmacokinetic-pharmacodynamic Modeling and Simulation
,
Springer
,
New York
.
31.
Mooney
,
C. Z.
, and
Duval
,
R. D.
, 1993,
Bootstrapping: A Nonparametric Approach to Statistical Inference
,
Sage Publications
,
Thousand Oaks, CA
.
32.
Liu
,
R. Y.
, and
Singh
,
K.
, 1988, “
Invited Discussion on Theoretical Comparison of Bootstrap Confidence Intervals
,”
Ann. Stat.
,
16
, pp.
978
979
.
33.
Genest
,
C.
, and
Favre
,
A -C.
, 2007, “
Everything You Always Wanted to Know About Copula Modeling but Were Afraid to Ask
,”
J. Hydrol. Eng.
,
12
(
4
), pp.
347
368
.
34.
Lee
,
I.
,
Choi
,
K. K.
,
Du
,
L.
, and
Gorsich
,
D.
, 2008, “
Inverse Analysis Method Using MPP-Based Dimension Reduction for Reliability-Based Design Optimization of Nonlinear and Multi-Dimensional Systems
,”
Comput. Methods Appl. Mech. Eng.
,
198
(
1
), pp.
14
27
.
35.
Swanson Analysis System Inc., ANSYS Engineering Analysis System User’s Manual, Vol.
I & II
,
Houston, PA
, 1989.
36.
Center for Computer-Aided Design
,
College of Engineering, DRAW Concept Manual
,
The University of Iowa
,
Iowa City, IA
, 1999.
37.
Center for Computer-Aided Design, College of Engineering, DRAW User Reference,
The University of Iowa
, Iowa City, IA, 1999.
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