This study presents a methodology for computing stochastic sensitivities with respect to the design variables, which are the mean values of the input correlated random variables. Assuming that an accurate surrogate model is available, the proposed method calculates the component reliability, system reliability, or statistical moments and their sensitivities by applying Monte Carlo simulation to the accurate surrogate model. Since the surrogate model is used, the computational cost for the stochastic sensitivity analysis is affordable compared with the use of actual models. The copula is used to model the joint distribution of the correlated input random variables, and the score function is used to derive the stochastic sensitivities of reliability or statistical moments for the correlated random variables. An important merit of the proposed method is that it does not require the gradients of performance functions, which are known to be erroneous when obtained from the surrogate model, or the transformation from X-space to U-space for reliability analysis. Since no transformation is required and the reliability or statistical moment is calculated in X-space, there is no approximation or restriction in calculating the sensitivities of the reliability or statistical moment. Numerical results indicate that the proposed method can estimate the sensitivities of the reliability or statistical moments very accurately, even when the input random variables are correlated.
1.
Yi
, K.
, Choi
, K. K.
, Kim
, N. H.
, and Botkin
, M. E.
, 2007, “Design Sensitivity Analysis and Optimization for Minimizing Springback of Sheet-Formed Part
,” Int. J. Numer. Methods Eng.
0029-5981, 71
, pp. 1483
–1511
.2.
Youn
, B. D.
, Choi
, K. K.
, and Yi
, K.
, 2005, “Performance Moment Integration (PMI) Method for Quality Assessment in Reliability-Based Robust Optimization
,” Mech. Based Des. Struct. Mach.
1539-7734, 33
(2
), pp. 185
–213
.3.
Youn
, B. D.
, Choi
, K. K.
, Yang
, R. J.
, and Gu
, L.
, 2004, “Reliability-Based Design Optimization for Crashworthiness of Vehicle Side Impact
,” Struct. Multidiscip. Optim.
1615-147X, 26
(3–4
), pp. 272
–283
.4.
Youn
, B. D.
, Choi
, K. K.
, and Tang
, J.
, 2005, “Structural Durability Design Optimization and Its Reliability Assessment
,” Int. J. Prod. Dev.
, 1
(3/4
), pp. 383
–401
.5.
Dong
, J.
, Choi
, K. K.
, Wang
, A.
, Zhang
, W.
, and Vlahopoulos
, N.
, 2005, “Parametric Design Sensitivity Analysis of High Frequency Structural-Acoustic Problems Using Energy Finite Element Method
,” Int. J. Numer. Methods Eng.
0029-5981, 62
, pp. 83
–121
.6.
Dong
, J.
, Choi
, K. K.
, Vlahopoulos
, N.
, Wang
, A.
, and Zhang
, W.
, 2007, “Design Sensitivity Analysis and Optimization of High Frequency Radiation Problems Using Energy Finite Element and Energy Boundary Element Methods
,” AIAA J.
0001-1452, 45
(6
), pp. 1187
–1198
.7.
Kim
, N. H.
, Wang
, H.
, and Queipo
, N. V.
, 2006, “Adaptive Reduction of Design Variables Using Global Sensitivity in Reliability-Based Optimization
,” International Journal of Reliability and Safety
, 1
(1/2
), pp. 102
–119
.8.
Acar
, E.
, and Haftka
, R. T.
, 2007, “Reliability-Based Aircraft Structural Design Pays, Even With Limited Statistical Data
,” J. Aircr.
0021-8669, 44
(3
), pp. 812
–823
.9.
Lee
, S. H.
, Chen
, W.
, and Kwak
, B. M.
, 2009, “Robust Design With Arbitrary Distribution Using Gauss-Type Quadrature Formula
,” Struct. Multidiscip. Optim.
1615-147X, 39
(3
), pp. 227
–243
.10.
Yu
, X.
, and Du
, X.
, 2006, “Reliability-Based Multidisciplinary Optimization for Aircraft Wing Design
,” Struct. Infrastruct. Eng.
1573-2479, 2
(3&4
), pp. 277
–289
.11.
Haldar
, A.
, and Mahadevan
, S.
, 2000, Probability, Reliability and Statistical Methods in Engineering Design
, Wiley
, New York
.12.
Madsen
, H. O.
, Krenk
, S.
, and Lind
, N. C.
, 1986, Methods of Structural Safety
, Prentice-Hall
, Englewood Cliffs, NJ
.13.
Hasofer
, A. M.
, and Lind
, N. C.
, 1974, “An Exact and Invariant First Order Reliability Format
,” J. Engrg. Mech. Div.
0044-7951, 100
(1
), pp. 111
–121
.14.
Ditlevsen
, O.
, and Madsen
, H. O.
, 1996, Structural Reliability Methods
, Wiley
, Chichester, UK
.15.
Hohenbichler
, M.
, and Rackwitz
, R.
, 1988, “Improvement of Second-Order Reliability Estimates by Importance Sampling
,” J. Eng. Mech.
0733-9399, 114
(12
), pp. 2195
–2199
.16.
Breitung
, K.
, 1984, “Asymptotic Approximations for Multinormal Integrals
,” J. Eng. Mech.
0733-9399, 110
(3
), pp. 357
–366
.17.
Rahman
, S.
, and Wei
, D.
, 2006, “A Univariate Approximation at Most Probable Point for Higher-Order Reliability Analysis
,” Int. J. Solids Struct.
0020-7683, 43
, pp. 2820
–2839
.18.
Wei
, D.
, 2006, “A Univariate Decomposition Method for Higher-Order Reliability Analysis and Design Optimization
,” Ph.D. thesis, University of Iowa, Iowa City, IA.19.
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.
0045-7825, 198
(1
), pp. 14
–27
.20.
Kalsi
, M.
, Hacker
, K.
, and Lewis
, K.
, 2001, “A Comprehensive Robust Design Approach for Decision Trade-Offs in Complex Systems Design
,” ASME J. Mech. Des.
0161-8458, 123
(1
), pp. 1
–10
.21.
Su
, J.
, and Renaud
, J. E.
, 1997, “Automatic Differentiation in Robust Optimization
,” AIAA J.
0001-1452, 35
(6
), pp. 1072
–1079
.22.
Rahman
, S.
, and Xu
, H.
, 2004, “A Univariate Dimension-Reduction Method for Multi-Dimensional Integration in Stochastic Mechanics
,” Probab. Eng. Mech.
0266-8920, 19
, pp. 393
–408
.23.
Xu
, H.
, and Rahman
, S.
, 2004, “A Generalized Dimension-Reduction Method for Multi-Dimensional Integration in Stochastic Mechanics
,” Int. J. Numer. Methods Eng.
0029-5981, 61
(12
), pp. 1992
–2019
.24.
Lee
, I.
, Choi
, K. K.
, Du
, L.
, and Gorsich
, D.
, 2008, “Dimension Reduction Method for Reliability-Based Robust Design Optimization
,” Comput. Struct.
0045-7949, 86
(13–14
), pp. 1550
–1562
.25.
Youn
, B. D.
, and Choi
, K. K.
, 2004, “A New Response Surface Methodology for Reliability-Based Design Optimization
,” Comput. Struct.
0045-7949, 82
(2–3
), pp. 241
–256
.26.
Zhang
, T.
, Choi
, K. K.
, Rahman
, S.
, Cho
, K.
, Perry
, B.
, Shakil
, M.
, and Heitka
, D.
, 2006, “A Response Surface and Pattern Search Based Hybrid Optimization Method and Application to Microelectronics
,” Struct. Multidiscip. Optim.
1615-147X, 32
(4
), pp. 327
–345
.27.
Kim
, C.
, and Choi
, K. K.
, 2008, “Reliability-Based Design Optimization Using Response Surface Method With Prediction Interval Estimation
,” ASME J. Mech. Des.
0161-8458, 130
(12
), p. 121401
.28.
Simpson
, T. W.
, Mauery
, T. M.
, Korte
, J. J.
, and Mistree
, F.
, 2001, “Kriging Models for Global Approximation in Simulation-Based Multidisciplinary Design Optimization
,” AIAA J.
0001-1452, 39
(12
), pp. 2233
–2241
.29.
Queipo
, N. V.
, Haftka
, R. T.
, Shyy
, W.
, Goel
, T.
, Vaidyanathan
, R.
, and Tucker
, P. K.
, 2005, “Surrogate-Based Analysis and Optimization
,” Prog. Aerosp. Sci.
0376-0421, 41
(1
), pp. 1
–28
.30.
Buranathiti
, T.
, Cao
, J.
, Chen
, W.
, Baghdasaryan
, L.
, and Xia
, Z. C.
, 2004, “Approaches for Model Validation: Methodology and Illustration on a Sheet Metal Flanging Process
,” ASME J. Manuf. Sci. Eng.
1087-1357, 126
, pp. 2009
–2013
.31.
Gu
, L.
, Yang
, R. J.
, Tho
, C. H.
, Makowskit
, M.
, Faruquet
, O.
, and Li
, Y.
, 2001, “Optimization and Robustness for Crashworthiness of Side Impact
,” Int. J. Veh. Des.
0143-3369, 26
(4
), pp. 348
–360
.32.
Rubinstein
, R. Y.
, 1981, Simulation and Monte Carlo Method
, Wiley
, New York
.33.
Noh
, Y.
, Choi
, K. K.
, and Du
, L.
, 2009, “Reliability Based Design Optimization of Problems With Correlated Input Variables Using Copulas
,” Struct. Multidiscip. Optim.
1615-147X, 38
(1
), pp. 1
–16
.34.
Noh
, Y.
, Choi
, K. K.
, and Lee
, I.
, 2009, “Reduction of Transformation Ordering Effect in RBDO Using MPP-Based Dimension Reduction Method
,” AIAA J.
0001-1452, 47
(4
), pp. 994
–1004
.35.
Noh
, Y.
, Choi
, K. K.
, and Lee
, I.
, 2010, “Identification of Marginal and Joint CDFs Using the Bayesian Method for RBDO
,” Struct. Multidiscip. Optim.
1615-147X, 40
(1–6
), pp. 35
–51
.36.
Annis
, C.
, 2004, “Probabilistic Life Prediction Isn’t As Easy As It Looks
,” J. ASTM Int.
1546-962X, 1
(2
), pp. 3
–14
.37.
Nelsen
, R. B.
, 1999, An Introduction to Copulas
, Springer
, New York
.38.
Rubinstein
, R. Y.
, and Shapiro
, A.
, 1993, Discrete Event Systems––Sensitivity Analysis and Stochastic Optimization by the Score Function Method
, Wiley
, New York
.39.
Rahman
, S.
, 2009, “Stochastic Sensitivity Analysis by Dimensional Decomposition and Score Functions
,” Probab. Eng. Mech.
0266-8920, 24
, pp. 278
–287
.40.
Zhao
, L.
, Choi
, K. K.
, Lee
, I.
, and Gorsich
, D.
, 2010, “A Metamodeling Method Using Dynamic Kriging and Sequential Sampling
,” The 13th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference
, Fort Worth, TX, Sept. 13–15.41.
McDonald
, M.
, and Mahadevan
, S.
, 2008, “Design Optimization With System-Level Reliability Constraints
,” ASME J. Mech. Des.
0161-8458, 130
(2
), p. 021403
.42.
Browder
, A.
, 1996, Mathematical Analysis: An Introduction
, Springer-Verlag
, New York
.43.
Hijab
, O.
, 1997, Introduction to Calculus and Classical Analysis
, Springer-Verlag
, New York
.44.
Lee
, I.
, Choi
, K. K.
, and Zhao
, L.
, 2010, “Sampling-Based RBDO Using the Dynamic Kriging (D-Kriging) Method and Stochastic Sensitivity Analysis
,” The 13th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference
, Fort Worth, TX, Sept. 13–15.45.
Viana
, A. C. F.
, Haftka
, R. T.
, and Steffen
, V.
, 2009, “Multiple Surrogates: How Cross-Validation Errors Can Help Us to Obtain the Best Predictor
,” Struct. Multidiscip. Optim.
1615-147X, 39
(4
), pp. 439
–457
.Copyright © 2011
by American Society of Mechanical Engineers
You do not currently have access to this content.
