Research Papers: Design Automation

Maximum Entropy Method-Based Reliability Analysis With Correlated Input Variables via Hybrid Dimension-Reduction Method

[+] Author and Article Information
Wanxin He

Department of Engineering Mechanics,
State Key Laboratory of Structural Analysis for Industrial Equipment,
Dalian University of Technology,
Dalian 116024, China
e-mail: hewanxin_dlut@mail.dlut.edu.cn

Gang Li

Department of Engineering Mechanics,
State Key Laboratory of Structural Analysis for Industrial Equipment,
Dalian University of Technology,
Dalian 116024, China
e-mail: ligang@dlut.edu.cn

Peng Hao

Department of Engineering Mechanics,
State Key Laboratory of Structural Analysis for Industrial Equipment,
Dalian University of Technology,
Dalian 116024, China
e-mail: haopeng@dlut.edu.cn

Yan Zeng

Department of Engineering Mechanics,
State Key Laboratory of Structural Analysis for Industrial Equipment,
Dalian University of Technology,
Dalian 116024, China
e-mail: zengyan@dlut.edu.cn

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the Journal of Mechanical Design. Manuscript received September 13, 2018; final manuscript received May 1, 2019; published online July 19, 2019. Assoc. Editor: Xiaoping Du.

J. Mech. Des 141(10), 101405 (Jul 19, 2019) (13 pages) Paper No: MD-18-1690; doi: 10.1115/1.4043734 History: Received September 13, 2018; Accepted May 04, 2019

The estimation of the statistical moments is widely involved in the industrial application, whose accuracy affects the reliability analysis result considerably. In this study, a novel hybrid dimension-reduction method based on the Nataf transformation is proposed to calculate the statistical moments of the performance function with correlated input variables. Nataf transformation is intrinsically the Gaussian copula, which is commonly used to transform the correlated input variables into independent ones. To calculate the numerical integration of the univariate component function in the proposed method, a normalized moment-based quadrature rule is employed. According to the statistical moments obtained by the proposed method, the probability density function of the performance function can be recovered accurately via maximum entropy method. Six examples are tested to illustrate the accuracy and efficiency of the proposed method, compared with that of Monte Carlo simulation, the conventional univariate dimension-reduction method, and the bivariate dimension-reduction method. It is confirmed that the proposed method achieves a good tradeoff between accuracy and efficiency for structural reliability analysis with correlated input variables.

Copyright © 2019 by ASME
Your Session has timed out. Please sign back in to continue.


Meng, Z., Pu, Y., and Zhou, H., 2017, “Adaptive Stability Transformation Method of Chaos Control for First Order Reliability Method,” Eng. Comput., 34(11) pp. 1–13.
Dai, H., Zhang, H., and Wang, W., 2016, “A New Maximum Entropy-Based Importance Sampling for Reliability Analysis,” Struct. Saf., 63(1), pp. 71–80. [CrossRef]
Santos, S. R., Matioli, L. C., and Beck, A. T., 2012, “New Optimization Algorithms for Structural Reliability,” Comput. Model. Eng. Sci., 83(1), pp. 23–55.
Li, G., Li, B., and Hu, H., 2017, “A Novel First–Order Reliability Method Based on Performance Measure Approach for Highly Nonlinear Problems,” Struct. Multidisc. Optim., 57(4), pp. 1593–1610. [CrossRef]
Ghosn, M., Frangopol, D. M., Mcallister, T. P., Shah, M., Diniz, S. M. C., Ellingwood, B. R., Manuel, L., Biondini, F., Catbas, N., Strauss, A., and Zhao, X. L., 2016, “Reliability-Based Performance Indicators for Structural Members,” J. Struct. Eng., 142(9), p. F4016002. [CrossRef]
Meng, Z., Hu, H., and Zhou, H., 2018, “Super Parametric Convex Model and Its Application for Non-Probabilistic Reliability-Based Design Optimization,” Appl. Math. Model, 55(1), pp. 354–370. [CrossRef]
Meng, Z., and Zhou, H., 2018, “New Target Performance Approach for a Super Parametric Convex Model of Non-Probabilistic Reliability-Based Design Optimization,” Comput. Methods Appl. Mech. Eng., 399(1), pp. 644–662. [CrossRef]
Meng, Z., Yang, D., Zhou, H., and Wang, B., 2018, “Convergence Control of Single Loop Approach for Reliability-Based Design Optimization,” Struct. Multidisc. Optim., 57(3), pp. 1079–1091. [CrossRef]
Meng, Z., and Keshtegar, B., 2019, “Adaptive Conjugate Single-Loop Method for Efficient Reliability-Based Design and Topology Optimization,” Comput. Methods Appl. Mech. Eng., 344(1), pp. 95–119. [CrossRef]
Zhao, Y. G., and Ono, T., 2001, “Moment Methods for Structural Reliability,” Struct. Saf., 23(1), pp. 47–75. [CrossRef]
Du, X., 2010, “System Reliability Analysis With Saddlepoint Approximation,” Struct. Multidisc. Optim., 42(2), pp. 193–208. [CrossRef]
Acar, E., Rais-Rohani, M., and Eamon, C. D., 2010, “Reliability Estimation Using Univariate Dimension Reduction and Extended Generalised Lambda Distribution,” Int. J. Reliab. Saf., 4(2–3), pp. 166–187. [CrossRef]
Li, G., and Zhang, K., 2011, “A Combined Reliability Analysis Approach With Dimension Reduction Method and Maximum Entropy Method,” Struct. Multidisc. Optim., 43(1), pp. 121–134. [CrossRef]
Youn, B. D., Xi, Z., and Wang, P., 2008, “Eigenvector Dimension Reduction (EDR) Method for Sensitivity-Free Probability Analysis,” Struct. Multidisc. Optim., 37(1), pp. 13–28. [CrossRef]
Johnson, N., Kotz, S., and Balakrishnan, N., 1995, Continuous Univariate Distributions, Wiley-Interscience, New York.
Huang, B., and Du, X., 2006, “Uncertainty Analysis by Dimension Reduction Integration and Saddlepoint Approximations,” J. Mech. Design, 128(1), pp. 26–33. [CrossRef]
Jaynes, E. T., 1957, “Information Theory and Statistical Mechanics,” Phys. Rev., 108(2), pp. 171–190. [CrossRef]
Gzyl, H., and Tagliani, A., 2010, “Hausdorff Moment Problem and Fractional Moments,” Appl. Math. Comput., 216(11), pp. 3319–3328.
Shore, J., and Johnson, R., 1980, “Axiomatic Derivation of the Principle of Maximum Entropy and the Principle of Minimum Cross-Entropy,” IEEE. T. IT, 26(1), pp. 26–37. [CrossRef]
Xu, H., and Rahman, S., 2004, “A Generalized Dimension-Reduction Method for Multidimensional Integration in Stochastic Mechanics,” Int. J. Numer. Meth. Eng., 61(12), pp. 1992–2019. [CrossRef]
Rahman, S., and Xu, H., 2004, “A Univariate Dimension-Reduction Method for Multi-Dimensional Integration in Stochastic Mechanics,” Probab. Eng. Mech., 19(4), pp. 393–408. [CrossRef]
Li, D., Chen, Y., Lu, W., and Zhou, C., 2011, “Stochastic Response Surface Method for Reliability Analysis of Rock Slopes Involving Correlated Non-Normal Variables,” Comput. Geotech., 38(1), pp. 58–68. [CrossRef]
Lee, I., Choi, K. K., Noh, Y., Zhao, L., and Gorsich, D., 2011, “Sampling-Based Stochastic Sensitivity Analysis Using Score Functions for RBDO Problems With Correlated Random Variables,” J. Mech. Design, 133(2), p. 021003. [CrossRef]
Lee, I., Choi, K. K., Noh, Y., and Lamb, D., 2012, “Comparison Study Between Probabilistic and Possibilistic Methods for Problems Under a Lack of Correlated Input Statistical Information,” Struct. Multidisc. Optim., 47(2), pp. 175–189. [CrossRef]
Rackwitz, R., and Flessler, B., 1987, “Structural Reliability Under Combined Random Load Sequences,” Comp. Struct., 9(5), pp. 489–494. [CrossRef]
Rosenblatt, M., 1952, “Remarks on a Multivariate Transformation,” Ann. Math. Stat., 23(3), pp. 470–472. [CrossRef]
Piric, K., 2015, “Reliability Analysis Method Based on Determination of the Performance Function’s PDF Using the Univariate Dimension Reduction Method,” Struct. Saf., 57(1), pp. 18–25. [CrossRef]
Lebrun, R., and Dutfoy, A., 2009, “A Generalization of the Nataf Transformation to Distributions With Elliptical Copula,” Probab. Eng. Mech., 24(2), pp. 172–178. [CrossRef]
Lebrun, R., and Dutfoy, A., 2009, “An Innovating Analysis of the Nataf Transformation From the Copula Viewpoint,” Probab. Eng. Mech., 24(3), pp. 312–320. [CrossRef]
Lebrun, R., and Dutfoy, A., 2009, “Do Rosenblatt and Nataf Isoprobabilistic Transformations Really Differ?,” Probab. Eng. Mech., 24(4), pp. 577–584. [CrossRef]
Karanki, D. R., Jadhav, P. A., Chandrakar, A., Srividya, A., and Verma, A. K., 2010, “Uncertainty Analysis in PSA With Correlated Input Parameters,” Int. J. Syst. Assur. Eng. Manag., 1(1), pp. 66–71. [CrossRef]
Noh, Y., Choi, K. K., and Du, L., 2008, “Reliability-Based Design Optimization of Problems With Correlated Input Variables Using a Gaussian Copula,” Struct. Multidisc. Optim., 38(1), pp. 1–16. [CrossRef]
Mnatsakanov, R. M., 2008, “Hausdorff Moment Problem: Reconstruction of Probability Density Functions,” Stat. Probabil. Lett., 78(13), pp. 1869–1877. [CrossRef]
Gzyl, H., and Tagliani, A., 2010, “Stieltjes Moment Problem and Fractional Moments,” Appl. Math. Comput., 216(11), pp. 3307–3318.
Mead, L. R., and Papanicolaou, N., 1984, “Maximum Entropy in the Problem of Moments,” J. Math. Phys., 25(8), pp. 2404–2417. [CrossRef]
Zhang, X., and Pandey, M. D., 2013, “Structural Reliability Analysis Based on the Concepts of Entropy, Fractional Moment and Dimensional Reduction Method,” Struct. Saf., 43(1), pp. 28–40. [CrossRef]
Nelsen, R. B., 2006, An Introduction to Copulas, Springer, New York.
Goda, K., and Tesfamariam, S., 2015, “Multi-Variate Seismic Demand Modelling Using Copulas: Application to Non-Ductile Reinforced Concrete Frame in Victoria, Canada,” Struct. Saf., 56(1), pp. 39–51. [CrossRef]
Tang, X. S., Li, D. Q., Zhou, C. B., and Phoon, K. K., 2015, “Copula-Based Approaches for Evaluating Slope Reliability Under Incomplete Probability Information,” Struct. Saf., 52(1), pp. 90–99. [CrossRef]
Yang, R. J., and Gu, L., 2004, “Experience With Approximate Reliability-Based Optimization Methods,” Struct. Multidisc. Optim., 26(1–2), pp. 152–159. [CrossRef]
Gu, L., Yang, R. J., Cho, C. H., Makowski, M., Faruque, M., and Li, Y., 2001, “Optimisation and Robustness for Crashworthiness of Side Impact,” Int. J. Vehicle. Des., 26(4), pp. 348–360. [CrossRef]
Keshtegar, B., and Hao, P., 2018, “A Hybrid Descent Mean Value for Accurate and Efficient Performance Measure Approach of Reliability-Based Design Optimization,” Comput. Meth. Appl. Mech. Eng., 336(1), pp. 237–259. [CrossRef]
Hao, P., Wang, Y. T., Liu, C., Wang, B., Tian, K., Li, G., Wang, Q., and Jiang, L. L., 2018, “Hierarchical Nondeterministic Optimization of Curvilinearly Stiffened Panel With Multicutouts,” AIAA J., 56(10), pp. 4180–4194. [CrossRef]
Li, G., Zhou, C. X., Zeng, Y., He, W. X., Li, H. R., and Wang, R. Q., 2019, “New Maximum Entropy-Based Algorithm for Structural Design Optimization,” Appl. Math. Model., 66(1), pp. 26–40. [CrossRef]
McNeil, A. J., Frey, R., and Embrechts, P., 2005, Quantitative Risk Management: Concepts, Techniques and Tools, Princeton University Press, Princeton.


Grahic Jump Location
Fig. 1

Flowchart of reliability analysis via MEM based on the HDRM

Grahic Jump Location
Fig. 2

A cantilever beam under vertical and lateral bending

Grahic Jump Location
Fig. 3

Comparison of PDF of Y1 obtained by different methods: (a) ρ12 = 0.9 and (b) ρ12 = −0.9

Grahic Jump Location
Fig. 4

Comparison of PDF of Y2 obtained by different methods: (a) ρ56 = 0.9, (b) ρ56 = 0.5, (c) ρ56 = −0.5, and (d) ρ56 = −0.9

Grahic Jump Location
Fig. 5

Comparison of PDF of Y3 obtained by different methods: (a) ρ = 0.9 and (b) ρ = −0.9

Grahic Jump Location
Fig. 6

Comparison of PDF of Y4 obtained by different methods (ρ23 = 0.9)

Grahic Jump Location
Fig. 7

Schematic view of aircraft stiffened shell problem

Grahic Jump Location
Fig. 8

Comparison of PDF of critical buckling load obtained by different methods

Grahic Jump Location
Fig. 9

Simplified model of the underwater vehicle with two nonlinear connections

Grahic Jump Location
Fig. 10

Comparison of PDF of critical buckling load obtained by different methods



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In