An Adaptive Mixture of Normal-Inverse Gaussian Distributions for Structural Reliability Analysis
Publication: Journal of Engineering Mechanics
Volume 148, Issue 3
Abstract
Recovering the probability distribution of the limit state function is an effective method of structural reliability analysis, in which it still is challenging to balance the precision and computational efforts. This paper proposes an adaptive mixture of normal-inverse Gaussian distributions which exhibits high flexibility to deal with this issue. First, the mixture distributions with two components were revisited briefly, and the limitations are pointed out. Then the proposed mixture distribution was established. According to the limit condition, one or two components are employed in the proposed mixture distribution to represent the unknown distribution of the limit state function (LSF), which makes the mixture distribution adaptive. To specify the unknown parameters effectively, the Laplace transform at some discrete values is utilized, in which a set of nonlinear equations can be solved easily. An effective cubature rule is utilized to assess numerically the Laplace transform and the involved moments, which can guarantee the efficiency and precision for structural reliability computation. After the LSF’s distribution is attained, the failure probability can be evaluated readily via an integral over the distribution. Five numerical examples were provided to indicate the result of the proposed method.
Get full access to this article
View all available purchase options and get full access to this article.
Data Availability Statement
All data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.
Acknowledgments
The National Natural Science Foundation of China (Nos. 51978253, 51820105014, and U1934217), the Fundamental Research Funds for the Central Universities (No. 531107040224), the 111 Project (Grant No. D21001) and the Key Laboratory of Urban Security and Disaster Engineering of Ministry of Education are gratefully appreciated for the financial support of this research.
References
Au, S.-K., and J. L. Beck. 2001. “Estimation of small failure probabilities in high dimensions by subset simulation.” Probab. Eng. Mech. 16 (4): 263–277. https://doi.org/10.1016/S0266-8920(01)00019-4.
Barndorff-Nielsen, O. E. 1997. “Normal-inverse Gaussian distributions and stochastic volatility modelling.” Scand. J. Stat. 24 (1): 1–13. https://doi.org/10.1111/1467-9469.00045.
Barndorff-Nielsen, O. E., J. Kent, and M. Sørensen. 1982. “Normal variance-mean mixtures and distributions.” Int. Stat. Rev. 50 (Aug): 145. https://doi.org/10.2307/1402598.
Blatman, G., and B. Sudret. 2010. “An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis.” Probab. Eng. Mech. 25 (2): 183–197. https://doi.org/10.1016/j.probengmech.2009.10.003.
Bucher, C. G., and U. Bourgund. 1990. “A fast and efficient response surface approach for structural reliability problems.” Struct. Saf. 7 (1): 57–66. https://doi.org/10.1016/0167-4730(90)90012-E.
Chen, X., and Z. Lin. 2014. Structural nonlinear analysis program OpenSEES theory and tutorial. Beijing: China Architecture & Building Press.
Cheng, J., and Q. Li. 2008. “Reliability analysis of structures using artificial neural network based genetic algorithms.” Comput. Methods Appl. Mech. Eng. 197 (45): 3742–3750. https://doi.org/10.1016/j.cma.2008.02.026.
Dai, H., B. Zhang, and W. Wang. 2015. “A multiwavelet support vector regression method for efficient reliability assessment.” Reliab. Eng. Syst. Saf. 136 (Apr): 132–139. https://doi.org/10.1016/j.ress.2014.12.002.
Dang, C., and J. Xu. 2020a. “A mixture distribution with fractional moments for efficient seismic reliability analysis of nonlinear structures.” Eng. Struct. 208 (Apr): 109912. https://doi.org/10.1016/j.engstruct.2019.109912.
Dang, C., and J. Xu. 2020b. “Unified reliability assessment for problems with low- to high-dimensional random inputs using the Laplace transform and a mixture distribution.” Reliab. Eng. Syst. Saf. 204 (Dec): 107124. https://doi.org/10.1016/j.ress.2020.107124.
Der Kiureghian, A., et al. 2005. First-and second-order reliability methods. Boca Raton, FL: CRC Press. https://doi.org/10.1201/9780203483930.
Der Kiureghian, A., and M. De Stefano. 1988. “Efficient algorithm for second-order reliability analysis.” J. Eng. Mech. 117 (12): 2904–2923. https://doi.org/10.1061/(ASCE)0733-9399(1991)117:12(2904).
Echard, B., N. Gayton, and M. Lemaire. 2011. “AK-MCS: An active learning reliability method combining kriging and Monte Carlo simulation.” Struct. Saf. 33 (2): 145–154. https://doi.org/10.1016/j.strusafe.2011.01.002.
Echard, B., N. Gayton, M. Lemaire, and N. Relun. 2013. “A combined importance sampling and kriging reliability method for small failure probabilities with time-demanding numerical models.” Reliab. Eng. Syst. Saf. 111 (Mar): 232–240. https://doi.org/10.1016/j.ress.2012.10.008.
Faravelli, L. 1989. “Response-surface approach for reliability analysis.” J. Eng. Mech. 115 (12): 2763–2781. https://doi.org/10.1061/(ASCE)0733-9399(1989)115:12(2763).
Hanssen, A., and T. A. Oigard. 2001. “The normal-inverse Gaussian distribution: A versatile model for heavy-tailed stochastic processes.” In Proc., ICASSP IEEE Int. Conf. on Acoustics Speech Signal Processing, 3985–3988. New York: IEEE.
Huang, X., Y. Li, Y. Zhang, and X. Zhang. 2018. “A new direct second-order reliability analysis method.” Appl. Math. Modell. 55 (Mar): 68–80. https://doi.org/10.1016/j.apm.2017.10.026.
Karian, Z. A., and E. J. Dudewicz. 2000. Fitting statistical distributions: The generalized lambda distribution and generalized bootstrap methods. Boca Raton, FL: CRC Press.
Keshtegar, B., and Z. Meng. 2017. “A hybrid relaxed first-order reliability method for efficient structural reliability analysis.” Struct. Saf. 66 (May): 84–93. https://doi.org/10.1016/j.strusafe.2017.02.005.
Kleiber, C., and S. Kotz. 2003. Statistical size distributions in economics and actuarial sciences. New York: Wiley.
Liu, R., W. Fan, Y. Wang, A. H.-S. Ang, and Z. Li. 2019. “Adaptive estimation for statistical moments of response based on the exact dimension reduction method in terms of vector.” Mech. Syst. Sig. Process. 126 (Jul): 609–625. https://doi.org/10.1016/j.ymssp.2019.02.035.
Low, Y. M. 2013. “A new distribution for fitting four moments and its applications to reliability analysis.” Struct. Saf. 42 (3): 12–25. https://doi.org/10.1016/j.strusafe.2013.01.007.
Marelli, S., and B. Sudret. 2014. “UQLab: A framework for uncertainty quantification in Matlab.” In Vulnerability, uncertainty, and risk: Quantification, mitigation, and management, 2554–2563. Reston, VA: ASCE.
Melchers, R. E. 1987. Structural reliability analysis and prediction. New York: Wiley.
Pearson, K. 1998. “X. Contributions to the mathematical theory of evolution.—II. Skew variation in homogeneous material.” Philos. Trans. R. Soc. London, Ser. A 186 (Dec): 343–414. https://doi.org/10.1098/rsta.1895.0010.
Roy, A., R. Manna, and S. Chakraborty. 2019. “Support vector regression based metamodeling for structural reliability analysis.” Probab. Eng. Mech. 55 (Jan): 78–89. https://doi.org/10.1016/j.probengmech.2018.11.001.
Song, H., K. K. Choi, I. Lee, L. Zhao, and D. Lamb. 2013. “Adaptive virtual support vector machine for reliability analysis of high-dimensional problems.” Struct. Multidiscip. Optim. 47 (4): 479–491. https://doi.org/10.1007/s00158-012-0857-6.
Xiao, S., and Z. Lu. 2018. “Reliability analysis by combining higher-order unscented transformation and fourth-moment method.” J. Risk Uncertainty Eng. Syst. Part A: Civ. Eng. 4 (1): 04017034. https://doi.org/10.1061/AJRUA6.0000944.
Xu, H., and S. Rahman. 2004. “A generalized dimension-reduction method for multidimensional integration in stochastic mechanics.” Int. J. Numer. Methods Eng. 61 (12): 1992–2019. https://doi.org/10.1002/nme.1135.
Xu, J., and C. Dang. 2019a. “A new bivariate dimension reduction method for efficient structural reliability analysis.” Mech. Syst. Sig. Process. 115 (Jan): 281–300. https://doi.org/10.1016/j.ymssp.2018.05.046.
Xu, J., and F. Kong. 2018. “A cubature collocation based sparse polynomial chaos expansion for efficient structural reliability analysis.” Struct. Saf. 74 (Sep): 24–31. https://doi.org/10.1016/j.strusafe.2018.04.001.
Xu, J., and F. Kong. 2019b. “Adaptive scaled unscented transformation for highly efficient structural reliability analysis by maximum entropy method.” Struct. Saf. 76 (Jan): 123–134. https://doi.org/10.1016/j.strusafe.2018.09.001.
Xu, J., and D. Wang. 2019c. “Structural reliability analysis based on polynomial chaos, Voronoi cells and dimension reduction technique.” Reliab. Eng. Syst. Saf. 185 (May): 329–340. https://doi.org/10.1016/j.ress.2019.01.001.
Xu, J., Y. Zhang, and C. Dang. 2020. “A novel hybrid cubature formula with Pearson system for efficient moment-based uncertainty propagation analysis.” Mech. Syst. Sig. Process. 140 (Jun): 106661. https://doi.org/10.1016/j.ymssp.2020.106661.
Yang, Z., and J. Ching. 2019. “A novel simplified geotechnical reliability analysis method.” Appl. Math. Modell. 74 (Oct): 337–349. https://doi.org/10.1016/j.apm.2019.04.055.
Yun, W., Z. Lu, X. Jiang, L. Zhang, and P. He. 2020. “AK-ARBIS: An improved AK-MCS based on the adaptive radial-based importance sampling for small failure probability.” Struct. Saf. 82 (Jan): 101891. https://doi.org/10.1016/j.strusafe.2019.101891.
Zhang, D., N. Zhang, N. Ye, J. Fang, and X. Han. 2020. “Hybrid learning algorithm of radial basis function networks for reliability analysis.” IEEE Trans. Reliab. 99 (Jul): 1–14. https://doi.org/10.1109/TR.2020.3001232.
Zhao, Y.-G., and T. Ono. 2001. “Moment methods for structural reliability.” Struct. Saf. 23 (1): 47–75. https://doi.org/10.1016/S0167-4730(00)00027-8.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 148 • Issue 3 • March 2022
Copyright
© 2022 American Society of Civil Engineers.
History
Received: Aug 4, 2021
Accepted: Nov 29, 2021
Published online: Jan 13, 2022
Published in print: Mar 1, 2022
Discussion open until: Jun 13, 2022
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.