A Novel Structural Reliability Method on the Basis of Gaussian Mixture and Scaled Unscented Transformation
Publication: Journal of Engineering Mechanics
Volume 147, Issue 12
Abstract
This paper proposes a novel method to capture efficiently the probability distribution of the limit state function (LSF) for structural reliability analysis. This method works by decomposing the probability distribution of the LSF into a Gaussian mixture and applying the scaled unscented transformation to approximate the required statistics for each mixture component. The number of mixture components is specified according to Akaike’s information criterion. Because the required sample size grows linearly with the number of random inputs, little computational effort is required to recover the entire distribution of the LSF with precision. The effectiveness of the proposed method was validated through numerical examples, in which the results obtained from pertinent Monte Carlo simulations and other approaches are compared. The results demonstrated that the proposed method efficiently can yield accurate estimates of the probability distribution of the LSF in the entire range, and the probability of failure accordingly can be evaluated straightforwardly. Issues that need to be studied further were discussed.
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 (Grant No. 51978253) and the Fundamental Research Funds for the Central Universities (Grant No. 531107040224) are gratefully appreciated for support of the research.
References
Akaike, H. 1974. “A new look at the statistical model identification.” IEEE Trans. Autom. Control 19 (6): 716–723. https://doi.org/10.1109/TAC.1974.1100705.
Au, S. K., and J. L. Beck. 2003. “Important sampling in high dimensions.” Struct. Saf. 25 (2): 139–163. https://doi.org/10.1016/S0167-4730(02)00047-4.
Au, S. K., J. Ching, and J. L. Beck. 2007. “Application of subset simulation methods to reliability benchmark problems.” Struct. Saf. 29 (3): 183–193. https://doi.org/10.1016/j.strusafe.2006.07.008.
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.
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.
Bourinet, J. M. 2016. “Rare-event probability estimation with adaptive support vector regression surrogates.” Reliab. Eng. Syst. Saf. 150 (Jun): 210–221. https://doi.org/10.1016/j.ress.2016.01.023.
Bozdogan, H., and S. L. Sclove. 1984. “Multi-sample cluster analysis using Akaike’s information criterion.” Ann. Inst. Stat. Math. 36 (1): 163–180. https://doi.org/10.1007/BF02481962.
Cardoso, J. B., J. R. De Almeida, J. M. Dias, and P. G. Coelho. 2008. “Structural reliability analysis using monte carlo simulation and neural networks.” Adv. Eng. Software 39 (6): 505–513. https://doi.org/10.1016/j.advengsoft.2007.03.015.
Chen, J., J. Yang, and J. Li. 2016. “A GF-discrepancy for point selection in stochastic seismic response analysis of structures with uncertain parameters.” Struct. Saf. 59 (Mar): 20–31. https://doi.org/10.1016/j.strusafe.2015.11.001.
Choi, S.-K., R. Grandhi, and R. A. Canfield. 2006. Reliability-based structural design. New York: Springer.
Dai, H., H. Zhang, and W. Wang. 2016. “A new maximum entropy-based importance sampling for reliability analysis.” Struct. Saf. 63 (Nov): 71–80. https://doi.org/10.1016/j.strusafe.2016.08.001.
Daniels, H. E. 1954. “Saddlepoint approximations in statistics.” Ann. Math. Stat. 25 (4): 631–650. https://doi.org/10.1214/aoms/1177728652.
Dempster, A. P. 1977. “Maximum likelihood from incomplete data via the EM algorithm.” J. R. Stat. Soc. 39 (1): 1–38. https://doi.org/10.1111/j.2517-6161.1977.tb01600.x.
Der Kiureghian, A., H.-Z. Lin, and S.-J. Hwang. 1987. “Second-order reliability approximations.” J. Eng. Mech. 113 (8): 1208–1225. https://doi.org/10.1061/(ASCE)0733-9399(1987)113:8(1208).
Dubourg, V., B. Sudret, and F. Deheeger. 2013. “Metamodel-based importance sampling for structural reliability analysis.” Probab. Eng. Mech. 33 (1): 47–57. https://doi.org/10.1016/j.probengmech.2013.02.002.
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).
Gaspar, B., A. P. Teixeira, and C. G. Soares. 2014. “Assessment of the efficiency of Kriging surrogate models for structural reliability analysis.” Probab. Eng. Mech. 37 (4): 24–34. https://doi.org/10.1016/j.probengmech.2014.03.011.
Ghanem, R. G., and P. D. Spanos. 2003. Stochastic finite elements: A spectral approach. North Chelmsford, MA: Courier.
Hasofer, A. M., and N. C. Lind. 1974. “An exact and invariant first order reliability format.” J. Eng. Mech. 100 (1): 111–121.
Huang, J., and D. V. Griffiths. 2011. “Observations on FORM in a simple geomechanics example.” Struct. Saf. 33 (1): 115–119. https://doi.org/10.1016/j.strusafe.2010.10.001.
Ibrahim, Y. 1991. “Observations on applications of importance sampling in structural reliability analysis.” Struct. Saf. 9 (4): 269–281. https://doi.org/10.1016/0167-4730(91)90049-F.
Julier, S. J. 2002. “The scaled unscented transformation.” In Vol. 6 of Proc., 2002 American Control Conf., 4555–4559. New York: IEEE.
Julier, S. J. 2003. “The spherical simplex unscented transformation.” In Vol. 3 of Proc., 2003 American Control Conf., 2430–2434. New York: IEEE.
Julier, S. J., and J. K. Uhlmann. 2004. “Unscented filtering and nonlinear estimation.” Proc. IEEE 92 (3): 401–422. https://doi.org/10.1109/JPROC.2003.823141.
Kaymaz, I. 2005. “Application of kriging method to structural reliability problems.” Struct. Saf. 27 (2): 133–151. https://doi.org/10.1016/j.strusafe.2004.09.001.
Kushary, D. 1997. “The EM algorithm and extensions.” Technometrics 40 (3): 260. https://doi.org/10.1080/00401706.1998.10485534.
Li, J. 2013. “An unscented Kalman smoother for volatility extraction: Evidence from stock prices and options.” Comput. Stat. Data Anal. 58 (1): 15–26. https://doi.org/10.1016/j.csda.2011.06.001.
Li, J., and J. Chen. 2009. Stochastic dynamics of structures. New York: Wiley.
Li, J., J. Chen, W. Sun, and Y. Peng. 2012. “Advances of the probability density evolution method for nonlinear stochastic systems.” Probab. Eng. Mech. 28 (4): 132–142. https://doi.org/10.1016/j.probengmech.2011.08.019.
Low, Y. 2013. “A new distribution for fitting four moments and its applications to reliability analysis.” Struct. Saf. 42 (May): 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 Proc., Vulnerability, uncertainty, and risk: Quantification, mitigation, and management, 2554–2563. Reston, VA: ASCE.
McLachlan, G., and D. Peel. 2000. Finite mixture models. New York: Wiley.
Melchers, R. E. 1994. “Structural system reliability assessment using directional simulation.” Struct. Saf. 16 (1–2): 23–37. https://doi.org/10.1016/0167-4730(94)00026-M.
Pearson, K. 1894. “Contributions to the mathematical theory of evolution.” Philos. Trans. R. Soc. London, Ser. A 185 (Jan): 71–110. https://doi.org/10.1098/rsta.1894.0003.
Pradlwarter, H. J., G. I. Schueller, P. S. Koutsourelakis, and D. C. Charmpis. 2007. “Application of line sampling simulation method to reliability benchmark problems.” Struct. Saf. 29 (3): 208–221. https://doi.org/10.1016/j.strusafe.2006.07.009.
Rahman, S. 2008. “A polynomial dimensional decomposition for stochastic computing.” Int. J. Numer. Methods Eng. 76 (13): 2091–2116. https://doi.org/10.1002/nme.2394.
Rahman, S., and H. Xu. 2004. “A univariate dimension-reduction method for multi-dimensional integration in stochastic mechanics.” Probab. Eng. Mech. 19 (4): 393–408. https://doi.org/10.1016/j.probengmech.2004.04.003.
Richter, J. H. 2011. “Reliability estimation using unscented transformation.” In Proc., 2011 3rd Int. Workshop on Dependable Control of Discrete Systems, 102–107. New York: IEEE.
Rossner, N., T. Heine, and R. King. 2010. “Quality-by-design using a Gaussian mixture density approximation of biological uncertainties.” IFAC Proc. Volumes 43 (6): 7–12. https://doi.org/10.3182/20100707-3-BE-2012.0035.
Sanseverino, C. M. R., and J. E. Ramirez-Marquez. 2014. “Uncertainty propagation and sensitivity analysis in system reliability assessment via unscented transformation.” Reliab. Eng. Syst. Saf. 132 (132): 176–185. https://doi.org/10.1016/j.ress.2014.07.024.
Sclove, S. L. 1987. “Application of model-selection criteria to some problems in multivariate analysis.” Psychometrika 52 (3): 333–343. https://doi.org/10.1007/BF02294360.
Shields, M. D. 2016. “Refined Latinized stratified sampling: A robust sequential sample size extension methodology for high-dimensional Latin hypercube and stratified designs.” Int. J. Uncertainty Quantif. 6 (1): 79–97. https://doi.org/10.1615/Int.J.UncertaintyQuantification.2016011333.
Shields, M. D., and J. Zhang. 2016. “The generalization of Latin hypercube sampling.” Reliab. Eng. Syst. Saf. 148 (Apr): 96–108. https://doi.org/10.1016/j.ress.2015.12.002.
Silva, D. S. F., and C. V. Deutsch. 2018. “Multivariate data imputation using Gaussian mixture models.” Spatial Stat. 27 (Oct): 74–90. https://doi.org/10.1016/j.spasta.2016.11.002.
Song, S., Z. Lu, and H. Qiao. 2009. “Subset simulation for structural reliability sensitivity analysis.” Reliab. Eng. Syst. Saf. 94 (2): 658–665. https://doi.org/10.1016/j.ress.2008.07.006.
Tagliani, A. 1999. “Hausdorff moment problem and maximum entropy: A unified approach.” Appl. Math. Comput. 105 (2): 291–305. https://doi.org/10.1016/S0096-3003(98)10084-X.
Wallace, D. L. 1958. “Asymptotic approximations to distributions.” Ann. Math. Stat. 29 (3): 635–654. https://doi.org/10.1214/aoms/1177706528.
Winterstein, S. R. 1988. “Nonlinear vibration models for extremes and fatigue.” J. Eng. Mech. 114 (10): 1772–1790. https://doi.org/10.1061/(ASCE)0733-9399(1988)114:10(1772).
Xi, Z., C. Hu, and B. D. Youn. 2012. “A comparative study of probability estimation methods for reliability analysis.” Struct. Multidiscip. Optim. 45 (1): 33–52. https://doi.org/10.1007/s00158-011-0656-5.
Xiao, S., and Z. Lu. 2016. “Structural reliability analysis using combined space partition technique and unscented transformation.” J. Struct. Eng. 142 (11): 04016089. https://doi.org/10.1061/(ASCE)ST.1943-541X.0001553.
Xu, J., and C. Dang. 2019. “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., C. Dang, and F. Kong. 2017. “Efficient reliability analysis of structures with the rotational quasi-symmetric point- and the maximum entropy methods.” Mech. Syst. Sig. Process. 95 (Oct): 58–76. https://doi.org/10.1016/j.ymssp.2017.03.019.
Xu, J., and F. Kong. 2018a. “An efficient method for statistical moments and reliability assessment of structures.” Struct. Multidiscip. Optim. 58 (5): 2019–2035. https://doi.org/10.1007/s00158-018-2015-2.
Xu, J., and F. Kong. 2018b. “A new unequal-weighted sampling method for efficient reliability analysis.” Reliab. Eng. Syst. Saf. 172 (Apr): 94–102. https://doi.org/10.1016/j.ress.2017.12.007.
Xu, J., and J. Li. 2016. “Stochastic dynamic response and reliability assessment of controlled structures with fractional derivative model of viscoelastic dampers.” Mech. Syst. Sig. Process. 72 (May): 865–896. https://doi.org/10.1016/j.ymssp.2015.11.016.
Xu, J., and Z.-H. Lu. 2017. “Evaluation of moments of performance functions based on efficient cubature formulation.” J. Eng. Mech. 143 (8): 06017007. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001248.
Xue, G., H. Dai, H. Zhang, and W. Wang. 2017. “A new unbiased metamodel method for efficient reliability analysis.” Struct. Saf. 67 (Jul): 1–10. https://doi.org/10.1016/j.strusafe.2017.03.005.
Yang, J., J. Tao, B. Sudret, and J. Chen. 2020. “Generalized F-discrepancy-based point selection strategy for dependent random variables in uncertainty quantification of nonlinear structures.” Int. J. Numer. Methods Eng. 121 (7): 1507–1529. https://doi.org/10.1002/nme.6277.
Zhang, D., and X. Han. 2020. “Kinematic reliability analysis of robotic manipulator.” J. Mech. Des. 142 (4): 044502. https://doi.org/10.1115/1.4044436.
Zhang, D., N. Zhang, N. Ye, J. Fang, and X. Han. 2021. “Hybrid learning algorithm of radial basis function networks for reliability analysis.” IEEE Trans. Reliab. 70 (3): 887–900. https://doi.org/10.1109/TR.2020.3001232.
Zhou, C., Z. Lu, L. Li, J. Feng, and B. Wang. 2013. “A new algorithm for variance based importance analysis of models with correlated inputs.” Appl. Math. Modell. 37 (3): 864–875. https://doi.org/10.1016/j.apm.2012.03.017.
Information & Authors
Information
Published In
Copyright
© 2021 American Society of Civil Engineers.
History
Received: Jul 24, 2020
Accepted: Aug 12, 2021
Published online: Sep 30, 2021
Published in print: Dec 1, 2021
Discussion open until: Feb 28, 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.