In the engineering practice of system reliability analysis, the strong nonlinearity of the limit state function is likely to cause large calculation error of the system failure probability and low efficiency. To solve the problems above, the Tree Markov Chain (TMC) algorithm and the system reliability analysis method based on TMC algorithm are proposed. The TMC is an improvement of the original Markov chain, and its state transition process is more flexible, with local multi-chain parallelism and adaptive exploration of the boundary of the failure domain. Applying multiple candidate states sampling, the TMC obtains as much additional failure domain information as possible, extracting the samples that fully reflect the failure distribution characteristics. The adaptive kernel density estimation of the sample is approximately optimal, improving the accuracy of the calculation results. At the end of the paper, two sets of numerical examples are used to verify the performance of the proposed algorithm. The calculation results show that the algorithm is not sensitive to the location of the design point and the sampling starting point. When dealing with strong nonlinear and complex series of system problems, the proposed method can get highly accurate calculation results with small sample size. When the sample size changes, the calculation result is relatively stable and reliable. The efficiency of the proposed method under practical problems is calculated in engineering examples, demonstrating its engineering application value.
[1] 贾利民, 林帅. 系统可靠性方法研究现状与展望[J]. 系统工程与电子技术, 2015, 37(12):2887-2893. JIA L M, LIN S. Current status and prospect for the methods of system reliability[J]. Systems Engineering and Electronics, 2015, 37(12):2887-2893(in Chinese).
[2] MELCHERS R E. Importance sampling in structural systems[J]. Structural Safety, 1989, 6(1):3-10.
[3] TOKDAR S T, KASS R E. Importance sampling:A review[J]. Wiley Interdisciplinary Reviews:Computational Statistics, 2010, 2(1):54-60.
[4] 马纪明, 詹晓燕, 曾声奎. 基于自适应重要抽样的可靠性分析方法[J]. 北京航空航天大学学报, 2011, 37(9):1142-1146. MA J M, ZHAN X Y, ZENG S K. Reliability analysis method based on adaptive importance sampling[J]. Journal of Beijing University of Aeronautics and Astronautics, 2011, 37(9):1142-1146(in Chinese).
[5] 张峰, 吕震宙. 可靠性灵敏度分析的自适应重要抽样法[J]. 工程力学, 2008, 25(4):80-84. ZHANG F, LV Z Z. An adaptive importance sampling method for estimation of reliability sensitivity[J]. Engineering Mechanics, 2008, 25(4):80-84(in Chinese).
[6] BOTEV Z I, L'ECUYER P, TUFFIN B. An importance sampling method based on a one-step look-ahead density from a Markov chain[C]//Simulation Conference (WSC), Proceedings of the 2011 Winter.Piscataway, NJ:IEEE Press, 2011:528-539.
[7] 侯本伟, 李小军, 刘爱文, 等. 网络可靠性评估的演化过程重要度抽样模拟方法[J]. 系统工程理论与实践, 2016, 36(7):1837-1847. HOU B W, LI X J, LIU A W, et al. Evolution process based importance sampling model for network reliability evaluation[J]. System Engineering-Theory & Practice, 2016, 36(7):1837-1847(in Chinese).
[8] 陈向前, 董聪, 闫阳. 自适应重要抽样方法的改进算法[J]. 工程力学, 2012, 29(11):123-128. CHEN X Q, DONG C, YAN Y. Improved adaptive importance sampling algorithm[J]. Engineering Mechanics, 2012, 29(11):123-128(in Chinese).
[9] 唐承, 郭书祥, 莫延彧. 基于灵敏度分析的系统可靠性稳健分配优化方法[J]. 系统工程理论与实践, 2017, 37(7):1903-1909. TANG C, GUO S X, MO Y Y. Robust allocation optimiza tion method for system reliability based on sensitivity analysis[J]. System Engineering-Theory & Practice, 2017, 37(7):1903-1909(in Chinese).
[10] AU S K, BECK J L. A new adaptive importance sampling scheme for reliability calculations[J]. Structural Safety, 1999, 21(2):135-158.
[11] 戴鸿哲, 赵威, 王伟. 结构可靠性分析高效自适应重要抽样方法[J]. 力学学报, 2011, 43(6):1133-1140. DAI H Z, ZHAO W, WANG W. An efficient adaptive importance sampling method for structural reliability analysis[J]. Chinese Journal of Theoretical and Applied Mechanics, 2011, 43(6):1133-1140(in Chinese).
[12] DAI H Z, ZHANG H, WANG W, et al. Structural reliability assessment by local approximation of limit state functions using adaptive Markov chain simulation and support vector regression[J]. Computer-Aided Civil and Infrastructure Engineering, 2012, 27(9):676-686.
[13] METROPOLIS N, ROSENBLUTH A W, ROSENBLUTH M N, et al. Equation of state calculations by fast computing machines[J]. The Journal of Chemical Physics, 1953, 21(6):1087-1092.
[14] ROSENBLATT M. Remarks on some nonparametric estimates of a density function[J]. Annals of Mathematics Society, 1956, 27:832-837.
[15] PARZEN E. On estimation of a probability density function and the mode[J]. Annals of Mathematics Statistics, 1962, 33:1065-1076.
[16] ANG G L, ANG A H S, TANG W H. Optimal importance-sampling density estimator[J]. Journal of Engineering Mechanics, 1992, 118(6):1146-1163.
[17] EPANECHNIKOV V A. Non-parametric estimation of a multivariate probability density[J]. Theory of Probability & Its Applications, 1969, 14(1):153-158.
[18] ABRAMSON I S. On bandwidth variation in kernel estimates-a square root law[J]. The Annals of Statistics, 1982, 10(4):1217-1223.
[19] 邵苗苗, 顾晓辉. BAT子弹药气动仿真分析[J]. 弹箭与制导学报, 2014, 34(1):118-122. SHAO M M, GU X H. The analysis of aerodynamic simulation of BAT submunition[J]. Journal of Projectiles, Rockets, Missiles and Guidance, 2014, 34(1):118-122(in Chinese).
[20] ZHOU C, LU Z, YUAN X. Use of relevance vector machine in structural reliability analysis[J]. Journal of Aircraft, 2013, 50(6):1726-1733.
[21] 袁修开, 吕震宙, 许鑫. 基于马尔科夫链模拟的支持向量机可靠性分析方法[J]. 工程力学, 2011, 28(2):36-43. YUAN X K, LV Z Z, XU X. Support vector machine reliability analysis method based on markov chain simulation[J]. Engineering Mechanics, 2011, 28(2):36-43(in Chinese).
[22] 赵翔, 李洪双. 基于交叉熵和空间分割的全局可靠性灵敏度分析[J]. 航空学报, 2018, 39(2):179-189. ZHAO X, LI H S. Global reliability sensitivity analysis using cross entropy method and space partition[J]. Acta Aeronautica et Astronautica Sinica, 2018, 39(2):179-189(in Chinese).
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