基于树形马氏链模型的可靠性分析方法
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摘要
复杂系统的极限状态函数非线性程度较高,在进行可靠性分析时,易导致失效概率的计算误差大、效率低,针对上述问题,提出了树形马氏链(TMC)算法和基于该算法的可靠性分析方法。树形马氏链是对原始马尔可夫链的改进,其状态转移过程更加灵活,具备局部多链并行和自适应探索失效域边界的特性。树形马氏链通过多候选状态点扩大对失效域信息的收集,生成能充分反映失效分布特征的样本,对该样本进行自适应核密度估计得到近似最优的重要抽样分布密度函数,从而提高计算结果的准确度。文末的数值算例和工程算例验证了算法性能,计算结果表明算法对设计点、抽样起点的位置不敏感,处理强非线性及复杂串联系统问题时,能在少样本量下得到相对高准确度的计算结果,且在样本量改变时,计算结果相对稳定可靠;工程算例给出了所提方法在实际问题下的效率,体现了所提方法的工程应用价值。
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.
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.
引文
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[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.
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[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).
[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:IEEEPress,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,ROSEN-BLUTH 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).