失效概率矩独立全局灵敏度分析的高效算法
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摘要
失效概率矩独立全局灵敏度分析对指导可靠性优化至关重要,本文从乘法降维及Edgeworth级数展开的角度提出一种失效概率矩独立全局灵敏度指标分析的高效算法。所提算法通过Edgeworth级数展开将失效概率矩独立全局灵敏度指标的求解转化为输出无条件及条件前四阶整数矩的求解。对于输出的无条件及条件四阶矩的计算,基于乘法降维的思想,本文推导了重复利用计算输出的无条件矩的输入输出样本来计算条件矩以及外层关于输入变量一维积分的计算公式,使得仅需重复利用输出无条件前四阶矩求解产生的积分网格内的数据即可同时求得输出的前四阶条件矩以及外层关于输入变量的一维积分。所提算法大大提高了失效概率矩独立全局灵敏度的分析效率。航空发动机涡轮盘以及汽车前轴的分析结果验证了所提算法的高效性及准确性。
The moment-independent global reliability sensitivity analysis can provide useful information for guiding the reliability-based design optimization.This paper proposes an efficient algorithm for moment-independent global reliability sensitivity analysis based on the multiplicative dimensional reduction technique and the Edgeworth expansion.By adopting Edgeworth expansion,the estimation of the moment-independent global reliability sensitivity index is approximately converted into the estimations of the unconditional and the conditional first four-order moments of the model output and is efficiently estimated by employing the multiplicative dimensional reduction technique.Based on this technique,this paper derives the algorithms of the conditional first four-order moments and the outer expectation of the sensitivity index by repeatedly assembling the information in the integration grid obtained from the estimation process of the unconditional first four-order moments of model output.The proposed algorithm improves the efficiency for analyzing the moment-independent global reliability sensitivity.The analyses of an aeroengine turbine disk and an automobile front axle demonstrate the efficiency and accuracy of the proposed method in estimating the moment-independent global reliability sensitivity index.
The moment-independent global reliability sensitivity analysis can provide useful information for guiding the reliability-based design optimization.This paper proposes an efficient algorithm for moment-independent global reliability sensitivity analysis based on the multiplicative dimensional reduction technique and the Edgeworth expansion.By adopting Edgeworth expansion,the estimation of the moment-independent global reliability sensitivity index is approximately converted into the estimations of the unconditional and the conditional first four-order moments of the model output and is efficiently estimated by employing the multiplicative dimensional reduction technique.Based on this technique,this paper derives the algorithms of the conditional first four-order moments and the outer expectation of the sensitivity index by repeatedly assembling the information in the integration grid obtained from the estimation process of the unconditional first four-order moments of model output.The proposed algorithm improves the efficiency for analyzing the moment-independent global reliability sensitivity.The analyses of an aeroengine turbine disk and an automobile front axle demonstrate the efficiency and accuracy of the proposed method in estimating the moment-independent global reliability sensitivity index.
引文
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[17]张磊刚,吕震宙,陈军.基于失效概率的矩独立重要性测度的高效算法[J].航空学报,2014,35(8):2199-2206.ZHANG L G,LYU Z Z,CHENG J.An efficient method for failure probability-based moment-independent importance measure[J].Acta Aeronautica et Astronautica Sinica,2014,35(8):2199-2206(in Chinese).
[18]SUDRET B.Global sensitivity analysis using polynomial chaos expansion[J].Reliability Engineering&System Safety,2008,93:964-979.
[19]SHAN S,YOUNES A,MARWAN F,et al.Bayesian sparse polynomial chaos expansion for global sensitivity analysis[J].Computer Methods in Applied Mechanics&Engineering,2017,318:474-496.
[20]WU Z P,WANG D H,OKOLO N P,et al.Global sensitivity analysis using a Gaussian radial basis function metamodel[J].Reliability Engineering and System Safety,2016,154:171-179.
[21]WEI P F,LU Z Z,HAO W R,et al.Efficient sampling methods for global reliability sensitivity analysis[J].Computation Physics Communication,2012,183(8):1728-1743.
[22]YUN W Y,LU Z Z,JIANG X,et al.An efficient method for estimating global sensitivity indices[J].International Journal for Numerical Methods in Engineering,2016,108:1275-1289.
[23]NOH Y G,CHOI K K,DU L.Reliability-based design optimization of problems with correlated input variables using a Gaussian copula[J].Structural and Multidisciplinary Optimization,2009,38:1-16.
[24]ASSAF S A,ZIRKLE L D.Approximate analysis of nonlinear stochastic systems[J].International Journal of Control,1976,23(4):477-492.
[2]TARANTOLA S,KOPUSTINSKAS V,BOLADO-LAVIN R,et al.Sensitivity analysis using contribution to sample variance plot:Application to a water hammer model[J].Reliability Engineering and System Safety,2012,99:62-73.
[3]WEI P F,LU Z Z,RUAN W B,et al.Regional sensitivity analysis using revised mean and variance ratio functions[J].Reliability Engineering and System Safety,2014,121:121-135.
[4]冯凯旋,吕震宙,蒋献.基于偏导数的全局灵敏度指标的高效求解方法[J].航空学报,2018,39(3):221699.FENG K X,LYU Z Z,JIANG X.Efficient algorithm for estimating derivative-based global sensitivity index.[J].Acta Aeronautica et Astronautica Sinica,2018,39(3):221699(in Chinese).
[5]BORGONOVO E,PLISCHKE E.Sensitivity analysis:Areview of recent advances[J].European Journal of Operational Research,2016,3(1):869-887.
[6]SALTELLI A,MARIVOET J.Non-parametric statistics in sensitivity analysis for model output:A comparison of selected techniques[J].Reliability Engineering and System Safety,1900,28(2):229-253.
[7]SOBOL I M.Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates[J].Mathematics and Computers in Simulation,2011,55:271-280.
[8]SALTELLI A,ANNONI P,AZZINI I,et al.Variance based sensitivity analysis of model output:Design and estimator for the total sensitivity index[J].Computation Physics Communication,2010,181(2):259-270.
[9]巩祥瑞,吕震宙,左键巍.两种基于方差的全局灵敏度分析W指标改进算法[J].航空学报,2016,37(6):1888-1898.GONG X R,LYU Z Z,ZUO J W.Two importance methods for variance-based global sensitivity analysisW-indices[J].Acta Aeronautica et Astronautica Sinica,2016,37(6):1888-1898(in Chinese).
[10]BORGONOVO E.A new uncertainty importance measure[J].Reliability Engineering and System Safety,2007,92(6):771-784.
[11]CUI L J,LU Z Z,ZHAO X P.Moment-independent importance measure of basic random variable and its probability density evolution solution[J].Science China Technological Sciences,2010,53(4):1138-1145.
[12]LI L Y,LU Z Z,FENG J,et al.Moment-independent importance measure of basic variable and its state dependent parameter solution[J].Structural Safety,2012,38:40-47.
[13]YUN W Y,LU Z Z,ZHANG K C,et al.An efficient sampling method for variance-based sensitivity analysis[J].Structural Safety,2017,65:74-83.
[14]YUN W Y,LU Z Z,JIANG X.An efficient sampling approach for variance-based on the law of total variance in the successive intervals without overlapping[J].Mechanical Systems and Signal Processing,2018,106:495-510.
[15]员婉莹,吕震宙,蒋献,等.可靠性全局灵敏度指标的空间分割高效方法[J].北京航空航天大学学报,2017,43(6):1199-1207.YUAN W Y,LYU Z Z,JIANG X,et al.An efficient method for reliability global sensitivity index by space partition[J].Journal of Beijing University of Aeronautics and Astronautics,2017,43(6):1199-1207(in Chinese).
[16]ZHANG X F,PANDEY M D.An effective approximation for variance-based global sensitivity analysis[J].Reliability Engineering System Safety,2014,121:164-174.
[17]张磊刚,吕震宙,陈军.基于失效概率的矩独立重要性测度的高效算法[J].航空学报,2014,35(8):2199-2206.ZHANG L G,LYU Z Z,CHENG J.An efficient method for failure probability-based moment-independent importance measure[J].Acta Aeronautica et Astronautica Sinica,2014,35(8):2199-2206(in Chinese).
[18]SUDRET B.Global sensitivity analysis using polynomial chaos expansion[J].Reliability Engineering&System Safety,2008,93:964-979.
[19]SHAN S,YOUNES A,MARWAN F,et al.Bayesian sparse polynomial chaos expansion for global sensitivity analysis[J].Computer Methods in Applied Mechanics&Engineering,2017,318:474-496.
[20]WU Z P,WANG D H,OKOLO N P,et al.Global sensitivity analysis using a Gaussian radial basis function metamodel[J].Reliability Engineering and System Safety,2016,154:171-179.
[21]WEI P F,LU Z Z,HAO W R,et al.Efficient sampling methods for global reliability sensitivity analysis[J].Computation Physics Communication,2012,183(8):1728-1743.
[22]YUN W Y,LU Z Z,JIANG X,et al.An efficient method for estimating global sensitivity indices[J].International Journal for Numerical Methods in Engineering,2016,108:1275-1289.
[23]NOH Y G,CHOI K K,DU L.Reliability-based design optimization of problems with correlated input variables using a Gaussian copula[J].Structural and Multidisciplinary Optimization,2009,38:1-16.
[24]ASSAF S A,ZIRKLE L D.Approximate analysis of nonlinear stochastic systems[J].International Journal of Control,1976,23(4):477-492.