模糊分布参数的全局灵敏度分析新方法
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摘要
为合理度量随机输入变量分布参数的模糊性对输出性能统计特征的影响,提出了模糊分布参数的全局灵敏度效应指标,并研究了指标的高效求解方法。首先,分析了不确定性从模糊分布参数至模型输出响应统计特征的传递机理,以输出性能期望响应为例,利用输出均值的无条件隶属函数与给定模糊分布参数取值条件下的隶属函数的平均差异来度量模糊分布参数的影响,建立了模糊分布参数的全局灵敏度效应指标。其次,为减少所提指标的计算成本、提高计算效率,采用了扩展蒙特卡罗模拟法(EMCS)来估算输入变量分布参数与模型输出响应统计特征的函数关系。最后通过对算例的计算,验证该文所提方法的准确性和高效性。
To measure the contribution of the fuzzy distribution parameters of random model inputs to the statistical characters of model output reasonably, the global sensitivity effect index of the fuzzy distribution parameters is proposed, and the efficient solution is studied. Firstly, according to the propagation of uncertainties from the fuzzy distribution parameters to the statistical characters of output, the mean of model output for instance is taken, and the contribution of the fuzzy distribution parameters is measured by using the average difference between the unconditional membership function of the mean of output and the membership function under the condition that one distribution parameter is fixed, and then the definition of the global sensitivity effect index is given. Secondly, to reduce the computational cost of the proposed index and improving the computation efficiency, the extended Monte Carlo simulation(EMCS) is applied for estimating the function relationship between the distribution parameters of model inputs and the statistical characters of model output. Finally, two analytical examples and the cylindrical pressure vessel engineering example are used to verify the accuracy and efficiency of the proposed method.
To measure the contribution of the fuzzy distribution parameters of random model inputs to the statistical characters of model output reasonably, the global sensitivity effect index of the fuzzy distribution parameters is proposed, and the efficient solution is studied. Firstly, according to the propagation of uncertainties from the fuzzy distribution parameters to the statistical characters of output, the mean of model output for instance is taken, and the contribution of the fuzzy distribution parameters is measured by using the average difference between the unconditional membership function of the mean of output and the membership function under the condition that one distribution parameter is fixed, and then the definition of the global sensitivity effect index is given. Secondly, to reduce the computational cost of the proposed index and improving the computation efficiency, the extended Monte Carlo simulation(EMCS) is applied for estimating the function relationship between the distribution parameters of model inputs and the statistical characters of model output. Finally, two analytical examples and the cylindrical pressure vessel engineering example are used to verify the accuracy and efficiency of the proposed method.
引文
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[23]Wei P,Lu Z,Song J.Extended monte carlo simulation for parametric global sensitivity analysis and optimization[J].AIAA Journal,2014,52(4):867―878.
[2]Sobol’I M.Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates[J].Mathematics and Computers in Simulation,2001,55(1/2/3):271―280.
[3]Yao W,Chen X,Luo W,et al.Review of uncertainty-based multidisciplinary design optimization methods for aerospace vehicles[J].Progress in Aerospace Sciences,2011,47(6):450―479.
[4]Helton J C,Davis F J.Latin hypercube sampling and the propagation of uncertainty in analyses of complex systems[J].Reliability Engineering&System Safety,2003,81(1):23―69.
[5]Saltelli A,Marivoet J.Non-parametric statistics in sensitivity analysis for model output:A comparison of selected techniques[J].Reliability Engineering&System Safety,1990,28(90):229―253.
[6]Saltelli A.Sensitivity analysis for importance assessment[J].Risk Analysis,2002,22(3):579―590.
[7]Iman R L,Hora S C.A robust measure of uncertainty importance for use in fault tree system analysis[J].Risk Analysis,2006,26(5):1349―1361.
[8]Chun M H,Han S J,Tak N I L.An uncertainty importance measure using a distance metric for the change in a cumulative distribution function[J].Reliability Engineering&System Safety,2000,70(3):313―321.
[9]Liu H,Chen W,Sudjianto A.Relative entropy based method for probabilistic sensitivity analysis in engineering design[J].Journal of Mechanical Design,2006,128(2):326―336.
[10]Borgonovo E.A new uncertainty importance measure[J].Reliability Engineering&System Safety,2007,92(6):771―784.
[11]Xu C G,George Z G.Uncertainty and sensitivity analysis for models with correlated parameters[J].Reliability Engineering and System Safety,2008,93(6):1563―1573.
[12]Mara T A,Tarantola S.Variance-based sensitivity indices for models with dependent inputs[J].Reliability Engineering&System Safety,2012,107:115―121.
[13]Krzykacz-Hausmann B.An approximate sensitivity analysis of results from complex computer models in the presence of epistemic and aleatory uncertainties[J].Reliability Engineering&System Safety,2006,91(10):1210―1218.
[14]Wang P,Lu Z,Tang Z.An application of the Kriging method in global sensitivity analysis with parameter uncertainty[J].Applied Mathematical Modelling,2013,37(9):6543―6555.
[15]Sankararaman S,Mahadevan S.Separating the contributions of variability and parameter uncertainty in probability distributions[J].Reliability Engineering&System Safety,2012,112:187―199.
[16]王攀,吕震宙,程蕾.不确定概率分布下的重要性测度及其稀疏网格解[J].中国科学:技术科学,2013,43(10):1094―1100.Wang Pan,LüZhenzhou,Cheng Lei.Importance measures for imprecise probability distributions and their sparse grid solutions[J].Science China Technological Sciences,2013,56(7):1733―1739.(in Chinese)
[17]Li Luyi,LüZhenzhou.Importance analysis for model with mixed uncertainties[J].Fuzzy Sets and System,2014,250(1):69―89.
[18]Cheng L,Lu Z,Li L.Global sensitivity analysis of fuzzy distribution parameter on failure probability and its single-loop estimation[J].Journal of Applied Mathematics,2014:1―11.
[19]吕媛波,吕震宙,唐樟春.模糊分布参数情况下可靠性模型及其矩估计方法[J].工程力学,2010,27(12):96―101.LüYuanbo,LüZhenzhou,Tang Zhangchun.Reliability model involving fuzzy distribution parameters and its moment estimation[J].Engineering Mechanics,2010,27(12):96―101.(in Chinese)
[20]李璐祎,吕震宙.模糊可靠度隶属函数求解的迭代线抽样法[J].工程力学,2011,28(5):13―20.Li Luyi,LüZhenzhou.Membership function of fuzzy reliability analysis by iteration based line samping[J].Engineering Mechanics,2011,28(5):13―20.(in Chinese)
[21]Rubinstein R Y,Kroese D P.Simulation and the monte carlo method[M].2nd ed.New York,USA:John Wiley&Sons,2011:1―372.
[22]Norouzi M,Nikolaidis E.Integrating subset simulation with probabilistic re-analysis to estimate reliability of dynamic systems[J].Structural and Multidisciplinary Optimization,2013,48(3):533―548.
[23]Wei P,Lu Z,Song J.Extended monte carlo simulation for parametric global sensitivity analysis and optimization[J].AIAA Journal,2014,52(4):867―878.