-
摘要:
可靠性全局灵敏度指标能够有效地分析输入变量的不确定性对结构系统失效概率的影响程度，为提高该灵敏度指标求解数字模拟法的效率，提出了一种基于密度权重及连续无重叠区间全方差公式的空间分割高效方法。所提方法通过连续无重叠区间上的全方差公式来加快该指标计算的收敛速度，利用密度权重法在输入变量可能的取值区间内进行均匀抽样，并以均匀样本点的联合概率密度函数的权重来保证计算的等价性，这使得所构造的方法不需要寻找失效域的设计点，因此其可以有效解决非线性程度较高难以找到设计点及多设计点的问题。除此之外，应用空间分割技术，使得本文所提方法仅需重复利用一组样本点，就可同时得到各个输入变量的可靠性全局灵敏度指标，消除了计算量与输入变量维数的相关性，大大地提高了样本的利用率和计算效率。验证算例的计算结果，说明了本文方法对计算功能函数非线性程度较高及多设计点问题的高效性。
-
关键词:
- 可靠性全局灵敏度指标 /
- 连续无重叠区间上的全方差公式 /
- 密度权重 /
- 空间分割 /
- 非线性 /
- 多设计点
Abstract:The reliability sensitivity index well analyzes how the failure probability of a model is affected by the different sources of uncertainty in the model inputs. In order to improve the efficiency of digital simulation in estimating this index, a method was proposed based on the weighted density, the law of total variance in the successive intervals without overlapping and the space partition. To accelerate the speed of convergence, the law of total variance in the successive intervals without overlapping was proved and used subsequently. The weighted density method generates uniform samples in the possible interval of model inputs, and it can ensure the equivalence of estimation by the weighted density indices. The proposed method can avoid searching the design point; therefore, for the highly nonlinear problem which is difficult to find the design point and the problem of multiple design points, the proposed method can well deal with. In addition, by the idea of space-partition, the dependence of the computational cost on the input dimensionality is removed, and the proposed method only requires one set of input-output samples to obtain all the sensitivity indices, which greatly improves the utilization of samples and computational efficiency. Examples illustrate that the proposed method has higher efficiency, accuracy, convergence and robustness than the existing methods for the problems of high nonlinearity and multiple design points.
-
表 1 算例1的计算结果
Table 1. Calculation results for example 1
样本量 Si X1 X2 本文方法 S-MCS 本文方法 S-MCS 512 0.032 9[0.011 0] N/A 0.043 4[0.013 5] N/A 1 024 0.032 8[0.005 9] N/A 0.043 7[0.005 6] N/A 2 048 0.033 7[0.002 7] N/A 0.044 6[0.004 3] N/A 4 096 0.034 7[0.001 4] 0.029 0[0.013 4] 0.045 0[0.002 7] 0.043 0[0.024 1] 8 192 0.036 0[0.001 0] 0.030 5[0.007 1] 0.044 5[0.001 3] 0.039 6[0.009 1] D-MCS(2×104×104) 0.036 2 0.044 7 
下载: 导出CSV
表 2 算例2中α=0的Si计算结果
Table 2. Calculation results of Si when α=0 for example 2
样本量 Si X1 X2 X3 本文方法 IS S-MCS 本文方法 IS S-MCS 本文方法 IS S-MCS 512(640) 0.612 0 0.624 0 N/A 0.003 5 0.001 8 N/A 0.003 4 0.002 3 N/A 1 024(1 280) 0.601 2 0.614 1 N/A 0.003 0 0.002 8 N/A 0.002 9 0.002 0 N/A 2 048(2 560) 0.656 2 0.616 1 0.624 0 0.002 6 0.002 1 0 0.002 7 0.002 1 -0.013 3 4 096(5 120) 0.621 1 0.612 0 0.621 7 0.002 7 0.001 9 0 0.002 7 0.002 8 0.003 3 8 192(10 240) 0.610 7 0.609 9 0.607 9 0.002 7 0.002 6 0.001 3 0.002 6 0.002 3 0.000 5 D-MCS(3×104×104) 0.611 21 0.002 7 0.002 7
下载: 导出CSV
表 3 算例2中α=0时100次Si计算结果的标准差(SD)
Table 3. Standard deviation (SD) of calculation results of Si by 100 iterations when α=0 for example 2
样本量 SD X1 X2 X3 本文方法 IS S-MCS 本文方法 IS S-MCS 本文方法 IS S-MCS 512(640) 0.060 3 0.118 5 N/A 0.001 1 0.006 9 N/A 0.001 5 0.007 0 N/A 1 024(1 280) 0.043 2 0.077 4 N/A 0.000 5 0.009 8 N/A 0.000 6 0.005 0 N/A 2 048(2 560) 0.036 3 0.052 1 0.120 8 0.000 3 0.005 1 0 0.000 3 0.003 7 0.044 7 4 096(5 120) 0.024 0 0.038 2 0.039 1 0.000 2 0.002 8 0 0.000 2 0.007 1 0.002 8 8 192(1 0240) 0.014 6 0.023 1 0.018 2 0.000 1 0.005 2 0.001 3 0.000 1 0.003 7 0.001 2
下载: 导出CSV
表 4 算例2中α=1的Si计算结果
Table 4. Calculation results of Si when α=1 for example 2
样本量 Si X1 X2 X3 本文方法 IS S-MCS 本文方法 IS S-MCS 本文方法 IS S-MCS 512(640) 0.508 8 0.054 5 N/A 0.011 8 0.014 1 N/A 0.012 1 0.022 8 N/A 1 024(1 280) 0.532 4 0.324 7 0.535 7 0.010 4 0.009 2 0.010 0 0.010 6 0.017 7 0.010 0 2 048(2 560) 0.544 2 0.199 6 0.520 0 0.010 0 0.008 6 -0.012 0 0.009 9 0.011 2 0.014 7 4 096(5 120) 0.545 6 0.204 5 0.506 7 0.010 0 0.006 2 0.003 1 0.009 9 0.008 0 0.010 0 8 192(10 240) 0.548 0 0.300 8 0.485 3 0.009 6 0.006 2 0.009 6 0.009 6 0.007 8 0.011 1 D-MCS(3×104×104) 0.542 8 0.009 6 0.009 6
下载: 导出CSV
表 5 算例2中α=1时100次Si计算结果的标准差
Table 5. Standard deviation of calculation results of Si by 100 iterations when α=1 for example 2
样本量 SD X1 X2 X3 本文方法 IS S-MCS 本文方法 IS S-MCS 本文方法 IS S-MCS 512(640) 0.065 2 0.144 8 N/A 0.004 2 0.039 5 N/A 0.005 9 0.117 4 N/A 1 024(1 280) 0.049 8 2.668 8 0.264 9 0.002 2 0.019 6 0.010 0 0.002 7 0.062 3 0.010 0 2 048(2 560) 0.029 0 1.000 2 0.066 8 0.001 7 0.020 2 0.045 7 0.001 9 0.030 6 0.007 5 4 096(5 120) 0.021 9 0.534 0 0.022 2 0.001 1 0.008 4 0.002 7 0.001 4 0.015 7 0.002 0 8 192(10 240) 0.008 0 1.314 7 0.009 5 0.000 7 0.018 3 0.001 0 0.000 8 0.021 4 0.000 9
下载: 导出CSV
-
[1] SALTELLI A. Sensitivity analysis for importance assessment[J].Risk Analysis, 2002, 22(3):579-590. doi: 10.1111/risk.2002.22.issue-3 [2] BORGONOVO E, APOSTOLAKIS G E.A new importance measure for risk-informed decision-making[J].Reliability Engineering and System Safety, 2001, 72(2):193-212. doi: 10.1016/S0951-8320(00)00108-3 [3] BORGONOVO E, APOSTOLAKIS G E, TARANTOLA S, et al.Comparison of local and global sensitivity analysis techniques in probability safety assessment[J].Reliability Engineering and System Safety, 2003, 79(2):175-185. doi: 10.1016/S0951-8320(02)00228-4 [4] 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(2):62-73. https://www.researchgate.net/publication/251621927_Sensitivity_analysis_using_contribution_to_sample_variance_plot_Application_to_a_water_hammer_model [5] 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(1):121-135. https://www.researchgate.net/publication/270814729_Regional_sensitivity_analysis_using_revised_mean_and_variance_ratio_functions [6] SALTELLI A, RATTO M, ANDRES T, et al.Global sensitivity analysis[M].New York:John Wiley & Sons, 2008:115-174. [7] SALTELLI A, MARIVOET J.Non-parametric statistics in sensitivity analysis for model output:A comparison of selected techniques[J].Reliability Engineering and System Safety, 1990, 28(2):229-253. doi: 10.1016/0951-8320(90)90065-U [8] ZHANG X F, PANDEY M D.An effective approximation for variance-based global sensitivity analysis[J].Reliability Engineering and System Safety, 2014, 121(4):164-174. https://www.researchgate.net/publication/275180221_An_effective_approximation_for_variance-based_global_sensitivity_analysis [9] WEI P, LU Z Z, SONG J W.A new variance-based global sensitivity analysis technique[J].Computation Physics Communication, 2013, 184(11):2540-2551. doi: 10.1016/j.cpc.2013.07.006 [10] DEMAN G, KONAKLI K, SUDRET B, et al.Using sparse polynomial chaos expansions for the global sensitivity analysis of groundwater lifetime expectancy in multi-layered hydrogeological model[J].Reliability Engineering and System Safety, 2016, 147:156-169. doi: 10.1016/j.ress.2015.11.005 [11] PIANOSI F, WAGENER T.A simple and efficient method for global sensitivity analysis based on cumulative distribution functions[J].Environmental Modelling & Software, 2015, 67:1-11. https://www.researchgate.net/publication/271428340_A_simple_and_efficient_method_for_global_sensitivity_analysis_based_on_cumulative_distribution_functions [12] BORGONOVO E.A new uncertainty importance measure[J].Reliability Engineering and System Safety, 2007, 92(6):771-784. doi: 10.1016/j.ress.2006.04.015 [13] LIU Q, HOMMA T.A new importance measure for sensitivity analysis[J].Journal of Nuclear Science and Technology, 2010, 47(1):53-61. doi: 10.1080/18811248.2010.9711927 [14] 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. doi: 10.1007/s11431-009-0386-8 [15] 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. doi: 10.1016/j.strusafe.2012.04.001 [16] 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. http://www.cnki.com.cn/Article/CJFDTOTAL-HKXB201408013.htmZHANG L G, LYU Z Z, CHEN J.An efficient method of failure probability-based moment-independent importance measure[J].Acta Aeronautica et Astronautica Sinica, 2014, 35(8):2199-2206(in Chinese). http://www.cnki.com.cn/Article/CJFDTOTAL-HKXB201408013.htm [17] 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. doi: 10.1016/j.cpc.2012.03.014 [18] DITLEVSEN O, MADSEN H O.Structural reliability methods[M].Chichester:Wiley, 1996:102-109. [19] RASHKI M, MIRI M, MOGHADDAM M A.A new efficient simulation method to approximate the probability of failure and most probability point[J].Structural Safety, 2012, 39(4):22-29. https://www.researchgate.net/publication/257006965_A_new_efficient_simulation_method_to_approximate_the_probability_of_failure_and_most_probable_point [20] ZHAI Q Q, YANG J, ZHAO Y.Space-partition method for the variance-based sensitivity analysis:Optimal partition scheme and comparative study[J].Reliability Engineering and System Safety, 2014, 131:66-82. doi: 10.1016/j.ress.2014.06.013 [21] 吕召燕, 吕震宙, 李贵杰, 等.基于密度权重的可靠性灵敏度分析方法[J].航空学报, 2014, 35(1):179-186. http://www.cnki.com.cn/Article/CJFDTOTAL-ZJKN201701002.htmLV Z Y, LV Z Z, LI G J, et al.Reliability sensitivity analysis method based on weight index of density[J].Acta Aeronautica et Astronautica Sinica, 2014, 35(1):179-186(in Chinese). http://www.cnki.com.cn/Article/CJFDTOTAL-ZJKN201701002.htm [22] SOBOL I M.Uniformly distributed sequences with additional uniformity properties[J].USSR Computational Mathematics and Mathematical Physics, 1976, 16(5):236-242. doi: 10.1016/0041-5553(76)90154-3 -
下载:
点击查看大图
计量
- 文章访问数: 1004
- HTML全文浏览量: 166
- PDF下载量: 434
- 被引次数: 0
