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摘要:
在矩独立重要性分析过程中，重要性指标往往用于衡量结构系统输出不确定性向输入变量不确定性的逆向分配问题。假设输入参数的方差可以减缩一定比例因子，那么矩独立重要性指标可以定义为该缩减因子的函数。同时，假设输入参数的方差缩减因子为一随机变量，那么可以取矩独立重要性指标函数的均值定义一个新的平均矩独立重要性指标。由于使用Sobol方法计算平均矩独立重要性指标的模型需要循环抽样，计算量很高，故引入拒绝抽样（RS）方法，通过重复利用矩独立重要性分析中的一组输入输出样本，就可以额外计算得到矩独立指标函数和平均矩独立重要性指标，这大大节约了计算成本。本文所提指标函数及平均指标的有效性和RS方法的准确性、高效性通过数值和工程算例得以验证。
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关键词:
- 重要性分析 /
- 矩独立重要性指标函数 /
- 平均矩独立重要性指标 /
- Sobol方法 /
- 拒绝抽样(RS)
Abstract:In the process of moment-independent importance analysis, importance index are always used to quantify the inverse allocation of structural system output uncertainty to input uncertainty.An assumption is given that the variance of a given factor can be reduced by future research, which leads to the development the moment-independent index function. The moment-independent importance index function provides an index for a given factor as a function of the amount of variance of that factor can be reduced. Meanwhile, by assuming the reduction amount of a particular factor variance as a random variable, the average moment-independent importance index is defined by taking average of the moment-independent importance index function. Estimating the average moment-independent importance index involves a large amount of computation using Sobol's method, and thus rejection sampling (RS) method is introduced here with the generated samples used in Sobol's method. Consequently, RS can use the samples generated during Sobol's method to accurately estimate the moment-independent importance index function and the average moment-independent importance index without any further model evaluation, which greatly reduces the computational cost. Numerical and engineering examples are demonstrated to show the effectiveness of the proposed measures and the accuracy and availability of the RS method.
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图 1 Sobol方法求解的Ishigami测试函数重要性指标结果
Figure 1. Importance index results solved by Sobol's method for Ishigami test function
图 2 RS方法求解的Ishigami测试函数重要性指标结果
Figure 2. Importance index results solved by RS method for Ishigami test function
图 4 Sobol方法求解的屋架结构模型重要性指标结果
Figure 4. Importance index results solved by Sobol's method for roof truss structure model
图 5 RS方法求解的屋架结构模型重要性指标结果
Figure 5. Importance index results solved by RS method for roof truss structure model
图 6 平面十杆桁架结构模型示意图
Figure 6. Schematic diagram of a planar ten-bar truss structure model
图 7 Sobol方法求解的十杆桁架结构模型重要性指标结果
Figure 7. Importance index results solved by Sobol's method for ten-bar truss structure model
图 8 RS方法求解的十杆桁架结构模型重要性指标结果
Figure 8. Importance index results solved by RS method for ten-bar truss structure model
表 1 屋架结构模型中输入变量的分布参数
Table 1. Distribution parameters of input variables of roof truss structure model
分布参数 均值 变异系数 q 20 000 N/m 0.07 l 12 m 0.01 AS 9.82×10-4 m2 0.06 AC 0.04 m2 0.12 ES 1×1011 N/m2 0.06 EC 2×1010 N/m2 0.06 
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表 2 十杆桁架结构模型中输入变量的分布参数
Table 2. Distribution parameters of input variables of ten-bar truss structure model
分布参数 均值 变异系数 L 1 m 0.05 E 100 GPa 0.05 P1 80 kN 0.05 P2 10 kN 0.05 P3 10 kN 0.05 Ai 0.001 m2 0.15
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