Solution method of fractional moments involved in probability density estimation of structural output response
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摘要:
鉴于概率不确定性背景下基于分数矩极大熵准则的结构可靠性分析方法具有较大的效率与精度优势,综合研究并给出了可以用于极大熵准则中约束条件输出响应分数矩求解的3种分数矩求解方法,包括降维积分(DRI)方法、稀疏网格积分(SGI)方法和无迹变换(UT)方法。阐述了分数矩求解原理及过程,给出了方法的计算效率,并分析了方法的适用性。3种分数矩求解方法在确保计算精度的同时可以很大程度减少结构输入-输出模型的调用次数,大幅提高统计分析效率。通过与Monte Carlo仿真分析法对比,验证了3种分数矩求解方法的正确性与高效性。
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关键词:
- 极大熵准则 /
- 分数矩 /
- 降维积分(DRI) /
- 稀疏网格积分(SGI) /
- 无迹变换(UT)
Abstract:For the fact that the fractional moment based principle of maximum entropy for structural reliability analysis has some advantages in computational efficiency and precision, in this paper, three computational methods for accurately estimating the fractional moments of constraint condition output response involved in the principle of maximum entropy, are studied and presented, including the dimension reduction integration (DRI) method, the sparse gird integration (SGI) method and the unscented transformation (UT) method. The computational theory and process are expounded, the calculation efficiency of each method is given, and the applicability of each method is analyzed in the paper. The presented three methods can greatly reduce the number of structural input-output model estimates and ensure the accuracy of calculation at the same time, so the efficiency of statistical analysis can be greatly improved. Besides, compared with the Monte Carlo simulation method, the accuracy and efficiency of the presented methods are verified according to the applied examples.
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表 1 一元函数分数矩计算的高斯积分表达式[19]
Table 1. Gauss integral expression of one-variable function fractional moment computation[19]
分布类型 积分区域 高斯积分准则 数值积分表达式 均匀 [a, b] 高斯-
勒让德正态 (-∞, +∞) 高斯-
埃尔米特对数正态 (0, +∞) 高斯-
埃尔米特指数 (0, +∞) 高斯-
拉盖尔威布尔 (0, +∞) 高斯-
拉盖尔积分准则 积分权重与点 j=1 j=2 j=3 j=4 j=5 高斯-埃尔米特 wj 1.13×10-2 0.222 1 0.533 3 0.222 1 1.13×10-2 zj -2.857 0 -1.355 6 0 1.355 6 2.857 0 高斯-勒让德 wj 0.236 9 0.478 6 0.568 9 0.478 6 0.236 9 zj -0.906 2 -0.538 5 0 0.538 5 0.906 2 高斯-拉盖尔 wj 0.521 8 0.398 7 7.59×10-2 3.61×10-3 2.34×10-5 zj 0.263 6 1.413 4 3.596 4 7.085 8 12.641 输入变量 均值 误差因子 X1 2 2.0 X2 3 2.0 X3 1×10-3 2.0 X4 2×10-3 2.0 X5 4×10-3 2.0 X6 5×10-3 2.0 X7 3×10-3 2.0 注:误差因子表征对数正态分布的分散程度。 表 4 数值算例分数矩计算结果
Table 4. Calculation results of fractional moments of numerical example
α DRI SGI(k=2) UT SGI(k=3) Monte Carlo -0.3 13.228 13.228 13.825 13.243 13.208 -0.05 1.535 1.535 1.543 1.535 1.535 0.62 5.217×10-3 5.171×10-3 5.146×10-3 5.204×10-3 5.209×10-3 1.3 1.856×10-5 1.743×10-5 1.859×10-5 1.838×10-5 1.849×10-5 1 2.197×10-4 2.131×10-4 2.190×10-4 2.186×10-4 2.190×10-4 2 6.407×10-8 5.202×10-8 6.265×10-8 6.142×10-8 6.445×10-8 3 2.449×10-11 1.313×10-11 1.386×10-11 2.008×10-11 2.640×10-11 Ncall 36 15 15 113 104 表 5 十杆桁架结构输入变量信息
Table 5. Input variable information of ten-bar truss structure
输入变量 均值 变异系数 Ai/m2 0.001 0.15 E/GPa 100 0.05 L/m 1 0.05 P1/kN 80 0.05 P2/kN 10 0.05 P3/kN 10 0.05 表 6 十杆桁架结构分数矩计算结果
Table 6. Calculation results of fractional moments of ten-bar truss structure
方法 分数矩 Ncall d=-1.5 d=-0.47 d=0.08 d=1.8 d=2.5 DRI 0.787 4 0.925 0 1.013 7 1.382 8 1.585 6 61 SGI(k=2) 0.786 1 0.924 8 1.013 7 1.383 0 1.585 6 31 UT 0.792 9 0.926 3 1.013 5 1.382 1 1.586 9 31 Monte Carlo 0.786 3 0.924 6 1.013 7 1.384 0 1.587 2 104 -
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