﻿ 结构输出响应概率密度估计中分数矩求解方法<sup>*</sup>
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1. 南京航空航天大学 航空宇航学院, 南京 210016;
2. 中国运载火箭技术研究院, 北京 100076

Solution method of fractional moments involved in probability density estimation of structural output response
LI Baoyu1,2, ZHANG Leigang2, SHI Jiao2, YU Xiongqing1
1. College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China;
2. China Academy of Launch Vehicle Technology, Beijing 100076, China
Received: 2017-10-25; Accepted: 2017-12-15; Published online: 2018-03-07 10:52
Foundation item: Equipment Development Department "13th Five-year" Equipment Research Field Foundation of China Central Military Commission (6140244010216HT15001)
Corresponding author. ZHANG Leigang, E-mail:leigang_zhang@163.com
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.
Key words: principle of maximum entropy     fractional moments     dimension reduction integration (DRI)     sparse grid integration (SGI)     unscented transformation (UT)

1 分数矩概念

 (1)

 (2)

 (3)

 (4)

2 分数矩求解方法 2.1 降维积分方法

 (5)

φ(X)按Cut-HDMR[19]形式展开, 表达为

 (6)

 (7)

 (8)

 (9)

 (10)

 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

2.2 稀疏网格积分方法

 (11)

 (12)

 (13)

2.3 无迹变换方法

i=0时,

 (14)

i=1, 2, …, n时,

 (15)

 (16)
3 计算效率及适用性分析 3.1 计算效率

 (17)

SGI方法调用函数次数是与所选择的精度水平k以及输入变量维数n有关。根据被积分函数的复杂程度确定k的取值, 一般情况下取2或3即可满足要求, 相应的函数估计次数NSGI

 (18)

UT方法选取2n+1个Sigma点并计算相应的函数值, 因此UT方法的函数估计次数NUT

 (19)

3.2 适用性分析

DRI方法通过对输出响应函数的变换, 用多个一元函数分数矩乘积求解一个多元函数的分数矩, 解决了多维积分的“维数灾难”问题。然而, 通过式(10)的转换过程可以看出, 输入向量X的联合概率密度函数, 进而实现了多元函数积分向一元函数的转化, 此转化的前提是输入变量之间需相互独立, 因此也即决定了DRI方法仅适用于输入变量相互独立的分数矩求解问题, 对于相关问题需选择其他方法进行分数矩求解。

SGI方法被广泛地应用于随机不确定性的传递中, 同时在微分方程的求解以及数值积分、差值等方面发挥了很大作用, 是一种能够有效用于多维复杂问题的离散化方法[22]。SGI方法对多元积分问题具有很好的适用性, 针对不同的输入分布类型, 可以采用任何类型的一维积分点进行适应匹配, 实现过程灵活简单。此外, SGI方法的使用对与输入变量是否具有相关性无要求, 可根据问题的复杂程度及实际需求选择核实的精度水平, 因此, SGI方法对于大部分的分数矩求解问题普遍适用。

UT方法效率远高于Monte Carlo等一般数值模拟方法, 尤其针对非线性程度不高的问题, 可以高效地且仅采用2n+1次函数调用即能求得函数的低阶矩。然而, 其精度有限, 如果所研究问题的非线性程度较高, 则UT方法的精度水平会下降[24]。实际上, 若用UT方法进行分数矩计算, 尤其是式(1)中所述的极大熵准则分数矩约束计算时, UT方法的精度已能满足使用要求, 因为仅仅只需要几个低阶次的分数矩就可以提供输出响应函数的大量整数阶矩信息。然而, 若需要计算复杂函数的更高阶矩信息时, 则可采用高阶UT方法[25]。此外, 同SGI方法, UT方法也适用于结构输入变量相关时的矩信息计算。

4 算例分析

4.1 数值算例

 (20)

 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。

 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

4.2 工程算例

 图 1 平面十杆桁架结构示意图 Fig. 1 Schematic diagram of a plane 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

 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

5 结论

1) 本文研究了降维积分(DRI)方法、稀疏网格积分(SGI)方法及无迹变换(UT)方法3种分数矩求解方法, 介绍了分数矩求解原理及过程, 给出了各方法的计算效率, 并分析了各方法的适用性。

2) 通过数值与具有隐式输入-输出关系的工程算例验证了3种分数矩求解方法在计算变量分数矩时具有较大的优势, 验证了本文方法均只需要少量的函数估计次数即可获得较高精度的计算结果。

3) 选择性地应用3种分数矩求解方法进行结构系统分数矩的计算, 可有效推动极大熵等密度估计技术在工程结构可靠性分析中的推广应用, 进而促进结构可靠性分析技术的工程适用性。

 [1] 王晓军, 杨海峰, 邱志平, 等. 基于测量数据的不确定性结构分析的模糊理论[J]. 北京航空航天大学学报, 2010, 36 (8): 887–891. WANG X J, YANG H F, QIU Z P, et al. Fuzzy theory for uncertain structural analysis based on measurement data[J]. Journal of Beijing University of Aeronautics and Astronautics, 2010, 36 (8): 887–891. (in Chinese) [2] WANG X J, WANG L, ELISHAKOFF I, et al. Probability and convexity concepts are not antagonistic[J]. Acta Mechanica, 2011, 219 (1-2): 45–64. DOI:10.1007/s00707-010-0440-4 [3] QIU Z P, WANG L. The need for introduction of non-probabilistic interval conceptions into structural analysis and design[J]. Science China-Physics, Mechanics & Astronomy, 2016, 59 (11): 114632. [4] ZHAO Y G, ONO T. A general procedure for first/second-order reliability method (FORM/SORM)[J]. Structural Safety, 1999, 21 (2): 95–112. DOI:10.1016/S0167-4730(99)00008-9 [5] KIUREGHIAN A D. The geometry of random vibrations and solutions by FORM and SORM[J]. Probabilistic Engineering Mechanics, 2000, 15 (1): 81–90. [6] 吕震宙, 宋述芳, 李洪双, 等. 结构机构可靠性及可靠性灵敏度分析[M]. 北京: 科学出版社, 2009. LU Z Z, SONG S F, LI H S, et al. Reliability and reliability sensitivity analysis of structural and mechanism[M]. Beijing: Science Press, 2009. (in Chinese) [7] MELCHERS R E. Importance sampling in structural system[J]. Structural Safety, 1989, 6 (1): 3–10. DOI:10.1016/0167-4730(89)90003-9 [8] ZHANG L G, LU Z Z, WANG P. Efficient structural reliability analysis method based on advanced Kriging model[J]. Applied Mathematical Modelling, 2015, 39 (2): 781–793. DOI:10.1016/j.apm.2014.07.008 [9] ROCCO C, MORENO J. Fast Monte Carlo reliability evaluation using support vector machine[J]. Reliability Engineering and System Safety, 2002, 76 (3): 237–243. DOI:10.1016/S0951-8320(02)00015-7 [10] JAYNES E. Information theory and statistical mechanics[J]. Physical Review, 1957, 108 (2): 171–190. DOI:10.1103/PhysRev.108.171 [11] INVERARDI P, TAGLIANI A. Maximum entropy density estimation from fractional moments[J]. Communications in Statistics-Theory and Methods, 2003, 32 (2): 327–345. DOI:10.1081/STA-120018189 [12] ZHANG X F, PANDEY M D. Structural reliability analysis based on the concepts of entropy, fractional moment and dimensional reduction method[J]. Structural Safety, 2013, 43 : 28–40. DOI:10.1016/j.strusafe.2013.03.001 [13] ZHANG L G, LU Z Z, CHENG L, et al. A new method for eva-luating Borgonovo moment-independent importance measure with its application in an aircraft structure[J]. Reliability Engineering and System Safety, 2014, 132 : 163–175. DOI:10.1016/j.ress.2014.07.011 [14] 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, LU Z Z, CHEN 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) [15] LI B Y, ZHANG L G, ZHU X J, et al. Reliability analysis based on a novel density estimation method for structures with correlations[J]. Chinese Journal of Aeronautics, 2017, 30 (3): 1021–1030. DOI:10.1016/j.cja.2017.04.005 [16] 郭健彬, 曾声奎, 陈云霞. 稳健协同优化方法的改进和应用[J]. 火力与指挥控制, 2010, 35 (4): 32–35. GUO J B, ZENG S K, CHEN Y X. Improvement and application of multidisciplinary robust design optimization method[J]. Fire Control and Command Control, 2010, 35 (4): 32–35. (in Chinese) [17] GUO J B, ZHAO Z T, ZHAO J Y, et al. Integral robust design method based on maximum tolerance region[J]. Journal of Donghua University, 2014, 31 (6): 737–740. [18] LI G, ROSENTHAL C, RABITZ H. High dimensional model representations[J]. Journal of Physical Chemistry, 2001, 105 (33): 7756–7777. [19] 张旭方, PANDEYM D, 张义民. 结构随机响应计算的一种数值方法[J]. 中国科学:技术科学, 2012, 42 (1): 103–114. ZHANG X F, PANDEY M D, ZHANG Y M. A numerical method for calculating the random response of structures[J]. Scientia Sinica Technologica, 2012, 42 (1): 103–114. (in Chinese) [20] 张磊刚. 不确定性结构的局部和矩独立灵敏度方法研究[D]. 西安: 西北工业大学, 2015. ZHANG L G. Study of local and moment-independent sensitivity analysis method for structures with uncertainty[D]. Xi'an: Northwestern Polytechnical University, 2015(in Chinese). [21] SMOLYAK S A. Quadrature and interpolation formulas for tensor products of certain classes of functions[J]. Soviet Mathematics Doklady, 1963, 4 : 240–243. [22] GERSTNER T, GRIEBEL M. Numerical integration using sparse grids[J]. Number Algorithms, 1998, 18 (3-4): 209–232. [23] ZHANG L G, LU Z Z, CHENG L, et al. Moment-independent regional sensitivity analysis of the complicated models with great efficiency[J]. International Journal for Numerical Methods in Engineering, 2015, 103 (13): 996–1014. DOI:10.1002/nme.v103.13 [24] JULIER S J, UHLMANN J K. Unscented filtering and nonlinear estimation[J]. Proceedings of the IEEE, 2004, 92 (3): 401–422. DOI:10.1109/JPROC.2003.823141 [25] 张勇刚, 费玉龙, 武哲民, 等. 一种高阶无迹卡尔曼滤波方法[J]. 自动化学报, 2014, 40 (5): 838–848. ZHANG Y G, FEI Y L, WU Z M, et al. A high order unscented Kalman filtering method[J]. Acta Automatica Sinica, 2014, 40 (5): 838–848. (in Chinese) [26] IMAN R L. A matrix-based approach to uncertainty and sensitivity analysis for fault tree[J]. Risk Analysis, 1987, 7 (1): 21–33. DOI:10.1111/risk.1987.7.issue-1

文章信息

LI Baoyu, ZHANG Leigang, SHI Jiao, YU Xiongqing

Solution method of fractional moments involved in probability density estimation of structural output response

Journal of Beijing University of Aeronautics and Astronsutics, 2018, 44(6): 1156-1163
http://dx.doi.org/10.13700/j.bh.1001-5965.2017.0664