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结构输出响应概率密度估计中分数矩求解方法

李宝玉 张磊刚 师娇 余雄庆

李宝玉, 张磊刚, 师娇, 等 . 结构输出响应概率密度估计中分数矩求解方法[J]. 北京航空航天大学学报, 2018, 44(6): 1156-1163. doi: 10.13700/j.bh.1001-5965.2017.0664
引用本文: 李宝玉, 张磊刚, 师娇, 等 . 结构输出响应概率密度估计中分数矩求解方法[J]. 北京航空航天大学学报, 2018, 44(6): 1156-1163. doi: 10.13700/j.bh.1001-5965.2017.0664
LI Baoyu, ZHANG Leigang, SHI Jiao, et al. Solution method of fractional moments involved in probability density estimation of structural output response[J]. Journal of Beijing University of Aeronautics and Astronautics, 2018, 44(6): 1156-1163. doi: 10.13700/j.bh.1001-5965.2017.0664(in Chinese)
Citation: LI Baoyu, ZHANG Leigang, SHI Jiao, et al. Solution method of fractional moments involved in probability density estimation of structural output response[J]. Journal of Beijing University of Aeronautics and Astronautics, 2018, 44(6): 1156-1163. doi: 10.13700/j.bh.1001-5965.2017.0664(in Chinese)

结构输出响应概率密度估计中分数矩求解方法

doi: 10.13700/j.bh.1001-5965.2017.0664
基金项目: 

军委装备发展部“十三五”装备预研领域基金 6140244010216HT15001

详细信息
    作者简介:

    李宝玉  男, 博士研究生, 高级工程师。主要研究方向:飞行器设计、结构可靠性设计及结构优化

    张磊刚  男, 硕士, 工程师。主要研究方向:飞行器结构可靠性设计、重要性分析及结构优化

    通讯作者:

    张磊刚, E-mail:leigang_zhang@163.com

  • 中图分类号: TB114.3

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

Funds: 

Equipment Development Department "13th Five-year" Equipment Research Field Foundation of China Central Military Commission 6140244010216HT15001

More Information
  • 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。

     

  • 图 1  平面十杆桁架结构示意图

    Figure 1.  Schematic diagram of a plane ten-bar truss structure

    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
    下载: 导出CSV

    表  2  五点高斯积分准则的积分数据[20]

    Table  2.   Integral data of five-point Gauss integral criteria [20]

    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
    下载: 导出CSV

    表  3  数值算例输入变量信息[26]

    Table  3.   Input variable information of numerical example [26]

    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。
    下载: 导出CSV

    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
    下载: 导出CSV

    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
    下载: 导出CSV

    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
    下载: 导出CSV
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出版历程
  • 收稿日期:  2017-10-25
  • 录用日期:  2017-12-15
  • 网络出版日期:  2018-06-20

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