IMPROVED RELIABILITY METHOD BASED ON HLRF AND MODIFIED SYMMETRIC RANK 1 METHOD
-
摘要: 一次可靠度方法简单、高效，但在处理强非线性功能函数时存在较大误差；已有的二次可靠度方法在提高精度的同时往往降低了效率。为此，该文中在发展改进一次可靠度方法的同时提出了更好地兼顾精度与效率的改进二次可靠度方法。将修正对称秩1方法与HLRF法的步长确定策略相结合，提出了具有较好收敛性的改进一次可靠度方法，且在基本不增加计算量的前提下获得了功能函数的近似Hessian矩阵；结合坐标旋转、单变量降维近似和非中心卡方分布，提出了与改进一次可靠度方法同效率但具有更高精度的改进二次可靠度方法；通过数值算例和工程算例验证了建议方法的广泛适用性以及精度或效率上的优势。
-
关键词:
- 结构 /
- 可靠度方法 /
- 改进一次可靠度方法 /
- 改进二次可靠度方法 /
- 近似Hessian矩阵
Abstract: The first-order reliability method (FORM) is simple and efficient, but the error is significant when dealing with strong nonlinear functions. With existing second-order reliability methods (SORM), the calculation accuracy is improve, but the efficiency is reduced. In this research, an improved SORM, which can achieve better balance between accuracy and efficiency, is presented. The modified symmetric rank 1 method is combined with the determination of step length of the HLRF method, and an improved FORM with better convergence is proposed, in which the approximate Hessian matrix of performance function is obtained without increasing the amount of function evaluations. Combining the coordinate rotation with the univariate dimensional reduction approximation of the performance function according to its known gradient vector and Hessian matrix, and introducing the non-central chi-square distribution, the paper proposes an improved SORM with the same efficiency but higher accuracy. The wide applicability and advantages in both accuracy and efficiency of the proposed methods are verified by several numerical examples and engineering examples. -
表 1 常用分布的mi
Table 1. mi for common distribution
分布类型 mi 正态分布 1 极值I型分布 4 对数正态分布 $\max \{ 2,floor[2 + 20({\sigma _{\ln }} - 0.1)/3]\} $ 注：floor$ \left(A\right) $函数指小于$ A $的最大整数值。 
下载: 导出CSV
表 2 算例1中随机变量的统计特征
Table 2. Statistical characteristics of the random variables in Example 1
变量 概率分布 均值 变异系数 $ {X}_{1} $ 对数正态分布 1.0 0.1600 $ {X}_{2} $ 极值I型分布 20.0 0.1000 $ {X}_{3} $ 韦布尔分布 48.0 0.0625
下载: 导出CSV
表 3 算例1中可靠度计算结果
Table 3. computed reliability results of Example 1
方法 分析次数 可靠指标 失效概率/
(×10−2)误差/(%) MCS 1×108 2.9040 0.1842 − HLRF 21 3.0855 0.1016 44.84 HLRF-BFGS 13 3.0855 0.1016 44.84 Breitung 21+6 2.8972 0.1882 2.19 拟牛顿近似 13 2.8967 0.1886 2.36 建议方法1 11 3.0855 0.1016 44.84 建议方法2-1 11 2.8956 0.1892 2.71 建议方法2-2 11 2.8956 0.1892 2.71
下载: 导出CSV
表 4 算例2中可靠度计算结果
Table 4. computed reliability results of Example 2
方法 分析次数 可靠指标 失效概率/(×10−2) 误差/(%) MCS 1×108 2.1890 1.4301 − HLRF 8 2.0457 2.0392 42.59 HLRF-BFGS 8 2.0457 2.0392 42.59 Breitung 8+10 2.1644 1.5217 6.42 拟牛顿近似 8 2.1644 1.5217 6.42 建议方法1 8 2.0457 2.0392 42.59 建议方法2-1 8 2.2125 1.3466 5.82
下载: 导出CSV
表 5 确定性参数
Table 5. deterministic parameters
基础确定性参数 数值 宽度B/m 30 长度L/m 40 嵌入深度D/m 3 地层厚度H/m 10 角数m 4 影响因素I1 0.073 影响因素I2 0.089 影响因素IF 0.950
下载: 导出CSV
表 6 算例3中随机变量的统计特征
Table 6. Statistical characteristics of the random variables in Example 3
变量 概率分布 均值 标准差 均布荷载$ {q}_{0} $/kPa 对数正态分布 280.00 40.00 泊松比$ \nu $ 对数正态分布 0.25 0.08 弹性模量${E}_{{\rm{s}}}$/MPa 正态分布 70.00 2.50
下载: 导出CSV
表 7 算例3中可靠度计算结果
Table 7. computed reliability results of Example 3
方法 分析次数 可靠指标 失效概率/
(×10−2)误差/(%) MCS 1×108 3.0697 0.1071 − HLRF 52 3.0259 0.1239 15.69 HLRF-BFGS 20 3.0259 0.1239 15.69 Breitung 52+6 3.0668 0.1082 0.98 拟牛顿近似 20 3.0390 0.1187 10.78 建议方法1 20 3.0259 0.1239 15.69 建议方法2-1 20 3.0741 0.1056 1.46
下载: 导出CSV
表 8 算例4中随机变量的统计特征
Table 8. Statistical characteristics of the random variables in Example 4
随机变量 均值 标准差 分布类型 壁厚t/mm 4.0 0.04 正态分布 直径d/mm 40.0 0.40 正态分布 长度L1/mm 120.0 6.00 正态分布 长度L2/mm 60.0 3.00 正态分布 外力F1/kN 3.0 0.30 正态分布 外力F2/kN 3.0 0.30 正态分布 外力P/kN 12.0 1.20 正态分布 扭转T/Nm 90.0 9.00 正态分布 屈服强度Sy/MPa 350.0 50.00 正态分布
下载: 导出CSV
表 9 算例4中可靠度计算结果
Table 9. computed reliability results of Example 4
方法 分析次数 可靠指标 失效概率/
(×10−2)误差/(%) MCS 1×108 3.4027 0.033 36 − HLRF 40 3.4042 0.033 18 0.539 HLRF-BFGS 50 3.4042 0.033 18 0.539 Breitung 40+45 3.4033 0.033 29 0.209 拟牛顿近似 50 3.4015 0.033 51 0.449 建议方法1 40 3.4042 0.033 18 0.539 建议方法2-1 40 3.4025 0.033 38 0.059
下载: 导出CSV
-
[1] 赵国藩. 工程结构可靠度 [M]. 北京: 水利电力出版社, 1984.Zhao Guofan. Reliability theory for engineering structures [M]. Beijing: Water Resources and Electric Power Press, 1984. (in Chinese) [2] 贡金鑫. 工程结构可靠度计算方法[M]. 大连: 大连理工大学出版社, 2003.Gong Jinxin. Computational methods for reliability of engineering structures [M]. Dalian: Dalian University of Technology Press, 2003. (in Chinese) [3] Hasofer A M, Lind N C. Exact and invariant second-moment code format [J]. Journal of the Engineering Mechanics Division, 1974, 100(1): 111 − 121. doi: 10.1061/JMCEA3.0001848 [4] Tu J, Choi K K, Park Y H. Design potential method for robust system parameter design [J]. AIAA Journal, 2001, 39(4): 667 − 677. doi: 10.2514/2.1360 [5] Rackwitz R, Fiessler B. Structural reliability under combined random load sequences [J]. Computers & Structures, 1978, 9: 489 − 494. [6] 秦琪, 张玄一, 卢朝辉, 等. 简化三阶矩拟正态变换及其在结构可靠度分析中的应用[J]. 工程力学, 2020, 37(12): 78 − 86, 113. doi: 10.6052/j.issn.1000-4750.2020.01.0015Qin Qi, Zhang Xuanyi, Lu Zhaohui, et al. Simplified third-order normal transformation and its application in structural reliability analysis [J]. Engineering Mechanics, 2020, 37(12): 78 − 86, 113. (in Chinese) doi: 10.6052/j.issn.1000-4750.2020.01.0015 [7] 蒋水华, 姚池, 杨建华, 等. 基于模型修正的空间变异边坡可靠度分析方法[J]. 工程力学, 2018, 35(8): 154 − 161. doi: 10.6052/j.issn.1000-4750.2017.04.0308Jiang Shuihua, Yao Chi, Yang Jianhua, et al. Model correction factor method based approach for reliability analysis of spatially variable slopes [J]. Engineering Mechanics, 2018, 35(8): 154 − 161. (in Chinese) doi: 10.6052/j.issn.1000-4750.2017.04.0308 [8] Liu P L, Der Kiureghian A. Optimization algorithms for structural reliability [J]. Structural Safety, 1991, 9(3): 161 − 177. doi: 10.1016/0167-4730(91)90041-7 [9] Santosh T V, Saraf R K, Ghosh A K. Optimum step length selection rule in modified HL–RF method for structural reliability [J]. International Journal of Pressure Vessels & Piping, 2006, 83(10): 742 − 748. [10] Santos S R, Matioli L C, Beck A T. New optimization algorithms for structural reliability analysis [J]. Computer Modeling in Engineering & Sciences, 2012, 83(1): 23 − 55. [11] 亢战, 罗阳军. 计算结构可靠度指标的修正迭代算法[J]. 工程力学, 2008, 25(11): 20 − 26.Kang Zhan, Luo Yangjun. A modified iteration algorithm for structural reliability index evaluation [J]. Engineering Mechanics, 2008, 25(11): 20 − 26. (in Chinese) [12] Periçaro G A, Santos S R, Ribeiro A A, et al. HLRF–BFGS optimization algorithm for structural reliability [J]. Applied Mathematical Modelling, 2015, 39(7): 2025 − 2035. doi: 10.1016/j.apm.2014.10.024 [13] Mikosch T V, Resnick S I, Robinson S M. Numerical optimization [M]. New York: Springer, 2006. [14] Breitung K. Asymptotic approximations for multinormal integrals [J]. Journal of Engineering Mechanics, 1984, 110(3): 357 − 366. doi: 10.1061/(ASCE)0733-9399(1984)110:3(357) [15] Zhang J, Du X. A Second-order reliability method with first-order efficiency [J]. Journal of Mechanical Design, 2010, 132(10): 101006. doi: 10.1115/1.4002459 [16] Hohenbichler M, Rackwitz R. Improvement of second-order reliability estimates by importance sampling [J]. Journal of Engineering Mechanics, 1988, 114(12): 2195 − 2199. doi: 10.1061/(ASCE)0733-9399(1988)114:12(2195) [17] Zhang Z, Jiang C, Wang G G, et al. First and second order approximate reliability analysis methods using evidence theory [J]. Reliability Engineering & System Safety, 2015, 137: 40 − 49. [18] Tvedt L. Distribution of quadratic forms in normal space-application to structural reliability [J]. Journal of Engineering Mechanics, 1990, 116: 1183 − 1197. [19] Lim J, Lee B, Lee I. Second-order reliability method-based inverse reliability analysis using Hessian update for accurate and efficient reliability-based design optimization [J]. International Journal for Numerical Methods in Engineering, 2014, 100(10): 773 − 792. doi: 10.1002/nme.4775 [20] Kang S B, Park J W, Lee I. Accuracy improvement of the most probable point-based dimension reduction method using the hessian matrix [J]. International Journal for Numerical Methods in Engineering, 2017, 111(3): 203 − 217. doi: 10.1002/nme.5458 [21] Zeng P, Jimenez R, Li T. An efficient quasi-Newton approximation-based SORM to estimate the reliability of geotechnical problems [J]. Computers and Geotechnics, 2016, 76: 33 − 42. doi: 10.1016/j.compgeo.2016.02.003 [22] Modarres F, Malik A H, Leong W J. Improved Hessian approximation with modified secant equations for symmetric rank-one method [J]. Journal of Computational and Applied Mathematics, 2011, 235(8): 2423 − 2431. doi: 10.1016/j.cam.2010.10.042 [23] Zu Y F, Fan W L. Investigation of equivalent correlation coefficient based on the Mehler's formula [J]. Engineering Computations, 2019, 36(4): 1169 − 1200. doi: 10.1108/EC-07-2018-0316 [24] Wei Z, Li G, Qi L. New quasi-Newton methods for unconstrained optimization problems [J]. Applied Mathematics and Computation, 2006, 175(2): 1156 − 1188. doi: 10.1016/j.amc.2005.08.027 [25] 唐贞云, 王志宇, 杜修力. 时域稳定的基础频响连续有理近似参数识别方法[J]. 工程力学, 2022, 39(4): 29 − 38. doi: 10.6052/j.issn.1000-4750.2021.01.0108Tang Zhenyun, Wang Zhiyu, Du Xiuli. A stable parameter identification method in the time domain for the frequency response of foundations based on continuous-time rational approximation [J]. Engineering Mechanics, 2022, 39(4): 29 − 38. (in Chinese) doi: 10.6052/j.issn.1000-4750.2021.01.0108 [26] Modarres F, Hassan M A, Leong W J. A symmetric rank-one method based on extra updating techniques for unconstrained optimization [J]. Computers & Mathematics with Applications, 2011, 62(1): 392 − 400. [27] Adhikari S. Reliability analysis using parabolic failure surface approximation [J]. Journal of Engineering Mechanics, 2004, 130(12): 1407 − 1427. doi: 10.1061/(ASCE)0733-9399(2004)130:12(1407) [28] Lee I, Yoo D, Noh Y. A novel second-order reliability method (SORM) using non-central or generalized chi-squared distributions [J]. Journal of Mechanical Design, 2012, 134(10): 100912. doi: 10.1115/1.4007391 [29] Rahman S, Wei D. A univariate approximation at most probable point for higher-order reliability analysis [J]. International Journal of Solids and Structure, 2006, 43(9): 2820 − 2839. doi: 10.1016/j.ijsolstr.2005.05.053 [30] Provost S B, Rudiuk E M. The exact distribution of indefinite quadratic forms in noncentral normal vectors [J]. Annals of the Institute of Statistical Mathematics, 1996, 48(2): 381 − 394. doi: 10.1007/BF00054797 [31] Johnson N L, Kotz S, Balakrishnan N. Continuous univariate distributions, Volume 2 [M]. New Jersey: Wiley & Sons, 1994. [32] Chan C L, Low B K. Practical second-order reliability analysis applied to foundation engineering [J]. International Journal for Numerical and Analytical Methods in Geomechanics, 2012, 36(11): 1387 − 1409. doi: 10.1002/nag.1057 [33] Guo J, Du X. Reliability sensitivity analysis with random and interval variables [J]. International Journal for Numerical Methods in Engineering, 2009, 78(13): 1585 − 1617. doi: 10.1002/nme.2543 -
点击查看大图
图(1) / 表(9)
计量
- 文章访问数: 464
- HTML全文浏览量: 105
- PDF下载量: 140
- 被引次数: 0
