针对目前多维变量可靠度分析中广泛应用的均匀设计响应面法(RSM),指出了使用最小二乘(LS)法拟合拟线性回归模型时存在的局限性,并提出采用拟线性偏最小二乘(PLS)法来回归响应面系数。由于拟线性回归法限制了模型的形式,精度提高有限,结果也很不稳定,因此又提出用基于样条变换的偏最小二乘回归模型代替拟线性回归模型并用于结构失效概率的计算,既能处理最小二乘法无法解决的变量间多重相关性的问题,又能避开拟线性回归中预先对模型形式的假定。通过算例验证了基于样条变换的偏最小二乘法的适用性和有效性,尤其对于多维变量非线性程度较高的可靠度分析,与普通最小二乘法拟合的响应面相比,其模型更加精确,失效概率精度更高。
The response surface method (RSM) with uniform design is widely used for current structural reliability analysis of multi-dimensional variables. However, it has the limitation of fitting the regression models in the original quasi-linear least squares (LS) method. To deal with this limitation, a new approach—the partial least squares (PLS) method based on the traditional method is proposed for improvement. However, the quasi-linear regression method restricts the form of the model, thus the improvement in accuracy is limited and the result is unstable. So, this paper presents a partial least squares regression model to substitute the quasi-linear model for the calculation of reliability, which not only handles the correlation between the variables but also avoids the pre-assumptions for the form of the regression model. The results of several examples show that the method proposed in this paper can be used effectively to analyze structural reliability, especially in multi-dimensional and non-linear cases, and higher accuracy can be obtained as shown by comparing the results with those from the least squares regression method.
[1] Zhao G F. Structural reliability. Beijing: Hydraulic Press, 1984: 42-68. (in Chinese) 赵国藩. 工程结构可靠度. 北京: 水利出版社, 1984: 42-68.
[2] Gong J X, Chen X B, Zhao G F. Gauss-Hermite integration-based approach for structural reliability analysis. Journal of Shanghai Jiaotong University, 2002, 36(11): 1625-1629. (in Chinese) 贡金鑫, 陈晓宝, 赵国藩. 结构可靠度计算的Gauss-Hermite积分方法. 上海交通大学学报, 2002, 36(11): 1625-1629.
[3] Rubinstein R Y, Krosese D P. Simulation and the Monte Carlo method. New York: John Wiley & Sons, 2008.
[4] Haldar A, Mahadevan S. Reliability assessment using stochastic finite element analysis. New York: John Wiley & Sons, 2000: 197-262.
[5] Peng Z. Metamodel methodology for reliability analysis and its application in engineering. Changsha: School of Resource and Safety Engineering, Central South University, 2010. (in Chinese) 彭泽. 结构可靠度Metamodel方法及其工程应用研究. 长沙: 中南大学资源与安全工程学院, 2010.
[6] Rackwitz R. Reliability analysis—a review and some perspectives. Structural Safety, 2001, 23(4): 365-395.
[7] Bucher C G, Bourgund U. A fast and efficient response surface approach for structural reliability problems. Structural Safety, 1990, 7(1): 57-66.
[8] Faravelli L. Response-surface approach for reliability analysis. Journal of Engineering Mechanics, 1989, 115(12): 2763-2781.
[9] Kim S H, Na S W. Response surface method using vector projected sampling points. Structural Safety, 1997, 19(1): 3-19.
[10] Kaymaz L, McMahon C A. A response surface method based on weighted regression for structural reliability analysis. Probabilistic Engineering Mechanics, 2005, 20(1): 11-17.
[11] Gavin H P, Yau S C. High-order limit state functions in the response surface method for structural reliability analysis. Structural Safety, 2008, 30(2): 162-179.
[12] Bucher C, Most T. A comparison of approximate response function in structural reliability analysis. Probabilistic Engineering Mechanics, 2008, 23(2-3): 154-163.
[13] Li X, Li W J, Peng C Y. Response surface methodology based on uniform design and its application to complex engineering system optimization. Mechanical Science and Technology, 2005, 24(5): 575-577. (in Chinese) 李响, 李为吉, 彭程远. 基于均匀实验设计的响应面方法及其在无人机一体化设计中的应用. 机械科学与技术, 2005, 24(5): 575-577.
[14] Wang Y,Wang Z Y, Zeng Z H, et al. The application of the method of uniform design response surface in inclined shaft traversing engineering of pipeline. Acta Petrolei Sinica, 2010, 31(4): 645-648. (in Chinese) 王怡, 王芝银, 曾志华,等. 均匀设计响应面法在管道斜井穿越工程中的应用. 石油学报, 2010, 31(4): 645-648.
[15] Wang H W. Partial least squares regression method and application. Beijing: National Defense Industry Press, 1999: 67-88. (in Chinese) 王惠文. 偏最小二乘回归方法及应用. 北京: 国防工业出版社, 1999: 67-88.
[16] Wold H. Partial least squares. New York: John Wiley & Sons, 1985: 581-591.
[17] de Jong S. SIMPLS: an alternative approach to partial least squares regression. Chemometrics and Intelligent Laboratory Systems, 1993, 18(3): 251-263.
[18] Trygg J, Wold S. Orthogonal projections to latent structures O-PLS. Journal of Chemometrics, 2002, 16(3): 119-128.
[19] Bang Y H, Yoo C K, Lee I B. Nonlinear PLS modeling with fuzzy inference system. Chemometrics and Intelligent Laboratory Systems, 2003, 64(2):137-155.
[20] Li S A, Zhang H X, Guo J L, et al. A partial least squares regression method based on principal variables selection. Computer Engineering, 2005, 31(16): 7-9. (in Chinese) 李寿安, 张恒喜, 郭基联,等. 一种基于主元选择的偏最小二乘回归方法. 计算机工程, 2005, 31(16): 7-9.
[21] Wang Y. Uniform design—a method for experimental design. Science and Technology Review, 1994(5): 20-22. (in Chinese) 王元. 均匀设计——一种试验设计方法. 科技导报, 1994(5): 20-22.
[22] Cao H Y, Li L. Uniform design table in MATLAB. Statistics and Decision, 2008(6): 144-166. (in Chinese) 曹慧荣, 李莉. 均匀设计表的MATLAB实现. 统计与决策, 2008(6): 144-146.
[23] Wang H W, Wu Z B, Meng J. Partial least squares regression-linear and nonlinear methods. Beijing: National Defense Industry Press, 2006: 35-38. (in Chinese) 王惠文,吴载斌,孟洁. 偏最小二乘回归的线性与非线性方法. 北京: 国防工业出版社, 2006: 35-38.
[24] Li L Y, Lu Z Z, Li W. A new importance measure system for basic random variables. Acta Aeronautica et Astronautica Sinica, 2012. DOI:CNKI:11-1929/V.20111107. 1021. 006. (in Chinese) 李璐祎, 吕震宙, 李维. 一种新的基本随机变量重要性测度指标体系. 航空学报, 2012. DOI:CNKI:11-1929/V.20111107.1021.006.