随机参数系统不确定性量化的泛函观点与整体灵敏度分析

万志强; 陈建兵; Michael Beer

doi:10.6052/0459-1879-20-336

随机参数系统不确定性量化的泛函观点与整体灵敏度分析

万志强^*,+,
陈建兵^*,+,2), ,,
Michael Beer^**

^*同济大学土木工程防灾国家重点实验室, 上海 200092
^††同济大学土木工程学院, 上海200092
^**莱布尼茨汉诺威大学风险与可靠性研究所, 德国汉诺威 30167

详细信息

通讯作者:
陈建兵

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出版历程
- 收稿日期: 2020-09-20
- 刊出日期: 2021-03-09

FUNCTIONAL PERSPECTIVE OF UNCERTAINTY QUANTIFICATION FOR STOCHASTIC PARAMETRIC SYSTEMS AND GLOBAL SENSITIVITY ANALYSIS

^*State Key Laboratory of Disaster Reduction in Civil Engineering$, $ Tongji University$, $ Shanghai $200092$$, $ China
^††College of Civil Engineering$, $ Tongji University$, $ Shanghai $200092$$, $ China
^**Institute for Risk and Reliability$, $ Leibniz Universität Hannover$, $ Hannover $30167$$, $ Germany

摘要

摘要: 不确定性因素广泛存在于实际工程分析与设计之中. 例如,土木工程结构的力学性能参数可能存在不可忽略的随机性.故而随机系统参数灵敏度分析, 对合理的工程设计与决策具有相当重要的意义.本文首先论述了随机系统中不确定性量化与传播的泛函空间分析观点. 在此基础上,给出了由泛函Fréchet导数所定义的整体灵敏度指标,讨论了该整体灵敏度指标的一些基本数学物理性质,定义了该整体灵敏度指标所对应的工程实用的重要性测度与方向灵敏度,并分别解释了二者的几何物理意义. 然后, 具体论述了在$\varepsilon$-等效分布定义下该整体灵敏度指标的参数化表达形式.结合概率密度演化-概率测度变换方法,给出了该指标的具体数值求解算法及整体灵敏度分析流程. 通过4个算例分析,包括正态随机变量线性组合解析函数分析、隧道锚杆支护系统稳定性分析、大坝下稳态受限渗流量分析以及钢筋混凝土平面框架结构随机地震响应分析,具体说明了该指标的实用性及重要性.

Abstract: Uncertainty exists broadly in real engineering design and analysis. For instance, some mechanical parameters of structures in civil engineering may be of randomness and usually cannot be ignored. Therefore, the process of uncertainty quantification, e.g., the sensitivity analysis on parameters of stochastic systems is, of paramount significance to reasonable engineering design and decision-making. In the present paper, the perspective of functional space analysis on uncertainty quantification and propagation in stochastic systems is firstly stated. On this basis, the global sensitivity index (GSI) is introduced based on the functional Fréchet derivative, of which some basically mathematical and physical properties are studied. Besides, the correspondingly defined importance measure and direction sensitivity of the GSI are also discussed, in terms of their geometric and physical meanings. Moreover, based on the definition of $\varepsilon$-equivalent distribution, the parametric form of the proposed GSI is elaborated in detail. By incorporating the probability density evolution method (PDEM) and the change of probability measure (COM), the numerical algorithm of the GSI and the procedure of sensitivity analysis are illustrated. Four numerical examples, including the analytical function of the linear combination of normal random variables, stability analysis of the rock bolting system of tunnel, the analysis of steady-state confined seepage below the dam, and the stochastic structural analysis of the reinforced concrete frame, are analyzed to demonstrate the effectiveness and significance of the GSI.

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参考文献(38)

[1]	Soize C. Uncertainty Quantification: An Accelerated Course with Advanced Applications in Computational Engineering. Cham, Switzerland: Springer International Publishing AG, 2017
[2]	万志强, 陈建兵. 数据稀缺与更新条件下基于概率密度演化-测度变换的认知不确定性量化分析. 工程力学, 2020,37(1):34-42 (Wan Zhiqiang, Chen Jianbing. Quantification of epistemic uncertainty due to data sparsity and updating based on the framework via synthesizing probability density evolution method and change of probability measure. Engineering Mechanics, 2020,37(1):34-42 (in Chinese))
[3]	陈建兵, 万志强, 宋鹏彦. 相依随机变量的随机函数模型. 中国科学: 物理学力学天文学, 2018,61(1):014609 (Chen Jianbing, Wan Zhiqiang, Song Pengyan. Random function model for dependent random variables. Scientia Sinica$:$ Physica, Mechanica & Astronomica, 2018,61(1):014609 (in Chinese))
[4]	Li DQ, Zhang L, Tang XS, et al. Bivariate distribution of shear strength parameters using copulas and its impact on geotechnical system reliability. Computers and Geotechnics, 2015,68:184-195
[5]	Hong X, Li J. Stochastic Fourier spectrum model and probabilistic information analysis for wind speed processes. Journal of Wind Engineering and Industrial Aerodynamics, 2018,174:424-436
[6]	Saltelli A, Ratto M, Andres T, et al. Global Sensitivity Analysis. The Primer. Chichester: John Wiley & Sons, 2008
[7]	Morris MD. Factorial sampling plans for preliminary computational experiments. Technometrics, 1991,33(2):161-174
[8]	Sobol' IM, Kucherenko S. Derivative based global sensitivity measures and their link with global sensitivity indices. Mathematics and Computers in Simulation, 2009,79:3009-3017
[9]	Sobol' IM. Sensitivity estimates for nonlinear mathematical models. Mathematical Modeling and Computational Experiment, 1993,1:407-414
[10]	Borgonovo E. A new uncertainty importance measure. Reliability Engineering and System Safety, 2007,92:771-784
[11]	陈建兵, 李杰. 结构随机动力非线性反应的整体灵敏度分析. 计算力学学报, 2008,25(2):169-176 (Chen Jianbing, Li Jie. Global sensitivity in nonlinear stochastic dynamic response analysis of structures. Chinese Journal of Computational Mechanics, 2008,25(2):169-176 (in Chinese))
[12]	Chen JB, Yang JS, Jensen H. Structural optimization considering dynamic reliability constraints via probability density evolution method and change of probability measure. Structural and Multidisciplinary Optimization, 2020,62(5):2499-2516
[13]	Wu YT. Computational methods for efficient structural reliability and reliability sensitivity analysis. AIAA Journal, 1994,32(8):1717-1723
[14]	Valdebenito MA, Jensen HA, Hernández HB, et al. Sensitivity estimation of failure probability applying line sampling. Reliability Engineering and System Safety, 2018,171:99-111
[15]	Dubourg V, Sudret B. Meta-model-based importance sampling for reliability sensitivity analysis. Structural Safety, 2014,49:27-36
[16]	Chen JB, Wan ZQ, Beer M. A global sensitivity index based on Fréchet derivative and its efficient numerical analysis. Probabilistic Engineering Mechanics, 2020,62:103096
[17]	Chen JB, Wan ZQ. A compatible probabilistic framework for quantification of simultaneous aleatory and epistemic uncertainty of basic parameters of structures by synthesizing the change of measure and change of random variables. Structural Safety, 2019,78:76-87
[18]	Ang AH-S, Tang W. 工程中的概率概念. 陈建兵, 彭勇波, 刘威等, 译. 北京: 中国建筑工业出版社, 2017 (Ang AH-S, Tang W. Probability Concepts in Engineering. Transl. Chen Jianbing, Peng Yongbo, Liu Wei, et al. Hoboken: John Wiley & Sons, 2006 (in Chinese))
[19]	Papoulis A, Pillai SU. Probability, Random Variables, Stochastic Processes. New York: McGraw-Hill, 2002
[20]	Li J, Chen JB. The principle of preservation of probability and the generalized density evolution equation. Structural Safety, 2008,30:65-77
[21]	Li J, Chen JB. Stochastic Dynamics of Structures. Singapore: John Wiley & Sons, 2009
[22]	Bobrowski A.x Functional Analysis for Probability and Stochastic Processes. An introduction. New York: Cambridge University Press. 2005
[23]	李杰. 工程结构整体可靠性分析研究进展. 土木工程学报, 2018,51(8):1-10 (Li Jie. Advances in global reliability analysis of engineering structures. China Civil Engineering Journal, 2018,51(8):1-10 (in Chinese))
[24]	Atkinson K, Han WM. Theoretical Numerical Analysis: A Functional Analysis Framework. New York: Springer Science $+$ Business Media, 2009
[25]	Saltelli A, Tarantola S, Campolongo F, et al. Sensitivity Analysis in Practice. A Guide to Assessing Scientific Models. Chichester: John Wiley & Sons, 2004
[26]	Zhao YG, Zhang XY, Lu ZH. Complete monotonic expression of the fourth-moment normal transformation for structural reliability. Computers and Structures, 2018,196:186-199
[27]	Zhao YG, Zhang XY, Lu ZH. A flexible distribution and its application in reliability engineering. Reliability Engineering and System Safety, 2018,176:1-12
[28]	Melchers RE, Beck AT. Structural Reliability Analysis and Prediction. Chichester: John Wiley & Sons, 2018
[29]	Nelsen RB. An Introduction to Copulas. New York: Springer Science $+$ Business Media, 2006
[30]	Berberan-Santos MN. Expressing a probability density function in terms of another PDF: A generalized Gram-Charlier expansion. Journal of Mathematical Chemistry, 2007,42(3):585-594
[31]	蒋仲铭, 李杰. 三类随机系统广义概率密度演化方程的解析解. 力学学报, 2016,48(2):413-421 (Jiang Zhongming, Li Jie. Analytical solutions of the generalized probability density evolution equation of three classes stochastic systems. Chinese Journal of Theoretical and Applied Mechanics, 2016,48(2):413-421 (in Chinese))
[32]	Chen JB, Yang JY, Li J. A GF-discrepancy for point selection in stochastic seismic response analysis of structures with uncertain parameters. Structural Safety, 2016,59:20-31
[33]	Chen JB, Chan JP. Error estimate of point selection in uncertainty quantification of nonlinear structures involving multiple nonuniformly distributed parameters. International Journal for Numerical Methods in Engineering, 2019,118:536-560
[34]	Li J, Chen JB, Fan WL. The equivalent-value event and evaluation of the structural system reliability. Structural Safety, 2007,29:112-131
[35]	Grigoriu M. Stochastic Calculus. Applications in Science and Engineering. New York: Springer Science $+$ Business Media, 2002
[36]	Su YH, Li X, Xie ZY. Probabilistic evaluation for the implicit limit-state function of stability of a highway tunnel in China. Tunnelling and Underground Space Technology, 2011,26:422-434
[37]	Song JW, Valdebenito M, Wei PF, et al. Non-intrusive imprecise stochastic simulation by line sampling. Structural Safety, 2020,84:101936
[38]	中华人民共和国住房和城乡建设部组织. GB 50010 2010 混凝土结构设计规范. 北京: 中国建筑工业出版社, 2010 ( Ministry of Housing and Urban-Rural Development of the People's Republic of China. GB 50010 2010 Code for Design of Concrete Structures. Beijing: China Architecture & Building Press, 2010 (in Chinese))