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 引用本文: 万志强, 陈建兵, Michael Beer. 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[J]. 力学学报, 2021, 53(3): 837-854.
Wan Zhiqiang, Chen Jianbing, Michael Beer. FUNCTIONAL PERSPECTIVE OF UNCERTAINTY QUANTIFICATION FOR STOCHASTIC PARAMETRIC SYSTEMS AND GLOBAL SENSITIVITY ANALYSIS[J]. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(3): 837-854.
 Citation: Wan Zhiqiang, Chen Jianbing, Michael Beer. FUNCTIONAL PERSPECTIVE OF UNCERTAINTY QUANTIFICATION FOR STOCHASTIC PARAMETRIC SYSTEMS AND GLOBAL SENSITIVITY ANALYSIS[J]. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(3): 837-854.

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

• 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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