核函数的性质及其在灵敏度分析上的应用

Properties of Kernel Functions and Their Application in Sensitivity Analysis

推导了多维随机变量情况下不含交叉项的二次功能函数统计矩对基本变量分布参数灵敏度的通用解析表达式。基于核函数的性质,对正态分布情况给出了二次不含交叉项功能函数统计矩对基本变量分布参数灵敏度的理论解,并给出了该情况下失效概率对基本变量分布参数灵敏度的近似表达式。证明了三分布参数情况下前三阶统计参数之间的独立性,并在此基础上以三参数W eibull分布为例推导了其核函数的性质,将这些性质代入功能函数统计矩灵敏度的通用解析表达式,即可解析求得三参数W eibull分布情况下函数统计矩对基本变量分布参数的灵敏度。数字仿真算例结果表明:所推导的函数统计矩的灵敏度解析表达式是正确的,所推导的失效概率灵敏度近似表达式也具有较高的计算精度。

Starting from Ref.6,we study the properties of kernel functions further.In case that the performance function with multi-dimensional basic variables is expressed by a quadratic polynomial without cross-terms（QPWCT）,the universal sensitivities of the statistical moments of the performance function with respect to the distribution parameters of the basic variables are derived analytically.Based on the properties of kernel functions,the analytical sensitivity solutions of the statistical moments of the QPWCT with respect to the distribution parameters are derived for the normal basic variables,and the approximate sensitivities of the failure probability with respect to their distribution parameters are derived as well.Furthermore,in subsection 5.1 of the full paper,the independence of the first-,second-and third-order statistical moments is proved for three independent distribution parameters of the basic variables,on which the properties of the kernel functions are derived for the three-parameter Weibull distribution.By use of these derived properties,the sensitivities of the statistical moments of the performance function can be obtained respectively and analytically with respect to the distribution parameters of the basic variables.Three numerical simulation examples are analyzed;the analysis results,given respectively in Tables 1 through 3,demonstrate preliminarily that the derived analytical expressions of the sensitivities of the statistical moments are correct and that the approximate sensitivities of failure probability are precise enough.