Properties of kernel functions and their application in sensitivity analysis

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.