高阶核函数的性质及其在灵敏度分析中的应用

High Order Properties of Kernel Functions and Their Application in Sensitivity Analysis

灵敏度信息能够反映基本变量分布参数对结构系统失效概率及输出性能分布函数的影响程度信息,而核函数对于解析求得灵敏度具有很大的作用。因此为得到高精度的失效概率及分布函数灵敏度解析解,推导出正态变量情况下核函数的高阶性质,利用这些核函数的高阶性质以及结构失效概率与输出性能分布函数的关系,解析地求得了二次不含交叉项多项式功能函数在考虑前四阶矩条件下的失效概率和分布函数灵敏度。算例中数值仿真结果与解析结果的对比表明,利用核函数的高阶性质,采用四阶矩法比二阶矩法具有更高的灵敏度分析精度,算例结果同时也证明了所推导的失效概率及分布函数灵敏度表达式的正确性,同时也说明了所提方法具有一定的工程适用性。

Sensitivity analysis can reflect how the distribution parameters of basic variables affect the failure probability and the distribution function of the structure or system output, and the kernel functions play a significant role in getting the sensitivities. So in order to obtain more accurate analytical results of the sensitivities, the high order properties of the kernel functions for the normal variables are derived. Based on the properties of the kernel functions and the relationship between the failure probability and the distribution function, and by taking a quadratic polynomial without cross-terms as an example of a performance function,the analytical sensitivity solutions of the failure probability and the distribution function are derived when considering the first forth-order moments. Comparing the numerical simulation results with the analytical results, it demonstrates that the forth-order moment method is more precise than the second-order method in sensitivity analysis, and that the derived analytical sensitivity expressions are correct, besides, it well shows good application of the proposed method.