响应量分布函数对基本变量分布参数的灵敏度函数的新算法

New Algorithm to Solve the Sensitivity of Response's Cumulative Distribution Function with Respect to Distribution Parameters of Basic Variables

响应量的分布函数能够较好地描述其统计规律,通过研究响应量分布函数对基本变量分布参数的灵敏度函数,能够得到基本变量分布参数对响应量完整分布的影响信息。对于仅包含正态变量的线性功能函数,存在响应量分布函数对变量分布参数的灵敏度函数的解析表达式。为求解一般情况下响应量分布函数对变量分布参数的灵敏度函数,提出基于矩估计的求解算法。该算法利用可靠性灵敏度分析的矩估计方法,以及分布函数与失效概率的关系,将响应量分布函数对变量分布参数的灵敏度函数转化为阈值和响应功能函数各阶矩的近似表达式,通过点估计求得阈值处功能函数的前四阶矩,进而基于四阶矩方法求得阈值处的响应量分布函数对变量参数的灵敏度,通过遍历阈值的取值范围,最终可以得到响应量分布函数灵敏度随阈值变化的完整规律。相比于直接数字模拟法,所提算法的最大优点是在计算不同阈值处的分布函数灵敏度时,不需要重复调用功能函数,利用对一个阈值点处响应功能函数各阶矩和阈值的显式代数运算,即可得到不同阈值处的分布函数灵敏度,提高了计算效率。另外,所提算法不受基本变量分布类型的限制,且适用于多失效模式的情况,其主要缺点在于求解高维变量及非线性问题时的效率和精度尚有待提高。所讨论的数值和工程算例验证了所提算法的可行性。

To quantify effects of distribution parameters of the input variables, the sensitivity of the response＇s cumulative distribution function （CDF） with respect to those parameters is researched. For linear performance response functions solely involving normally distributed variables, analytical expressions of the CDF sensitivity are presented. As for a general structural application which probably involves non-normal variables, nonlinear performance functions or multiple response functions, this contribution puts forwards a new algorithm based on moment estimation to solve the CDF sensitivity. By using the relationship between the CDF and failure probability, and the reliability sensitivity based on moment estimation, the proposed algorithm establishes an analytical expression with the moments of performance function and threshold values for the CDF sensitivity. With the first four moments obtained by point estimate, the CDF sensitivity value at each threshold can be computed. Compared with numerical simulation methods, the most remarkable advantage of the proposed algorithm lies in that no additional evaluation of the performance function is needed at different thresholds, as the CDF sensitivity value at each threshold can be calculated using the moments of performance function obtained at one certain threshold. Furthermore, the proposed algorithm is not limited by the distribution type of basic variables, and is applicable to problems with multiple failure modes, yet its efficiency and precision when applied to problems involving highly dimensional variables or highly nonlinear performance functions still need to be improved. To illustrate the applicability and strength of the proposed algorithm, numerical and engineering examples are provided and discussed.