一种基于复数变量求偏导的随机有限元可靠度法

A Stochastic Finite Element Reliability Analysis Method Based on Complex variable Derivative Technique

摘要:

提出一种基于复数变量求偏导的随机有限元可靠度法.将工程中的随机因素设置为复数变量，通过复数函数的泰勒级数展式得到一阶导数的近似计算式.这种求导方法效率高，精度高，应用简单方便，只需在复数空间进行有限元计算，无需对有限元方程进行偏导计算，便可求出响应量的偏导数,进而求得响应量的方差.在随机有限元一次二阶矩的迭代格式中，取复数空间有限元计算结果的实部作为响应量的值，这样在求可靠度系数的迭代过程中，无需再在实数空间进行计算.复数变量法大大简化了随机有限元(SFEM)和随机有限元可靠度(SFEMR)的计算和编程过程，为工程应用提供了一种现实可行的途径.

Abstract:

A new approach is proposed for stochastic finite element method (SFEM) and reliability analysis by using a complex variable technique. The random factors in engineering are defined as complex variables, the first derivative formulation can be obtained by the Taylor’s series of a complex function. This derivative method is computationally very accurate, efficient, and very easily implemented. In SFEM, to get the variances of responses, it only needs to implement FEM in complex variables space, without a need of partial derivatives of FEM functions. In the iteration scheme of SFEM reliability analysis, the real parts of complex response is considered as the response value to simplify the process. The complex variable method greatly simplifies SFEM and reliability program, providing a feasible approach for engineering application.