基于二项级数近似的结构概率分析

PROBABILITY ANALYSIS OF STRUCTURES BASED ON BINOMIAL SERIES APPROXIMATIONS

结构稳健优化设计中,一个关键的环节是分析结构响应量的概率特性,即计算响应的均值和方差。常用的方法主要有泰勒级数法、蒙特卡洛法以及数值积分法等。其中泰勒级数法精度较差,不适用于参数方差较大的随机结构,而蒙特卡洛法和高斯积分法计算量又过大。为了提高结构稳健性分析的计算效率,将结构位移的二项级数近似技术引入到高斯积分方法之中,提出一种结构位移均值及方差的计算方法。同时,用伴随向量法推导了相关的灵敏度计算公式。通过一个算例与已有的方法进行了比较,表明该方法较大程度上减少了高斯积分法的计算量,而与泰勒级数法相比,该方法又具有较高的计算精度,并且其灵敏度计算不再需要重分析,计算量较少。

One of the key techniques for structural robust optimization is the probability characteristics analysis for structure responses such as calculations of their mean and variance values. Typically used methods include the Taylor series method, Monte Carlo simulation and numerical integration methods etc. The Taylor series method is not accurate when the variances of random parameters are large. On the other hand, Monte Carlo simulation and numerical integration methods are relatively accurate but computationally inefficient. To abate computational efforts, in this paper, the binomial series approximation for structural displacements is introduced into the Gauss-Hermite quadrate formula. By doing so, full finite element analyses are not needed at the integration points, thus the computational amount is reduced efficiently. The sensitivities of means and variances of displacements with respect to the mean values of input variables are also derived by the adjoint method. A numerical example is provided to demonstrate the effectiveness of the proposed method. The results are encouraging in terms of accuracy and efficiency.