预处理谱随机变分原理和随机有限元法

Preconditioned spectral stochastic variational principle and preconditioned finite element method

利用正交级数对材料和外荷载随机场进行离散;利用刚度矩阵在随机场均值处的逆矩阵作为预处理因子,定义一组随机子空间的新的基向量,将结构随机响应过程展开为该组随机基向量的线性表达式,首次提出了预处理谱随机变分原理,在此基础上建立了结构随机分析的预处理谱随机有限元法。该方法能够导出降阶的控制方程,因而与普通的谱随机有限元法相比计算量大大降低;而且该方法能够同时适用于小变异或大变异的随机结构的分析和计算。算例表明,预处理谱随机有限元法不仅对内存要求低,而且在梁板等结构的随机分析中具有较高精度和计算效率。

The random fields of material and external loads are discretized by means of Karhunen- Loeve series. A new basis vector is defined for the Krylov subspace, which is preconditioned by a preconditioning factor. Therefore, the stochastic process of displacements is expanded into a linear combination of the basis. The preconditioned spectral stochastic variational principle is proposed for the first time as well as the preconditioned spectral stochastic finite element method for analy- sis of stochastic structures. The presented method can achieve governing equations with reduced order, so that the EMS memory requirement is much fewer than the conventional spectral stochas- tic finite element method, which can educe governing equations with dilatation order. Examples in the paper show that the presented method is suitable for analysis of stochastic structures with ran- dom quantities with large or small variation coefficients, and yield results agreeing well with those from MCFEM.