This paper presents a weak-intrusive stochastic finite element method for solving stochastic structural dynamics equations. In this method, the stochastic solution is decomposed into the summation of a series of products of random variables, spatial vectors and temporal functions. An iterative algorithm is proposed to compute each triplet of the random variable, spatial vector and temporal function one by one. The original stochastic dynamics problem is firstly transformed into spatial–temporal coupled problems (i.e. deterministic structural dynamics equations), which can be solved efficiently by existing FEM solvers. Based on the solution of the spatial–temporal coupled problem, the original problem is then transformed into stochastic-temporal coupled problems (i.e. one-dimensional second-order stochastic ordinary differential equations), which are solved by a proposed sampling method. All random sources are embedded into the stochastic-temporal coupled problems. The proposed sampling method can solve the stochastic-temporal problems of hundreds of dimensions with low computational costs. Thus the curse of dimensionality in high-dimensional stochastic spaces is avoided with great success. Three numerical examples, including low- and high-dimensional stochastic problems, are used to demonstrate the good accuracy and the high efficiency of the proposed method.

本文提出了一种求解随机结构动力学方程的弱侵入随机有限元方法。在该方法中，随机解被分解为一系列随机变量、空间向量和时间函数的乘积之和。提出了一种迭代算法，将随机变量、空间向量和时间函数的每一个三元组逐一计算。原始随机动力学问题首先转化为时空耦合问题（即确定性结构动力学方程），可以通过现有的有限元求解器有效地求解。在求解时空耦合问题的基础上，将原问题转化为随机-时间耦合问题（即一维二阶随机常微分方程），通过所提出的采样方法求解。所有随机源都嵌入到随机时间耦合问题中。所提出的采样方法可以以较低的计算成本解决数百维的随机时间问题。因此，维度的诅咒在高维随机空间中成功地避免了。三个数值例子，包括低维和高维随机问题，被用来证明所提出方法的良好准确性和高效率。