We present an intrusive formulation for the dynamic stochastic finite‐element method to propagate the epistemic uncertainty in material properties into a finite‐element system over time. The stochastic finite‐element method, originally developed for the static case, uses generalized polynomial chaos (gPC) expansions to represent the uncertainty in both material/load fields and displacement fields and solves for the unknown PC coefficients of displacement at each degree of freedom of the finite‐element system. In this case, the gPC basis used to represent the solution is optimal and can be kept the same throughout the static simulation; however, when integrating a stochastic system over time, it is proven that using gPC tends to break down for integrations over long times. The reason is that the solution's complexity increases in time, and the set of polynomials used in the gPC expansion to represent the solution, therefore, does not stay optimal. In this formulation, new stochastic variables and orthogonal polynomials are constructed as time progresses. These variables are obtained as the Kahrunen‐Loeve expansion of the finite‐element solution at the times of update, thus, optimally representing the distribution of the solution using a minimal set of orthogonal polynomials. The result from the method is a time‐domain polynomial chaos representation of the entire finite‐element solution. A fast post‐processing phase can be used to (a) obtain the probability distribution of the solution at each degree of freedom and (b) generate any number of time series realizations of the solution, which correspond to the same time series that would be obtained from a set of simulations based on direct Monte‐Carlo simulations of the uncertain material properties.

中文翻译：

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基于时间依赖的广义多项式混沌的动态随机有限元方法

我们提出了一种动态随机有限元方法的侵入式公式，以随着时间的推移将材料属性中的认知不确定性传播到有限元系统中。最初为静态情况开发的随机有限元方法，使用广义多项式混沌（gPC）展开来表示材料/载荷场和位移场的不确定性，并求解每个自由度下的未知PC位移系数有限元系统。在这种情况下，用于表示解决方案的gPC基础是最佳的，并且可以在整个静态仿真中保持相同；但是，事实证明，随着时间的推移集成随机系统时，长时间使用gPC往往会导致集成失败。原因是解决方案的复杂性随时间增加，因此，在gPC扩展中用来表示解决方案的多项式集并不会保持最佳状态。在这种表述中，随着时间的推移，构造了新的随机变量和正交多项式。这些变量是在更新时作为有限元解的Kahrunen-Loeve展开而获得的，因此，使用最小的一组正交多项式来最优地表示解的分布。该方法的结果是整个有限元解的时域多项式混沌表示。快速后处理阶段可用于（a）获得每个自由度下解决方案的概率分布，以及（b）生成解决方案的任意数量的时间序列实现，