The Finite Element Method (FEM) is the gold standard for spatial discretization in numerical simulations for a wide spectrum of real-world engineering problems. Prototypical areas of interest include linear heat transfer and linear structural dynamics problems modeled with partial differential equations (PDEs). While different algorithms for direct integration of the equations of motion exist, exploring all feasible behaviors for varying loads, initial states and fluxes in models with large numbers of degrees of freedom remains a challenging task. In this article we propose a novel approach, based in set propagation methods and motivated by recent advances in the field of Reachability Analysis. Assuming a set of initial states and inputs, the proposed method consists in the construction of a union of sets (flowpipe) that enclose the infinite number of solutions of the spatially discretized PDE. We present the numerical results obtained in five examples to illustrate the capabilities of our approach, and compare its performance against reference numerical integration methods. We conclude that, for problems with single known initial conditions, the proposed method is accurate. For problems with uncertain initial conditions included in sets, the proposed method can compute all the solutions of the system more efficiently than numerical integration methods.

有限元方法 (FEM) 是在广泛的实际工程问题的数值模拟中空间离散化的黄金标准。感兴趣的原型领域包括线性​​传热和使用偏微分方程 (PDE) 建模的线性结构动力学问题。虽然存在用于直接积分运动方程的不同算法，但在具有大量自由度的模型中探索不同载荷、初始状态和通量的所有可行行为仍然是一项具有挑战性的任务。在本文中，我们提出了一种新方法，该方法基于集合传播方法，并受到可达性分析领域的最新进展的启发。假设一组初始状态和输入，所提出的方法包括构建一个集合（流管）的并集，该集合包含空间离散 PDE 的无限数量的解。我们展示了在五个示例中获得的数值结果，以说明我们方法的能力，并将其性能与参考数值积分方法进行比较。我们得出的结论是，对于具有单个已知初始条件的问题，所提出的方法是准确的。对于集合中包含不确定初始条件的问题，所提出的方法可以比数值积分方法更有效地计算系统的所有解。我们得出的结论是，对于具有单个已知初始条件的问题，所提出的方法是准确的。对于集合中包含不确定初始条件的问题，所提出的方法可以比数值积分方法更有效地计算系统的所有解。我们得出的结论是，对于具有单个已知初始条件的问题，所提出的方法是准确的。对于集合中包含不确定初始条件的问题，所提出的方法可以比数值积分方法更有效地计算系统的所有解。