  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.
