The temporal finite element method in structural dynamics

Abstract

This work introduces a novel temporal finite element methodology (TFEM) for dynamic problems in structural mechanics. The theoretical foundation of the method is Hamilton's law of varying action. In contrast to traditional finite element approaches, where only the spatial domain is interpolated via shape functions, the present method interpolates both spatial and temporal domains. Following a review of the temporal finite element literature, the general theoretical development and formulative aspects of the current methodology are presented. To illustrate the proposed approach, a number of numerical examples are presented involving multiple degree-of-freedom (DOF) structural systems built from simple rod type elements that can exhibit linear, nonlinear elastic, or elasto-plastic (bilinear) material response. The time marching algorithm that results from this approach is conditionally stable with a time step limit comparable to that of other conventional schemes, exhibits no algorithmic damping, and gives good accuracy versus time step solutions. By using this rudimentary element, the fundamentals of the methodology are readily exposed, yet the approach is extendable to different element types such as beams and continual elements.