A general framework for interpreting time finite element formulations

In this work, an attempt is made at filling the apparent gap existing between the two major approaches evolved in the literature towards formulating space-time finite element methods. The first assumes Hamilton's Law as underlying concept, while the second performs a weighted residual approach on the ordinary differential equations emanating from the semidiscretization in the space dimension. A general framework is proposed in the following pages, where the configuration space and the phase space forms of Hamilton's Law provide the general statements of the problem of motion. Within this framework, different families of integration algorithms are derived, according to different interpretations of the boundary terms. The bi-discontinuous form is obtained as the consequence of a consistent impulsive formulation of dynamics, while the discontinuous Galerkin form is obtained when the boundary terms at the end of the time interval are appropriately approximated.