Newmark's time integration method from the discretization of extended functionals

In this note we illustrate how to obtain the full family of Newmark's time integration algorithms within a rigorous variational framework, i.e., by discretizing suitably defined extended functionals, rather than by starting from a weak form (for instance, of the Galerkin type), as done in the past. The availability of functionals as a starting point is useful both as a tool to obtain new families of time integration methods, and as a theoretical basis for error estimates. To illustrate the first issue, here we provide some examples of how to obtain modified algorithms, in some cases significantly more accurate than the basic Newmark one despite having a comparable computational cost.