Variational Integrators, the Newmark Scheme, and Dissipative Systems

Matt West, Control and Dynamical Systems, Caltech

Variational integrators are numerical solvers for mechanical ODEs which are an example of a symplectic method. These schemes are designed to preserve the mechanical nature of the system and for this reason typically have excellent energy and momentum behavior. In this talk, we discuss two new uses of variational methods.

First, the classical Newmark integrator which is popular in finite element simulations is shown to be variational in a slightly subtle way. This allows analytical results known for variational methods to be used in explaining the otherwise mysterious behavior of Newmark.

Second, we extend the variational framework from conservative systems to include forcing and dissipation. This is done in such as way that the excellent energy and momentum properties are largely retained, making these methods very attractive for simulation of weakly forced or damped mechanical systems.