**P. Krysl, S. Lall and J. E. Marsden
**

A general approach to the dimensional reduction of non-linear finite element
models of solid dynamics is presented. For the Newmark implicit
time-discretization, the computationally most expensive phase is the repeated
solution of the system of linear equations for displacement increments.
To deal with this, it is shown how the problem can be formulated in an
approximation (Ritz) basis of much smaller dimension. Similarly, the explicit
Newmark algorithm can be also written in a reduced-dimension basis,
and the computation time savings in that case follow from an increase in the
stable time step length.

In addition, the empirical eigenvectors are proposed as the basis in which to expand the incremental problem. This basis achieves approximation optimality by using computational data for the response of the full model in time to construct a reduced basis which reproduces the full system in a statistical sense. Because of this `global' time viewpoint, the basis need not be updated as with reduced bases computed from a linearization of the full finite element model.

If the dynamics of a finite element model is expressed in terms of a small number of basis vectors, the asymptotic cost of the solution with the reduced model is lowered and optimal scalability of the computational algorithm with the size of the model is achieved. At the same time, numerical experiments indicate that by using reduced models, substantial savings can be achieved even in the pre-asymptotic range. Furthermore, the algorithm parallelizes very efficiently.

The method we present is expected to become a useful tool in applications requiring a large number of repeated non-linear solid dynamics simulations, such as convergence studies, design optimization, and design of controllers of mechanical systems.