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