Optimal spatiotemporal reduced order modeling, Part II: application to a nonlinear beam

A geometrically nonlinear, simply supported beam under the influence of time-dependent external forcing serves as a testbed to demonstrate application of the optimal spatiotemporal reduced order modeling (OPSTROM) framework proposed in Part I of this work. Fully resolved simulations, which are relatively expensive to perform, are used to accurately predict the beam response for a few forcing parameters. More affordable simulations are achieved with a conventional finite-difference scheme by coarsening the computational grid in space and time. Discretization errors are reduced with OPSTROM as subgrid-scale models are designed to account for the underlying space-time statistical structure using principles of mean-square error minimization, conditional expectations and stochastic estimation. When included in the under-resolved simulations, these optimal subgrid-scale models are shown to significantly improve the accuracy of predictions for both periodic and chaotic response types. This improved accuracy is further demonstrated through a set of numerical experiments designed to capture the complex bifurcation behavior of the beam response.