Optimal spatiotemporal reduced order modeling, Part I: proposed framework

An optimal spatiotemporal reduced order modeling framework is proposed for nonlinear dynamical systems in continuum mechanics. In this paper, Part I, the governing equations for a general system are modified for an under-resolved simulation in space and time with an arbitrary discretization scheme. Basic filtering concepts are used to demonstrate the manner in which subgrid-scale dynamics arise with a coarse computational grid. Models are then developed to account for the underlying spatiotemporal structure via inclusion of statistical information into the governing equations on a multi-point stencil. These subgrid-scale models are designed to provide closure by accounting for the interactions between spatiotemporal microscales and macroscales as the system evolves. Predictions for the modified system are based upon principles of mean-square error minimization, conditional expectations and stochastic estimation, thus rendering the optimal solution with respect to the chosen resolution. Practical methods are suggested for model construction, appraisal, error measure and implementation. The companion paper, Part II, is devoted to demonstrating the methodology through a computational study of a nonlinear beam.