Reduced order models for systems with disparate spatial and temporal scales

Abstract

Simulations and parametric studies of large-scale models can be facilitated by high-fidelity reduced order models (ROMs). One of the methods to obtain these ROMs, which is of considerable current interest, is using linear subspaces obtained from spatiotemporal decompositions including proper orthogonal decomposition (POD) and smooth orthogonal decomposition (SOD). Previous studies showed that SOD has advantages over POD in obtaining a lower dimensional ROM. However, in many dynamical models, the data matrices used in multivariate analysis can be ill-conditioned. This leads to non-optimal results with POD, since it only aims to find the subspace on which the data projection has the maximal variance. Therefore, POD is likely to overlook small-variance state-variables. A large-scale nonlinear system is used as the full-scale model of an ill-conditioned system, where some state variable have considerably smaller variations than others. Four different methods are applied to obtain ROMs. First, a conventional POD as well as SOD are applied to the whole state space data matrix of the model. Then, the position and velocity data of the full system are separated into two data matrices and POD and SOD are applied to them individually. Finally, the ROMs built based on these multivariate analysis are compared in terms of accuracy and stability. While separated POD-based ROM shows considerable improvement over the conventional POD ROM, SOD still provides lower-dimensional ROMs and it does not seem to benefit from the separated SOD reduction.