Robust and Dynamically Consistent Reduced Order Models

Abstract

The need for reduced order models (ROMs) has become considerable higher with the increasing technological advances that allows one to model complex dynamical systems. When using ROMs, the following two questions always arise: 1) “What is the lowest dimensional ROM?” and 2) “How well does the ROM capture the dynamics of the full scale system model?” This paper considers the newly developed concepts the authors refer to as subspace robustness — the ROM is valid over a range of initial conditions, forcing functions, and system parameters — and dynamical consistency — the ROM embeds the nonlinear manifold — which quanitatively answers each question. An eighteen degree-of-freedom pinned-pinned beam which is supported by two nonlinear springs is forced periodically and stochastically for building ROMs. Smooth and proper orthogonal decompositions (SOD and POD, respectively) based ROMs are dynamically consistent in four or greater dimensions. In the strictest sense POD-based ROMs are not considered coherent whereas, SOD-based ROMs are coherent in roughly five dimesions and greater. Is is shown that in the periodically forced case, the full scale dynamics are captured in a five-dimensional POD and SOD-based ROM. For the randomly forced case, POD and SOD-based ROMs need three dimensions but SOD captures the dynamics better in a lower-dimensional space. When the ROM is developed from a different set of initial conditions and forcing values, SOD outperforms POD in periodic forcing case and are equal in the random forcing case.