The importance of energy criteria for selecting modes in reduced order modeling

We study the performance of the proper orthogonal decomposition when used for model reduction of an Euler-Bernoulli beam subjected to periodic impulses. We assess the accuracy of reduced order models (ROMs) obtained using steady-state displacement time series. The spatiotemporal localization of the applied impulses is tuned to control the number of excited modes in, and hence the effective dimensionality of, the system’s response. We find that when the impacts are significantly localized (i.e., are more impulsive), the conventional variance-based mode selection criterion can lead to inaccurate ROMs. We show that this arises when the reduced subspace capturing a fixed amount (say, 99.9%) of the total data variance underestimates the energy input and/or dissipated in the ROM, leading to energy imbalance. We thus propose a new energy closure criterion that provides an improved method for generating ROMs. The energetics of the resulting ROMs properly reflect those of the full system, and yield simulations that accurately represent the system’s true behavior.