Experimental Implementation of Energy Closure Analysis for Reduced Order Modeling

Reduced order models (ROMs) provide an efficient, kinematically condensed representation of computationally expensive high-dimensional dynamical systems; however, their accuracy depends crucially on the accurate estimation of their dimension. We here demonstrate how the energy closure criterion, developed in our prior work, can be experimentally implemented to accurately estimate the dimension of ROMs obtained using the proper orthogonal decomposition (POD). We examine the effect of using discrete data with and without measurement noise, as will typically be gathered in an experiment or numerical simulation, on estimating the degree of energy closure on a candidate reduced subspace. To this end, we used a periodically kicked Euler–Bernoulli beam with Rayleigh damping as the model system and studied ROMs obtained by applying POD to discrete displacement field data obtained from simulated numerical experiments. An improved method for quantifying the degree of energy closure is presented: the convergence of energy input to or dissipated from the system is obtained as a function of the subspace dimension, and the dimension capturing a predefined percentage of either energy is selected as the ROM dimension. This method was found to be more robust to data discretization error and measurement noise. The data-processing necessary for the experimental application of energy closure analysis is discussed in detail. We show how ROMs formulated from the simulated data using our approach accurately capture the dynamics of the beam for different sets of parameter values.