‘On-the-fly’ snapshots selection for Proper Orthogonal Decomposition with application to nonlinear dynamics

Abstract Over the last few decades, Reduced Order Modeling (ROM) has slowly but surely inched towards widespread acceptance in computational mechanics, as well as other simulation-based fields. Projection-based Reduced Order Modeling (PROM) relies on the construction of an appropriate Reduced Basis (RB), which is typically a low-rank representation of a set of “observations” made using full-field simulations, usually obtained through truncated Singular Value Decomposition (SVD). However, SVD encounters limitations when dealing with a large number of high-dimensional observations, requiring the development of alternatives such as the incremental SVD. The key advantages of this approach are reduced computational complexity and memory requirement compared to a regular “single pass” spectral decomposition. These are achieved by only using relevant observations to enrich the low-rank representation as and when available, to avoid having to store them. In addition, the RB may be truncated ’on-the-fly’ so as to reduce the size of the matrices involved as much as possible and, by doing so, avoid the quadratic scale-up in computational effort with the number of observations. In this paper, we present a new error estimator for the incremental SVD, which is shown to be an upper bound for the approximation error, and propose an algorithm to perform the incremental SVD truncation and observation selection ’on-the-fly’, instead of using a prohibitively large number of frequently “hard to set” parameters. The performance of the approach is discussed on the reduced-order Finite Element (FE) model simulation of impact on a Taylor beam.