Application of DMD family methods and Higher Order DMD with Control for Reduced-Order Modelling of Nonlinear Elastic Structures

In recent years, data-driven algorithms have become a key part of engineering. The Dynamic Mode Decomposition is one of those algorithms, with many applications in mechanical engineering (i.e. fluid dynamics, robotics or thermodynamics). It uses time series data from a system to perform a spatio-temporal decomposition, extracting dynamic modes that characterise the system’s behaviour. In this paper, some of the most popular variations of the algorithm are thoroughly presented, together with a new one called Higher Order Dynamic Mode Decomposition with Control. It accounts for loading conditions, adding more physical insight to the original Higher Order Dynamic Mode Decomposition method. These algorithms are then used to develop different reduced-order models of a highly nonlinear elastic system. First, the system is modelled using the Finite Element Method, and a series of dynamic simulations are performed under time-varying loading conditions. The collected data serve as input for the application of the state-of-the-art algorithms, as well as for the new algorithm presented in this article. Based on the reduced-order models, the same full dynamic simulations previously done with the Finite Element Method are repeated. The results show that the Dynamic Mode Decomposition based models produce outcomes comparable to those of the Finite Element Method simulations, but with a significantly lower computational cost. Additionally, further simulations with new load scenarios are conducted. Once again, some of the reduced models delivered results close to those of the full models, while the computational cost is several orders of magnitude lower.