Piecewise Dynamic Mode Decomposition of Fluid Flow Simulations

The computational simulation of fluid flows over structures still is a major research area in fluid mechanics. Because of the multiscale nature of many of the application models and consequent large amount of data, these simulations are computationally expensive, but can benefit from modern Reduced Order Models and data-driven methods. In this work, Dynamic Mode Decomposition (DMD) and its recent variation, Piecewise DMD (pDMD), are presented and compared in terms of dynamic modes extracted from the data, accuracy in reconstructing an approximation for the original dataset as a reduced order model and, most importantly, computational cost. The pDMD method is shown to be a variation of the traditional DMD that aims to improve and overcome some of the caveats of the standard version. This variation consists in decomposing the entire data into smaller datasets and applying a linear mapping independently on each one of these subsets instead of calculating a global linear fitting. Basically, it is an application of multiple DMD on small subsets instead one DMD over the whole data. Piecewise DMD is a simple and elegant idea that is based on the "divide and conquer" approach well known in the numerical analysis literature. Even though pDMD is new and should be carefully and extensively tested, it can be considered as a promising improvement over the standard DMD. The preliminary results presented in this work show how DMD can capture the dynamics and accurately reconstruct the simulation data and how pDMD can provide more accurate results when traditional DMD reaches its limitations, capture the specific dynamics of different stages of transient flows, and reduce the computational cost by 90% for a two-dimensional flow over a cylinder when compared to standard DMD. Future applications of Reduced Order Models using both DMD and pDMD include future state predictions, computationally cheap parametric simulations, and qualitative dynamic analysis of fluid flows.