Cluster-Based Model Reduction with Application to Fluid Flows

The development of model reduction techniques for physical systems governed by partial differential equations (PDEs) continues to be an active research area. Accurately capturing the dynamics of these systems often requires a large number of states, making them impractical for control design. Therefore, the system’s order must be reduced before control strategies can be effectively implemented. This dissertation investigates new methods that generalize the popular proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) to nonlinear PDEs. The k-means and manifold clustering are combined with POD and DMD algorithms to create cluster-based reduced-order models, applied to the one- and two-dimensional Burgers' equations, which govern nonlinear convective flows. The Burgers’ equations are used as a surrogate for the Navier-Stokes equations. Each cluster represents similar dynamic behavior within itself, while being distinct from other clusters. Two different clustering schemes are proposed for time and space domains. In spatial clustering, the discontinuous Galerkin method is introduced to address the discontinuity of POD modes over the space domain. Following model reduction, linear quadratic regulators (LQRs) are designed for boundary control and applied to the full-order fluid flows.