Structure-Preserving Hyper-Reduction and Temporal Localization for Reduced Order Models of Incompressible Flows

A novel hyper-reduction method is proposed that conserves kinetic energy and momentum for reduced order models of the incompressible Navier-Stokes equations. The main advantage of conservation of kinetic energy is that it endows the hyper-reduced order model (hROM) with a nonlinear stability property. The new method poses the discrete empirical interpolation method (DEIM) as a minimization problem and subsequently imposes constraints to conserve kinetic energy. Two methods are proposed to improve the robustness of the new method against error accumulation: oversampling and Mahalanobis regularization. Mahalanobis regularization has the benefit of not requiring additional measurement points. Furthermore, a novel method is proposed to perform structure-preserving temporal localization with the principle interval decomposition: new interface conditions are derived such that energy and momentum are conserved for a full time-integration instead of only during separate intervals. The performance of the new structure-preserving hyper-reduction methods and the structure-preserving temporal localization method is analysed using two convection-dominated test cases; a shear-layer roll-up and two-dimensional homogeneous isotropic turbulence. It is found that both Mahalanobis regularization and oversampling allow hyper-reduction of these test cases. Moreover, the Mahalanobis regularization provides comparable robustness while being more efficient than oversampling.