Model reduction for nonlinear dynamical systems with parametric uncertainties

Abstract

Nonlinear dynamical systems are known to be sensitive to input parameters. In this thesis, we apply model order reduction to an important class of such systems -- one which exhibits limit cycle oscillations (LCOs) and Hopf-bifurcations. Highfidelity simulations for systems with LCOs are computationally intensive, precluding probabilistic analyses of these systems with uncertainties in the input parameters. In this thesis, we employ a projection-based model reduction approach, in which the proper orthogonal decomposition (POD) is used to derive the reduced basis while the discrete empirical interpolation method (DEIM) is employed to approximate the nonlinear term such that the repeated online evaluations of the reduced-order model (ROM) is independent of the full-order model (FOM) dimension. In problems where vastly different magnitudes exist in the unknowns variables, the original POD-DEIM approach results in large error in the smaller variables. In unsteady simulations, such error quickly accumulates over time, significantly reducing the accuracy of the ROM. The interpolatory nature of the DEIM also limits its accuracy in approximating highly oscillatory nonlinear terms. In this work, modifications to the existing methodology are proposed whereby scalar-valued POD modes are used in each variable of the state and the nonlinear term, and the pure interpolation of the DEIM approximation is also replaced by a regression via over-sampling of the nonlinear term. The modified methodology is applied to two nonlinear dynamical problems: a reacting flow model of a tubular reactor and an aeroelastic model of a cantilevered plate, both of which exhibit LCO and Hopf-bifurcation. Results indicate that in situations where the efficiency of the original POD-DEIM ROM is compromised by disparate magnitudes in unknown variables or by the need to include large sets of interpolation points, the modified POD-DEIM ROM accurately predicts the system responses in a small fraction of the FOM computational time.