Model reduction for nonlinear dynamical systems with parametric uncertainties
Model reduction for nonlinear dynamical systems with parametric uncertainties
复制标题
具有参数不确定性的非线性动力系统的模型简化
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发表时间：
2012
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通讯作者：
Yuxiang Zhou
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作者：
Yuxiang Zhou
已知非线性动力系统对输入参数敏感。在这篇论文中，我们将模型降阶应用于一类重要的此类系统-表现出极限环振荡（LCOs）和Hopf-分岔。高保真仿真系统与LCO的计算密集型，排除这些系统的概率分析与输入参数的不确定性。在这篇论文中，我们采用了一种基于投影的模型降阶方法，在该方法中，使用适当的正交分解（POD）来获得降阶基，而使用离散经验插值方法（DEIM）来近似非线性项，使得降阶模型（ROM）的重复在线评估与全阶模型（FOM）维数无关。在问题中存在着巨大的差异的幅度在未知变量，原来的POD-DEIM方法在较小的变量的结果在很大的误差。在非定常模拟中，这种误差随着时间的推移迅速积累，显着降低了ROM的精度。DEIM的插值性质也限制了其近似高度振荡的非线性项的精度。在这项工作中，修改现有的方法，提出了标量值POD模式中使用的每个变量的状态和非线性项，和纯插值的DEIM近似也被取代的回归通过过采样的非线性项。修正后的方法被应用到两个非线性动力学问题：反应流模型的管式反应器和悬臂板的气动弹性模型，这两个表现出LCO和Hopf分岔。结果表明，在原POD-DEIM ROM的效率是由不同的幅度在未知变量或需要包括大量的插值点的情况下，修改后的POD-DEIM ROM准确地预测系统响应的FOM计算时间的一小部分。导师：Karen E.威尔考克斯职称：航空航天学教授
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 redcution 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. Thesis Supervisor: Karen E. Willcox Title: Professor of Aeronautics and Astronautics