This paper considers the problem of parametric, nonlinear, projection-based model order reduction (PMOR) and makes three contributions to advance its state of the art. First, it presents a computational strategy for reducing the total offline cost of nonlinear PMOR by enabling hyperreduction to be performed adaptively within the typical greedy sampling procedure, rather than uniformly at each of its iteration. Then, it presents a more accurate and computationally efficient approach for training hyperreduction based on residual Jacobians rather than residuals. Finally, it demonstrates for a large-scale, industrial-grade, parametric, nonlinear application, that the two aforementioned contributions make both offline and online stages of nonlinear PMOR practical in today’s engineering environments. The paper also shows that despite the so-called Kolmogorov $n$-width barrier for the model reduction of convection-dominated transport problems, the current state of the art of nonlinear, Petrov–Galerkin PMOR equipped with the advances proposed in this paper is capable of accelerating by orders of magnitude the accurate solution of shape-parametric, high Reynolds number, viscous flow problems represented by the Reynolds-averaged Navier–Stokes equations augmented with turbulence modeling. All three contributions are presented and discussed in the contexts of steady-state CFD problems; however, they are equally applicable to unsteady CFD problems as well as problems in solid mechanics and structural dynamics.

本文考虑了参数化、非线性、基于投影的模型降阶 (PMOR) 问题，并为提升其最先进水平做出了三项贡献。首先，它提出了一种计算策略，通过在典型的贪婪采样过程中自适应地执行超缩减，而不是在每次迭代中统一执行，从而降低非线性 PMOR 的总离线成本。然后，它提出了一种更准确和计算效率更高的方法来训练基于残差的超归约雅可比而不是残差。最后，它证明了大规模、工业级、参数化、非线性应用，上述两个贡献使非线性 PMOR 的离线和在线阶段在当今的工程环境中都实用。该论文还表明，尽管所谓的 Kolmogorov$n$- 对流主导传输问题的模型减少的宽度障碍，非线性的当前状态，配备本文提出的进步的 Petrov-Galerkin PMOR 能够将形状参数的精确解加速几个数量级, 高雷诺数, 粘性流动问题, 由雷诺平均 Navier-Stokes 方程表示, 并增加了湍流建模。所有这三个贡献都在稳态 CFD 问题的背景下进行了介绍和讨论；但是，它们同样适用于非定常 CFD 问题以及固体力学和结构动力学问题。