公共核平滑本征正交分解（CKSPOD）：一种模拟时空演变流动动力学的有效降阶模型
Common kernel-smoothed proper orthogonal decomposition (CKSPOD): An efficient reduced-order model for emulation of spatiotemporally evolving flow dynamics
摘要：
在本研究中，我们提出了一种新的代理模型，称为共同核平滑适当的正交分解（CKSPOD），有效地模拟时空演化的流体流动动力学。所提出的代理模型集成并扩展了高斯过程学习、高保真模拟、基于投影的模型简化、不确定性量化和实验设计等方面的最新发展，形成了一个系统的、多学科的框架。CKSPOD仿真的新颖之处在于一个共同的革兰氏矩阵的建设，这是从所有观察到的设计设置的革兰氏矩阵的Hadamard产品的结果。Gram矩阵是空间平均的时间相关矩阵，包含相应采样点的时间动态。公共Gram矩阵通过在每个观察到的设计设置下将POD模式转换为空间函数来合成时间动态，这解决了在Chang等人（2020）提出的最新流体流动仿真器内核平滑POD（KSPOD）仿真中遇到的相位差问题。CKSPOD的方法是通过一个模型研究的旋流喷油器的流动动力学与三个设计参数。共包括30个培训设计设置和8个验证设计设置。定性和定量的结果表明，CKSPOD仿真优于KSPOD仿真的所有验证情况下，并能够捕捉小尺度上的液体膜表面的波动结构的忠实。使用CKSPOD的湍流动能预测揭示了比KSPOD更低的预测不确定性，从而允许更准确和精确的流动预测。CKSPOD仿真的周转时间比相应的高保真仿真快约5个数量级，这为设计探索和优化提供了一个高效且可扩展的框架。
英文摘要：
In the present study, we propose a new surrogate model, called common kernel-smoothed proper orthogonal decomposition (CKSPOD), to efficiently emulate the spatiotemporal evolution of fluid flow dynamics. The proposed surrogate model integrates and extends recent developments in Gaussian process learning, high-fidelity simulations, projection-based model reduction, uncertainty quantification, and experimental design, rendering a systematic, multidisciplinary framework. The novelty of the CKSPOD emulation lies in the construction of a common Gram matrix, which results from the Hadamard product of Gram matrices of all observed design settings. The Gram matrix is a spatially averaged temporal correlation matrix and contains the temporal dynamics of the corresponding sampling point. The common Gram matrix synthesizes the temporal dynamics by transferring POD modes into spatial functions at each observed design setting, which remedies the phase-difference issue encountered in the kernel-smoothed POD (KSPOD) emulation, a recent fluid flow emulator proposed in Chang et al. (2020). The CKSPOD methodology is demonstrated through a model study of flow dynamics of swirl injectors with three design parameters. A total of 30 training design settings and 8 validation design settings are included. Both qualitative and quantitative results show that the CKSPOD emulation outperforms the KSPOD emulation for all validation cases, and is capable of capturing small-scale wave structures on the liquid-film surface faithfully. The turbulent kinetic energy prediction using CKSPOD reveals lower predictive uncertainty than KSPOD, thereby allowing for more accurate and precise flow predictions. The turnaround time of the CKSPOD emulation is about 5 orders of magnitude faster than the corresponding high-fidelity simulation, which enables an efficient and scalable framework for design exploration and optimization.
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