流体流动的非线性模型简化

Samir Sahyoun, S. Djouadi
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引用次数: 10

摘要

模型简化是一种重要的工具,已应用于许多控制应用,如流体流动的控制。大多数模型约简算法都假定为线性模型,当应用于非线性高维系统,特别是高雷诺数的流体流动问题时,这些算法都失败了。例如,适当的正交分解(POD)无法捕获这些系统中的非线性自由度,因为它假设数据属于线性空间,因此依赖欧几里得距离作为最小化的度量。然而,由非线性偏微分方程(PDEs)生成的快照属于流形,其测地线通常不对应于欧几里得距离。测地线是一条曲线,它是点之间的局部最短路径。本文提出了一种模型约简方法,将POD推广到每个点都具有可微结构的非线性流形。此外,提出了一种构造边界控制下二维Burgers方程降阶模型的最优方法,并与POD降阶模型进行了比较。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Nonlinear model reduction for fluid flows
Model Reduction is an essential tool that has been applied in many control applications such as control of fluid flows. Most model reduction algorithms assume linear models and fail when applied to nonlinear high dimensional systems, in particular, fluid flow problems with high Reynolds numbers. For example, proper orthogonal decomposition (POD) fails to capture the nonlinear degrees of freedom in these systems, since it assumes that data belong to a linear space and therefore relies on the Euclidean distance as the metric to minimize. However, snapshots generated by nonlinear partial differential equations (PDEs) belong to manifolds for which the geodesies do not correspond in general to the Euclidean distance. A geodesic is a curve that is locally the shortest path between points. In this paper, we propose a model reduction method which generalizes POD to nonlinear manifolds which have a differentiable structure at each of their points. Moreover, an optimal method in constructing reduced order models for the two-dimensional Burgers' equation subject to boundary control is presented and compared to the POD reduced models.
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