Data-Driven Computation of Lagrangian Transport Structure in Realistic Flows

及时预测污染物在海洋、洪水或大气中的扩散是至关重要的。要在以后的时间里准确预测污染物的位置，需要一个真实的底层流体流动模型，以及污染物最初位置的知识。此外，为了帮助决策者，流体模拟必须进行得足够快。海洋和大气的运动，就像其他流体的运动一样，产生了某些模式，其形状受到特殊区域的影响，这些区域统称为拉格朗日输运结构，或LTS（以数学家和流体力学家约瑟夫·路易斯·拉格朗日命名）。LTS是一个组织流体运动方式的模板：一些结构是吸引附近流体的表面，一些是排斥附近流体的表面，还有一些形成蜿蜒的射流路径或漩涡状结构。关注LTS有助于减轻污染物预测对其初始位置的依赖，因为污染物倾向于遵循这些关键的组织特征。此外，将这些结构结合到一个更快的近似模型中，可以加快流体模拟本身的速度。然而，尽管它有潜在的用途——在搜索和救援场景中预测危险物质、碎片或失踪人员的扩散——但由于计算成本高，LTS目前没有被使用。该项目旨在开发一种新的框架，使移动平台上的实时、强大的LTS计算成为可能，从而为实时决策提供信息（例如，直接在载人或自动驾驶车辆上进行侦察和传感）。这种对现实的水生和大气环境的有效、及时的计算可以提供信息，以防止生命损失，减轻环境破坏，并避免巨大的经济成本。本课题研究了一种新颖的拉格朗日数据驱动的降阶建模和空间滤波框架，用于流体输运模拟。这个新框架旨在将当前算法的计算成本降低几个数量级，并产生准确而稳健的LTS近似，相对于现实流动中固有的数值不准确性。在计算流体力学和非线性动力学——混沌理论背后的数学理论——中发展几个相互交织的方法将会取得进展。该项目的主要新颖之处在于连接欧拉算法（用于速度场计算）和拉格朗日算法（用于LTS计算）。这使得新的拉格朗日数据驱动的降阶模型（rom）和空间滤波器的发展成为可能。新的拉格朗日数据驱动ROM是基于一种新颖的拉格朗日内积，使得精确和有效地逼近平均LTS成为可能。相比之下，标准欧拉rom产生不准确的LTS结果。研究了一种新的拉格朗日数据驱动空间滤波器，用于粗糙逼真网格的LTS计算。这种新的滤波器相对于实际流动中固有的数值不准确性来说是稳定、准确、高效和稳健的。虽然这项工作将侧重于环境流动，但预计其结果将适用于各种流体应用。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Forecasting the spread of contaminants in the ocean, floodwaters, or atmosphere in a timely manner is of critical importance. Accurate prediction of the contaminant location at a later time requires a realistic model of the underlying fluid flow, as well as knowledge of where the contaminant was initially located. Furthermore, to aid decision-makers, the fluid simulations must proceed sufficiently rapidly. The movement of the ocean and atmosphere, like that of other fluids, creates certain patterns whose shape is influenced by special regions, known collectively as the Lagrangian transport structure, or LTS (named after the mathematician and fluid mechanist Joseph-Louis Lagrange). LTS is a template that organizes how the fluid moves: some structures are surfaces that attract nearby fluid, some repel nearby fluid, and some form meandering jet pathways or vortex-like structures. Focusing on the LTS helps alleviate the dependence of contaminant forecast on its initial location, as contaminants tend to follow these key organizing features. Furthermore, incorporating these structures into a faster, approximate model, the fluid simulation itself can be sped up. However, despite its potential usefulness -- to predict the spread of hazardous material, debris, or missing individuals in a search-and-rescue scenario -- LTS is not currently used because of the high computational cost. This project aims to develop a new framework to make possible real-time, robust LTS computation on mobile platforms, to inform real-time decision-making (for instance, directly onboard the manned or autonomous vehicles doing reconnaissance and sensing). Such effective, timely computations for realistic aquatic and atmospheric environments can provide information to prevent the loss of lives, mitigate environmental damage, and avoid enormous financial cost.This project studies a novel Lagrangian data-driven reduced-order modeling and spatial filtering framework for fluid transport simulation. This new framework in intended to decrease the computational cost of current algorithms by orders of magnitude and yield LTS approximations that are accurate and robust with respect to the numerical inaccuracies that are inherent in realistic flows. Progress will be made by developing several intertwined approaches in computational fluid mechanics and nonlinear dynamics -- the mathematical theory underlying chaos theory. The main novelty of the project is bridging Eulerian algorithms (used in the velocity field computation) and Lagrangian algorithms (used in the LTS computation). This makes possible the development of novel Lagrangian data-driven reduced-order models (ROMs) and spatial filters. The new Lagrangian data-driven ROM is based on a novel Lagrangian inner product that makes possible the accurate and efficient approximation of average LTS. In contrast, standard Eulerian ROMs produce inaccurate LTS results. A novel Lagrangian data-driven spatial filter for LTS computation on coarse realistic meshes is also studied. This new filter is stable, accurate, efficient, and robust with respect to the numerical inaccuracies that are inherent in realistic flows. While the work will focus on environmental flows, the results are expected to apply to a wide variety of fluid applications.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.