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至关重要。 准确预测污染物的位置在以后的时间需要一个现实的模型的基础流体流动，以及知识的污染物最初位于。 此外，为了帮助决策者，流体模拟必须足够快地进行。海洋和大气的运动，就像其他流体的运动一样，会产生特定的模式，其形状受到特殊区域的影响，统称为拉格朗日传输结构，或LTS（以数学家和流体力学家约瑟夫-路易斯拉格朗日的名字命名）。LTS是一个组织流体如何移动的模板：一些结构是吸引附近流体的表面，一些结构排斥附近流体，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，因为污染物往往遵循这些关键的组织特征。 此外，将这些结构合并到更快的近似模型中，可以加速流体模拟本身。 然而，尽管LTS在预测搜索和救援情景中危险材料、碎片或失踪人员的扩散方面具有潜在的用处，但由于计算成本高，目前尚未使用。 该项目旨在开发一个新的框架，使移动的平台上的实时，强大的LTS计算成为可能，以告知实时决策（例如，直接在载人或自动驾驶车辆上进行侦察和传感）。 本计画研究一种新颖的拉格朗日资料驱动降阶模型与空间滤波架构，以应用于流体输运模拟。 这种新的框架旨在降低当前算法的计算成本的数量级和产量LTS近似是准确的和强大的数值不准确性，是固有的现实流。通过发展计算流体力学和非线性动力学（混沌理论的数学理论）中几种相互交织的方法，将取得进展。该项目的主要新奇是桥接欧拉算法（用于速度场计算）和拉格朗日算法（用于LTS计算）。这使得开发新的拉格朗日数据驱动的降阶模型（ROM）和空间滤波器成为可能。新的拉格朗日数据驱动的ROM是基于一种新的拉格朗日内积，使得可能的平均LTS的准确和有效的近似。 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的空间滤波器，用于粗糙真实网格上的LTS计算。这种新的过滤器是稳定的，准确的，高效的，和强大的数值不准确性，在现实的流动。虽然这项工作将集中在环境流量，其结果预计将适用于各种各样的流体应用。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.