Shape-Morphing Modes for Efficient Computation of Multiscale Evolution Partial Differential Equations with Conserved Quantities

用于高效计算具有守恒量的多尺度演化偏微分方程的形状变形模式

基本信息

批准号：
2208541
负责人：
Mohammad Farazmand
金额：
$ 19.63万
依托单位：
North Carolina State University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-08-15 至 2025-07-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2208541&HistoricalAwards=false
关键词：
Shape Morphing Modes Efficient Computation

项目摘要

Large-scale computations are needed in many areas of science and engineering, such as climate modeling, weather forecast, and design of sustainable structures. The corresponding mathematical models often involve a wide range of time and spatial scales which the simulations need to resolve. Efficiently resolving these multiscale structures has been a long-standing challenge in scientific computing. This project proposes shape-morphing modes as a new computational method that will drastically reduce the computational time and memory requirements of simulating multiscale systems. Shape-morphing modes are computational elements that adaptively change their shape and location to efficiently capture various temporal and spatial scales. The resulting computational speedup will enable us to perform real-time prediction, optimization, and control tasks that had been inaccessible to previous methods. The dynamics of spatiotemporal systems are routinely described by time-dependent partial differential equations (PDEs). The solutions of these PDEs often exhibit time-varying localized structures, with sharp gradients, surrounded by regions of large-scale motion. Such multiscale PDEs arise in numerous applications, such as aircraft design, weather prediction, ocean and climate modeling, where resolving small scale structures remains a major challenge. Currently, there are two broad classes of methods for addressing this challenge: 1. Adaptive methods which dynamically evolve the spatial discretization so that the computational grid is refined around the localized structure and less so in the quiescent regions. 2. Multiresolution methods, such as wavelets, which encode various scales in the basis instead of the discretization. This project will develop a new and computationally efficient method called shape-morphing modes. The main idea behind this method is to use a time-dependent basis of functions that automatically morph their shapes over time and space in order to efficiently resolve all scales. Being mesh-free, the proposed method substantially reduces the computational cost as compared to existing adaptive methods. Furthermore, since the modes adapt themselves to the solution of the PDE, far fewer modes are needed to resolve all scales. This significantly reduces the memory requirements, thus outperforming the existing multiresolution methods.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.

在许多科学和工程领域（例如气候建模，天气预报和可持续结构设计）需要进行大规模计算。相应的数学模型通常涉及模拟需要解决的广泛时间和空间尺度。有效解决这些多尺度结构一直是科学计算中的长期挑战。该项目提出了形状模式作为一种新的计算方法，该方法将大大减少模拟多尺度系统的计算时间和内存要求。形状修复模式是计算元素，可自适应地改变其形状和位置，以有效地捕获各种时间和空间尺度。由此产生的计算速度将使我们能够执行对以前方法无法访问的实时预测，优化和控制任务。时空系统的动力学通常由时间依赖性的部分微分方程（PDE）来描述。这些PDE的溶液通常表现出较时变的局部结构，并具有尖锐的梯度，周围是大规模运动区域。这种多尺度PDE在许多应用中都出现，例如飞机设计，天气预测，海洋和气候建模，在此解决小规模结构仍然是一个主要挑战。当前，有两种针对这一挑战的方法：1。动态发展空间离散化的自适应方法，以使计算网格围绕局部结构进行完善，而在静态区域中则较少。 2。多解决方法，例如小波，该方法在基础上而不是离散化编码各种量表。该项目将开发一种称为Shape-Molphing模式的新的且计算高效的方法。这种方法背后的主要思想是使用函数的时间相关基础，该功能会自动在时间和空间中自动变形其形状，以便有效地解决所有量表。与现有的自适应方法相比，提出的无网状方法大大降低了计算成本。此外，由于模式适应了PDE的解决方案，因此解决所有尺度所需的模式要少得多。这大大降低了内存要求，从而超过了现有的多分辨率方法。该奖项反映了NSF的法定任务，并且使用基金会的知识分子优点和更广泛的影响评估标准，被认为值得通过评估来获得支持。