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Fast Spectral Solvers for Partial Differential Equations in General Domains

一般域中偏微分方程的快速谱求解器

基本信息

批准号：
1714169
负责人：
Oscar Bruno
金额：
$ 14.71万
依托单位：
California Institute of Technology
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-08-15 至 2021-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1714169&HistoricalAwards=false
关键词：
Fast Spectral Solvers Partial Differential

项目摘要

1714169Bruno This project concerns development and analysis of computational methods enabling prediction of the behavior of physical systems including fluid flow, solid mechanics, acoustics, and electromagnetism. The new computational methods seek to efficiently predict the time-dependent behavior of physical and engineering systems, with application in areas such as medicine, seismology, aerodynamics, and stealth. A common thread in all of these areas relates to the complexity of the time evolution inherent in these systems. The flow of blood from a beating heart, the seismic waves arising from an earthquake, the fluid flow around an aircraft, and the behavior of radar technologies all share some of the same elements: the time evolution of the observed phenomena. It is this complexity in time-dependence that is addressed by this project: the Fourier Continuation (FC) and spectral time methodologies seek to reproduce the outstanding performance of certain spectral solvers, which are unfortunately restricted to simple geometric configurations, for simulations in a much more general realm, with greatly reduced computer memory and processing time -- thus, effectively enabling the solution of previously intractable problems in areas of highly significant engineering and scientific interest. A graduate student and an undergraduate student participate in the project. From a mathematical perspective, the effort seeks to produce solvers for partial differential equations (PDEs) in the time domain that can deliver, in fast computing times, nearly dispersionless (spectral) PDE solutions for general complex three-dimensional structures. The project thus extends to general engineering and scientific configurations the high quality otherwise inherent in spectral solvers for spatially periodic structures. The FC and related frequency-domain approaches have already been highly successful in a wide range of important problems in science and engineering. The extensions studied here seek to bring time-domain solvers to the realm of general applicability, thus extending to the time domain the previous successes in the area of spectral solvers in the frequency domain. The resulting methods are applied to important problems in the fields of materials science, applied physics, electrical engineering, geophysics, and aerospace engineering. A graduate student and an undergraduate student participate in the project.

1714169Bruno 该项目涉及计算方法的开发和分析，能够预测物理系统的行为，包括流体流动、固体力学、声学和电磁学。新的计算方法旨在有效地预测物理和工程系统的时间相关行为，并应用于医学、地震学、空气动力学和隐形等领域。所有这些领域的共同点都与这些系统固有的时间演化的复杂性有关。跳动的心脏的血液流动、地震产生的地震波、飞机周围的流体流动以及雷达技术的行为都具有一些相同的要素：观察到的现象的时间演变。该项目所解决的正是这种时间依赖性的复杂性：傅里叶连续（FC）和谱时间方法试图再现某些谱求解器的出色性能，不幸的是，这些求解器仅限于简单的几何配置，用于更一般领域的模拟，大大减少了计算机内存和处理时间，从而有效地解决了具有高度重要工程和科学意义的领域中以前棘手的问题。一名研究生和一名本科生参与了该项目。从数学角度来看，这项工作旨在为时域中的偏微分方程（PDE）生成求解器，该求解器可以在快速计算时间内为一般复杂的三维结构提供几乎无色散（谱）PDE 解决方案。因此，该项目将空间周期性结构的光谱解算器固有的高质量扩展到一般工程和科学配置。 FC 和相关的频域方法已经在科学和工程领域的许多重要问题上取得了巨大成功。这里研究的扩展旨在将时域求解器带到普遍适用的领域，从而将之前在频域频谱求解器领域取得的成功扩展到时域。由此产生的方法应用于材料科学、应用物理学、电气工程、地球物理学和航空航天工程领域的重要问题。一名研究生和一名本科生参与该项目。