Numerical Methods for Multiple Scale Problems in Wave Propagation: Efficient Approximation of Integral Operators in the Time Domain

波传播中多尺度问题的数值方法：时域积分算子的有效逼近

• 批准号: 0306285
• 负责人: Thomas Hagstrom
• 金额: $ 12.82万
• 依托单位: University of New Mexico
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-06-01 至 2006-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0306285&HistoricalAwards=false
• 关键词: Numerical; Methods; Multiple; Scale; Problems

Many of the main obstacles to the development of efficient and reliable computational tools for simulating waves are rooted in the multiple spatial scales which are universally present. The focus of this project is the detailed study of select questions which are relevant for overcoming these obstacles. A unifying feature of the questions addressed is that all involve the efficient approximate evaluation of integral operators in space and time. Although the work, if successful, has the potential to impact numerous scientific and engineering disciplines, the efforts will be directed towards problems in aeroacoustics and electromagnetics. Precisely, the following will be developed: (i) Accurate, efficient and reliable computational domain truncation methods, allowing the direct simulations to take place only in regions where the medium is complex or where nonlinear effects are important; (ii) Efficient time-stepping methods allowing the simple treatment of concentrated regions of high resolution or geometric detail.Wave propagation problems are of fundamental importance in many areas of applied science and technology. They encompass a wide range of physics (electromagnetics, fluid and solid mechanics), but share essential mathematical properties. The defining characteristic of a wave is its ability to travel long distances relative to its basic dimension, the wavelength, carrying detailed information about the medium through which it has traveled. For this reason, waves are the primary method by which we probe nature and communicate. A consequence of this fundamental characteristic is that wave propagation problems typically involve disparate spatial scales - from the geometrical details of scatterers through a range of wavelengths to the propagation distances. These multiple scales, in turn, lead to difficulties in computational analysis. In particular, their uniform resolution would lead to a prohibitive number of degrees of freedom. Thus methods must be developed which can concentrate computational resources only where they are needed, providing the primary motivation for the problems we consider. In addition to this analysis, the plan is to collaborate with researchers who are building software for simulating jet noise, electromagnetic scattering in complex structures, as well as general-purpose wave propagation problems. Thus any positive developments from the research can come into use as rapidly as possible.

开发高效可靠的波浪模拟计算工具的许多主要障碍都源于普遍存在的多个空间尺度。该项目的重点是对与克服这些障碍相关的选定问题进行详细研究。所讨论问题的一个统一特征是，所有问题都涉及空间和时间积分算子的有效近似评估。尽管这项工作如果成功，有可能影响许多科学和工程学科，但其努力将针对气动声学和电磁学问题。具体来说，将开发以下内容：（i）准确、高效和可靠的计算域截断方法，允许仅在介质复杂或非线性效应重要的区域进行直接模拟； (ii) 有效的时间步进方法可以对高分辨率或几何细节的集中区域进行简单处理。波传播问题在应用科学和技术的许多领域中具有根本重要性。它们涵盖了广泛的物理学（电磁学、流体力学和固体力学），但具有共同的基本数学特性。波的定义特征是它能够相对于其基本尺寸（波长）传播长距离，并携带有关其传播的介质的详细信息。因此，波是我们探索自然和交流的主要方法。这一基本特征的结果是，波传播问题通常涉及不同的空间尺度——从散射体的几何细节到一系列波长再到传播距离。这些多重尺度反过来又导致计算分析的困难。特别是，它们的统一分辨率将导致令人望而却步的自由度。因此，必须开发能够将计算资源集中在需要的地方的方法，为我们考虑的问题提供主要动机。除了这一分析之外，该计划还计划与正在构建软件的研究人员合作，以模拟喷射噪声、复杂结构中的电磁散射以及通用波传播问题。因此，研究的任何积极进展都可以尽快投入使用。