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.

开发有效且可靠的计算工具用于模拟波的许多主要障碍植根于普遍存在的多个空间尺度。该项目的重点是针对克服这些障碍有关的精选问题的详细研究。所解决的问题的一个统一特征是，所有问题都涉及对空间和时间中整体运营商的有效近似评估。尽管这项工作（如果成功）有可能影响众多科学和工程学科，但这些工作将针对航空声学和电磁学方面的问题。确切地说，将开发以下内容：（i）准确，有效且可靠的计算域截断方法，允许直接模拟仅在培养基复杂或非线性效应很重要的区域进行； （ii）有效的时间步变方法，允许对高分辨率或几何细节的集中区域进行简单处理。在应用科学和技术的许多领域，波浪传播问题至关重要。它们包括广泛的物理学（电磁，流体和固体力学），但具有基本的数学特性。波的定义特征是它相对于其基本维度（波长）的长距离行驶的能力，并带有有关其传播介质的详细信息。因此，波是我们探测性质和交流的主要方法。这种基本特征的结果是，波传播问题通常涉及不同的空间尺度 - 从散射器的几何细节到一系列波长到传播距离。这些多个量表反过来又导致了计算分析的困难。特别是，它们统一的决议将导致一定数量的自由度。因此，必须开发只能在需要的情况下集中计算资源的方法，从而为我们考虑的问题提供主要动机。除了此分析外，该计划是与正在建立软件的研究人员合作，以模拟喷气噪声，复杂结构中的电磁散射以及通用波波传播问题。因此，研究的任何积极发展都可以尽快使用。