Arbitrarily Wide Angle Wave Equations: New Constructs for Subsurface Imaging, Unbounded Domain Analysis and Multiscale Modeling of Solids

任意广角波动方程：地下成像、无界域分析和固体多尺度建模的新结构

• 批准号: 1016514
• 负责人: Murthy Guddati
• 金额: $ 24.4万
• 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: 2010
• 资助国家: 美国
• 起止时间: 2010-09-15 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1016514&HistoricalAwards=false
• 关键词: Arbitrarily; Wide; Angle; Wave; Equations

One-way wave equations are mathematical constructs that allow the propagation of waves in a specified direction, while suppressing the propagation in the opposite direction, i.e. they have a 180-degree range of propagation angles as opposed to the 360-degree range of full wave equations. Due to this special property, they are being used in various application areas including wave-based imaging algorithms (seismic imaging and nondestructive testing), ocean acoustics (modeling of long-range propagation), wave propagation modeling in unbounded domains, and multi-scale modeling of solids (phonon-absorbing boundary conditions for coupling molecular dynamics with continuum models). While the existing one-way wave equations are well developed for simple acoustic media, they are not as robust and efficient for more complicated, elastic, media. To cater to this need, the PI and his coworkers have recently developed a new series of one-way wave equations called the Arbitrarily Wide-angle Wave Equations (AWWEs). Unlike the existing one-way wave equations which are derived only for acoustics and special cases of elasticity, AWWEs can be derived for complicated media where the full wave equation has second-order derivatives in space (this includes wave propagation in general anisotropic, viscous and porous elastic media). Furthermore, AWWEs have simple form and are easy to implement. They are highly efficient and have the flexibility to treat various types of propagating and evanescent waves. The current limitation is that a straightforward design of AWWE leads to instabilities for complicated media (this is similar to many existing one-way wave equations). Stability of an AWWE is application-dependent and the proposed effort is aimed at devising stable AWWEs that can be used for various application areas including, (a) imaging in heterogeneous and anisotropic elastic media, (b) analysis of wave propagation in unbounded elastic domains that are heterogeneous and/or anisotropic, and (c) phonon-absorbing boundary conditions for molecular dynamics. Stabilization procedures will be developed by building on existing wellposedness and stability theory for linear hyperbolic systems in the contexts of absorbing boundary conditions, perfectly matched layers, and ocean acoustics. The resulting stabilized AWWE would be implemented and tested in various settings to ensure their robustness.The proposed work is aimed at developing new mathematical constructs that transmit waves in a specified direction while suppressing them in the other direction. Due to the ubiquitous nature of wave phenomenon in physics, successful completion of the proposed project would facilitate the solution of several important problems related to: (a) seismic inversion - locating hidden oil reservoirs; (b) seismology - modeling of wave scattering and focusing in complex geological basins; (c) soil-structure interaction - simulation of complex response of structures embedded in unbounded soil during earthquakes; (d) nanomechanics - understanding the failure of materials at nanometer level; (e) nondestructive evaluation - characterizing hidden cracks for strength assessment; (f) military applications - detection and characterization of buried mines. The proposed work also has applications in many other areas such as modeling optical circuits, synthetic aperture sonar and medical imaging. Finally, the project includes a graduate education component (thus contributing to the human resources development for computational mathematics), and the development of instructional modules for wave propagation and multiscale modeling (thus contributing to broader education in mechanics).

单向波方程是允许波在指定方向上传播，同时抑制相反方向上的传播的数学构造，即它们具有180度范围的传播角，而不是全波方程的360度范围。由于这种特殊的属性，它们被用于各种应用领域，包括基于波的成像算法（地震成像和无损检测），海洋声学（长距离传播的建模），无限域中的波传播建模，以及固体的多尺度建模（用于耦合分子动力学与连续模型的声子吸收边界条件）。虽然现有的单程波方程对于简单的声学介质是很好的，但是对于更复杂的弹性介质，它们不是那么鲁棒和有效。为了满足这一需求，PI和他的同事最近开发了一系列新的单向波方程，称为广义广角波方程（AWWE）。不同于现有的单程波方程，仅用于声学和弹性的特殊情况下，AWWE可以推导出复杂的介质，其中全波方程在空间中具有二阶导数（这包括一般各向异性，粘性和多孔弹性介质中的波传播）。此外，AWWE具有简单的形式和易于实现。它们是高效的，并具有灵活性，以处理各种类型的传播和倏逝波。目前的限制是，AWWE的简单设计会导致复杂介质的不稳定性（这类似于许多现有的单程波方程）。AWWE的稳定性取决于应用，并且所提出的努力旨在设计可用于各种应用领域的稳定的AWWE，所述应用领域包括（a）在非均质和各向异性弹性介质中的成像，（B）在非均质和/或各向异性的无界弹性域中的波传播的分析，以及（c）用于分子动力学的声子吸收边界条件。稳定程序将建立在现有的适定性和稳定性理论的线性双曲系统的背景下吸收边界条件，完全匹配层，海洋声学。由此产生的稳定AWWE将在各种设置中实施和测试，以确保其鲁棒性。拟议的工作旨在开发新的数学结构，在指定方向上传输波，同时在另一个方向上抑制波。由于波动现象在物理学中无处不在，拟议项目的成功完成将有助于解决与下列方面有关的几个重要问题：（a）地震反演-确定隐藏的储油层的位置;（B）地震学-模拟复杂地质盆地中的波散射和聚焦;（c）土壤-结构相互作用-模拟地震期间嵌入无限土壤中的结构的复杂反应;（d）地震学-模拟地震中嵌入无限土壤中的结构的复杂反应。（d）纳米力学-在纳米一级了解材料的破坏情况;（e）非破坏性评估-确定隐藏裂缝的特征，以评估强度;（f）军事应用-探测埋在地下的地雷并确定其特征。 所提出的工作也有许多其他领域的应用，如建模光学电路，合成孔径声纳和医学成像。最后，该项目包括研究生教育部分（从而有助于计算数学的人力资源开发），以及波传播和多尺度建模的教学模块的开发（从而有助于更广泛的力学教育）。