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

单向波动方程是一种数学构造，它允许波在指定方向上传播，同时抑制相反方向的传播，即它们具有180度范围的传播角度，而不是360度范围的全波方程。由于这种特殊的性质，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(远程传播模型)、无界域中的波传播模型以及固体的多尺度模型(用于耦合分子动力学和连续介质模型的声子吸收边界条件)。虽然现有的单程波动方程对于简单的声学介质发展得很好，但对于更复杂的弹性介质，它们并不是那么健壮和有效。为了满足这一需要，PI和他的同事们最近发展了一系列新的单向波动方程，称为任意广角波动方程(AWWES)。与现有的单程波动方程仅针对声学和弹性力学的特殊情况不同，AWWE可用于全波动方程在空间具有二阶导数的复杂介质中(包括波在一般各向异性、粘性和多孔弹性介质中的传播)。此外，AWWE具有形式简单、易于实现的特点。它们的效率很高，可以灵活地处理各种类型的传播和消逝的波。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向波动方程)。AWWE的稳定性取决于应用，建议的工作旨在设计稳定的AWWE，可用于各种应用领域，包括：(A)在非均匀和各向异性弹性介质中成像，(B)分析非均匀和/或各向异性无限弹性区域中的波传播，以及(C)分子动力学的声子吸收边界条件。在吸收边界条件、完美匹配层和海洋声学的背景下，通过建立现有的线性双曲型系统的适定性和稳定性理论，将开发稳定化程序。由此产生的稳定AWWE将在不同的环境中实施和测试，以确保其健壮性。拟议的工作旨在开发新的数学构造，在指定方向发射波，同时在另一个方向抑制波。由于物理学中普遍存在波动现象，因此，拟议项目的成功完成将有助于解决与以下几个重要问题有关的问题：(A)地震反演--寻找隐藏的油气藏；(B)地震学--波散射的建模和在复杂地质盆地中的聚焦；(C)土壤-结构相互作用--模拟地震期间埋藏在无限土壤中的结构的复杂反应；(D)纳米力学--在纳米层面了解材料的失效情况；(E)无损评估--表征隐藏的裂缝以进行强度评估；(F)军事应用--探测和表征埋藏的地雷。所提出的工作在许多其他领域也有应用，如光路建模、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)，以及开发波传播和多尺度建模的教学模块(从而有助于更广泛的力学教育)。

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).