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

North Carolina State University, Raleigh NC

Investigators

Abstract

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).