We propose a new approach based on the direct solution of the high frequency (WKB) approximation of the Helmholtz equation defined on a complicated domain, such as a large and intricate 3D environment, typical of a game or virtual environment. This allows us to recover the complete geometric acoustic field, including multiple arrivals. The crux of our method is a compact, high-order, semi-Lagrangian direct solver for the eikonal equation, which describes the phase of rays propagating in an environment. We couple this solver with techniques from physical optics and high frequency acoustic diffraction theory to compute the phase and amplitude of all rays, accounting for reflection and diffraction.
Since our efforts are focused on large-scale acoustic modeling, applying our algorithms to realistic (large) problems is crucial. We present additional results related to modeling sound propagation in built environments, including how to practically adapt our solver to large, unstructured tetrahedron meshes. We show how we can use the eikonal equation itself to adaptively place sampling probes. Additionally, we describe how our work relates to existing time-domain algorithms, and a design for a memory-efficient parallel pipelined encoder.
We refer to our combined set of techniques as numerical geometric acoustics, and our high-order method for solving the eikonal equation as a jet marching method. We discuss how to design jet marching methods for more general static Hamilton-Jacobi equations, of which the eikonal equation is a special case. Furthermore, our numerical algorithms constitute a development that stands independently of the room acoustics thrust of our proposal; in particular, they relate to the broader field of Eulerian geometric optics. We discuss how our results complement and improve on existing results in this field.
Examining Committee:
Dept rep: Dr. Ming Lin
Members: Dr. Ramani Duraiswami
Dr. Howard Elman
Sam Potter is a fourth year PhD student supervised by Drs. Maria Cameron and Ramani Duraiswami. He recently received his MS in electrical engineering from UMD’s Department of Electrical and Computer Engineering. Before coming to Maryland, he received his BS in Mathematics at the University of Washington in Seattle, where he hails from, and worked for two years as a software engineer in computer graphics. In 2018, he did an internship at NASA Goddard Space Flight Center developing fast algorithms for radiosity in the context of planetary science, and currently works part time at the Planetary Science Institute on a continuation of this project. He works on fast direct solvers for partial differential equations and integral equations and their application to large and practical problems.