Research

Overview
My research covers mathematical modeling and numerical simulation of acoustic systems. My approach to modeling and simulation is guided by several core concepts in differential geometry, including conformal invariance, differential forms, and geometric variational principles. These geometric tools can be used to overcome many difficulties and limitations that exist in standard Newtonian models and coordinate-based Euclidean spaces. Currently, I am utilizing this geometric approach in three areas of acoustic simulation: boundary modeling, non-spherical harmonics, and cochlear mechanics. My research has potential applications in a variety of fields including music, architecture, seismology, and hearing science.


Geometric Boundary Modeling for Acoustic Wave Simulation
I have developed mathematical and computational methods for handling open exterior boundaries in acoustic wave simulation, including a differential geometric method that solves the wave equation in infinite spacetime and a discrete perfectly matched layer that exhibits machine-zero reflections. Future directions of my research involve developing boundary flattening methods for curved obstacles and formulating a finite-element method version of the reflectionless perfectly matched layer. A common theme in this work is the discovery of conformally invariant transformations of wave propagation, an approach that derives from Felix Klein's Erlangen program. Application areas may include architectural acoustic design, personalized binaural audio, and audio device prototyping. Image caption (left to right): solution to the Helmhotz equation in infinite space, (next two images) solution to the wave equation in infinite spacetime, local flattening of a circular boundary, and the frequency-domain reflectionless perfectly matched layer.




Non-Spherical Harmonics for Acoustic Simulation, Capture, and Rendering
Below is an image of four of the harmonic modes of the Stanford bunny mesh. From left to right, I've plotted the \(0^{th}\), \(1^{st}\), \(2^{nd}\), and \(6^{th}\) modes. In this methodology, I compute the displacement of each mode by numerically solving an eigen value problem at every vertex on the triangulated mesh. The purple portions represent regions where the displacement is maximally negative, and the red portions represent where it is maximally positive. Notably, the solution is computed using coordinate-free differential forms, which allow one to more easily model dynamics on arbitrary geometries. I am interested in using this methodology to calculate acoustic transfer functions involving complex rigid obstacle geometries, such as head-related transfer functions, and to capture and render sounds using non-spherical microphone arrays.




Physical Simulation of Elastic and Fluid Dynamics in the Human Cochlea
Physical simulation is the computational emulation of natural phenomena using physics-based models. Physical simulation is useful for producing very accurate emulations, offering intuitive user control of modeling parameters, and validating experimental measurements. I am especially interested in using a physics-based approach to simulate hydroelastic waves and other elastic-fluid dynamics in the human cochlea. Of particular interest is the use of geometric variational principles to simplify the elastic-fluid models. Currently, I am building foundational elastic and fluid simulations using arbitrary geometries. The next phase of the research is to simulate passive surface waves in an elastic membrane with fluid chambers above and below the membrane. Application areas may include psychoacoustics, music cognition, and otology. Image caption (left to right): elastic deformation of a 3D geometry, fluid flow in a 2D plane, (next two images) longitudinal section of the human cochlea and section of the organ of Corti (Gray's Anatomy, 1918).