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 micromechanics. My research has potential
applications in a variety of fields including music, architecture, 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. A promising future
direction of my research involves developing boundary flattening methods for curved obstacles. 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.
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 Cochlear Micromechanics
Physical simulation is the computational
emulation of natural phenomena using physics-based models. Physical simulation is useful for producing
very accurate emulations and offering intuitive user control of modeling parameters. In the past, I have
developed physics-based modal synthesis and finite-difference time-domain simulations of musical instruments
and human speech. Currently, I am interested in using geometric variational principles to simulate
fluid-solid interactions in the human cochlea. Application areas may include music cognition, psychoacoustics,
and otology.
