Research
Overview
My research covers mathematical modeling and numerical simulation of physical systems, with an emphasis on wave dynamics. 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 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 mathematical approach in three areas of acoustic wave simulation:
geometric boundary modeling, non-spherical harmonics, and elastic wave propagation. Past projects include modeling, simulating,
and fabricating optical standing-wave traps. My research has potential
applications in a variety of fields including acoustics, computer graphics, optics, and biophysics.
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 scalar 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, audio device prototyping, and seismic wave simulation. 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.
Variational Principles for Elastic Wave Simulation
A variational principle is a method used to model the state or dynamics of a physical system. The
physical model which it derives is a function that optimizes the value of a physical quantity of the system. For
example, one can use the stationary action variational principle to model a physical system by minimizing a quantity
known as the action between two states of the system. The action is the accumulated value of a so-called Lagrangian,
which for many systems is equal to the difference between the kinetic and potential energies of the system. Currently,
I am developing 3D elastic simulations based on a physical model derived using the first Piola-Kirchoff stress
tensor and its corresponding Lagrangian. The next phase of the project is to simulate passive surface waves in an
elastic membrane with homogeneous fluid chambers above and below the membrane. Of particular interest is the use of
geometric variational principles to simplify models of fluid-solid coupling in the inner ear. Besides this
application in auditory biophysics, my research in elastic wave simulation may have applications in other fields such
as music technology, music cognition, seismology, and ultrasound technology. 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).
Past Projects
Optical Standing-Wave Traps
Optical trapping is a useful tool for manipulating
microscopic particles and probing the physical interactions of matter. However, early optical trapping techniques
introduced complications for analyzing Brownian particle diffusion in viscous media because they either restricted the
particle motion or trapped the particles too close to a surface. My work presents the first known
realization of two-dimensional tracking of Brownian microparticles in multiple, surface-isolated optical traps.
The fabricated optical traps allow tight z-axis confinement and loose transverse (xy-plane) confinement of microparticles in the central maximum
of a standing-wave Bessel beam (note: the Bessel beam propagates along the z-axis).
As part of this project, I modeled and simulated the normalized field irradiance of the standing-wave Bessel beam,
as shown below. My model is based on the electric and magnetic field solutions to Maxwell's free-space scalar wave equation. Image caption
(left to right): contour plot of the field irradiance in the xz plane, grayscale plot of the field irradiance in the
xz plane, and contour plot of the field irradiance in the xy plane.
