Research

Overview
My research covers mathematical modeling and numerical simulation of physical systems with an emphasis on differential geometry and wave motion. I am especially interested in applications of physics simulation in music technology and auditory biophysics. My approach to modeling and simulation takes advantage of mathematical tools such as conformal symmetry, differential forms, and variational principles. These concepts can be used to overcome many difficulties and limitations that exist in standard Newtonian models and coordinate-based Euclidean spaces. Currently, I am utilizing these tools in three areas of wave simulation: geometric boundary modeling, non-spherical harmonics, and elastic wave propagation. Past projects include underwater acoustic sensing and optical standing-wave trapping.

Geometric Boundary Modeling for Wave Simulation
I have developed mathematical and computational methods for handling open exterior boundaries in wave simulations, 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 of interest include outdoor architectural acoustics and personalized binaural audio. 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 Wave 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 a variational principle 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 variational principles to simulate elastic waves in mammalian hearing organs and nonlinear shells. 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.