Manuscripts in Progress

Wave simulations in infinite spacetime

Chad McKell, Mohammad Sina Nabizadeh, Stephanie Wang, Albert Chern

Under review

Solving the wave equation on an infinite domain has been an ongoing challenge in scientific computing. Conventional approaches to this problem only generate numerical solutions on a small subset of the infinite domain. In this paper, we present a method for solving the wave equation on the entire infinite domain using only finite computation time and memory. Our method is based on the conformal invariance of the scalar wave equation under the Kelvin transformation in Minkowski spacetime. As a result of the conformal invariance, any wave problem with compact initial data contained in a causality cone is equivalent to a wave problem on a bounded set in Minkowski spacetime. We use this fact to perform wave simulations in infinite spacetime using a finite discretization of the bounded spacetime with no additional loss of accuracy introduced by the Kelvin transformation.

arXiv preprint

Acoustic Wave Simulation using the Minkowski-Kelvin Transform

Chad McKell

Ph.D. Dissertation (defense planned for Spring 2024)

In this work, I demonstrate that the conformal invariance of the wave equation under a Kelvin transform in Minkowski spacetime allows for the efficient handling of complex geometric boundaries in acoustic wave simulations. In particular, I use the transform to convert infinite domains into bounded domains and curved boundaries into flat boundaries. I then solve the transformed wave problems using standard numerical methods with no additional loss of accuracy introduced by the transform.

Published Papers and Theses

Optical corral using a standing-wave Bessel beam

Chad McKell, Keith Bonin

Journal of the Optical Society of America B (2018)

Here we create a series of optical corrals and calculate their potential energy profile. A standing-wave Bessel beam is used to form traps in 1D (along the optical axis) and corrals in 2D, in planes perpendicular to the optical axis at the antinodal regions of the standing waves. These optical corrals are formed by an axicon-generated Bessel beam that is retro-reflected back onto itself. We report on Mie calculations of the 2D optical corrals and then compare the resulting probability distributions to those observed for latex particles of diameters 100, 200, and 300 nm. The experimental radial probability density function of tracked particles closely mimics the theoretical optical structure of a Bessel standing-wave corral. The Bessel standing-wave corrals we have characterized are being developed to measure rotational diffusion and torques on micro- and nanorods to help understand microfluidic behavior. The maximum forces on our small beads in the diffraction-free central zone of the Bessel beam standing wave are 𝐹||=0.5  pN and 𝐹⊥=0.1  pN.


Real-Time Physical Modeling for Haptic Feedback Rendering

Chad McKell

Master's Thesis, The University of Edinburgh (2017)

Physical modeling is an effective signal processing method for enhancing realism in haptic feedback rendering. In this study, real-time physics-based simulations of linear elastic vibrations were developed for use in vibrotactile displays. Applications to a variety of consumer use cases were explored. A discussion of future work on nonlinear modeling is included. Advisor: Stefan Bilbao. (Edinburgh Seal: By Source, Fair use,

(Publication of thesis restricted by NDA).

Sonification of optically-ordered Brownian motion

Chad McKell

International Computer Music Conference (2016)

In this paper, a method is outlined for the sonification of experimentally-observed Brownian motion organized into optical structures. Sounds were modeled after the tracked, three-dimensional motion of Brownian microspheres confined in the potential wells of a standing-wave laser trap. Stochastic compositions based on freely-diffusing Brownian particles are limited by the indeterminacy of the data range and by constraints on the data size and dimensions. In this study, these limitations are overcome by using an optical trap to restrict the random motion to an ordered stack of two-dimensional regions of interest. It is argued that the confinement of the particles in the optical lattice provides an artistically appealing geometric landscape for constructing digital audio effects and musical compositions based on experimental Brownian motion. A discussion of future work on data mapping and computational modeling is included. The present study finds relevance in the fields of stochastic music and sound design.

Paper     Video     Audio

Confinement and Tracking of Brownian Particles in a Bessel Beam Standing Wave

Chad McKell

Master's Thesis, Wake Forest University (2015)

Optical trapping is a useful tool for manipulating microscopic particles and probing the physical interactions of matter. However, previous optical trapping techniques introduced complications for analyzing Brownian particle diffusion in viscous media because they either restricted the particles' motion or trapped the particles too close to a surface. To our knowledge, this thesis presents the first realization of two-dimensional, transverse tracking of Brownian microparticles in multiple, surface-isolated traps. To accomplish this, we used an axicon-generated, zeroth-order Bessel beam standing wave whose parameters were adjusted to allow tight axial confinement and loose transverse confinement of microscopic-sized particles in the central maximum of the Bessel beam. We chose a Bessel beam because its unique non-diffracting and self-healing properties provided distinct advantanges over a Gaussian beam. In particular, a Bessel beam standing wave was shown to produce optical potential wells that are more abundant, uniform, and stable than those of a Gaussian standing wave. Advisor: Keith Bonin. (WFU Seal: By Source, Fair use,


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