2023 Faculty and Student Summer Talks

Samantha Evans

A Fast Mesh-Free Boundary Integral Method for Two Phase Flow with Soluble Surfactant

We develop a fast and accurate boundary integral method to simulate the two-phase flow of a viscous drop immersed in a surrounding viscous fluid which contains dissolved soluble surfactant. The drop is stretched by an imposed extensional flow. Surfactant is a surface active agent important in microfluidic applications that lowers surface tension and introduces a Marangoni force that typically opposes the imposed outer flow. However, it is difficult to adapt boundary integral methods to problems with soluble surfactant as the equation for the bulk surfactant does not have a Green’s function formulation. In addition, at large Peclet numbers that are characteristic of real physical systems, the concentration of soluble surfactant develops a transition or boundary layer adjacent to the drop interface which is difficult to resolve using traditional numerical methods. To address these difficulties, we further develop a hybrid numerical method that was first introduced in Booty and Siegel (2010) and extended to two-phase flow with soluble surfactant in Xu, Booty, and Siegel (2013). The hybrid method is based on an asymptotic reduction of the surfactant dynamics in the transition layer in the limit Pe → ∞. This reduced advection-diffusion equation for the concentration of bulk soluble surfactant has a Green’s function formulation. We introduce a fast method for computing a time-convolution integral that arises in the Green’s function formulation of the advection-diffusion equation. The fast algorithm is based on a method originally introduced for Abel integral equations by Johannes Tausch. Results illustrating the speed and accuracy of the fast method will be presented.

Thi Phong Nguyen

Inverse Scattering Problems and Application to the Maging of Local Defects in Periodic Media

Inverse scattering problems arise from many real life applications such as non-destructive evaluation, medical imaging, geophysical exploration. In general, such problems aim to determine information about an object (scatterer) from measurements of waves scattered by that object. 

In this talk, I will briefly introduce inverse scattering problems and the use of Sampling Methods for solving them. I will then discuss some ideas on how they can be applied to determine local defects in periodic media when only minimal measurements are needed. Some numerical results will be provided.

Lauren Barnes

The Role of Concentration-dependent viscosity on the dynamics of colloid-polymer mixtures

Phase transitions in colloid-polymer mixtures give valuable insight into atomic-scale phase transitions. While colloidal suspensions exhibit fluid-solid coexistence, the addition of polymer leads to three-phase coexistence. We here conduct a numerical exploration of the effects of fluid viscosity on the phase behavior of colloid-polymer suspensions in a microgravity environment. Specifically, we numerically simulate a phase-field model, which consists of a Cahn-Hilliard equation for the colloid concentration. Colloidal particles are advected by the surrounding Stokes flow, the model for which incorporates the dependence of viscosity on the local colloid concentration. The results from numerical simulations of our model will be compared against experimental data from BCAT experiments performed on the International Space Station. 

Jose Pabon

Reduced-order Models of Hydrodynamically Interacting Flapping Airfoils

Fish schools are examples of collective motion in which hydrodynamic interactions play a role in dynamical behavior and self-organization principles. However, the long-time evolution of hydrodynamically interacting collectives is challenging to investigate due to the persistent influence of long-lived vortical structures, and the high-resolution requirements of direct numerical simulation at large Reynolds numbers. Reduced-order models have therefore played an important role in theoretical investigations. We introduce a new reduced-order model of swimmers that self-propel by flapping, i.e., executing a prescribed periodic rigid body motion. The model is extension of a discrete-time dynamical system developed by Oza, Ristroph and Shelley in which flapping swimmers interact through periodically shed vortices. Our extensions include allowing a variable separation distance between swimmers, and more faithful modeling of wing-wing interactions by enforcing the Kutta condition and Kelvin Circulation Theorem to our modeling setup. The models are used to investigate conditions under which hydrodynamic interactions lead to stable swimming configurations with an optimized swimming speed. Preliminary numerical results are presented.

Nicholas Dubicki

A Micromagnetic Study of Skyrmions in Ferromagnetic Thin Film Bilayers

Phase transitions in colloid-polymer mixtures give valuable insight into atomic-scale phase transitions. While colloidal suspensions exhibit fluid-solid coexistence, the addition of polymer leads to three-phase coexistence. We here conduct a numerical exploration of the effects of fluid viscosity on the phase behavior of colloid-polymer suspensions in a microgravity environment. Specifically, we numerically simulate a phase-field model, which consists of a Cahn-Hilliard equation for the colloid concentration. Colloidal particles are advected by the surrounding Stokes flow, the model for which incorporates the dependence of viscosity on the local colloid concentration. The results from numerical simulations of our model will be compared against experimental data from BCAT experiments performed on the International Space Station. 

Moshe Silverstein

Calcium Signaling: Stochastic Models, Large Deviation Theory, and Statistical Analysis

This talk focuses on the stochastic modeling and application of large deviation theory (LDT), of calcium signaling in immune cells. We begin by presenting experimental results that shed light on the intricacies of calcium signaling. Building upon these findings, we introduce a stochastic model to capture the dynamic nature of calcium signaling processes. LDT provides insights into the most likely path taken by the dynamics of our model, offering a deeper understanding of the probabilistic nature of calcium signaling.  By integrating statistical analysis, stochastic modeling, and LDT, we aim to enhance our understanding of calcium signaling dynamics and pave the way for further advancements in this field.

David Mazowiecki

Numerical Simulation of Particles Undergoing Quincke Rotation

We consider the behavior of many identical rigid spheres in a Stokes flow fluid driven by an applied electric field in the electrohydrodynamic limit.  The governing equations follow the Melcher-Taylor leaky dielectric model, in which the applied field induces a surface charge at a solid-liquid interface.

The ratio of electrical permittivity to conductivity is a time scale for charge relaxation.  When the time scale of the solid particles is greater than the scale of the host fluid, the induced polarization in a particle is opposite to the applied field.  When the applied field is strong enough, a particle can undergo rotation about an axis orthogonal to the applied field, which is referred to as Quincke rotation.  We investigate this for multiple interacting spheres.

Professor Kristina Wicke

Exploring Phylogenetic Networks and Their Properties

Traditionally, phylogenetic trees have been used to depict the evolutionary relationships among different species. However, it is now widely acknowledged that evolution is not always tree-like and phylogenetic networks have come to the fore as a generalization of phylogenetic trees.

In this talk, I will give a broad introduction into the area of mathematical phylogenetics with a focus on phylogenetic networks. I will then illustrate some specific problems related to my own research interests, for instance measuring the tree-likeness of phylogenetic networks and moving through phylogenetic network space.

Connor Greene

Efficient Points for Polynomial Interpolation on the Unit Interval

Polynomial interpolation is a crucial topic in numerical methods. It is a central aspect of many spectral PDE solvers. Here, I present a simple framework for devising good points for polynomial interpolation both from a purely theoretical perspective and with some numerical optimization. These methods have the advantage of producing points which can be easily scaled to higher resolution and allow for forming and evaluating interpolants in O(n log n) time.

Kosuke Sugita

Scattering Matrix Computations for Integrated Photonics

We present an accurate and efficient numerical method for large-scale simulations of integrated photonic devices.

The governing equations for light propagating through these devices are the Helmholtz equation in two dimensions, and Maxwell’s equations in three dimensions. The problem is numerically challenging to simulate owing to the size of the object as measured in wavelengths of the propagating modes – the devices are typically of millimeter scale, while the wavelength of the propagating modes is on the micrometer scale.

This difficulty results in a large number of degrees of freedom required to discretize the problem which in turn leads to prohibitively large simulation times for each iteration of the design loop.

We present a divide-and-conquer approach to enable computer-aided design of photonic devices which relies on the devices consisting of simple function modules with straight waveguides albeit with different numbers of input and output channels. Since each component only supports a small number of propagating modes, we construct scattering matrices which completely characterize the input/output behavior of these components using state-of-art boundary integral equation methods coupled with high-order quadrature rules, and the recursively compressed inverse preconditioning method.

Rituparna Basak

Application of Computational Topology to Particulate Systems In Intermittent Flow Regime

We discuss the properties of force networks in a two-dimensional annular Couette geometry for both experiments and simulations in which a small intruder is pulled by a spring. In particular, the connection between topological features of the force network and fluctuations of the intruder velocity is studied in the stick-slip regime. The force networks are analyzed using persistent homology methods, focusing on the statistics of clusters and loops composed of particles experiencing strong forces. We find that the networks
evolve in a nontrivial manner as the system approaches a slip event. The presentation will discuss this evolution for systems of disks and pentagonal particles at several different packing fractions.

Philip Zaleski

Convergence Rates for Stochastic Gradient Descent Markov Operators

We formulate Stochastic gradient descent (SGD) as a Markov chain. We provide conditions for the existence and uniqueness of an Invariant measure, to which measures converge at a geometric rate with respect to the Wasserstein one metric. We also provide upper bounds on the geometric convergence rates. In the one-dimensional case, our work applies to C^2 functions with small step sizes giving rise to monotone maps. In the n-dimensional case, our work applies to a restricted class of functions having non-negative Hessian, which give rise to maps that preserve monotone paths. This work provides convergence proofs for SGD on specific non-convex functions in R^n. 

Joseph D'Addesa

Phase Separation of Two-Fluid Mixtures using Surface Acoustic Waves

In this talk we will present some theoretical and computational aspects of flow of thin fluid films. The theoretical part involves basic fluid mechanics and presents a brief derivation of the thin film equation including gravity and acoustics using the long wave approximation. A simplified version of this equation is then analyzed numerically using the software Comsol Multiphysics and compared to experimental results. The mechanism which makes such separation possible is based on the application of surface acoustic waves (SAWs), which propagate in the solid substrate in contact with oil-water-surfactant mixtures. Reaching the proposed goals will lead to establishment of proof-of-principle for use of SAW for the purpose of phase separation.

Nicholas Harty

Time-Domain Multiple Scattering Using Convolution Quadrature

Chaff is a fibrous substance with special electromagnetic properties used to prevent radar detection. Given a random distribution of chaff, one wants to know how an incident electromagnetic field, described by the wave equation in an initial boundary value problem, is scattered by the chaff cloud. In this talk, we will present two methods to compute the time-domain scattered fields – one using a Fourier transform (FT) formalism, and the other using convolution quadrature (CQ). We will start by presenting the multiple scattering formalism for the wave equation for various boundary conditions. We then describe and compare both the FT and CQ methods to compute the time-domain scattered field. Lastly, we present numerical experimental results comparing these two techniques. 

Michael Luo

Modeling SCN Neuron Dynamics

An organism's circadian cycle is regulated by a group of neurons in the brain known as the suprachiasmatic nucleus (SCN). Most of our knowledge of the SCN comes from nocturnal mice. In our study, we have an experimental data set from a day-active rodent species. The modeling process consists of finding a Hodgkin-Huxley type model that can accurately match the features of the experimental data. From here, it becomes a parameter estimation problem where we employ machine learning methods to determine optimal parameter distributions. 

Mark Fasano

Forced Phase Separation in a Closed Cell

In this presentation, I will discuss a model derived from first principles that describes a binary mixture undergoing spinodal decomposition due to a conservative volume force in a closed cell geometry. By assuming the mass of each component is conserved, general transport equations for the component concentrations can be found. Then a thermodynamic argument is used to find constitutive equations for the component velocities and associated diffusive fluxes; it is in this derivation where the boundary conditions necessary for the model arise. The system is closed by utilizing the Cahn-Hilliard free energy functional. Full 3D simulations are carried out and the results are presented. 

Zhiwen Wang

Intensity Estimation in Spatial Analysis

This talk focuses on the problem of intensity estimation in spatial analysis. We discuss traditional intensity estimation methods and introduce new approaches based on deep learning. The talk explores the strengths and limitations of both traditional and deep learning-based methods for intensity estimation. By comparing and discussing these approaches, we aim to provide insights into the advancements in intensity estimation techniques in spatial analysis.

Zheng Zhang

Consistent Estimation of the Number of Communities in Nonuniform Hypergraph Model

We propose an algorithm based on cross-validation to estimate the number of communities in a general non-uniform hypergraph model. The algorithm proceeds by (1) randomly dividing the set of hyperedges into a training set and a testing set, (2) for each candidate number of communities, constructing spectral estimation of community labels and least square estimation of the hyperedge probabilities based on training set, and (3) constructing cross validation scores involving testing set. The proposed algorithm is proven consistent when number of vertices tends to infinity. We examine the performance of our method through a simulation study and analysis of coauthorship data.

Justin Maruthanal

Oscillations of the Mitotic Spindle in C. Elegans During Asymmetric Cell Division

The one cell embryo of the C. elegans, a species of nematodes, undergoes asymmetric cell division. Astral Microtubules from the spindle poles attach to force generators on the cortex, which exert pulling forces on the spindle poles. Due to greater forces being applied to the posterior spindle pole, the spindle ends up moving towards the posterior half of the embryo. While the spindle moves towards the posterior half, oscillations of the spindle are observed. The purpose of this talk is to discuss 2 simplified models describing these oscillations caused by pulling forces coming from force generators on the cell cortex.

Atul Anurag

Generalization of Leapfrogging Orbits of Point Vortices

Point vortex motion arises in the study of concentrated vorticity in an ideal, incompressible fluid described by Euler’s equations. The two-dimensional Euler equations of fluid mechanics, a partial differential equation (PDE) system, support a solution where the vorticity is concentrated at a single point. Helmholtz derived a system of ordinary differential equations (ODEs) that describe the motion of a set of interacting vortices that behave as discrete particles, which approximates the fluid motion in the case that the vorticity is concentrated in very small regions. This system of equations has continued to provide interesting questions for over 150 years.

We will discuss the special class of relative periodic orbits known as the leapfrogging orbits. The relative periodic of a four-point vortex problem, two positive and two negative point vortices, all of the same absolute circulation arranged as co-axial vortex pairs, is known as the leapfrogging orbit.  This dissertation will present the generalizations to the leapfrogging motion of point vortices and vortex rings, including their stability and dynamics. More specifically,  we will study the leapfrogging motion of 2N vortices, with circulations half positive and half negative.

Nastaran Rezaei

Interfacial Instability of Two-Layer Flows of Newtonian Liquids

In this talk, I will present an overview of the interface instability in two-layer Poiseuille flows. Due to their industrial applications, multi-layer flows have been a classical topic for decades. The main issue that the industry tries to avoid is interfacial instability between layers. The wave evolution between layers is analyzed by using the VOF (volume of fluid) method. The results were compared to the available experimental data to ensure the method's efficiency.

Furthermore, this approach lets us predict the cases for which there is no information due to the limitation of the tools and methods for measuring growth rate and wave evolution at the interface. Changes in the governing equations cause the interface amplitude to decay or grow. Some parameters, including the viscosity ratio, density ratio, thickness ratio, and Reynolds numbers, are highly effective in amplitude growth rate. The wave behavior is analyzed by tracking the interface movement and evolution in different cases with the above-mentioned parameters. Viscosity plays a key role in Newtonian fluids, while in viscoelastic fluids, elasticity has a destabilizing effect as well as viscosity. This study points out the importance of the role played by viscosity and provides numerical results.

Patrick Grice

Detecting Unexploded Ordnances

Unexploded ordnances (UXO) are explosive weapons (bombs, bullets, shells, grenades, mines, etc.) that did not explode when they were employed and still pose a risk of detonation. Ocean disposal of munitions was also an accepted international practice until 1970, when it was prohibited by the Department of Defense. In 1972 Congress also passed the Marine Protection, Research, and Sanctuaries Act banning ocean disposal of munitions and other pollutants. This dataset represents known or possible former explosive dumping areas and UXOs. This is NOT a complete collection of unexploded ordnances on the seafloor, nor are the locations considered to be accurate. Two related datasets should be viewed in tandem: Unexploded Ordnance Locations displays known/possible individual or tightly grouped unexploded ordnances on the ocean floor and Formerly Used Defense Sites (FUDS) displays areas identified by the United States Army Corps of Engineers where unexploded ordnances may exist.

Andrew White

Effect of Calcium Buffer Properties on Cell Calcium Dynamics

Synaptic neurotransmitter release is caused by calcium ions entering the cell during an action potential and the increase in efficiency of neurotransmitter release is called synaptic facilitation.  Synaptic facilitation is caused by some form of calcium accumulation and thus depends on calcium buffers that regulate the concentration of calcium in the cell. Buffer saturation/depletion is one mechanism proposed to explain synaptic facilitation. Another mechanism that can cause synaptic facilitation is buffer dislocation. The goal of this study is to investigate the interplay between these two effects with the focus on complex buffers that have two calcium binding sites. 

Prianka Bose

Analyzing Son Clave Rhythmic Patterns

This talk will focus on the analysis of Son Clave, a prominent Afro-Cuban rhythmic pattern known for its cultural significance and captivating musical essence. Through a unique combination of musical insights and mathematical techniques, this investigation aims to uncover the intricate mathematical structure underlying Son Clave. We introduce a novel model that is a combination of biophysical mathematical Beat Generator model and a modified Morris Lecar Tapping Neuron model, which replicate the mesmerizing Son Clave rhythm.  Additionally, through rigorous statistical analysis and the integration of advanced machine learning techniques, we explore the temporal dynamics and rhythmic intricacies of Son Clave. 

Professor Timothy Faver

Title: TBA

Abstract: TBA