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Department of Mathematics

Upcoming Events

Branwen Purdy at her stall during OMSI meet-a-scientist day.

Branwen Purdy prepares hands-on activities for kids at the OMSI Meet-A-Scientist Day in Portland, to share hands-on learning experiences about her research in topological data analysis.

Join us for these events hosted by the Department of Mathematics, including colloquia, seminars, graduate student defenses and outreach, or of interest to Mathematicians hosted by other groups on campus.

Access our archive of events

Exploring Granular Materials: Role of Particle Shape and Modelling with Superellipsoids

STAG 111
Applied Mathematics and Computation Seminar

Speaker: Yuxuan Wen

ABSTRACT:Granular materials, including sands, clays, and mineral rocks, are ubiquitous in nature and critical to research in geotechnical engineering, granular physics, mining, and mechanical engineering. These materials exhibit diverse states: solid (embankments), liquid (landslides), and gas-like (dust storms) state. In civil engineering, the focus often lies on their behavior in the solid state. The particle morphology, such as shape and grain size distribution, plays a critical role in influencing both micro- and macro-scale behavior. Various shape descriptors, such as sphericity, roundness, and aspect ratio, have been proposed to correlate shape parameters with mechanical properties. This study investigates the influence of particle shape on the mechanical behavior of granular soils using discrete element method (DEM) simulations. By employing superellipsoids in DEM, we systematically assess superellipsoids’ capability to control particle shape parameters and examine the… Read more.

Finite element methods for the Landau-de Gennes minimization problem of nematic liquid crystals

Colloquium

Speaker: Ruma Maity

Nematic liquid crystals represent a transitional state of matter between liquid and crystalline phases that combine the fluidity of liquids with the ordered structure of crystalline solids. These materials are widely utilized in various practical applications, such as display devices, sensors, thermometers, nanoparticle organizations, proteins, and cell membranes. In this talk, we discuss finite element approximation of the nonlinear elliptic partial differential equations associated with the Landau-de Gennes model for nematic liquid crystals. We establish the existence and local uniqueness of the discrete solutions, a priori error estimates, and a posteriori error estimates that steer the adaptive refinement process. Additionally, we explore Ball and Majumdar's modifications of the Landau-de Gennes Q- tensor model that enforces the physically realistic values of the Q tensor eigenvalues. We discuss some numerical experiments that corroborate the theoretical estimates, and adaptive… Read more.

Numerical analysis of coupled free-flow and porous media flow problems

STAG 111
Applied Mathematics and Computation Seminar

Speaker: Aycil Cesmelioglu

ABSTRACT: In this talk, I will focus on the numerical analysis of several coupled problems involving both free-flow and porous media flow. These problems are commonly encountered in applications such as groundwater contamination, where contaminants spread through both surface and subsurface flows. I will present numerical discretizations based on strongly conservative hybridizable discontinuous Galerkin (HDG) methods, emphasizing well-posedness and error estimates. Finally, I will present results from various computational experiments. BIO:Dr. Cesmelioglu is a Professor in the Department of Mathematics and Statistics at Oakland University, where she has been a faculty member since 2012. She holds a Ph.D. in Computational and Applied Mathematics from Rice University (2010) and held a postdoctoral position at the Institute for Mathematics and its Applications (2010-2012). Her research focuses on numerical methods for partial differential equations with specific interest in fluid flow… Read more.

Sink or Soar: the interplay between buoyant bubbles and sinking sediments inenergizing turbulence near the ice-ocean boundary

STAG 111
Applied Mathematics and Computation Seminar

Speaker: Megan Wengrove

ABSTRACT: At the terminus of tidewater glaciers an interplay of connected processes result in the melt of ice. From both field and laboratory observations, it has been suggested that both bubbles and sediments could be important yet neglected contributors to ice melt at the submarine tidewater glacier terminus. In the laboratory it has been shown that as glacier ice melts, air trapped in pores inside of the ice is released creating flow transpiration at the boundary and buoyant bubble rise at the ice-ocean interface, leading to increased melt (Wengrove et al.,2023). Additionally, during separate laboratory experiments, sediments entrained in the subglacial discharge plume are shown to increase the entrainment of warm ocean water toward the ice leading to higher melt rates (McConnochie andCenedese, 2023). In July 2024, we made the first ever video observations of both bubbles rising and sediments mixing and falling from a stationary-bolted platform to an Alaskan tidewater glacier… Read more.

The role of boundary constraints in simulating biological systems with nonlocal dispersal

STAG 111
Applied Mathematics and Computation Seminar

Speaker: Gabriela Jaramillo

ABSTRACT:Population and vegetation models often use nonlocal forms of dispersal to describe the spread of individuals and plants. When these long-range effects are modeled by spatially extended convolution kernels, the mathematical analysis of solutions can be simplified by posing the relevant equations on unbounded domains. However, in order to numerically validate these results, these same equations then need to be restricted to bounded sets. Thus, it becomes important to understand what effects, if any, do the different boundary constraints have on the solution. To address this question we present a quadrature method valid for convolution kernels with finite second moments. This scheme is designed to approximate at the same time the convolution operator together with the prescribed nonlocal boundary constraints, which can be Dirichlet, Neumann, or what we refer to as free boundary constraints. We then apply this scheme to study pulse solutions of an abstract 1-d nonlocal Gray-Scott… Read more.

The effect of Anderson acceleration on the convergence order of superlinear and sublinear nonlinear solvers

TBA
Applied Mathematics and Computation Seminar

Speaker: Leo Rebholz

We consider the effect of Anderson acceleration (AA) on the convergence order of nonlinear solvers in fixed point form $x_{k+1}=g(x_k)$, that are looking for a fixed point of g. While recent work has answered the fundamental question of how AA affects the convergence rate of linearly converging fixed point iterations (at a single step), no analytical results exist (until now) for how AA affects the convergence order of solvers that do not converge linearly. We first consider AA applied to general methods with convergence order r, and show that AA changes the convergence order to (at least) (r+1)/2 for depth m=1; a more complicated expression for the order is found for the case of larger m. This result is valid for superlinearly converging methods and also locally for sublinearly converging methods where r<1 locally but r$\rightarrow$1 as the iteration converges, revealing that AA asymptotically slows convergence for superlinearly converging methods but (locally) accelerates it for… Read more.

Accelerating solvers for fluids with (continuous) data assimilation

TBA
Colloquium

Speaker: Leo Rebholz

We consider nonlinear solvers for the incompressible, steady Navier-Stokes equations in the setting where partial solution data is available, e.g. from physical measurements or sampled solution data from a (too big to send) very high-resolution computation. The measurement data is incorporated/assimilated into the solver through a nudging term addition that penalizes at each iteration the difference between the coarse mesh interpolants of the true solution and solver solution, analogous to how continuous data assimilation (CDA) is implemented at each time step for time dependent dissipative PDEs. For a Picard solver, we quantify the acceleration provided by the data in terms of the density of the measurement locations and the level of noise in the data. For Newton, we show how the convergence basin for the initial condition is expanded as more data is assimilated.Numerical tests illustrate the results. While the setting is for Navier-Stokes, the ideas are applicable to solvers for a… Read more.