With the assistance of the SIAM Student Chapter Advisors: Michael Benzi, Chris Cox, Suzanne Lenhart, A.J. Meier, Shari Moskow, James Nagy, Paul Schmidt, Michael Shearer and Ralph Smith.
2:15-2:40 Beyza C. Aslan, Department of Mathematics, University of Florida
A continuous approach to the lightning discharge
In earlier work Dr. William Hager developed the following approach for modeling a lightning discharge: The continuous partial differential equation describing the electric potential was discretized, and integrated forward in time. Whenever the electric field reached the breakdown threshhold in some region of the atmosphere, the associated conductivity parameter in the discrete equation was taken to infinity. An explicit formula for the limiting potential was obtained. In this talk, I will introduce a continuous version of this model. I will first focus on a one-dimensional analogue of the continuous three-dimensional equation, and obtain an explicit formula for the limit of the potential as the conductivity in a subinterval of the domain tends to infinity. Then, as the time permits, I will focus on the three-dimensional eqauation in spherical coordinates. In this case, assuming complete symmetry, I obtain an explicit formula for the potential as the conductivity in a small neihborhood of the origin tends to infinity.
2:45-3:10 Xiaoming Chen, Department of Chemical Engineering, University of South Carolina, with Thanasis D. Papathanasiou
A Statistical Geometrical Correlation for the Transverse Permeability of Disordered Fiber Arrays
In the porous media literature, unidirectional fibrous systems are broadly categorized as ordered or disordered. The former class, easily tractable for analysis purposes but severely limited in its relation to reality, involves square, hexagonal and various staggered arrays. The latter class involves everything else. While the hydraulic permeability (K) of ordered media is known to be a deterministic function of their porosity, the parameters affecting the permeability of random fiber arrays are poorly understood. The objective of this study is to computationally investigate flow across many unidirectional arrays of randomly placed fibers and, in the process, explain the wide scatter in experimentally observed permeability values.
This task is made possible by a parallel implementation of the Boundary Element Method (BEM). A large number of simulations were carried out in model systems generated by a Monte Carlo (MC) procedure, in which porosity (φ = 0.45, 0.5, 0.6 and 0.7) and the minimum inter-fiber spacing (δmin = 0.1 ~ 1.0 R) are variable. The results indicate a drop in hydraulic permeability as the degree of local aggregation increases. Analysis of the permeability data resulted in an empirical relationship between permeability and the mean closest spacing of fibers (δ1), valid in the range of porosity and δ1 examined.
3:15-3:40 Sarah Grove, Department of Mathematics, North Carolina State University
HIV Modeling and Inference
During this talk we will briefly explain Monte Carlo Markov Chains (MCMC). Also we will consider Gibb's method which uses Metropolis-Hastings to create these chains. We will examine how these techniques are used to better understand the distribution of parameters across our population. We will be using the Bayesian approach to update our model on both the individual as well as the population level. Specifically the HIV model will use viral load, uninfected CD4 counts, and immune indicator data to characterize an individuals dynamics.
3:45-4:10 Volodymyr Hrynkiv, University of Tennessee at Knoxville and Oak Ridge National Laboratory, with Suzanne Lenhart and Vladimir Protopopescu
Optimal Control of Convective Boundary Condition in a Thermistor Problem
Optimal control of a thermistor problem is considered. The heat transfer coefficient is taken as the control. The state system has two nonlinear elliptic partial differential equations. We establish the existence of an optimal control that minimizes the objective functional which involves keeping the temperature and the heat transfer coefficient low (which gives low temperature variation). An optimality system is derived.
8:45-9:10 Julianne Chung, Emory University
SVD Based Filtering Methods for Image Restoration
Image restoration is the process of removing blur and noise from degraded images to recover an approximation of the original image. The mathematical model is an ill-posed inverse problem, and thus regularization must be used. In this talk we consider SVD based regularization (filtering) methods, and various schemes for choosing the regularization parameter. Matrix structure is exploited to reduce the computational complexity of the problem. Numerical experiments compare effectiveness and efficiency, and also reveal emerging relationships among the methods.
9:15-9:40 Jason Howell, Department of Mathematical Sciences, Clemson University
Applying a defect correction method to viscoelastic fluid flow
The numerical simulation of viscoelastic fluid flow becomes more difficult as a physical parameter, the Weissenberg number, increases. Specifically, at Weissenberg numbers larger than a critical value, the iterative nonlinear solver fails to converge. In this work we apply a defect correction method to the nonlinear Johnson-Segalman model for steady viscoelastic fluid flow and simulate flow through a four-to-one contraction channel. In the defect step, we artificially reduce the Weissenberg number to solve a stable nonlinear problem, then we determine the residual correction by solving a linearized version of the problem.
9:45-10:10 Kang Jin, Department of Mathematics, Auburn University
The Lattice Gas Model and Lattice BGK Model on a 2-D Hexagonal Grid
We present an overview of the HPP model and the Lattice BGK model. Details regarding to boundary condition and initial condition is discussed through implementations on driven cavity flow, Couette flow, and flow pass a plate. We give the results and analysis in different Reynolds numbers.