Math 621 - Numerical Methods for Partial Differential Equations
Spring 2013 - Matthias K. Gobbert
Presentations of the Class Projects
Friday, May 17, 2013, 01:00 p.m., MP 104
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01:05-01:20
A Two-Dimensional Model for Calcium Flow in a Heart Cell
Using COMSOL
Xuan Huang, Department of Mathematics and Statistics
A model for the flow of calcium on the scale of one heart cell is given by a
time-dependent reaction-diffusion equation. Calcium ions enter into the cell
at release units distributed throughout the cell and then diffuse.
At each release unit, the probability for calcium to be released
increases along with the concentration of calcium.
Calcium release units are modeled as point sources using COMSOL.
With different initial condition, induced and spontaneous flows are captured.
This work is collaborative with my advisor Dr. Matthias K. Gobbert.
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01:20-01:35
Efficiency Improvements in Numerical Methods for Studying a Model
of a Pancreatic Islet with Automatic Differentiation Using ADiMat
Samuel Khuvis, Department of Mathematics and Statistics
The pancreas contains clusters of beta-cells called islets of Langerhans.
The dynamics of beta-cells are modeled with a seven variable and a
three variable model which consist of coupled ordinary differential equations.
A computational islet, a three-dimensional cube of beta-cells,
is used to simulate an islet.
We use Matlab's ode15s in this simulation and demonstrate several orders
of magnitude improvement in runtimes for both models by providing a
Jacobian generated by automatic differentiation with the ADiMat software tool.
Through the use of ADiMat we demonstrate a method
to improve the efficiency of a simulation without the
need to manually derive and code a potentially complex function
to generate the Jacobian of a system of ODEs.
We also compare the performance of this code with code which provides a
manually coded Jacobian to Matlab's ode15s.
The work is collaborative with advisor Dr. Matthias Gobbert
and Dr. Bradford Peercy.
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01:35-01:50
Solving the Diffusion Term in "Stable Fluids" Using GPUs
Yu Wang, Department of Computer Science and Electrical Engineering
In computer graphics, we are interested in recreating natural phenomena
such as cloud, smoke, fire and water. By modeling the mechanics and
allowing user interaction with a virtual fluid in real-time, orchestrating
the effects becomes easier for artists. In order to model the mechanics to
suit the needs in computer graphics, a fluid solver should be fast, produce
visually convincing effects, and easy to code. Historically fluids in
computer graphics involves simple primitives and the clever usage of
texture maps, but to handle the dynamics accurately, using physical
equations such as the Navier-Stokes equations is a better choice. One of
the terms in the Navier-Stokes equations is the diffusion term, and it
requires a stable Poisson solver. Gauss-Seidel relaxation is used in this
project, also accelerated using the Graphics Processing Units (GPUs), which
proves to be faster than the CPU solution by up to a magnitude of 2. This
work is collaborative with advisor Dr. Marc Olano.
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01:50-02:05
Kinetic Solidification of Silicon Film in the Ribbon Growth
on Substrate System
Samin Askarian, Department of Mechanical Engineering
Ribbon growth on substrate (RGS) has emerged as an attractive method for
growing silicon films by contact solidification. Its great potential to reduce
the cost of silicon wafers for photovoltaic applications is one of its major
advantages. Thermal conditions play an important role in determining the
thickness and quality of the as-grown films. In this study, we developed a
mathematical model for heat transfer and kinetic solidification in the RGS
process, and conducted computational study of silicon film growth during
kinetic solidification to predict solidified layer thickness and the duration
of this process. The effects of important operational parameters, such as
pulling speed, preheating temperature and thermal properties of the substrate
material, have been examined. The results show that the solidification rate is
not significant during kinetic solidification. The rate of solidification and
film thickness are very sensitive to both the thermal conductivity and preheat
temperature of the substrate. The numerical model and the theoretical solution
provide an important tool for thermal design and optimization of the RGS
system. This work is collaborative with advisor Dr. Ronghui Ma.
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02:05-02:20
Nanoparticle Infusion of a 2D Spherical Heterogeneous Tumor during
Magnetic Fluid Hyperthermia: A Computational Study
Timothy Munuhe, Department of Mechanical Engineering
Nanoparticle infusion of a cancerous tumor is modeled using
computational software with the goal of identifying reliable tools
for the computational modeling of the Magnetic Fluid Hyperthermia
procedure. We first develop a theoretical framework for the
nanoparticle transport during the infusion process and analyze the
equations produced to infer certain properties of the system. The
infusion process is then modeled in COMSOL 4.2 and MATLAB for a
homogeneous tumor. Results indicate that while COMSOL is faster it
produces less realistic results when modeling convection-dominated
transport. MATLAB's higher level of control necessitates a strong
understanding of finite difference and finite volume methods and
tight control of basic parameters, such as grid size. This work was
conducted under the partial supervision of Dr. Ronghui Ma.
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02:20-02:35
Comparison of MATLAB's PDE Toolbox and COMSOL
Multiphysics Applied to Sample Problems
Bryce Carey, Department of Mathematics and Statistics
The PDE Toolbox for use with MATLAB is an utility for solving elliptic,
parabolic, and hyperbolic partial differential equations on two-dimensional
domains using the finite element method with linear basis functions.
The software COMSOL Multiphysics can also perform these tasks,
but with a lot more flexibility.
We examine a sample problem and conduct separate convergence
studies using the PDE Toolbox and COMSOL.
The convergence study demonstrates that both utilities obtain comparable
results, but COMSOL achieves a slightly better convergence order.
Further, we consider a heat transfer problem
involving various materials in a three-dimensional setting. A
two-dimensional cross-section of this problem is implemented
and numerically solved in both PDE Toolbox and COMSOL, and the results
from both applications are in agreement. However, the restriction of
PDE Toolbox to two-dimensional geometries leaves us uniformed
about potential issues in the three-dimensional setting.
Copyright © 2001-2013 by Matthias K. Gobbert. All Rights Reserved.
This page version 1.0, May 2013.