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

  1. 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.

  2. 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.

  3. 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.

  4. 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.

  5. 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.

  6. 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.


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