Differential Equations Seminar: undergraduate students
Adaku Uchendu, Teddy Weinberg, Dongli Deng
Monday, April 30, 2018 · 11:45 AM - 12:45 PM
Speaker: Adaku Uchendu
Title: Numerical Simulation of Vibrations of Mechanical Structures
Abstract: We develop an implementation of finite element method to simulate vibrations of mechanical structures. Specifically, we use a 2D frame model and corresponding stiffness, mass and damping matrices to set up a system of ordinary differential equations, which is solved in Matlab. We also consider uncertainties in the model parameters by taking the Young's modulus as a random variable. We use Monte Carlo simulation, and the effect of uncertainties is studied by numerical experiments.
Speaker: Teddy Weinberg
Title: Fast Implementation of Mixed Finite Elements and Applications to Flow in Porous Media Using MATLAB
Abstract: Understanding and modeling flow in porous media is important in many areas including managing groundwater reserves, maintaining CO2 storage facilities, and simulating petroleum reservoirs. This has created a growing need to efficiently describe flow in porous media. Models are typically described by partial differential equations (PDEs). We have developed an efficient implementation of the mixed finite element method for the lowest order Raviart-Thomas elements (RT0), which can be used to discretize problems related to flow in porous media. This implementation was created in MATLAB. As MATLAB is inefficient with iterative operations, the code had to be vectorized, replacing loops with array operations. In other words, instead of interacting with one element at a time the code interacts with all the elements simultaneously. The code supports two-dimensional and three-dimensional domains, and the finite elements can be triangles, quadrilaterals, tetrahedrons, or blocks. Based on numerical experiments, we have shown that our implementation is significantly more efficient than the standard approach.
Speaker: Dongli Deng
Title: Numerical Optimization and its application to Molecular Conformation
Abstract: We give a brief overview of classic algorithms for unconstrained optimization. The emphasis is put on algorithms using derivatives such as Steepest Descent, Conjugate gradients and Newton's method including a quasi-Newton version. The methods were implemented in Matlab, and we study their performance on several small testing problems. In the second step, we apply the methods to study molecular conformation given by minimization of Lennard-Jones potential.
Acknowledgement: The research was supported in part by the National Science Foundation (DMS-1521563), and Adaku and Teddy were supported by URA.