ESM 696 - Computational and Numerical Methods in Materials Science

Course Description:

This course is intended as a graduate level introduction to the use of computational and numerical methods to solve materials related problems. The course will cover topics such as data analysis, curve fitting, numerical solution to differential equations, calculations of phase diagrams for binary alloys, band gap calculations and introduce Monte Carlo and Molecular Dynamics methods.

The course will consist of theory lectures and computer lab tutorials. Assignments will be given regularly through the semester, culminating in a final project due at the end of the semester. All students are required to have a basic knowledge of a computer programming language, either FORTAN, C or PASCAL.

Brief Syllabus

Introduction

• Importance of Computers in Materials Science
• The Nature of Computer Simulation
• Programming Languages -- Review

Data Analysis

• Curve Fitting Techniques
• Accuracy and Stability
• Simple Linear Least Squares Fit
• Solution of set of linear equations

Finding the root of an equation

• Bracketing and Bisection
• Newton Rhapson method using the derivative

Introduction to Random Walks

• Problems in Probability
• Random Number Sequences
• Random Walks and the Diffusion Equation
• Self-Avoiding Walks and the Polymer problem

The Percolation Problem

• Introduction
• The Percolation Threshold
• Cluster Labeling
• Critical Exponents and Finite Size Scaling

Fractals

• Fractal Dimension
• Regular Fractals
• Fractal Growth Processes
• Fractals and Chaos

Numerical Integration and Monte Carlo Methods

• Calculation of

• Numerical Integration Methods in One Dimension
• Simple Monte Carlo Evaluation of Integrals
• Numerical Integration of MultiDimensional Integrals
• Monte Carlo Error Analysis
• Importance Sampling
• Metropolis Monte Carlo Method

Monte Carlo Simulation of the Canonical Ensemble

• The Canonical Ensemble
• The Metropolis Algorithm
• Verification of the Boltzmann Distribution
• The Ising Model
• The Ising Phase Transition
• Other Applications of the Ising Model
• Simulation of Classical Fluids
• Other Ensembles
• Other Applications

Molecular Dynamics Methods

• Introduction
• NVT, NPT Ensembles
• Simulation of classical fluids
• Force fields and intermolecular potentials