Haroon Stephen Sahotra

Numerical Methods in Mechanics
In this course, students will learn about numerical analysis of engineering problems through class lessons, labs exercises, research articles, and term projects. The course is divided into three parts. In the first part, students will be introduced/refreshed about basic numerical methods by finding solutions of linear and non-linear equations in single and multiple variables. The solution error and its stability/convergence in iterative methods will be explained. In the second part, students will learn to perform numerical integration and differentiation and to find numerical solutions of ordinary differential equations (ODE). Initial and boundary value problems will be solved. In the third part, students will learn to find numerical solutions of partial differential equations (PDE). Throughout the course, Matlab will be used to conduct hands-on exercises. All lessons and labs cover numerical analysis with examples from civil engineering (water, environment, structures, transportation, and geotech) such as sediment transport, surface flooding, groundwater flow, traffic network, pollute dispersion, and shock wave propagation.

Lab01: Introduction

• Matlab Desktop Environment
• Command Line
• Matlab Variables
• Scripting
• Matlab Functions
• Example: Functions
• Function to Calculate Free Fall Velocity
• Function to Calculate Actual Velocity
• Script to Simulate
• Error Analysis
• Visualization

Lab02: Root Finding

• Matlab Scalars and Arrays
• Graphical Method
• Bisection Method
• False Position Method
• Newton-Raphson Method
• Secant Method
• Brent Method (Matlab fzero)
• Matlab Printing Graphics

Lab03: Solution of System of Linear Equations

• Solving Ax=b
• Condition Number
• Great Lakes Problem
• Special Matrices
• Ax=b with Many Unknowns

Lab04: Optimization

• Code and plot math functions
• Unconstrained Optimization 1D
• Unconstrained Optimization 2D
• Constrained Optimization 2D
• Application Problem

Lab05: Curve Fitting

• Line Fitting
• Quadratic Fitting
• Polynomial Fitting
• General Nonlinear Fitting
• Problem 20.28
• Problem 20.58

Lab06: Integration and Differentiation

• Rectangular Rule
• Trapezoidal Rule
• Simpson’s 1/3 Rule
• 1-Point Gauss Quadrature
• 2-Point Gauss Quadrature
• First Order Difference
• Second Order Difference
• Integration Example

Lab07: Ordinary Differential Equations (1st Order)

• Euler’s Method – Free falling object
• Euler’s Method – Exponential Curve
• ode23 Function

Lab08: Ordinary Differential Equations (2nd Order)

• Euler’s Method – Free falling object
• Free falling object 2D
• Free falling object with Drag
• ode45: Predator Prey Model
• Implicit Method: Heat Transfer
• Shooting Method: Heat Transfer

Lab09: Partial Differential Equations (Laplace Equation)

• Scalar Field
• Vector Field
• Laplace Equation 1
• Laplace Equation 2

Lab10: Partial Differential Equations (Diffusion Equation)

• Diffusion Equation 1D No Source
  
• Diffusion Equation 1D with Source
• Diffusion Equation 2D with Source
• Gradient of Scalar Field
• Diffusion Equation 3D with Source (Part 1)
• Diffusion Equation 3D with Source (Part 2)

Lab11: Partial Differential Equations (Burger’s Equation)

• Advection Equation 1D
• Burger’s Equation 1D
• Burger’s Equation 2D (Part 1)
• Burger’s Equation 2D (Part 2)

Lab12: Partial Differential Equations (Wave Equation)

• Wave Equation 1D
• Wave Equation 2D
• Creating Movie File