Numerical, Analytical and Approximate Approaches
Dr Ben Thornber (Room N312, firstname.lastname@example.org
This course aims to give you:
- Knowledge of the mathem
Foundations of Stress Analysis
This chapter aims to give you the capability to
- Understand how to formulate an Finite Element Analysis Problem
- How to implement the weak form of the governing equations for a one
Solution of the Heat Equation
How do we get to the heat equation?
How can I solve it analytically?
- Diffusion problems
- Transient heat transfer
- Concentration in fluids
- Drug absorption
- Electric potentials
What am I?
What do I represent?
A few useful translations from Maths Maths to Engineering Maths
- For a scalar property u, gradients and Laplacian:
2 2 2
= = 2 +
The Laplace and Poisson Equations
This chapter aims to give you:
- Intro to Laplace & Poisson Equation
- Where they apply
- How to solve them
Pierre-Simon Marquis de Laplace
- Formulated the Laplace
The Wave Equation
What processes are represented by the wave equation?
How do derive the wave equation?
How can I solve it analytically? Applications in:
Surface waves, compression waves, vibrations, shear waves,
Fundamentals of Numerical
Analysis for PDEs
We can solve the heat equation analytically for simple cases
What about complex 3D geometries, multiple heat sources,
multiple materials, heat loss due to radiation and convection?
Numerical Solution of the Wave Equation
As with the heat equation we first discretise the equation:
- It must be accurate enough for your needs
- It must be consistent
- It must be stable
Finally we will solve this algorithm o
Weeks 8: Laplace and Fourier Transforms
1. These problems can be easily solved using the properties of the Laplace transform (i.e.
linearity) and the table solutions to generalised forms. It is highly recommended you know
how to o
12.1 What have we learnt?
Methods of analysing engineering systems:
- Qualitative classification (diffusive, wave like, steady diffusion)
- Diffusion of heat in 1D (unsteady) and 2D (steady)
- Diffusion of species in 1D (unstead
The Finite Element Method
8.1 Examples: Saab Crash
8.1 Examples: F16 Standard Dynamic Model
8.1 Examples: Hypersonic A2
8.1 Examples: NASA Research Aircraft
8.1 Examples: Generic Large Aircraft
8.1 Examples: Sensorcraft Structure
Introduction to PDEs and the Heat Equation
What is a PDE?
Why do we use PDEs?
How do we classify them?
Complementary reading: Kreyszig p582 onwards.
2.2 What is a PDE?
You are familiar with ODEs e.g.
Fourier Integrals and Laplace Transforms
Many analytical problems can be solved more easily using Fourier or
Very useful for problems on infinite or semi-infinite domains, e.g.:
We will cover
- Fourier Inte
Week 1 A Primer in MATLAB
This course will require students to make extensive use of MATLAB programming in order to complete
course material. Our first tutorial will give you a fly by tour of the programming environment in
order to get you started. When y
Week 7: Fourier Integrals and Transforms
Part 1- Diffusion of a Toxic Release
Yt = c2 Yxx
Y (x, 0) =1 for 1 < x < 1
Y (x, 0) =0 for |x| 1
The analytical solution for this problem may be found by way of a Fourie
Weeks 1 & 2: Interpolations, Taylor Series and PDEs
The main aim of this tutorial is to familiarise yourselves with Matlab by implementing some
numerical methods. I will assume that you are familiar with the basics of Matlab.
Week 3: Heat/Diffusion Equation
This tutorial is split into two tasks. The first section is the investigation of an industrial safety
problem. The second section has a series of practice questions solving the heat equation.
Task 1: Sp
Week 13: Ice Road Safety
Figure 1: An ice road in Canada and the consequences of hot weather
In some parts of the world the only way into and out of a region is by driving over a frozen lake.
The image above shows an ice road in Canada along which trucks
Week 7: Fourier Integrals and Transforms
This week we are going to explore the solutions to two typical engineering problems. The first is
the diffusion of a toxic gas release. The second is to calculate the spectrum of sound levels in a
Week 10: Extending Pollutant Tracking Capability
In Week 4 you wrote a numerical scheme to solve for the release of nitrogen from a sealed tank.
The company you work for has decided to extend their capabilities - they now want to be able to
offer a prod
Week 12: Multiphysics!
Figure 1: Types of physical effects which can be coupled in commercial solvers [LEAP Australia
The big strength in numerical methods is the ability to start with a validated simple solver, and then
combine capabilities to
Week 4: Finite Difference Methods
In last weeks tutorial you solved the analytical solution to the Diffusion equation for the problem
of species diffusion along a corridor. In this weeks tutorial you will solve the same problem
numerically using the For