Module overview
This module provides an in depth coverage of key numerical methods to solve practical mathematical problems that occur throughout engineering. The module consists of two main parts:
1. The course demonstrates the use of numerical analysis as a powerful problem solving tool in engineering. The course encompasses Numerical Analysis, Regression Analysis, Taylor Series and Applications, Numerical Integration and Solutions to Ordinary Differential Equations, with applications to engineering problems through computational simulations using a commercial package, MATLAB.
2. The course also provides an introduction to the theory underlying the finite element (FE) method, with applications through computational simulations for a range of engineering problems using a commercially available general-purpose finite element package, ABAQUS.
Computer lab exercises and assignments will give opportunity for students to analyse and solve a number of practical engineering problems.
Linked modules
Pre-requisite: MATH1054
Aims and Objectives
Learning Outcomes
Subject Specific Practical Skills
Having successfully completed this module you will be able to:
- Write, compile, execute and test programs using MATLAB to solve a range of problems numerically.
- Utilise user manuals and online help pages/tutorials to learn and gain familiarity with commercial software platforms.
- Use a commercial FE package to solve practical 1D and 2D engineering problems (including understanding of the pre-processing, solution and post-processing phases of the process).
- Use MATLAB built-in functions in a range of contexts relating to problem solving in numerical analysis.
Transferable and Generic Skills
Having successfully completed this module you will be able to:
- Time management and independent learning.
- Critical analysis and judgment as to the quality of computer analysis output.
- Logical thinking and conceptualisation of a problem for solution by computer algorithm.
- Learning the use of commercial FE software.
- Ability to use the MATLAB programming platform including built-in functions.
- Presentation of data and analysis results.
Knowledge and Understanding
Having successfully completed this module, you will be able to demonstrate knowledge and understanding of:
- The significance, calculation and interpretation of numerical errors and methods to eliminate or mitigate their effects.
- The key steps required to complete a FE simulation.
- Numerical analysis techniques available to solve a range of mathematical problems encountered in engineering (root finding, regression analysis, Taylor series, differentiation and integration, solution of ODEs)
- The limitations of the FE method and appropriate choice of element type to suit the problem.
- The theory of the formulation and solution of finite element models.
- The fundamental concepts of the FE method.
Subject Specific Intellectual and Research Skills
Having successfully completed this module you will be able to:
- Select the appropriate numerical solution technique to solve the problem.
- Develop and solve FE simulations.
- Recognise engineering problems that may be solved numerically.
- Apply concepts of numerical analyses and FE models for analysis and solving of real-life engineering problems.
- Demonstrate the accuracy of typical numerical and FE results.
Syllabus
Numerical analysis
1. Introduction to numerical analysis and the use of MATLAB as a numerical tool (1L)
2. Root finding (2L)
- Direct and iterative methods
- Newton–Raphson method
- Round-off and Truncation errors
- MATLAB in-built functions
3. Regression analysis (2L)
Polynomials, Least squares analysis, Chebyshev polynomials, MATLAB in-built functions
4. Taylor series and applications (1L)
5. Numerical differentiation and integration (2L)
- Motivation and objectives
- Numerical differentiation (first derivative, second derivative)
- Numerical integration (Trapezoidal rule, Mid-point rule)
- MATLAB in-built functions
6. Solution to Ordinary differential equations (ODE) (2L)
- Introduction to ODE and their occurrence in engineering problems
- Finite difference methods
- Euler Method, Runge–Kutta methods
- Applications using MATLAB
MATLAB computer lab (Weeks 3-6)
Weeks 3, 4 & 5 – Programming using MATLAB
Week 6 – Assignment
Finite element analysis ( lectures /tutorials/ computer labs)
1. Introduction (1L)
2. Finite Element Method (1L)
- Problem clarification, modelling and discretisation
- Elements, nodes and degrees of freedom, meshes
3. Getting started with ABAQUS/Standard (1L)
4. Elastic rods and beams (5L)
- Truss elements
- Beam elements
- Element stiffness
- Assembly of elements
- Global stiffness matrix and its characteristics
- Direct formulation (stiffness method)
- Solution of structure equations
5. Concept generalization to two dimensions (6L)
- Preliminaries (Stress–strain relationship, Stress–displacement relationship, Compatibility, Equilibrium equations, Boundary conditions, Exact and approximate solutions)
- Interpolation and shape functions
- Formulation of element matrices (Linear (constant strain) triangle, Quadratic rectangle)
- Isoparametric element formulations (Bilinear quadrilateral, Quadratic quadrilateral)
- Limitations associated with different types of element
ABAQUS computer labs and tutorials (Weeks 7-10)
- Week 7, 8, 9 – Problem solving using ABAQUS
- Week 10 – Assignment
Worked examples (Computer Labs):
1. 2D truss
2. 2D plane stress analysis
Learning and Teaching
Teaching and learning methods
- Lectures and tutorials
- Computer lab supervisions and Computer lab classes
- Tutorials and worked examples
- Problem assignments
- Private study
PowerPoint slides available from Blackboard
Type | Hours |
---|---|
Follow-up work | 24 |
Completion of assessment task | 26 |
Revision | 24 |
Tutorial | 4 |
Wider reading or practice | 20 |
Preparation for scheduled sessions | 12 |
Supervised time in studio/workshop | 16 |
Lecture | 24 |
Total study time | 150 |
Resources & Reading list
General Resources
ABAQUS. User manual
Internet Resources
Mathworks references (Mathworks is the official owner of Matlab).
Assessment
Summative
This is how we’ll formally assess what you have learned in this module.
Method | Percentage contribution |
---|---|
Continuous Assessment | 100% |
Referral
This is how we’ll assess you if you don’t meet the criteria to pass this module.
Method | Percentage contribution |
---|---|
Set Task | 100% |
Repeat
An internal repeat is where you take all of your modules again, including any you passed. An external repeat is where you only re-take the modules you failed.
Method | Percentage contribution |
---|---|
Set Task | 100% |
Repeat Information
Repeat type: Internal & External