Numerical Methods

Learning Outcomes

Knowledge and Understanding

Having successfully completed this module, you will be able to demonstrate knowledge and understanding of:

• The fundamental concepts of the FE method.
• The theory of the formulation and solution of finite element models.
• 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 significance, calculation and interpretation of numerical errors and methods to eliminate or mitigate their effects.

Transferable and Generic Skills

Having successfully completed this module you will be able to:

• Critical analysis and judgment as to the quality of computer analysis output.
• Logical thinking and conceptualisation of a problem for solution by computer algorithm.
• Time management and independent learning.
• Presentation of data and analysis results.
• Ability to use the MATLAB programming platform including built-in functions.
• Learning the use of commercial FE software.

Subject Specific Intellectual and Research Skills

Having successfully completed this module you will be able to:

• 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.
• Select the appropriate numerical solution technique to solve the problem.
• Develop and solve FE simulations.

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 MATLAB built-in functions in a range of contexts relating to problem solving in numerical analysis.
• 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).

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

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

Resources & Reading list

General Resources

ABAQUS. User manual

Internet Resources

Mathworks references (Mathworks is the official owner of Matlab).

Summative

This is how we’ll formally assess what you have learned in this module.

Referral

This is how we’ll assess you if you don’t meet the criteria to pass this module.

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.

Repeat Information

Repeat type: Internal & External