The University of Southampton
Courses

# CENV2026 Numerical Methods

## 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.

### Aims and Objectives

#### Module Aims

• To give students a thorough understanding of key numerical analysis methods and their application to solve common engineering problems. • To provide the students with skills to understand the significance, calculation and interpretation of numerical errors and convergence criteria associated with numerical analysis. • To train students to write, implement, test and evaluate computer programs (using the MATLAB platform) to solve a range of problems numerically. • To provide students with a knowledge and understanding of the fundamental theoretical basis of the FE method and its application in analysis of 1D and 2D engineering problems.

#### Learning Outcomes

##### Knowledge and Understanding

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

• 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 significance, calculation and interpretation of numerical errors and methods to eliminate or mitigate their effects.
• 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.
• The limitations of the FE method and appropriate choice of element type to suit the problem.
##### Subject Specific Intellectual and Research Skills

Having successfully completed this module you will be able to:

• Recognise engineering problems that may be solved numerically.
• Select the appropriate numerical solution technique to solve the problem.
• Apply concepts of numerical analyses and FE models for analysis and solving of real-life engineering problems.
• Develop and solve FE simulations.
• Demonstrate the accuracy of typical numerical and FE results.
##### Transferable and Generic Skills

Having successfully completed this module you will be able to:

• Logical thinking and conceptualisation of a problem for solution by computer algorithm.
• Ability to use the MATLAB programming platform including built-in functions.
• Presentation of data and analysis results.
• Critical analysis and judgment as to the quality of computer analysis output.
• Learning the use of commercial FE software.
• Time management and independent learning.
##### 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.
• 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).
• Utilise user manuals and online help pages/tutorials to learn and gain familiarity with commercial software platforms.

### 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

TypeHours
Revision24
Follow-up work24
Tutorial4
Supervised time in studio/workshop16
Lecture24
Preparation for scheduled sessions12
Total study time150

R. J. Schilling and S. L Harris (2000). Applied Numerical Methods for Engineers using MATLAB and C.

ABAQUS. User manual

D. Faires and R. Burden. Numerical Methods.

K.J. Bathe (1996). Finite Element Procedures in Engineering Analysis.

R.D. Cook (1981). Concepts and Applications of Finite Element Analysis.

### Assessment

#### Summative

MethodPercentage contribution
Assignment 20%
Assignment 20%
Examination  (120 minutes) 60%

#### Referral

MethodPercentage contribution
Assignment 20%
Assignment 20%
Examination  (120 minutes) 60%

#### Repeat Information

Repeat type: Internal & External

Pre-requisite/s: MATH1054 Mathematics for Engineering and The Environment.

#### Co-requisites

To study this module, you will need to also study the following module(s):

CodeModule
MATH2048Mathematics for Engineering and the Environment Part II

### Costs

#### Costs associated with this module

Students are responsible for meeting the cost of essential textbooks, and of producing such essays, assignments, laboratory reports and dissertations as are required to fulfil the academic requirements for each programme of study.

In addition to this, students registered for this module typically also have to pay for:

##### Printing and Photocopying Costs

Students are expected to cover the costs associated with the printing of drawings and graphic presentations. These are typically expected to be of the order of £50 per group (typically five students per group), also depending on the quality of printing chosen. Two CDs to submit computer codes to be covered by the each student.

Please also ensure you read the section on additional costs in the University’s Fees, Charges and Expenses Regulations in the University Calendar available at www.calendar.soton.ac.uk.

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