Postgraduate Course: The Finite Element Method (PGEE11046)
Course Outline
School  School of Engineering 
College  College of Science and Engineering 
Credit level (Normal year taken)  SCQF Level 11 (Postgraduate) 
Availability  Not available to visiting students 
SCQF Credits  10 
ECTS Credits  5 
Summary  The finite element method is an indispensable tool for engineers in all disciplines. This course introduces students to the fundamental theory of the finite element method as a general tool for numerically solving differential equations for a wide range of engineering problems. A range of field problems described by the Laplace, Poisson and Fourier equations is presented first and all steps of the FE formulation is described. Specific applications in heat transfer and flow in porous media are demonstrated with associated tutorials. The application of the method to elasticity problems is then developed from fundamental principles. Specific classes of problem are then discussed based on abstractions and idealisations of 3D solids, such as plane stress and strain, EulerBernoulli and Timoshenko beams and Kirchoff and MindlinReissner plates and shells. 
Course description 
Lectures: Titles & Contents
L1 Introduction
Structure of the course. Aims of the course. References with comments. Recap of Direct Stiffness Method for frame type structures (members/elements, joint/nodes, joint or nodal dofs, free and restrained dofs, element stiffness matrix, assembly into structure stiffness matrix, rearranging of structures stiffness matrix into free and restrained parts, solution for free doffs, calculation of reactions at restrained dofs, calculation of member forces). Recap of the virtual work formulation based finite element formulation for framed structures and continua. Introduction to computing tutorial problem (as follows).
Analysis of a plate with hole and a hot disk in hole. This tutorial introduces nonrectangular elements in such a way that the orientation of the main stresses is understood in advance. This ensures that the student examines the principal stress as a means of understanding the behaviour. The stress concentration around the hole will require judicious mesh refinement to capture and provide useful experience. Assuming that the hole contains a disc of material at high temperature, the heat conduction into the plate will be analysed. The effect of thermally induced stresses caused by the thermally expanding disc will also be analysed.
L26 Mathematical foundations of the finite element method and application to field problems
The finite element concept and its history. Mathematical preliminaries (Equations of calculus describing physical phenomena, exact solutions and approximate solutions). Strong and weak formulations of a problem. The finite element method will be introduced as a tool for discretising continuum equations of physics describing a problem of interest in engineering. A number of common types of differential equations of interest primarily in civil and mechanical engineering will be presented and their applications discussed. The methods of FEM used to achieve discretisation (variational and Galerkin weighted residual approaches) will be introduced and demonstrated using problems described by Laplace and Poisson equations (this includes steady heat conduction, flow in porous media etc.).
L78 FEM for continuum elasticity problems and thermomechanics
Concepts developed in the previous lectures will be applied to continuum elasticity problems. The discretisation process will be described first for general 3D solids and then specialised to 2D idealisations of plane stress, plane strain and axial symmetry. This will be extended to show how thermomechanical effects may be accommodated in the formulations.
L9 Finite element method programming
Students will be introduced to a number of matlab based programmes for developing and solving finite element application problems in steady and transient heat conduction, plane stress and plane strain elasticity and thermomechanics problems. Introduction to programming project (as follows).
Based on one (or more) of the Matlab programmes introduced in the lecture, a programming exercise will be set for the students as a project. The exercise will require the students to develop some part of the theory taught in the lectures of this course and implement the enhancement made to the theory into the program and then solve a practical problem using this modified programme. The solution of the problem will be checked against an independent calculation using ABAQUS. Students will be required to submit their theory enhancement work in a report which will need to contain appendices providing a listing of the programme changes and the results from solving the problem.
L1013 FEM for structural engineering idealisations
The specialist concepts and formulations required for structural engineering applications of FEM will be formally developed. Starting with a brief recap of 1D bars and beams and extending to plates and shells covering EulerBernoulli and Timoshenko beams and Kirchoff and MindlinReissner plates and shells including issues such as membrane and shear locking.
L1415 Introduction to time dependent problems
The complete Fourier equation for transient heat transfer will be introduced. Finite difference based approaches for solving transient field problems will be described (forward, backward and central difference methods). Implicit and explicit time integration and stability of various time integration schemes will be discussed. Equations time discretisation for structural dynamics and vibration problems.
L1617 Special topics
A number of special topics, such as skew boundary conditions, multiple point constraints and substructuring for large problems will be introduced with examples. Introduction to mixed formulations for solids and structures.
L18 FEM formulation for incompressible flow problems
Euler and NavierStokes equations will be introduced and commonly used formulations presented.
This is a postgraduate level finite element course which builds on the introductory course "FEM for Solids and Structures". The subject is approached in a more general sense in a relatively more mathematical framework. Many topics from the rich FEM literature are presented preferring breadth over depth. The course is primarily intended for MSc students and those undergraduates who are fascinated by the subject and would like to pursue higher degrees in the field of computational mechanics.

Entry Requirements (not applicable to Visiting Students)
Prerequisites 
Students MUST have passed:

Corequisites  
Prohibited Combinations  
Other requirements  None 
Course Delivery Information

Academic year 2016/17, Not available to visiting students (SS1)

Quota: None 
Course Start 
Semester 2 
Timetable 
Timetable 
Learning and Teaching activities (Further Info) 
Total Hours:
100
(
Lecture Hours 18,
Seminar/Tutorial Hours 9,
Formative Assessment Hours 1,
Summative Assessment Hours 2,
Programme Level Learning and Teaching Hours 2,
Directed Learning and Independent Learning Hours
68 )

Assessment (Further Info) 
Written Exam
60 %,
Coursework
40 %,
Practical Exam
0 %

Additional Information (Assessment) 
The assessment will be made on the basis of: Intermittent assessment 40%. Degree examination 60% 
Feedback 
Verbal feedback on the tutorials at the student's request.
Written feedback on the coursework. 
Exam Information 
Exam Diet 
Paper Name 
Hours & Minutes 

Main Exam Diet S2 (April/May)  The Finite Element Method  2:00  
Learning Outcomes
On completion of this course, the student will be able to:
 Produce FEM based numerical discretisations of mathematical descriptions (differential equations) of simple problems in continuum mechanics;
 Use FEM for solving simple steady and transient field problems using a standard software package;
 Use FEM to produce a reliable prediction of displacements and stresses in linear elastic bodies of relevance to engineering practice using a standard software package;
 Make a critical assessment of FEM calculations.

Reading List
Recommended reading:
J. N. Reddy, An Introduction to the Finite Element Method, 3rd ed., McGrawHill, 2005
E. G. Thompson, Introduction to the Finite Element Method  Theory, Programming and Applications, John Wiley and Sons, 2004.
Background reading:
O. C. Zienkiewicz and R. L. Taylor, The Finite Element Method Set, 7th ed., ButterworthHeinemann, 2013.
K. J. Bathe, Finite Element Procedures, Prentice Hall, 1996.

Additional Information
Graduate Attributes and Skills 
Not entered 
Keywords  Numerical methods,computational mechanics 
Contacts
Course organiser  Dr Stefanos Papanicolopulos
Tel: (0131 6)50 7214
Email: S.Papanicolopulos@ed.ac.uk 
Course secretary  Mr Craig Hovell
Tel: (0131 6)51 7080
Email: c.hovell@ed.ac.uk 

