Postgraduate Course: Finite Element Method and Implementation (PGEE11086)
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  20 
ECTS Credits  10 
Summary  The finite element method is an indispensable tool for engineers in all disciplines. Part 1 of this course is an introduction to FEM as applied to elasticity problems in solid and structural mechanics. The mathematical equations are developed using the virtual work basis of FEM and this is used to develop equations for one, two and three dimensional elements. As FEM is a computational tool this course includes practical exercises using the commercial package ABAQUS. A number of tutorials involving hand calculations are provided to aid understanding of the technique. Part 2 of the 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. Field problems described by the Laplace, and Poisson equations are presented first and all steps of the FE formulation are 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 problems 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. Time dependent problems and time integraton schemes are presented. Special topics such as multiple constraints, mixed formulations and substructuring are introduced. Finite element formulation for incompressible flow problems is introduced through discretisations of Euler and NavierStokes equations. 
Course description 
Lectures
L1 Introduction
Course outline; areas of application of the finite element (FE) method; examples of some problems for which FE analysis has been used.
L2 FE terminology and steps
Introduction to FE terminology; steps of the analysis using an assumed displacement field approach for linear elastic analysis of structures.
L3 Input to and Output from a FE program 1
Feeding a finite element program (ABAQUS) with geometric, physical and loading information.
L4 Input to and Output from a FE program 2
Understanding and interpreting results from a FE program.
L5 FE Modelling
Introduction to plane stress, plane strain, axisymmetric, and plate bending problems; degrees of freedom; stressstrain and straindisplacement relations.
L6 Virtual Work Basis of Finite Element Method: 1
Definition of generic displacements, body forces, nodal displacements, and nodal actions; displacement shape functions with simple examples; relating generic displacements, strains, and stresses to nodal displacements.
L7 Virtual Work Basis of Finite Element Method: 2
Derivation of FE equilibrium equations using the virtual work principle; examples of derivation of stiffness and equivalent load vector for a two node truss element.
L8 Quadrilateral Elements 1
Normalised coordinates; shape functions for the bilinear and quadratic elements; Isoparametric concept; examples
L9 Quadrilateral Elements 2
Evaluation of element matrices; the Jacobian matrix; examples of specific cases.
L10 Quadrilateral Elements 3
Numerical integration; examples of numerical evaluation of element matrices
L11 Quadrilateral Elements 4
Examples of numerical and closed form evaluation of stiffness and load matrix terms
L12 Triangular elements 1
Natural coordinates; shape functions of constant and linear strain triangular elements; isoparametric mapping; examples
L13 Triangular elements 2
Evaluation of element matrices; the Jacobian matrix; examples of specific cases.
L14 Triangular elements 3
Numerical integration; examples of numerical evaluation of element matrices
L15 Beam elements 1
FE basis of Euler Bernoulli beam elements; FE matrices and shape functions
L16 Beam elements 2
Straindisplacement and stressstrain relations for Euler Bernoulli beams; evaluation FE matrices; limitations; examples
L17 Beam elements 3
FE basis of thick (Timoshenko) beam elements; shape functions
L18 Beam elements 4
Generalised straindisplacement and generalised stressstrain relations; evaluation of FE matrices; reduced integration
L19 Revision
L20 Recap and introduction to mathematical FEM
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.
L2125 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.).
L2627 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.
L28 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.
L2932 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.
L3334 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.
L3536 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.
L3738 FEM formulation for incompressible flow problems
Euler and NavierStokes equations will be introduced and commonly used formulations presented.

Entry Requirements (not applicable to Visiting Students)
Prerequisites 

Corequisites  
Prohibited Combinations  
Other requirements  None 
Additional Costs  0 
Course Delivery Information
Not being delivered 
Learning Outcomes
By the end of the course, the student should be able to:
¿ describe the analytical methods and procedures which the finite element programs use to analyse elastic solid structures;
¿ be able to use the computer based finite element methods to solve simple problems by hand calculations
¿ identify and understand all the various matrix operations involved in the process;
¿ use computer programs to analyse elastic structures, present results in appropriate graphical formats, carry out checks to assess the correctness of the output, and interpret results properly.
¿ demonstrate the ability to produce FEM based numerical discretisations of mathematical descriptions (differential equations) of simple problems in continuum mechanics;
¿ demonstrate the ability to use FEM for solving simple steady and transient field problems using a standard software package;
¿ demonstrate the ability to 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;
¿ demonstrate the ability to make a critical assessment of the calculation.

Reading List
Cook, RD; Malkus, DS; Plesha, ME; Witt, RJ. Concepts and Applications of Finite Element Analysis, Wiley, 2002.
Zienkiewicz, OC; Taylor, RL. The Finite Element Method for Solid and Structural Mechanics, ButterworthHeinemann, 2005.
Bathe, KJ. Finite Element Procedures, Prentice Hall, 1996.
Smith, IM; Griffiths, DV. Programming the Finite Element Method, Wiley, 2004.
Introduction to the Finite Element Method ¿ Theory, Programming and Applications, Erik G. Thompson, John Wiley and Sons, 2005.
Finite Element Analysis  From Concepts To Applications, D.S. Burnett, AddisonWesley 1988.
http://www.see.ed.ac.uk/~asif/Protected/CVFEM
http://homepage.usask.ca/~ijm451/finite/fe_resources/fe_resources.html
http://www.colorado.edu/engineering/CAS/Felippa.d/FelippaHome.d/Home.html
http://en.wikipedia.org/wiki/Finite_element_method 
Additional Information
Graduate Attributes and Skills 
Not entered 
Keywords  Finite element method, structural engineering 
Contacts
Course organiser  Dr Asif Usmani
Tel: (0131 6)50 5789
Email: Asif.Usmani@ed.ac.uk 
Course secretary  Mr Craig Hovell
Tel: (0131 6)51 7080
Email: c.hovell@ed.ac.uk 

