# DEGREE REGULATIONS & PROGRAMMES OF STUDY 2018/2019

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# Undergraduate Course: The Finite Element Method 5 (CIVE11029)

 School School of Engineering College College of Science and Engineering Credit level (Normal year taken) SCQF Level 11 (Year 5 Undergraduate) Availability Available to all 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. 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, Euler-Bernoulli and Timoshenko beams and Kirchoff and Mindlin-Reissner 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 Navier-Stokes equations. 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 non-rectangular 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. L2-6 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.). L7-8 FEM for continuum elasticity problems and thermo-mechanics 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 thermo-mechanical 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 thermo-mechanics 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. L10-13 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 Euler-Bernoulli and Timoshenko beams and Kirchoff and Mindlin-Reissner plates and shells including issues such as membrane and shear locking. L14-15 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. L16-17 Special topics A number of special topics, such as skew boundary conditions, multiple point constraints and sub-structuring 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 Navier-Stokes 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. Accreditation of Higher Education Programmes Learning Outcomes: SM2m, SM5m, EA1b, EA2, G1 (Definite); EA3m, P4 (Possible)
 Pre-requisites Students MUST have passed: Finite Element Methods for Solids and Structures 4 (CIVE10022) Co-requisites Prohibited Combinations Other requirements None
 Pre-requisites This course can only be taken by students with prior experience of Advanced Structural Analysis. Visiting students must discuss their experience with the Course Organiser before they will be permitted to enroll on the course High Demand Course? Yes
 Academic year 2018/19, Available to all students (SV1) 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 examples at the student's request. Written feedback on the coursework. Exam Information Exam Diet Paper Name Hours & Minutes Main Exam Diet S2 (April/May) 2:00
 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.
 Recommended reading: J. N. Reddy, An Introduction to the Finite Element Method, 3rd ed., McGraw-Hill, 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., Butterworth-Heinemann, 2013. K. J. Bathe, Finite Element Procedures, Prentice Hall, 1996.
 Graduate Attributes and Skills Application of Mathematical concepts, Computer Modelling skills, Interpreting design problems. Keywords Numerical methods,computational mechanics
 Course organiser Dr Stefanos Papanicolopulos Tel: (0131 6)50 7214 Email: S.Papanicolopulos@ed.ac.uk Course secretary Miss Margaret Robertson Tel: (0131 6)50 5565 Email: margaret.robertson@ed.ac.uk
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