# Biomedical Engineering (MEng)

## Finite Element Analysis and Applications

### Module aims

The aim of the course is to teach the students to use Finite Element programs in a practical way to solve problems in linear elastic stress analysis. A student who has studied the course should be able, in a later industrial setting, to undertake the analysis of real problems with a fair understanding of sensible modelling procedures. In support of this, the course is split into two stages:

1. Theoretical study of the Finite Element method, with emphasis on understanding what goes on inside a typical, modern, commercial program
2. Practical experience in analysis using an industry-standard, interactive, Finite Element program.

ECTS units:    6
Contributing to Course Elements: 6 to ME3-LCTVS or ME4-LCTVS

### Learning outcomes

On successfully completing this module, students will be able to:

• Explain the basic theoretical principles of the Finite Element method
• Employ industry-standard software for interactive FE model generation, analysis and the post-processing of results
• Interpret the output from the software critically and intelligently in order to yield the required information
• Formulate the boundary conditions of a problem in a suitable form for correct analysis
• Assess alternative strategies (of element type, mesh design, boundary condition definition etc.) for economical and accurate FE modelling of specific 2D, 3D and axisymmetric structural problems

### Module syllabus

• Introduction: with application to static stress analysis. Context and history.
• Truss frame example: division into elements, selection of variables and shape functions, stiffness derivation, assembly of stiffness equations, application of boundary conditions, solution for displacements, computation of element stresses and strains.
• The constant strain triangle: Continuum model and role of continuum elements; Geometry of constant strain triangle (CST), nodal variables and shape functions, stiffness derivation (plane stress and plane strain), element stress and strain computation.
• Element forumulation: the need for more advanced and generalised procedures; element stiffness by virtual work; general procedure applied to CST; detailed derivation of linear rectangular element (Gauss quadrature, location of element integration points etc.); quadratic rectangular element, axisymmetric elements, 3-dimensional solid elements, truss and beam elements, membrane, plate and shell elements. Isoparametric elements, Jacobian mapping for arbitrarily shaped elements. Element performance: Stiffness and accuracy considerations. Nonconforming elements, reduced integration.
• Element libraries: elements offered by commercial programs (shapes, nodes, degrees of freedom, allowable load types, etc.). Materials, loads, supports and solution: analysis procedure for modern commercial programs (definition of structure and loads, supports and other constraints, solution, post-processing); material property definitions and matrices; geometric properties (thicknesses, cross-sectional areas etc.); load types (point forces and moments, pressure, body forces, thermal) and internal conversion to nodal loads; supports, prescribed displacements, rigid links; symmetric and antisymmetric boundary conditions and their application to reduce model size; stiffness transformations to model supports or loads at arbitrary angels; stiffness matrix assembly and solution, bandwidth and its minimisation; Other types of solution (structural dynamics, material plasticity, large deflections, contact problems, fracture mechanics).
• Guide to good modelling: Identification of appropriate domain of solution (2-/3-dimensional, axisymmetry, beams/shells etc.). Selection of elements, degrees of freedom, stress assumptions etc. Creation of mesh (refinement, shape, aspect ratios, curvature); Definition of material and geometric properties; application of loads and supports; pre-analysis checks; post-processing results - typical options; importance of verification, development of checking strategies; sources of inaccuracies and errors.

### Pre-requisites

ME1-HSAN; ME2-HSAN

### Teaching methods

• Duration: Spring term (11 weeks)
• Lectures: 5 x 3hr lectures covering the theoretical material listed in the syllabus
• Tutorials: 6 x 1.5hr sessions in the computer room, devoted mainly to tackling the four tasks listed in the syllabus, but including some general tutorial time. Students work in groups of 2 or 3. Results of practicals are reported on brief structured forms for assessment.
• Projects: Two tutorial sheets (not assessed), short project, both to be done mainly in own time.

 Summary of student timetabled hours Autumn Spring Summer Lectures 0 15 0 Tutorials Details of tutorials to be advised by the course leader during the course. Other (computing) 0 9 0 Total 24 Expected private study time 3 hr per week, plus exam revision

### Assessments

 Written examinations: Date (approx.) Max. mark Pass mark Finite Element Analysis & Applications (3h) A Data and Formulae handbook is provided. This is a CLOSED BOOK Examination. April/ May 160 n/a Coursework (including progress tests, oral presentations etc.) Submission date Max. mark Pass mark Submission Feedback Practical task 1 Returned, with grade and written comments for discussion in tutorial Jan-March 10 n/a Practical task 2 Ditto Jan-March 10 n/a Practical task 3 Ditto Jan-March 10 n/a Practical task 4 Ditto Jan-March 10 n/a Total 200 n/a