## Finite Element Analysis

NB: FURTHER INFORMATION TO BE ADDED
These notes provide some background to finite element analysis (FEA).

Originally most of these notes were writen by Graham Roberts and David Grieve. Later additions are by David Grieve

i) Introduction to Numerical Techniques
For stress, thermal and many other types of analysis, only the simplest of configurations can be solved by exact analytic methods. For the vast majority of engineering components some alternative method is needed and the usual approach involves a method of approximation requiring some numerical technique to solve, which in turn requires the use of a digital computer.

ii) Numerical Techniques
There are 3 main methods of numerical approximation:

Finite difference (FD) method
Finite element analysis (FEA)
Boundary element method (BEM)
These methods all have some advantages and disadvantages. FEA is a variant of some classical approximation methods, whereas the FD method is an older procedure that has been used in one form or another almost since the inception of differential calculus. FEA was first adapted for stress analysis problem solving by Southwell. Ever since FEA were first used, there have been arguments about their relative merits versus the longer established FD method.

FEA
FEA involves mathematical approximations associated with the displacement function of a particular element and geometric approximations to the component shape. An advantage of FEA is that it can be used to analyse virtually any structural problem in a routine manner, hence non-homogeneous material can be accommodated as the assembly of elements with different properties is straightforward.

FD Method
The FD method uses a finite difference approximation to the (partial) differential equations that describe the behaviour of the system. A significant difficulty in solving the equations is that with fine meshes, needed for adequate accuracy, there may be instability in determining the solution. Discontinuous interfaces, such as abrupt changes in material properties or geometry require special treatment. FD tend to be used more for Fluids problems than for Mechanics problems.
The two applets on the introductory web page, for computing the load carrying capacity and pressure distribution of a hydrodynamic tilt pad thrust bearing and for a journal bearing, are both solved by 2 dimensional FD methods using rectangular meshes with an iterative solver.
Finite difference method example

BEM
The BEM method reduces the dimensionality of the problem by 1, so the overall size of the computation the smaller than in FEA or FD, but the matrices produced are assymetric and densely packed and may require just as much computational effort to solve as when using FEA or FD. The method is particularly suited to problems with high stress gradients such as fracture mechanics. Boundaries that are nominally at 'infinity' may readily by handled whereas such boundaries increase the size of FEA and FD problems. The underlying mathematics is however more complex than the other two methods.

(N.B. Information about two dimensional stress and strain transformations can be found in the the Mechanics and Engineering Science section).

The notes below relate to FEA.

 Topic Derivation of Axial Tension - Compression Linear Elastic Element Axial Tension - Compression - Displacement Function Local and Global Co-ordinates - Transformation 2.3 The Constant Strain Continuum Element 3 A General Approach to Derive the Finite Element Stiffness Matrices for Structural Problems Mapping and shape functions Potential energy in structural FEA 4 Modal analysis ..4.2 Sinusoidal Response ..4.3 Transient Response Errors arising in FEA ...Example of reducing errors by increasing the number of elements, increasing element order and by gauss integration 5 Direct solution methods for FE equations, restraints and loads ..5.6 Iterative solution methods ..5.7 Solution of equations in dynamics problems Non linear effects and dealing with them Vibration - Excitation, Damping and Control ...

References:

'Finite Elements - A Gentle Introduction', by D Henwood and J Bonet, Macmillan, 1996.

'Basic Principles of the Finite Element Method', by K M Entwistle, Maney Publishing, 2001. ISBN: 978-1902653532.

'Roarks Formulas for Stress and Strain', 6th Ed., Ed by W C Young, McGraw-Hill International Editions, 1989.

'Stress Concentration Factors', by R E Peterson, John Wiley and Sons, 1974.

'Shock and Vibration Handbook', Ed. C M Harris, McGraw-Hill, 1988.

David J Grieve, 6th December 2012.