1. Finite Element Analysis

2. What is FEA ?  
The FEA method gives an approximate solution for certain problems in engineering and science
It is mainly used for problems for which no exact solution is available
It is a numerical rather than an analytical method
One of the first applications of FEA was to find the stresses and strains in engineering components under load
FEA requires large computation resources when applied to any realistic model of an engineering component

3. What is FEA ?  
The development of the FE method has depended on the availability of suitable digital computers for it to run on and graphics technology to display the results
The FE method can be applied to a wide range of material properties, linear-elastic (Hookean) behaviour and behaviour involving deviation from Hooke’s law (plasticity or rubber elasticity)
Many comprehensive general-purpose FE systems are now available

4. What is FEA ?  
There are more specialised packages for particular applications (e.g. steel framework, piping)
Depending on the type and complexity of the analysis, large analyses may benefit from multiple processors
FEA can be applied to one-dimensional (line), 2-D (area) or 3-D (volume) problems

5. What is FEA ?  
The FE method is now applied to problems involving a wide range of phenomena
 structural and stress analysis
dynamics and vibrations
heat conduction
fluid mechanics
electrostatic

6. What is FEA ?  
Any continuous problem can be divided up into a number of smaller areas or volumes which are called finite elements. The process is called discretisation and the assembly of elements is called the mesh

7. Element types Elements can be various shapes some examples are
In two dimensions, quadrilateral or triangular
In three-dimensions, brick-shaped, wedge-shaped or tetrahedral

8. Linear Elastic Analysis
The displacement is the quantity that is first found in the analysis at a series of points called nodes
Nodes are placed at the corners of the elements and often at the midside – depending on the element formulation
Nodes on the boundaries of adjacent elements must be common to the elements that meet there

9. Linear Elastic Analysis
There are a number of incorrect ways of specifying a finite element mesh

10. Linear Elastic Analysis
The analysis calculates the displacement at the nodes for the particular loading applied to the FE model
The displacements are found for a finite number of points, namely, the nodes
The displacement of each particular node is defined in terms of the displacements of the other nodes of the element

11. Linear Elastic Analysis
The displacement of each node in a 2-D element has two components:
one parallel to a reference x axis
one parallel to the y axis
these are called degrees of freedom
Each node has two degrees of freedom associated with it in a 2-D model and three for a 3-D bricked-shaped element
The computer time and the cost of the analysis increases as the number of degrees of freedom in the model is increased

12. Linear Elastic Analysis
The solution procedure for the displacement finite element method allows the user to determine a number of unknowns, these are obtained in the following order:
Firstly calculate the nodal displacements & reactions
the program finds the corresponding strains
from the strains the stresses are computed

13. Pre-processing  
Pre-processing is concerned with the creation of the finite element model and the definition of the way in which it is to be constrained and loaded
The model can be built up within the pre-processor itself
The success of the analysis depends on:
The simplifications introduced into the model as compared with the real thing
The choice of the mesh and type of elements to be used

14. Pre-processing The data for the model must contain information about:
 Appropriate mechanical or physical properties must be allocated to the elements – e.g. the material of which the structure is made, plate thickness etc.
Restrictions to the movement of certain nodes (constraints)
The geometry and finite element mesh
Control information for the analysis (type of analysis, etc)

15. Pre-processing  
Additional information in the input file specifies the type of analysis, the way it is to be carried out and the output required
e.g. static analysis
solver to be used
Output required - displacements, stresses at nodes, etc
 With some systems, an input file can be produced without the use of a pre-processor if the model is particularly simple

16. Analysis The FE system reads in the data from the input file
It carries out certain checks on this information
If there are no errors in the input file, the analysis is carried out and output is produced
These files can be examined and the relevant information extracted
The post-processor can be used to present this information in a user-friendly manner

17. Post-processing  
The post-processor takes in the information from the output files and presents it in a range of different graphical and tabular forms
Colour bands may be used to indicate the value of stress on the surface of the component
Contour lines of equal stress may be drawn
The model may be rotated and examined from different viewpoints
Point values may be queried
Graphs may be plotted

18. The FEA Developer’s View
The starting point is represented by the differential equations that govern the phenomena to be modelled
The FEA is then one method of providing numerical solutions to these equations when analytical solutions are not available
Much of the advanced work in this field is not accessible to the non-expert
Once the equations for the FE formulation of a problem have been developed, efficient procedures (algorithms) and programmes have to be developed