1. Variational and Weighted Residual Methods

2. FE Modification of the Rayleigh-Ritz Method
In the Rayleigh-Ritz method
A single trial function is applied throughout the entire region
Trial functions of increasing complexity are required to model all but the simplest problems
The FE approach
uses comparatively simple trial functions that are applied piece-wise to parts of the region
These subsections of the region are then the finite elements

3. FE Modification of the Rayleigh-Ritz Method
Consider the problem of 1-D heat flow, the functional to be extremised is
where the integral over W corresponds to the length of the region and Neumann boundary conditions are specified at one end, G,of the region

4. FE Modification of the Rayleigh-Ritz Method
The length over which the solution is required, is divided up into finite elements
In each element the value of f is found at certain points called nodes
Two nodes will mark the extremities of the element
Other nodes may occur inside the element

5. FE Modification of the Rayleigh-Ritz Method
Let the unknown temperatures at the nodes of the element e be
where n+1 is the number of nodes in each element.

6. FE Modification of the Rayleigh-Ritz Method
The temperature at any other position in the element is represented in terms of the nodal values {f}e and shape functions associated with each node
where Nb is the shape function associated with the node b and b=i ... i+n and [N] is the corresponding row matrix.

7. FE Modification of the Rayleigh-Ritz Method
Let us write the trial function f over the entire region Ω in the form
where the summation is over all the nodes in W.

8. FE Modification of the Rayleigh-Ritz Method
The global shape functions have been used to take into account the contribution from fa to f over the entire region W
The global shape functions over much of W will be zero
For interior nodes of an element will be non-zero only within that element
End nodes of an element will have non-zero values over the two elements sharing the node.

9. FE Modification of the Rayleigh-Ritz Method
For example :
is non-zero only in elements e and e+1.
will be non-zero only in element e.

10. FE Modification of the Rayleigh-Ritz Method
Neglecting for the moment, consideration of the first and last elements of the region
Write the Rayleigh-Ritz statement in which the nodal values are the adjustable parameters.
Consider the nodes i...i+n belonging to element e

11. FE Modification of the Rayleigh-Ritz Method
where for example òelement e stands for
over the element e

12. FE Modification of the Rayleigh-Ritz Method
Since
is an expression involving {f}e-1
involves {f}e
and there is no relationship between {f}e-1 and {f}e ,both expressions must be equal to zero

13. FE Modification of the Rayleigh-Ritz Method
Let us
focus on the terms containing an integral over the element e
Drop the superscript g on the shape functions
Suppose that the element extends from x=xe to x=xe+h
No loss in generality is incurred if we
Shift the origin to x=xe
Take the element to extend rather from 0 to h

14. FE Modification of the Rayleigh-Ritz Method
The function can be written as,
where
Note that
a = i ...i+n

15. FE Modification of the Rayleigh-Ritz Method
Also, noting
Since
Hence

16. FE Modification of the Rayleigh-Ritz Method
So, differentiating under the integral sign, we have
Hence

17. FE Modification of the Rayleigh-Ritz Method
This equation is one in the set of n+1 simultaneous equations obtained by letting a run through the values i...i+n :

18. FE Modification of the Rayleigh-Ritz Method
where
and
In the end elements, where Neumann boundary conditions may have to be considered, there is an additional term
where Nar is the value of Na on the boundary G

19. FE Modification of the Rayleigh-Ritz Method
If there are two 2-noded elements, labelled m and n, with nodes i, i+1 and i+2, assembly of the element matrices is as before. Then
 for the first element m
and similarly for element n

20. FE Modification of the Rayleigh-Ritz Method
By combining these two matrix equations
The global assembly matrix is built up in this way
The boundary conditions on the extreme elements are inserted
The set of equations is solved for the unknown values of f

21. Example 3  
Find an approximate solution to Example 1 for the rod of constant cross section using three linear elements of equal length.

22. FE Modification of the Rayleigh-Ritz Method
All elements will have the same stiffness matrix
The coordinate origin is to be at node 1
The shape function in element 1 for node 1 is
For node 2

23. FE Modification of the Rayleigh-Ritz Method
From the trial function
and

24. FE Modification of the Rayleigh-Ritz Method
For element 1 we have
where N1,r is the value of N1 (the value is 1) at node 1
For element 2

25. FE Modification of the Rayleigh-Ritz Method
For element 3
where N4,r is the value of N4 at node 4 (N is equal to 1)

26. FE Modification of the Rayleigh-Ritz Method
Assembling the matrices, we have

27. FE Modification of the Rayleigh-Ritz Method
Given f1 = f4 = f0 we can solve for f2 and f3 from rows 2 and 3 
from which
since

28. Comparison of FE and Exact Solution

29. Comparison of FE and Exact Solution
It can be seen that,
This is the same as the exact solution for the nodal values
The finite-element approximation deviates from the exact solution between the nodes
As the number of elements is increased, the deviation from the exact results at the non-nodal positions decreases

30. Natural Coordinates and Quadratic Shape Functions
For convenience, a dimensionless coordinate x is used rather than x so that over the length of a 1-D element the value of x runs from +1 to -1
In the previous example if x is measured from node 1, then in element 1,
The shape functions become

31. Natural Coordinates and Quadratic Shape Functions
Since
The trial function becomes
and

32. Natural Coordinates and Quadratic Shape Functions
Higher-order shape functions allow the variable to alter in a more complicated fashion within an element (fewer quadratic than linear elements are required but with a higher amount of computation per element)
There are two methods used to obtain good precision in FE packages
 p-approach: better precision by using shape functions of increasing complexity
h-approach: better precision is obtained by mesh refinement

33. Natural Coordinates and Quadratic Shape Functions
The shape functions for a Quadratic 1-D element, which has three nodes, are

34. Stiffness Matrix for 1-D Quadratic Element (HC)
From the trial function, the components of the 3x3 element stiffness matrix satisfy the condition
 Now

35. Stiffness Matrix for 1-D Quadratic Element (HC)
Hence
and so on, so that
Note that the matrix is symmetrical

36. Stiffness Matrix for 1-D Quadratic Element (HC)
To evaluate
With Q a constant

37. Stiffness Matrix for 1-D Quadratic Element (HC)
If and are the values of at x = ± L, then the equations to be solved

38. Stiffness Matrix for 1-D Quadratic Element (HC)
With f1 = f3= f0 expanding row 2 and solving for f2 gives
Since h=2L which is the exact solution, can be found by substituting the value of f2 and expanding row 1
 Again, the exact value is obtained
 The estimate of approaches the exact value as the number of elements is increased