Variational and Weighted Residual Methods

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 regionΩin the form • where the summation is over all the nodes inW.

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 exampleòelementestands 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 Naris the value of Naon 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