Variational and Weighted Residual Methods

1. Variational and Weighted Residual Methods

2. The Weighted Residual Method • The governing equation for 1-D heat conduction • A solution to this equation for specific boundary conditions was sought in terms of extremising a functional • A solution can be found by making use of a trial function which contains a number of parameters to be determined

3. The Weighted Residual Method • In general, the trial function will not satisfy at all points in the region in which the solution is sought and that where r(x) is the residual at the point x in the region

4. The Weighted Residual Method • If : • Φ produces the exact solution as the number of parameters α in Φ is increased indefinitely. • Wiare linearly independent functions of x (weighting functions) • It can be shown that Φ→u, the exact solution, if for all Wi

5. The Weighted Residual Method • Noting that the values of Wi are linearly independent if none of them can be expressed as a linear combination of the other . No set of numbers bj exists such that

6. The Weighted Residual Method • In the FE version of the weighted method, the trial function is expressed in terms of its nodal values Φi=1....M , where M is the total number of nodes • where are the global shape functions

7. The Weighted Residual Method • For the series of linear elements ..e-1, e, e+1, e+2 ...

8. The Weighted Residual Method • is as shown in elements e and e+1, but is zero in all other elements. • The parameters to be determined are the nodal values, Φi • The weighting functions are chosen the same as the shape functions (Galerkin method)

9. The Weighted Residual Method • The weighted residual statement is • for i= 1....M • The method fails with piecewise linear shape functions, since is everywhere zero (except at nodes where is discontinuous)

10. The Weighted Residual Method • Integrating the above equation by parts • The right-hand side second term is zero unless both and belong to the same finite element. • The first term on the right-hand side is also zero, apart from the two elements at the extremities of Ω.

11. The Weighted Residual Method • For the first element • Since • Then

12. The Weighted Residual Method • This can be written as • Again , since • This becomes

13. The Weighted Residual Method • Similarly for the last element • Hence •  gives the component Kiαof the global stiffness matrix [K]g and

14. The Weighted Residual Method • where

15. The Weighted Residual Method • In the j-th row of the matrix product [K]g {Φ}, the only non-zero terms are

16. The Weighted Residual Method • Reverting to the shape functions within the individual elements, these three non-zero terms can be expressed as • This is an integral over element e, since is zero for elements numbers less than e and is zero for element numbers greater than e

17. The Weighted Residual Method • Similarly the second term can be expressed as • and the third as

18. The Weighted Residual Method • Hence we can express the global stiffness matrix by assembling the element stiffness matrices [K]e, where • α and β could each take on the values j-1 and j for element e and the superscript has been removed from the element shape functions.

19. Review of results • The result obtained by Galerkin weighted residual method is exactly the same as that obtained by the piecewise application of the Variational approach • The stiffness matrices are symmetrical (advantageous in reducing the computation in solving the system simultaneous equations) • The Variational methods are extremely powerful in engineering mathematics

20. Review of results • If a variational formulation of a certain problem is possible the same results can be obtained by the Galerkin weighted residual method. • The Galerkin weighted residual method can be used when no variational formulation is available. • The Galerkin weighted residual method is the most commonly used of the weighted residual methods.