EEE 431 Computational Methods in Electrodynamics

1. EEE 431Computational Methods in Electrodynamics Lecture 18 By Dr. Rasime Uyguroglu Rasime.uyguroglu@emu.edu.tr

2. Variational Methods

3. Variational Methods/Weighted Residual Method • The name Method of Moments is derived from the original terminology that • Is the nth moment of f. When is replaced by an arbitrary , we continue to call the integral a moment of f.

4. Variational Methods/Weighted Residual Method • The name method of weighted residuals is derived from the following interpretation:

5. Variational Methods/Weighted Residual Method • Consider again the operator equation: • Linear Operator. • Known function, source. • Unknown function. • The problem is to find g from f.

6. Variational Methods/Weighted Residual Method • Let f be represented by a set of functions • scalar to be determined (unknown expansion coefficients. • expansion functions or basis functions.

7. Variational Methods/Weighted Residual Method • Now, substitute (2) into (1): • Since L is linear:

8. Variational Methods/Weighted Residual Method • Now define a set of testing functions or weighting functions • Define the inner product (usually an integral). Then take the inner product of (3) with each and use the linearity of the inner product:

9. Variational Methods/Weighted Residual Method • If (3) represents an approximate equality, then the difference between the exact and approximate is: • R, is the error in the equation.

10. Variational Methods/Weighted Residual Method • The inner products are called the weighted residuals. • In the weighted residual method, the weighting functions are chosen such that the integral of a weighted residual of the approximation is zero.

11. Variational Methods/Weighted Residual Method • Equation (4) can be obtained by setting all weighted residuals to zero. • Which is equation (4).

12. Variational Methods/Weighted Residual Method • A system of linear equations can be written in matrix form as:

13. Variational Methods/Weighted Residual Method • Where:

14. Variational Methods/Weighted Residual Method • Solving for and substituting for in Eq. 2, gives an approximate solution to Eq. 1. However, there are different ways of choosing the weighting functions

15. Variational Methods/Weighted Residual Method • Selection of basis and weighting functions: • There are infinitely many possible sets of basis and weighing functions. Although the choice of these is specific to each problem, we can state rules that can be applied generally to optimize the change of success of obtaining accurate results in a minimum time and computer memory storage.

16. Variational Methods/Weighted Residual Method • Selection of basis and weighting functions: • They should form a set of linearly independent functions. • should approximate the (expected) function reasonably well.

17. Variational Methods/Weighted Residual Method • Keep the following in mind in the selection of basis and weighting functions: • The desired accuracy of the solution, • The size of the matrix [A] to be inverted, • The realization of a well-behaved matrix [A], • The easy of evaluation of the inner products.

18. Variational Methods/Weighted Residual Method • Methods used for choosing the weighting functions: • Collocation (or point matching) method, • Subdomain method, • Galerkin Method, • Least squares method.

19. Variational Methods/Weighted Residual Method • Let us discuss the point matching Method: • Collocation (or point matching) method: • It is the simplest method for choosing the weighting functions: • It basically involves satisfying the approximate representation: at discrete points in the region of interest.

20. Variational Methods/Weighted Residual Method • Collocation (or point matching) method: • In terms of the MoM this is equivalent to choosing the testing functions to be Dirac delta functions. i.e.,

21. Variational Methods/Weighted Residual Method • Substituting Eq. 9 into • Results • We select as many matching points in the interval as there are unknown coefficients and make the residual zero at those points.

22. Variational Methods/Weighted Residual Method • The integrations represented by the inner products now become trivial, i.e..

23. Variational Methods/Weighted Residual Method • Although the point matching method is the simplest specialization for the computation, it is not possible to determine in advance for the particular operator equation what weighting functions would be suitable.

24. Variational Methods/Weighted Residual Method • Example: Find an approximate solution to: • Using the method of weighted residuals.

25. Variational Methods/Weighted Residual Method • Let the approximate solution be: • Select to satisfy . So a reasonable choice is

26. Variational Methods/Weighted Residual Method • Now select: • If i=2 the approximate solution is: • Where the expansion functions are to be determined.

27. Variational Methods/Weighted Residual Method • To find the residual R:

28. Variational Methods/Weighted Residual Method • Point Matching Method: • Since we have two unknowns • We select • And set the residual equal to zero at those points. i.e:

29. Variational Methods/Weighted Residual Method • Point Matching Method: • Solving these equations, • And substituting:

30. Variational Methods/Weighted Residual Method • Point Matching Method: • Select • As the match points. Then:

31. Variational Methods/Weighted Residual Method • Point Matching Method: • Solving these equations: • With the approximate solution: