1 EEE 431 Computational Methods in Electrodynamics Lecture 18 By Dr. Rasime Uyguroglu [email protected]

1

EEE 431Computational Methods in

Electrodynamics

Lecture 18By

Dr. Rasime [email protected]

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.

( )nx f x dxnx

nw

4


The name method of weighted residuals is derived from the following interpretation:

5


Consider again the operator equation:

Linear Operator. Known function, source. Unknown function. The problem is to find g from f.

...........................................................(1)Lf g

:L:g:f

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Let f be represented by a set of functions

scalar to be determined (unknown expansion coefficients.

expansion functions or basis functions.

1

.................................................(2)N

i ii

f f

1 2 3,..., ,f f f

i

if

7


Now, substitute (2) into (1):

Since L is linear: 1

N

i ii

L f g

1

.....................................(3)N

i ii

Lf g

8


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:

1 2, ,..., Nw w w

jw

1

, , , 1, 2,....., .......(4)N

i i j ji

Lf w g w j M

9


If (3) represents an approximate equality, then the difference between the exact and approximate is:

R, is the error in the equation.

'Lf s

1

.............(7)N

i ii

Lf g R residual

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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.

,jw R

jw

0 , 0j jw Rdv or w R

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Equation (4) can be obtained by setting all weighted residuals to zero.

Which is equation (4).

1

1

, 0, 1,2,...,

, , 0

, , 1, 2,...,

j

N

j i j ii

N

i j i ji

w R j N

g w w Lf

w Lf g w j N

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A system of linear equations can be written in matrix form as:

A g

13


Where:

,

,

j i

i

j

A w Lf

g w g

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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

i

jw

15


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.

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Selection of basis and weighting functions: They should form a set of linearly independent

functions. should approximate the (expected) function

reasonably well.

f'if s

17


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.

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Methods used for choosing the weighting functions:

Collocation (or point matching) method, Subdomain method, Galerkin Method, Least squares method.

19


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.

i ii

Lf g

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.,

1( ) ........(8)

0j

j jj

r rw r r

r r

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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.

, 0jw R 0R

j

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The integrations represented by the inner products now become trivial, i.e..

{ }i

i

ij i r r

j r r

A Lf

g g

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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.

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Example: Find an approximate solution to:

Using the method of weighted residuals.

'' 2

'

4 0, 0 1,

( ) 0, (1) 1

x x

x

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Let the approximate solution be:

Select to satisfy . So a reasonable choice is

01

N

i in

f f f

0f

'(1) 1

0f x

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Now select:

If i=2 the approximate solution is:

Where the expansion functions are to be determined.

0 1 1 2 2

21 2( 2) ( 3 / 2)

f f f f

f x x x x x

1( ) ( )ii

if x x x

i

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To find the residual R:

22

2

2 3 2 21 2

( 4)

(4 8 2) (4 6 6 3) 4

R Lf g

dR f x

dx

R x x x x x x x

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Point Matching Method: Since we have two unknowns We select And set the residual equal to zero at

those points. i.e:

1 2, 1/ 3, 2 / 3x x

1 2

1 2

(1/ 3) 0 6 41 33

(2 / 3) 0 42 13 60

R

R

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Point Matching Method: Solving these equations,

And substituting:

1 2

677 342,

548 548

2 31.471 0.2993 0.6241f x x x

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Point Matching Method: Select As the match points. Then:

1/ 4, 3 / 4x x

1 2

1 2

(1/ 4) 0 4 29 15

(3/ 4) 0 28 3 39

R

R

31


Point Matching Method: Solving these equations:

With the approximate solution:

1 2

543 228,

412 412

2 31.636 0.2694 0.699f x x x

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1 EEE 431 Computational Methods in Electrodynamics Lecture 18 By Dr. Rasime Uyguroglu [email protected]