1 EEE 431 Computational Methods in Electrodynamics Lecture 18 By Dr. Rasime Uyguroglu...
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Transcript of 1 EEE 431 Computational Methods in Electrodynamics Lecture 18 By Dr. Rasime Uyguroglu...
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
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.
...........................................................(1)Lf g
:L:g: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.
1
.................................................(2)N
i ii
f f
1 2 3,..., ,f f f
i
if
7
Variational Methods/Weighted Residual Method
Now, substitute (2) into (1):
Since L is linear: 1
N
i ii
L f g
1
.....................................(3)N
i ii
Lf g
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:
1 2, ,..., Nw w w
jw
1
, , , 1, 2,....., .......(4)N
i i j ji
Lf w g w j M
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.
'Lf s
1
.............(7)N
i ii
Lf g R residual
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.
,jw R
jw
0 , 0j jw Rdv or w R
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Variational Methods/Weighted Residual Method
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
12
Variational Methods/Weighted Residual Method
A system of linear equations can be written in matrix form as:
A g
13
Variational Methods/Weighted Residual Method
Where:
,
,
j i
i
j
A w Lf
g w g
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
i
jw
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.
f'if s
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.
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Variational Methods/Weighted Residual Method
Methods used for choosing the weighting functions:
Collocation (or point matching) method, Subdomain method, Galerkin Method, Least squares method.
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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.
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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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.
, 0jw R 0R
j
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Variational Methods/Weighted Residual Method
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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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.
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Variational Methods/Weighted Residual Method
Example: Find an approximate solution to:
Using the method of weighted residuals.
'' 2
'
4 0, 0 1,
( ) 0, (1) 1
x x
x
25
Variational Methods/Weighted Residual Method
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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Variational Methods/Weighted Residual Method
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
27
Variational Methods/Weighted Residual Method
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
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:
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
29
Variational Methods/Weighted Residual Method
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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Variational Methods/Weighted Residual Method
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
Variational Methods/Weighted Residual Method
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