{} = {α 0 , α 1 , β1 , α 2 ,
β2 , ... , α m , βm }
T
(50)
1M
where M = 2m +1. The integral I3 can now be rewritten as
1
{}[ 3 ] }
K {
T
I3 =
(51)
2
where [K3] is a diagonal matrix of dimension M by M:
{
}
[ 3 ]= πkR~
diag 2H '0 H 0 , H 1H 1 , H 1H 1 , ... , H 'm H m , H 'm H m
'
'
K
a
(52)
The integral I4 in Equation 45 is
η
$
I4 = ⌠ ~η S ds
a$
⌡G
n
~ ⌠
P P
NΓ
m
(
)
N i ηi + N j η j α 0 H 0 + ∑ H'n (α n cosnθP + βn sinnθP )ds
≅ ka ∑
P P
'
$
$
P=1 ⌡ segment P
n =0
k~ NΓ P P
P
m
(
)
a
L ηi + η j α 0 H 0 + ∑ H'n (α n cosnθP + βn sinnθP )
∑
=
'
$
$
2 P=1
n =0
(53)
where LP is the length of segment P and ΝΓ is the total number of segments (= total
number of nodes) along the circular boundary Γ (Figure 5). In Equation 53, the value of
η S/ n is approximated by its value at the center of segment P. Equation 53 may be
$
written in matrix form :
{ }[ ]}
K {
T
I4 = η Γ
^
(54)
4
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