1
I2 = ⌠ α~η2ds
a$
⌡B 2
(42)
ηP
1 Nb - 1 P P
$
} ]
[
{
= ∑ η
$ i , η j K 2 iP
P
$
η j
$
2 P=1
where Nb is the total number of nodes along boundary B and
K 2,i, j = α~bP ∫point i) N P N P N Pds
(point j)
P
a (
b
i
j
1
(~
)
+ ~jP
αLP
for i ≠ j
P
a
a
12
i
(43)
=
1
(3~
)
+ ~jP
αLP
for i = j ≡ i
P
a
a
i
12
Assembling all segments on coastal boundary B, we have
ηiP 1 B
1 Nb - 1 P P
$
} ]
[
{
{ } [ ]{η }
I2 = ∑ η
T
$ i , ηj K 2 P =
η
P
B
$
$
$
K
(44)
2
η j 2 1Nb
$
2 P=1
N b N b N
b1
The third part integral in Equation 19 is
( ηS + ηI )
η
⌠ ~ 1
$
$
$
$
a ηS + ηI S - η
$
$
ds = I3 + I 4 + I5 + I6
2
n
n
⌡G
η
1⌠ ~
$
I3 = a η
$ S S ds
2 ⌡G
n
η
$
I 4 = -⌠ ~η S ds ⋅⋅⋅
a$
(45)
I4
⌡G
n
⌠ ~η ηI ds ⋅⋅⋅
$
I5 = - a
$
I5
⌡G
n
η
$
I6 = + ⌠ ~ηI S ds ⋅⋅⋅
a$
I6
⌡G
n
20
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