By combining the expressions for η at nodes i, j and k, the second and third terms
$
of Equation 91 for element e can be represented by means of matrix,
] ([ ] [ ]) {η }
[I
e
+ IIIe = A e + Be
e
( e = 1, 2, ..., E )
(103)
$
I
where
{ }{
}
T
ηe = ηe , ηe , ηe
$
$i $ j $k
(104)
[][]
[]
1
An expression for the first term of Equation 91 may be obtained by applying
certain boundary condition and also using Equations 87 through 90. This gives the
following relationship when node i is at coastal boundary,
(
)
I P i = - α ∫N iP ~ i N iP + ~ j N P N iP ηiP + N P ηP ds
a
a
$
$j
C
j
j
P
= C Pi ηiP + CiPj ηP
(105)
$
$j
i
Similarly for node j,
I P j = C Pi ηP + C Pj ηP
(106)
$i
$j
C
j
j
Therefore, for segment P with node i and j, we have the matrix formula
[ ]= [ ] η }
C {$
P
P
P
I
(107)
C
33
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