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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