therefore,

1 ~ P 1

[][]

IΓ3 = ΓP = -

P

(118)

a gL

3

1

2

[]

The function η will be obtained by first computing the element matrix A e and

$

[ ] for elements e = 1,... E,

[]

and the boundary matrix C P , [ΓP ], [Γ2P ], [Γ3P ] for

e

B

1

segments P = 1,... , N P . These matrices are assembled to obtain an NN system of

equations,

∑ ([ ]+ [ ]{η }+ ∑ [ ] η }+ ∑ ([ ]+ [ ]{η }+ ∑ [ ]+ ∑ [ ]= 0

B )$

Γ )$

C {$

Γ

Γ

e

e

e

P

P

P

P

P

P

P

A

D

Γ

C

1

2

3

Ω

Γ

Γ

Γ

C

Notice that

T

~ qr 2 η , -

η

~ qr 2 0 η , η

1

$

$

∑[ ]

{

}

T

= αa

=- a

P

(119)

D

0 1 $ A1 $ A2

s A2

s A1

Γ

where A1 and A2 are two points that connect the open boundary and coastal boundary and

∑ [ ] can

η / s. Therefore, term

P

D

the wall boundary condition, (9), applies to

be

$

Γ

∑ [ ] η }and then the assembled equation becomes

C {$

P

P

included into

C

C

[ 1 ] η} [ 2 ] ηC } [ 3 ] ηΓ} {f }

K {$ + K {$ + K {$ =

(120)

or

[ ] η} {f }

A {$ =

(121)

This linear system of equations may be solved to obtain η.

36

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