~

η

$

~

∫ ~∇ η∇ N i dΩ +

- ∫ +

∫ N i a

∫a $

∫ b η∇ N i dΩ = 0

∫ $

ds +

(78)

n

C

Γ

Ω

Ω

where C and Γ denotes coastal boundaries and open boundaries respectively, and η / n is

$

the normal derivative of η .

$

1

for (x, y) at node i

N i (x, y) = 0 to 1

(79)

for (x, y) within elements surrounding node i

0

for (x, y) outside elements surrounding node i

Therefore, Ni can be represented as

N i (x, y) =

N e (x, y)

∑

(80)

i

ei

function corresponding to an element e and one of its node i. When (x, y) is at boundary,

Equation 80 becomes

N i (x, y) =

N P (x, y) = N iP1 (x, y)+ N iP2 (x, y)

∑

(81)

i

Pi

where P1 and P2 are the boundary segments to either side on node i. The function NiP≡0

for all other segments.

Within an element e, the value of η is obtained by substituting (80) into (76),

$

η(x, y) = ηi N e + η j N ej + ηk N e

(82)

$

$ i $

$ k

29

Integrated Publishing, Inc. |