~
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
Previous Page
