In the finite-element method, we first discretize the computational domain Ω into a
network of simple triangular elements. The size of these elements should be much smaller
than both the local wavelength and the scale of local bathymetric variation. Finer
resolution is also desirable at places where the change of amplitude in space is rapid (e.g.
near "caustics"). Over each triangular element, the wave potential η is approximated by
$
the following linear two-dimensional function η e ,
$
η1
$e
e
$2
3
[
]
η =∑
N e ηe = N1 + N e + N 3 ηe
e
e
$
$i
(20)
i
2
ηe
i =1
$ 3
where η ie represent the wave potentials at the corners (nodes) of the element e (Figure 3)
$
ai + bi x + ci y
Ne =
(21)
i
2∆e
i
with
e
ai = x jy k - y jxk
j
bi = y j - y k
(22)
k
ci = x k - x j
Fig. 3 A typical element
and
∆e = area of element e
1 x1e
y1e
(23)
1
e
e
= 1 x2
y2
2
e
e
1 x3
y3
14
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