where i(xi,yi), j(xj,yj), k(xk,yk) are the three nodes of element e (see Figure 3) and function
Nie(x,y) is given by Equations 21 through 24.
When (x, y) is on a boundary segment with nodes i and j at the ends, Equation 82
is simplified to:
η(x, y) = ηi N iP + η j N P
(83)
$
$
$ j
where N iP = (s j - s) / LP , N P = (si - s) / LP , and s is the relative coordinate along P and
j
has values of si and sj at the two endpoints. The length of the segment is L = |si - sj|.
P
relations referring to linear function Nie(x,y) and NiP(x,y) are
The following
developed for later use:
Ne ∧
Ne ∧
1
b ∧ + c ∧j
∇N =
i +
e
ii
i
i
i
j=
(84)
i
2∆e
x
y
⌠ ⌠ N edxdy = ∆
e
(85)
i
⌡⌡
3
e
6∆e / 60
for i = j = k
∫∫ i j k
N e N e N e dxdy = 2∆e / 60
(86)
for i = j or j = k or k = i
∆e / 60
e
for i ≠ j ≠ k
NP
N iP
1
1
j
=- P ,
=
(87)
LP
s
L
s
30
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