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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