Civil Engineering Reference
In-Depth Information
such that the temperature distribution in the element is described by
M
T
(
x
,
y
)
=
N
i
(
x
,
y
)
T
i
=
[
N
]
{
T
}
(7.24)
i
=
1
where
N
i
(
x
,
y
)
is the interpolation function associated with nodal temperature
T
i
,[
N
]
is the row matrix of interpolation functions, and
{
T
}
is the column matrix
(vector) of nodal temperatures.
Applying Galerkin's finite element method, the residual equations corre-
sponding to Equation 7.23 are
N
i
(
x
,
y
)
∂
∂
tk
x
∂
tk
y
∂
T
a
)
d
A
T
∂
∂
T
+
+
Qt
−
2
h
(
T
−
=
0
x
∂
x
y
∂
y
A
1,
M
(7.25)
where thickness
t
is assumed constant and the integration is over the area of the
element. (Strictly speaking, the integration is over the volume of the element,
since the volume is the domain of interest.) To develop the finite element
equations for the two-dimensional case, a bit of mathematical manipulation is
required.
Consider the first two integrals in Equation 7.25 as
t
i
=
∂
∂
k
x
∂
N
i
+
k
y
∂
N
i
d
A
T
∂
∂
T
x
∂
x
y
∂
y
A
∂
N
i
d
A
t
q
x
∂
N
i
+
∂
q
y
∂
=−
(7.26)
x
y
A
and note that we have used Fourier's law per Equation 7.21. For illustration, we
now assume a rectangular element, as shown in Figure 7.7a, and examine
t
y
2
x
2
∂
q
x
∂
∂
q
x
∂
N
i
d
A
=
t
N
i
d
x
d
y
(7.27)
x
x
A
y
1
x
1
y
(
x
1
,
y
2
)
(
x
2
,
y
2
)
a
b
x
q
x
(
x
1
,
y
)
q
x
(
x
2
,
y
)
(
x
1
,
y
1
)
(
x
2
,
y
1
)
a
b
(a)
(b)
Figure 7.7
Illustration of boundary heat flux in
x
direction.
