Civil Engineering Reference
In-Depth Information
the triangular element become (assuming
Q
∗
to be constant)
Q
∗
N
1
(
x
0
,
y
0
)
N
2
(
x
0
,
y
0
)
N
3
(
x
0
,
y
0
)
f
(
e
Q
=
d
A
(7.57)
A
For a three-node triangular element, the interpolation functions (from Chapter 6)
are simply the area coordinates, so we now have
Q
∗
L
1
(
x
0
,
y
0
)
L
2
(
x
0
,
y
0
)
L
3
(
x
0
,
y
0
)
L
1
(
x
0
,
y
0
)
L
2
(
x
0
,
y
0
)
L
3
(
x
0
,
y
0
)
f
(
e
Q
=
Q
∗
A
d
A
=
(7.58)
A
Now consider the “behavior” of the area coordinates as the position of the inte-
rior point
P
varies in the element. As
P
approaches node 1, for example, area
coordinate
L
1
approaches unity value. Clearly, if the source is located at node 1,
the entire source value should be allocated to that node. A similar argument can
be made for each of the other nodes. Another very important point to observe
here is that the total heat generation as allocated to the nodes by Equation 7.58 is
equivalent to the source. If we sum the individual nodal contributions given in
Equation 7.58, we obtain
3
3
L
(
e
1
L
(
e
3
Q
∗
A
Q
∗
(
e
)
i
L
(
e
2
Q
∗
A
=
+
+
=
(7.59)
i
=
1
i
=
1
3
since
L
i
=
1
is known by the definition of area coordinates.
i
=
1
The foregoing approach using logic and our knowledge of interpolation
functions is without mathematical rigor. If we approach the situation of a line
source mathematically, the result is exactly the same as that given by Equa-
tion 7.58 for the triangular element. For any element chosen, the force vector
corresponding to a line source (keep in mind that, in two-dimensions, this looks
like a point source) the nodal force contributions are
f
(
e
Q
=
Q
∗
{
N
(
x
0
,
y
0
)
}
d
A
(7.60)
A
Thus, a source of internal heat generation is readily allocated to the nodes of a
finite element via the interpolation functions of the specific element applied.
7.5 HEAT TRANSFER WITH MASS TRANSPORT
The finite element formulations and examples previously presented deal with
solid media in which heat flows as a result of conduction and convection. An ad-
ditional complication arises when the medium of interest is a flowing fluid.
In such a case, heat flows by conduction, convection, and via motion of the
media. The last effect, referred to as
mass transport,
is considered here for the
