Civil Engineering Reference
In-Depth Information
y
z
1
2
x
(a)
q
x
q
x
1
2
x
n
1
(1, 0, 0)
n
2
(
1, 0, 0)
(b)
Figure 7.17
(a) Common face in two 3-D elements. (b) Edge view of
common face, illustrating cancellation of conduction
gradient terms.
conditions. For such convection boundary conditions, the flux conditions of
Equation 7.82 must be in balance with the convection from the area of concern.
Mathematically, the condition is expressed as
A
A
f
(
e
q
=−
q
z
n
z
)[
N
]
T
q
n
n
[
N
]
T
(
q
x
n
x
+
q
y
n
y
+
d
A
=−
d
A
A
h
T
(
e
)
T
a
[
N
]
T
=−
−
d
A
(7.83)
where
q
n
is the flux normal to the surface area
A
of a specific element face on the
global boundary and
n
is the unit outward normal vector to that face. As in two-
dimensional analysis, the convection term in the rightmost integral of Equation
7.83 adds to the stiffness matrix when the expression for
T
(
e
)
in terms of inter-
polation functions and nodal temperatures is substituted. Similarly, the ambient
temperature terms add to the forcing function vector.
In most commercial finite element software packages, the three-dimensional
heat transfer elements available do
not
explicitly consider the gradient force vec-
tor represented by Equation 7.82. Instead, such programs compute the system
(global) stiffness matrix on the basis of conductance only and rely on the user to
specify the flux or convection boundary conditions (and the specified tempera-
ture conditions, of course) as part of the loading (input) data.
