Civil Engineering Reference
In-Depth Information
the element force vector representing internal heat generation is
f
(
e
Q
=
Q
[
N
]
T
d
V
(7.81)
V
and the element nodal force vector associated with heat flux across the element
surface area is
A
f
(
e
q
q
z
n
z
)[
N
]
T
(7.82)
=−
(
q
x
n
x
+
q
y
n
y
+
d
A
7.6.1 System Assembly and Boundary Conditions
The procedure for assembling the global equations for a three-dimensional
model for heat transfer analysis is identical to that of one- and two-dimensional
problems. The element type is selected (tetrahedral, brick, quadrilateral solid,
for example) based on geometric considerations, primarily. The volume is then
divided into a mesh of elements by first defining nodes (in the global coordinate
system) throughout the volume then each element by the sequence and number
of nodes required for the element type. Element-to-global nodal correspondence
relations are then determined for each element, and the global stiffness (con-
ductance) matrix is assembled. Similarly, the global force vector is assembled
by adding element contributions at nodes common to two or more elements.
The latter procedure is straightforward in the case of internal generation, as
given by Equation 7.81. However, in the case of the element gradient terms,
Equation 7.82, the procedure is best described in terms of the global boundary
conditions.
In the case of three-dimensional heat transfer, we have the same three types
of boundary conditions as in two dimensions: (1) specified temperatures,
(2) specified heat flux, and (3) convection conditions. The first case, specified
temperatures, is taken into account in the usual manner, by reducing the system
equations by simply substituting the known nodal temperatures into the system
equations. The latter two cases involve only elements that have surfaces (element
faces) on the outside surface of the global volume. To illustrate, Figure 7.17a
shows two brick elements that share a common face in an assembled finite ele-
ment model. For convenience, we take the common face to be perpendicular to
the
x
axis. In Figure 7.17b, the two elements are shown separately with the asso-
ciated normal vector components identified for the shared faces. For steady-state
heat transfer, the heat flux across the face is the same for each element and, since
the unit normal vectors are opposite, the gradient force terms cancel. The result
is completely analogous to internal forces in a structural problem via Newton's
third law of action and reaction. Therefore, on interelement boundaries (which
are areas for three-dimensional elements), the element force terms defined by
Equation 7.82 sum to zero in the global assembly process.
What of the element surface areas that are part of the surface area of the vol-
ume being modeled? Generally, these outside areas are subjected to convection
