Civil Engineering Reference
In-Depth Information
Plane of
symmetry
(a)
2
6
4
T
180
F
1
2
1
5
3
(b)
Figure 7.11
Model of Example 7.5,
showing (a) the plane of symmetry and
(b) a two-element model with adjusted
boundary conditions.
7.4.3 Symmetry Conditions
As mentioned previously in connection with Example 7.5, symmetry conditions
can be used to reduce the size of a finite element model (or any other computa-
tional model). Generally, the symmetry is observed geometrically; that is, the
physical domain of interest is symmetric about an axis or plane. Geometric sym-
metry is not, however, sufficient to ensure that a problem is symmetric. In addi-
tion, the boundary conditions and applied loads must be symmetric about the
axis or plane of geometric symmetry as well. To illustrate, consider Figure 7.12a,
depicting a thin rectangular plate having a heat source located at the geometric
center of the plate. The model is of a heat transfer fin removing heat from a cen-
tral source (a pipe containing hot fluid, for example) via conduction and convec-
tion from the fin. Clearly, the situation depicted is symmetric geometrically. But,
is the situation a symmetric problem? The loading is symmetric, since the heat
source is centrally located in the domain. We also assume that
k
x
k
y
so that the
material properties are symmetric. Hence, we must examine the boundary condi-
tions to determine if symmetry exists. If, for example, as shown in Figure 7.12b,
the ambient temperatures external to the fin are uniform around the fin and the
convection coefficients are the same on all surfaces, the problem is symmetric
about both
x
and
y
axes and can be solved via the model in Figure 7.12c. For this
situation, note that the heat from the source is conducted radially and, conse-
quently, across the
x
axis, the heat flux
q
y
is zero and, across the
y
axis, the heat
flux
q
x
must also be zero. These observations reveal the boundary conditions for
the quarter-symmetry model shown in Figure 7.12d and the internal forcing
function is taken as
Q
/
4
. On the other hand, let us assume that the upper edge of
the plate is perfectly insulated, as in Figure 7.12e. In this case, we do not have
=
