Civil Engineering Reference
In-Depth Information
via the constraint equations. The left-hand side of Equation 7.126 is now, however,
quite different, in that the unknowns at each step
T
i
(
t
+
t
)
appear in both capac-
itance and conductance terms. Multiplying by
t
and rearranging Equation 7.126,
we obtain
[
C
]
[
K
]
t
+
{
T
(
t
+
t
)
}
2
[
C
]
F
Q
(
t
F
g
(
t
[
K
]
t
2
+
t
)
+
F
Q
(
t
)
+
t
)
+
F
g
(
t
)
=
−
{
T
(
t
)
}+
+
2
2
(7.127)
Equation 7.127 can be solved for the unknown nodal temperatures at time
t
+
t
and the “marching” solution can progress in time until a steady state is reached.
The central difference methods is, in general, more accurate than the forward or
backward difference method, in that it does not give preference to either temper-
atures at
t
or
t
+
t
but, rather, gives equal credence to both.
In finite difference methods, the key parameter governing solution accuracy
is the selected time step
t
. In a fashion similar to the finite element method, in
which the smaller the elements are, physically, the better is the solution, the finite
difference method converges more rapidly to the true solution as the time step
is decreased. These ideas are amplified in Chapter 10, when we examine the
dynamic behavior of structures.
7.9 CLOSING REMARKS
In Chapter 7, we expand the application of the finite element method into two-
and three-dimensional, as well as axisymmetric, problems in heat transfer. While
the majority of the chapter focuses on steady-state conditions, we also present
the finite difference methods commonly used to examine transient effects. The
basis of our approach is the Galerkin finite element method, and this text stays
with that procedure, as it is so general in application. As we proceed into appli-
cations in fluid mechanics, solid mechanics, and structural dynamics in the fol-
lowing chapters, the Galerkin method is the basis for the development of many
of the finite element models.
REFERENCES
1.
Huebner, K. H., and E. A. Thornton.
The Finite Element Method for Engineers,
2nd
ed. New York: John Wiley and Sons, 1982.
2.
Incropera, F. P., and D. P. DeWitt.
Introduction to Heat Transfer,
3rd ed. New York:
John Wiley and Sons, 1996.
