Game Development Reference
In-Depth Information
Figure 12-3.
The error function profile
For the 1-D head conduction equation, two boundary conditions are required. As an example,
let's assume we want to solve the 1-D heat conduction equation over a metal rod. One possible
boundary condition would be to maintain a constant temperature,
T
1
, at the end of the rod
where
x
= 0. If
x
= 0, then the argument to the error function in Equation (12.18) is zero, meaning
that the value of the error function itself is zero. The first term in Equation (12.18) disappears,
and the constant
B
is equal to
T
1
.
Tt
(0, )
==
T B
(12.20)
1
Another boundary condition can be obtained from the initial condition of the metal rod.
At time
t
= 0, the argument to the error function in Equation (12.18) is infinity, which means
that the value of the error function is 1. If the entire rod initially has a constant temperature
T
0
,
then Equation (12.18) reduces to the following:
Tx
(,0)
==
T
A
π
K
+
B
(12.21)
0
If the boundary conditions shown in Equations (12.20) and (12.21) are used together, an
expression can be found for the
A
constant.
AKTT
π
=−
(12.22)
0
1
Many other boundary condition combinations are possible depending on the situation
that is being modeled. The temperature at both ends of a rod might be specified. The temper-
ature derivative with respect to
x
might be specified at one or both ends. The temperature might
be held constant at some intermediate location along the length of the rod.
