Game Development Reference
In-Depth Information
The 1-D heat conduction equation can be written in terms of the variable
u
as a second-
order ODE.
2
udT
d T
−=
K
(12.15)
2
2
du
du
Without going into all of the details, the solution to Equation (12.15) is an integral equation
in terms of
u
and the thermal diffusivity,
K
.
u
2
u
2
−
∫
T x t
(,)
=
A
e
du
+
B
4
K
(12.16)
0
The quantities
A
and
B
are constants that are determined according to the boundary
conditions that are applied to the system. We'll talk about boundary conditions a bit more later
on in the section. At first glance, it doesn't seem as if we have made any progress, because we
still have to solve a nasty-looking integral. Fortunately for us, we can make use of a similarly
nasty-looking but well-known integral known as the
error function
.
x
2
2
∫
−
x
erf
()
x
=
e
dx
(12.17)
π
0
The error function is important in this case because it can be used to solve the integral
expression shown in Equation (12.16). With a little rearranging, Equation (12.16) can be expressed
in terms of the error function,
erf
.
⎛
⎞
⎛
⎞
u
x
⎟
⎜
⎟
⎜
T x t
(,)
=
A
π
Kerf
+
B
=
A
π
Kerf
+
B
⎟
⎟
⎜
⎜
(12.18)
⎟
⎟
⎜
⎜
⎟
⎝
⎠
⎝
⎠
2
K
2
Kt
The error function itself has been calculated and tabulated since the mid-1800s. It can be
approximated by a series equation.
⎛
⎞
n
()
(
21
n
+
2
∞
−
1
x
⎟
⎜
∑
⎟
=+
erf
()
x
x
⎟
(12.19)
⎜
⎟
⎜
)
21!
nn
+
⎟
π
⎝
⎠
n
=
1
The error function can be obtained by computing a certain number of the terms from
Equation (12.19), although higher values of
x
(1.5-2.0) will require the evaluation of at least
eight terms in the series. As an alternative, the error function could be tabulated for incre-
mental values of
x
, and you could use a table lookup method to find the value for a given
x
.
The error function profile is shown in Figure 12-3. The curve has an exponential-type
shape. At values of
x
above 2.0, the value of the error function is very close to 1.
The only thing left to do is to determine the constants
A
and
B
. These will depend on what
boundary conditions are applied to the system. If you recall from Chapter 2, oftentimes there
will be an infinite number of possible solutions to a differential equation. In order to “pin
down” the solution to the one that we want, the conditions need to be specified at one or both
ends of the object being analyzed.
