Temperature Diffusion
The unknown temperature
is a function of its location in the domain and time
. The governing equation
in the interior of the domain is given by
 |
(14) |
where
and
are given material constants. In case of a composite
material the parameters depend on their location in the domain.
is
a heat source (or sink) within the domain. We are using the Einstein summation convention
as introduced in Chapter 1.2. In our case we assume
to be equal to a constant heat production rate
on a circle or sphere with center
and radius
and 0 elsewhere:
 |
(15) |
for all
in the domain and all time
.
On the surface of the domain we are
specifying a radiation condition
which prescribes the normal component of the flux
to be proportional
to the difference of the current temperature to the surrounding temperature
:
 |
(16) |
is a given material coefficient depending on the material of the block and the surrounding medium.
is the
-th component of the outer normal field
at the surface of the domain.
To solve the time-dependent Equation (1.15) the initial temperature at time
has to be given. Here we assume that the initial temperature is the surrounding temperature:
 |
(17) |
for all
in the domain. It is pointed out that
the initial conditions satisfy the
boundary condition defined by Equation (1.17).
The temperature is calculated at discrete time nodes
where
and
where
is the step size which is assumed to be constant.
In the following the upper index
refers to a value at time
. The simplest
and most robust scheme to approximate the time derivative of the the temperature is
the backward Euler
scheme. The backward Euler
scheme is based
on the Taylor expansion of
at time
:
 |
(18) |
This is inserted into Equation (1.15). By separating the terms at
and
one gets for
 |
(19) |
where
is taken form the initial condition given by Equation (1.18).
Together with the natural boundary condition
 |
(20) |
taken from Equation (1.17)
this forms a boundary value problem that has to be solved for each time step.
As a first step to implement a solver for the temperature diffusion problem we will
first implement a solver for the boundary value problem that has to be solved at each time step.
esys@esscc.uq.edu.au