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%{\Large Delft University of Technology}\\
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{\large Faculty of Electrical Eng, Mathematics and Computer Science}\\
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        {\bf Computer Assignment 1.1.1} \\
        {\bf Numerical Analysis CII (wi4014)} \\
        {\bf Numerical Methods PDE (wi3001)} 
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The sun shines on the toroidal object in the figure causing a heat flow
into the object. The objective of this assignment is to calculate the
thermal equilibrium in the object. Except for the solar radiation there
are no heat sources.\\
The mathematical formulation of this problem is: 
\begin{equation}
- \Delta T = 0
\label{een}
\end{equation}
\begin{equation}
\frac{\displaystyle \partial T}{\displaystyle \partial n} + \alpha T =
f\;\; \;
\mbox{\rm \ on the boundary}
\label{twee}
\end{equation}

In this equation $T = (T_{torus} - T_{env}) * $ radiation constant, in
which $T_{env}$ is the environment temperature.\\
$f$ represents the heat source and $\alpha$ is a material
constant.\\
The main radius of the torus is $R$, the secondary radius $r$.\\
The source term $f$ is given by:
\[f = \left\{\begin{array}{ccc}
\sin \phi &\mbox{\rm \ if} &0 < \phi < \pi\\
0 &\mbox{\rm \ if} &\pi < \phi < 2\pi
\end{array}
\right.\]
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\\[2ex]
Assignment:
\begin{itemize}
\item[a.] Derive the minimization problem for (\ref{een}) with boundary
condition (\ref{twee}).
\item[b.] Which symmetries can be used in solving this problem?
\item[c.]
To approximate the solution to this minimization problem we use the
Finite Element Method. The elements in the cross section have triangular
shape and the approximation is linear. Derive the element matrices and vectors for the
internal elements. \\
Derive element matrices and vectors for the boundary elements..
\item[d.]Solve the problem with the SEPRAN package.\\
Use $r = 1, R = 2.$\\
Choose $\alpha = 1$ and solve the problem for two different grid
sizes.\\
Print the temperatures and make a contour plot.
\item[e.] Explain qualitatively the main characteristics of the contour
plot.\\
\item[f.] Perform the same calculations with $\alpha = 100$.\\
Explain why temperatures are lower in this case compared to d.
\end{itemize}

{\bf Remark}\\
The assignment has to be submitted in the form of a report
providing mathematical background, derivation and results. The above 
questions  may be used as guidance to make the report, but a report should
contain more than merely a list of answers to the questions.

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