\documentstyle[11pt,amssymb]{article}
\hoffset-1.5cm
\voffset-1.0cm
\textwidth16cm
\textheight22cm
\pagestyle{empty}

\begin{document}

\bibliographystyle{abbrv}

\begin{center}
{\large \bf
Efficient storage techniques for the radiation heat transfer equation}
\end{center}

\begin{center}
 {\small Mario Bebendorf,\ \underline{Sergej Rjasanow},\\
 Saarbr\"ucken, Germany}
\end{center}

The use of catalytic converters is an important method of reducing the
$CO, NO_x$ and various hydrocarbons in the exhaust of an
automobile engine. Due to current and certainly stricter future
government regulations for emission control
there is a continuous urgency to improve the efficiency.
Since this is excellent
if the temperature of the converter is high enough, the so-called warm-up
cycle should be reduced to obtain
more performance. A new concept of exhaust pipes using
insulated construction with a double wall has been in use for this purpose
during the last few years.
Practical experiments to find the optimal geometric
layout of exhaust construction are very expensive and rather difficult.
Mathematical modelling and effective numerical tools
can therefore be a solution for such complicated technical problems.

There are several physical processes in the exhaust pipe which must be modelled:
the flow of the exhaust gas in the inner pipe of the construction, heat transfer
due to conduction within the steel walls and heat transfer due to
radiation in the insulating split.

In \cite{1} we considered this problem for a piece of straight pipe.
This was a necessary simplification at this time to check the
functionality of the model. Now we
consider the much more complicated 3D geometry of the exhaust construction
and develop the corresponding numerical tools.
One of the most complicated problems here is the numerical
solution for the boundary integral equation of radiation heat transfer in the
insulating split.  The boundary element method (BEM) produces large
dense matrices especially for
complex geometries in 3D, where a large number of panels is needed to obtain
a sufficiently accurate approximation of the boundary.

For some special surfaces in 3D, for example for surfaces of
revolution, the BEM matrices take an exceptional, very special
circulant-block \cite{2} or Toeplitz-block structure \cite{1}. This structure
can be used efficiently for saving computer memory as well
as for the numerical treatment of the problem. In the above
application we find surfaces containing parts
which are rotational or which are close to rotational surfaces.
In such situations we can expect increase in efficiency
if we approximate the corresponding blocks by the sum of a
structured matrix and some rest, which can be approximated
by a low-rank matrix \cite{3}.
In some examples we show that the success of this approximation
particularly depends on the type of clustering of the panels.
A new hierarchical block-clustering will be introduced and discussed.

Finally we will present the idea of approximating a block of the
matrix using the sum of a low-rank matrix and a Toeplitz matrix.

\begin{thebibliography}{1}
\bibitem{1}
S.~Rjasanow.
\newblock Heat transfer in an insulated exhaust pipe.
\newblock {\em  Journal of Engineering Mathematics}
29:33-49, 1995
\bibitem{2}
S.~Rjasanow.
\newblock Effective algorithms with block circulant matrices.
\newblock {\em  Linear Algebra and its Applications}
202:55-69, 1994
\bibitem{3}
S.~A.~Goreinov, E.~E.~Tyrtyshnikov and A.~Yu.~Yeremin.
\newblock Matrix-free iteration solution strategies for large
dense linear systems.
\newblock {\em Numer. Linear Algebra Appl.} 4(5):1--22, 1996
\end{thebibliography}

\end{document}