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{\large\bf Numerical Simulation of}\\[0.1cm]
{\large\bf Coupled Fluid-Solid Problems}\\[0.6cm]
{\em Michael~Sch\"afer}\\[0.4cm]
Department of Numerical Methods in Mechanical Engineering\\
Darmstadt University of Technology\\
Petersenstr.~30, D-64287 Darmstadt, Germany\\[0.8cm]
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The investigation and optimization of products or processes
in engineering practice by numerical simulation in most cases is
related to the solution of problems from structural mechanics, fluid mechanics or
heat transfer in the framework of continuum mechanics.
In many applications one is faced with a coupling
of phenomena from two or all three of these fields.
Examples of such (thermally) coupled fluid-solid problems
can be found, for instance, in
machine and plant building, turbomachinery, chemical engineering,
microsystem techniques, biology or medicine to mention only a few of
them.

A general view of the possible physical coupling
mechanisms is indicated schematically
in Fig.~\ref{fig1}.
In applications, four important levels of coupling can be
distinguished:
\begin{itemize}
\item deformations and stresses of solids induced by fluid forces
(pressure and friction),
\item fluid flow induced by change of flow geometry by prescribed
deformation,
\item deformations of solids induced by fluid forces interacting
with induced fluid flow by change of geometries by deformations of
solids
\item thermal coupling by temperature-dependent material properties,
buoyancy, convective heat transfer, thermal stresses
and mechanical dissipation.
\end{itemize}

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\put(50,54){Forces} \put(43,42){Deformations}
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\put(16,16){\shortstack{convective\\heat transfer}}
\put(83,16){\shortstack{thermal\\stresses}}
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\caption{Coupled fluid-solid problems.}
\label{fig1}
\end{figure}

Usually, already the numerical solution of individual fluid and
structural problems is a difficult task, such that
a realistic simulation of coupled processes is highly
pretentious to the underlying numerical methods and computational
resources.
Therefore, in order to allow for a reliable and efficient simulation
for such kind of applications, it is in particular important
to consider advanced numerical techniques
in combination with an efficient exploitation of computing power
provided by modern parallel computers.

In this lecture general concepts for the numerical treatment of
coupled problems are presented especially taking into
account aspects of accuracy and efficiency.
Of course, a crucial issue within any solution procedure for
coupled problems are the coupling mechanisms along the interfaces
between fluid and solid parts.
Efficient solvers for the fluid and solid subtasks, which due to
recent intensive research in this direction are available,
have to be combined to an overall coupled solution procedure
showing a similar performance.

While for fluid flow problems finite-volume methods are well established and have proven their efficiency,
their application for structural mechanics problems,
where finite-element approaches dominate, is very limited so far.
In the present contribution special emphasis is given to an
unified finite-volume approach for both the fluid and solid parts,
which appears to constitute a well suited base for designing efficient coupling algorithms.
Concerning the underlying continuum-mechanical model we consider
the fluid part to be described by the incompressible
Navier-Stokes equations together with the energy equation
and the solid part to be described by the equations of linear
thermoelasticity.

Coupling mechanisms can be invoked at different levels within
the numerical scheme resulting either in a more weakly or more
strongly coupled procedure.
In our combined finite-volume approach a strong coupling is
employed, where the thermal coupling is done within
the linear system solver by exchange of interfacial boundary data,
whereas deformations and fluid flow forces are coupled within
a pressure-correction method acting as a smoother
within a global nonlinear multigrid method.

Quite often different properties of problem regions also
result in different requirements with respect to numerical accuracy
for the individual problem variables.
In such cases it can be advantageous to consider different
numerical techniques for different subregions.
As an example of such an approach a combined
spectral/finite-volume method is presented,
where the fluid part is treated with a
spectral collocation method and the solid part with a finite-volume
method.

By considering some representative examples
with different levels of physical coupling mechanisms the
capabilities of the considered approaches
with respect to accuracy and efficiency are illustrated.
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