Rechnen für langlebige Produkte : Optimierung von mechanischen Schädigungsprozessen
Mechanische Bauteile unterliegen äußeren Einwirkungen – Druck- und Zugkräfte verändern ihre Form und Eigenschaften. Neben Verformungen kann es dabei auch zu Schädigungen des Materials kommen. In der Praxis wird häufig ein gewisser Schädigungsgrad hingenommen, da schädigungsfreie Bauteile zu schwer, zu teuer oder nicht realisierbar sind, wie beispielsweise im Flugzeugbau. Umso wichtiger ist es, den Umfang der sich einstellenden Schädigung genau vorhersagen und kontrollieren zu können, um ein gefährliches Versagen des Bauteils auszuschließen.
Mechanical devices are deformed
by forces. Forces cause elastic and
plastic deformations and damage
of the material. In our project
“Optimal control of mechanical
damage processes” we investigated
the optimization of loading forces
for mechanical processes in a new
damage model.
The damage is described by two
variables. The first one describes the
spatial behavior of the damage and
the second one the temporal behavior. Both variables are coupled in
an objective by a penalty term. An
essential feature of damage models
is that healing of the material is not
possible, i.e., damage is a monotone
increasing function in time.
The monotonicity requirement
of the damage variable leads to theoretical
and numerical challenges. The
main problem is that the dependence
of the solution of the damage model
with respect to optimization variables
is not differentiable. The optimization
theory and also the numerical
approach can only work with
directional derivatives.
Nevertheless, the new model can
be formulated in different mathematical
ways. It allows a rigorous
mathematical theory and the development
of fast and stable numerical
methods. The finite dimensional
optimization problems appearing
after discretization are large scaled,
i.e., 106 to 109 optimization variables.
Solving of such huge optimization
problems requires modern
and innovative techniques which are
developed in our research groups.
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