H. Gottschalk (Uni. Wuppertal)
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Mechanical components fail under cyclic loading due to fatigue of the
materials. 
This process is highly stochastic as the time to the formation of mesoscopic 
cracks is subject to a statistical scatter of one order of magniture in lab 
experiments. In this talk, the probabilistic crack formation is modeled by 
spatio-temporal point processes with intensity depending locally on the
mechanical load situation. This allows to compute failure probabilities at a 
given number of load cycles. A survey is given about experimental calibration 
and validation of these models, numerical implementation and their applications to gas turbine design.
In mechanical design, reliability is one of the main design objectives. The
mathematical quest thus is to find a shape for the component such that the
failure probability is minimized. This optimization task can be understood as
an optimal control problem with the elasticity PDE as the state equation and
the boundary of the shape as control parameter. A proof for the existence of
optimal shapes within certain regularity classes of the boundary is given using
strong solutions and uniform Schauder estimates.