Abstract

This talk discusses the optimization for elastic materials and elastic microstructures under different and in particular stochastic loading scenarios.
To this end, on the one hand we transfers concepts from finite-dimensional stochastic programming to elastic shape optimization.
Thereby, the paradigm of stochastic dominance allows for flexible risk aversion via comparison with benchmark random variables,
Rather than handling risk aversion in the objective, this enables
risk aversion by including dominance constraints that single out subsets of
nonanticipative shapes which compare favorably to a chosen stochastic benchmark.

On the other hand, we investigate multiscale shape optimization using mechanically simple, parametrized microscopic
supporting structure those parameters have to be optimized.
An posteriori analysis of the discretization error and the modeling error is investigated
for a compliance cost functional in the context of the optimization of composite elastic materials
and a two-scale linearized elasticity model. This error analysis includes a control of the
modeling error caused when replacing an optimal nested laminate microstructure by this considerably simpler microstructure.

Furthermore, an elastic shape optimization problem with simultaneous and competitive optimization of domain and complement
is discussed. Such a problem arises in biomechanics where a bioresorbable polymer scaffold is implanted in
place of lost bone tissue and in a regeneration phase new bone tissue grows in the scaffold complement via osteogenesis.
In fact, the polymer scaffold should be mechanically stable to bear loading in the early stage regeneration phase
and at the same time the new bone tissue grown in the complement of this scaffold should as well bear the loading.

The talk is based on joint work with Sergio Conti, Patrick Dondl, Benedikt Geihe, Harald Held, Rüdiger Schultz,
Stefan Simon, and Sascha Tölkes.