DFG
FAU Erlangen-Nuremberg

Multi-Scale Shape Optimization under Uncertainty

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Project leaders

Institut für Angewandte Mathematik
Universität Bonn

Tel: +49 (0)228 73-62211
Fax:

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Institut für Numerische Simulation
Uni Bonn

Tel: 0228 73-7866
Fax: 0228 73-9015

Email: martin.rumpf@ins.uni-bonn.de

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FB Mathematik, Gebäude LE
Uni Duisburg

Tel: 0203 379-1898
Tel: 0203 379-2687 (Sekr.)
Fax: 0203 379-3139

Email: schultz@math.uni-duisburg.de

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DFG funded assistants

Martin Pach (Uni Duisburg-Essen)
pach@math.uni-duisburg.de


Pradeep Atwal (Uni Bonn)
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Benedict Geihe (Uni Bonn)
benedict.geihe@ins.uni-bonn.de

Description

Risk aversion by stochastic dominance

Different concepts of stochastic dominance are becoming increasingly attractive in decision making under uncertainty. They allow for flexible risk aversion via comparison with benchmark random variables. Rather than handling risk aversion in the objective this enables risk aversion in the constraints. In our project this is another conceptual transfer of a paradigm from finite dimensional stochastic programming. Dominance constraints single out subsets of nonanticipative shapes which compare favorably to a chosen stochastic benchmark. The new class of stochastic shape optimization problems we plan to investigate arises by optimizing over such feasible sets.






Uncertain realization of the reference geometry


In many real world applications of stochastic shape optimization such as clothes design a nonanticipative prototype shape has to adapt to an uncertain realization of an elastic body. This is quantifiable with various risk measures which go beyond expected value minimization. We intend to apply a principal component analysis to reduce the computational complexity. Our implementation is based on a phase field description of the prototype shape.






Two scale shape optimization and microscopic topological modifications

So far we have studied stochastic shape optimization based on a full resolution of the geometry. In this setting topological modification turned out to be crucial which led us to adapting the concept of topological derivatives for stochastic cost functionals. We intend to advance this research in order to include correlation between separate nearby holes.

The same correlations will be studied by a two scale method in which they appear on the micro–scale. We will consider rigorous analysis of this approach and develop an efficient numerical method based on boundary elements on a parametric microscopic shape and macroscopic finite elements. Microstructures will consist of a regular lattice of holes with varying size and orientation. Concrete examples are two shells with perforated connecting material and a hard shell filled with porous material such as bones.

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