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Two-Stage Stochastic Optimization Meets Two-Scale Simulation

  • Sergio Conti
  • Benedict GeiheEmail author
  • Martin Rumpf
  • Rüdiger Schultz
Chapter
Part of the International Series of Numerical Mathematics book series (ISNM, volume 165)

Abstract

Risk averse stochastic optimization is investigated in the context of elastic shape optimization, allowing for microstructures in the admissible shapes. In particular, a two-stage model for shape optimization under stochastic loading with risk averse cost functionals is combined with a two-scale approach for the simulation of microstructured materials. The microstructure is composed of an elastic material with geometrically simple perforations located on a regular periodic lattice. Different types of microscopic geometries are investigated and compared to each other. In addition they are compared to optimal nested laminates, known to realize the optimal lower bound of compliance cost functionals. We combine this two-scale approach to elastic shapes with a two-stage stochastic programming approach to risk averse shape optimization, dealing with risk neutral and risk averse cost functionals in the presence of stochastic loadings.

Keywords

Two-stage stochastic programming Risk averse optimization Two-scale elastic shape optimization Microstructure optimization Finite element method Boundary element method 

Mathematics Subject Classification (2010)

90C15 74B05 74P05 74Q05 74S05 74S15 49M29 

Notes

Acknowledgements

The authors would like to thank Martin Lenz for help with the boundary element method applied to microscopic cell problems. This work was supported by the Deutsche Forschungsgemeinschaft through the Schwerpunktprogramm 1253 Optimization with Partial Differential Equations.

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Sergio Conti
    • 1
  • Benedict Geihe
    • 2
    Email author
  • Martin Rumpf
    • 2
  • Rüdiger Schultz
    • 3
  1. 1.Institut für Angewandte MathematikUniversität BonnBonnGermany
  2. 2.Institut für Numerische SimulationUniversität BonnBonnGermany
  3. 3.Fakultät für MathematikUniversität Duisburg-EssenEssenGermany

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