Mathematical Programming

, Volume 141, Issue 1–2, pp 383–403 | Cite as

Risk averse elastic shape optimization with parametrized fine scale geometry

  • Benedict Geihe
  • Martin Lenz
  • Martin Rumpf
  • Rüdiger SchultzEmail author
Full Length Paper Series A


Shape optimization of the fine scale geometry of elastic objects is investigated under stochastic loading. Thus, the object geometry is described via parametrized geometric details placed on a regular lattice. Here, in a two dimensional set up we focus on ellipsoidal holes as the fine scale geometric details described by the semiaxes and their orientation. Optimization of a deterministic cost functional as well as stochastic loading with risk neutral and risk averse stochastic cost functionals are discussed. Under the assumption of linear elasticity and quadratic objective functions the computational cost scales linearly in the number of basis loads spanning the possibly large set of all realizations of the stochastic loading. The resulting shape optimization algorithm consists of a finite dimensional, constraint optimization scheme where the cost functional and its gradient are evaluated applying a boundary element method on the fine scale geometry. Various numerical results show the spatial variation of the geometric domain structures and the appearance of strongly anisotropic patterns.

Mathematics Subject Classification

90C15 Stochastic Programming 65N38 Boundary element methods 65K10 Optimization and variational techniques 


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Copyright information

© Springer and Mathematical Optimization Society 2012

Authors and Affiliations

  • Benedict Geihe
    • 1
  • Martin Lenz
    • 1
  • Martin Rumpf
    • 1
  • Rüdiger Schultz
    • 2
    Email author
  1. 1.Institute for Numerical SimulationRheinische Friedrich-Wilhelms-Universität BonnBonnGermany
  2. 2.Fakultät für MathematikUniversität Duisburg-EssenDuisburgGermany

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