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Constrained Optimization and Optimal Control for Partial Differential Equations pp 215–243Cite as

Birkhäuser

On Shape Optimization with Stochastic Loadings

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Part of the book series: International Series of Numerical Mathematics ((ISNM,volume 160))

Abstract

This article is concerned with different approaches to elastic shape optimization under stochastic loading. The underlying stochastic optimization strategy builds upon the methodology of two-stage stochastic programming. In fact, in the case of linear elasticity and quadratic objective functionals our strategy leads to a computational cost which scales linearly in the number of linearly independent applied forces, even for a large set of realizations of the random loading. We consider, besides minimization of the expectation value of suitable objective functionals, also two different risk averse approaches, namely the expected excess and the excess probability. Numerical computations are performed using either a level set approach representing implicit shapes of general topology in combination with composite finite elements to resolve elasticity in two and three dimensions, or a collocation boundary element approach, where polygonal shapes represent geometric details attached to a lattice and describing a perforated elastic domain. Topology optimization is performed using the concept of topological derivatives. We generalize this concept, and derive an analytical expression which takes into account the interaction between neighboring holes. This is expected to allow efficient and reliable optimization strategies of elastic objects with a large number of geometric details on a fine scale.

Mathematics Subject Classification (2000). 90C15, 74B05, 65N30, 65N38, 34E08, 49K45.

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Author information

Authors and Affiliations

  1. Institut für Angewandte Mathematik, Universität Bonn, Endenicher Allee 60, D-53115, Bonn, Germany

    Pradeep Atwal & Sergio Conti

  2. Institut für Numerische Simulation, Universität Bonn, Endenicher Allee 60, D-53115, Bonn, Germany

    Benedict Geihe & Martin Rumpf

  3. Martin Pach and Rüdiger Schultz Fakultät für Mathematik, Universität Duisburg-Essen, Forsthausweg 2, D-47057, Duisburg, Germany

    Martin Pach & Rüdiger Schultz

Authors
  1. Pradeep Atwal
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  2. Sergio Conti
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  3. Benedict Geihe
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  4. Martin Pach
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  5. Martin Rumpf
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  6. Rüdiger Schultz
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Editor information

Editors and Affiliations

  1. , Department Mathematik, Universität Erlangen-Nürnberg, Martensstr. 3, Erlangen, 91058, Germany

    Günter Leugering

  2. , Fakultät Bio- und Chemieingenieurwesen, Technische Universität Dortmund, Geschossbau 2, Dortmund, 44221, Germany

    Sebastian Engell

  3. , Institut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25, Berlin, 12489, Germany

    Andreas Griewank

  4. , Fachbereich Mathematik, Universität Hamburg, Bundesstr. 55, Hamburg, 20146, Germany

    Michael Hinze

  5. , Institut für Angewandte Mathematik, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 293, Heidelberg, 69120, Germany

    Rolf Rannacher

  6. , FB 4 - Department Mathematik, Universität Trier, Trier, 54286, Germany

    Volker Schulz

  7. Fakultät für Mathematik, M1, Lehrstuhl für Mathematische Optimierung, Technische Universität München, Boltzmannstr. 3, Garching bei München, 85748, Germany

    Michael Ulbrich

  8. , Fachbereich Mathematik, Technische Universität Darmstadt, Dolivostrasse 15, Darmstadt, 64293, Germany

    Stefan Ulbrich

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Atwal, P., Conti, S., Geihe, B., Pach, M., Rumpf, M., Schultz, R. (2012). On Shape Optimization with Stochastic Loadings. In: Leugering, G., et al. Constrained Optimization and Optimal Control for Partial Differential Equations. International Series of Numerical Mathematics, vol 160. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0133-1_12

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