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Shape Optimization via Control of a Shape Function on a Fixed Domain: Theory and Numerical Results

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Numerical Methods for Differential Equations, Optimization, and Technological Problems

Part of the book series: Computational Methods in Applied Sciences ((COMPUTMETHODS,volume 27))

Abstract

We present a fixed-domain approach for the solution of shape optimization problems governed by linear or nonlinear elliptic partial differential state equations with Dirichlet boundary conditions, where shape optimization is facilitated via optimal control of a shape function. The method involves extending the state equation to a larger domain using regularization. Results regarding the convergence to the original problem are provided as well as differentiability properties of the control-to-state mappings. An algorithm for the numerical implementation of the method is stated and, in a series of numerical shape optimization experiments, the algorithm’s behavior is studied with regard to varying the regularization parameter and initial conditions.

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Notes

  1. 1.

    High Temperature Numerical Induction Heating Simulator; pronunciation: ∼hit-nice.

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Acknowledgements

The work of Dan Tiba was supported by CNCS Romania under Grant ID-PCE-2011-3-0211.

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Authors and Affiliations

  1. Department of Mathematics, Ludwig-Maximilians University (LMU) Munich, Theresienstrasse 39, 80333, Munich, Germany

    Peter Philip

  2. Institute of Mathematics, Romanian Academy, PO Box 1-764, 014700, Bucharest, Romania

    Dan Tiba

  3. Academy of Romanian Scientists, Bucharest, Romania

    Dan Tiba

Authors
  1. Peter Philip
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  2. Dan Tiba
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Correspondence to Peter Philip .

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Editors and Affiliations

  1. Mathematical Information Technology, University of Jyväskylä, Mattilanniemi 2, Jyväskylä, 40100, Finland

    Sergey Repin

  2. Mathematical Information Technology, University of Jyväskylä, Mattilanniemi 2, Jyväskylä, 40100, Finland

    Timo Tiihonen

  3. Mathematical Information Technology, University of Jyväskylä, Mattilanniemi 2, Jyväskylä, 40100, Finland

    Tero Tuovinen

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Philip, P., Tiba, D. (2013). Shape Optimization via Control of a Shape Function on a Fixed Domain: Theory and Numerical Results. In: Repin, S., Tiihonen, T., Tuovinen, T. (eds) Numerical Methods for Differential Equations, Optimization, and Technological Problems. Computational Methods in Applied Sciences, vol 27. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5288-7_16

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  • DOI: https://doi.org/10.1007/978-94-007-5288-7_16

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-007-5287-0

  • Online ISBN: 978-94-007-5288-7

  • eBook Packages: EngineeringEngineering (R0)

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