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Algorithms for Optimal Control of Elastic Contact Problems with Finite Strain

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Non-Smooth and Complementarity-Based Distributed Parameter Systems

Part of the book series: International Series of Numerical Mathematics ((ISNM,volume 172))

Abstract

Optimal control of hyperelastic contact problems in the regime of finite strains combines various severe theoretical and algorithmic difficulties. Apart from being large scale, the main source of difficulties is the high nonlinearity and non-convexity of the elastic energy functional, which precludes uniqueness of solutions and simple local sensitivity results. In addition, the contact conditions add non-smoothness to the overall problem.

In this chapter, we discuss algorithmic approaches to address these issues. In particular, the non-smoothness is tackled by a path-following approach, whose theoretical properties are reviewed. The subproblems are highly nonlinear optimal control problems, which can be solved by an affine invariant composite step method. For increased robustness and efficiency, this method has to be adapted to the particular problem, taking into account its large-scale nature, its function space structure and its non-convexity.

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Acknowledgements

This work was supported by the DFG grant SCHI 1379/2-1 within the priority programme SPP 1962 (Non-smooth and Complementarity-Based Distributed Parameter Systems: Simulation and Hierarchical Optimization).

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Authors and Affiliations

  1. Mathematisches Institut, Universität Bayreuth, Bayreuth, Germany

    Anton Schiela & Matthias Stöcklein

Authors
  1. Anton Schiela
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  2. Matthias Stöcklein
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Corresponding author

Correspondence to Anton Schiela .

Editor information

Editors and Affiliations

  1. Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany

    Michael Hintermüller

  2. Institute for Applied Mathematics, University of Heidelberg, Heidelberg, Germany

    Roland Herzog

  3. Lehrstuhl für Mathematik VII, Julius-Maximilians-Universität Würzburg, Würzburg, Germany

    Christian Kanzow

  4. Department of Mathematics, Technical University of Munich, Garching b. München, Germany

    Michael Ulbrich

  5. Fachbereich Mathematik, Technische Universitaet Darmstadt, Darmstadt, Germany

    Stefan Ulbrich

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Schiela, A., Stöcklein, M. (2022). Algorithms for Optimal Control of Elastic Contact Problems with Finite Strain. In: Hintermüller, M., Herzog, R., Kanzow, C., Ulbrich, M., Ulbrich, S. (eds) Non-Smooth and Complementarity-Based Distributed Parameter Systems. International Series of Numerical Mathematics, vol 172. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-79393-7_14

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