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Approximation of an Elastic Rod with Self-Contact

Impenetrable Elastic Rod

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Abstract

The variational formulation of an elastic rod with an impenetrable surface surrounding the centerline corresponds to a nonsmooth optimization of cost functional \(\mathcal {J}\) with nonconvex inequality constraints and so presents many analytical and computational challenges in approximating the minima. We construct a sequence of approximate variational cost functionals \(\mathcal {J}_k\), corresponding to elastic rods with infinite energy barriers that enforce impenetrability constraints. Using this construction, we show strong convergence of minimizing configurations of \(\mathcal {J}_k\) to the minimizer of \(\mathcal {J}\) and weak* convergence in \([C(0,1)]^*\) of contact forces induced by the repulsive potential to the contact forces of the minimizing configurations of \(\mathcal {J}\).

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Acknowledgements

The authors thank Stuart Antman and Boris Mordukhovich for fruitful conversations.

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The authors did not receive support from any organization for the submitted work.

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Authors and Affiliations

  1. University of Maryland, Baltimore County, Baltimore, MD, 21250, USA

    Kathleen A. Hoffman & Thomas I. Seidman

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  1. Kathleen A. Hoffman
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  2. Thomas I. Seidman
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Correspondence to Kathleen A. Hoffman.

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Communicated by Irena Lasiecka.

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Hoffman, K.A., Seidman, T.I. Approximation of an Elastic Rod with Self-Contact. J Optim Theory Appl 192, 1001–1021 (2022). https://doi.org/10.1007/s10957-022-02002-5

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  • DOI: https://doi.org/10.1007/s10957-022-02002-5

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