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Phase Field Models for Thin Elastic Structures with Topological Constraint

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Abstract

This article is concerned with the problem of minimising the Willmore energy in the class of connected surfaces with prescribed area which are confined to a small container. We propose a phase field approximation based on De Giorgi’s diffuse Willmore functional to this variational problem. Our main contribution is a penalisation term which ensures connectedness in the sharp interface limit. The penalisation of disconnectedness is based on a geodesic distance chosen to be small between two points that lie on the same connected component of the transition layer of the phase field. We prove that in two dimensions, sequences of phase fields with uniformly bounded diffuse Willmore energy and diffuse area converge uniformly to the zeros of a double-well potential away from the support of a limiting measure. In three dimensions, we show that they converge \({\mathcal{H}^1}\)-almost everywhere on curves. This enables us to show \({\Gamma}\)-convergence to a sharp interface problem that only allows for connected structures. The results also imply Hausdorff convergence of the level sets in two dimensions and a similar result in three dimensions. Furthermore, we present numerical evidence of the effectiveness of our model. The implementation relies on a coupling of Dijkstra’s algorithm in order to compute the topological penalty to a finite element approach for the Willmore term.

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Authors and Affiliations

  1. Abteilung für Angewandte Mathematik, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Str. 10, 79104, Freiburg i. Br., Germany

    Patrick W. Dondl

  2. Université Paris Diderot, Paris 7, U.F.R de Mathématiques, Bâtiment Sophie Germain, 75205, Paris Cedex 13, France

    Antoine Lemenant

  3. Department of Mathematical Sciences, Durham University, Durham, DH1 1PT, UK

    Stephan Wojtowytsch

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  1. Patrick W. Dondl
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  2. Antoine Lemenant
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  3. Stephan Wojtowytsch
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Correspondence to Patrick W. Dondl.

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Communicated by G. Friesecke

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Dondl, P.W., Lemenant, A. & Wojtowytsch, S. Phase Field Models for Thin Elastic Structures with Topological Constraint. Arch Rational Mech Anal 223, 693–736 (2017). https://doi.org/10.1007/s00205-016-1043-6

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