Phase Field Models for Thin Elastic Structures with Topological Constraint
Résumé
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 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. For sequences of phase fields with bounded diffuse Willmore energy and bounded area term, we prove uniform convergence in two ambient space dimensions and a certain weak mode of convergence on curves in three dimensions. This enables us to show Γ-convergence to a sharp interface problem that only allows for connected structures. The topological contribution 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 furthermore 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.
Domaines
Mathématiques générales [math.GM]
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