  This paper addresses the approximation of the mean curvature flow of thin
structures for which classical phase field methods are not suitable. By thin
structures, we mean surfaces that are not domain boundaries, typically higher
codimension objects such as 1D curves in 3D, i.e. filaments, or soap films
spanning a boundary curve. To approximate the mean curvature flow of such
surfaces, we consider a small thickening and we apply to the thickened set an
evolution model that combines the classical Allen-Cahn equation with a penalty
term that takes on larger values around the skeleton of the set. The novelty of
our approach lies in the definition of this penalty term that guarantees a
minimal thickness of the evolving set and prevents it from disappearing
unexpectedly. We prove a few theoretical properties of our model, provide
examples showing the connection with higher codimension mean curvature flow,
and introduce a quasi-static numerical scheme with explicit integration of the
penalty term. We illustrate the numerical efficiency of the model with accurate
approximations of filament structures evolving by mean curvature flow, and we
also illustrate its ability to find complex 3D approximations of solutions to
the Steiner problem or the Plateau problem.
