  This article is concerned with the problem of minimising the Willmore energy
in the class of \emph{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.
  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.
