Confined elastic curves
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Date
2010-10-08
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Abstract
We consider the problem of minimizing Euler's elastica energy for
simple closed curves confined to the unit disk. We approximate a simple closed
curve by the zero level set of a function with values +1 on the inside and -1 on the
outside of the curve. The outer container now becomes just the domain of the phase
field. Diffuse approximations of the elastica energy and the curve length are well
known. Implementing the topological constraint thus becomes the main difficulty
here. We propose a solution based on a diffuse approximation of the winding
number, present a proof that one can approximate a given sharp interface using
a sequence of phase fields, and show some numerical results using finite elements
based on subdivision surfaces.