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Confined elastic curves. (English) Zbl 1246.49036

Summary: 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.

MSC:

49Q10 Optimization of shapes other than minimal surfaces
74G65 Energy minimization in equilibrium problems in solid mechanics
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