Dondl, Patrick W.; Mugnai, Luca; Röger, Matthias Confined elastic curves. (English) Zbl 1246.49036 SIAM J. Appl. Math. 71, No. 6, 2205-2226 (2011). 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. Cited in 14 Documents MSC: 49Q10 Optimization of shapes other than minimal surfaces 74G65 Energy minimization in equilibrium problems in solid mechanics Keywords:elastica energy; topological constraint; phase field; calculus of variations; subdivision finite elements PDFBibTeX XMLCite \textit{P. W. Dondl} et al., SIAM J. Appl. Math. 71, No. 6, 2205--2226 (2011; Zbl 1246.49036) Full Text: DOI arXiv