Confined elastic curves

TL;DR

This paper develops a phase field method to approximate and minimize Euler's elastica energy for closed curves confined within a disk, addressing topological constraints with diffuse winding number approximation.

Contribution

It introduces a diffuse approximation of the winding number to handle topological constraints in elastica energy minimization for closed curves.

Findings

Successful numerical implementation using finite elements and subdivision surfaces.

Approximation of sharp interfaces by phase fields with proven convergence.

Effective handling of topological constraints in curve optimization.

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.