Convergence of the penalty method applied to a constrained curve straightening flow

We apply the penalty method to the curve straightening flow of inextensible planar open curves generated by the Kirchhoff bending energy. Thus we consider the curve straightening flow of extensible planar open curves generated by a combination of the Kirchhoff bending energy and a functional penalizing deviations from unit arc-length.

We start with the governing equations of the explicit parametrization of the curve and derive an equivalent system for the two quantities indicatrix and arc-length. We prove existence and regularity of solutions and use the indicatrix/arc-length representation to compute the energy dissipation. We prove its coercivity and conclude exponential decay of the energy.

Finally, by an application of the Lions-Aubin Lemma, we prove convergence of solutions to a limit curve which is the solution of an analogous gradient flow on the manifold of inextensible open curves. This procedure also allows us to characterize the Lagrange multiplier in the limit model as a weak limit of force terms present in the relaxed model.

Keywords

curve straightening flow, energy dissipation, elastic regularization, curvature flow, penalty method

2010 Mathematics Subject Classification

35K91, 53C44, 92C10

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Published 7 February 2014