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

Oelz, Dietmar (2014) Convergence of the penalty method applied to a constrained curve straightening flow. Communications in Mathematical Sciences, 12 4: 601-621. doi:10.4310/CMS.2014.v12.n4.a1


Author Oelz, Dietmar
Title Convergence of the penalty method applied to a constrained curve straightening flow
Journal name Communications in Mathematical Sciences   Check publisher's open access policy
ISSN 1539-6746
1945-0796
Publication date 2014-01-01
Sub-type Article (original research)
DOI 10.4310/CMS.2014.v12.n4.a1
Open Access Status Not yet assessed
Volume 12
Issue 4
Start page 601
End page 621
Total pages 21
Place of publication Somerville, MA, United States
Publisher International Press
Language eng
Abstract 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.
Keyword Curve straightening flow
Energy dissipation
Elastic regularization
Curvature flow
Penalty method
Q-Index Code C1
Q-Index Status Provisional Code
Institutional Status Non-UQ

Document type: Journal Article
Sub-type: Article (original research)
Collection: School of Mathematics and Physics
 
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Created: Sat, 14 Jan 2017, 02:12:18 EST by Kay Mackie on behalf of School of Mathematics & Physics