Elastic flow with junctions: Variational approximation and applications to nonlinear splines

摘要

We consider stable semidiscrete approximations of parameterized curve networksudfor gradient flows of elastic type functionals. Here meaningful and relevant conditionsudat junction points, such as double and triple junctions, need to be derivedudand suitably discretized. Examples for double junction types are C0 attachmentudand C1 continuity. We develop strong and weak formulations for the elastic flow forudcurve networks with such junction points. For junctions with three or more curvesudthe conditions at the junctions are derived here for the first time. Possible applicationsudinclude a simplified one-dimensional model of geometric biomembranes, asudwell as nonlinear splines in two and higher dimensions. The numerical results presentedudin this paper demonstrate the practicality of the introduced finite elementudapproximations.