Stable approximations for axisymmetric Willmore flow for closed and open surfaces

TL;DR

This paper introduces stable numerical methods for simulating axisymmetric Willmore flow of surfaces, including boundary conditions and generalized energies, with proven stability and demonstrated effectiveness through numerical experiments.

Contribution

The paper develops novel stable numerical approximations for axisymmetric Willmore flow, including boundary conditions and generalized energy models, with theoretical stability proofs and practical demonstrations.

Findings

Proved stability of semidiscrete schemes.

Effective handling of boundary conditions.

Numerical experiments show robustness and efficiency.

Abstract

For a hypersurface in ${\mathbb R}^3$, Willmore flow is defined as the $L^2$--gradient flow of the classical Willmore energy: the integral of the squared mean curvature. This geometric evolution law is of interest in differential geometry, image reconstruction and mathematical biology. In this paper, we propose novel numerical approximations for the Willmore flow of axisymmetric hypersurfaces. For the semidiscrete continuous-in-time variants we prove a stability result. We consider both closed surfaces, and surfaces with a boundary. In the latter case, we carefully derive weak formulations of suitable boundary conditions. Furthermore, we consider many generalizations of the classical Willmore energy, particularly those that play a role in the study of biomembranes. In the generalized models we include spontaneous curvature and area difference elasticity (ADE) effects, Gaussian curvature and line energy contributions. Several numerical experiments demonstrate the efficiency and robustness of our developed numerical methods.