Stable approximations for axisymmetric Willmore flow for closed and open surfaces

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.

52 pages, 19 figures

References in corpus (5)

• Parametric finite element approximations of curvature driven interface evolutions
• Finite element methods for fourth order axisymmetric geometric evolution equations
• Variational discretization of axisymmetric curvature flows
• Numerical approximation of curve evolutions in Riemannian manifolds
• Stable discretizations of elastic flow in Riemannian manifolds

Cited by in corpus (5)

• Volume-preserving parametric finite element methods for axisymmetric geometric evolution equations
• Numerical approximation of boundary value problems for curvature flow and elastic flow in Riemannian manifolds
• On the convergence of the Willmore flow with Dirichlet boundary conditions
• On the numerical approximation of hyperbolic mean curvature flows for surfaces
• Singularities of the hyperbolic elastic flow: Convergence, quantization and blow-ups