Stable approximations for axisymmetric Willmore flow for closed and open surfaces

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