Stable approximations for axisymmetric Willmore flow for closed and open
surfaces
surfaces
File(s)
Working paper
Author(s)
Barrett, John W
Garcke, Harald
Nürnberg, Robert
Type
Working Paper
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 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.
$L^2$--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 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.
Date Issued
2019-11-04
Citation
2019
Publisher
arXiv
Copyright Statement
© 2019 The Author(s)
Identifier
http://arxiv.org/abs/1911.01132v1
Subjects
math.NA
math.NA
cs.NA
65M60, 65M12, 35K55, 53C44
Notes
66 pages, 13 figures
Publication Status
Published