Codes for all publications are available via GitLab, or upon request.
Mean curvature flow
Numerical experiments for mean curvature flow algorithm from our paper.
A dumbbell-shaped surface developing a pinch singularity along the flow. Comparing Dziuk's algorithm to ours (without and with normalisation of the normal vector, and a zoom-in on the singularity):
An Angenent torus remaining self-similar as it numerically evolves under the mean curvature flow:
Forced mean curvature flow
Numerical experiments for a tumour growth model using the algorithm from the paper (with two different reaction parameters \(\gamma=30\) and \(300\), left and right):
Willmore flow
Some numerical experiments for Willmore flow using the algorithm analysed in the paper.
Topologically spherical objects are known to converge to a sphere, while minimising the Willmore energy \( ( W(\Gamma) \to 8\pi ) \):
Surfaces of genus 1 are converging to a Clifford torus minimising the Willmore energy \( ( W(\Gamma) \to 4\pi^2 ) \) (top and side view):
Generalised mean curvature flows
Numerical experiments for an algorithm for generalised mean curvature flows, i.e. the surface velocity is given by \(v = - V(H) \nu_\Gamma\).
A non-convex dumbbell-shaped surface evolving towards a singularity along inverse mean curvature flow (\(V(H) = - 1 / H\)):
The same surface evolving towards a pinch singularity along powers of mean curvature flow (\(V(H) = H^2\)):
A genus-5 surface evolving under powers of inverse mean curvature flow (\(V(H) = - 1 / H^2 \)):
Mean curvature flow in higher codimension
Numerical experiments for an algorithm for mean curvature flow in codimension two.
A trefoil knot evolving under mean curvature flow in codimension 2 (with and without rescaling):
Mean curvature flow interacting with diffusion
Numerical experiments for an algorithm for the interaction of mean curvature flow and diffusion on closed surfaces. This model was first developed and studied by Bürger (UR).
Mean curvature flow interacting with diffusion on an elongated ellipsoid, and a comparison with pure mean curvature flow.
Numerical solutions for the mean curvature flow interacting with diffusion illustrating slow diffusion through a thin neck (left), and the possibility for self-intersecting solutions (right).
Numerical surgery for mean curvature flow
Inspired by surgery process of Huisken & Sinestrari and Brendle & Huisken, this paper proposes a numerical mean curvature flow algorithm with surgery for surfaces.
Resolving a pinch singularity of a dumbbell-shaped surface, and a more complicated example for a torus and a sphere joined by a thin neck.