Codes for all publications are available via GitLab, or upon request.
Mean curvature flow
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
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 \( ( W(\Gamma) \to 8\pi ) \):
Generalised mean curvature flows
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
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.