Geometric flows and surface PDEs

Mean curvature flow (MCF)	Forced mean curvature flow	Willmore flow
Generalised mean curvature flow	MCF in in higher codimension	MCF–diffusion
Numerical surgery for MCF

PDEs with dynamic boundary conditions

Cahn-Hilliard equation with dynamic boundary conditions

Mean curvature flow

Numerical experiments for mean curvature flow algorithm from our paper.

A dumbbell-shaped surface developing a pinch singularity along the flow (with a zoom-in on the singularity).

Comparison of algorithms for mean curvature flow

Comparing mean curvature flow algorithms

A self-similar solution to mean curvature flow — Angenent's doughnut

Self-similar mean curvature flow for the Angenent torus

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):

Forced mean curvature flow modeling tumour growth (\gamma = 30)

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).

Willmore flow of a red blood cell

Willmore flow of a perturbed tours (top view)

Willmore flow of a perturbed torus towards a Clifford torus

Generalised mean curvature flow

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).

Inverse mean curvature flow of a dumbbell

A non-convex dumbbell-shaped surface evolving towards a pinch singularity along powers of mean curvature flow (V(H) = H^2)

Generalised mean curvature flow of a dumbbell

A genus-5 surface evolving under powers of inverse mean curvature flow (V(H) = - 1/H^2)

Generalised inverse mean curvature flow of a genus-5 surface

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.

Codimension 2 mean curvature flow of a trefoil knot (rescaled)

A trefoil knot evolving under mean curvature flow in codimension 2 (without rescaling)

Codimension 2 mean curvature flow of a trefoil knot

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 (University of Regensburg).

Mean curvature flow interacting with diffusion on an elongated ellipsoid.

Mean curvature flow interacting with diffusion - a singular flow

Mean curvature flow interacting with diffusion compared to pure mean curvature flow

Comparing mean curvature flow with diffusion-forced mean curvature flow

Numerical solutions for the mean curvature flow interacting with diffusion illustrating slow diffusion through a thin neck

Mean curvature flow interacting with diffusion - Ecker's example

Numerical solutions for the mean curvature flow interacting with diffusion illustrating the possibility for self-intersecting solutions

Mean curvature flow interacting with diffusion - a self intersecting flow

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.