Diffusion-redistanciation schemes for 2D and 3D constrained Willmore flow: application to the equilibrium shapes of vesicles

In this paper we present a novel algorithm for simulating the Willmore flow with conservation of volume and area. The idea is to adapt the class of diffusion-redistanciation algorithms to the Willmore flow or high-order geometrical flows and extend it in dimension three. These algorithms rely on alternating a diffusion of the signed distance function to the interface and a redistanciation step, following Merriman, Bence, and Osher's ideas. The constraints are enforced between the diffusion and redistanciation steps via a simple rescaling method. The energy globally decreases at the end of each global step. The algorithm features the computational efficiency of thresholding methods without requiring adaptive remeshing thanks to the use of a signed distance function to describe the interface. This opens its application to dynamic fluid-structure simulations. The methodology is validated by computing the equilibrium shapes of two and three-dimensional vesicles, as well as the Clifford torus.