  In this paper we present a novel algorithm for simulating geometrical flows,
and in particular the Willmore flow, with conservation of volume and area. The
idea is to adapt the class of diffusion-redistanciation algorithms to the
Willmore flow in both two and three dimensions. These algorithms rely on
alternating diffusions of the signed distance function to the interface and a
redistanciation step, and with careful choice of the applied diffusions, end up
moving the zero level-set of the distance function by some geometrical quantity
without resorting to any explicit transport equation. 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
algorithms feature the computational efficiency of thresholding methods without
requiring any adaptive remeshing thanks to the use of a signed distance
function to describe the interface. This opens their application to dynamic
fluid-structure simulations for large and realistic cases. The methodology is
validated by computing the equilibrium shapes of two- and three-dimensional
vesicles, as well as the Clifford torus.
