Optimal control of volume-preserving mean curvature flow

Abstract

We develop a framework and numerical method for controlling the full space-time tube of a geometrically driven flow. We consider an optimal control problem for the mean curvature flow of a curve or surface with a volume constraint, where the control parameter acts as a forcing term in the motion law. The control of the trajectory of the flow is achieved by minimizing an appropriate tracking-type cost functional. The gradient of the cost functional is obtained via a formal sensitivity analysis of the space-time tube generated by the mean curvature flow. We show that the perturbation of the tube may be described by a transverse field satisfying a parabolic equation on the tube. We propose a numerical algorithm to approximate the optimal control and show several results in two and three dimensions demonstrating the efficiency of the approach.

Pre-shape calculus and its application to mesh quality optimization

“…Several strategies for increasing mesh quality, not based on mesh deformation methods mentioned or Laplacian smoothing, exist. In the context of shape optimization and -morphing, these include correcting for errors, frequently arising in discretizations of Hadamard's theorem (Etling et al, 2018), adding non-linear advection terms in shape gradient representations (Onyshkevych and Siebenborn, 2020), approximating shape morphing by volume-preserving mean-curvature flows (Laurain and Walker, 2020), and use of techniques related to centroidal Voronoi reparameterization in combination with eikonal equation based non-linear advection terms for representation of shape gradients (Schmidt, 2014).…”

Pre-Shape Calculus: Foundations and Application to Mesh Quality Optimization

“…Several strategies for increasing mesh quality not based on mesh deformation methods mentioned or Laplacian smoothing exist. In the context of shape optimization and -morphing, these include correcting for errors in Hadamard's theorem due to discretization [16], adding non-linear advection terms in shape gradient representations [36], approximating shape morphing by volume-preserving mean-curvature flows [28], and use of techniques related to Centroidal Voronoi Reparameterization in combination with Eikonal equation based non-linear advection terms for representation of shape gradients [40].…”