Introduction

The Willmore energy of a closed embedded surface is given by the surface integral of the square of its mean curvature vector. It has been studied as an invariant in conformal geometry, the rate of change of area under mean curvature flow, and as a model for the bending energy of thin sheets. I am primarily interested in minimizing Willmore's energy in the class of surfaces satisfying an area constraint which are embedded into a given domain. This problem is motivated from biological observations on mitochondria where a membrane with large surface area bends to fit into a small container. The membranes are liquid (so no reference metric is given) but inextensible (so the area constraint is enforced) and connected. 

Phase-field Models

The minimization problem is difficult because of the geometric nature of the energy which does not control reparametrization of surfaces, and because of the non-local surface area and embeddedness constraints. The extrinsic nature of the energy suggests an extrinsic approach, and the existence of minimizers in a class of generalized surfaces known as varifolds can be proved using geometric measure theory.

Mimicking the extrinsic view point in the computational approach, we choose a phase-field approximation of Willmore's energy which has the added benefit of converting a quasi-linear geometrically degenerate problem into a family of semi-linear uniformly parabolic ones. Essentially, this means that we approximate the characteristic function of the set enclosed by a surface by smooth functions which make the transition from 0 to 1 on a small length-scale ε>0 and find suitable approximations of the energy and area functionals. The surface can be thought of as the level set of 1/2. A naive gradient flow approach, however, fails, since the topological constraint of remaining connected is not incorporated in the energy:

gradient flow without topological constraint in 3d

In this simulation, we included a non-zero spontaneous curvature to force the break up into two components - other biological processes incorporated into the model may have the same effect, and biological membranes are expected to have spontaneous curvature. The surface is coloured by mean curvature. Similar problems occur in two dimensions without extra contributions to the energy. Here, the colours denote the phase parameter.

gradient flow without topological constraint in 2d

Together with Patrick Dondl I developed a quantitative notion of path-connectedness for the transition layers of phase-fields which could be used to penalize membranes breaking up into several component [1]. The quantity is given by a weighted double integral of a geodesic distance function where both the weights in the integral and the metric for computing the geodesic distance are related to the phase parameter. An efficient implementation using Dijkstra's algorithm for the computation of the distance function is described in [6].

To prove that the penalization method asymptotically enforces connectedness, we need a strong kind of convergence of the phase function away from the support of a limiting measure. While uniform convergence is sufficient, phase fields do not necessarily converge uniformly in this problem, and convergence in all finite Lp-spaces is not sufficient. In [2] we introduce a new kind of convergence which we call essentially uniform convergence and prove that phase functions converge essentially uniformly to the potential wells away from the support of the limiting measure. Basically, this means that for any given positive amount 𝛿, uniform convergence can only fail by 𝛿 close to finitely many points (depending on 𝛿).

Simulations confirm the effectiveness of our method and demonstrate that little extra time is spent enforcing the topological constraint:

gradient flow with topological constraint in 3d

gradient flow with topological constraint in 2d

We obtained further results on this and related topics.
  • In [3], we show that boundary values play a central role when using the phase-field approximation for Willmore's energy and construct unphysical energy minimizers if the boundary values are not strongly controlled.
  • In [4], I show that it is not possible to control the genus of energy minimizers from energy bounds alone even in the sharp interface problem and thus that the topological control we obtained is optimal.
  • In [5], we modify the method of enforcing connectedness to approximate the relaxation of the perimeter functional under a connectedness constraint in two dimensions and apply it to image segmentation under connectedness constraints.
  • In [7], I obtain an energy expansion of the same bending energy functional on curves which are confined to the unit disk and have length slightly larger than the unit circle. I prove that the highest order term can be obtained by solving a fourth order semi-linear obstacle type problem on the real line and obtain an explicit solution. I then determine (to highest order) the explicit excess length where cylindrical shells in a too small container are expected to buckle rather than compress.
Much of this project is joint work with Patrick Dondl. Several other people were involved in various parts of the work (see articles below for more precise information).

Articles

  1. Phase Field Models for Thin Elastic Structures with Topological Constraint
    (with P. Dondl and A. Lemenant), Arch Rational Mech Anal (2017) 223:693 link ArXiv
  2. Uniform Regularity and Convergence of Phase Fields for Willmore's Energy
    (with P. Dondl), Calc. Var. (2017) 56: 90 link ArXiv 
  3. On the Boundary Regularity of Phase-Fields for Willmore's Energy
    (with P. Dondl), accepted for publication in Proc A Royal Soc of Edinburgh (2017) ArXiv
  4. Helfrich's Energy and Constrained Minimisation
    Comm in Math Sci (2017) 15:8 link ArXiv 
  5. Approximation of the relaxed perimeter functional under a connectedness constraint by phase-fields
    (with P. Dondl, M. Novaga and B. Wirth), 2018 ArXiv
  6. Keeping it together: a phase field version of path-connectedness and its implementation
    (with P. Dondl), 2018 ArXiv
  7. Confined elasticae and the buckling of cylindrical shells
    Adv Calc Var (2020) link ArXiv

Further Sources