Consistent and convergent discretizations of Helfrich-type energies on general meshes

TL;DR

This paper develops a new method for discretizing Helfrich-type curvature energies on surfaces using triangular meshes, ensuring consistency and convergence of the approximation.

Contribution

It introduces a novel discretization approach for integral curvature energies on surfaces that is both consistent and convergent on general meshes.

Findings

Discrete energies converge to the continuous energy as mesh size decreases.

The method applies to general triangular complexes, not just regular meshes.

A new asymptotic lower bound and recovery sequence are established.

Abstract

We show that integral curvature energies on surfaces of the type $E_0(M) := \int_M f(x,n_M(x),D n_M(x))\,d\mathcal{H}^2(x)$ have discrete versions for triangular complexes, where the shape operator $D n_M$ is replaced by the piecewise gradient of a piecewise affine edge director field. We combine an ansatz-free asymptotic lower bound for any uniform approximation of a surface with triangular complexes and a recovery sequence consisting of any regular triangulation of the limit sequence and an almost optimal choice of edge director.