Calculus of Variations and Geometric Measure Theory

P. Gladbach - H. Olbermann

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

created by olbermann on 21 Jul 2024

[BibTeX]

preprint

Inserted: 21 jul 2024

Year: 2023

ArXiv: 2302.01705 PDF

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.