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

TL;DR

This work develops a convergent and consistent discretization for Helfrich-type energies on general triangular meshes by introducing an edge-director based discretization $E({\mathscr T},n)$ with a piecewise affine gradient $D n_\kappa$. The authors establish Γ-convergence to the continuum energy $E_0(M)$ for surfaces that are graphs, proving compactness, a liminf inequality, and a recovery sequence, and they extend the framework to pseudo-unit edge directors for computational efficiency. The results link discrete normals with the continuous surface normals via a Cr-like interpolation, ensuring that minimizers of the discrete problem converge to minimizers of the continuous problem. Numerically, the method is demonstrated on graphical domains, with runtime and convergence behavior analyzed under mesh refinement and model variants (full vs reduced edge-direction freedom). The approach provides a robust, geometry-aware discretization suitable for simulations of elastic plates, shell regularization, and surface PDEs on meshes.

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.

Figures (7)

• Figure 1: While the triangle normals $\bar{n}(\kappa)$, $\bar{n}(\kappa')$ are predetermined by the triangle’s vertices, edge directors have one additional degree of freedom. A possible unit edge director is shown in green, and a pseudo-unit edge director $n$ is shown in black.
• Figure 2: On the left, the 2D domain constructed from 3rd order B-splines with its associated nodes. On the right, the target function $\sin(2x)\cos(2y)$ evaluated on that domain.
• Figure 3: Surfaces minimizing the discrete energy for increasing refinement. The meshes are respectively made of $50$, $1\, 000$, and $15\, 000$ triangles.
• Figure 4: Mean (left) and Gaussian (right) curvatures of the optimal surface on the $15\,000$ triangles mesh.
• Figure 5: Computational cost and accuracy versus mesh size $h$: time scales as $\mathcal{O}(h^3)$, the objective value stabilizes with mesh refinement while the full reduced solutions $u$ and $\tilde{u}$ converge towards distinct optima.

Theorems & Definitions (16)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Theorem 2.5
• Remark 2.6
• Lemma 3.1
• proof
• proof : Proof of Theorem \ref{['thm: main']} (i)
• proof : Proof of Theorem \ref{['thm: main']} (ii)