Variational Convergence of Discrete Elasticae

TL;DR

This work addresses the convergence of discrete elasticae to smooth Euler elasticae under polygonal discretization. It develops a pair of paired operators—reconstruction from discrete data to smooth curves and sampling from smooth curves to discrete data—together with Newton–Kantorovich-based restoration to enforce exact constraints. The main result establishes Hausdorff convergence of discrete almost-minimizers to the smooth minimizers in $W^{2,p}$ for $p\in[2,\infty)$ and in $W^{1,\infty}$ for piecewise-linear interpolants as the mesh size $h\to0$, without relying on full $\Gamma$-convergence. This yields a quantitative, topology-refined link between discrete polygonal models and smooth elasticae, with explicit control of energy, curvature, and higher-order regularity across reconstruction and sampling steps.

Abstract

We discuss a discretization by polygonal lines of the Euler-Bernoulli bending energy and of Euler elasticae under clamped boundary conditions. We show Hausdorff convergence of the set of almost minimizers of the discrete bending energy to the set of smooth Euler elasticae under mesh refinement in (i) the $W^{1,\infty}$-topology for piecewise-linear interpolation and in (ii) the $W^{2,p}$-topology, $p \in{[2,\infty[}$, using a suitable smoothing operator to create $W^{2,p}$-curves from polygons.

Figures (6)

• Figure 1: An elastica in ${\mathbb{R}}^2$ with clamped ends (blue, obtained from Jacobi elliptic functions) compared to discrete energy minimizers (orange) for various resolutions.
• Figure 2: Two polygonal curves (orange) that have exactly the same length and approximate a smooth elastic curve (blue) for vertically clamped edges. The energies of the discrete curves, however, differ by a factor of three, independent of their resolution. Small perturbations of vertex positions may thus lead to drastic changes in elastic energy.
• Figure 3: The almost minimizing sets$\mathop{\mathrm{arg\,min}}\nolimits^\frac{3}{n}({\mathcal{F}}_n)$ of the tilted potentials ${\mathcal{F}}_n(x) = (1-\lvert{x}\rvert^2)^2 - (-1)^n \frac{1}{n} \frac{x_1}{1+\lvert{x}\rvert^2}$ converge (as sets) to the minimizer of the potential ${\mathcal{F}}(x) = (1-\lvert{x}\rvert^2)^2$ (the unit circle). In contrast, while ${\mathcal{F}}_n$$\Gamma$-converges to ${\mathcal{F}}$, the respective minimizers ($\set{(1,0)}$ for even $n$ and $\set{(-1,0)}$ for odd $n$) do not converge to the minimizers of ${\mathcal{F}}$.
• Figure 4: Sketch of the notation used for polygonal lines.
• Figure 5: (a) A discrete curve $P$ (orange) is smoothened by a piecewise circular curve ${\tilde{\mathcal{R}}_{{T}}}(P)$ (blue) such that tangents of $P$ at edge midpoints agree with tangents of ${\tilde{\mathcal{R}}_{{T}}}(P)$ where circular arcs meet (white points). These curves have the same length, leading to differences between their end points and boundary tangents (gray), which can be controlled. (b) These differences can be repaired by applying the restoration operator $\mathcal{P}$, leading to the curve ${\mathcal{R}_{{T}}}(P) = \mathcal{P} \circ {\tilde{\mathcal{R}}_{{T}}}(P)$ (blue).

Theorems & Definitions (37)

• Theorem 1.1: Convergence Theorem
• Proof 1
• Lemma 1: Norm Equivalences
• Lemma 1: Right Inverse of $D\varPhi$
• Proof 2
• Theorem 1: Regularity of Minimizers
• Proof 3
• Remark 1
• Lemma 2: Right Inverse of $D\varPhi_{T}$
• Proof 4