Conservation, convergence, and computation for evolving heterogeneous elastic wires

TL;DR

This work develops and analyzes a planar, heterogeneous elastic wire model where a curve γ is endowed with a density ρ, and the energy $\,\mathcal{E}_μ(θ,ρ)=\frac{1}{2}\int_0^L [β(ρ)(∂_sθ-c_0)^2+μ(∂_sρ)^2] ds$ couples geometry and material distribution via a density-dependent bending stiffness $β(ρ)$. Using an angle-based formulation, the authors derive a constrained, nonlocal $L^2$-gradient flow for $(θ,ρ)$, establish global existence and convergence to stationary constrained elasticae, and investigate permanence properties such as convexity, density sign, embeddedness, and symmetry. They prove that under growth conditions on $β$ or in the large‐μ or large‐time regimes, the flow converges to homogeneous elasticae (circle-like states for $ω≠0$ and figure-eight states for $ω=0$) with sometimes exponential rates, and they provide a detailed asymptotic classification in several parameter regimes. Complementing the theory, extensive numerical experiments using a Newton-based, energy‐stable scheme and symmetry-projected iterations illustrate loss of convexity/embeddedness, energy dynamics, and parameter-driven transitions among symmetric, circle, and figure-eight configurations, validating the analytical findings and highlighting practical implications for material-geometry coupling in evolving elastic interfaces.

Abstract

The elastic energy of a bending-resistant interface depends both on its geometry and its material composition. We consider such a heterogeneous interface in the plane, modeled by a curve equipped with an additional density function. The resulting energy captures the complex interplay between curvature and density effects, resembling the Canham-Helfrich functional. We describe the curve by its inclination angle, so that the equilibrium equations reduce to an elliptic system of second order. After a brief variational discussion, we investigate the associated nonlocal $L^2$-gradient flow evolution, a coupled quasilinear parabolic problem. We analyze the (non)preservation of quantities such as convexity, positivity, and symmetry, as well as the asymptotic behavior of the system. The results are illustrated by numerical experiments.

Figures (13)

• Figure 1: Curve with dents.
• Figure 2: Cigar-shaped curve \ref{['fig:cigarre:A']} with linear density on $[a,b]$\ref{['fig:cigarre:B']}.
• Figure 3: Example of a $3$-fold rotationally symmetric configuration $(\theta,\rho)$.
• Figure 4: Example of an axially symmetric configuration $(\theta,\rho)$.
• Figure 5: Loss of convexity of a stadium with sides parallel to the $x$-axis. Here and in subsequent figures, the width of the stroke increases with $|\hat{\rho}|$. For the sake of readability, the y-scale is amplified 10 times and the curve is shown with constant width in the inset. The gray lines are the tangents parallel to the $x$-axis, for reference. Here and in all the following figures, positive values of $\rho$ are shown in blue, negative values in red. It is not shown here, but the curve becomes convex again at later times.

Theorems & Definitions (58)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3: Preservation of convexity for $c_0=0$
• Theorem 1.4
• Theorem 1.5: Asymptotic behavior under growth assumptions on $\beta$
• Theorem 1.6: Asymptotic behavior for large $\mu$
• Proposition 2.1: Existence of a minimizer
• Proposition 2.2: Smoothness of critical points
• proof
• Definition 2.3