Conservation, convergence, and computation for evolving heterogeneous elastic wires

TL;DR

This paper studies the equilibrium and evolution of heterogeneous elastic curves influenced by geometry and material composition, analyzing their properties and behavior through variational methods, PDE analysis, and numerical experiments.

Contribution

It introduces a model for elastic interfaces with variable density, deriving equilibrium equations and analyzing a nonlocal gradient flow for such heterogeneous curves.

Findings

Equilibrium equations reduce to an elliptic system involving curvature and density.

The nonlocal gradient flow preserves some quantities but not others, like convexity or symmetry.

Numerical experiments illustrate the theoretical results and system behavior.

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.