Heterogeneity and incompressibility in the evolution of elastic wires

Abstract

Elastic wires are mathematical curves composed of matter. They have no thickness but exhibit bending stiffness. Although elastic wires do not exist in the real world, they are widely used in modeling. Examples include the modeling of plant stems, polymers such as DNA, marine cables, and hair movement in digital simulations.

The elastic energy of a sufficiently smooth regular curve describing an elastic wire is defined as the integral of the squared curvature. In the last decades, several authors have studied the L2-gradient flow of the elastic energy in different variants. In this monograph, we briefly present the underlying theory and background, summarize their results, and then focus on two new variants.

First, we consider elastic wires with a heterogeneity described by a density function. We define a generalization of the elastic energy, which depends on material parameters, captures the interplay between curvature and density effects and resembles the Canham-Helfrich functional. Describing the closed planar curve by its inclination angle, the L2-gradient flow of this energy is a nonlocal coupled parabolic system of second order. We discuss local well-posedness, global existence and convergence. Then, we show that the (non)preservation of quantities such as convexity as well as the asymptotic behavior of the system depend delicately on the choice of material parameters.

Second, we study the evolution of elastic wires under the assumption of incompressiblity and derive a gradient flow of the elastic energy which preserves the enclosed area of the evolving planar curves. Contrary to an earlier approach based on a fourth order equation featuring nonlocal Lagrange multipliers, we give priority to the locality of the evolution and propose a sixth order gradient flow equation with no nonlocal terms. For this flow, we establish a global existence result. When including an additional term penalizing length, we prove convergence to an area constrained critical point of the elastic energy.