TY - JOUR

T1 - On the geometric flow of kirchhoff elastic rods

AU - Lin, Chun Chi

AU - Schwetlick, Hartmut R.

PY - 2005

Y1 - 2005

N2 - Recently, rod theory has been applied to the mathematical modeling of bacterial fibers and biopolymers (e.g., DNA) to study their mechanical properties and shapes (e.g., supercoiling). In static rod theory, an elastic rod in equilibrium is the critical point of an elastic energy. This induces a natural question of how to find elasticae. In this paper, we focus on how to find the critical points by means of gradient flows. We relate a geometric function of curves to the Isotropic Kirchhoff elastic energy of rods so that the generalized elastic curves are the centerlines of elastic rods in equilibrium. Thus, the variational problem for rods is formulated in curve geometry. This problem turns out to be a generalization of curve-straightening flows, which induce nonlinear fourth-order evolution equations. We establish the long time existence of length-preserving gradient flow for the geometric energy. Furthermore, by studying the asymptotic behavior, we show that the limit curves are the centerlines of the Kirchhoff elastic rods in equilibrium.

AB - Recently, rod theory has been applied to the mathematical modeling of bacterial fibers and biopolymers (e.g., DNA) to study their mechanical properties and shapes (e.g., supercoiling). In static rod theory, an elastic rod in equilibrium is the critical point of an elastic energy. This induces a natural question of how to find elasticae. In this paper, we focus on how to find the critical points by means of gradient flows. We relate a geometric function of curves to the Isotropic Kirchhoff elastic energy of rods so that the generalized elastic curves are the centerlines of elastic rods in equilibrium. Thus, the variational problem for rods is formulated in curve geometry. This problem turns out to be a generalization of curve-straightening flows, which induce nonlinear fourth-order evolution equations. We establish the long time existence of length-preserving gradient flow for the geometric energy. Furthermore, by studying the asymptotic behavior, we show that the limit curves are the centerlines of the Kirchhoff elastic rods in equilibrium.

KW - Fourth order

KW - Geometric flows

KW - Kirchhoff elastic rods

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U2 - 10.1137/S0036139903431713

DO - 10.1137/S0036139903431713

M3 - Article

AN - SCOPUS:19944376166

VL - 65

SP - 720

EP - 736

JO - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

SN - 0036-1399

IS - 2

ER -