A discrete mechanics approach to the Cosserat rod theory—Part 1: static equilibria

Abstract

A theory of discrete Cosserat rods is formulated in the language of discrete Lagrangian mechanics. By exploiting Kirchhoff's kinetic analogy, the potential energy density of a rod is a function on the tangent bundle of the configuration manifold and thus formally corresponds to the Lagrangian function of a dynamical system. The equilibrium equations are derived from a variational principle using a formulation that involves null‐space matrices. In this formulation, no Lagrange multipliers are necessary to enforce orthonormality of the directors. Noether's theorem relates first integrals of the equilibrium equations to Lie group actions on the configuration bundle, so‐called symmetries. The symmetries relevant for rod mechanics are frame‐indifference, isotropy, and uniformity. We show that a completely analogous and self‐contained theory of discrete rods can be formulated in which the arc‐length is a discrete variable ab initio. In this formulation, the potential energy density is defined directly on pairs of points along the arc‐length of the rod, in analogy to Veselov's discrete reformulation of Lagrangian mechanics. A discrete version of Noether's theorem then identifies exact first integrals of the discrete equilibrium equations. These exact conservation properties confer the discrete solutions accuracy and robustness, as demonstrated by selected examples of application. Copyright © 2010 John Wiley & Sons, Ltd.

Numerical aspects in the dynamic simulation of geometrically exact rods

“…The choice of a discrete curvature measure is by no means unique. We propose several choices from literature [6,7,10,24,27,32,40], which can be expressed in terms of the material unit axis u n and the material angle ϕ n of difference rotation. Typical examples are…”

“…For thorough discussions on that topic, we refer to [31]. Approaches that use Cayley transformation instead of interpolation are as well possible [24]. For each of the proposed curvatures in (12), it is possible to show that they are objective/frame-indifferent.…”

“…They solve the full index-1 problem with u = (q, v, λ). For solvers that may handle linearly implicit systems (24), the accelerations w =v must be introduced as additional primary unknowns. Again, one has to solve the index-1 problem with u = (q, v, w, λ) with appropriate rescaling in the error estimator [22].…”

“…We distinguish three basic types of classical rod models. In hierarchical descending order, these are the Cosserat, the extensible Kirchhoff and the inextensible Kirchhoff model [1,2,3,9,13,14,21,23,24,25,26,27,28,32,38,39]. Table 1 presents a short overview, including numerical problems in time integration to be discussed in this article.…”

Conformal Invariance and Conserved Quantities for Lagrange Equation of Thin Elastic Rod

“…Fu et al [39] studied the Noether symmetry of a superlong elastic rod in the Hamilton form. Jung et al [40] studied a discrete method for special Cosserat elastic rod statics and gave the related Noether theorem. Xue et al [41] studied the the conserved quantities in general theorems of elastic rod dynamics.…”