Explicit symplectic integration of rod dynamics

Abstract

In this paper, we derive an explicit and symplectic integration scheme for elastic rods. To do so, we first discuss various spatial discretizations that yield an appropriate finite-dimensional truncation of the Simo-Marsden-Krishnaprasad formulation of rod dynamics [1]. Then, similar to rigid body dynamics [2], we formulate an explicit time-integrator by splitting the Hamiltonian in an appropriate way. The resulting scheme is 2nd order, explicit, symplectic, and preserves the underlying symmetries of rod dynamics. We also consider the case of an unshearable and unextensible rod. It is demonstrated that this leads to a Hamiltonian formulation with holonomic constraints that can be integrated by an appropriate modification of the constrained schemes considered in [3]. Application of our scheme also includes continuum models of DNA [4].