Multi-body dynamics simulation of geometrically exact Cosserat rods

Abstract

In this paper, we present a viscoelastic rod model that is suitable for fast and accurate dynamic simulations. It is based on Cosserats geometrically exact theory of rods and is able to represent extension, shearing (stiff dof), bending and torsion (soft dof). For inner dissipation, a consistent damping potential proposed by Antman is chosen. We parametrise the rotational dof by unit quaternions and directly use the quaternionic evolution differential equation for the discretisation of the Cosserat rod curvature. The discrete version of our rod model is obtained via a finite difference discretisation on a staggered grid. After an index reduction from three to zero, the right-hand side function f and the Jacobian f/(q,v,t) of the dynamical system q=v,v=f(q,v,t) is free of higher algebraic (e.g. root) or transcendental (e.g. trigonometric or exponential) functions and, therefore, cheap to evaluate. A comparison with Abaqus finite element results demonstrates the correct m echanical behaviour of our discrete rod model. For the time integration of the system, we use well established stiff solvers like Radau5 or Daspk. As our model yields computational times within milliseconds, it is suitable for interactive applications in virtual reality as well as for multi-body dynamics simulation

Numerical aspects in the dynamic simulation of geometrically exact rods

“…In the present paper, we combine an objective/frame-indifferent geometrically exact space discretisation of Kirchhoff and Cosserat rods with standard methods for the time integration of the equations of motion for constrained mechanical systems [4,17,22]. Following the method of lines, the equations of motion for Kirchhoff rods and Cosserat rods are discretised first in space by finite differences on a staggered grid [27]. The rotations are parametrised by unit quaternions resulting in constraints to guarantee the normalisation of the quaternions.…”

“…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.…”

“…In section 2, we shortly introduce these three basic classical models in the continuum, where we concentrate on the kinematics -especially kinematic restrictions for the Kirchhoff models -, the strain measures and the internal energies. For the method of lines, in section 3, we summarise a recently introduced new spatial discretisation approach, which is based on geometric finite differences on a staggered grid [26,27]. Our ansatz generalises a discrete differential geometric approach in [7,10], where inextensible Kirchhoff rods have been examined.…”

“…This just simplifies the exposition. For the handling of precurvature, see [26,27,38]. We remark that the choice of quadratic potentials V SE resp.…”

“…An equivalent formulation of (8) in 'quaternion language', derived in an alternative fashion, can be found in [26,27]. By the quaternionic parametrisation of rotations, which involves the constraint of unit length in any case, (8) becomes a partial differential-algebraic equation.…”