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Multi-body dynamics simulation of geometrically exact Cosserat rods

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Abstract

In this paper, we present a viscoelastic rod model that is suitable for fast and accurate dynamic simulations. It is based on Cosserat’s 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 \(\dot{q}=v\), \(\dot{v}=f(q,v,t)\) 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 mechanical 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.

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Authors and Affiliations

  1. Fraunhofer Institute for Industrial Mathematics, Fraunhofer Platz 1, 67663, Kaiserslautern, Germany

    Holger Lang & Joachim Linn

  2. Institute for Mathematics, Martin-Luther-Universität Halle-Wittenberg, 06099, Halle (Saale), Germany

    Martin Arnold

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  1. Holger Lang
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  2. Joachim Linn
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Correspondence to Holger Lang.

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Lang, H., Linn, J. & Arnold, M. Multi-body dynamics simulation of geometrically exact Cosserat rods. Multibody Syst Dyn 25, 285–312 (2011). https://doi.org/10.1007/s11044-010-9223-x

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