Numerical aspects in the dynamic simulation of geometrically exact rods

Abstract

Classical geometrically exact Kirchhoff and Cosserat models are used to study the nonlinear deformation of rods. Extension, bending and torsion of the rod may be represented by the Kirchhoff model. The Cosserat model additionally takes into account shearing effects. Second order finite differences on a staggered grid define discrete viscoelastic versions of these classical models. Since the rotations are parametrisecl by unit quaternions, the space discretisation results in differential-algebraic equations that are solved numerically by standard techniques like index reduction and projection methods. Using absolute coordinates, the mass and constraint matrices are sparse and this sparsity may be exploited to speed-up time integration. Further improvements are possible in the Cosserat model, because the constraints are just the normalisation conditions for unit quaternions such that the null space of the constraint matrix can be given analytically. The results of the theoretical investigations are illustrated by numerical tests

Discrete variational Lie group formulation of geometrically exact beam dynamics

“…Approaches to dynamic beam simulation being inherently different from Lie group methods, are already mentioned in the introduction. In particular, [5,34,35] use redundant coordinates like quaternions or director triads. To describe the rotational degrees of freedom, they need to be subject to constraints, leading to differential algebraic equations of motion that are treated, e.g., with index reduction or projection methods.…”

“…This choice is consistent with the approximation we made for the potential term. We now give the discrete Lagrange-d'Alembert equations (35), for a given diffeomorphism τ : se(3) → S E(3) in a neighborhood of the origin such that τ (0) = e.…”

“…The constrained formulation is particularly popular when the beam is part of a multibody system, where further constraints representing the connection to other (rigid or flexible) components are naturally present. Additional popular formulations are the so called absolute nodal coordinates formulation based on works of [57,58] or formulations in terms of unit quaternions [34,35]. Recently, Lie group formulations are becoming more and more important in multibody dynamics; see, e.g., [12,13].…”

Geometrically exact Cosserat rods with Kelvin–Voigt type viscous damping

“…(1) for our discrete 5 Cosserat model 5 Practical applications of our Cosserat rod model with KelvinVoigt damping in Multibody System Dynamics are reported in our formulated with unit quaternions as explained in detail by Lang et al (2011) and investigated further in Lang and Arnold (2012) w.r.t. numerical aspects, we do not make use of this particular formulation here, as it is more practical to work with the directors associated to SO(3) frames for the vector-algebraic calculations which we have to carry out within our derivations of one-dimensional rod functionals from three-dimensional continuum formulation.…”

“…Our recent articles (Lang et al, 2011;Lang and Arnold, 2012) provide one concrete example of an implementation of a discrete version of our constitutive model (1), as an integral part of (and taylored to) our specific continuum formulation of the Cosserat rod model using unimodular quaternions, our specific spatial discretization approach -finite differences for the centerline, finite quotients for the quaternion field, both on a staggered grid -applied on the level of the stored energy (10), kinetic energy (see Sect. 3.6) and the dissipation function (11), the specific formulation of the resulting semidiscrete system as a first order DAE or ODE (depending on the kind of internal kinematical constraints and their treatment), and the class of time integration methods we choose to solve the semidiscrete equations for various initial-boundary value problems.…”