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Research Papers

Implementation Details of a Generalized-α Differential-Algebraic Equation Lie Group Method

[+] Author and Article Information
Martin Arnold

Institute of Mathematics,
Martin Luther University Halle-Wittenberg,
Halle (Saale) 06099, Germany
e-mail: martin.arnold@mathematik.uni-halle.de

Stefan Hante

Institute of Mathematics,
Martin Luther University Halle-Wittenberg,
Halle (Saale) 06099, Germany
e-mail: stefan.hante@mathematik.uni-halle.de

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received September 22, 2015; final manuscript received April 13, 2016; published online December 2, 2016. Assoc. Editor: Andreas Mueller.

J. Comput. Nonlinear Dynam 12(2), 021002 (Dec 02, 2016) (8 pages) Paper No: CND-15-1301; doi: 10.1115/1.4033441 History: Received September 22, 2015; Revised April 13, 2016

Configuration spaces with Lie group structure display kinematical nonlinearities of mechanical systems. In Lie group time integration, this nonlinear structure is also considered at the time-discrete level using nonlinear updates of the configuration variables. For practical implementation purposes, these update formulae have to be adapted to each specific Lie group setting that may be characterized from the algorithmic viewpoint by group operation, exponential map, tilde, and tangent operator. In this paper, we discuss these practical aspects for the time integration of a geometrically exact Cosserat rod model with rotational degrees-of-freedom being represented by unit quaternions. Shearing and longitudinal extension of the Cosserat rod may be neglected using suitable constraints that result in a differential-algebraic equation (DAE) formulation of the beam structure. The specific structure of unconstrained systems and constrained systems is exploited by tailored algorithms for the corrector iteration of the generalized-α Lie group integrator.

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Figures

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Fig. 1

Flying spaghetti benchmark [18]: endpoint trajectories and snapshots at t=0 s,2 s,3 s,…,14 s

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Fig. 2

Cosserat model: angular velocity Ω7−1/2 of the central beam slice, N = 15

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Fig. 3

Global errors in q and v for the Cosserat model: maximum of the 2-norm for t∈[0 s,15 s], N = 15

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Fig. 4

Kirchhoff model: angular velocity Ω7−1/2 of the central beam slice, N = 15

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Fig. 5

Kirchhoff model: Lagrange multipliers λ7−1/2 corresponding to the central beam slice, N = 15

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Fig. 6

Global errors in q, v, and λ for the index-3 formulation of the Kirchhoff model: maximum of the 2-norm for t∈[0 s,15 s], N = 15

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Fig. 7

Global errors in q, v, and λ for the stabilized index-2 formulation of the Kirchhoff model: maximum of the 2-norm for t∈[0 s,15 s], N = 15

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Fig. 8

Stabilized index-2 formulation of the Kirchhoff model: auxiliary variable η7−1/2,1 for h = 2.0 ms (dashed line) and h = 1.0 ms (solid line)

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