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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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References

Géradin, M. , and Cardona, A. , 2001, Flexible Multibody Dynamics: A Finite Element Approach, Wiley, Chichester, UK.
Betsch, P. , and Steinmann, P. , 2001, “ Constrained Integration of Rigid Body Dynamics,” Comput. Methods Appl. Mech. Eng., 191(3–5), pp. 467–488. [CrossRef]
Romero, I. , 2004, “ The Interpolation of Rotations and Its Application to Finite Element Models of Geometrically Exact Rods,” Comput. Mech., 34(2), pp. 121–133. [CrossRef]
Betsch, P. , and Siebert, R. , 2009, “ Rigid Body Dynamics in Terms of Quaternions: Hamiltonian Formulation and Conserving Numerical Integration,” Int. J. Numer. Methods. Eng., 79(4), pp. 444–473. [CrossRef]
Brüls, O. , and Cardona, A. , 2010, “ On the Use of Lie Group Time Integrators in Multibody Dynamics,” ASME J. Comput. Nonlinear Dyn., 5(3), p. 031002. [CrossRef]
Celledoni, E. , and Säfström, N. , 2010, “ A Hamiltonian and Multi-Hamiltonian Formulation of a Rod Model Using Quaternions,” Comput. Methods Appl. Mech. Eng., 199(45–48), pp. 2813–2819. [CrossRef]
Müller, A. , and Terze, Z. , 2014, “ The Significance of the Configuration Space Lie Group for the Constraint Satisfaction in Numerical Time Integration of Multibody Systems,” Mech. Mach. Theory, 82, pp. 173–202. [CrossRef]
Crouch, P. , and Grossman, R. , 1993, “ Numerical Integration of ODEs on Manifolds,” J. Nonlinear Sci., 3(1), pp. 1–33. [CrossRef]
Munthe-Kaas, H. , 1998, “ Runge–Kutta Methods on Lie Groups,” BIT Numer. Math., 38(1), pp. 92–111. [CrossRef]
Arnold, M. , Brüls, O. , and Cardona, A. , 2011, “ Convergence Analysis of Generalized-α Lie Group Integrators for Constrained Systems,” Multibody Dynamics 2011 ECCOMAS Thematic Conference, J. Samin and P. Fisette , eds.
Arnold, M. , Brüls, O. , and Cardona, A. , 2015, “ Error Analysis of Generalized-α Lie Group Time Integration Methods for Constrained Mechanical Systems,” Numerische Math., 129(1), pp. 149–179. [CrossRef]
Hairer, E. , Lubich, C. , and Wanner, G. , 2006, Geometric Numerical Integration. Structure–Preserving Algorithms for Ordinary Differential Equations, 2nd ed., Springer-Verlag, Berlin.
Brüls, O. , Cardona, A. , and Arnold, M. , 2012, “ Lie Group Generalized-α Time Integration of Constrained Flexible Multibody Systems,” Mech. Mach. Theory, 48, pp. 121–137. [CrossRef]
Brüls, O. , Arnold, M. , and Cardona, A. , 2011, “ Two Lie Group Formulations for Dynamic Multibody Systems With Large Rotations,” ASME Paper DETC2011-48132.
Lang, H. , Linn, J. , and Arnold, M. , 2009, “ Multibody Dynamics Simulation of Geometrically Exact Cosserat Rods,” Fraunhofer ITWM, Kaiserslautern, Germany, Report No. 159.
Lang, H. , Linn, J. , and Arnold, M. , 2011, “ Multibody Dynamics Simulation of Geometrically Exact Cosserat Rods,” Multibody Syst. Dyn., 25(3), pp. 285–312. [CrossRef]
Lang, H. , and Arnold, M. , 2012, “ Numerical Aspects in the Dynamic Simulation of Geometrically Exact Rods,” Appl. Numer. Math., 62(10), pp. 1411–1427. [CrossRef]
Simo, J. , and Vu-Quoc, L. , 1988, “ On the Dynamics in Space of Rods Undergoing Large Motions—A Geometrically Exact Approach,” Comput. Methods Appl. Mech. Eng., 66(2), pp. 125–161. [CrossRef]
Hairer, E. , and Wanner, G. , 1996, Solving Ordinary Differential Equations—II: Stiff and Differential-Algebraic Problems, 2nd ed., Springer-Verlag, Berlin.
Chung, J. , and Hulbert, G. , 1993, “ A Time Integration Algorithm for Structural Dynamics With Improved Numerical Dissipation: The Generalized-α Method,” ASME J. Appl. Mech., 60(2), pp. 371–375. [CrossRef]
Bottasso, C. , Bauchau, O. , and Cardona, A. , 2007, “ Time-Step-Size-Independent Conditioning and Sensitivity to Perturbations in the Numerical Solution of Index Three Differential Algebraic Equations,” SIAM J. Sci. Comp., 29(1), pp. 397–414. [CrossRef]
Deuflhard, P. , 2004, Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms, Springer, Berlin.
Kelley, C. , 2003, Solving Nonlinear Equations With Newton's Method, SIAM, Philadelphia, PA.
Arnold, M. , and Brüls, O. , 2007. “ Convergence of the Generalized-α Scheme for Constrained Mechanical Systems,” Multibody Syst. Dyn., 18(2), pp. 185–202. [CrossRef]
Hante, S. , 2015, “ A General Purpose Lie Group Generalized-α Time Integrator Applied to Nonlinear Flexible Geometrically Exact Beam Models,” Master's thesis, Martin Luther University Halle-Wittenberg, Halle, Germany.
Arnold, M. , 1995, “ A Perturbation Analysis for the Dynamical Simulation of Mechanical Multibody Systems,” Appl. Numer. Math., 18(1–3), pp. 37–56. [CrossRef]

Figures

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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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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. 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. 5

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

Grahic Jump Location
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

Grahic Jump Location
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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