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Discrete variational Lie group formulation of geometrically exact beam dynamics. (English) Zbl 1315.53090

Summary: The goal of this paper is to derive a structure preserving integrator for geometrically exact beam dynamics, by using a Lie group variational integrator. Both spatial and temporal discretization are implemented in a geometry preserving manner. The resulting scheme preserves both the discrete momentum maps and symplectic structures, and exhibits almost-perfect energy conservation. Comparisons with existing numerical schemes are provided and the convergence behavior is analyzed numerically.

MSC:

53D05 Symplectic manifolds (general theory)
65P10 Numerical methods for Hamiltonian systems including symplectic integrators
74B20 Nonlinear elasticity
74H15 Numerical approximation of solutions of dynamical problems in solid mechanics
37M15 Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems
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[1] Abraham, R., Marsden, J.E.: Foundations of Mechanics, 2nd edn, revised and enlarged. With the assistance of Tudor Ratiu and Richard Cushman. Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, MA (1978) · Zbl 0393.70001
[2] Antman, S.S.: Kirchhoff’s problem for nonlinearly elastic rods. Q. J. Appl. Math. 32, 221-240 (1974) · Zbl 0302.73031
[3] Antmann, S.S.: Nonlinear Problems in Elasticity. Springer, Berlin (1995) · Zbl 0820.73002 · doi:10.1007/978-1-4757-4147-6
[4] Betsch, P., Menzel, A., Stein, E.: On the parametrization of finite rotations in computational mechanics: a classification of concepts with application to smooth shells. Comput. Methods Appl. Mech. Eng. 155, 273-305 (1998) · Zbl 0947.74060 · doi:10.1016/S0045-7825(97)00158-8
[5] Betsch, P., Steinmann, P.: Frame-indifferent beam finite elements based upon the geometrically exact beam theory. Int. J. Numer. Methods Eng. 54, 1775-1788 (2002) · Zbl 1053.74041 · doi:10.1002/nme.487
[6] Betsch, P., Steinmann, P.: Constrained dynamics of geometrically exact beams. Comput. Mech. 31, 49-59 (2003) · Zbl 1038.74580 · doi:10.1007/s00466-002-0392-1
[7] Bobenko, A.I., Suris, Y.B.: Discrete time Lagrangian mechanics on Lie groups, with an application to the Lagrange top. Commun. Math. Phys. 204, 147-188 (1999a) · Zbl 0945.70010 · doi:10.1007/s002200050642
[8] Bobenko, A.I., Suris, Y.B.: Discrete Lagrangian reduction, discrete Euler-Poincaré equations, and semidirect products. Lett. Math. Phys. 49, 79-93 (1999b) · Zbl 0965.70025 · doi:10.1023/A:1007654605901
[9] Bottasso, C., Borri, M., Trainelli, L.: Geometric invariance. Comput. Mech. 29, 163-169 (2002) · Zbl 1024.74036 · doi:10.1007/s00466-002-0329-8
[10] Bou-Rabee, N., Marsden, J.E.: Hamilton-Pontryagin integrators on Lie groups Part I: introduction and structure-preserving properties. Found. Comput. Math. 9, 197-219 (2009) · Zbl 1221.37166 · doi:10.1007/s10208-008-9030-4
[11] Bou-Rabee, N., Owhadi, H.: Stochastic variational integrators. IMA J. Numer. Anal. 29, 421-443 (2008) · Zbl 1171.37027 · doi:10.1093/imanum/drn018
[12] Brüls, O., Cardona, A.: On the use of Lie group time integrators in multibody dynamics. J. Comput. Nonlinear Dyn. 5(3), 031002 (2010). Special issue on Multi- disciplinary High-Performance Computational Multibody Dynamics, edited by Dan Negrut and Olivier Bauchau. doi:10.1115/1.4001370 · Zbl 1060.70500
[13] Brüls, O., Cardona, A., Arnold, M.: Lie group generalized-\[ \alpha\] α time integration of constrained flexible multibody systems. Mech. Mach. Theory 48, 1212-137 (2012) · doi:10.1016/j.mechmachtheory.2011.07.017
[14] Crisfield, M.A., Jelenić, G.: Objectivity of strain measures in the geometrically exact three-dimensional beam theory and its finite-element implementation. Proc. R. Soc. Lond. A 455, 1125-1147 (1999) · Zbl 0926.74062
[15] de León, M., Marrero, J.C., Martín de Diego, D.: Some applications of semi-discrete variational integrators to classical field theories. Qual. Theory. Dyn. Syst. 7(1), 195-212 (2008) · Zbl 1168.37024
[16] Demoures, F., Gay-Balmaz, F., Kobilarov, M., Ratiu, T.S.: Multisymplectic Lie group variational integrators for a geometrically exact beam in \[{\mathbb{R}}^3\] R3. Commun. Nonlinear Sci. Numer. Simul. 19(10), 3492-3512 (2014). http://arxiv.org/pdf/1403.5410v1 · Zbl 1473.70029
[17] Dichmann, D.J., Li, Y., Maddocks, J.H.: Hamiltonian formulations and symmetries in rod mechanics. In: Mathematical Approaches to Biomolecular Structure and Dynamics, vol. 82, pp. 71-113. Springer, New York (1996) · Zbl 0864.92004
[18] Ellis, D., Gay-Balmaz, F., Holm, D.D., Putkaradze, V., Ratiu, T.S.: Symmetry reduced dynamics of charged molecular strands. Arch. Rat. Mech. Anal. 197(2), 811-902 (2010) · Zbl 1333.70029 · doi:10.1007/s00205-010-0305-y
[19] Fetecau, R.C., Marsden, J.E., Ortiz, M., West, M.: Nonsmooth Lagrangian mechanics and variational collision integrators. SIAM J. Appl. Dyn. Syst. 2(3), 381-416 (2003) · Zbl 1088.37045 · doi:10.1137/S1111111102406038
[20] Gay-Balmaz, F., Holm, D.D., Ratiu, T.S.: Variational principles for spin systems and the Kirchhoff rod. J. Geom. Mech. 1(4), 417-444 (2009) · Zbl 1288.70012 · doi:10.3934/jgm.2009.1.417
[21] Gay-Balmaz, F., Marsden, J.E., Ratiu, T.S.: Reduced variational formulations in free boundary continuum mechanics. J. Nonlinear Sci. 22(4), 463-497 (2012) · Zbl 1260.37031
[22] Gonzalez, O.: Time integration and discrete Hamiltonian systems. J. Nonlinear Sci. 6, 449-467 (1996) · Zbl 0866.58030 · doi:10.1007/BF02440162
[23] Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations. Springer, Berlin (2006) · Zbl 1094.65125
[24] Ibrahimbegović, A., Frey, F., Kozar, I.: Computational aspects of vector-like parametrization of three-dimensional finite rotations. Int. J. Numer. Methods Eng. 38, 3653-3673 (1995) · Zbl 0835.73074 · doi:10.1002/nme.1620382107
[25] Ibrahimbegović, A., Mamouri, S.: Finite rotations in dynamics of beams and implicit time-stepping schemes. Int. J. Numer. Methods Eng. 41, 781-814 (1998) · Zbl 0902.73045 · doi:10.1002/(SICI)1097-0207(19980315)41:5<781::AID-NME308>3.0.CO;2-9
[26] Iserles, A., Munthe-Kaas, H.Z., Nørsett, S.P., Zanna, A.: Lie-group methods. Acta Num. 9, 215-365 (2000) · Zbl 1064.65147
[27] Jelenić, G., Crisfield, M.: Interpolation of rotational variables in non-linear dynamics of \[33\] D beams. Int. J. Numer. Methods Eng. 43, 1193-1222 (1998) · Zbl 0939.74068 · doi:10.1002/(SICI)1097-0207(19981215)43:7<1193::AID-NME463>3.0.CO;2-P
[28] Jelenić, G., Crisfield, M.: Geometrically exact \[33\] D beam theory: implementation of a strain-invariant finite element for statics and dynamics. Comput. Methods Appl. Mech. Eng. 171, 141-171 (1999) · Zbl 0962.74060 · doi:10.1016/S0045-7825(98)00249-7
[29] Jelenić, G., Crisfield, M.: Problems associated with the use of Cayley transform and tangent scaling for conserving energy and momenta in the Reissner-Simo beam theory. Commun. Numer. Methods Eng. 18, 711-720 (2002) · Zbl 1022.74019 · doi:10.1002/cnm.531
[30] Jung, P., Leyendecker, S., Linn, J., Ortiz, M.: A discrete mechanics approach to Cosserat rod theory. Part 1: static equilibria. Int. J. Numer. Methods Eng. 85, 31-60 (2010) · Zbl 1217.74069 · doi:10.1002/nme.2950
[31] Kane, C., Marsden, J.E., Ortiz, M., West, M.: Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems. Int. J. Numer. Methods Eng. 49(10), 1295-1325 (2000) · Zbl 0969.70004 · doi:10.1002/1097-0207(20001210)49:10<1295::AID-NME993>3.0.CO;2-W
[32] Kobilarov, M., Marsden, J.E.: Discrete geometric optimal control on Lie groups. IEE Trans. Robot. 27, 641-655 (2011) · doi:10.1109/TRO.2011.2139130
[33] Kobilarov, M., Marsden, J.E., Sukhatme, G.S.: Geometric discretization of nonholonomic systems with symmetries. Discrete Contin. Dyn. Syst. Ser. S 3(1), 61-84 (2010) · Zbl 1197.37071
[34] Lang, H., Linn, J., Arnold, M.: Multi-body dynamics simulation of geometrically exact Cosserat rods. Multibody Syst. Dyn. 25(3), 285-312 (2011) · Zbl 1271.74264 · doi:10.1007/s11044-010-9223-x
[35] Lang, H., Arnold, M.: Numerical aspects in the dynamic simulation of geometrically exact rods. Appl. Numer. Math. 62(10), 1411-1427 (2012) · Zbl 1295.70004 · doi:10.1016/j.apnum.2012.06.011
[36] Lee, T., Leok, M., McClamroch, N.H.: Dynamics of a \[33\] D elastic string pendulum. In: Proceedings of the IEEE Conference on Decision and Control (2009) · Zbl 1280.70002
[37] Lee, T.: Computational geometric mechanics and control of rigid bodies. PhD Thesis, University of Michigan (2008) · Zbl 0618.73100
[38] Lew, A., Marsden, J.E., Ortiz, M., West, M.: Asynchronous variational integrators. Arch. Ration. Mech. Anal. 167(2), 85-146 (2003) · Zbl 1055.74041 · doi:10.1007/s00205-002-0212-y
[39] Lew, A., Marsden, J.E., Ortiz, M., West, M.: Variational time integrators. Int. J. Numer. Methods Eng. 60(1), 153-212 (2004) · Zbl 1060.70500 · doi:10.1002/nme.958
[40] Lew, A., Marsden, J.E., Ortiz, M., West, M.: An overview of variational integrators. In: Franca, L.P., Tezduyar, T.E., Masud, A. (eds.) Finite Element Methods: 1970’s and Beyond, CIMNE, pp. 98-115 (2004) · Zbl 1024.74036
[41] Leyendecker, S., Betsch, P., Steinmann, P.: Objective energy-momentum conserving integration for the constrained dynamics of geometrically exact beams. Comput. Methods Appl. Mech. Eng. 195, 2313-2333 (2006) · Zbl 1142.74045 · doi:10.1016/j.cma.2005.05.002
[42] Leyendecker, S., Betsch, P., Steinmann, P.: The discrete null space method for the energy consistent integration of constrained mechanical systems. Part III: flexible multibody dynamics. Multibody Syst. Dyn. 19, 45-72 (2008) · Zbl 1200.70003 · doi:10.1007/s11044-007-9056-4
[43] Leyendecker, S., Marsden, J.E., Ortiz, M.: Variational integrators for constrained dynamical systems. J. Appl. Math. Mech. 88(9), 677-708 (2008) · Zbl 1153.70004
[44] Leyendecker, S., Ober-Blöbaum, S.: A variational approach to multirate integration for constrained systems. In: Fisette, P., Samin, J.-C. (eds.) ECCOMAS Thematic Conference: Multibody Dynamics: Computational Methods and Applications, Brussels, Belgium, 4-7 July 2011 (2011) · Zbl 1311.70026
[45] Leyendecker, S., Ober-Blöbaum, S., Marsden, J.E., Ortiz, M.: Discrete mechanics and optimal control for constrained systems. Optim. Control Appl. Methods 31(6), 505-528 (2010) · Zbl 1211.49039 · doi:10.1002/oca.912
[46] Marsden, J.E., Hughes, J.R.: Mathematical Foundations of Elasticity. Dover, New York (1994) · Zbl 0545.73031
[47] Marsden, J.E., Patrick, G., Shkoller, S.: Multisymplectic geometry, variational integrators, and nonlinear PDEs. Commun. Math. Phys. 199(2), 351-395 (1998) · Zbl 0951.70002 · doi:10.1007/s002200050505
[48] Marsden, J.E., Pekarsky, S., Shkoller, S.: Discrete Euler-Poincaré and Lie-Poisson equations. Nonlinearity 12(6), 1647-1662 (1999) · Zbl 0978.37045 · doi:10.1088/0951-7715/12/6/314
[49] Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry. Springer, Berlin (1999) · Zbl 0933.70003 · doi:10.1007/978-0-387-21792-5
[50] Marsden, J.E., West, M.: Discrete mechanics and variational integrators. Acta Numer. 10, 357-514 (2001) · Zbl 1123.37327 · doi:10.1017/S096249290100006X
[51] Moser, J., Veselov, A.P.: Discrete versions of some classical integrable systems and factorization of matrix polynomials. Commun. Math. Phys. 139, 217-243 (1991) · Zbl 0754.58017 · doi:10.1007/BF02352494
[52] Ober-Blöbaum, S., Junge, O., Marsden, J.E.: Discrete mechanics and optimal control: an analysis. Control Optim. Calc. Var. 17(2), 322-352 (2011) · Zbl 1357.49120 · doi:10.1051/cocv/2010012
[53] Reissner, E.: On one-dimensional finite strain beam theory: the plane problem. J. Appl. Math. Phys. 23, 795-804 (1972) · Zbl 0248.73022 · doi:10.1007/BF01602645
[54] Reissner, R.: On a one-dimensional, large-displacement, finite-strain beam-theory. Stud. Appl. Math. 52, 87-95 (1973) · Zbl 0267.73032
[55] Romero, I., Armero, F.: An objective finite element approximation of the kinematics of geometrically exact rods and its use in the formulation of an energy-momentum scheme in dynamics. Int. J. Numer. Methods Eng. 54, 1683-1716 (2002) · Zbl 1098.74713 · doi:10.1002/nme.486
[56] Romero, I.: The interpolation of rotations and its application to finite element models of geometrically exact rods. Comput. Mech. 34, 121-133 (2004) · Zbl 1138.74406 · doi:10.1007/s00466-004-0559-z
[57] Shabana, A.: Dynamics of Multibody Systems. Cambridge University Press, Cambridge (1998) · Zbl 0932.70002
[58] Shabana, A., Yacoub, R.Y.: Three dimensional absolute nodal coordinate formulation for beam elements: theory. ASME J. Mech. Des. 123, 606-613 (2001) · doi:10.1115/1.1410100
[59] Simo, J.C.: A finite strain beam formulation. The three-dimensional dynamic problem. Part I. Comput. Meth. Appl. Mech. Eng. 49, 79-116 (1985)
[60] Simo, J.C., Marsden, J.E., Krishnaprasad, P.S.: The Hamiltonian structure of nonlinear elasticity: the material, spatial and convective representations of solids, rods and plates. Arch. Ration. Mech. Anal. 104, 125-183 (1988) · Zbl 0668.73014 · doi:10.1007/BF00251673
[61] Simo, J.C., Vu-Quoc, L.: A three-dimensional finite-strain rod model. Part II: computational aspects. Comput. Meth. Appl. Mech. Eng. 58, 55-70 (1986) · Zbl 0608.73070 · doi:10.1016/0045-7825(86)90079-4
[62] Simo, J.C., Vu-Quoc, L.: On the dynamics in space of rods undergoing large motions—a geometrically exact approach. Comput. Meth. Appl. Mech. Eng. 66, 125-161 (1988) · Zbl 0618.73100 · doi:10.1016/0045-7825(88)90073-4
[63] Tao, M., Owhadi, H., Marsden, J.E.: Nonintrusive and structure preserving multiscale integration of stiff ODEs, SDEs, and Hamiltonian systems with hidden slow dynamics via flow averaging. Multiscale Model. Simul. 8(4), 1269-1324 (2010) · Zbl 1215.65187 · doi:10.1137/090771648
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