A position-based time-stepping algorithm for vibro-impact problems with a moving set of constraints
Confluentes Mathematici, Volume 3 (2011) no. 2, pp. 263-290.

We consider a second-order measure differential inclusion describing the dynamics of a mechanical system subjected to time-dependent frictionless unilateral constraints and we assume inelastic collisions when the contraints are saturated. For this model of impact, we propose a time-stepping algorithm formulated at the position level and we establish its convergence to a solution of the Cauchy problem.

Published online:
DOI: 10.1142/S179374421100031X
Laetitia Paoli 1

1
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Laetitia Paoli. A position-based time-stepping algorithm for vibro-impact problems with a moving set of constraints. Confluentes Mathematici, Volume 3 (2011) no. 2, pp. 263-290. doi : 10.1142/S179374421100031X. https://cml.centre-mersenne.org/articles/10.1142/S179374421100031X/

[1] P. Ballard, The dynamics of discrete mechanical systems with perfect unilateral con- straints, Arch. Rational Mech. Anal. 154 (2000) 199–274.

[2] F. Bernicot and A. Lefebvre-Lepot, Existence results for nonsmooth second-order differential inclusions, convergence result for a numerical scheme and application to the modelling of inelastic collisions, Confluentes Mathematici 2 (2010) 445–471.

[3] F. Bernicot and J. Venel, Existence of solutions for second-order differential inclusions involving proximal normal cones, arXiv:1006.2292v1.

[4] F. Bernicot and J. Venel, Stochastic perturbation of sweeping process and a conver- gence result for an associated numerical scheme, arXiv:1001.3128v1.

[5] A. Bressan, Questioni di regolarita e di unicita del moto in presenza di vincoli olonomi unilaterali, Rend. Sem. Mat. Univ. Padova 29 (1959) 271–315.

[6] R. Dzonou and M. D. P. Monteiro Marques, Sweeping process for inelastic impact problem with a general inertia operator, Eur. J. Mech. A/Solids 26 (2007) 474– 490.

[7] R. Dzonou, M. D. P. Monteiro Marques and L. Paoli, A convergence result for a vibro-impact problem with a general inertia operator, Nonlinear Dynamics 58 (2009) 361–384.

[8] M. Mabrouk, A unified variational model for the dynamics of perfect unilateral con- straints, Eur. J. Mech. A/Solids 17 (1998) 819–842.

[9] B. Maury, A time-stepping scheme for inelastic collisions, numerical handling of the nonoverlapping constraint, Numer. Math. 102 (2006) 649–679.

[10] M. D. P. Monteiro Marques, Differential Inclusions in Nonsmooth Mechanical Prob- lems (Birkhäuser, 1993).

[11] J. J. Moreau, Les liaisons unilatérales et le principe de Gauss, C. R. Acad. Sci. Paris 256 (1963) 871–874.

[12] J. J. Moreau, Un cas de convergence des itérés d’une contraction d’un espace hilber- tien, C. R. Acad. Sci. Paris 286 (1978) 143–144.

[13] J. J. Moreau, Standard inelastic shocks and the dynamics of unilateral constraints, in Unilateral Problems in Structural Analysis, eds. G. Del Piero and F. Maceri, CISM courses and lectures, No. 288 (Springer-Verlag, 1985), pp. 173–221.

[14] L. Paoli, Analyse numérique de vibrations avec contraintes unilatérales, Ph.D. thesis, Université Lyon I (1993).

[15] L. Paoli, Continuous dependence on data for vibro-impact problems, Math. Models Methods Appl. Sci. 15 (2005) 53–93.

[16] L. Paoli, An existence result for non-smooth vibro-impact problems, J. Diff. Eqns. 211 (2005) 247–281.

[17] L. Paoli, Time discretization of rigid body dynamics with perfect unilateral constraints I and II, Arch. Rational Mech. Anal. 198 (2010) 457–503; 505–568.

[18] L. Paoli, A proximal-like algorithm for vibro-impact problems with a non smooth set of constraints, J. Diff. Eqns. 250 (2011) 476–514.

[19] L. Paoli and M. Schatzman, Schéma numérique pour un modèle de vibrations avec contraintes unilatérales et perte d’énergie aux impacts, en dimension finie, C. R. Acad. Sci. Paris, Série I 317 (1993) 211–215.

[20] L. Paoli and M. Schatzman, Approximation et existence en vibro-impact, C. R. Acad. Sci. Paris, Série I 329 (1999) 1103–1107.

[21] L. Paoli and M. Schatzman, Ill-posedness in vibro-impact and its numerical conse- quences, in Proc. European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS ), CD Rom (2000).

[22] L. Paoli and M. Schatzman, Penalty approximation for nonsmooth constraints in vibro-impact, J. Diff. Eqns. 177 (2001) 375–418.

[23] L. Paoli and M. Schatzman, A numerical scheme for impact problems I and II, SIAM J. Numer. Anal. 40 (2002) 702–733; 734–768.

[24] M. Schatzman, A class of nonlinear differential equations of second order in time, Nonlinear Anal. 2 (1978) 355–373.

[25] M. Schatzman, Penalty method for impact in generalized coordinates, Phil. Trans. Roy. Soc. London A 359 (2001) 2429–2446.

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