On the constraints formulation in the nonsmooth generalized-alpha method
Bruls, Olivier; Acary, Vincent; Cardona, Alberrto
2018 • In Leine, Remco I.; Acary, Vincent; Bruls, Olivier (Eds.) Advanced Topics in Nonsmooth Dynamics, Transactions of the European Network for Nonsmooth Dynamics
Impact; Time integration; Differential-algebraic equation
Abstract :
[en] The simulation of flexible multibody systems with unilateral contact conditions and impacts requires advanced numerical methods. The nonsmooth generalized-alpha method was developed in order to combine an accurate and second-order time discretization of the smoother part of the dynamics and a consistent but first-order time discretization of the impulsive contributions. Compared to the Moreau-Jean scheme, this approach improves the quality of the numerical solution, especially for the representation of the vibrating response of flexible bodies. It entirely relies on the formal definition of a so-called smooth motion that captures a non impulsive part of the total nonsmooth motion. This definition may account for some contributions of the bilateral constraints and/or of the active unilateral constraints at the velocity or at the acceleration level. This chapter shows that the formulation of the constraints strongly influences the numerical stability and the computational cost of the method. A strategy for enforcing the bilateral and unilateral constraints simultaneously at the position, velocity and acceleration levels is also established, with a careful formulation of the activation criteria based on augmented Lagrange multipliers. In the special case of smooth systems, a comparison is made with more standard solvers for differential-algebraic equations. The properties of this method are demonstrated using illustrative numerical examples of smooth and nonsmooth mechanical systems.
Disciplines :
Mechanical engineering
Author, co-author :
Bruls, Olivier ; Université de Liège - ULiège > Département d'aérospatiale et mécanique > Laboratoire des Systèmes Multicorps et Mécatroniques
Acary, Vincent
Cardona, Alberrto
Language :
English
Title :
On the constraints formulation in the nonsmooth generalized-alpha method
Publication date :
2018
Main work title :
Advanced Topics in Nonsmooth Dynamics, Transactions of the European Network for Nonsmooth Dynamics
Author, co-author :
Leine, Remco I.
Acary, Vincent
Bruls, Olivier ; Université de Liège - ULiège > Département d'aérospatiale et mécanique
Acary, V., Projected event-capturing time-stepping schemes for nonsmooth mechanical systems with unilateral contact and Coulomb's friction (2013) Comput Methods Appl Mech Eng, 256 (10), pp. 224-250
Acary, V., Numerical methods for nonsmooth dynamical systems - applications in mechanics and electronics (2008) Lecture notes in applied and computational mechanics, 35. , Springer, Berlin
Alart, P., Curnier, A., A mixed formulation for frictional contact problems prone to Newton like solution methods (1991) Comput Methods Appl Mech Eng, 92, pp. 353-375
Arnold, M., Numerical methods for simulation in applied dynamics (2008) Simulation techniques in applied dynamics - CISM Lecture notes, pp. 191-246. , In: Schiehlen W, Arnold M (eds) Springer, Wien
Arnold, M., Brüls, O., Convergence of the generalized-a scheme for constrained mechanical systems (2007) Multibody Syst Dyn, 18 (2), pp. 185-202
Arnold, M., Brüls, O., Cardona, A., Convergence analysis of generalized-a Lie group integrators for constrained systems (2011) Proceedings of multibody dynamics ECCOMAS thematic conference, , Brussels
Arnold, M., Brüls, O., Cardona, A., Error analysis of generalized-a Lie group time integration methods for constrained mechanical systems (2015) Numerische Mathematik, 129, pp. 149-179
Baumgarte, J., Stabilization of constraints and integrals of motion in dynamical systems (1972) Comput Methods Appl Mech Eng, 1, pp. 1-16
Ben Gharbia, I., Gilbert, J., Nonconvergence of the plain Newton-min algorithm for linear complementarity problems with a P-matrix (2012) Math Program Ser A, 134, pp. 349-364
Bottasso, C., Bauchau, O., Cardona, A., Time-step-size-independent conditioning and sensitivity to perturbations in the numerical solution of index three differential algebraic equations (2007) SIAM J Sci Comput, 29 (1), pp. 397-414
Brüls, O., Acary, V., Cardona, A., Simultaneous enforcement of constraints at position and velocity levels in the nonsmooth generalized-a scheme (2014) Comput Methods Appl Mech Eng, 281, pp. 131-161
Brüls, O., Cardona, A., Arnold, M., Lie group generalized-a time integration of constrained flexible multibody systems (2012) Mech Mach Theory, 48, pp. 121-137
Chen, Q., Acary, V., Virlez, G., Brüls, O., A nonsmooth generalized-a scheme for flexible multibody systems with unilateral constraints (2013) Int J Numer Methods Eng, 96, pp. 487-511
Chung, J., Hulbert, G., A time integration algorithm for structural dynamics with improved numerical dissipation: The generalized-a method (1993) ASME J Appl Mech, 60, pp. 371-375
Doyen, D., Ern, A., Piperno, S., Time-integration schemes for the finite element dynamic Signorini problem (2011) SIAM J Sci Comput, 33 (1), pp. 223-249
Flores, P., Machado, M., Seabra, E., Tavares da Silva, M., A parametric study on the Baumgarte stabilization method for forward dynamics of constrained multibody systems (2010) ASME J Comput Nonlinear Dyn, 6, pp. 011, 019-011 and 019-9
Gear, C., Leimkuhler, B., Gupta, G., Automatic integration of Euler-Lagrange equations with constraints (1985) J Comput Appl Math, 12-13, pp. 77-90
Géradin, M., Cardona, A., (2001) Flexible multibody dynamics: A finite element approach, , Wiley, Chichester
Haddouni, M., Acary, V., Garreau, S., Beley, J.D., Brogliato, B., Comparison of several formulations and integration methods for the resolution of DAEs formulations in event-driven simulation of nonsmooth frictionless multibody dynamics (2017) Multibody Syst Dyn, 41, pp. 201-231
Hilber, H., Hughes, T., Taylor, R., Improved numerical dissipation for time integration algorithms in structural dynamics (1977) Earthq Eng Struct Dyn, 5, pp. 283-292
Hintermüller, M., Ito, K., Kunish, K., The primal-dual active set strategy as a semismooth Newton method (2003) SIAM J Optim, 13 (3), pp. 865-888
Hüeber, S., Stadler, G., Wohlmuth, B.I., A primal-dual active set algorithm for threedimensional contact problems with Coulomb friction (2008) SIAM J Sci Comput, 30 (2), pp. 572-596
Hüeber, S., Wohlmuth, B., A primal-dual active set strategy for non-linear multibody contact problems (2005) Comput Methods Appl Mech Eng, 194 (2729), pp. 3147-3166
Jean, M., The non-smooth contact dynamics method (1999) Comput Methods Appl Mech Eng, 177, pp. 235-257
Moreau, J., Bounded variation in time (1988) Topics in nonsmooth mechanics, pp. 1-74. , In: Moreau J, Panagiotopoulos P, Strang G (eds) Birkhäuser, Basel
Moreau, J.J., Unilateral contact and dry friction in finite freedom dynamics (1988) Non-smooth mechanics and applications, vol, pp. 1-82. , In: Moreau JJ, Panagiotopoulos P (eds) 302. Springer, New York
Newmark, N., A method of computation for structural dynamics (1959) ASCE J Eng Mech Div, 85, pp. 67-94
Paoli, L., Schatzman, M., A numerical scheme for impact problems I: The one-dimensional case (2002) SIAM J Numer Anal, 40, pp. 702-733
Paoli, L., Schatzman, M., A numerical scheme for impact problems II: The multidimensional case (2002) SIAM J Numer Anal, 40, pp. 734-768
Pfeiffer, F., On non-smooth dynamics (2008) Meccanica, 43, pp. 533-554
Pfeiffer, F., Foerg, M., Ulbrich, H., Numerical aspects of non-smooth multibody dynamics (2006) Comput Methods Appl Mech Eng, 195, pp. 6891-6908
Pfeiffer, F., Glocker, C., Multibody dynamics with unilateral contacts (2004) Wiley series in nonlinear science, , Wiley-VCH, Weinheim
Schindler, T., Acary, V., Timestepping schemes for nonsmooth dynamics based on discontinuous galerkin methods: Definition and outlook (2014) Math Comput Simul, 95, pp. 180-199
Schindler, T., Rezaei, S., Kursawe, J., Acary, V., Half-explicit timestepping schemes on velocity level based on time-discontinuous Galerkin methods (2015) Comput Methods Appl Mech Eng, 290, pp. 250-276
Schoeder, S., Ulbrich, H., Schindler, T., Discussion on the Gear-Gupta-Leimkuhler method for impacting mechanical systems (2013) Multibody Syst Dyn, 31, pp. 477-495
Studer, C., Leine, R.I., Glocker, C., Step size adjustment and extrapolation for time-stepping schemes in non-smooth dynamics (2008) Int J Numer Methods Eng, 76 (11), pp. 1747-1781
