The nonsmooth generalized-α method for flexible and rigid multibody system dynamics

GALVEZ, JAVIER; BRÜLS, OLIVIER; CARDONA, ALBERTO; COSIMO, ALEJANDRO; CAVALIERI, FEDERICO J.

Congreso; ECCOMAS Multibody Dynamics Conference 2019; 2019

The aim of this work is the development of methods for the numerical simulation of the nonsmooth dynamics of multibody systems involving rigid and/or flexible elements which can be subject to frictionless contacts and impacts. These systems are characterized by bilateral constraints which are associated to kinematic joints interconnecting the bodies, and by unilateral constraints stemming from the frictionless contacts and impacts. An additional difficulty comes from the presence of flexible elements, with vibration effects that need to be efficiently captured by the numerical scheme.For the robust and accurate simulation of such systems, special attention must be paid to the adopted time integration scheme as it not only has to deal successfully with the nonsmooth character of the problem but also with the vibration effects. Time integrators for nonsmooth dynamics can be classified in two main groups: event-driven and time-stepping integrators. The former are based on the exact detection of impacts by accordingly adapting the time step size. However, they become inefficient in situations involving a large number of impact events. A different strategy is adopted in this work, which falls under the category of timestepping integrators. These techniques share the common feature that the time step size does not need to be adapted to impact events. The most widespread time-stepping integrators for nonsmooth dynamical systems are the Schatzman?Paoli scheme [1, 2], which is based on a central difference scheme, and the Moreau?Jean scheme [3, 4, 5], which is based on a q-method. Despite their robustness for dealing with problems involving a large number of impacts, they generally lead to poor approximations of vibration phenomena. Additionally, in the Moreau?Jean scheme, the constraints are only imposed at velocity level which leads to the violation of the constraints at position level and, in consequence, a drift phenomenon is observed [6].These problems have been studied in the literature of nonsmooth dynamics, mainly dealing with the use of higher order integrators for the smooth or free-flight part of the motion [6, 7, 8, 9, 10], and with the simultaneous imposition of the constraints at position and at velocity levels [11, 12, 6]. The imposition of the constraints at acceleration level was also analyzed recently by Brüls et al. [10]. The current work takes as starting point the nonsmooth generalized-a (NSGA) introduced by Brüls et al. in [6] and proposes a modification to improve its robustness for problems with nonlinear bilateral constraints and/or flexible components.The NSGA deals with the transient simulation of nonsmooth dynamical systems comprised of rigid and/or flexible bodies, kinematic joints and frictionless contacts. It is characterized by the splitting of the involved fields into a smooth and a (nonsmooth) impulsive contribution, where the former is integrated with second order accuracy by means of the generalized-a scheme and the latter with first-order accuracy. Also, the involved unilateral and bilateral constraints are exactly satisfied both at position and at velocity levels. This results in a numerical scheme which involves three coupled subsets of equations or sub-problems to be solved at each time step: one for the smooth prediction of the motion, and two others for correcting that prediction at position and at velocity levels with the nonsmooth contributions. The existing coupling stems from the adopted splitting in which the smooth sub-problem depends on the position correction and on the velocity jump. Therefore, if a semi-smooth Newton approach is used to solve the derived formulation without making any additional assumption, at each nonlinear iteration the method would have to deal monolithically with the complete set of unknowns. In order to avoid this issue, Brüls et al. [6] proposed to neglect the terms coupling the smooth sub-problem with the other ones from the tangent matrix. The advantage of this procedure is that the algorithm can be described as a sequence of three sub-problems, instead of having to solve the complete set of equations monolithically. This approximation is fully justified when the adopted step size tends to zero. However, for problems with flexible bodies and nonlinear bilateral constraints, this approximation led to a slow convergence of the global scheme, or even to the divergence of the scheme if a small enough step size was not adopted. In order to overcome this difficulty, the current work proposes to modify the way to do the splitting in order to ensure a full decoupling of the different subsets of equations, so that the three sub-problems can be processed in a sequential decoupled manner without any approximation. Here, the adjective decoupled is used because the resulting numerical scheme involves the sequential solution of three sub-problems without neglecting any term in the discrete problem. This implies that the proposed decoupled scheme results in a robust alternative especially for problems characterized by nonlinear bilateral constraints and flexible elements.Examples of application to impacts with and without friction, and also simultaneous impact problems with rigid and flexible bodies are presented.[1] Paoli, L., and Schatzman, M., 2002. ?A numerical scheme for impact problems I: The one-dimensional case?. SIAM Journal on Numerical Analysis, 40(2), pp. 702?733.[2] Paoli, L., and Schatzman, M., 2002. ?A numerical scheme for impact problems II: The multidimensional case?. SIAM Journal on Numerical Analysis, 40(2), pp. 734?768.[3] Jean, M., and Moreau, J. J., 1987. ?Dynamics in the presence of unilateral contacts and dry friction: A numerical approach?. In Unilateral Problems in Structural Analysis ? 2. Springer Vienna, pp. 151?196.[4] Moreau, J. J., 1988. ?Unilateral contact and dry friction in finite freedom dynamics?. In Nonsmooth Mechanics and Applications. Springer Vienna, pp. 1?82.[5] Jean, M., 1999. ?The non-smooth contact dynamics method?. Computer Methods in Applied Mechanics and Engineering, 177(3-4), pp. 235?257.[6] Brüls, O., Acary, V., and Cardona, A., 2014. ?Simultaneous enforcement of constraints at position and velocity levels in the nonsmooth generalized-alpha scheme?. Computer Methods in Applied Mechanics and Engineering, 281, pp. 131 ? 161.[7] Chen, Q., Acary, V., Virlez, G., and Brüls, O., 2013. ?A nonsmooth generalized-a scheme for flexible multibody systems with unilateral constraints?. International Journal for Numerical Methods in Engineering, 96(8), pp. 487?511.[8] Schindler, T., and Acary, V., 2014. ?Timestepping schemes for nonsmooth dynamics based on discontinuous galerkin methods: Definition and outlook?. Mathematics and Computers in Simulation, 95, pp. 180?199.[9] Schindler, T., Rezaei, S., Kursawe, J., and Acary, V., 2015. ?Half-explicit timestepping schemes on velocity level based on time-discontinuous galerkin methods?. Computer Methods in Applied Mechanics and Engineering, 290, pp. 250?276.[10] Brüls, O., Acary, V., and Cardona, A., 2018. ?On the constraints formulation in the nonsmooth generalized-a method?. In Advanced Topics in Nonsmooth Dynamics. Springer International Publishing, pp. 335?374.[11] Acary, V., 2013. ?Projected event-capturing time-stepping schemes for nonsmooth mechanical systems with unilateral contact and coulomb?s friction?. Computer Methods in Applied Mechanics and Engineering, 256, pp. 224?250.[12] Schoeder, S., Ulbrich, H., and Schindler, T., 2013. ?Discussion of the gear-gupta-leimkuhler method for impacting mechanical systems?. Multibody System Dynamics, 31(4), pp. 477?495.