Comparison of several formulations and integration methods for the resolution of DAEs formulations in event-driven simulation of nonsmooth frictionless multibody dynamics

Abstract

International audienceThis article is devoted to the comparison of numerical integration methods for nonsmooth multibody dynamics with joints, unilateral contacts and impacts in an industrial context. With an event–driven strategy, the smooth dynamics, which is integrated between two events, may be equivalently formulated as a Differential Algebraic Equation (DAE) of index 1, 2 or 3. It is well-known that these reformulations are no longer equivalent when a numerical time–integration technique is used. The drift-off effect and the stability of the numerical scheme strongly depend on the index of the formulation. But, besides the standard properties of accuracy and stability of the DAE solvers, the event–driven context imposes some further requirements that are crucial for a robust and efficient event-driven strategy. In this article, several state–of–the–art numerical time integration methods for each formulation are compared: the generalized-α scheme for index-3 formulation and stabilized index-2 formulation, (Partitioned) Runge–Kutta Half–Explicit Method of order 5 (HEM5 and PHEM56) for index-2 DAEs with projection techniques, and Runge-Kutta explicit scheme of order 5, the Dormand-Prince scheme (DOPRI5), for index-1 DAEs with projection techniques (MDOP5). We compare these schemes in terms of efficiency, violation of the constraints and the way they handle stiff dynamics on numerous industrial benchmarks, where a CAD software is in this loop. One of the major conclusions is that the index-2 DAEs solvers prove to be better than other schemes to maintain low violations at position and acceleration levels. The best compromise allows us to design efficient event–driven solvers. When the dynamics is stiff, implicit schemes outperform explicit and half-explicit methods whichare sometimes unable to compute the dynamics when the system’s frequency range is wide. Furthermore, in industrial context, some solvers fail to reproduce the properties that they enjoy in theory. This is particularly true for half-explicit schemes when the Jacobian of the constraints has not full rank

An algorithmic approach to the multiple impact of a disk in a corner

“…Moreau, 1988;Lötstedt, 1982), the DAE and HHT methods (see e.g. Haddouni et al, 2017;Negrut et al, 2007), the differential inclusions and measure derivative methods (Kiseleva et al, 2018;Monteiro Marques, 1993), analytical or numerical integration methods (Moreau, 1999;Paoli and Schatzman, 2002;Liu et al, 2008). The wide bibliographies of the books of Brogliato (2016) and Pfeiffer and Glocker (2000) give a clear idea of the hugeness of the state of the art about these methods.…”

Dynamic modeling and simulation of rigid-flexible coupling cable system by absolute nodal coordinate formulation

“…Compared with the explicit algorithm, the implicit algorithm is more stable with higher calculation efficiency. Therefore, the implicit integral method is preferred for the multi-body system dynamics 23 . Runge Kutta algorithm is often used to solve nonlinear ordinary differential equations, however, the Explicit Runge Kutta method has limitations of stabilization regions, so rigid body dynamics equations are generally solved by Implicit Runge Kutta, which requires more computation cost.…”

Dynamics Modeling of Gear Transmissions with Asymmetric Load-Dependent Friction

“…To deal with such problems, the implicit (backward) Euler discretization method has been known to be useful, especially for the time integration of systems involving Coulomb friction [24,[36][37][38][39]. A higher integration accuracy might be achieved by more sophisticated integration schemes such as event-driven schemes [40] and implicit Runge-Kutta schemes [41], but it is left outside the scope of this paper.…”