Geometric Numerical Integration of Inequality Constrained, Nonsmooth Hamiltonian Systems

Abstract

Abstract. We consider the geometric numerical integration of Hamiltonian systems subject to both equality and "hard" inequality constraints. As in the standard geometric integration setting, we target long-term structure preservation. Additionally, however, we also consider invariant preservation over persistent, simultaneous, and/or frequent boundary interactions. Appropriately formulating geometric methods for these cases has long remained challenging due the inherent nonsmoothness and one-sided conditions that they impose. To resolve these issues we thus focus both on symplectic-momentum preserving behavior and the preservation of additional structures, unique to the inequality constrained setting. Toward these goals we introduce, for the first time, a fully nonsmooth, discrete Hamilton's principle and obtain an associated framework for composing geometric numerical integration methods for inequality-equality-constrained systems. Applying this framework, we formulate a new family of geometric numerical integration methods that, by construction, preserve momentum and equality constraints and are observed to retain good long-term energy behavior. Along with these standard geometric properties, the derived methods also enforce multiple simultaneous inequality constraints, obtain smooth unilateral motion along constraint boundaries, and allow for both nonsmooth and smooth boundary approach and exit trajectories. Numerical experiments are presented to illustrate the behavior of these methods on difficult test examples where both smooth and nonsmooth active constraint modes persist with high frequency.Key words. geometric integration, Hamiltonian systems, inequality constraints, nonsmooth systems, impact, contact dynamics AMS subject classifications. 65P10, 65L05, 82B80 DOI. 10.1137/100800105 1. Introduction. Recent attention has focused on geometric numerical integration methods that closely preserve invariants of continuous systems (Sanz-Serna and Calvo (1994);Hairer, Lubich, and Wanner (2002); Leimkuhler and Reich (2004)). In particular, symplectic-momentum preserving integration schemes have been successfully applied over a wide range of applications. In the unconstrained setting, these methods exactly preserve the symplectic form and momenta and maintain good long-term energetic behavior by approximately conserving energy (up to an additive constant) over long spans of simulation.While extensions of symplectic methods to equality-constrained systems have been extensively investigated (Hairer, Lubich, and Wanner (2002); Leimkuhler and Reich (2004)), appropriately formulating symplectic methods to include inequality constraints continues to remain a challenging problem. The difficulty in treating these latter cases stems from the inherent nonsmoothness imposed by such conditions. Stewart (2000), in particular, notes that, despite good conservative properties in the smooth setting, symplectic methods do not necessarily translate easily into nonsmooth problems. Direct extensions of existing symplectic methods (e....

Using Nesterov's Method to Accelerate Multibody Dynamics with Friction and Contact

“…Nonetheless, this has been the approach of choice in the vast majority of numerical studies of the dynamics of large rigid body systems; see, for instance, Cundall [1971Cundall [ , 1988, Cundall and Strack [1979], Jaeger et al [1996], Brilliantov et al [1996], Vu-Quoc and Zhang [1999], Vu-Quoc et al [2004], Luding [2005], and Pöschel and Schwager [2005]. Finally, in terms of (iii), the randomness can be tied back to the undesired influence that the integration step size has on the surrogate deformation at the next time step, (see discussion in Kaufman and Pai [2012]). …”

“…A mass splitting procedure was embedded into a DVI formulation to improve numerical stability and reduce jitter for piles and stacks of objects in Tonge et al [2012]. The overall solution methodology adopted herein shares several traits with the one proposed in Kaufman and Pai [2012]: both impose complementarity between normal force and the distance gap instead of bringing into the discussion the relative velocities at the contact point. Furthermore, like in Kaufman and Pai [2012], symplectic integration is considered for numerical discretization of the equations of motion.…”

“…The overall solution methodology adopted herein shares several traits with the one proposed in Kaufman and Pai [2012]: both impose complementarity between normal force and the distance gap instead of bringing into the discussion the relative velocities at the contact point. Furthermore, like in Kaufman and Pai [2012], symplectic integration is considered for numerical discretization of the equations of motion. While here the discretization is based on a half-implicit Euler method, Kaufman and Pai [2012] present an approach that accommodates a spectrum of formulas: implicit midpoint, Newmark, and an implicit-explicit formula due to Kane et al [1999].…”

“…Furthermore, like in Kaufman and Pai [2012], symplectic integration is considered for numerical discretization of the equations of motion. While here the discretization is based on a half-implicit Euler method, Kaufman and Pai [2012] present an approach that accommodates a spectrum of formulas: implicit midpoint, Newmark, and an implicit-explicit formula due to Kane et al [1999]. In handling impact and constraint drift, the approach of Kaufman and Pai [2012] is more versatile compared to ours, which in the current implementation handles inelastic collision and relies on a Baumgarte-style penalty to prevent constraint violation or drift.…”

“…While here the discretization is based on a half-implicit Euler method, Kaufman and Pai [2012] present an approach that accommodates a spectrum of formulas: implicit midpoint, Newmark, and an implicit-explicit formula due to Kane et al [1999]. In handling impact and constraint drift, the approach of Kaufman and Pai [2012] is more versatile compared to ours, which in the current implementation handles inelastic collision and relies on a Baumgarte-style penalty to prevent constraint violation or drift. However, on the upside, the approach herein handles friction and is guaranteed to produce a global solution for systems with friction and contact.…”