Geometric Numerical Integration of Inequality Constrained, Nonsmooth Hamiltonian Systems

TL;DR

This paper develops geometric numerical methods for Hamiltonian systems with inequality constraints, ensuring structure preservation even during frequent boundary interactions and nonsmooth dynamics, demonstrated through challenging numerical examples.

Contribution

It introduces a family of variational integrators that handle inequality constraints and nonsmooth behavior while preserving key geometric properties.

Findings

Methods preserve momentum and approximate energy conservation.

They handle multiple simultaneous inequality constraints effectively.

Numerical experiments demonstrate robustness in complex nonsmooth scenarios.

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. We additionally, however, also consider invariant preservation over persistent, simultaneous and/or frequent boundary interactions. Appropriately formulating geometric methods to include such conditions has long-remained challenging due to the inherent nonsmoothness 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. Leveraging discrete variational techniques, we construct a family of geometric numerical integration methods that not only obtain the usual desirable properties of momentum preservation, approximate energy conservation and equality constraint preservation, but 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.