[Paper Review] A numerical scheme for impact problems

Lætitia Paoli, Michelle Schatzman|ArXiv.org|Oct 4, 1999

Dynamics and Control of Mechanical Systems14 references20 citations

TL;DR

This paper presents a novel numerical scheme for solving mechanical impact problems with n degrees of freedom, where motion is constrained within a C³-regular closed set K. The scheme avoids explicit impact time detection by using a localized finite difference approach with a coefficient of restitution e ∈ [0,1], and proves convergence to a solution, establishing both local and global existence under a priori estimates.

ABSTRACT

We consider a mechanical system with impact and n degrees of freedom, written in generalized coordinates. The system is not necessarily Lagrangian. The representative point of the system must remain inside a set of constraints K; the boundary of K is three times differentiable. At impact, the tangential component of the impulsion is conserved, while its normal coordinate is reflected and multiplied by a given coefficient of restitution e between 0 and 1. The orthognality is taken with respect to the natural metric in the space of impulsions. We define a numerical scheme which enables us to approximate the solutions of the Cauchy problem: this is an ad hoc scheme which does not require a systematic search for the times of impact. We prove the convergence of this numerical scheme to a solution, which yields also an existence result. Without any a priori estimates, the convergence and the existence are local; with some a priori estimates, the convergence and the existence are proved on intervals depending exclusively on these estimates. This scheme has been implemented with a trivial and a non trivial mass matrix.

Motivation & Objective

• To develop a numerical method for mechanical systems with impacts that avoids explicit detection of impact times.
• To handle non-Lagrangian systems with general dissipative forces and constraints defined by a C³-regular set K.
• To establish convergence of the numerical scheme to a solution of the continuous-time impact problem.
• To prove local and global existence of solutions using a priori estimates and scheme convergence.
• To rigorously address the challenge of boundary localization in finite difference schemes with quadratic errors from boundary straightening.

Proposed method

• The scheme uses generalized coordinates and a mass matrix M(u) to model dynamics, with impulsive reactions μ represented as λdϕ(u) on the boundary ∂K.
• It employs a finite difference discretization in time with a local reconstruction near the boundary ∂K to handle the non-smooth impact dynamics.
• The method enforces conservation of tangential impulse and reflection of normal impulse with coefficient of restitution e via a cotangent metric-based orthogonality condition.
• A key innovation is the use of a localized scheme that approximates the dynamics near ∂K while controlling quadratic errors introduced by straightening the boundary.
• The convergence proof relies on a priori estimates and compactness arguments, showing that the discrete solution sequence converges to a continuous solution.
• The scheme is proven convergent in the sense of essential supremum of velocity norm, using estimates on the discrete velocity norm |V^m|_{U^m}^2.

Experimental results

Research questions

• RQ1 Can a numerical scheme be constructed for impact problems without requiring explicit detection of impact times?
• RQ2 How can the non-smooth dynamics at the boundary ∂K be approximated using finite differences when the boundary is curved?
• RQ3 What conditions ensure the convergence of the numerical scheme to a true solution of the impact problem?
• RQ4 How do a priori estimates on momentum and position influence the global existence of solutions?
• RQ5 What role does the coefficient of restitution e play in the stability and convergence of the numerical scheme?

Key findings

• The numerical scheme converges to a solution of the continuous impact problem, proving the existence of solutions for initial conditions in the admissible set.
• Convergence and existence are established locally without a priori estimates, and globally when such estimates are available.
• The scheme is proven to converge in the sense that the discrete velocity norm |V^m|_{U^m}^2 converges to the essential supremum of the continuous velocity norm.
• A contradiction is derived if the scheme fails to converge beyond a time τ(R), which is determined by the initial momentum and the geometry of K.
• The proof relies on a localization technique near ∂K, where boundary straightening introduces quadratic errors that are carefully controlled via estimates.
• The global existence result is achieved by combining the local convergence with a priori bounds on momentum and position, ensuring the scheme can be extended to time τ(R) depending only on these bounds.