A numerical scheme for impact problems

TL;DR

This paper introduces a new numerical scheme for simulating impact problems in mechanical systems with constraints, ensuring convergence and existence of solutions without needing impact time searches.

Contribution

The paper presents an ad hoc numerical scheme for impact problems that does not require systematic impact time search and proves its convergence and existence of solutions.

Findings

Scheme successfully approximates impact solutions

Convergence proven under local and a priori estimate conditions

Implemented with different mass matrices

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.