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A numerical scheme for impact problems

Laetitia Paoli, Michelle Schatzman

TL;DR

This work presents a projection-based numerical scheme for finite-dimensional dynamics with unilateral constraints and impact, modeled in a Riemannian setting with mass matrix M(u). It proves local (and under a priori bounds, global) existence by showing convergence of the scheme to a solution of the constrained, measure-driven dynamics, and it derives the energy and impulse transmission laws at impact: the tangential impulse is conserved while the normal component is reflected by a coefficient of restitution e. The analysis combines BV compactness, measure convergence, and a careful handling of the constraint boundary via local coordinates and straightening, yielding both a rigorous existence theory and a numerically advantageous method that avoids explicit event-detection. The results apply to both trivial and nontrivial mass matrices and provide a framework for understanding energy transfer across impacts in constrained mechanical systems.

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.

A numerical scheme for impact problems

TL;DR

This work presents a projection-based numerical scheme for finite-dimensional dynamics with unilateral constraints and impact, modeled in a Riemannian setting with mass matrix M(u). It proves local (and under a priori bounds, global) existence by showing convergence of the scheme to a solution of the constrained, measure-driven dynamics, and it derives the energy and impulse transmission laws at impact: the tangential impulse is conserved while the normal component is reflected by a coefficient of restitution e. The analysis combines BV compactness, measure convergence, and a careful handling of the constraint boundary via local coordinates and straightening, yielding both a rigorous existence theory and a numerically advantageous method that avoids explicit event-detection. The results apply to both trivial and nontrivial mass matrices and provide a framework for understanding energy transfer across impacts in constrained mechanical systems.

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.

Paper Structure

This paper contains 9 sections, 19 theorems, 387 equations.

Table of Contents

  1. Introduction
  2. The heart of the estimates
  3. Existence of $(U^m)_{0 \le m \le \lfloor \tau/h \rfloor}$ for some $\tau>0$
  4. Estimates on the acceleration
  5. Variational properties of the limit of the numerical scheme
  6. Transmission of energy during impact
  7. Initial conditions
  8. A priori estimates
  9. Global results

Key Result

lemma 1

Let the real-valued sequence $\bigl(y^m\bigr)_m$ satisfy the following recurrence relation for all $m\ge 1$: Then, for all $m\ge 2$, the discrete velocity satisfies the estimate

Theorems & Definitions (37)

  • lemma 1
  • proof
  • lemma 2
  • proof
  • lemma 3
  • proof
  • theorem 1
  • proof
  • theorem 2
  • proof
  • ...and 27 more