# An algorithmic approach to the multiple impact of a disk in a corner

## TL;DR

This paper develops and analyzes algorithmic procedures to determine the impulsive behavior of a rigid disk impacting with two walls forming a corner, covering ideal and non-ideal frictionless impacts, with theoretical and numerical insights.

## Contribution

It introduces two theoretical algorithms for impact analysis and a robust numerical version, providing termination criteria and behavior predictions for ideal and non-ideal impacts.

## Key findings

- The ideal impact algorithm always terminates with the disk exiting the corner.
- In the non-ideal case, the disk's velocity decreases to zero, but finite termination isn't guaranteed.
- The numerical algorithms are more robust to noisy data and floating point errors.

## Abstract

We present the algorithmic procedure determining the impulsive behavior of a rigid disk having a single or possibly multiple frictionless impact with two walls forming a corner. The algorithmic procedure represents an application of the general theory of multiple impacts as presented in \cite{Pasquero2016Multiple} for the ideal case. In the first part, two theoretical algorithms are presented for the cases of ideal impact and Newtonian frictionless impact with global dissipation index. The termination analysis of the algorithms differentiates the two cases: in the ideal case, we show that the algorithm always terminates and the disk exits from the corner after a finite number of steps independently of the initial impact velocity of the disk and the angle formed by the walls; in the non--ideal case, although is not proved that the disk exits from the corner in a finite number of steps, we show that its velocity decreases to zero and the termination of the algorithm can be fixed through an "almost at rest" condition. In the second part, we present a numerical version of both the theoretical algorithms that is more robust than the theoretical ones with respect to noisy initial data and floating point arithmetic computation. Moreover, we list and analyze the outputs of the numerical algorithm in several cases.