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

The paper presents and analyzes the iterative rules determining the impulsive behavior of a rigid disk having a single or possibly multiple frictionless impact with two walls forming a corner. In the first part, two theoretical iterative rules are presented for the cases of ideal impact and Newtonian frictionless impact with global dissipation index. In the second part, it is presented a numerical version of both the theoretical algorithms. The termination analysis of the algorithms differentiates the twocases: in the ideal case, it is shown 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, it is shown that its velocity decreases to zero, so that the termination of the algorithm can be fixed through an ``almost at rest'' condition. It is shown that the stable version of the algorithm is more robust than the theoretical ones with respect to noisy initial data and floating point arithmetic computation. The outputs of the stable and theoretical versions of the algorithms are compared, showing that they are similar, even if not coincident, outputs. Moreover, the outputs of the stable version of the algorithm in some meaningful cases are graphically presented and discussed. The paper clarifies the applicability of theoretical methods presented in Pasquero, S. (2018): "Ideal characterizations of multiple impacts: A frame–independent approach by means of jet–bundle geometry." Quarterly of Applied Mathematics, 76(3), by analyzing the paradigmatic case of the disk in the corner