A discrete contact model for crowd motion

TL;DR

This paper introduces a mathematical model for crowd motion in dense situations, using a projection-based velocity approach and a sweeping process framework, supported by a numerical scheme and illustrative simulations.

Contribution

It develops a novel discrete contact model for crowd dynamics based on sweeping processes and provides a numerical algorithm for simulation.

Findings

Model effectively handles high-density crowd scenarios.

Numerical scheme successfully simulates crowd movement.

Framework grounded in recent mathematical advances on sweeping processes.

Abstract

The aim of this paper is to develop a crowd motion model designed to handle highly packed situations. The model we propose rests on two principles: We first define a spontaneous velocity which corresponds to the velocity each individual would like to have in the absence of other people; The actual velocity is then computed as the projection of the spontaneous velocity onto the set of admissible velocities (i.e. velocities which do not violate the non-overlapping constraint). We describe here the underlying mathematical framework, and we explain how recent results by J.F. Edmond and L. Thibault on the sweeping process by uniformly prox-regular sets can be adapted to handle this situation in terms of well-posedness. We propose a numerical scheme for this contact dynamics model, based on a prediction-correction algorithm. Numerical illustrations are finally presented and discussed.