Estimating contact forces and pressure in a dense crowd: Microscopic and macroscopic models

This paper deals with the estimation of pressure at collisions times during the movement of a dense crowd. Through the non-smooth contact dynamics approach for rigid and deformable solids, proposed by Frémond and his collaborators, the value of pressure and contact forces at collisions points, generated through congestion or panic situation are estimated. Firstly, we propose a second-order microscopic model, in which the crowd is treated as a system of rigid solids. Contact forces are rigorously defined by taking into account multiple simultaneous contacts and the non-overlapping condition between pedestrians. We show that for a dense crowd, percussions can be seen as contact forces. Secondly, in order to overcome the restrictive hypothesis related to the geometric form adapted to model the pedestrian, a continuous equivalent approach is proposed where the crowd is modeled as a deformable solid, the pressure is then defined by the divergence of the stress tensor and calculated according to volume and surface constraints. This approach makes it possible to retain an admissible right-velocity, including both the non-local interactions between non-neighbor pedestrians and the choice of displacement strategy of each pedestrian. Finally, the comparison between the two proposed approaches and some other existing approaches are presented on several illustrative examples to estimate the contact forces between pedestrians.