Handling of Contacts in Crowd Motion Simulations

This paper is concerned with the modelling of crowd motion in highly packed configurations. We are especially interested in describing evacuation situations: people want to escape from a room, a building, a railway station or a plane, that may contain obstacles (walls, seats, tables, ...). Many strategies have been followed to model crowd motions: macroscopic models [14, 9, 15], Cellular Automata [2, 16, 3], queuing models [17, 23] and microscopic models [8, 11]. The possibility that people may actually get into contact is usually handled indirectly (repulsive forces or adapted cut-off for microscopic models, exclusion principle for Cellular Automata or queuing models). We propose here to integrate a strong non-overlapping constraint in a ODE framework, in the spirit of granular flow models. Our approach rests on two principles. On the one hand, we define a spontaneous velocity, which corresponds to the velocity each individual would like to have in the absence of other people. On the other hand, individuals (which are identified to rigid discs) must obey a non-overlapping constraint. Those two principles lead us to define the actual velocity field as the projection of the spontaneous velocity over the set of admissible velocities (regarding the nonoverlapping constraints). To perform this projection, we put the problem in a saddle-point form, which leads us to introduce a collection of Lagrange multipliers. Those Lagrange multipliers can be interpreted as interaction pressure between people which are in contact.