Bertrand Maury

Modelling of crowd motion in panic situation.

B. Maury, J. Venel

We propose a microscopic model for crowd motion. We are especially interested in describing panic situations : people want to leave a room, building, railway station or a plane, that may contain obstacles. Our model 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 to the non-overlapping constraints). The model takes the form of a differential inclusion, for which well-posedness can be established by means of recent abstract results in convex analysis.
The non-overlapping constraints are  handled by means of  Lagrange multipliers that can be interpreted as pressure forces applied on each individual by its neighbors. We propose a time discretization scheme based on granular flow principles, which makes it possible to  simulate the evacuation of thousands of  highly-packed individuals. Simulations of some typical situations will be presented.
We shall also propose a macroscopic version of this model, and present the theoretical and modelling issues that it raises.