Optimal location of exit doors for efficient evacuation of crowds at gathering places

Abstract

This work deals with the optimal design for the location of the exit doors at meeting places (such as sports centers, public squares, street markets, transport stations, etc.) to guarantee a safer emergency evacuation in events of a sporting, social, entertainment or religious type. This problem is stated as an optimal control problem of nonlinear partial differential equations, where the state system is a reformulation of the Hughes model (coupling the eikonal equation for a density-weighted walking velocity of pedestrians and the continuity equation for conservation of the pedestrian density), the control is the location of the exit doors at the domain boundary (subject to several geometric constraints), and the cost function is related to the evacuation rate. We provide a full numerical algorithm for solving the problem (a finite element technique for the discretization and a gradient-free procedure for the optimization), and show several numerical results for a realistic case.