Discrete kinetic theory for 2D modeling of a moving crowd: Application to the evacuation of a non-connected bounded domain

Abstract

This paper concerns the mathematical modeling of the motion of a crowd in a non connected bounded domain, based on the kinetic and stochastic game theories. The proposed model is derived by a hybrid approach at an intermediate mesoscopic representation between micro and macro-features; pedestrians interactions with various obstacles being managed from a probabilistic perspective. A proof of the existence and uniqueness of the proposed mathematical models solution is given for large times. A numerical resolution scheme based on the Splitting method is implemented and then applied to the crowd evacuation in a non connected bounded domain with one rectangular obstacle. The evacuation time of the room is then calculated by our technique, according to the dimensions and position of a square-shaped obstacle, and finally compared to that obtained by a deterministic approach by randomly varying some of its parameters.

Vehicular traffic, crowds, and swarms: From kinetic theory and multiscale methods to applications and research perspectives

“…Discrete velocity models have been applied in 94 to compute the evacuation time from a venue that includes internal obstacles. The authors show how the qualitative analysis, proposed in 17 , for the solutions to the initial vale problem in unbounded domains can be technically generalized to include interactions with boundaries.…”

Cluster Risk of Walking Scenarios Based on Macroscopic Flow Model and Crowding Force Analysis

“…Afterwards, the Equations (2)-(4) are applied to obtain the (u, v) n+1 . Calculate ϕ n+1 by using the Godunov scheme to solve the Eikonal equation, which is the second equation of Equation (8). Stop the iteration of the fast sweeping method if the convergence of ϕ n+1 is satisfied.…”

“…As for the ϕ| (x,y)∈Γ i = 0, it changes with the ρ 0 and is calculated by Equation (8). According to the definition of potential ϕ, the exit's potential is the smallest (set to zero in this paper), and it is satisfied in the whole walking process.…”

“…He also calibrated the simulation model for emergency door by observing the children, adults and the elderly's walking behaviors. Recently, a mesoscopic approach based on the kinetic theory has been applied to model crowd dynamics [7][8][9]. It's proved to be able to reproduce the complex crowd behaviors such as avoiding obstacles, evacuating in a bounded domain.…”

Particle Methods Simulations by Kinetic Theory Models of Human Crowds Accounting for Stress Conditions

“…The development of the computational approach first accounts for the requirements presented in the previous section, namely dealing with a crowd with large size, high performance computing suitable to keep the computational time lower that the real simulated time, and accounting for the level of stress in the crowd. Some interesting results have been proposed in the literature concerning simulations of kinetic type models [19,20]. Our computational experiments have suggested the use of Monte Carlo particle methods introduced by Bird [21], within the general framework of computational methods for Boltzmann type equations [22], and subsequently extended by various authors by modifying the method depending on the specific system under consideration, for instance [23,24], see also Section 6 of [3] which is specifically focused on crowd and swarm dynamics, while the survey [25] exhaustively covers this challenging field of scientific computing.…”