Pedestrian flows in bounded domains with obstacles

Piccoli, Benedetto; Tosin, Andrea · 2009 · Crossref

DOI: 10.1007/s00161-009-0100-x

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Summary

This paper presents a macroscopic mathematical model for pedestrian flows in bounded domains containing obstacles, utilizing a framework of time-evolving measures. The research is motivated by the need for efficient modeling tools to address complex crowd dynamics, such as those observed in infrastructure failures like the London Millennium Footbridge or disasters at the Jamarat Bridge. Existing microscopic models, which track individual agents via ordinary differential equations, are computationally intensive and difficult to analyze globally. Conversely, traditional macroscopic models based on nonlinear hyperbolic conservation laws face significant analytical and numerical challenges, particularly in two-dimensional settings and when handling boundary conditions with obstacles. The authors aim to overcome these limitations by applying a measure-theoretical approach that simplifies the handling of multi-dimensional applications and nonlocal interactions. The proposed method is a discrete-time Eulerian model where space occupancy is described by a sequence of positive Radon measures. The dynamics are governed by a push-forward recursive relation, where the pedestrian distribution at time $n+1$ is derived from the distribution at time $n$ via a motion mapping. This mapping incorporates two fundamental behavioral aspects: the desire to reach specific targets, which determines the primary direction of motion, and the tendency to avoid crowding, which introduces interactions among individuals. The model assumes pedestrians evaluate crowding primarily within their frontal visibility field. Mathematically, the approach ensures well-posedness, including existence, uniqueness, and a priori estimates of solutions, provided the velocity field satisfies specific Lipschitz continuity conditions. This framework allows for a straightforward transition from Lagrangian to Eulerian descriptions without heavy reliance on regularity issues, facilitating easier numerical implementation compared to hyperbolic PDEs. The study demonstrates the model’s capability to reproduce experimental evidences of pedestrian behavior through numerical simulations. Key findings include the successful reproduction of lane formation in parallel flows and the emergence of alternate lanes in crossing flows, phenomena characteristic of self-organization in pedestrian crowds. The model effectively handles motion in complex domains scattered with obstacles, such as pedestrians navigating toward escalators or exiting rooms with pillars. Additionally, the authors show that by modifying the visibility assumption to include a rear area, the model can qualitatively reproduce flocking behaviors, though this is noted as beyond the primary scope. The measure-theoretical approach proves capable of capturing these macroscopic patterns while remaining analytically and numerically more tractable than existing hyperbolic models. The significance of this work lies in providing a robust, efficient mathematical framework for simulating pedestrian flows in realistic, obstacle-rich environments. By avoiding the complexities of nonlinear hyperbolic conservation laws, the model offers a practical tool for engineering applications, such as optimizing walkway infrastructure and improving safety in crowded spaces. The approach bridges the gap between microscopic individual behaviors and macroscopic crowd dynamics, offering a clear method for analyzing crowd-structure coupling and predicting critical circumstances that may lead to accidents. This contributes to the broader field of multi-agent system modeling by demonstrating the utility of time-evolving measures in capturing complex, self-organizing behaviors.

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• pedestrian behavior perception