Networks and Heterogeneous Media
September 2011 , Volume 6 , Issue 3
Special issue on Crowd Dynamics: Results and Perspectives
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The research about crowd dynamics has undergone a dramatic development in the recent years. This fast advancement made it rather difficult for researchers in applied mathematics to keep contacts with the variety of analytical and numerical techniques recently introduced, as well as with the new problems being considered. Indeed, Crowd Dynamics is of interest to disciplines ranging from pure mathematical analysis to operation research, from numerical analysis to computer graphics, from model theory to statistics. The variety of MSC classifications labeling the papers of this special issue testifies the broadness of the subjects covered hereafter and, hence, also of this whole field.
This special issue of Networks and Heterogeneous Media aims at bridging several different research directions of interest to applied mathematicians. Each of the present papers describes key problems of particular interest for the authors, points at the related most relevant techniques and includes the corresponding main results. The common spirit is to share, also with non specialists of the very same field, achieved results in their full depth.
We extend the Aw-Rascle macroscopic model of car traffic into a two-way multi-lane model of pedestrian traffic. Within this model, we propose a technique for the handling of the congestion constraint, i.e. the fact that the pedestrian density cannot exceed a maximal density corresponding to contact between pedestrians. In a first step, we propose a singularly perturbed pressure relation which models the fact that the pedestrian velocity is considerably reduced, if not blocked, at congestion. In a second step, we carry over the singular limit into the model and show that abrupt transitions between compressible flow (in the uncongested regions) to incompressible flow (in congested regions) occur. We also investigate the hyperbolicity of the two-way models and show that they can lose their hyperbolicity in some cases. We study a diffusive correction of these models and discuss the characteristic time and length scales of the instability.
This paper presents a critical overview on the modeling of crowds and swarms and focuses on a modeling strategy based on the attempt to retain the complexity characteristics of systems under consideration viewed as an assembly of living entities characterized by the ability of expressing heterogeneously distributed strategies.
A flow composed of two populations of pedestrians moving in different directions is modeled by a two-dimen\-sional system of convection-diffusion equations. An efficient simulation of the two-dimensional model is obtained by a finite-volume scheme combined with a fully adaptive multiresolution strategy. Numerical tests show the flow behavior in various settings of initial and boundary conditions, where different species move in countercurrent or perpendicular directions. The equations are characterized as hyperbolic-elliptic degenerate, with an elliptic region in the phase space, which in one space dimension is known to produce oscillation waves. When the initial data are chosen inside the elliptic region, a spatial segregation of the populations leads to pattern formation. The entries of the diffusion-matrix determine the stability of the model and the shape of the patterns.
Force-based models describe the interactions of pedestrians in terms of physical and social forces. We discuss some intrinsic problems of this approach, like penetration of particles, unrealistic oscillations and velocities as well as conceptual problems related to violations of Newton's laws. We then present the generalized centrifugal force model which solves some of these problems. Furthermore we discuss the problem of choosing a realistic driving force to an exit. We illustrate this problem by investigating the behaviour of pedestrians at bottlenecks.
In this article, an evacuation model describing the egress in case of danger is considered. The underlying evacuation model is based on continuous network flows, while the spread of some gaseous hazardous material relies on an advection-diffusion equation. The contribution of this work is twofold. First, we introduce a continuous model coupled to the propagation of hazardous material where special cost functions allow for incorporating the predicted spread into an optimal planning of the egress. Optimality can thereby be understood with respect to two different measures: fastest egress and safest evacuation. Since this modeling approach leads to a pde/ode-restricted optimization problem, the continuous model is transferred into a discrete network flow model under some linearity assumptions. Second, it is demonstrated that this reformulation results in an efficient algorithm always leading to the global optimum. A computational case study shows benefits and drawbacks of the models for different evacuation scenarios.
It is suggested that flows of pedestrians on curved paths, such as the recirculating flow that occur around the Kaaba in Mecca, continually stratify themselves according to tolerance of crowd density with those pedestrians who are more tolerant of high densities taking a path of shorter length. Such stratification occurs over a distance, referred to here as the ``stratification distance scale" and is generally of the order of the radius of curvature of the flow. Once stratified a pedestrian crowd flows smoothly around an obstacle with a distance scale greater than the stratification distance scale. However, flow past a smaller obstacle with a distance scale less than the stratification distance scale, leads to some temporary breakdown in this stratification, with the flow developing patches of turbulent-like behavior with different pedestrian types responding differently to the obstacle. The flow between the nearby Safa and Marwa Hills is poorly stratified because of the lack of curvature of the flow between the Hills even though the flow is recirculating. At the start of each turning point by each Hill, the change in curvature leads to turbulence-like behavior as the stratification reforms, after its breakdown between the Hills, for the flow around the Hills.
We address here the issue of congestion in the modeling of crowd motion, in the non-smooth framework: contacts between people are not anticipated and avoided, they actually occur, and they are explicitly taken into account in the model. We limit our approach to very basic principles in terms of behavior, to focus on the particular problems raised by the non-smooth character of the models. We consider that individuals tend to move according to a desired, or spontaneous, velocity. We account for congestion by assuming that the evolution realizes at each time an instantaneous balance between individual tendencies and global constraints (overlapping is forbidden): the actual velocity is defined as the closest to the desired velocity among all admissible ones, in a least square sense. We develop those principles in the microscopic and macroscopic settings, and we present how the framework of Wasserstein distance between measures allows to recover the sweeping process nature of the problem on the macroscopic level, which makes it possible to obtain existence results in spite of the non-smooth character of the evolution process. Micro and macro approaches are compared, and we investigate the similarities together with deep differences of those two levels of description.
Mobile on-body sensing has distinct advantages for the analysis and understanding of crowd dynamics: sensing is not geographically restricted to a specific instrumented area, mobile phones offer on-body sensing and they are already deployed on a large scale, and the rich sets of sensors they contain allows one to characterize the behavior of users through pattern recognition techniques.
In this paper we present a methodological framework for the machine recognition of crowd behavior from on-body sensors, such as those in mobile phones. The recognition of crowd behaviors opens the way to the acquisition of large-scale datasets for the analysis and understanding of crowd dynamics. It has also practical safety applications by providing improved crowd situational awareness in cases of emergency.
The framework comprises: behavioral recognition with the user's mobile device, pairwise analyses of the activity relatedness of two users, and graph clustering in order to uncover globally, which users participate in a given crowd behavior. We illustrate this framework for the identification of groups of persons walking, using empirically collected data.
We discuss the challenges and research avenues for theoretical and applied mathematics arising from the mobile sensing of crowd behaviors.
The current status of empirical results for pedestrian dynamics is reviewd. Suprisingly even for basic quantities like the flow-density relation there is currently no consensus since the results obtained in various empirical and experimental studies deviate substantially. We report results from recent large-scale experiments for pedestrian flow in simple scenarios like long corridors and bottlenecks which have been performed under controlled laboratory conditions that are easily reproducible. Finally the implications of the unsatisfactory empirical situation for the modeling of pedestrian dynamics is discussed.
In this paper we consider first order differential models of collective behaviors of groups of agents, based on the mass conservation equation. Models are formulated taking the spatial distribution of the agents as the main unknown, expressed in terms of a probability measure evolving in time. We develop an existence and approximation theory of the solutions to such models and we show that some recently proposed models of crowd and swarm dynamics fit our theoretic paradigm.
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