PHYSICAL REVIEW E 97, 032319 (2018) Effect of form of obstacle on speed of crowd evacuation Ryosuke Yano * Tokio, Marine and Nichido Risk Consulting Co. Ltd., 1-5-1 Otemachi, Chiyoda-ku, Tokyo, Japan (Received 3 July 2017; revised manuscript received 16 January 2018; published 29 March 2018) This paper investigates the effect of the form of an obstacle on the time that a crowd takes to evacuate a room, using a toy model. Pedestrians are modeled as active soft matter moving toward a point with intended velocities. An obstacle is placed in front of the exit, and it has one of four shapes: a cylindrical column, a triangular prism, a quadratic prism, or a diamond prism. Numerical results indicate that the evacuation-completion time depends on the shape of the obstacle. Obstacles with a circular cylinder (C.C.) shape yield the shortest evacuation-completion time in the proposed model. DOI: 10.1103/PhysRevE.97.032319 I. INTRODUCTION The study of collective motion [1,2] in biological systems has attracted physicists’ and mathematicians’ attention, be- cause models of these phenomena require rich physical and mathematical insights. In particular, Vicsek’s pioneering work on collective motion [3,4] implemented insights from physics in order to model a biological collective motion system. The clustering of agents’ velocity-vector angles toward a local angle, averaged over those of agents with a distance less than a threshold, yields a characteristic spatial structure [3,5–10]. This convergence toward an averaged value over a finite range is also implemented in the Hegselmann-Krause model of opin- ion formation [11]. Similarly, several mathematical models such as the Cucker-Smale model [12], the self-propulsion– repulsive-attractive model with drag force [13,14], and the Kuramoto-Vicsek model [15,16] have been used in order to model biological swarming. Pedestrian collective motion has been studied by Helbing et al. [17,18], who used the social-force model [19] to investigate the key characteristics of pedestrian collective motion. Recently, Dietrich and Köster [20] proposed the gradient-navigation model for pedestrian collective motion. The attractive-repulsive force and accel- eration or deceleration due to the relaxation of pedestrian velocities toward their intended velocities [21] are consid- ered to be significant characteristics of pedestrian collective motion. Many discrete element method (DEM) simulations [22–25] have been performed which implemented Helbing and Molnar’s social-force model [21], whereas cellular automata [26,27] have been also considered as an application of game theory in pedestrian dynamics. Meanwhile, Helbing proposed a collective-motion model that uses a hydrodynamic equation derived from the Boltz- mann equation [28–30]. This equation includes a distribution function with five dimensions, namely, f (t,v,v in , x )(t ∈ R + : time; x ∈ R 2 : physical space; v ∈ R 2 : velocity space; v in ∈ R 2 : intended velocity space). The derivation of hydrodynamic equations [31–34] such as the Navier-Stokes (NS) equation * ryosuke.yano@tokiorisk.co.jp has been considered to model phase transitions in biolog- ical collective-motion systems. We consider, however, that collective phenomena of pedestrians are always beyond the Grad-Boltzmann or Grad-Enskog limit, because the number of pedestrians in a typical crowd is much smaller than the Avogadro number, as observed in the flow of granular materials [35]. Therefore, numerical modeling of pedestrian motion can offer insights beyond the hydrodynamic description de- rived from the Boltzmann-Enskog equation [22]. The network structure among destinations also has a significant effect on collective motion when movements of pedestrians occur among cities or countries [36], but this work focuses on a finer-grained level of analysis. In this paper, we focus on how the shape of an obstacle affects the speed with which a crowd of pedestrians evacuates a room. Evacuation is an important application of collective- motion modeling [37] because excessive jamming in the flow of pedestrians can generate extremely hazardous, even fatal, conditions when a large crowd must quickly evacuate a room. The following discussion assumes that the forces between pedestrians can be modeled as spring-mass systems. Therefore, pedestrian dynamics are similar to those observed in active soft matter such as granular particles or soft disks [38] that have intended velocities. In order to realistically model pedestrian dynamics, the model must include representations of the following dynamic phenomena: herding (swarming) among neighboring pedestrians [39], the effect of visibility of vacant spaces or obstacles [39], route-choice strategies [40], avoiding injured or dead pedestrians as obstacles [41], the distribution of intended velocities in accordance with the age [42] or (competitive) character of pedestrians [43], the relationship between pedestrian density and intended velocity, the convergence rate toward the intended velocities [44], and the effect of polydisperse noncircular areas of personal space [45]. Obviously, these phenomena cannot all be repre- sented with spring-mass interaction forces. A toy model with spring-mass forces, while not being totally realistic, can offer basic data about how the shape of an obstacle affects the evacuation-completion time. The spring constant in our model corresponds to the strength of an individual’s repulsion from other pedestrians that infringe upon this individual’s personal 2470-0045/2018/97(3)/032319(9) 032319-1 ©2018 American Physical Society