Crowd evacuation of pairs of pedestrians

Ryosuke Yano

Tokio Marine dR Co.ltd.,1-5-1,Otemachi,Chiyoda-ku,Tokyo,100-0004,Japan

Abstract The crowd evacuation of pairs of pedestrians(i.e.pairs consisting of a parent and a child)is numerically investigated.Here,it is assumed that all pedestrians have their own partners,and move randomly inside the bounded domain of the right-hand room as an initial state.All pedestrians start their evacuations after they contact their partners.The evacuations are completed by the transfer of all the pairs from the right-hand room to the left-hand room through an exit.A frozen swarm tends to appear in the right-hand room as the total number of pedestrians increases.The frozen swarm moves without changing its form,unless it is dissolved by a strong collision with a pair of pedestrians that comes back from the left-hand room by accident.Finally,the evacuation speed also depends on the area of the Escape Zone,whereas an obstacle placed in front of an exit also changes the speed of the evacuation in accordance with the type of motion of the children.

Keywords:Evacuation of paired pedestrians,force chain network,escape from crowd

1.Introduction

The crowd evacuation of pedestrians sometimes causes a serious incident which leads to the death by crushing of pedestrians[1].The social force model of pedestrians proposed by Helbing[2]has been widely used to simulate the motions of pedestrians.So far,a significant number of textbooks and papers have been published on the topics of pedestrian dynamics,crowd dynamics,and evacuations[3–8].Additionally,experiments(i.e.data obtained from evacuation drills)[9,10]and numerical simulations[11,12]have enabled us to understand the characteristics of the collective motions of pedestrians in a crowd in detail.In recent years,the evacuations of some groups have been investigated by Krüchten and Schadschneider[13](an experimental study of the group evacuation of pupils),Collins and Frydenlund[14](a numerical analysis of refugee groups’ migrations using game theory),Rozanet al[15](a numerical analysis of group evacuations using the social force model),and Nicolas and Hassan[16](a review of previous studies of group evacuations).Meanwhile,the characteristics of the crowd evacuation of pairs of pedestrians(i.e.husband–wife,parent–child,etc.)have not been adequately understood.In this study,the characteristics of crowd evacuation are numerically investigated for the case in which all pedestrians have their own partners.

Now,let a pair of pedestrians be composed of a parent(mother or father)and child.The author considers that the difference between the evacuation of a parent–child pair and that of unpaired pedestrians is primarily demonstrated by the human characteristic that most of the pedestrians,namely,parents and children,first begin by seeking their own partners,and then start their evacuations after they contact their own partners.If the pair of pedestrians were composed of two friends or husband and wife,the pedestrians might begin to evacuate before they contacted their partners,because they would predict that their partners would finish their evacuations in the same way and expect to contact their partners after the completion of evacuation.Consequently,it can readily be conjectured that the process of search and contact with one’s partner surely delays the completion of the evacuation.Indeed,such a search for his/her child by a parent is essential,when his/her child is bewildered or young enough to be unable to seek his/her parent or a path to an exit.Thus,the study of the evacuation of paired parents and children is significant as a special and critical case of group evacuation.As with the evacuation of children in the classroom in the training school with the help of their pickers(i.e.parents),Tang and his coworkers performed the pioneering work via experiments and cellular automaton simulation[17,18].

The situation in which all the pedestrians are randomly placed inside the specific domain of a right-hand room is considered to be the initial state.Here,all the children do not always stay near their parents but move inside the right-hand room as they like,before the alarm rings att=0.The effects of the geometry of the domain in which pedestrians are initially distributed on the evacuation process are not considered,because they are the subject of a future study.First,parents and children try to find their partners when the alarm rings att=0.Once he/she contacts his/her partner,a pair of pedestrians starts the evacuation by passing through an exit from the right-hand room to the left-hand room.Two types of child motion are considered.In type I,children can move toward the locations of their parents in addition to the parents’movement,because they can see the locations of their partners all the time.In type II,only the parents can move toward the locations of their children,who never move until their parents contact them.The latter case is plausible when the ages of the children are in the periods of infancy or early childhood.Of course,consideration of the motions of wandering panicked children is necessary when the ages of the children are in the period of infancy or early childhood.However,such a consideration of wandering panicked children is the subject of a future study.Readers are referred to the experiments and simulations by Chenet al[19]for an understanding of the realistic motions of children under nonemergent evacuation.Additionally,the consideration of evacuation by a family,which is composed of more than two members(i.e.a group consisting of father,mother,and child),is also the subject of a future study.It is easily conjectured that the time required to find family members is prolonged by the increase in the number of family members because a larger swarm is formed by families with more members.

Numerical results indicate that a swarm of pedestrians is formed while they move toward their partners.Provided that pairs of pedestrians can escape from the swarm in order to successfully move toward an exit,the volume of the swarm temporally decreases,and all the pairs of pedestrians are able to evacuate from the right-hand room to the left-hand room through an exit.Meanwhile,a frozen state of the swarm[20]appears when a stable force chain network[21]is formed and most pedestrians cannot reach their partners inside the swarm.In short,the stability of the swarm determines whether all the pedestrians can successfully evacuate from the right-hand room to the left-hand room or not.The numerical results certainly indicate that the dependency of the stability of the swarm on the initial locations of the pedestrians(i.e.the initial distribution of the distances between a parent–child pair)becomes significant as the total number of pedestrians increases.Additionally,the dynamics of the velocity vector seem to be similar to that of spin glass,in which the glassy state is characterized by the direction of the velocity vector of the pedestrian inside the swarm.It is readily understood that the process of searching for their partners corresponds to the long-range interaction of particles,as with spin glass[22].

As an interaction model between two pedestrians,the mass–spring model[23]is applied in order to simplify our discussions as much as possible.Actually,Zuriguel and his coworkers[24,25]investigated crowd evacuation using granular particles(i.e.those of a silo),which fall by the gravitational force inside the funnel and clog when they go through the throat of the funnel.The author then models the dynamics of pedestrians using active soft matter with the intended velocity,whose interaction characteristics are similar to that of the granular matter.Meanwhile,the motion of pedestrians,whose strategies are determined on the basis of their future cost functions,is modeled using the mean field games approach on the basis of the Hamilton–Jacobi–Bellman equation coupled with the Kolmogorov–Fokker–Planck equation[26,27].The simulation of the motion of pedestrians using advanced intelligence,including route choices,is,however,outside the scope of the present study.The equation of motion of pedestrians can be modeled by that of a disc with a constant mass,whose domain corresponds to the effective domain in which pedestrians sense the social force.The velocities of all the discs(i.e.pedestrians)always relax toward their intended velocities[28].Therefore,a paired parent and child,a parent,and a child are expressed in terms of active soft matter(i.e.discs)[23,29]with a common mass and three types of radius,respectively.In short,the three types of disc with different radii correspond to the parent,child,and paired parent and child,respectively.The radius of the effective domain(disc)of the parent is set toRp,that of the child toRcand that of the paired parent and child toRpc=Rp+Rc.Therefore,the situation in which parents evacuate by holding their children by their arms,is not assumed.The parents evacuate by pulling the hands of their children because the children can move with the same velocity as their parents.As a result of this condition imposed on the motion of the children,the age range of children is restricted(i.e.to children in elementary school).The total area of the effective domain before contactincreases by 2πRpRcafter contact,because the parent becomes more sensitive to other pedestrians while he/she is pulling the hand of his/her child.Meanwhile,the masses of the three types of disc are assumed to be equal in order to simplify our discussion.As an additional condition,we investigate the effects of an obstacle placed in front of an exit on the evacuation process.Here,four types of the obstacle’s form,namely,a cylindrical column(C.C.),a triangular prism(T.P.),a quadratic prism(Q.P.),and a diamond prism(D.P.)are considered.

This paper is organized as follows.The mass–spring model,which models the dynamics of pedestrians,is discussed in section 2.The numerical setting of the evacuation of paired pedestrians is demonstrated in section 3,and the numerical results are discussed in section 4.Finally,concluding remarks are made in section 5.

2.Numerical method used to calculate the evacuation of paired pedestrians

The dynamics of pedestrians are demonstrated using the spring-mass model[23].As mentioned above,the equation of motion of pedestrians is represented by the motions of active soft discs with the intended velocity.Here,the direction of the intended velocity is assumed to coincide with the direction of the location of the partner,before he/she(the parent)contacts his/her own partner(the child).Once the pairing is completed by contact,the direction of the intended velocity is changed to accomplish evacuation from the right-hand room to the lefthand room through an exit.The interaction among contacting unpaired pedestrians is expressed using the repulsive(social)force[30],which is modeled by the linear spring force.As a result,the type of interaction is similar to that of 2D granular disc.The area of the disc represents the effective domain in which the pedestrian experiences the social force.Since the pair of one parent and child is focused,three types of radius,Rpfor the parent,Rcfor the child,andRpcfor the paired parent and child are considered,while the masses of the three types of disc are equal tom.The relationRp=2Rc=2Ris set.The modeling of the tangential force,which considers the realistic cohesion[31]between two interacting pedestrians,is outside the scope of this study.Meanwhile,the spin of the disc owing to the tangential force is neglected in order to simplify our discussion.Finally,the set of parameters used in our numerical model must be modified by the future experimental results of the evacuation drill of paired pedestrians[32,33].

The interaction between a pedestrian and a wall or obstacle is modeled in a similar way to equation(1).The spring constant(k),which represents the force between the pedestrian and wall(or obstacle),is set tokpw(kpo),where the friction coefficient in the interaction between the pedestrian and the wall(or obstacle)is defined by μpw(μpo).On the other hand,the velocities of the pedestrians are never relaxed toward their intended velocities and the decay of velocity is neglected while pedestrians are in contact with the wall or obstacle.

3.Physical setting of parameters

A schematic of the intended velocities of paired parents and children is shown at τip≤tin the right-hand frame of figure 3.Pairs of parents and children are confirmed in domains I and II;domain IV shows movement toward an exit via the shortest path to an exit,as shown in the right-hand frame of figure 3.The above model surely does not include any of the effects of the route choices of pedestrians[36]who try to move to a more vacant space to avoid collisions with other pedestrians[37],whereas the paired parents and children inside domain III strategically intend to move in the vertical(Y-axis)direction in order to escape from the swarm which is formed from unpaired parents and children who are seeking their partners.Such a swarm will be confirmed in the next section.Of course,′L,which characterizes the size of domain III,must be originally determined in accordance with the size of the swarm and the local density of pedestrians,whereas such a sophisticated determination ofL′ is the subject of a future study,because the route choice model that reflects the local density of the pedestrians is outside the scope of the present study.In summary,the characteristics of the risk aversion of the pedestrians to the swarm are modeled using ′Las an initial study of the evacuation of paired pedestrians.Therefore,domain III can be seen as theEscape Zone.Once the paired pedestrians reach domain IV,they aim to go to an exit via the shortest path to an exit.The effects of changingLʹ,the area of domain III,on the evacuation speed are investigated in section 4.1.The velocity of the paired parent and child,who move to the right-hand room,never relaxes toward the intended velocity in domain V.In previous studies[28,38],ζ=2 s−1,|vin|=1.34ms−1,the contact distanceR∼0.5±0.2[m],β=1 andkpp∼2 kNm−1,(when the repulsive force is defined by an exponential function of the distance between two pedestrians[38]).Therefore,the set of parameters is similar to that used in previous studies[28],although the value ζ=1.7×10−1s−1used in our study is smaller than ζ=2 s−1.β,D,μpi(i=p,c,w),andkpi(i=w)are,however,artificially fixed.Indeed,a more realistic set of parameters must be obtained using(video)data of a real evacuation or evacuation drill.

The normalized time interval is set as 5×10−5and the contour of unfilled circles in figure 3 corresponds to the angleIn figure 3,the evacuations of 1382 pedestrians(691 parent–child pairs)from the right-hand room to the left-hand room through an exit are considered.Therefore,the evacuations occur in a large space(i.e.two rooms with 2×104m2),which usually has multiple evacuation exits[38,39].

4.Numerical results

The evacuations of the paired pedestrians are investigated on the basis of the physical setting discussed in section 3.This section is composed of three subsections.First,the effects of the area of theEscape Zone(domain III)on the evacuation speed are numerically investigated in section 4.1.Second,the effects of the initial locations of the pedestrians and the total number of pedestrians on the evacuation process are numerically investigated in section 4.2.Finally,the effects of the form of an obstacle placed in front of an exit on the evacuation speed are numerically investigated in section 4.3.

Table 1.Set of parameters used in the simulation of equation(2).

4.1.Dependency of evacuation speed on the area of domain III

4.2.Effects of the initial locations and the total number of pedestrians on evacuation speed

Figure 10 shows snapshots of the pedestrians and the force chain network att=0.72,1.44,2.16,6.48,and 9.36 min in case I(left-half frames)and those att=0.72,1.44,2.16,6.48,and 36 min in case II(right-half frames),whenN=1382,L′ =60[m],and the type I motion of children are used.The scale of the force chain network in case I does not markedly differ from that in case II att=0 andt= 1.44 min.The size of the swarm in case I,which is primarily composed of parents and children seeking their partners,seems to be similar to that in case II att= 2.16 min.It is,however,confirmed that the number of links in the force chain network in case I is much smaller than that in case II att= 2.16 min.Finally,the force chain network in case I collapses att=6.48 and 9.36 min,whereas the links at the boundary inside the force chain network are closed att=6.48 and 36 min in case II.In particular,it is interesting that the swarm collectively moves in the positiveYdirection att= 6.48 min and toward the negativeXdirection att=36 min in case II.The velocities of these collective motions of the swarm att=6.48 and 36 min are much slower thanV∞.This type of collective motion of the swarm is also obtained as the solution to the self-propulsion,friction,and attraction–repulsion model described by Carrilloet al[40].Provided that the normalized velocity vectors are regarded as the direction of the spin,the dynamics of the swarm are similar to that of spin glass[41].

Figure 1.Schematic of pairing of parent and child.

Figure 2.Schematic of the interaction between pedestrians,when ip=kc and jp=lc.

Figure 3.Left-hand frame:schematic of the numerical domain and the initial states of pedestrians at t=0.Filled discs,unfilled larger and smaller discs correspond to a pair of parent and child,a parent,and a child,respectively.Domains I-V,used to determine the intended velocity of the paired parent and child,are shown.Right-hand frame:schematic of the intended velocities of paired parents and children in each domain of domains I-IV,when τip ≤t.

Figure 4.versus t(left-hand frame)in test 1 and versus t(right-hand frame)in the cases of L′=0,10,20,40,60,and 80[m],when N=900 and the type I motion of children are used.

Figure 5.versus t for the cases of L′=0,10,20,40,60,and 80[m],when N=900 and the type I motion of children are used.

Figure 6.Snapshots of pedestrians at t=2,3,and 4 min obtained using L′=0,10,20,40,60,and 80[m]in test 1,when N=900 and the type I motion of children are used.

Figure 7.Max versus t obtained from Nt=72 tests,when N=900,L′=60[m]and the type I motion of children are used,together withversus Nev at t=1,2,and 3 min.

Figure 8.versus ℓ with error bars(left-hand frame)and versus τp with error bars,when N=900,L′=60[m],and the type I motion of children are used.

Figure 9.Nev versus t for cases I,II,and III when N=1382,L′=60[m],and the type I motion of children are used.

Figure 10.Snapshots of pedestrians and the force chain network at t=0.72,1.44,2.16,6.48,and 9.36 min in case I(left-half frames)and those at t=0.72,1.44,2.16,6.48,and 36 min in case II(right-half frames),when N=1382,L′=60[m]and the type I motion of children are used.

Figure 11.Snapshots of pedestrians and the force chain network at t=0.72,1.44,2.16,6.48,27.72,28,28.8,29.7,31,and 36 min in case III,when N=1382,L′=60[m],and the type I motion of children are used.

Figure 11 shows snapshots of pedestrians and the force chain network att=0.72,1.44,2.16,6.48,27.72,28,28.8,29.7,31,and 36 min in case III,whenN=1382,L′=60[m],and the type I motion of children are used.It can be observed that a parent–child pair with a large velocity(see a disc filled with red)comes back from the left-hand room into the right-hand room through an exit and nearly hits a stable swarm att= 27.72 min.This pair hits the swarm and collapses a collisional part of the force chain network att=28 min.This collapse of the force chain network is accelerated byt= 29.7 min and all the parent–child pairs successfully evacuate from the right-hand room to the left-hand room through an exit.Meanwhile,this specific type of force chain network collapse is only obtained for one test out of 70.The probability of the return of a pair with a large velocity from the left-hand room to the right-hand room through an exit is markedly low.Meanwhile,it is understood that the force chain network is sometimes vulnerable to an external force with a large magnitude.This characteristic is similar to the avalanches of a(wet)sand pile on an inclined plate caused by shooting a bullet into the(wet)sand pile[42].As with the stability of the swarm,a new measure is introduced to evaluate the stability of the force chain network.In particular,the symmetric distribution of the attractive forces between the parent–child pairs inside the swarm is presumably significant for the stability of the force chain network of the swarm.

Figure 12.1versus,when N=1382,L′=60[m]and the type I motion of children are used(left-hand frame)and S(N)versus N,when L′=60[m]and the type I motion of children are used(right-hand frame).

Figure 13.Nev versus t for the cases of N.O.,the C.C.,the Q.P.,the T.P.,and the D.P.,when N=900,and the type I motion of the children are used(left-hand frame).for the cases of N.O.,the C.C.,the Q.P.,the T.P.,and the D.P.(right-hand frame).

Figure 14.Snapshots of pedestrians at t=1,2,3,and 4 min for the cases of the C.C.,the T.P.,the D.P.,the Q.P.,and N.O.,when N=900 and the type I motion of children are used.

Figure 15.Nev versus t for the cases of the C.C.and the T.P.,when N=900 and the type II motion of children are used,together with those obtained using Type I.

Figure 16.Snapshots of pedestrians at t=2,3,and 4 min for the cases of the C.C.and the T.P.,when N=900,and the type II motion of children are used.

4.3.Effects of obstacles on evacuation speed

5.Concluding remarks

In this paper,the crowd evacuation of pairs of pedestrians was considered.The evacuation of pedestrians from the right-hand room to the left-hand room through an exit was numerically analyzed.The parent–child pairs aim to escape from the swarm formed by unpaired parents and children seeking their partners.Two types of motion of the unpaired children,namely types I and II,were considered.The area of theEscape Zonestrongly affects the evacuation speed.An increase in the area of theEscape Zonerelaxes the excessive concentration of parent–child pairs in front of an exit.On the contrary,a marked increase in the area of theEscape Zoneincreases the evacuation time.Therefore,there is an optimalEscape Zonearea that minimizes the evacuation time.An increase in the total number of parents and children promotes the freezing of the swarm owing to the stable force chain network and decreases the achievability of the evacuation.Additionally,the achievability of the evacuation also depends on the initial locations of the parents and children.The longer the mean value of the distance between parent–child pairs becomes,the more the achievability of the evacuation decreases.In our numerical test,the effects of the form of an obstacle placed in front of an exit on the evacuation time were coupled with the effects of theEscape Zone’sarea.Provided that the unpaired children move toward their parents,the effects of the form of the obstacle on the evacuation time are significant.On the other hand,the effects of the form of the obstacle on the evacuation time are negligible when the unpaired children never move from their initial locations(i.e.the type II motion of children),because a marked concentration of the parent–child pairs in front of an exit is decreased not by an increase in the area of theEscape Zone,but by the disappearance of the swarm for the type II motion of the children.

Author’s contribution statement

The manuscript was written by the author(R.Y.)alone,and all the numerical results were obtained by the author.