Modeling Crowd Turbulence by Many-Particle Simulations

Modeling Crowd Turbulence by Many-Particle Simulations
Modeling Crowd Turbulence by Many-Particle Simulations

a r X i v :0708.3282v 1 [p h y s i c s .s o c -p h ] 24 A u g 2007

Modeling Crowd Turbulence by Many-Particle Simulations

Wenjian Yu and Anders Johansson

Institute for Transport &Economics,Dresden University of Technology,

Andreas-Schubert Str.23,01062Dresden,Germany

Department of Humanities and Social Sciences,ETH Zurich,UNO D 11Universit¨a tstrasse 41,8092Zurich,Switzerland

A recent study [D.Helbing,A.Johansson and H.Z.Al-Abideen,Phys.Rev.E 75,046109(2007)]has revealed a “turbulent”state of pedestrian ?ows,which is characterized by sudden displacements and causes the falling and trampling of people.However,turbulent crowd motion is not reproduced well by current many-particle models due to their insu?cient representation of the local interactions in areas of extreme densities.In this contribution,we extend the repulsive force term of the social force model to reproduce crowd turbulence.We perform numerical simulations of pedestrians moving through a bottleneck area with this new model.The transitions from laminar to stop-and-go and turbulent ?ows are observed.The empirical features characterizing crowd turbulence,such as the structure function and the probability density function of velocity increments are reproduced well,i.e.they are well compatible with an analysis of video data during the annual Muslim pilgrimage.

I.INTRODUCTION

Pedestrian dynamics[1,2,3]has been described by physicists through various macroscopic and microscopic models.Macroscopic models,predominantly ?uid-dynamic models [4,5],have the advantage of describing the large-scale dynamics of crowds,especially depicting intermittent ?ows and stop-and-go ?ows.These features are basically understood as the e?ect of a hybrid continuity equation [6]with two regimes:forward pedestrian motion and backward gap propagation.Details of pedestrian interactions are neglected in these models.In contrast,microscopic models can be used to describe the details of pedestrian behavior.Previously proposed pedestrian models include many-particle force models[7,8],CA models [9,10]and others,e.g.multi-agent approaches [11],which have received an increasing attention among physicists in the past.In recent years,the interest has turned to empirical or experimental studies [12,13,14,15,16,17,18]of pedestrian ?ows based on video analysis [1,13,17,19,28].This has contributed to the calibration of current models [20,25]and the discovery of new phenomena such as crowd turbulence [1],which can help

to understand many crowd disasters.

Turbulent motion of pedestrians occurs,when the crowd is extremely compressed,and people attempt to gain space by pushing others,which leads to irregular displacements,or even the falling of people.If the fallen pedestrians do not manage to stand up quickly enough,they will become obstacles and cause others to fall as well.Such dynamics can eventually spread over a large area and result in a crowd disaster.

However,crowd turbulence is not well reproduced and understood by pedestrian models yet, which challenges current many-particle models.Their shortcoming is due to the underestimation of the local interactions triggered by high densities.In the following sections,we will extend the repulsive force of the social force model[3,7,15],which has successfully depicted many observed self-organized phenomena,such as lane formation in counter?ows and oscillatory?ows at bottle-necks[21].

II.MODEL OF CROWD TURBULENCE

Previous empirical studies[1,26]have revealed that people are involuntarily moved when they are densely packed,and as a consequence,the interactions increases in areas of extreme densities, which leads to an instability of pedestrian?ows.When the average density is increasing,sudden transitions from laminar to stop-and-go and turbulent?ows are observed.Moreover,the average ?ow does not reach zero.We will now show how the turbulent?ows can be modeled,by a small extension of the social force model[3,7,15].What we do is to add an extra term to the repulsive force,and show how this will give qualitatatively di?erent dynamics,leading to turbulent?ows.

The social force model assumes that a pedestrian i tries to move in a desired direction e i with desired speed v0i,and adapt the actual velocity v i to the desired velocity v0i e i within the relaxation timeτ.The velocity v i=d r i/dt,i.e.the temporal change of the position r i is also a?ected by repulsive forces.

The social force model is given by

d v i(t)

m i

(v0i e i? v i)+ j(=i) f ij(t).(2)

τ

The term f ij(t)denotes the repulsive force,which represents both the attempt of pedestrian i

to keep a certain safety distance to other pedestrians j and the desire to gain more space in very crowded situations.

Instead of introducing an additional force term,one may re?ect the desire to gain more space under crowded conditions,by a local interaction range in the repulsive pedestrian force f ij ,which is proposed as follows,

f ij =F Θ(?ij )exp[?d ij /D 0+(D 1/d ij )k ]

e ij ,(3)

where F is the maximum repulsive force (assuming there is no overlapping/compression );d ij is the distance between center of masses of pedestrians;k ,D 0,and D 1are constants; e ij is the normalized vector pointing from pedestrian j to pedestrian i ,?ij is the angle between e ji and the desired walking direction e i of pedestrian i ,i.e.cos(?ij )= e i · e ji .

In normal situations,the function Θ(?ij )re?ects the fact that pedestrians react much stronger to what happens in front of them,and it has been suggested [21]to have the form,

Θ(?)=

λ+(1?λ)

1+cos(?)

In our simulations,the desired speeds v0are assumed to be Gaussian distributed with the mean value v0=1.34m/s and the standard deviation0.26m/s[22,23,24].The relaxation time,τis set to0.5s.The average mass of a pedestrian is set to60kg with a standard deviation of10kg. Assuming that pedestrians have a strong desire to gain more space in dense areas,the maximum repulsive force F is set to160N.Note that,when pedestrians are compressed,the maximum force can be signi?cantly larger than F.The other parameters are set to,λ=0.25,k=2,D0=0.31 and D1=0.45.

The bottleneck area(see Fig.1)contains two pedestrian sources,A and B,where A denotes those walking from left to right,while B represents those walking from the bottom,and then turn right to join those coming from left.These two sources give an increasing number of pedestrians in time,since we use both pedestrian sorces as well as periodic boundary conditions.With this scheme we can see how the transition from laminar to turbulent?ow occur when the density is growing.The whole area is40m×20m.The reason for this setup,is to get a situation that is comparable to the one in the empirical study[1].

The preferred direction is de?ned as follows:

If the vertical position of a pedestrian is above

ˉ

the corner,he/she will walk in the right direction.Otherwise,he/she will?rst walk upwards until he/she is above the corner,and then turn to the right and keep walking straight ahead.

Note that interactions increase greatly when those two crowds intersect,especially in high density situations.

The“Crowd pressure”[1],re?ecting the irregular/chaotic motion of people,is given by

p=ρi Var( v i),(5) where Var( v i)is the local velocity variance.The local densityρi,

ρi= j1

FIG.1:(Color online)The bottleneck area of the simulation.Two groups of people set out respectively from A and B,and intersect,causing pedestrians to be highly compressed,which eventually results in turbulent ?ows.

FIG.2:(Color online)The crowd pressure as a function of local density.The turbulence starts when the pressure exceeds a value of0.02/s2[1].

If the?ow is reduced to zero,which means all the people stop moving,then there is no turbulence. Therefore it is essential to have nonvanishing?ow for high densities.This is compatible with the empirical study[1].Note that,at this point,the?ow is no longer laminar.Therefore,the strong interactions between pedestrians are potentially more dangerous for the crowd.The motion of

pedestrians become turbulent,and people are pushed into all possible directions.As people are pushed by those behind,the fallen people will be trampled,if they do not get back on their feet quickly enough.However,in our simulations,we assume that pedestrians will never fall,since we are focussing on the dynamics of the crowd during a high level of crowdedness.These conditions can potentially lead to an accident,but we do not focus on the dynamics of the accident itself.

FIG.3:(Color online)The average of the local?ow[1]as a function of the local density.The empirical data is represented by stars and the simulation results are represented by circles connected by a solid line. The dashed line shows the fundamental diagram,from the original social force model.

With the increment of density,the pedestrian?ow suddenly turns into stop-and-go?ow(see Fig.4),which is characterized by temporarily interrupted and longitudinally unstable?ow.This phenomenon is also predicted by a recent theoretical approach[6],which suggests that intermittent ?ows at bottlenecks can be triggered when the in?ow exceeds the out?ow.

Further increment of the density will lead to turbulent?ows.Figure5(a)shows a typical trajectory of the turbulent motion from our simulations.We can see that,?rst,the curve is smooth, which represents a laminar?ow,then suddenly,vibrations occur due to the turbulence.Also,note that the pedestrian is sometimes even pushed backwards.The turbulent motion does not vanish until an individual walks out of the extremely dense central area,where the two streams intersect. Figure5(b)is an example of the temporal evolution of an individual’s velocity components v y and v x.One can clearly see the irregular motion into all directions.Although no large eddies are observed,as in turbulent?uids,there is still an analogy to the turbulence of the currency exchange market[27].This can be characterized by the probability density function of velocity increments

FIG.4:(Color online)Simulation results.During high crowd densities,the smooth,laminar?ow will turn into stop-and-go?ow.Here the average density is4.3/m2.

and the so-called structure function,Eq.8.

The shape of the probability density function of velocity increment is a typical indicator for turbulence,if the time shiftτ′is small enough,and is given by

vτ′x=v x( r,t+τ′)?v x( r,t).(7) The structure function,

S(△ r)= v( r+△ r,t)? v( r) 2 r,t,(8) re?ects the dependence of the relative speed on the distance.

We?nd that the probability functions of velocity increment is sharply peaked for turbulence, while it is parabolic-like for laminar?ow(see Fig.6),when the time shiftτ′is small enough.The structure function has a slope of2.0,when the distance is small,while at large steps,the slope turns to0.18due to the increased interactions in crowded regions.Both of the functions are compatible to the analysis of the video recordings of the Jamarat Bridge[1].

IV.SUMMARY AND OUTLOOK

In summary,an added social force component,re?ecting the strong interactions in the extremely crowded areas is proposed for the simulation of crowd turbulence,which questions current many-particle models.The transition from laminar to stop-and-go and turbulent?ows is observed in the

FIG.5:(Color online)(a)A representative pedestrian trajectory in laminar and turbulent?ow.(b)An example of velocity components in the turbulent motion.Here the average density is9.0/m2.

simulations.The fundamental diagram is reproduced and demonstrates the e?ects of the extended repulsive forces at highly dense situations,i.e.the average local?ow is not reduced to zero.A typical turbulent trajectory and velocity components are presented.Functions like the probability density function of velocity increment and the structure function,characterizing the features of turbulence are simulated,and the results are compatible to the empirical studies.

FIG.6:(Color online)(a)the probability density function of the velocity increment in the laminar and the turbulent region,determined over many locations r for R=

Acknowledgments

The authors are grateful to the German Research Foundation for funding(DFG project He 2789/7-1).

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5. _______ (final), wait for the cake to cool and eat it. Keys:1. bake 2. speaking 3. shopping 4. free 5. Finally (四) 1. I'm looking forward_______ (visit) you soon. 2. He would rather_______ (have) some tea instead of soft drinks. 3. What ingredients do we need_______ (buy) to make a chocolate cake? 4. We_______ (plan) to have a party for Kitty now. 5. Let's_______ (spread) the icing on top of the cake. Keys: 1. to visiting 2. have 3. to buy 4. are planning 5. spread (五) 1. I'm looking forward(visit) you soon. 2. He would rather__________________ (have) some tea instead of soft drinks. 3. What ingredients do we need_______________ (buy) to make a chocolate cake? 4. We _________________ (plan) to have a party for Kitty now. 5. Let's___________________ (spread) the icing on top of the cake. 6. Alice prefers to do some___________. (shop) 7. Mary is my ___________ sister and she is 2 years ___________ than I. (old) 8. Excuse me, what is the ____________ of this word? (mean) Keys:1. to visiting 2. have 3. to buy 4. are planning 5. spread 6. shopping 7.elder…older 8. meaning (六) 1. ___________ read and green together and they will become yellow. (mixture )

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第19 讲期末冲刺 重点单词及其变换 Unit 1 How can we become good learners 1. n.发音,读音,→ v.发音 2. adj.有耐心的→ n.耐心 3. n.表情,表示,表达方式→ v.表达 4. n.能力,才能→ adj. 能够的,能干的 5. v.创造,创建→ n.创造,创造力→ n.创造者→ adj.有创造力的 6. adj.活跃的,积极的→ n.活动 7. v.(使)连接,与……有联系→ n.联系 8. adj.明智的,聪明的→ adv. 明智地,聪明地 Unit 2 I think that mooncakes are delicious! 1. n.陌生人→ adj.奇怪的,陌生的 2. v.偷,窃取→(过去式)→(过去分词) 3. v.放置,下蛋→(过去式)→(过去分词)→(现在分词) 4. v.平躺,处于→(过去式)→(过去分词)→(现在分词) 5. adj.死的,失去生命的→ v. 死→ n.死亡

6. n.生意,商业→ adj.忙碌的 7. v.处罚,惩罚→ n.处罚,惩罚 8. n.温暖,暖和→ adj.温暖的 9. v.传播,展开→(过去式)→(过去分词) Unit 3 Could you please tell me where the restrooms are 1. n.建议,提议→ v.建议,提议 2. adj.中心的,中央的→ n.中心,中央 3. n.东方,东部→ adj.东方的,东部的 4. adj.便利的,方便的→ n.便利,方便 5. adj.有礼貌的,客气的→ adj.不礼貌的,粗鲁的→ adv.礼貌地,客气地→ adv.不礼貌地,不客气地 6. adj.直接的,直率的v.指引,指导→ adj.间接的→ adv.直接地→adv.间接地→ n.导演→ n.方向 Unit 4 I used to be afraid of the dark. 1. adj.幽默的→ n.幽默 2. adj.沉默的→ n.沉默 3. adj.有用的,有帮助的→ n./v.帮助 4. adj.亚洲的,亚洲人→ n.亚洲 5. n.欧洲→ adj.欧洲的,欧洲人的

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动词变名词1.v+ment结尾 achieve---achievement成就advertise---advertisement广告agree---agreement同意 apartment公寓 amusement娱乐 argue---argument争吵 commit---commitment奉献compliment称赞,恭维 develop---development发展 disgree---disagreement不赞同department局,部 experiment实验,试验 equip装备---equipment装备,器材govern统治---government政府manage---management经营管理 2.V+tion结尾 admit---admission承认 attract---attraction吸引

conclude---conclusion结论compete---competition竞争,比赛discuss---discussion讨论 educate---education教育 decide---decision决定 describe---description描写,描绘express----expression词语;表达graduate---graduation毕业 operate---operation操作,动手术organize---organization imagine---imagination想象力introduce---introduction介绍instruct---instruction指导,介绍invent---inventor/invention发明illustrate---illustration阐明,举例说明invite---invitation邀请 inspire---inspiration灵感,鼓舞人心的pollute---pollution污染 predict---prediction预言pronounce---pronunciation resolve---resolution决心 impress---impression印象

初中英语词性转换

初中英语词性转换 1.v-n 1.clean— 2. cook— 3. dance-- 4. drive— 5. jump-- 6. lead— 7. own-- 8. play— 9.read-- 10.run— 11.sell-- 12.sing— 13.speak-- 14.swim— 15.teach-- 16.travel—17.wait-- 18.wash— 19.win-- 20.work— 21.write-- 22.act— 23.direct-- 24.invent—25.visit-- 26.act— https://www.360docs.net/doc/013462550.html,municate-- 28.introduce—29.produce—30.pronounce—31.suggest-- 32.agree—33.develop-- 34.discuss—35.decide-- 37.begin—38.breathe-- 39.build—40.cross-- 41.feel—

45.park-- 46.shop— 47.skate-- 48.train— 49.turn-- 50.turn— 51.advise-- 52.enter—53.serve-- 54.know—55.live-- 56.die—57.succeed-- 58.discover—59.please-- 60. sit—61.thank-- 62.marry—63.weigh— 2.v-adj 1.close-- 2. excite— 3. interest-- 4. follow— 5. miss-- 6.please— 7.surprise-- 8.relax— https://www.360docs.net/doc/013462550.html,e-- 10.sleep-- 11.worry—12.wake-- 13.die—14.worry-- 15.wake—16.break-- 17.enjoy—18.lose— 3. n-adj

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(一)用所给词的正确形式填空 A 1. The Greens are _____ (happy) to live in this _____(noise) street. They have decided to move to another place. 2. The panda has been ______ (die) for about two months. 3. I like her dress. It looks very ______ (beauty). 4. Don’t feel ______ (worry) about your child. The whole class would be ______ (friend) to the new classmate. 5. The ______ (finally) exams usually take place at the end of June. 6. It’s _____ (possible) for an ordinary plane to fly to the moon. 7. It’s a _____ (please) trip for all of us. 8. The children in China are living a _____ (color) life. 9. It was an _____ (amaze) match. It amazed us. 10.He felt very _____ (sleep) and fell ____ (sleep) soon when he lay in bed. 11.We all had a very _____ (enjoy) time at the party. B 1.I don’t like him and it would be ______________ of me to do the opposite. (honest) 2.Our holidays in Thailand were really ______________ and unforgettable. (enjoy) 3.The government is doing everything possible to help those ______________ people after the earthquake. (home) 4.Put on the glasses so that you can see the word on the blackboard ______________. (clear) 5.The war made millions of people ______________. (home) 6.What is the most ______________ animal in the forest? (power) 7.It is ______________ to travel by train than by air. (cheap) 8.It is ______________ for him to finish the job in two hours. He is so slow. (possible) 9._________, over 28,000 civilians (平民) lost their lives during the three-year war in Iraq. (sad) 10.I feel quite comfortable at home when it is raining ______________ outside. (heavy) 11.Jack is such a ______________ fellow, for he always leaves this or that at the office. (forget) 12.It is not ______________ to eat food with Sudan I (苏丹红1号). (safely) 13.Lily is ______________ enough to get the prize in the 2006 Shanghai Junior High School English Contest. (luck) 14.We’d better drive our cars ______________ on rainy days. (slow) 15.One starry night, when the seven little dwarfs went back home from work, they found something ______________ at once. (usual)

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