With the continuous development of modern economic society, massive crowds are inevitable in public places, such as subway stations and airports. In recent years, the study of pedestrian flow dynamics has attracted more and more attention from worldwide scholars.^[1] In order to study the complex pedestrian behaviors to alleviate the breakdown phenomenon^[2] and also to reduce the occurrence of accidents, pedestrian models have been emerging since the early 1970s.

Some self-organization phenomena such as lane formation in the bidirectional flows,^[3] strips in the crossing flows,^[4] and “zipper” effects^[5] have already been discovered and represented by using a variety of pedestrian models. Generally, pedestrian models can be divided into macroscopic models and microscopic models.^[6,7] The macroscopic model mainly focuses on studying the whole moving trend of the crowd described by average speed, density, location, and time. The typical macroscopic model usually means the fluid dynamic model which treats the pedestrian flow as fluid or gas. In 1971, Henderson was the first to apply the idea of gas dynamics and fluid dynamics to the actual data of the crowd,^[8,9] but the research was based on the traditional theory of an ordinary fluid, which required the conservation of momentum and energy. In 1992, Helbing et al. developed a kind of fluid dynamic pedestrian model based on the Boltzmann-like gas kinetic equation, which overcame the requirement of the momentum and energy conservation in Henderson’s theory.^[10] In 2002, Hughes applied the LWR model which could describe the vehicle flow to represent and predict the two-dimensional crowd.^[11] Since the emergence of the Hughes model, the continuity theory has been paid more attention in the field of pedestrian dynamics.

The microscopic model, however, considers pedestrians’ detailed behaviors such as avoiding collisions and overlapping. The typical microscopic pedestrian model contains the cellular automata model,^[12] the social force model,^[13] the agent-based model,^[14] and the game theory model.^[15] Microscopic cellular automata model is a grid-based discrete model, which is more suitable to describe pedestrian dynamics in the complex environment because of its simplicity and efficiency.^[16] Social force model is a continuous force-driven model, and its first application mainly focused on emergency evacuation from buildings.^[17,18] Agent-based model which goes back to Reynolds–Boids model^[19] usually uses the virtual agents to develop the social structure; this approach provides an innovative perspective to pedestrian dynamics.^[20] The principle of the game theory model is to minimize the total cost such as the total escape time which is at the global navigation level by maximizing each pedestrian’s utility. In Ref. [21], pedestrians selected the exit based on the game theory with the assumption that each pedestrian would select the “best” exit, and the evacuation strategy was considered to be optimal when pedestrians and the congestion state of the route achieved the Nash equilibrium.^[22]

With the development of modernization, more and more public buildings are established which inevitably produce the bottlenecks. According to Kretz et al.,^[23] a bottleneck typically denominates a limited area of reduced capacity or increased demand, and they studied the dependence of total time, flux, and time gap from the bottleneck width through experiments. In recent years, much attention has been paid to the research on pedestrian dynamics by conducting experiments. Daamen et al. conducted the narrow bottleneck experiments for the unidirectional crowd, and found the bottleneck was used differently for the free flow and the congestion flow conditions.^[24] Hoogendoorn et al. further observed the “zipper” effect of the unidirectional crowd in the bottleneck through experiments, and found that the number of layers had the relationship with the width of the bottleneck.^[5] Helbing et al. performed the bidirectional crowd experiments for bottleneck areas, and found that the oscillation effect was more pronounced for the longer narrow bottleneck which could bring risks.^[3] Seyfried et al. also focused on doing experiments under the laboratory environment to study the unidirectional crowd through the bottleneck, and found a linear growth of the flow with the bottleneck width.^[25] Zhang et al. further experimentally investigated the ordering in the bidirectional crowd in the straight corridor and its influence on the fundamental diagram.^[26] In addition to using the experiments to investigate pedestrian dynamics, there is also research on theoretical analysis and modeling of the crowd in the bottleneck. Jiang et al.^[27] developed a macroscopic type model of the unidirectional pedestrian flow moving through a bottleneck based on two-dimensional Lighthill–Whitham–Richards model. Dai et al. proposed an agent-based model which combined the agent and force to study the bidirectional flow evacuation through the bottleneck.^[20]

The current study about flow dynamics in the bottleneck mainly focuses on the unidirectional pedestrian flow. According to our investigation, there is little literature studying the bidirectional crowd dynamics in the bottleneck whether from the experimental view or from the view of theoretical analysis.^[20] However, this is also a necessary research issue that can provide some support for designers to assess the layout of corridors. In this paper, we mainly focus on investigating the bidirectional crowd dynamics, including the free flow dynamics and breakdown phenomenon in different shapes of corridors with the bottlenecks. We choose to use the social force model to study the problem of flow dynamics. One reason is that the social force model can represent some self-organization phenomena such as lane formation in the bidirectional flow. Another reason is that the social force model, as a continuous microscopic pedestrian model, not only considers the physical forces but also takes the motivation forces into consideration.

This article mainly presents the bottleneck effects on the bidirectional crowd dynamics. It continues with the introduction of the social force model in Section 2. Section 3 presents the simulation setup. The effects of computation time step, free flow dynamics, breakdown phenomenon, effects of different ratios of inflows from both sides, and effects of the shapes of corridors are studied by numerical simulations in Section 4. After the simulation results are analyzed, the key discoveries are reviewed in Section 5.

The social force model was first proposed by Helbing et al.^[18] in 2000. In the social force model, each pedestrian is driven by three forces, namely the desired force, ; the interaction force between pedestrians i and j, f_ij; and the interaction force between pedestrian i and walls w, f_iw. Generally, the social force model is expressed by

Here, is the desired speed, and is the desired walking direction. τ_i is the adaptation time to adjust the current velocity to the desired velocity.

The f_ij mainly contains the pedestrian’s social-psychological force and the physical force . The former force reflects a tendency to move away from others, and the latter force occurs only when the distance between two pedestrians’ centers d_ij is less than the sum of these two pedestrians’ radii r_ij = r_i + r_j:

The f_iw is modeled similar to f_ij, which is expressed by

The specific parameters in the social force model are listed in Table 1.^[18]

Table 1.

Table 1.

Table 1. Parameters of the social force model. . Symbol Meaning Value m Pedestrian mass 80 kg A Strength of social repulsive force 2000 N B Characteristic distance of social repulsive force 0.08 m κ Coefficient of sliding friction 2.4 × 105 kg · m−1 · s−1 k Body compression coefficient 1.2 × 105 kg · s−2

Table 1. Parameters of the social force model. .

Some self-organization phenomena characterized by the “zipper” effect, lane formation in the bidirectional flow, strips in the crossing flow, and others can be represented by using the social force model.^[18] Take lane formation in the bidirectional flow as an example, the detailed lane formation behavior is shown in Fig. 1. The red empty circles stand for pedestrians walking from the left to the right, while the blue filled circles stand for pedestrians walking from the right to the left throughout this paper. Note that the corridor in Fig. 1 is a typical straight corridor, and there has already been a lot of research on the bidirectional crowd dynamics in the straight corridor.^[28]

Fig. 1.

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	Fig. 1. Lane formation of the bidirectional flow in the straight corridor.

According to Ref. [29], a pedestrian in the high density crowd appears to vibrate continuously when encountering the bottleneck. Pelechano et al. proposed a “stopping rule” to avoid the behavior of vibration, in which the personality of a pedestrian, the walking directions of others, and pedestrian’s current situation were all considered.^[29] Besides, a “respect” mechanism, a self-stopping mechanism of the social force model, was introduced by Parisi et al. that can reproduce the experimental data and also avoid the vibration of pedestrians.^[30] In this paper, we also adopt the same “respect” mechanism in Ref. [30] to simulate the movement of pedestrian flow with high density. The respect distance D_R for pedestrian i is D_Ri = R_F · r_i, where R_F is the respect factor. Once any other pedestrian hits the respect area of pedestrian i which is , the desired speed will be set to 0 until the respect area is free. For more details, we refer the reader to Ref. [30].

In some public places such as the subway stations and airports, it is very common to see some corridors with width variations as figure 2 shows. Generally, the narrow corridor which connects with the wide corridor can usually be seen as the bottleneck according to Ref. [23]. In Fig. 2, the left part of the corridor can be seen as the bottleneck. The set method of the bottleneck in this paper is similar to that in Ref. [5] where the unidirectional crowd experiment through the bottleneck was carried out. The shape of the corridor shown in Fig. 2 is marked by shape 1 for the convenience of mentioning. It is worth noting that the unidirectional pedestrian flow, moving from the right side to the left side of the corridor in Fig. 2, usually forms layers or trails inside bottlenecks effectively, which refers to “zipper” effect.^[5]

Fig. 2.

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	Fig. 2. The diagram of the corridor with the bottleneck.

When using the social force model to simulate the bidirectional flow in a straight corridor shown in Fig. 1, the desired walking directions of pedestrians are usually set to point to both ends of the corridor, such as the set of the desired direction of pedestrian i or j in Fig. 2. However, pedestrians would change their desired directions when encountering the bottleneck, in order to go through the narrow bottleneck as soon as possible or to avoid collisions with “wall 1” or “wall 2”. In this paper, we consider the entrance of the bottleneck as a wide exit for pedestrians walking from the right side to the left side. Their desired directions will point to the center between the endpoints “a” and “b” as Refs. [18] and [29] mentioned once pedestrians enter the influence area of the entrance of the bottleneck. Generally, the influence area of an exit with a certain radius R_inf is larger than the pedestrian’s vision field with a certain radius R_visual because of the phenomenon of flow with the stream.^[31] In this paper, we set R_inf = 1.1 · R_visual,^[32] R_visual randomly distributes between 10 m and 15 m for convenience. When pedestrians from the right side are through the cross-section of “a” and “b” meaning that pedestrians are in the bottleneck, their desired walking directions will recover the original ones. Note that we do not consider the effect of cultural bias on bidirectional crowd dynamics in this paper.

In Fig. 2, the length of the entire corridor with a bottleneck is 60 m, the wider width of the corridor is 8 m, and the narrower width of the corridor is 4 m. The inflows are from the outside of the corridor which are controllable, and we set the initial speeds for pedestrians as 0.5 m/s in order that they could reach the normal walking speed which is usually greater than 1 m/s as soon as possible. Besides, the distance between the place where pedestrians come from and the entrance of the corridor is 1 m, in order that pedestrians can achieve the normal walking states in the corridor and we can also eliminate the impact of loading profiles at both ends of the corridor. Once pedestrians go through the corridor, they can still keep walking forward based on the social force model until totally passing the place where the inflows come from. Assume that the inflow from each side of the corridor maintains a constant value in each simulation, and are respectively the inflows from the left side and the right side of the corridor. Besides, there are no pedestrians inside the corridor at the initial simulation time.^[33]

According to Ref. [2], pedestrians’ reaction time τ uniformly distributes between the minimum value 0.15 s and the maximum value 0.50 s with the step 0.05 s, and radius r uniformly distributes between 0.250 m and 0.300 m with the step 0.025 m. Pedestrians’ desired speed , however, meets the normal distribution, and the mean value is 1.35 m/s with standard deviation 0.02.^[2] In this paper, we set R_F = 0.7.^[30]

Before studying the important issue of the bottleneck effects on the bidirectional flow in the corridor shown in Fig. 2, we will first verify the model even though the presented model has already been calibrated with the fundamental diagram. The experimental case of bottleneck in Ref. [25] is adopted in this paper to compare the relevant results of the unidirectional pedestrian flow between simulations and experiments. Figure 3(a) shows the diagram of the corridor in the experimental case, and pedestrians walk from the right to the left. At the beginning of each experiment, 60 tested pedestrians were placed in the holding area, and the speeds of pedestrians inside the measurement section of the bottleneck were recorded with time. In order to test the validity of the presented social force model, we also set the same simulation scenario with the experimental case. The snapshot of the flow dynamics during a simulation process is presented in Fig. 3(b), which could represent the unidirectional flow characteristics in the bottleneck. The simulation result of the speed decreasing with time is presented in Fig.4. Due to the limit of unavailable experimental data in Ref. [25], we here cannot give the specific comparison results, but the general variation trend and range of the simulation data are consistent with the experimental results. Considering the particularity of pedestrian movement and also the keystone of the study in this paper, some small differences are acceptable, thus the presented model using the same parameters with the references is feasible and effective to investigate the bottleneck problem. Note that pedestrians in different countries may have different body parameters and walking behaviors, the heterogeneity effect on the bidirectional flow dynamics has already been investigated in Ref. [33], and this paper will no longer focus on it.

Fig. 3.

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	Fig. 3. The simulation of the referenced experiment case. (a) The diagram of the referenced experiment corridor. (b) The snapshot of a simulation in the referenced experiment corridor.

Fig. 4.

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	Fig. 4. The simulation result of speed with time in the referenced experiment case.

According to Ref. [34], the bidirectional pedestrian flow has three walking phases: the freely moving phase, the coexisting phase and the jamming phase. Jamming will start when pedestrians cannot keep their speeds in the desired directions, which may further lead to a temporary or total standstill which implies breakdown.^[1,2,33] In this section, effects of computation time step on the bidirectional crowd dynamics are first focused on, meanwhile the free flow dynamics and breakdown phenomenon for the bidirectional crowd in the corridor with shape 1 are investigated to have an in-depth understanding of pedestrian behaviors. We further study effects of different ratios of inflows from both sides of the corridor with shape 1 on both the free flow dynamics and breakdown phenomenon. Also, effects of two different shapes of corridors with bottlenecks are compared and analyzed in detail.

4.1. Effects of computation time step

Two different values of the computation time step Δt which are 0.1 s and 0.01 s respectively are set in the simulation to study its effects on the bidirectional crowd dynamics in the corridor with shape 1. The performance of the influence of Δt on the free flow is reflected by the mean number of pedestrians in the corridor N_in and the total number of leaving pedestrians N_leave, and the performance of its influence on the jamming flow is indicated by the start time of breakdown. In this paper, the total breakdown is defined as that at least 360 pedestrians walk very slowly in the five consecutive seconds, and this slow velocity has the maximum value of 0.2 m/s.^[2,33]

Figure 5 shows the effects of two different values of Δt on the free bidirectional crowd dynamics indicated by N_in and N_leave, and Table 2 gives their corresponding mean values and standard deviations σ after 20 times repeated simulations. Take σ_Nin for example, it means the standard deviation of the number of pedestrians in the corridor N_in. Through comparing the results in Fig. 5 and Table 2, we can conclude that the computation time step does affect the free flow, but its influence is not very significant.

Fig. 5.

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	Fig. 5. Effects of computation time step on the free bidirectional pedestrian flow . (a) The number of pedestrians in the corridor with shape 1. (b) The total number of leaving pedestrians from the corridor with shape 1.

Table 2.

Table 2.

Table 2. Effects of computation time step on the mean numbers of pedestrians in the corridor and leaving the corridor with shape 1 . . Δt Mean Nin σNin Mean Nleave σNleave 0.1 s 99.3 6.2 300.7 6.2 0.01 s 104.3 5.3 295.7 5.3

Table 2. Effects of computation time step on the mean numbers of pedestrians in the corridor and leaving the corridor with shape 1 . .

We further investigate the start time of breakdown for different inflows affected by different computation time steps, and the simulation results are shown in Fig. 6 after 50 times repeated simulations. It is worth noting that the probability of breakdown is 1 for all mentioned situations in Fig. 6, in which the inflows are respectively 1.25 p/s, 1.5 p/s, and 2 p/s from each side of the corridor under different computation time steps. From Fig. 6, we can find that the start time of breakdown when Δt = 0.1 s is a little bit earlier than that when Δt = 0.01 s under the same inflow; this is because of more collisions between each other. However, this difference is not very significant to affect our analysis of the bottleneck effects on the bidirectional crowd dynamics. In this paper, we set the time step Δt to be 0.1 s considering many factors, such as avoiding overlapping and improving the calculation speed.

Fig. 6.

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	Fig. 6. Box-plots of the start time of breakdown with different inflows for the corridor with shape 1. (a) The start time of breakdown under different inflows when Δt=0.1 s. (b) The start time of breakdown under different inflows when Δt=0.01 s.

4.2. Free flow dynamics

In this section, we respectively set for the corridor with shape 1. During the simulation, we can find that pedestrians walk freely inside of the corridor as figure 7 shows, and lane formation which is a very typical phenomenon of self-organization is also represented. Specifically, pedestrians walking from left to right usually develop a middle lane in the right part of the corridor, which can also be reflected from Fig. 8. Figure 8 presents the trajectory of the pedestrian flow from the beginning time to 200 s. It can be observed that the red trajectory lines are very clear throughout the middle of the right part of the corridor, while the blue trajectory lines are clear on both sides of the red ones, which further illustrates the same conclusion as that from Fig. 7. This phenomenon is a direct consequence of the steering adjustment between each other. Pedestrians coming from the left side do not require any goal adjustment, while pedestrians from the right side need to adjust their goals when seeing the bottleneck. Pedestrians from the left side push pedestrians from the other side towards the outside, which is also clearly reflected from the blue trajectory lines near the bottleneck in Fig. 8.

Fig. 7.

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	Fig. 7. The snapshot of the free flow dynamics .

Fig. 8.

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	Fig. 8. Trajectory of the bidirectional flow in the corridor at t = 200 s .

Figure 9 shows the variation of the number of the leaving pedestrians over time, which reflects the leaving efficiency of pedestrians from the corridor. From this figure, we can observe that the number of pedestrians leaving from each side is similar to each other even though the bottleneck which has the relatively narrow width and small capacity exists. This result, however, is from just one simulation. Table 3 further lists the mean numbers of the leaving pedestrians from the right side N_R and from the left side N_L, and their corresponding standard deviations σ after 20 times repeated simulations. From this table, we can also obtain the similar result that no matter if for the case in the straight corridor or in the corridor with shape 1, N_R and N_L are close to each other with relatively small standard deviations. Therefore, the bottleneck cannot reduce the number of the leaving pedestrians from the narrow side for the free bidirectional crowd. Furthermore, as imagined, the bottleneck shown in Fig. 7 cannot significantly decrease N_R or N_L in the free flow compared with the corresponding results of the straight corridor. Note that the shape of the straight corridor mentioned as the benchmark corridor in Table 3 is similar to that in Fig. 1, and the length and width of this straight corridor are respectively 60 m and 8 m. Besides, we will not give the mean values and their standard deviations for the free bidirectional pedestrian flow in the later sections because of the small fluctuations of the numbers of leaving pedestrians.

Fig. 9.

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	Fig. 9. The number of the leaving pedestrians from both sides .

Table 3.

Table 3.

Table 3. The numbers of leaving pedestrians from the straight corridor and the corridor with shape 1 . . Mean NR σR Mean NL σL Mean Nleave σNleave Straight corridor 153.2 1.5 151.4 3.2 304.6 4.1 Corridor with shape 1 150.9 3.4 149 2.6 299.9 3.9

Table 3. The numbers of leaving pedestrians from the straight corridor and the corridor with shape 1 . .

4.3. Breakdown phenomenon study

In this section, we set for the corridor with shape 1. Under this inflow, the probability of total breakdown phenomenon is 1, and the median of the start time of total breakdown is under 200 s reflected from Fig. 6(a).

Figure 10 shows the snapshot of the coming total breakdown phenomenon of the bidirectional pedestrian flow at t = 60 s. From Fig. 10(a), we can observe that breakdown phenomenon occurs beginning from the bottleneck. Figure 10(b) further indicates the occurrence of the coming total breakdown phenomenon. In Fig. 10(b), the red trajectory lines and blue trajectory lines are distinguished obviously at both sides of the entire corridor, and the red lines and blue lines intersect seriously with each other in the right part of the bottleneck, which means it is very difficult for pedestrians to pass this jamming area. When t = 200 s, the breakdown phenomenon spreads to more regions as figure 11 shows because of the continuous inflows.

Fig. 10.

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	Fig. 10. Breakdown phenomenon of the bidirectional pedestrian flow at t = 60 s . (a) Snapshot of the movement of the bidirectional flow. (b) Trajectory of the bidirectional flow.

Fig. 11.

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	Fig. 11. Trajectory of the bidirectional pedestrian flow at t = 200 s .

When the breakdown phenomenon occurs especially when it is very serious, there will be few pedestrians going out from both sides of the corridor inevitably.

Figure 12 shows the variation of the number of the leaving pedestrians over time for two different inflows. It is very distinct that the total numbers of the leaving pedestrians increase slowly or even keep unchanged as time increases, which is a direct result of the total breakdown phenomenon. By comparing Figs. 12 and 9, we can observe that the total numbers of leaving pedestrians in Fig. 12 are lower than that when , which is also because of the occurrence of breakdown. Moreover, we can observe that N_R is higher than N_L, the reason may be the bottleneck with relatively wider “exit” to the right part of the infrastructure. These conclusions can also be reflected from Tables 6 and 7, in which the corresponding mean values of the numbers of leaving pedestrians after 20 times simulations are given. In addition, because of relatively high inflows from both sides of the corridor, the number of leaving pedestrians is easier to reach the constant value under compared with that under .

Fig. 12.

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	Fig. 12. The number of the leaving pedestrians for different values of inflows. (a) . (b) .

4.4. Effects of different ratios of inflows from both sides

When inflows from both sides of the corridor with a bottleneck are different, the bidirectional crowd dynamics may change accordingly. In this section, we set six cases listed in Table 4 to investigate this issue in the corridor with shape 1. In the cases 1 and 2 of Table 4, pedestrians in the bidirectional flow can walk freely, because pedestrians can still walk freely when in Subsection 4.2. In the cases 3, 4, 5, and 6, we are uncertain about the moving states of bidirectional flows in the corridor with shape 1.

Table 4.

Table 4.

Table 4. The set of inflows in the investigated cases. . Case 1 0.5 1 2 1 0.5 3 1 2 4 2 1 5 0.5 1.5 6 1.5 0.5

Table 4. The set of inflows in the investigated cases. .

In the case 1, is twice of , and the case 2 is just the opposite case. Figure 13 shows the bidirectional free flow dynamics in the cases 1 and 2 respectively. By observing Figs. 13(a) and 13(c), we can find that the typical self-organization phenomenon characterized by the lane formation here is as obvious as that in Fig. 7. Besides, the pedestrians from left to right are mainly in the middle layer, this phenomenon, however, is obtained just from two snapshots in Figs. 13(a) and 13(c). In order to figure out whether or not this phenomenon is the normal state of the bidirectional flow despite different inflows from both sides of the corridor, we further investigate the trajectory lines of both cases as shown in Figs. 13(b) and 13(d). From these two figures, we can observe that the red trajectory lines mainly concentrate in the middle, which also verifies the conclusion obtained from Figs. 13(a) and 13(c). Therefore, we can conclude that the pedestrians coming from the bottleneck shown in Fig. 2 are more likely to distribute in the middle of the right part of the corridor to develop the middle layer in the free bidirectional flow, no matter whether the inflows from both sides are the same or not. Figures 13(e) and 13(f) show the numbers of the pedestrians leaving from both sides of the corridor with shape 1. The number of the leaving pedestrians which are from a side with larger inflow is almost twice that from the other side with smaller inflow, this further reflects that the pedestrians can walk freely in the corridor.

Fig. 13.

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Fig. 13. Pedestrian flow dynamics in the cases 1 and 2. (a) The snapshot of pedestrian flow at 180 s in the case 1. (b) Trajectory of pedestrian flow at 200 s in the case 1. (c) The snapshot of pedestrian flow at 180 s in the case 2. (d) Trajectory of pedestrian flow at 200 s in the case 2. (e) The number of the leaving pedestrians in the case 1. (f) The number of the leaving pedestrians in the case 2.

In the cases 3 and 4, breakdown phenomenon occurs. By comparing Figs. 14(a)–14(d), it can be found that breakdown phenomenon occurs basically beginning from the bottleneck and then spreads to more regions. By comparing Figs.14(e) and 14(f), it can be observed that pedestrians coming from the left side of the corridor in the case 4 find it easier to leave the corridor from the right side, and there are more pedestrians leaving from the corridor in the case 4 during the same time period. This can also be reflected from Table 5, in which the results come from 20 times repeated simulations.

Fig. 14.

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Fig. 14. Pedestrian flow dynamics in the cases 3 and 4. (a) The snapshot of pedestrian flow at 90 s in the case 3. (b) Trajectory of pedestrian flow at 200 s in the case 3. (c) The snapshot of pedestrian flow at 90 s in the case 4. (d) Trajectory of pedestrian flow at 200 s in the case 4. (e) The number of the leaving pedestrians in the case 3. (f) The number of the leaving pedestrians in the case 4.

Table 5.

Table 5.

Table 5. The comparison of the numbers of leaving pedestrians from the corridors in the cases 3 and 4. . Mean NR σR Mean NL σL Mean Nleave σNleave The case 3 55.1 16.3 91.4 23.0 146.5 36.7 The case 4 139.2 32.3 36.3 20.0 175.5 51.3

Table 5. The comparison of the numbers of leaving pedestrians from the corridors in the cases 3 and 4. .

In Table 5, the mean N_R is nearly four times as much as the mean N_L in the case 4, while the mean N_L is only nearly twice the mean N_R in the case 3. Relatively speaking, if the inflows from both sides of the corridor with the bottleneck are respectively 1 p/s and 2 p/s, the design of the case 4, meaning the larger inflow 2 p/s comes into the corridor from the narrow side, is much better for the evacuation than that in the case 3.

In the cases 5 and 6, the difference of the values of inflows from both sides of the corridor with shape 1 is larger than that in the cases 1, 2, 3, and 4. If , breakdown phenomenon will occur around 250 s as Subsection 4.1 gives. However, breakdown phenomenon does not occur in the cases 5 and 6 before the critical time 1500 s we set in this paper, when the inflow from one side of the corridor decreases to 0.5 p/s. Besides, by comparing Figs. 15(e) and 15(f), the number of the leaving pedestrians who come from a side of the corridor with larger inflow is nearly three times as much as that coming from the other side.

Fig. 15.

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Fig. 15. Pedestrian flow dynamics in the cases 5 and 6. (a) The snapshot of pedestrian flow at 180 s in the case 5. (b) Trajectory of pedestrian flow at 200 s in the case 5. (c) The snapshot of pedestrian flow at 180 s in the case 6. (d) Trajectory of pedestrian flow at 200 s in the case 6. (e) The number of the leaving pedestrians in the case 5. (f) The number of the leaving pedestrians in the case 6.

4.5. Effects of different shapes of corridors with bottlenecks

In this section, we propose another shape of the corridor with the bottleneck shown in Fig. 16(a) to analyze its effects on pedestrian dynamics. The widths of the entire corridor on both sides are also 4 m and 8 m, which are the same with those in Fig. 2. Note that the shape of the corridor shown in Fig. 16(a) is marked by shape 2, and the left half of the corridor is considered as the bottleneck. Besides, the setting method of the desired walking direction is similar to that in section 3.

Fig. 16.

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	Fig. 16. Pedestrian flow dynamics in the corridor with shape . (a) The snapshot of pedestrian flow at 180 s. (b) Trajectory of pedestrian flow at 200 s. (c) The number of the leaving pedestrians.

From Figs. 16(a) and 16(b), we can observe that the pedestrian flow coming from the left side of the corridor mainly develops an upper lane, while the flow coming from the right side mainly forms a lower lane because of the steering adjustment between each other. This distribution of lane formation is a little different from that in Figs. 7 and 8. However, by comparing the simulation results in Figs. 16(c) and 9, we can observe that the numbers of the leaving pedestrians from the corridors with different shapes 1 and 2 are close to each other. Note that for the corridor with shape 2. Therefore, we can further conclude that the shapes of corridors shown in Figs. 1, 2, and 16(a) have little influence on the free flow efficiency.

Figure 17 shows breakdown phenomenon occurs in the corridor with shape 2, and also gives the number of leaving pedestrians over time. From Fig. 17(a), we can observe that breakdown phenomenon begins to develop from the bottleneck, this is because the width of bottleneck cannot meet the requirement of the leaving pedestrians. By comparing Figs. 17(b) and 12(a), we can find a larger number of pedestrians leaving from the corridor with shape 2 within the same time period. In order to find whether or not this difference really exists, we, therefore, compare the mean results of the number of leaving pedestrians by running 20 times repeated simulations listed in Table 6. From Table 6, we can observe the corridor with shape 2 can let more pedestrians leave when , meanwhile σ_Nleave is very large, which means the degrees of breakdown phenomena in different simulations are greatly different. In order to study whether or not the corridor with shape 2 can still result in more pedestrians to leave, we further run 20 times simulations where , and the comparison results are listed in Table 7. From Table 7, we also obtain the same conclusion that the corridor with shape 2 can improve the leaving efficiency. It can be comprehended from the snapshots of the bidirectional flow in these two corridors with shapes 1 and 2, such as Figs. 7 and 16(a). As shown in these snapshots, pedestrians mostly form into three lanes in the corridor with shape 1, while forming into two lanes in the corridor with shape 2 for the free bidirectional flow. Then, the number of pedestrians who require steering away pedestrians from the other side is relatively few in the corridor with shape 2 because of only one contact surface. Once inflows are large enough, breakdown will occur. Thus, the start time of breakdown may be earlier for the flow in the corridor with shape 1 as relatively more conflicts, which can also be reflected by comparing the results in Figs. 16(a) and 18. In Fig. 6(a), the medians of the start time of breakdown are 379 s, 261 s, and 175 s under inflows 1.25 p/s, 1.5 p/s, and 2 p/s respectively. In Fig. 18, the corresponding values are 419 s, 272 s, and 182 s, respectively. In addition, figure 19 shows the simulation result of the corresponding bidirectional crowd dynamics in the corridor with shape 2 in a simulation. It is worth noting that figure 19(a) shows the trajectory of the pedestrian flow in a simulation, in which the blue and red trajectory areas are separated except some blue trajectory lines through the red trajectory area. This further reflects the occurrence of breakdown and its severity.

Table 6.

Table 6.

Table 6. The comparison of the numbers of leaving pedestrians from the corridors with shapes 1 and . . Mean NR σR Mean NL σL Mean Nleave σNleave Corridor with shape 1 89.9 27.7 49.5 18.2 139.4 42.8 Corridor with shape 2 113.5 38.9 74.6 26.8 188.1 64.6

Table 6. The comparison of the numbers of leaving pedestrians from the corridors with shapes 1 and . .

Table 7.

Table 7.

Table 7. The comparison of the numbers of leaving pedestrians from the corridors with shapes 1 and . . Mean NR σR Mean NL σL Mean Nleave σNleave Corridor with shape 1 46.2 6.3 17.1 4.7 63.3 9.0 Corridor with shape 2 50 6.9 22.5 6.8 72.5 13.0

Table 7. The comparison of the numbers of leaving pedestrians from the corridors with shapes 1 and . .

Fig. 17.

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	Fig. 17. Pedestrian flow dynamics in the corridor with shape . (a) The snapshot of pedestrian flow at 60 s. (b) The number of the leaving pedestrians.

Fig. 18.

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	Fig. 18. Box-plot of the start time of breakdown with different inflows for the corridor with shape 2.

Fig. 19.

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	Fig. 19. Pedestrian flow dynamics in the corridor with shape . (a) Trajectory of pedestrian flow at 200 s. (b) The number of the leaving pedestrians.

This paper investigated the bottleneck effect on the bidirectional crowd dynamics based on the famous social force model. As an intuitive mimic of our daily observations, the dynamics of pedestrians is described in a novel way that: in order to leave the corridor as soon as possible, pedestrians will re-aim their desired walking directions once the entrance of the bottleneck is observed. Simulation results indicate the existence of lane formation of free bidirectional flow in the corridor with the bottleneck, simulations further indicate that: the composition of lanes formed by pedestrians is related to the shape of the corridor, but does not relate to the magnitude of the inflow. The leaving efficiency of pedestrians in the free bidirectional flow is relatively stable in different simulations and is related to the magnitude of the inflow. Moreover, breakdown phenomenon is also observed in the simulations, once it occurs, it begins from the bottleneck and gradually spreads to both sides. The leaving efficiency will accordingly reduce when breakdown phenomenon occurs, which is related to the shape of the bottleneck. Based on these findings, the designing of infrastructures in the future is possibly to become more accessible.