Article Contents
Article Contents

Micro- and macroscopic modeling of crowding and pushing in corridors

• Experiments with pedestrians revealed that the geometry of the domain, as well as the incentive of pedestrians to reach a target as fast as possible have a strong influence on the overall dynamics. In this paper, we propose and validate different mathematical models at the micro- and macroscopic levels to study the influence of both effects. We calibrate the models with experimental data and compare the results at the micro- as well as macroscopic levels. Our numerical simulations reproduce qualitative experimental features on both levels, and indicate how geometry and motivation level influence the observed pedestrian density. Furthermore, we discuss the dynamics of solutions for different modeling approaches and comment on the analysis of the respective equations.

Mathematics Subject Classification: 91C20, 68Q80, 35Q84.

 Citation:

• Figure 1.  Left: Sketch of experimental setup at the University of Wuppertal, showing the corridor width $w_E$ in the experiments, the exit $\Gamma_E$ and the measurement area. Right: computational domain with adapted width $w_{CA}$ to ensure a consistent discretization of the exit and an increased length $l_{CA} = 9.6$m

Figure 2.  Cellular automaton: transition rules

Figure 3.  Discretization of the exit. In the case of an even number of cells, a central positioned agent will leave the corridor faster than in the case of three cells

Figure 4.  (A) distribution of $n = 40$ agents at time $t = 0$ and $t = 16 \Delta t$, (B) the simulated density, (C) densities of exit times when increasing the number of Monte-Carlo runs

Figure 5.  Influence of the scaling parameter $\beta$ on individual dynamics

Figure 6.  Average exit time as a function of $\beta$ and $p_{ex}$, or $p_{ex}$ only

Figure 7.  Impact of the corridor width on the maximum density. The CA approach yields comparable results for high density regimes and low motivation level

Figure 8.  Impact of the motivation level on the maximum pedestrian density: experimental (A) vs. microscopic simulations (B)

Figure 9.  Comparison of the potentials $\phi_E$ and $\phi_L$ for a convex obstacle

Figure 10.  Comparison of the potentials $\phi_E$ and $\phi_L$ for a non-convex obstacle

Figure 11.  Left: Bifurcation diagram detailing the behavior of the solution to (12)-(13). The behavior along the interface lines is identical as in the bottom right corner. Right: exit time corresponding to $L = 1$

Figure 12.  Simulations for $n = 60$, $\beta = 3.84$, $\mu = 1$. We observe a good agreement between the CA (microscopic) and PDE (macroscopic) solutions. The effect of higher densities for wider corridors also occur on a macroscopic scale. There is a clear difference in behavior between narrow and wide corridors

Figure 13.  Effects of pushing

Figure 14.  Comparison of the congestion at the exit in case of pushing (bottom row) and no-pushing (top row) for $60$ individuals: We observe that people move faster towards the exit and the formation of a larger congested area in front of it

Table 1.  exit times for different runs and different motivations from [3]. Run 01 is used to set the desired maximum velocity $v_{max}$

 Run $\mu=1$ $\mu_0$ 01, $n=1$ $8\mathrm{s}$ 02, $n=63, w=1.2\mathrm{m}$ $53\mathrm{s}$ $64\mathrm{s}$ 03, $n=67, w=3.4\mathrm{m}$ $60\mathrm{s}$ $68\mathrm{s}$ 04, $n=57, w=5.6\mathrm{m}$ $55\mathrm{s}$ $57\mathrm{s}$
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