September  2020, 15(3): 405-426. doi: 10.3934/nhm.2020025

Micro- and macroscopic modeling of crowding and pushing in corridors

1. 

Radon Institute for Applied and Computational Mathematics, Altenbergerstr. 69, 4040, Linz, Austria

2. 

University of Warwick, Mathematics Institute, Gibbet Hill Road, CV47AL Coventry, UK

Received  November 2019 Revised  July 2020 Published  September 2020

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.

Citation: Michael Fischer, Gaspard Jankowiak, Marie-Therese Wolfram. Micro- and macroscopic modeling of crowding and pushing in corridors. Networks & Heterogeneous Media, 2020, 15 (3) : 405-426. doi: 10.3934/nhm.2020025
References:
[1]

Crowding and queuing at entrances, (2018), http://ped.fz-juelich.de/da/doku.php?id=wuptmp. Google Scholar

[2]

Jupedsim, (2019), https://www.jupedsim.org. Google Scholar

[3]

J. Adrian, M. Boltes, S. Holl, A. Sieben and A. Seyfried, Crowding and queuing in entrance scenarios: Influence of corridor width in front of bottlenecks, arXiv e-prints, arXiv: 1810.07424. Google Scholar

[4]

A. A. AlmetM. PanB. Hughes and K. Landman, When push comes to shove: Exclusion processes with nonlocal consequences, Physica A, 437 (2015), 119-129.  doi: 10.1016/j.physa.2015.05.031.  Google Scholar

[5]

M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1997. doi: 10.1007/978-0-8176-4755-1.  Google Scholar

[6]

M. BurgerS. HittmeirH. Ranetbauer and M.-T. Wolfram, Lane formation by side-stepping, SIAM Journal on Mathematical Analysis, 48 (2016), 981-1005.  doi: 10.1137/15M1033174.  Google Scholar

[7]

M. Burger and J.-F. Pietschmann, Flow characteristics in a crowded transport model, Nonlinearity, 29 (2016), 3528-3550.  doi: 10.1088/0951-7715/29/11/3528.  Google Scholar

[8]

C. BursteddeK. KlauckA. Schadschneider and J. Zittartz, Simulation of pedestrian dynamics using a two-dimensional cellular automaton, Physica A: Statistical Mechanics and its Applications, 295 (2001), 507-525.  doi: 10.1016/S0378-4371(01)00141-8.  Google Scholar

[9]

L. A. Caffarelli and M. G. Crandall, Distance functions and almost global solutions of eikonal equations, Communications in Partial Differential Equations, 35 (2010), 391-414.  doi: 10.1080/03605300903253927.  Google Scholar

[10]

A. Corbetta, J. A. Meeusen, C.-m. Lee, R. Benzi and F. Toschi, Physics-based modeling and data representation of pairwise interactions among pedestrians, Physical Review E, 98 (2018), 062310. doi: 10.1103/PhysRevE.98.062310.  Google Scholar

[11]

E. Cristiani, B. Piccoli and A. Tosin, Multiscale Modeling of Pedestrian Dynamics, MS & A, Modeling, Simulation and Applications, 12. Springer, Cham, 2014. doi: 10.1007%2F978-3-319-06620-2.  Google Scholar

[12]

M. Di FrancescoP. MarkowichJ.-F. Pietschmann and M.-T. Wolfram, On the Hughes' model for pedestrian flow: The one-dimensional case, Journal of Differential Equations, 250 (2011), 1334-1362.  doi: 10.1016/j.jde.2010.10.015.  Google Scholar

[13]

D. C. Duives, W. Daamen and S. Hoogendoorn, Trajectory analysis of pedestrian crowd movements at a Dutch music festival, Pedestrian and Evacuation Dynamics 2012, Springer, (2014), 151–166. doi: 10.1007/978-3-319-02447-9_11.  Google Scholar

[14]

S. N. GomesA. M. Stuart and M.-T. Wolfram, Parameter estimation for macroscopic pedestrian dynamics models from microscopic data, SIAM Journal on Applied Mathematics, 79 (2019), 1475-1500.  doi: 10.1137/18M1215980.  Google Scholar

[15]

B. Hein, Agent-Based Modelling for Crowding and Queuing in Front of Bottlenecks, Bachelor's Thesis, University of Wuppertal, 2019. Google Scholar

[16]

D. Helbing and P. Molnár, Social force model for pedestrian dynamics, Physical Review E, 51 (1995), 4282. doi: 10.1103/PhysRevE.51.4282.  Google Scholar

[17]

K. Hirai and K. Tarui, A simulation of the behavior of a crowd in panic, Systems and Control. Google Scholar

[18]

R. Hughes, A continuum theory for the flow of pedestrians, Transportation Research Part B: Methodological, 36 (2002), 507-535.  doi: 10.1016/S0191-2615(01)00015-7.  Google Scholar

[19]

A. Johansson and D. Helbing, Analysis of empirical trajectory data of pedestrians, Pedestrian and Evacuation Dynamics 2008, (2009), 203–214. doi: 10.1007/978-3-642-04504-2_15.  Google Scholar

[20]

A. Jüngel, The boundedness-by-entropy method for cross-diffusion systems, Nonlinearity, 28 (2015), 1963. Google Scholar

[21]

A. Kirchner and A. Schadschneider, Simulation of evacuation processes using a bionics-inspired cellular automaton model for pedestrian dynamics, Physica A: Statistical Mechanics and its Applications, 312 (2002), 260-276.  doi: 10.1016/S0378-4371(02)00857-9.  Google Scholar

[22]

C. Koutschan, H. Ranetbauer, G. Regensburger and M.-T. Wolfram, Symbolic derivation of mean-field PDEs from lattice-based models, 2015 17th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC), 27–33. doi: 10.1109/SYNASC.2015.14.  Google Scholar

[23]

O. Ladyzhenskaya, Linear and Quasilinear Elliptic Equations, ISSN, Elsevier Science, 1968. Google Scholar

[24]

P. LeFloch, Explicit formula for scalar nonlinear conservation laws with boundary condition, Mathematical Methods in the Applied Sciences, 10 (1988), 265-287.  doi: 10.1002/mma.1670100305.  Google Scholar

[25]

P. LeFloch and J.-C. Nédélec, Explicit formula for weighted scalar nonlinear hyperbolic conservation laws, Transactions of the American Mathematical Society, 308 (1988), 667-683.  doi: 10.1090/S0002-9947-1988-0951622-X.  Google Scholar

[26]

C. Lehrenfeld, On a Space-Time Extended Finite Element Method for the Solution of a Class of Two-Phase Mass Transport Problems, PhD thesis, RWTH Aachen, 2015, http://publications.rwth-aachen.de/record/462743. Google Scholar

[27]

A. Y. Leroux, Approximation de Quelques Problèmes Hyperboliques Non-Linéaires, Thèse d'état, Rennes, 1979. Google Scholar

[28]

B. Maury and S. Faure, Crowds in Equations. An Introduction to the Microscopic Modeling of Crowds, Advanced Textbooks in Mathematics. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2019.  Google Scholar

[29]

M. MoussaïdD. HelbingS. GarnierA. JohanssonM. Combe and G. Theraulaz, Experimental study of the behavioural mechanisms underlying self-organization in human crowds, Proceedings of The Royal Society B: Biological Sciences, 276 (2009), 2755-2762.  doi: 10.1098/rspb.2009.0405.  Google Scholar

[30]

S. Nowak and A. Schadschneider, Quantitative analysis of pedestrian counterflow in a cellular automaton model, Physical Review E, 85 (2012), 066128. doi: 10.1103/PhysRevE.85.066128.  Google Scholar

[31]

S. Okazaki, A study of pedestrian movement in architectural space, Trans. of A.I.J., 283. Google Scholar

[32]

B. Piccoli and A. Tosin, Time-evolving measures and macroscopic modeling of pedestrian flow, Archive for Rational Mechanics and Analysis, 199 (2011), 707-738.  doi: 10.1007/s00205-010-0366-y.  Google Scholar

[33]

J. QianY.-T. Zhang and H.-K. Zhao, Fast sweeping methods for eikonal equations on triangular meshes, SIAM J. Numerical Analysis, 45 (2007), 83-107.  doi: 10.1137/050627083.  Google Scholar

[34]

C. Rudloff, T. Matyus and S. Seer, Comparison of different calibration techniques on simulated data, Pedestrian and Evacuation Dynamics 2012, Springer, (2013), 657–672. doi: 10.1007/978-3-319-02447-9_55.  Google Scholar

[35]

A. Schadschneider, C. Eilhardt, S. Nowak and R. Will, Towards a calibration of the floor field cellular automaton, Pedestrian and Evacuation Dynamics, (2011), 557–566. doi: 10.1007/978-1-4419-9725-8_50.  Google Scholar

[36]

A. Schadschneider, H. Klpfel, T. Kretz, C. Rogsch and A. Seyfried, Fundamentals of pedestrian and evacuation dynamics, IGI Global, (2009), 124–154. Google Scholar

[37]

M. TwarogowskaP. Goatin and R. Duvigneau, Comparative study of macroscopic pedestrian models, Transportation Research Procedia, 2 (2014), 477-485.  doi: 10.1016/j.trpro.2014.09.063.  Google Scholar

[38]

U. Weidmann, Transporttechnik der Fussgänger: Transporttechnische Eigenschaften des Fussgängerverkehrs, Schriftenreihe des IVT, IVT, 1993. Google Scholar

[39]

W. G. Weng, T. Chen, H. Y. Yuan and W. C. Fan, Cellular automaton simulation of pedestrian counter flow with different walk velocities, Physical Review. E, 74 (2006), 036102. doi: 10.1103/PhysRevE.74.036102.  Google Scholar

[40]

C. A. Yates, A. Parker and R. E. Baker, Incorporating pushing in exclusion-process models of cell migration, Physical Review E, 91 (2015), 052711. doi: 10.1103/PhysRevE.91.052711.  Google Scholar

show all references

References:
[1]

Crowding and queuing at entrances, (2018), http://ped.fz-juelich.de/da/doku.php?id=wuptmp. Google Scholar

[2]

Jupedsim, (2019), https://www.jupedsim.org. Google Scholar

[3]

J. Adrian, M. Boltes, S. Holl, A. Sieben and A. Seyfried, Crowding and queuing in entrance scenarios: Influence of corridor width in front of bottlenecks, arXiv e-prints, arXiv: 1810.07424. Google Scholar

[4]

A. A. AlmetM. PanB. Hughes and K. Landman, When push comes to shove: Exclusion processes with nonlocal consequences, Physica A, 437 (2015), 119-129.  doi: 10.1016/j.physa.2015.05.031.  Google Scholar

[5]

M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1997. doi: 10.1007/978-0-8176-4755-1.  Google Scholar

[6]

M. BurgerS. HittmeirH. Ranetbauer and M.-T. Wolfram, Lane formation by side-stepping, SIAM Journal on Mathematical Analysis, 48 (2016), 981-1005.  doi: 10.1137/15M1033174.  Google Scholar

[7]

M. Burger and J.-F. Pietschmann, Flow characteristics in a crowded transport model, Nonlinearity, 29 (2016), 3528-3550.  doi: 10.1088/0951-7715/29/11/3528.  Google Scholar

[8]

C. BursteddeK. KlauckA. Schadschneider and J. Zittartz, Simulation of pedestrian dynamics using a two-dimensional cellular automaton, Physica A: Statistical Mechanics and its Applications, 295 (2001), 507-525.  doi: 10.1016/S0378-4371(01)00141-8.  Google Scholar

[9]

L. A. Caffarelli and M. G. Crandall, Distance functions and almost global solutions of eikonal equations, Communications in Partial Differential Equations, 35 (2010), 391-414.  doi: 10.1080/03605300903253927.  Google Scholar

[10]

A. Corbetta, J. A. Meeusen, C.-m. Lee, R. Benzi and F. Toschi, Physics-based modeling and data representation of pairwise interactions among pedestrians, Physical Review E, 98 (2018), 062310. doi: 10.1103/PhysRevE.98.062310.  Google Scholar

[11]

E. Cristiani, B. Piccoli and A. Tosin, Multiscale Modeling of Pedestrian Dynamics, MS & A, Modeling, Simulation and Applications, 12. Springer, Cham, 2014. doi: 10.1007%2F978-3-319-06620-2.  Google Scholar

[12]

M. Di FrancescoP. MarkowichJ.-F. Pietschmann and M.-T. Wolfram, On the Hughes' model for pedestrian flow: The one-dimensional case, Journal of Differential Equations, 250 (2011), 1334-1362.  doi: 10.1016/j.jde.2010.10.015.  Google Scholar

[13]

D. C. Duives, W. Daamen and S. Hoogendoorn, Trajectory analysis of pedestrian crowd movements at a Dutch music festival, Pedestrian and Evacuation Dynamics 2012, Springer, (2014), 151–166. doi: 10.1007/978-3-319-02447-9_11.  Google Scholar

[14]

S. N. GomesA. M. Stuart and M.-T. Wolfram, Parameter estimation for macroscopic pedestrian dynamics models from microscopic data, SIAM Journal on Applied Mathematics, 79 (2019), 1475-1500.  doi: 10.1137/18M1215980.  Google Scholar

[15]

B. Hein, Agent-Based Modelling for Crowding and Queuing in Front of Bottlenecks, Bachelor's Thesis, University of Wuppertal, 2019. Google Scholar

[16]

D. Helbing and P. Molnár, Social force model for pedestrian dynamics, Physical Review E, 51 (1995), 4282. doi: 10.1103/PhysRevE.51.4282.  Google Scholar

[17]

K. Hirai and K. Tarui, A simulation of the behavior of a crowd in panic, Systems and Control. Google Scholar

[18]

R. Hughes, A continuum theory for the flow of pedestrians, Transportation Research Part B: Methodological, 36 (2002), 507-535.  doi: 10.1016/S0191-2615(01)00015-7.  Google Scholar

[19]

A. Johansson and D. Helbing, Analysis of empirical trajectory data of pedestrians, Pedestrian and Evacuation Dynamics 2008, (2009), 203–214. doi: 10.1007/978-3-642-04504-2_15.  Google Scholar

[20]

A. Jüngel, The boundedness-by-entropy method for cross-diffusion systems, Nonlinearity, 28 (2015), 1963. Google Scholar

[21]

A. Kirchner and A. Schadschneider, Simulation of evacuation processes using a bionics-inspired cellular automaton model for pedestrian dynamics, Physica A: Statistical Mechanics and its Applications, 312 (2002), 260-276.  doi: 10.1016/S0378-4371(02)00857-9.  Google Scholar

[22]

C. Koutschan, H. Ranetbauer, G. Regensburger and M.-T. Wolfram, Symbolic derivation of mean-field PDEs from lattice-based models, 2015 17th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC), 27–33. doi: 10.1109/SYNASC.2015.14.  Google Scholar

[23]

O. Ladyzhenskaya, Linear and Quasilinear Elliptic Equations, ISSN, Elsevier Science, 1968. Google Scholar

[24]

P. LeFloch, Explicit formula for scalar nonlinear conservation laws with boundary condition, Mathematical Methods in the Applied Sciences, 10 (1988), 265-287.  doi: 10.1002/mma.1670100305.  Google Scholar

[25]

P. LeFloch and J.-C. Nédélec, Explicit formula for weighted scalar nonlinear hyperbolic conservation laws, Transactions of the American Mathematical Society, 308 (1988), 667-683.  doi: 10.1090/S0002-9947-1988-0951622-X.  Google Scholar

[26]

C. Lehrenfeld, On a Space-Time Extended Finite Element Method for the Solution of a Class of Two-Phase Mass Transport Problems, PhD thesis, RWTH Aachen, 2015, http://publications.rwth-aachen.de/record/462743. Google Scholar

[27]

A. Y. Leroux, Approximation de Quelques Problèmes Hyperboliques Non-Linéaires, Thèse d'état, Rennes, 1979. Google Scholar

[28]

B. Maury and S. Faure, Crowds in Equations. An Introduction to the Microscopic Modeling of Crowds, Advanced Textbooks in Mathematics. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2019.  Google Scholar

[29]

M. MoussaïdD. HelbingS. GarnierA. JohanssonM. Combe and G. Theraulaz, Experimental study of the behavioural mechanisms underlying self-organization in human crowds, Proceedings of The Royal Society B: Biological Sciences, 276 (2009), 2755-2762.  doi: 10.1098/rspb.2009.0405.  Google Scholar

[30]

S. Nowak and A. Schadschneider, Quantitative analysis of pedestrian counterflow in a cellular automaton model, Physical Review E, 85 (2012), 066128. doi: 10.1103/PhysRevE.85.066128.  Google Scholar

[31]

S. Okazaki, A study of pedestrian movement in architectural space, Trans. of A.I.J., 283. Google Scholar

[32]

B. Piccoli and A. Tosin, Time-evolving measures and macroscopic modeling of pedestrian flow, Archive for Rational Mechanics and Analysis, 199 (2011), 707-738.  doi: 10.1007/s00205-010-0366-y.  Google Scholar

[33]

J. QianY.-T. Zhang and H.-K. Zhao, Fast sweeping methods for eikonal equations on triangular meshes, SIAM J. Numerical Analysis, 45 (2007), 83-107.  doi: 10.1137/050627083.  Google Scholar

[34]

C. Rudloff, T. Matyus and S. Seer, Comparison of different calibration techniques on simulated data, Pedestrian and Evacuation Dynamics 2012, Springer, (2013), 657–672. doi: 10.1007/978-3-319-02447-9_55.  Google Scholar

[35]

A. Schadschneider, C. Eilhardt, S. Nowak and R. Will, Towards a calibration of the floor field cellular automaton, Pedestrian and Evacuation Dynamics, (2011), 557–566. doi: 10.1007/978-1-4419-9725-8_50.  Google Scholar

[36]

A. Schadschneider, H. Klpfel, T. Kretz, C. Rogsch and A. Seyfried, Fundamentals of pedestrian and evacuation dynamics, IGI Global, (2009), 124–154. Google Scholar

[37]

M. TwarogowskaP. Goatin and R. Duvigneau, Comparative study of macroscopic pedestrian models, Transportation Research Procedia, 2 (2014), 477-485.  doi: 10.1016/j.trpro.2014.09.063.  Google Scholar

[38]

U. Weidmann, Transporttechnik der Fussgänger: Transporttechnische Eigenschaften des Fussgängerverkehrs, Schriftenreihe des IVT, IVT, 1993. Google Scholar

[39]

W. G. Weng, T. Chen, H. Y. Yuan and W. C. Fan, Cellular automaton simulation of pedestrian counter flow with different walk velocities, Physical Review. E, 74 (2006), 036102. doi: 10.1103/PhysRevE.74.036102.  Google Scholar

[40]

C. A. Yates, A. Parker and R. E. Baker, Incorporating pushing in exclusion-process models of cell migration, Physical Review E, 91 (2015), 052711. doi: 10.1103/PhysRevE.91.052711.  Google Scholar

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} $
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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