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A kinetic theory approach to model pedestrian dynamics in bounded domains with obstacles
Department of Mathematics, University of Houston, 3551 Cullen Blvd, Houston, TX 77204, USA |
We consider a kinetic theory approach to model the evacuation of a crowd from bounded domains. The interactions of a person with other pedestrians and the environment, which includes walls, exits, and obstacles, are modeled by using tools of game theory and are transferred to the crowd dynamics. The model allows to weight between two competing behaviors: the search for less congested areas and the tendency to follow the stream unconsciously in a panic situation. For the numerical approximation of the solution to our model, we apply an operator splitting scheme which breaks the problem into two pure advection problems and a problem involving the interactions. We compare our numerical results against the data reported in a recent empirical study on evacuation from a room with two exits. For medium and medium-to-large groups of people we achieve good agreement between the computed average people density and flow rate and the respective measured quantities. Through a series of numerical tests we also show that our approach is capable of handling evacuation from a room with one or more exits with variable size, with and without obstacles, and can reproduce lane formation in bidirectional flow in a corridor.
References:
[1] |
J. P. Agnelli, F. Colasuonno and D. Knopoff,
A kinetic theory approach to the dynamics of crowd evacuation from bounded domains, Mathematical Models and Methods in Applied Sciences, 25 (2015), 109-129.
doi: 10.1142/S0218202515500049. |
[2] |
G. Antonini, M. Bierlaire and M. Weber,
Discrete choice models of pedestrian walking behavior, Transportation Research Part B: Methodological, 40 (2006), 667-687.
doi: 10.1016/j.trb.2005.09.006. |
[3] |
M. Asano, T. Iryo and M. Kuwahara,
Microscopic pedestrian simulation model combined with a tactical model for route choice behaviour, Transportation Research Part C: Emerging Technologies, 18 (2010), 842-855.
doi: 10.1016/j.trc.2010.01.005. |
[4] |
S. Bandini, S. Manzoni and G. Vizzari, Agent based modeling and simulation: An informatics perspective, Journal of Artificial Societies and Social Simulation, 12 (2009). |
[5] |
N. Bellomo and A. Bellouquid,
On the modeling of crowd dynamics: Looking at the beautiful shapes of swarms, Networks and Heterogeneous Media, 6 (2011), 383-399.
doi: 10.3934/nhm.2011.6.383. |
[6] |
N. Bellomo, A. Bellouquid and D. Knopoff,
From the microscale to collective crowd dynamics, Society for Industrial and Applied Mathematics Multiscale Modeling and Simulation, 11 (2013), 943-963.
doi: 10.1137/130904569. |
[7] |
N. Bellomo, A. Bellouquid, L. Gibelli and N. Outada, A Quest Towards a Mathematical Theory of Living Systems, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser/Springer, Cham, 2017.
doi: 10.1007/978-3-319-57436-3. |
[8] |
N. Bellomo and C. Dogbe,
On the modeling of traffic and crowds: A survey of models, speculations, and perspectives, Society for Industrial and Applied Mathematics Review, 53 (2011), 409-463.
doi: 10.1137/090746677. |
[9] |
N. Bellomo and L. Gibelli,
Toward a mathematical theory of behavioral-social dynamics for pedestrian crowds, Mathematical Models and Methods in Applied Sciences, 25 (2015), 2417-2437.
doi: 10.1142/S0218202515400138. |
[10] |
N. Bellomo and L. Gibelli,
Behavioral crowds: Modeling and Monte Carlo simulations toward validation, Computers and Fluids, 141 (2016), 13-21.
doi: 10.1016/j.compfluid.2016.04.022. |
[11] |
N. Bellomo, L. Gibelli and N. Outada,
On the interplay between behavioral dynamics and social interactions in human crowds, Kinetic and Related Models, 12 (2019), 397-409.
doi: 10.3934/krm.2019017. |
[12] |
N. Bellomo, D. Knopoff and J. Soler,
On the difficult interplay between life, "Complexity", and mathematical sciences, Mathematical Models and Methods in Applied Sciences, 23 (2013), 1861-1913.
doi: 10.1142/S021820251350053X. |
[13] |
N. Bellomo, B. Piccoli and A. Tosin, Modeling crowd dynamics from a complex system viewpoint, Mathematical Models and Methods in Applied Sciences, 22 (2012), 1230004, 29 pp.
doi: 10.1142/S0218202512300049. |
[14] |
V. J. Blue and J. L. Adler,
Cellular automata microsimulation of bidirectional pedestrian flows, Journal of Transportation Research Record, 1678 (1999), 135-141.
doi: 10.3141/1678-17. |
[15] |
V. J. Blue and J. L. Adler,
Cellular automata microsimulation for modeling bi-directional pedestrian walkways, Transportation Research Part B: Methodological, 35 (2000), 293-312.
doi: 10.1016/S0191-2615(99)00052-1. |
[16] |
C. Burstedde, K. Klauck, A. 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. |
[17] |
N. Chooramun, P. J. Lawrence and E. R. Galea,
An agent based evacuation model utilising hybrid space discretisation, Safety Science, 50 (2012), 1685-1694.
doi: 10.1016/j.ssci.2011.12.022. |
[18] |
M. Chraibi, U. Kemloh, A. Schadschneider and A. Seyfried,
Force-based models of pedestrian dynamics, Networks and Heterogeneous Media, 6 (2011), 425-442.
doi: 10.3934/nhm.2011.6.425. |
[19] |
M. Chraibi, A. Tordeux, A. Schadschneider and A. Seyfried, Modelling of pedestrian and evacuation dynamics, Encyclopedia of Complexity and Systems Science Series, (2019), 649–669. |
[20] |
J. C. Dai, X. Li and L. Liu,
Simulation of pedestrian counter flow through bottlenecks by using an agent-based model, Physica A: Statistical Mechanics and its Applications, 392 (2013), 2202-2211.
doi: 10.1016/j.physa.2013.01.012. |
[21] |
J. Dijkstra, J. Jessurun and H. Timmermans, A multi-agent cellular automata model of pedestrian movement, Pedestrian and Evacuation Dynamics, (2001), 173–181. |
[22] |
B. Einarsson, Accuracy and Reliability in Scientific Computing, Environments, and Tools, 18. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2005.
doi: 10.1137/1.9780898718157. |
[23] |
R. Glowinski,
Finite element Methods for incompressible viscous flow, Handbook of Numerical Analysis, Handb. Numer. Anal., IX, North-Holland, Amsterdam, 9 (2013), 3-1176.
|
[24] |
D. Helbing,
A mathematical model for the behavior of pedestrians, Behavioral Science, 36 (1991), 298-310.
doi: 10.1002/bs.3830360405. |
[25] |
D. Helbing, I. J. Farkas, P. Molnar and T. Vicsek,
Simulation of pedestrian crowds in normal and evacuation situations, Pedestrian and Evacuation Dynamics, 21 (2002), 21-58.
|
[26] |
D. Helbing and P. Molnar,
Social force model for pedestrian dynamics, Physical Review E, 51 (1998), 4282-4286.
doi: 10.1103/PhysRevE.51.4282. |
[27] |
D. Helbing and T. Vicsek, Optimal self-organization, New Journal of Physics, 1 (1999).
doi: 10.1088/1367-2630/1/1/313. |
[28] |
R. L. 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. |
[29] |
A. Johansson, D. Helbing and P. K. Shukla,
Specification of the social force pedestrian model by evolutionary adjustment to video tracking data, Advances in Complex Systems, 10 (2007), 271-288.
doi: 10.1142/S0219525907001355. |
[30] |
R. J. LeVeque, Numerical Methods for Conservation Laws, Second edition, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1992.
doi: 10.1007/978-3-0348-8629-1. |
[31] |
X. M. Li, X. D. Yan, X. G. Li and J. F. Wang, Using cellular automata to investigate pedestrian conflicts with vehicles in crosswalk at signalized intersection, Discrete Dynamics in Nature and Society, 2012 (2012), 287502, 16 pp.
doi: 10.1155/2012/287502. |
[32] |
S. B. Liu, S. M. Lo, J. Ma and W. L. Wang,
An agent-based microscopic pedestrian flow simulation model for pedestrian traffic problems, IEEE Transactions on Intelligent Transportation Systems, 15 (2014), 992-1001.
doi: 10.1109/TITS.2013.2292526. |
[33] |
J. Moussaïd, D. Helbing, S. Garnier, A. Johansson, M. 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.
|
[34] |
A. Schadschneider, W. Klingsch, H. Kluepfel, T. Kretz, C. Rogsch, A. Seyfried, Evacuation dynamics: Empirical results, modeling and applications, Extreme Environmental Events: Complexity in Forecasting and Early Warning, (2011), 517–550. |
[35] |
A. Schadschneider and A. Seyfried,
Empirical results for pedestrian dynamics and their implications for modeling, Networks and Heterogeneous Media, 6 (2011), 545-560.
doi: 10.3934/nhm.2011.6.545. |
[36] |
A. Seyfried, O. Passon, B. Steffen, M. Boltes, T. Rupprecht and W. Klingsch,
New insights into pedestrian flow through bottlenecks, Transportation Science, 43 (2009), 267-406.
doi: 10.1287/trsc.1090.0263. |
[37] |
A. Shende, M. P. Singh and P. Kachroo,
Optimization-Based feedback control for pedestrian evacuation from an exit corridor, IEEE Transactions on Intelligent Transportation Systems, 12 (2011), 1167-1176.
doi: 10.1109/TITS.2011.2146251. |
[38] |
B. Steffen and A. Seyfried,
Methods for measuring pedestrian density, flow, speed and direction with minimal scatter, Physica A: Statistical Mechanics and its Applications, 389 (2009), 1902-1910.
doi: 10.1016/j.physa.2009.12.015. |
[39] |
A. Turner and A. Penn,
Encoding natural movement as an agent-based system: An investigation into human pedestrian behaviour in the built environment, Environment and Planning B: Urban Analytics and City Science, 29 (2002), 473-490.
doi: 10.1068/b12850. |
[40] |
A. U. Kemloh Wagoum, A. Tordeux and W. Liao, Understanding human queuing behaviour at exits: An empirical study, Royal Society Open Science, 4 (2017), 160896, 13 pp.
doi: 10.1098/rsos.160896. |
[41] |
J. A. Ward, A. J. Evans and N. S. Malleson, Dynamic calibration of agent-based models using data assimilation, Royal Society Open Science, 3 (2016), 150703, 17 pp.
doi: 10.1098/rsos.150703. |
[42] |
J. Zhang, W. Klingsch, A. Schadschneider and A. Seyfried, Transitions in pedestrian fundamental diagrams of straight corridors and T-junctions, Journal of Statistical Mechanics: Theory and Experiment, 2011 (2011).
doi: 10.1088/1742-5468/2011/06/P06004. |
[43] |
B. Zhou, X. Wang and X. Tang, Understanding collective crowd behaviors: Learning a mixture model of dynamic pedestrian-Agents, 2012 IEEE Conference on Computer Vision and Pattern Recognition, (2012), 2871–2878. |
show all references
References:
[1] |
J. P. Agnelli, F. Colasuonno and D. Knopoff,
A kinetic theory approach to the dynamics of crowd evacuation from bounded domains, Mathematical Models and Methods in Applied Sciences, 25 (2015), 109-129.
doi: 10.1142/S0218202515500049. |
[2] |
G. Antonini, M. Bierlaire and M. Weber,
Discrete choice models of pedestrian walking behavior, Transportation Research Part B: Methodological, 40 (2006), 667-687.
doi: 10.1016/j.trb.2005.09.006. |
[3] |
M. Asano, T. Iryo and M. Kuwahara,
Microscopic pedestrian simulation model combined with a tactical model for route choice behaviour, Transportation Research Part C: Emerging Technologies, 18 (2010), 842-855.
doi: 10.1016/j.trc.2010.01.005. |
[4] |
S. Bandini, S. Manzoni and G. Vizzari, Agent based modeling and simulation: An informatics perspective, Journal of Artificial Societies and Social Simulation, 12 (2009). |
[5] |
N. Bellomo and A. Bellouquid,
On the modeling of crowd dynamics: Looking at the beautiful shapes of swarms, Networks and Heterogeneous Media, 6 (2011), 383-399.
doi: 10.3934/nhm.2011.6.383. |
[6] |
N. Bellomo, A. Bellouquid and D. Knopoff,
From the microscale to collective crowd dynamics, Society for Industrial and Applied Mathematics Multiscale Modeling and Simulation, 11 (2013), 943-963.
doi: 10.1137/130904569. |
[7] |
N. Bellomo, A. Bellouquid, L. Gibelli and N. Outada, A Quest Towards a Mathematical Theory of Living Systems, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser/Springer, Cham, 2017.
doi: 10.1007/978-3-319-57436-3. |
[8] |
N. Bellomo and C. Dogbe,
On the modeling of traffic and crowds: A survey of models, speculations, and perspectives, Society for Industrial and Applied Mathematics Review, 53 (2011), 409-463.
doi: 10.1137/090746677. |
[9] |
N. Bellomo and L. Gibelli,
Toward a mathematical theory of behavioral-social dynamics for pedestrian crowds, Mathematical Models and Methods in Applied Sciences, 25 (2015), 2417-2437.
doi: 10.1142/S0218202515400138. |
[10] |
N. Bellomo and L. Gibelli,
Behavioral crowds: Modeling and Monte Carlo simulations toward validation, Computers and Fluids, 141 (2016), 13-21.
doi: 10.1016/j.compfluid.2016.04.022. |
[11] |
N. Bellomo, L. Gibelli and N. Outada,
On the interplay between behavioral dynamics and social interactions in human crowds, Kinetic and Related Models, 12 (2019), 397-409.
doi: 10.3934/krm.2019017. |
[12] |
N. Bellomo, D. Knopoff and J. Soler,
On the difficult interplay between life, "Complexity", and mathematical sciences, Mathematical Models and Methods in Applied Sciences, 23 (2013), 1861-1913.
doi: 10.1142/S021820251350053X. |
[13] |
N. Bellomo, B. Piccoli and A. Tosin, Modeling crowd dynamics from a complex system viewpoint, Mathematical Models and Methods in Applied Sciences, 22 (2012), 1230004, 29 pp.
doi: 10.1142/S0218202512300049. |
[14] |
V. J. Blue and J. L. Adler,
Cellular automata microsimulation of bidirectional pedestrian flows, Journal of Transportation Research Record, 1678 (1999), 135-141.
doi: 10.3141/1678-17. |
[15] |
V. J. Blue and J. L. Adler,
Cellular automata microsimulation for modeling bi-directional pedestrian walkways, Transportation Research Part B: Methodological, 35 (2000), 293-312.
doi: 10.1016/S0191-2615(99)00052-1. |
[16] |
C. Burstedde, K. Klauck, A. 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. |
[17] |
N. Chooramun, P. J. Lawrence and E. R. Galea,
An agent based evacuation model utilising hybrid space discretisation, Safety Science, 50 (2012), 1685-1694.
doi: 10.1016/j.ssci.2011.12.022. |
[18] |
M. Chraibi, U. Kemloh, A. Schadschneider and A. Seyfried,
Force-based models of pedestrian dynamics, Networks and Heterogeneous Media, 6 (2011), 425-442.
doi: 10.3934/nhm.2011.6.425. |
[19] |
M. Chraibi, A. Tordeux, A. Schadschneider and A. Seyfried, Modelling of pedestrian and evacuation dynamics, Encyclopedia of Complexity and Systems Science Series, (2019), 649–669. |
[20] |
J. C. Dai, X. Li and L. Liu,
Simulation of pedestrian counter flow through bottlenecks by using an agent-based model, Physica A: Statistical Mechanics and its Applications, 392 (2013), 2202-2211.
doi: 10.1016/j.physa.2013.01.012. |
[21] |
J. Dijkstra, J. Jessurun and H. Timmermans, A multi-agent cellular automata model of pedestrian movement, Pedestrian and Evacuation Dynamics, (2001), 173–181. |
[22] |
B. Einarsson, Accuracy and Reliability in Scientific Computing, Environments, and Tools, 18. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2005.
doi: 10.1137/1.9780898718157. |
[23] |
R. Glowinski,
Finite element Methods for incompressible viscous flow, Handbook of Numerical Analysis, Handb. Numer. Anal., IX, North-Holland, Amsterdam, 9 (2013), 3-1176.
|
[24] |
D. Helbing,
A mathematical model for the behavior of pedestrians, Behavioral Science, 36 (1991), 298-310.
doi: 10.1002/bs.3830360405. |
[25] |
D. Helbing, I. J. Farkas, P. Molnar and T. Vicsek,
Simulation of pedestrian crowds in normal and evacuation situations, Pedestrian and Evacuation Dynamics, 21 (2002), 21-58.
|
[26] |
D. Helbing and P. Molnar,
Social force model for pedestrian dynamics, Physical Review E, 51 (1998), 4282-4286.
doi: 10.1103/PhysRevE.51.4282. |
[27] |
D. Helbing and T. Vicsek, Optimal self-organization, New Journal of Physics, 1 (1999).
doi: 10.1088/1367-2630/1/1/313. |
[28] |
R. L. 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. |
[29] |
A. Johansson, D. Helbing and P. K. Shukla,
Specification of the social force pedestrian model by evolutionary adjustment to video tracking data, Advances in Complex Systems, 10 (2007), 271-288.
doi: 10.1142/S0219525907001355. |
[30] |
R. J. LeVeque, Numerical Methods for Conservation Laws, Second edition, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1992.
doi: 10.1007/978-3-0348-8629-1. |
[31] |
X. M. Li, X. D. Yan, X. G. Li and J. F. Wang, Using cellular automata to investigate pedestrian conflicts with vehicles in crosswalk at signalized intersection, Discrete Dynamics in Nature and Society, 2012 (2012), 287502, 16 pp.
doi: 10.1155/2012/287502. |
[32] |
S. B. Liu, S. M. Lo, J. Ma and W. L. Wang,
An agent-based microscopic pedestrian flow simulation model for pedestrian traffic problems, IEEE Transactions on Intelligent Transportation Systems, 15 (2014), 992-1001.
doi: 10.1109/TITS.2013.2292526. |
[33] |
J. Moussaïd, D. Helbing, S. Garnier, A. Johansson, M. 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.
|
[34] |
A. Schadschneider, W. Klingsch, H. Kluepfel, T. Kretz, C. Rogsch, A. Seyfried, Evacuation dynamics: Empirical results, modeling and applications, Extreme Environmental Events: Complexity in Forecasting and Early Warning, (2011), 517–550. |
[35] |
A. Schadschneider and A. Seyfried,
Empirical results for pedestrian dynamics and their implications for modeling, Networks and Heterogeneous Media, 6 (2011), 545-560.
doi: 10.3934/nhm.2011.6.545. |
[36] |
A. Seyfried, O. Passon, B. Steffen, M. Boltes, T. Rupprecht and W. Klingsch,
New insights into pedestrian flow through bottlenecks, Transportation Science, 43 (2009), 267-406.
doi: 10.1287/trsc.1090.0263. |
[37] |
A. Shende, M. P. Singh and P. Kachroo,
Optimization-Based feedback control for pedestrian evacuation from an exit corridor, IEEE Transactions on Intelligent Transportation Systems, 12 (2011), 1167-1176.
doi: 10.1109/TITS.2011.2146251. |
[38] |
B. Steffen and A. Seyfried,
Methods for measuring pedestrian density, flow, speed and direction with minimal scatter, Physica A: Statistical Mechanics and its Applications, 389 (2009), 1902-1910.
doi: 10.1016/j.physa.2009.12.015. |
[39] |
A. Turner and A. Penn,
Encoding natural movement as an agent-based system: An investigation into human pedestrian behaviour in the built environment, Environment and Planning B: Urban Analytics and City Science, 29 (2002), 473-490.
doi: 10.1068/b12850. |
[40] |
A. U. Kemloh Wagoum, A. Tordeux and W. Liao, Understanding human queuing behaviour at exits: An empirical study, Royal Society Open Science, 4 (2017), 160896, 13 pp.
doi: 10.1098/rsos.160896. |
[41] |
J. A. Ward, A. J. Evans and N. S. Malleson, Dynamic calibration of agent-based models using data assimilation, Royal Society Open Science, 3 (2016), 150703, 17 pp.
doi: 10.1098/rsos.150703. |
[42] |
J. Zhang, W. Klingsch, A. Schadschneider and A. Seyfried, Transitions in pedestrian fundamental diagrams of straight corridors and T-junctions, Journal of Statistical Mechanics: Theory and Experiment, 2011 (2011).
doi: 10.1088/1742-5468/2011/06/P06004. |
[43] |
B. Zhou, X. Wang and X. Tang, Understanding collective crowd behaviors: Learning a mixture model of dynamic pedestrian-Agents, 2012 IEEE Conference on Computer Vision and Pattern Recognition, (2012), 2871–2878. |















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