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An adaptive finite-volume method for a model of two-phase pedestrian flow
Force-based models of pedestrian dynamics
1. | Jülich Supercomputing Centre, Forschungszentrum Jülich, 52425 Jülich, Germany, Germany, Germany |
2. | Institute for Theoretical Physics, University of Cologne, 50937 Köln, Germany |
References:
[1] |
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.
doi: 10.1016/j.trc.2010.01.005. |
[2] |
J. V. Berg, M. Lin and D. Manocha, Reciprocal velocity obstacles for real-time multi-agent navigation,, in, (2008). Google Scholar |
[3] |
P. Bourke, Minimum distance between a point and a line, accessed 24, December 2010., Available from: , (). Google Scholar |
[4] |
M. Chraibi and A. Seyfried, Pedestrian dynamics with event-driven simulation,, in [27], (): 713. Google Scholar |
[5] |
M. Chraibi, A. Seyfried and A. Schadschneider, Generalized centrifugal force model for pedestrian dynamics,, Phys. Rev. E, 82 (2010).
doi: 10.1103/PhysRevE.82.046111. |
[6] |
M. Chraibi, A. Seyfried, A. Schadschneider and W. Mackens, "Quantitative Description of Pedestrian Dynamics with a Force-Based Model,", IEEE/WIC/ACM International Joint Conference on Web Intelligence and Intelligent Agent Technology, 3 (2009), 583. Google Scholar |
[7] |
M. Chraibi, A. Seyfried, A. Schadschneider and W. Mackens, "Quantitative Verification of a Force-Based Model for Pedestrian Dynamics,", Traffic and Granular Flow '09, (2009). Google Scholar |
[8] |
Z. Fang, J. P. Yuan, Y. C. Wang and S. M. Lo, Survey of pedestrian movement and development of a crowd dynamics model,, Fire Safety Journal, 43 (2008), 459.
doi: 10.1016/j.firesaf.2007.12.005. |
[9] |
E. R. Galea, ed., "Pedestrian and Evacuation Dynamics 2003,", CMS Press, (2003). Google Scholar |
[10] |
C. Gloor, L. Mauron and K. A. Nagel, "Pedestrian Simulation for Hiking in the Alps,", Proceedings of Swiss Transport Research Conference (STRC), (2003). Google Scholar |
[11] |
, GNU General public license,, , (). Google Scholar |
[12] |
D. Helbing, Collective phenomena and states in traffic and self-driven many-particle systems,, Computational Materials Science, 30 (2004), 180.
doi: 10.1016/j.commatsci.2004.01.026. |
[13] |
D. Helbing and P. Molnár, Social force model for pedestrian dynamics,, Phys. Rev. E, 51 (1995), 4282.
doi: 10.1103/PhysRevE.51.4282. |
[14] |
D. Helbing, I. J. Farkas, P. Molnár and T. Viscek, Simulation of pedestrian crowds in normal and evacuation situations,, in [44], (): 21. Google Scholar |
[15] |
M. Höcker, V. Berkhahn, A. Kneidl, A. Borrmann and W. Klein, "Graph-based Approaches for Simulating Pedestrian Dynamics in Building Models,", 8th European Conference on Product & Process Modelling (ECPPM), (2010). Google Scholar |
[16] |
S. Holl and A. Seyfried, Hermes - an evacuation assistant for mass events,, inSiDe, 7 (2009), 60. Google Scholar |
[17] |
S. P. Hoogendoorn, "Walking Behavior in Bottlenecks and its Implications for Capacity,", TRB 2004 Annual Meeting, (2004). Google Scholar |
[18] |
S. P. Hoogendoorn and W. Daamen, A novel calibration approach of microscopic pedestrian models,, in, (2009). Google Scholar |
[19] |
S. P. Hoogendoorn and W. Daamen, Pedestrian behavior at bottlenecks,, Transportation Science, 39 (2005), 147.
doi: 10.1287/trsc.1040.0102. |
[20] |
S. P. Hoogendoorn, W. Daamen and P. H. L Bovy, "Extracting Microscopic Pedestrian Characteristics from Video Data,", TRB 2004 Annual Meeting Washington DC: National Academy Press, (2004). Google Scholar |
[21] |
S. P. Hoogendoorn, W. Daamen and P. H. L. Bovy, Microscopic pedestrian traffic data collection and analysis by walking experiments: Behaviour at bottlenecks,, in [9], (): 89. Google Scholar |
[22] |
S. P. Hoogendoorn, W. Daamen and R. Landman, Microscopic calibration and validation of pedestrian models - Cross-comparison of models using experimental data,, in [58], (). Google Scholar |
[23] |
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.
doi: 10.1142/S0219525907001355. |
[24] |
A. Kirchner, "Modellierung und Statistische Physik biologischer und sozialer Systeme,", Ph.D thesis, (2003). Google Scholar |
[25] |
A. Kirchner and A. Schadschneider, Simulation of evacuation processes using a bionics-inspired cellular automaton model for pedestrian dynamics,, Physica A, 312 (2002), 260.
doi: 10.1016/S0378-4371(02)00857-9. |
[26] |
A. Kirchner, K. Nishinari and A. Schadschneider, Friction effects and clogging in a cellular automaton model for pedestrian dynamics,, Phys. Rev. E, 67 (2003).
doi: 10.1103/PhysRevE.67.056122. |
[27] |
W. Klingsch, C. Rogsch, A. Schadschneider and M. Schreckenberg, eds., "Pedestrian and Evacuation Dynamics 2008,", Springer-Verlag, (2010). Google Scholar |
[28] |
T. Kretz, A. Grünebohm and M. Schreckenberg, Experimental study of pedestrian flow through a bottleneck,, J. Stat. Mech., 10 (2006). Google Scholar |
[29] |
T. Kretz, S. Hengst and P. Vortisch, Pedestrian flow at bottlenecks - validation and calibration of Vissim's social force model of pedestrian traffic and its empirical foundations,, in, (2008). Google Scholar |
[30] |
T. I. Lakoba, D. J. Kaup and N. M. Finkelstein, Modifications of the Helbing-Molnár-Farkas-Vicsek social force model for pedestrian evolution,, Simulation, 81 (2005), 339.
doi: 10.1177/0037549705052772. |
[31] |
K. Lewin, "Field Theory in Social Science,", Greenwood Press Publishers, (1951). Google Scholar |
[32] |
J. Liddle, A. Seyfried, T. Rupprecht, W. Klingsch, A. Schadschneider and A. Winkens, "An Experimental Study of Pedestrian Congestions: Influence of Bottleneck Width and Length,", Traffic and Granular Flow '09, (2009). Google Scholar |
[33] |
R. Löhner, On the modelling of pedestrian motion,, Applied Mathematical Modelling, 34 (2010), 366.
doi: 10.1016/j.apm.2009.04.017. |
[34] |
K. Müller, "Zur Gestaltung und Bemessung von Fluchtwegen für die Evakuierung von Personen aus Bauwerken auf der Grundlage von Modellversuchen,", Ph.D thesis, (1981). Google Scholar |
[35] |
D. R. Parisi and C. O. Dorso, Morphological and dynamical aspects of the room evacuation process,, Physica A, 385 (2007), 343.
doi: 10.1016/j.physa.2007.06.033. |
[36] |
D. R. Parisi, M. Gilman and H. Moldovan, A modification of the social force model can reproduce experimental data of pedestrian flows in normal conditions,, Physica A, 388 (2009), 3600.
doi: 10.1016/j.physa.2009.05.027. |
[37] |
J. Ondřej, J. Pettré, A. Olivier and S. Donikian, "A Synthetic-Vision-Based Steering Approach for Crowd Simulation,", SIGGRAPH '10: ACM SIGGRAPH, (2010). Google Scholar |
[38] |
A. Schadschneider, I'm a football fan ... get me out of here,, Physics World, 21 (2010). Google Scholar |
[39] |
A. Schadschneider, W. Klingsch, H. Klüpfel, T. Kretz, C. Rogsch and A. Seyfried, Evacuation dynamics: Empirical results, modeling and applications,, Encyclopedia of Complexity and System Science, (2009). Google Scholar |
[40] |
A. Schadschneider, H. Klüpfel, T. Kretz, C. Rogsch and A. Seyfried, Fundamentals of pedestrian and evacuation dynamics,, in, (2009), 124. Google Scholar |
[41] |
A. Schadschneider and A. Seyfried, Empirical results for pedestrian dynamics and their implications for modelling,, Networks and Heterogeneous Media, 3 (2011), 545. Google Scholar |
[42] |
A. Schadschneider and A. Seyfried, "Modeling Pedestrian Dynamics - From Experiment to Theory and Back,", Traffic and Granular Flow '09, (2009). Google Scholar |
[43] |
V. Schneider and R. Könnecke., Simulating evacuation processes with ASERI,, in [44], (): 303. Google Scholar |
[44] |
M. Schreckenberg and S. D. Sharma, eds., "Pedestrian and Evacuation Dynamics 2001,", Springer, (2002). Google Scholar |
[45] |
W. Schroeder, K. Martin and B. Lorense, "Visualization Toolkit: An Object-Oriented Approach to 3D Graphics,", 4th edition, (2006). Google Scholar |
[46] |
A. Seyfried, M. Boltes, J. Kähler, W. Klingsch, A. Portz, T. Rupprecht, A. Schadschneider, B. Steffen and A. Winkens, Enhanced empirical data for the fundamental diagram and the flow through bottlenecks,, in [27], (). Google Scholar |
[47] |
A. Seyfried, O. Passon, B. Steffen, M. Boltes, T. Rupprecht and W. Klingsch, New insights into pedestrian flow through bottlenecks,, Transportation Science, 43 (2009), 395.
doi: 10.1287/trsc.1090.0263. |
[48] |
A. Seyfried, A. Portz and A. Schadschneider, Phase coexistence in congested states of pedestrian dynamics cellular automata,, in, (6350), 496. Google Scholar |
[49] |
A. Seyfried and A. Schadschneider, Validation of cellular automata models of pedestrian dynamics using controlled large-scale experiments,, Cybernetics and Systems, 40 (2009). Google Scholar |
[50] |
A. Seyfried, B. Steffen and T. Lippert, Basics of modelling the pedestrian flow,, Physica A, 368 (2006), 232.
doi: 10.1016/j.physa.2005.11.052. |
[51] |
W. Shao and D. Terzopoulos, Autonomous pedestrians,, in, (2005). Google Scholar |
[52] |
B. Steffen and A. Seyfried, Modelling of pedestrian movement around $90^\circ$ and $180^\circ$ bends,, in, (2009). Google Scholar |
[53] |
B. Steffen and A. Seyfried, The repulsive force in continous space models of pedestrian movement,, , (2008). Google Scholar |
[54] |
A. Steiner, M. Philipp and A. Schmid, "Parameter Estimation for a Pedestrian Simulation Model,", Swiss Transport Research Conference, (2007). Google Scholar |
[55] |
P. A. Thompson and E. W. Marchant, A computer model for the evacuation of large building populations,, Fire Safety Journal, 24 (1995), 131.
doi: 10.1016/0379-7112(95)00019-P. |
[56] |
P. A. Thompson and E. W. Marchant, Testing and application of the computer model 'SIMULEX',, Fire Safety Journal, 24 (1995), 149.
doi: 10.1016/0379-7112(95)00020-T. |
[57] |
H. Timmermans, ed., "Pedestrian Behavior,", Emerald, (2009). Google Scholar |
[58] |
N. Waldau, P. Gattermann, H. Knoflacher and M. Schreckenberg, eds., "Pedestrian and Evacuation Dynamics 2005,", Springer, (2007). Google Scholar |
[59] |
D. Yanagisawa, A. Kimura, A. Tomoeda, N. Ryosuke, Y. Suma, Kazumichi Ohtsuka and Katsuhiro Nishinari, Introduction of frictional and turning function for pedestrian outflow with an obstacle,, Phys. Rev. E, 80 (2009).
doi: 10.1103/PhysRevE.80.036110. |
[60] |
W. J. Yu, L. Y. Chen, R. Dong and S. Q. Dai, Centrifugal force model for pedestrian dynamics,, Phys. Rev. E, 72 (2005).
doi: 10.1103/PhysRevE.72.026112. |
[61] |
X. Liu, W. Song and J. Zhang, Extraction and quantitative analysis of microscopic evacuation characteristics based on digital image processing,, Physica A, 388 (2009), 2717.
doi: 10.1016/j.physa.2009.03.017. |
[62] |
X. Zheng and P. Palffy-Muhoray, Distance of closest approach of two arbitrary hard ellipses in two dimensions,, Phys. Rev. E, 75 (2007).
doi: 10.1103/PhysRevE.75.061709. |
show all references
References:
[1] |
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.
doi: 10.1016/j.trc.2010.01.005. |
[2] |
J. V. Berg, M. Lin and D. Manocha, Reciprocal velocity obstacles for real-time multi-agent navigation,, in, (2008). Google Scholar |
[3] |
P. Bourke, Minimum distance between a point and a line, accessed 24, December 2010., Available from: , (). Google Scholar |
[4] |
M. Chraibi and A. Seyfried, Pedestrian dynamics with event-driven simulation,, in [27], (): 713. Google Scholar |
[5] |
M. Chraibi, A. Seyfried and A. Schadschneider, Generalized centrifugal force model for pedestrian dynamics,, Phys. Rev. E, 82 (2010).
doi: 10.1103/PhysRevE.82.046111. |
[6] |
M. Chraibi, A. Seyfried, A. Schadschneider and W. Mackens, "Quantitative Description of Pedestrian Dynamics with a Force-Based Model,", IEEE/WIC/ACM International Joint Conference on Web Intelligence and Intelligent Agent Technology, 3 (2009), 583. Google Scholar |
[7] |
M. Chraibi, A. Seyfried, A. Schadschneider and W. Mackens, "Quantitative Verification of a Force-Based Model for Pedestrian Dynamics,", Traffic and Granular Flow '09, (2009). Google Scholar |
[8] |
Z. Fang, J. P. Yuan, Y. C. Wang and S. M. Lo, Survey of pedestrian movement and development of a crowd dynamics model,, Fire Safety Journal, 43 (2008), 459.
doi: 10.1016/j.firesaf.2007.12.005. |
[9] |
E. R. Galea, ed., "Pedestrian and Evacuation Dynamics 2003,", CMS Press, (2003). Google Scholar |
[10] |
C. Gloor, L. Mauron and K. A. Nagel, "Pedestrian Simulation for Hiking in the Alps,", Proceedings of Swiss Transport Research Conference (STRC), (2003). Google Scholar |
[11] |
, GNU General public license,, , (). Google Scholar |
[12] |
D. Helbing, Collective phenomena and states in traffic and self-driven many-particle systems,, Computational Materials Science, 30 (2004), 180.
doi: 10.1016/j.commatsci.2004.01.026. |
[13] |
D. Helbing and P. Molnár, Social force model for pedestrian dynamics,, Phys. Rev. E, 51 (1995), 4282.
doi: 10.1103/PhysRevE.51.4282. |
[14] |
D. Helbing, I. J. Farkas, P. Molnár and T. Viscek, Simulation of pedestrian crowds in normal and evacuation situations,, in [44], (): 21. Google Scholar |
[15] |
M. Höcker, V. Berkhahn, A. Kneidl, A. Borrmann and W. Klein, "Graph-based Approaches for Simulating Pedestrian Dynamics in Building Models,", 8th European Conference on Product & Process Modelling (ECPPM), (2010). Google Scholar |
[16] |
S. Holl and A. Seyfried, Hermes - an evacuation assistant for mass events,, inSiDe, 7 (2009), 60. Google Scholar |
[17] |
S. P. Hoogendoorn, "Walking Behavior in Bottlenecks and its Implications for Capacity,", TRB 2004 Annual Meeting, (2004). Google Scholar |
[18] |
S. P. Hoogendoorn and W. Daamen, A novel calibration approach of microscopic pedestrian models,, in, (2009). Google Scholar |
[19] |
S. P. Hoogendoorn and W. Daamen, Pedestrian behavior at bottlenecks,, Transportation Science, 39 (2005), 147.
doi: 10.1287/trsc.1040.0102. |
[20] |
S. P. Hoogendoorn, W. Daamen and P. H. L Bovy, "Extracting Microscopic Pedestrian Characteristics from Video Data,", TRB 2004 Annual Meeting Washington DC: National Academy Press, (2004). Google Scholar |
[21] |
S. P. Hoogendoorn, W. Daamen and P. H. L. Bovy, Microscopic pedestrian traffic data collection and analysis by walking experiments: Behaviour at bottlenecks,, in [9], (): 89. Google Scholar |
[22] |
S. P. Hoogendoorn, W. Daamen and R. Landman, Microscopic calibration and validation of pedestrian models - Cross-comparison of models using experimental data,, in [58], (). Google Scholar |
[23] |
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.
doi: 10.1142/S0219525907001355. |
[24] |
A. Kirchner, "Modellierung und Statistische Physik biologischer und sozialer Systeme,", Ph.D thesis, (2003). Google Scholar |
[25] |
A. Kirchner and A. Schadschneider, Simulation of evacuation processes using a bionics-inspired cellular automaton model for pedestrian dynamics,, Physica A, 312 (2002), 260.
doi: 10.1016/S0378-4371(02)00857-9. |
[26] |
A. Kirchner, K. Nishinari and A. Schadschneider, Friction effects and clogging in a cellular automaton model for pedestrian dynamics,, Phys. Rev. E, 67 (2003).
doi: 10.1103/PhysRevE.67.056122. |
[27] |
W. Klingsch, C. Rogsch, A. Schadschneider and M. Schreckenberg, eds., "Pedestrian and Evacuation Dynamics 2008,", Springer-Verlag, (2010). Google Scholar |
[28] |
T. Kretz, A. Grünebohm and M. Schreckenberg, Experimental study of pedestrian flow through a bottleneck,, J. Stat. Mech., 10 (2006). Google Scholar |
[29] |
T. Kretz, S. Hengst and P. Vortisch, Pedestrian flow at bottlenecks - validation and calibration of Vissim's social force model of pedestrian traffic and its empirical foundations,, in, (2008). Google Scholar |
[30] |
T. I. Lakoba, D. J. Kaup and N. M. Finkelstein, Modifications of the Helbing-Molnár-Farkas-Vicsek social force model for pedestrian evolution,, Simulation, 81 (2005), 339.
doi: 10.1177/0037549705052772. |
[31] |
K. Lewin, "Field Theory in Social Science,", Greenwood Press Publishers, (1951). Google Scholar |
[32] |
J. Liddle, A. Seyfried, T. Rupprecht, W. Klingsch, A. Schadschneider and A. Winkens, "An Experimental Study of Pedestrian Congestions: Influence of Bottleneck Width and Length,", Traffic and Granular Flow '09, (2009). Google Scholar |
[33] |
R. Löhner, On the modelling of pedestrian motion,, Applied Mathematical Modelling, 34 (2010), 366.
doi: 10.1016/j.apm.2009.04.017. |
[34] |
K. Müller, "Zur Gestaltung und Bemessung von Fluchtwegen für die Evakuierung von Personen aus Bauwerken auf der Grundlage von Modellversuchen,", Ph.D thesis, (1981). Google Scholar |
[35] |
D. R. Parisi and C. O. Dorso, Morphological and dynamical aspects of the room evacuation process,, Physica A, 385 (2007), 343.
doi: 10.1016/j.physa.2007.06.033. |
[36] |
D. R. Parisi, M. Gilman and H. Moldovan, A modification of the social force model can reproduce experimental data of pedestrian flows in normal conditions,, Physica A, 388 (2009), 3600.
doi: 10.1016/j.physa.2009.05.027. |
[37] |
J. Ondřej, J. Pettré, A. Olivier and S. Donikian, "A Synthetic-Vision-Based Steering Approach for Crowd Simulation,", SIGGRAPH '10: ACM SIGGRAPH, (2010). Google Scholar |
[38] |
A. Schadschneider, I'm a football fan ... get me out of here,, Physics World, 21 (2010). Google Scholar |
[39] |
A. Schadschneider, W. Klingsch, H. Klüpfel, T. Kretz, C. Rogsch and A. Seyfried, Evacuation dynamics: Empirical results, modeling and applications,, Encyclopedia of Complexity and System Science, (2009). Google Scholar |
[40] |
A. Schadschneider, H. Klüpfel, T. Kretz, C. Rogsch and A. Seyfried, Fundamentals of pedestrian and evacuation dynamics,, in, (2009), 124. Google Scholar |
[41] |
A. Schadschneider and A. Seyfried, Empirical results for pedestrian dynamics and their implications for modelling,, Networks and Heterogeneous Media, 3 (2011), 545. Google Scholar |
[42] |
A. Schadschneider and A. Seyfried, "Modeling Pedestrian Dynamics - From Experiment to Theory and Back,", Traffic and Granular Flow '09, (2009). Google Scholar |
[43] |
V. Schneider and R. Könnecke., Simulating evacuation processes with ASERI,, in [44], (): 303. Google Scholar |
[44] |
M. Schreckenberg and S. D. Sharma, eds., "Pedestrian and Evacuation Dynamics 2001,", Springer, (2002). Google Scholar |
[45] |
W. Schroeder, K. Martin and B. Lorense, "Visualization Toolkit: An Object-Oriented Approach to 3D Graphics,", 4th edition, (2006). Google Scholar |
[46] |
A. Seyfried, M. Boltes, J. Kähler, W. Klingsch, A. Portz, T. Rupprecht, A. Schadschneider, B. Steffen and A. Winkens, Enhanced empirical data for the fundamental diagram and the flow through bottlenecks,, in [27], (). Google Scholar |
[47] |
A. Seyfried, O. Passon, B. Steffen, M. Boltes, T. Rupprecht and W. Klingsch, New insights into pedestrian flow through bottlenecks,, Transportation Science, 43 (2009), 395.
doi: 10.1287/trsc.1090.0263. |
[48] |
A. Seyfried, A. Portz and A. Schadschneider, Phase coexistence in congested states of pedestrian dynamics cellular automata,, in, (6350), 496. Google Scholar |
[49] |
A. Seyfried and A. Schadschneider, Validation of cellular automata models of pedestrian dynamics using controlled large-scale experiments,, Cybernetics and Systems, 40 (2009). Google Scholar |
[50] |
A. Seyfried, B. Steffen and T. Lippert, Basics of modelling the pedestrian flow,, Physica A, 368 (2006), 232.
doi: 10.1016/j.physa.2005.11.052. |
[51] |
W. Shao and D. Terzopoulos, Autonomous pedestrians,, in, (2005). Google Scholar |
[52] |
B. Steffen and A. Seyfried, Modelling of pedestrian movement around $90^\circ$ and $180^\circ$ bends,, in, (2009). Google Scholar |
[53] |
B. Steffen and A. Seyfried, The repulsive force in continous space models of pedestrian movement,, , (2008). Google Scholar |
[54] |
A. Steiner, M. Philipp and A. Schmid, "Parameter Estimation for a Pedestrian Simulation Model,", Swiss Transport Research Conference, (2007). Google Scholar |
[55] |
P. A. Thompson and E. W. Marchant, A computer model for the evacuation of large building populations,, Fire Safety Journal, 24 (1995), 131.
doi: 10.1016/0379-7112(95)00019-P. |
[56] |
P. A. Thompson and E. W. Marchant, Testing and application of the computer model 'SIMULEX',, Fire Safety Journal, 24 (1995), 149.
doi: 10.1016/0379-7112(95)00020-T. |
[57] |
H. Timmermans, ed., "Pedestrian Behavior,", Emerald, (2009). Google Scholar |
[58] |
N. Waldau, P. Gattermann, H. Knoflacher and M. Schreckenberg, eds., "Pedestrian and Evacuation Dynamics 2005,", Springer, (2007). Google Scholar |
[59] |
D. Yanagisawa, A. Kimura, A. Tomoeda, N. Ryosuke, Y. Suma, Kazumichi Ohtsuka and Katsuhiro Nishinari, Introduction of frictional and turning function for pedestrian outflow with an obstacle,, Phys. Rev. E, 80 (2009).
doi: 10.1103/PhysRevE.80.036110. |
[60] |
W. J. Yu, L. Y. Chen, R. Dong and S. Q. Dai, Centrifugal force model for pedestrian dynamics,, Phys. Rev. E, 72 (2005).
doi: 10.1103/PhysRevE.72.026112. |
[61] |
X. Liu, W. Song and J. Zhang, Extraction and quantitative analysis of microscopic evacuation characteristics based on digital image processing,, Physica A, 388 (2009), 2717.
doi: 10.1016/j.physa.2009.03.017. |
[62] |
X. Zheng and P. Palffy-Muhoray, Distance of closest approach of two arbitrary hard ellipses in two dimensions,, Phys. Rev. E, 75 (2007).
doi: 10.1103/PhysRevE.75.061709. |
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