American Institute of Mathematical Sciences

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December  2019, 12(6): 1273-1296. doi: 10.3934/krm.2019049

A kinetic theory approach to model pedestrian dynamics in bounded domains with obstacles

Daewa Kim and  Annalisa Quaini

Department of Mathematics, University of Houston, 3551 Cullen Blvd, Houston, TX 77204, USA

Received  January 2019 Revised  May 2019 Published  September 2019

Fund Project: This work was partially supported by NSF grant DMS-1620384.

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.

Keywords: Crowd dynamics, Boltzmann-type kinetic model, nonlinear interactions, stochastic games, operator-splitting scheme.
Mathematics Subject Classification: Primary: 35Q91, 91C99; Secondary: 65C20, 65M06.
Citation: Daewa Kim, Annalisa Quaini. A kinetic theory approach to model pedestrian dynamics in bounded domains with obstacles. Kinetic & Related Models, 2019, 12 (6) : 1273-1296. doi: 10.3934/krm.2019049
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[21]

J. Dijkstra, J. Jessurun and H. Timmermans, A multi-agent cellular automata model of pedestrian movement, Pedestrian and Evacuation Dynamics, (2001), 173–181. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[24]

D. Helbing, A mathematical model for the behavior of pedestrians, Behavioral Science, 36 (1991), 298-310.  doi: 10.1002/bs.3830360405.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[27]

D. Helbing and T. Vicsek, Optimal self-organization, New Journal of Physics, 1 (1999). doi: 10.1088/1367-2630/1/1/313.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[21]

J. Dijkstra, J. Jessurun and H. Timmermans, A multi-agent cellular automata model of pedestrian movement, Pedestrian and Evacuation Dynamics, (2001), 173–181. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[24]

D. Helbing, A mathematical model for the behavior of pedestrians, Behavioral Science, 36 (1991), 298-310.  doi: 10.1002/bs.3830360405.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[27]

D. Helbing and T. Vicsek, Optimal self-organization, New Journal of Physics, 1 (1999). doi: 10.1088/1367-2630/1/1/313.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

Figure 1.  (A) Dependence of the dimensionless velocity modulus $ v $ on the dimensionless density $ \rho $ for different values of the parameter $ \alpha $, which represents the quality of the environment. (B) Sketch of computational domain $ \Omega $ with exit $ E $ and a pedestrian located at $ {\boldsymbol{x}} $, moving with direction $ \theta_h $. The pedestrian should choose direction $ {\boldsymbol u}_E $ to reach the exit, while direction $ {\boldsymbol u}_W $ is to avoid collision with the wall. The distances form the exit and from the wall are $ d_E $ and $ d_W $, respectively
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Figure 2.  Definition of $ {\boldsymbol u}_W $ and $ {\boldsymbol u}_E $ with respect to the effective area
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Figure 3.  Evacuation process of 46 pedestrians grouped into two clusters with opposite initial directions $ \theta_3 $ and $ \theta_7 $ using the medium mesh and $ \Delta t_{medium} $ for time $ t = 0, 1.5, 3, 6, 10.5, 13.5 $ s. The color refers to density
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Figure 4.  (A) Number of pedestrians inside the room over time computed with six different combinations of mesh and time step. For ease of comparison, (B) shows only the curves in (A) obtained with simultaneous refinements of mesh and time step
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1]">Figure 5.  Computed evacuation time from the room with one exit versus the exit size: (A) our results and (B) results from [1]
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40] and initial density and direction (i.e., $ \theta_1 $) for the experiment with 138 pedestrians">Figure 6.  Computational domain corresponding to the experimental set-up in [40] and initial density and direction (i.e., $ \theta_1 $) for the experiment with 138 pedestrians
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40]">Figure 7.  Computed (A) mean density $ D_{V} $ and (B) mean flow rate $ F_{V} $ as defined in (15), and measured (C) mean density and (D) mean flow rate from [40]
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Figure 8.  (A) Different velocity moduli under consideration and (B) corresponding number of pedestrians in the room versus time for the 138 pedestrian case
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Figure 9.  Density (top) and velocity magnitude with selected velocity vectors (bottom) for the evacuation process of 138 pedestrians with the purple (left), orange (middle), and blue (right) velocity moduli at time $ t = 15 $ s
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Figure 10.  (A) Widths of exit 2 under consideration and corresponding width ratios and (B) evacuation time versus width ratios for two different scenarios
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Figure 11.  Computational domain with effective area for (A) an obstacle placed in the middle of the room, towards the exit, and (B) two obstacles placed symmetrically with respect to the exit
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Figure 12.  Configuration 1 with $ \alpha = 1 $ in the effective area: computed density for $ t $ = 0, 3, 6, 7.5, 9, 13.5 s. The small square within the effective area represents the real obstacle
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Figure 13.  Configuration 1 with $ \alpha = 0 $ in the effective area: computed density for $ t $ = 0, 3, 6, 7.5, 9, 13.5 s. The small rectangle within the effective area represents the real obstacle
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Figure 14.  Configuration 2 with $ \alpha = 1 $ in the effective area: computed density for $ t $ = 0, 3, 6, 7.5, 10.5, 13.5 s. The small square within the effective area represents the real obstacle
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Figure 15.  Configuration 2 with $ \alpha = 0 $ in the effective area: computed density for $ t $ = 0, 3, 6, 7.5, 10.5, 13.5 s s. The small rectangle within the effective area represents the real obstacle
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Figure 16.  Evacuation times for the room with no obstacles ($ \alpha = 1 $ everywhere in the domain), for room with one and two obstacles with $ \alpha = 1 $ and $ \alpha = 0 $ in the effective area
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Figure 17.  The movement process of 98 pedestrians grouped into four clusters with opposite initial direction $ \theta_1 $ and $ \theta_5 $ in the periodic corridor for $ t = 0, 4.2, 12.3, 19.8, 33.9, 50.7, 72.3 $ s, respectively
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Figure 18.  The movement process of 188 pedestrians grouped into four clusters with initial opposite direction $ \theta_1 $ and $ \theta_5 $ in the periodic corridor for $ t = 0, 4.5, 12.6, 19.8, 33.9, 50.7, 89.7 $ s, respectively
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