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An anisotropic interaction model with collision avoidance
University of Mannheim, School of Business Informatics and Mathematics, 68159 Mannheim, Germany |
In this article an anisotropic interaction model avoiding collisions is proposed. Starting point is a general isotropic interacting particle system, as used for swarming or follower-leader dynamics. An anisotropy is induced by rotation of the force vector resulting from the interaction of two agents. In this way the anisotropy is leading to a smooth evasion behaviour. In fact, the proposed model generalizes the standard models, and compensates their drawback of not being able to avoid collisions. Moreover, the model allows for formal passage to the limit 'number of particles to infinity', leading to a mesoscopic description in the mean-field sense. Possible applications are autonomous traffic, swarming or pedestrian motion. Here, we focus on the latter, as the model is validated numerically using two scenarios in pedestrian dynamics. The first one investigates the pattern formation in a channel, where two groups of pedestrians are walking in opposite directions. The second experiment considers a crossing with one group walking from left to right and the other one from bottom to top. The well-known pattern of lanes in the channel and travelling waves at the crossing can be reproduced with the help of this anisotropic model at both, the microscopic and the mesoscopic level. In addition, the 'right-before-left' and 'left-before-right' rule appear intrinsically for different anisotropy parameters.
References:
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G. Albi, M. Bongini, E. Cristiani and D. Kalise,
Invisible control of self-organizing agents leaving unknown environments, SIAM J. Appl. Math., 76 (2016), 1683-1710.
doi: 10.1137/15M1017016. |
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G. Albi and L. Pareschi,
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doi: 10.1016/j.aml.2012.10.011. |
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C. Appert-Rolland, J. Cividini, H. J. Hilhorst and P. Degond,
Pedestrian flows: From individuals to crowds, Transportation Research Procedia, 2 (2014), 468-476.
doi: 10.1016/j.trpro.2014.09.062. |
[4] |
R. Bailo, J. A. Carrillo and P. Degond, Pedestrian models based on rational behaviour, in Crowd Dynamics, Volume 1: Theory, Models, and Safety Problems (eds. L. Gibelli and N. Bellomo), Springer Internat. Publishing, (2018), 259–292. |
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N. Bellomo, P. Degond and E. Tadmor, Active Particles, Volume 1, Birkhäuser, Basel, 2017.
doi: 10.1007/978-3-319-49996-3. |
[6] |
N. W. F. Bode and E. A. Codling, Statistical models for pedestrian behaviour in front of bottlenecks, in Traffic and Granular Flow'15 (eds. V. Knoop and W. Daamen) Springer, Cham (2016), 81–88.
doi: 10.1007/978-3-319-33482-0_11. |
[7] |
N. W. F. Bode and E. Ronchi,
Statistical model fitting and model selection in pedestrian dynamics research, Collective Dynamics, 4 (2019), 1-32.
doi: 10.17815/CD.2019.20. |
[8] |
R. Borsche and A. Meurer,
Microscopic and macroscopic models for coupled car traffic and pedestiran flow, Journal Computational and Applied Mathematics, 348 (2019), 356-382.
doi: 10.1016/j.cam.2018.08.037. |
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R. Borsche, A. Klar, S. Kühn and A. Meurer,
Coupling traffic flow networks to pedestrian motion, Math. Mod. Meth. Appl. Sci., 24 (2014), 359-380.
doi: 10.1142/S0218202513400113. |
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M. Burger, M. Di Francesco, P. A. Markowich and M.-T. Wolfram,
Mean field games with nonlinear mobilities in pedestrian dynamics, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1311-1333.
doi: 10.3934/dcdsb.2014.19.1311. |
[11] |
M. Burger, B. Düring, L. M. Kreusser, P. A. Markowich and C.-B. Schönlieb,
Pattern formation of a nonlocal, anisotropic interaction model, Math. Mod. Meth. Appl. Sci., 28 (2018), 409-451.
doi: 10.1142/S0218202518500112. |
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M. Burger, S. Hittmeir, H. Ranetbauer and M.-T. Wolfram,
Lane formation by side-stepping, SIAM J. Math. Anal., 48 (2016), 981-1005.
doi: 10.1137/15M1033174. |
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M. Burger, R. Pinnau, C. Totzeck, O. Tse and A. Roth, Instantaneous Control of interacting particle systems in the mean-field limit, J. Comp. Phys., 405 (2020), 109181.
doi: 10.1016/j.jcp.2019.109181. |
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J. A. Carrillo, Y.-P. Choi and M. Hauray, The derivation of swarming models: Mean-field limit and Wasserstein distances, in Collective dynamics from bacteria to crowds (eds. A. Muntean, F. Toschi), Springer, (2014), 1–46.
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Stability analysis of line patterns of an anisotropic interaction model, SIAM J. Appl. Dyn. Sys., 18 (2019), 1798-1845.
doi: 10.1137/18M1181638. |
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J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming, in Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences (eds. G. Naldi, L. Pareschi, G. Toscani), Birkhäuser Boston, (2010), 297–336.
doi: 10.1007/978-0-8176-4946-3_12. |
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J. A. Carrillo, A. Klar and A. Roth,
Single to double mill small noise transition via semi-lagrangian finite volume methods, Comm. Math. Sci., 14 (2016), 1111-1136.
doi: 10.4310/CMS.2016.v14.n4.a12. |
[19] |
J. A. Carrillo, S. Martin and M.-T. Wolfram,
An improved version of the Hughes model for pedestrian flow, Math. Models Meth. Appl. Sci., 26 (2016), 671-697.
doi: 10.1142/S0218202516500147. |
[20] |
J. Cividini, C. Appert-Rolland and H. J. Hilhorst, Diagonal patterns and chevron effect in intersecting traffic flows, EPL, 102 (2013), 20002.
doi: 10.1209/0295-5075/102/20002. |
[21] |
E. Cristiani and D. Peri,
Handling obstacles in pedestrian simulations: Models and optimization, Appl. Math. Mod., 45 (2017), 285-302.
doi: 10.1016/j.apm.2016.12.020. |
[22] |
E. Cristiani, B. Piccoli and A. Tosin, Multiscale Modeling of Pedestrian Dynamics, Springer International Publishing, Switzerland, 2014.
doi: 10.1007/978-3-319-06620-2. |
[23] |
E. Cristiani, F. S. Priuli and A. Tosin,
Modeling rationality to control self-organization of crowds: An environmental approach, SIAM J. Appl. Math., 75 (2015), 605-629.
doi: 10.1137/140962413. |
[24] |
P. Degond, C. Appert-Rolland, M. Moussaid, J. Pettré and G. Theraulaz,
A hierarchy of heuristic-based models of crowd dynamics, J. Statistical Physics, 152 (2013), 1033-1068.
doi: 10.1007/s10955-013-0805-x. |
[25] |
P. Degond, C. Appert-Rolland, J. Pettreé and G. Theraulaz,
Vision-based macroscopic pedestrian models, Kin. Rel. Models, 6 (2013), 809-839.
doi: 10.3934/krm.2013.6.809. |
[26] |
P. Degond, A. Frouvelle and S. Merino-Aceituno,
A new flocking model through body attitude coordination, Math. Mod. Meth. Appl. Sci., 27 (2017), 1005-1049.
doi: 10.1142/S0218202517400085. |
[27] |
P. Degond and S. Motsch,
Continuum limit of self-driven particles with orientation interaction, Math. Mod. Meth. Appl. Sci., 18 (2008), 1193-1215.
doi: 10.1142/S0218202508003005. |
[28] |
M. Di Francesco, P. A. Markowich, J.-F. Pietschmann and M.-T. Wolfram,
On the Hughes' model for pedestrian flow: The one dimensional case, J. Differential Equations, 250 (2011), 1334-1362.
doi: 10.1016/j.jde.2010.10.015. |
[29] |
M. R. D'Orsogna, Y. L. Chuang, A. L. Bertozzi and L. S. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability, and collapse, Phys. Rev. Lett., 96 (2006), 104302.
doi: 10.1103/PhysRevLett.96.104302. |
[30] |
B. Düring, C. Gottschlich, S. Huckemann, L. M. Kreusser and C. -B.Schönlieb,
An anisotropic interaction model for simulating fingerprints, J. Math. Biol., 78 (2019), 2171-2206.
doi: 10.1007/s00285-019-01338-3. |
[31] |
J. H. M. Evers, R. C. Fetecau and L. Ryzhik, Anisotropic interactions in a first-order aggregation model, Nonlinearity, 28 (2015), 2847.
doi: 10.1088/0951-7715/28/8/2847. |
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J. H. M. Evers, R. C. Fetecau and W. Sun,
Small intertia regularization of an anisotropic aggregation model, Math. Mod. Meth. Appl. Sci., 27 (2017), 1795-1842.
doi: 10.1142/S0218202517500324. |
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A. Festa, A. Tosin and M.-T. Wolfram,
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doi: 10.3934/krm.2018022. |
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L. Gibelli and N. Bellomo, Crowd Dynamics, Volume 1, Birkhäuser, Cham, 2018.
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F. Golse, The mean-field limit for the dynamics of large particle systems, Journées équations aux dérivées partielles, 9 (2003), 1–47. |
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S. N. Gomes, A. M. Stuart and M.-T. Wolfram,
Parameter estimation for macroscopic pedestrian dynamics models from microscopic data, SIAM J. Appl. Math., 79 (2019), 1475-1500.
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A mathematical model for the behavior of pedestrians, Behav. Sci., 36 (1991), 298-310.
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doi: 10.1287/trsc.1040.0108. |
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D. Helbing, I. J. Farkas and T. Vicsek,
Simulating dynamical features of escape panic, Nature, 407 (2000), 487-490.
doi: 10.1038/35035023. |
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D. Helbing, A. Johansson and H. Z. Al-Abideen, Dynamics of crowd disasters: An empirical study, Phys. Rev. E, 75 (2007), 046109.
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Social force model for pedestiran dynamics, Phys. Rev. E, 51 (1995), 4282-4286.
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S. Hittmeir, H. Ranetbauer, C. Schmeiser and M.-T. Wolfram,
Derivation and analysis of continuum models for crossing pedestrian traffic, Math. Meth. Appl. Sci., 27 (2017), 1301-1325.
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show all references
References:
[1] |
G. Albi, M. Bongini, E. Cristiani and D. Kalise,
Invisible control of self-organizing agents leaving unknown environments, SIAM J. Appl. Math., 76 (2016), 1683-1710.
doi: 10.1137/15M1017016. |
[2] |
G. Albi and L. Pareschi,
Modeling self-organized systems interacting with few individuals: From microscopic to macroscopic dynamics, Appl. Math. Letters, 26 (2013), 391-401.
doi: 10.1016/j.aml.2012.10.011. |
[3] |
C. Appert-Rolland, J. Cividini, H. J. Hilhorst and P. Degond,
Pedestrian flows: From individuals to crowds, Transportation Research Procedia, 2 (2014), 468-476.
doi: 10.1016/j.trpro.2014.09.062. |
[4] |
R. Bailo, J. A. Carrillo and P. Degond, Pedestrian models based on rational behaviour, in Crowd Dynamics, Volume 1: Theory, Models, and Safety Problems (eds. L. Gibelli and N. Bellomo), Springer Internat. Publishing, (2018), 259–292. |
[5] |
N. Bellomo, P. Degond and E. Tadmor, Active Particles, Volume 1, Birkhäuser, Basel, 2017.
doi: 10.1007/978-3-319-49996-3. |
[6] |
N. W. F. Bode and E. A. Codling, Statistical models for pedestrian behaviour in front of bottlenecks, in Traffic and Granular Flow'15 (eds. V. Knoop and W. Daamen) Springer, Cham (2016), 81–88.
doi: 10.1007/978-3-319-33482-0_11. |
[7] |
N. W. F. Bode and E. Ronchi,
Statistical model fitting and model selection in pedestrian dynamics research, Collective Dynamics, 4 (2019), 1-32.
doi: 10.17815/CD.2019.20. |
[8] |
R. Borsche and A. Meurer,
Microscopic and macroscopic models for coupled car traffic and pedestiran flow, Journal Computational and Applied Mathematics, 348 (2019), 356-382.
doi: 10.1016/j.cam.2018.08.037. |
[9] |
R. Borsche, A. Klar, S. Kühn and A. Meurer,
Coupling traffic flow networks to pedestrian motion, Math. Mod. Meth. Appl. Sci., 24 (2014), 359-380.
doi: 10.1142/S0218202513400113. |
[10] |
M. Burger, M. Di Francesco, P. A. Markowich and M.-T. Wolfram,
Mean field games with nonlinear mobilities in pedestrian dynamics, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1311-1333.
doi: 10.3934/dcdsb.2014.19.1311. |
[11] |
M. Burger, B. Düring, L. M. Kreusser, P. A. Markowich and C.-B. Schönlieb,
Pattern formation of a nonlocal, anisotropic interaction model, Math. Mod. Meth. Appl. Sci., 28 (2018), 409-451.
doi: 10.1142/S0218202518500112. |
[12] |
M. Burger, S. Hittmeir, H. Ranetbauer and M.-T. Wolfram,
Lane formation by side-stepping, SIAM J. Math. Anal., 48 (2016), 981-1005.
doi: 10.1137/15M1033174. |
[13] |
M. Burger, R. Pinnau, C. Totzeck, O. Tse and A. Roth, Instantaneous Control of interacting particle systems in the mean-field limit, J. Comp. Phys., 405 (2020), 109181.
doi: 10.1016/j.jcp.2019.109181. |
[14] |
J. A. Carrillo, Y.-P. Choi and M. Hauray, The derivation of swarming models: Mean-field limit and Wasserstein distances, in Collective dynamics from bacteria to crowds (eds. A. Muntean, F. Toschi), Springer, (2014), 1–46.
doi: 10.1007/978-3-7091-1785-9_1. |
[15] |
J. A. Carrillo, Y.-P. Choi, C. Totzeck and O. Tse,
An analytical framework for consensus-based global optimization method, Math. Mod. Meth. Appl. Sci., 28 (2018), 1037-1066.
doi: 10.1142/S0218202518500276. |
[16] |
J. A. Carrillo, B. Düring, L. M. Kreusser and C.-B. Schönlieb,
Stability analysis of line patterns of an anisotropic interaction model, SIAM J. Appl. Dyn. Sys., 18 (2019), 1798-1845.
doi: 10.1137/18M1181638. |
[17] |
J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming, in Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences (eds. G. Naldi, L. Pareschi, G. Toscani), Birkhäuser Boston, (2010), 297–336.
doi: 10.1007/978-0-8176-4946-3_12. |
[18] |
J. A. Carrillo, A. Klar and A. Roth,
Single to double mill small noise transition via semi-lagrangian finite volume methods, Comm. Math. Sci., 14 (2016), 1111-1136.
doi: 10.4310/CMS.2016.v14.n4.a12. |
[19] |
J. A. Carrillo, S. Martin and M.-T. Wolfram,
An improved version of the Hughes model for pedestrian flow, Math. Models Meth. Appl. Sci., 26 (2016), 671-697.
doi: 10.1142/S0218202516500147. |
[20] |
J. Cividini, C. Appert-Rolland and H. J. Hilhorst, Diagonal patterns and chevron effect in intersecting traffic flows, EPL, 102 (2013), 20002.
doi: 10.1209/0295-5075/102/20002. |
[21] |
E. Cristiani and D. Peri,
Handling obstacles in pedestrian simulations: Models and optimization, Appl. Math. Mod., 45 (2017), 285-302.
doi: 10.1016/j.apm.2016.12.020. |
[22] |
E. Cristiani, B. Piccoli and A. Tosin, Multiscale Modeling of Pedestrian Dynamics, Springer International Publishing, Switzerland, 2014.
doi: 10.1007/978-3-319-06620-2. |
[23] |
E. Cristiani, F. S. Priuli and A. Tosin,
Modeling rationality to control self-organization of crowds: An environmental approach, SIAM J. Appl. Math., 75 (2015), 605-629.
doi: 10.1137/140962413. |
[24] |
P. Degond, C. Appert-Rolland, M. Moussaid, J. Pettré and G. Theraulaz,
A hierarchy of heuristic-based models of crowd dynamics, J. Statistical Physics, 152 (2013), 1033-1068.
doi: 10.1007/s10955-013-0805-x. |
[25] |
P. Degond, C. Appert-Rolland, J. Pettreé and G. Theraulaz,
Vision-based macroscopic pedestrian models, Kin. Rel. Models, 6 (2013), 809-839.
doi: 10.3934/krm.2013.6.809. |
[26] |
P. Degond, A. Frouvelle and S. Merino-Aceituno,
A new flocking model through body attitude coordination, Math. Mod. Meth. Appl. Sci., 27 (2017), 1005-1049.
doi: 10.1142/S0218202517400085. |
[27] |
P. Degond and S. Motsch,
Continuum limit of self-driven particles with orientation interaction, Math. Mod. Meth. Appl. Sci., 18 (2008), 1193-1215.
doi: 10.1142/S0218202508003005. |
[28] |
M. Di Francesco, P. A. Markowich, J.-F. Pietschmann and M.-T. Wolfram,
On the Hughes' model for pedestrian flow: The one dimensional case, J. Differential Equations, 250 (2011), 1334-1362.
doi: 10.1016/j.jde.2010.10.015. |
[29] |
M. R. D'Orsogna, Y. L. Chuang, A. L. Bertozzi and L. S. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability, and collapse, Phys. Rev. Lett., 96 (2006), 104302.
doi: 10.1103/PhysRevLett.96.104302. |
[30] |
B. Düring, C. Gottschlich, S. Huckemann, L. M. Kreusser and C. -B.Schönlieb,
An anisotropic interaction model for simulating fingerprints, J. Math. Biol., 78 (2019), 2171-2206.
doi: 10.1007/s00285-019-01338-3. |
[31] |
J. H. M. Evers, R. C. Fetecau and L. Ryzhik, Anisotropic interactions in a first-order aggregation model, Nonlinearity, 28 (2015), 2847.
doi: 10.1088/0951-7715/28/8/2847. |
[32] |
J. H. M. Evers, R. C. Fetecau and W. Sun,
Small intertia regularization of an anisotropic aggregation model, Math. Mod. Meth. Appl. Sci., 27 (2017), 1795-1842.
doi: 10.1142/S0218202517500324. |
[33] |
A. Festa, A. Tosin and M.-T. Wolfram,
Kinetic description of collision avoidance in pedestrian crowds by sidestepping, Kinet. Relat. Models, 11 (2018), 491-520.
doi: 10.3934/krm.2018022. |
[34] |
L. Gibelli and N. Bellomo, Crowd Dynamics, Volume 1, Birkhäuser, Cham, 2018.
doi: 10.1007/978-3-030-05129-7. |
[35] |
F. Golse, The mean-field limit for the dynamics of large particle systems, Journées équations aux dérivées partielles, 9 (2003), 1–47. |
[36] |
S. N. Gomes, A. M. Stuart and M.-T. Wolfram,
Parameter estimation for macroscopic pedestrian dynamics models from microscopic data, SIAM J. Appl. Math., 79 (2019), 1475-1500.
doi: 10.1137/18M1215980. |
[37] |
D. Helbing,
A mathematical model for the behavior of pedestrians, Behav. Sci., 36 (1991), 298-310.
doi: 10.1002/bs.3830360405. |
[38] |
D. Helbing, L. Buzna, A. Johansson and T. Werner,
Self-organized pedestrian crowd dynamics: Experiments, simulations, and design solutions, Transp. Sci., 39 (2005), 1-24.
doi: 10.1287/trsc.1040.0108. |
[39] |
D. Helbing, I. J. Farkas and T. Vicsek,
Simulating dynamical features of escape panic, Nature, 407 (2000), 487-490.
doi: 10.1038/35035023. |
[40] |
D. Helbing, A. Johansson and H. Z. Al-Abideen, Dynamics of crowd disasters: An empirical study, Phys. Rev. E, 75 (2007), 046109.
doi: 10.1103/PhysRevE.75.046109. |
[41] |
D. Helbing and P. Molnár,
Social force model for pedestiran dynamics, Phys. Rev. E, 51 (1995), 4282-4286.
doi: 10.1103/PhysRevE.51.4282. |
[42] |
S. Hittmeir, H. Ranetbauer, C. Schmeiser and M.-T. Wolfram,
Derivation and analysis of continuum models for crossing pedestrian traffic, Math. Meth. Appl. Sci., 27 (2017), 1301-1325.
doi: 10.1142/S0218202517400164. |
[43] |
P.-E. Jabin,
A review of the mean field limits for Vlasov equations, Kin. Rel. Mod., 7 (2014), 661-711.
doi: 10.3934/krm.2014.7.661. |
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