April  2013, 6(2): 547-565. doi: 10.3934/dcdss.2013.6.547

The non-smooth view for contact dynamics by Michel Frémond extended to the modeling of crowd movements

1. 

Laboratoire Navier (ENPC/IFSTTAR/CNRS), École des Ponts ParisTech, Université Paris-Est, 6 & 8 av Blaise Pascal, 77455 Marne-la-Vallée, France, France

2. 

IFSTTAR, Université Paris-Est, 58 boulevard Lefebvre, 75732 Paris, France

3. 

EGIS Industries, 4 rue Dolores Ibarruri, 91188 Montreuil, France

Received  July 2011 Revised  February 2012 Published  November 2012

The non-smooth view of Michel Frémond has already been proven successful in managing collisions between rigid particles and in this paper, it will be adapted so as to represent pedestrians and their strategy of displacement. The developed discrete approach applies a rigorous thermodynamic framework in which the local interactions between particles are managed by the use of pseudopotentials of dissipation. It handles local interactions such as pedestrian-pedestrian and pedestrian-obstacle in order to reproduce the global and real dynamics of pedestrian traffic. Social forces are introduced and implemented in order to simulate the behavior of pedestrians and subgroups of pedestrians. The numerical implementation allows us to perform simulations in various situations so that the safety and comfort of public spaces can be enhanced.
Citation: Philippe Pécol, Pierre Argoul, Stefano Dal Pont, Silvano Erlicher. The non-smooth view for contact dynamics by Michel Frémond extended to the modeling of crowd movements. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 547-565. doi: 10.3934/dcdss.2013.6.547
References:
[1]

M. P. Allen and D. J. Tildesley, "Computer Simulation of Liquids,", Oxford University Press, (1987).   Google Scholar

[2]

D. Barese, "A New Discrete Model for Simulating Crowd-Structure Interaction: General Formulation and Application to the Millenium Bridge,", Tesi di Laurea in Tecnicadelle Costruzioni, (2011).   Google Scholar

[3]

V. Blue and J. Adler, Cellular automata microsimulation of bi-directional pedestrian flows,, Journal of the Transportation Research Board, 1678 (2000), 135.  doi: 10.3141/1678-17.  Google Scholar

[4]

J. Bodgi, S. Erlicher and P. Argoul, Lateral vibration of footbridges under crowd - loading : Continuous crowd modelling approach,, Key Engineering Materials, 347 (2007), 685.  doi: 10.4028/www.scientific.net/KEM.347.685.  Google Scholar

[5]

J. Bodgi, S. Erlicher and P. Argoul, Pedestrians-footbridge synchronization: Analytical study of a macroscopic model,, Journal of Sound and Vibration, (2011).   Google Scholar

[6]

C. Chalons, "La méthode Fast-Marching Pour la Propagation de Fronts,", cours ENSTA, (2009).   Google Scholar

[7]

C. Cholet, "Chocs de Solides Rigides,", Ph. D thesis, (1998).   Google Scholar

[8]

P. A. Cundall, A computer model for simulating progressive large scale movements of blocky rock systems,, in, 1 (1971), 132.   Google Scholar

[9]

P. A. Cundall and O. D. L. Strack, A discrete numerical model for granular assemblies,, Geotechnique, 29 (1979), 47.  doi: 10.1680/geot.1979.29.1.47.  Google Scholar

[10]

S. Dal Pont and E. Dimnet, A theory for multiple collisions of rigid solids and numerical simulation of granular flow,, Int. J. Solids and Structures, 43 (2006), 6100.   Google Scholar

[11]

S. Dal Pont and E. Dimnet, Theoretical approach to instantaneous collisions and numerical simulation of granular media using the A-$CD^2$ method,, Communications in Applied Mathematics and Computational Science -Berkeley, 3 (2008), 1.   Google Scholar

[12]

E. Dimnet, "Mouvement et Collisions de Solides Rigides ou Déformables,", Ph. D thesis, (2002).   Google Scholar

[13]

C. Ericson, "Real Time Collision Detection,", Morgan Haufmann Publishers, (2004).   Google Scholar

[14]

M. Frémond, Rigid bodies collisions,, Physics Letters A, 204 (1995), 33.  doi: 10.1016/0375-9601(95)00418-3.  Google Scholar

[15]

M. Frémond, "Collisions,", Edizioni del Dipartimento di Ingegneria Civile dell' Universita di Roma Tor Vergata, (2007).   Google Scholar

[16]

J. J. Fruin, Designing for pedestrians: A level of service concept,, Highway Research Record, (1971), 1.   Google Scholar

[17]

B. D. Hankin and R. A. Wright, Passenger flow in subways,, Oper. Res., 9 (1958), 81.   Google Scholar

[18]

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

[19]

D. Helbing, I. Farkas and T. Vicsek, Simulating dynamic features of escape panic,, Nature, 407 (2000), 487.  doi: 10.1038/35035023.  Google Scholar

[20]

D. Helbing, Traffic and related self-driven many-particle systems,, Reviews of Modern Physics, 73 (2002), 1067.  doi: 10.1103/RevModPhys.73.1067.  Google Scholar

[21]

D. Helbing, M. Isobe, T. Nagatani and K. Takimoto, Lattice gas simulation of experimentally studied evacuation dynamics,, Physical review E, 67 (2003).  doi: 10.1103/PhysRevE.67.067101.  Google Scholar

[22]

D. Helbing, L. Buzna, A. Johansson and T. Werner, Self-organized pedestrian crowd dynamics: Experiments, simulations, and design solutions,, Transportation Science, 39 (2005), 1.   Google Scholar

[23]

L. F. Henderson, Thestatistics of crowd fluids,, Nature, 229 (1971), 381.  doi: 10.1038/229381a0.  Google Scholar

[24]

S. P. Hoogendoorn, P. H. L. Bovy and W. Daamen, Microscopic pedestrian wayfinding and dynamics modelling,, Pedestrian and Evacuation Dynamics, (2001), 123.   Google Scholar

[25]

M. Jean and J. J. Moreau, Unilaterality and dry friction in the dynamics of rigid bodies collection,, Contact Mechanics International Symposium, (1992), 31.   Google Scholar

[26]

M. Jean, The non smooth contact dynamics method,, Compt. Methods Appl. Math. Engrg., 177 (1999), 235.  doi: 10.1016/S0045-7825(98)00383-1.  Google Scholar

[27]

G. Keith Still, "Crowd Dynamics,", Ph. D thesis, (2000).   Google Scholar

[28]

R. Kimmel and J. A. Sethian, Fast marching methods for computing distance maps and shortest paths,, Technical Report 669, (1996).   Google Scholar

[29]

Y. Kishino, Disk model analysisof granular media,, Micromechanics of Granular Materials, (1988), 143.   Google Scholar

[30]

H. Klüpfel, "A Cellular Automaton Model for Crowd Movement and Egress Simulation,", Ph. D thesis, (2003).   Google Scholar

[31]

B. Maury, A time-stepping scheme for inelastic collisions,, Numerische Mathematik, 102 (2006), 649.  doi: 10.1007/s00211-005-0666-6.  Google Scholar

[32]

J. J. Moreau, Décomposition orthogonale d'un espace hilbertien selon deux cones mutuellement polaires,, C. R. Acad. Sci, 255 (1962), 238.   Google Scholar

[33]

J. J. Moreau, Sur les lois dufrottement, de la viscosité et de la plasticité,, Comptesrendus de l'Académie des Sciences de Paris, 271 (1970), 608.   Google Scholar

[34]

J. J. Moreau, Unilateral contactand dry friction in finite freedom dynamics,, in, 302 (1988), 1.   Google Scholar

[35]

J. J. Moreau, New computation methods in granular dynamics,, Thornton, (1993), 227.   Google Scholar

[36]

J. J. Moreau, Some numerical methods in multibody dynamics: Application to granular materials,, Eur. J. Mech. A/Solids, 13 (1994), 93.   Google Scholar

[37]

M. Moussaïd, N. Perozo, S. Garnier, D. Helbing and G. Theraulaz, The walking behaviour of pedestrian social groups and its impact on crowd dynamics,, PLoS ONE, 5 (2010).   Google Scholar

[38]

S. R. Musse, C. R. Jung, J. C. S. Jacques Jr. and A. Braun, Using computer vision to simulate the motion of virtual agents,, Computer Animation and Virtual Worlds, 18 (2007), 83.  doi: 10.1002/cav.163.  Google Scholar

[39]

L. Paoli, Time discretization of vibro-impact,, Phil. Trans. R. Soc. A, 359 (2001), 2405.  doi: 10.1098/rsta.2001.0858.  Google Scholar

[40]

S. Paris, "Characterisation of Levels of Services and Modelling of Flows of People Inside Exchange Areas,", Ph. D thesis, (2007).   Google Scholar

[41]

S. Paris, J. Pettrï and S. Donikian, Pedestrian reactive navigation for crowd simulation: A predictive approach,, Computer Graphics Forum, 26 (2007), 665.  doi: 10.1111/j.1467-8659.2007.01090.x.  Google Scholar

[42]

P. Pécol, S. Dal Pont, S. Erlicher and P. Argoul, Modelling crowd-structure interaction,, Mécanique $&$ Industries, 11 (2010), 495.   Google Scholar

[43]

P. Pécol, S. Dal Pont, S. Erlicher and P. Argoul, Discrete approaches for crowd movement modelling,, European Journal of Computational Mechanics, 20 (2011), 189.   Google Scholar

[44]

P. Pécol, S. Dal Pont, S. Erlicher and P. Argoul, Smooth/non-smooth contact modeling of human crowds movement: numerical aspects and application to emergency evacuations,, Ann. Solid Struct. Mech., 2 (2011), 69.   Google Scholar

[45]

P. Pécol, "Modélisation 2D Discrète du Mouvement des Piétons - Applicationà L'évacuation des Structures du Génie Civil et Àl'interaction Foule-Passerelle,", Ph.D thesis, (2011).   Google Scholar

[46]

P. Pécol, S. Dal Pont, S. Erlicher, J. Bodgi and P. Argoul, A 2D discrete model for crowd-structure interaction,, in Proc. of the fourth international conference Footbridge 2011, (2011), 6.   Google Scholar

[47]

N. Pelechano, J. M. Addler and N. I. Badler, Controlling individual agents in high-density crowd simulation,, in Proc. of the 2007 ACM SIGGRAPH/ Eurographics Symposium on Computer Animation, (2007), 99.   Google Scholar

[48]

F. Radjai, M. Jean, J. J. Moreau and S. Roux, Force distributions in dense two-dimensional granular systems,, Phys. Rev. Lett., 77 (1996), 264.  doi: 10.1103/PhysRevLett.77.274.  Google Scholar

[49]

F. Radjai and V. Richefeu, Mechanics of Materials,, Contact Dynamics as a Nonsmooth Discrete Element Method, 41 (2009), 715.   Google Scholar

[50]

S. Reicher, The St. Pauls riotan explanation of the limits of crowd action in terms of asocial identity model,, EJSP, 14 (1984), 1.   Google Scholar

[51]

M. Renouf, "Optimisationnumérique et Calcul Parallèle Pour L'étude des Milieux Divisés Bi- ettri Dimensionnels,", Ph. D thesis, (2004).   Google Scholar

[52]

C. Reynolds, Flocks, herds, and schools: A distributed behavioral model,, Computer Graphics, 21 (1987), 25.  doi: 10.1145/37402.37406.  Google Scholar

[53]

G. Saussine, C. Cholet, P. E. Gautier, F.Dubois, C. Bohatier and J. J. Moreau, Modelling ballast behaviour under dynamic loading. Part 1: a 2D polygonal discrete element method approach,, Comput. Methods Appl. Mech. Engrg, 195 (2006), 2841.  doi: 10.1016/j.cma.2005.07.006.  Google Scholar

[54]

J. C. Simo and T. J. R. Hughes, "Elastoplasticity and Viscoplasticity Computational Aspects,", Springer, (1996).   Google Scholar

[55]

H. Singh, R. Arter, L. Dodd and J. Drury, Modelling subgroup behavior in crowd dynamics DEM simulation,, Applied Mathematical Modelling, 33 (2009), 4408.  doi: 10.1016/j.apm.2009.03.020.  Google Scholar

[56]

M. Sung, M. Gleicher and S. Chenney, Scalable behaviors for crowd simulation,, Eurographics, 23 (2004), 519.   Google Scholar

[57]

K. Teknomo, Application of microscopic pedestrian simulation model,, Transportation Research Part F, 9 (2006), 15.   Google Scholar

[58]

J. Venel, "Modélisation Mathématique des Mouvements de Foule,", Ph. D thesis, (2008).   Google Scholar

[59]

W. Yu and A. Johansson, Modelling crowd turbulence by many-particle simulations,, Physical Review E, 76 (2007).  doi: 10.1103/PhysRevE.76.046105.  Google Scholar

show all references

References:
[1]

M. P. Allen and D. J. Tildesley, "Computer Simulation of Liquids,", Oxford University Press, (1987).   Google Scholar

[2]

D. Barese, "A New Discrete Model for Simulating Crowd-Structure Interaction: General Formulation and Application to the Millenium Bridge,", Tesi di Laurea in Tecnicadelle Costruzioni, (2011).   Google Scholar

[3]

V. Blue and J. Adler, Cellular automata microsimulation of bi-directional pedestrian flows,, Journal of the Transportation Research Board, 1678 (2000), 135.  doi: 10.3141/1678-17.  Google Scholar

[4]

J. Bodgi, S. Erlicher and P. Argoul, Lateral vibration of footbridges under crowd - loading : Continuous crowd modelling approach,, Key Engineering Materials, 347 (2007), 685.  doi: 10.4028/www.scientific.net/KEM.347.685.  Google Scholar

[5]

J. Bodgi, S. Erlicher and P. Argoul, Pedestrians-footbridge synchronization: Analytical study of a macroscopic model,, Journal of Sound and Vibration, (2011).   Google Scholar

[6]

C. Chalons, "La méthode Fast-Marching Pour la Propagation de Fronts,", cours ENSTA, (2009).   Google Scholar

[7]

C. Cholet, "Chocs de Solides Rigides,", Ph. D thesis, (1998).   Google Scholar

[8]

P. A. Cundall, A computer model for simulating progressive large scale movements of blocky rock systems,, in, 1 (1971), 132.   Google Scholar

[9]

P. A. Cundall and O. D. L. Strack, A discrete numerical model for granular assemblies,, Geotechnique, 29 (1979), 47.  doi: 10.1680/geot.1979.29.1.47.  Google Scholar

[10]

S. Dal Pont and E. Dimnet, A theory for multiple collisions of rigid solids and numerical simulation of granular flow,, Int. J. Solids and Structures, 43 (2006), 6100.   Google Scholar

[11]

S. Dal Pont and E. Dimnet, Theoretical approach to instantaneous collisions and numerical simulation of granular media using the A-$CD^2$ method,, Communications in Applied Mathematics and Computational Science -Berkeley, 3 (2008), 1.   Google Scholar

[12]

E. Dimnet, "Mouvement et Collisions de Solides Rigides ou Déformables,", Ph. D thesis, (2002).   Google Scholar

[13]

C. Ericson, "Real Time Collision Detection,", Morgan Haufmann Publishers, (2004).   Google Scholar

[14]

M. Frémond, Rigid bodies collisions,, Physics Letters A, 204 (1995), 33.  doi: 10.1016/0375-9601(95)00418-3.  Google Scholar

[15]

M. Frémond, "Collisions,", Edizioni del Dipartimento di Ingegneria Civile dell' Universita di Roma Tor Vergata, (2007).   Google Scholar

[16]

J. J. Fruin, Designing for pedestrians: A level of service concept,, Highway Research Record, (1971), 1.   Google Scholar

[17]

B. D. Hankin and R. A. Wright, Passenger flow in subways,, Oper. Res., 9 (1958), 81.   Google Scholar

[18]

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

[19]

D. Helbing, I. Farkas and T. Vicsek, Simulating dynamic features of escape panic,, Nature, 407 (2000), 487.  doi: 10.1038/35035023.  Google Scholar

[20]

D. Helbing, Traffic and related self-driven many-particle systems,, Reviews of Modern Physics, 73 (2002), 1067.  doi: 10.1103/RevModPhys.73.1067.  Google Scholar

[21]

D. Helbing, M. Isobe, T. Nagatani and K. Takimoto, Lattice gas simulation of experimentally studied evacuation dynamics,, Physical review E, 67 (2003).  doi: 10.1103/PhysRevE.67.067101.  Google Scholar

[22]

D. Helbing, L. Buzna, A. Johansson and T. Werner, Self-organized pedestrian crowd dynamics: Experiments, simulations, and design solutions,, Transportation Science, 39 (2005), 1.   Google Scholar

[23]

L. F. Henderson, Thestatistics of crowd fluids,, Nature, 229 (1971), 381.  doi: 10.1038/229381a0.  Google Scholar

[24]

S. P. Hoogendoorn, P. H. L. Bovy and W. Daamen, Microscopic pedestrian wayfinding and dynamics modelling,, Pedestrian and Evacuation Dynamics, (2001), 123.   Google Scholar

[25]

M. Jean and J. J. Moreau, Unilaterality and dry friction in the dynamics of rigid bodies collection,, Contact Mechanics International Symposium, (1992), 31.   Google Scholar

[26]

M. Jean, The non smooth contact dynamics method,, Compt. Methods Appl. Math. Engrg., 177 (1999), 235.  doi: 10.1016/S0045-7825(98)00383-1.  Google Scholar

[27]

G. Keith Still, "Crowd Dynamics,", Ph. D thesis, (2000).   Google Scholar

[28]

R. Kimmel and J. A. Sethian, Fast marching methods for computing distance maps and shortest paths,, Technical Report 669, (1996).   Google Scholar

[29]

Y. Kishino, Disk model analysisof granular media,, Micromechanics of Granular Materials, (1988), 143.   Google Scholar

[30]

H. Klüpfel, "A Cellular Automaton Model for Crowd Movement and Egress Simulation,", Ph. D thesis, (2003).   Google Scholar

[31]

B. Maury, A time-stepping scheme for inelastic collisions,, Numerische Mathematik, 102 (2006), 649.  doi: 10.1007/s00211-005-0666-6.  Google Scholar

[32]

J. J. Moreau, Décomposition orthogonale d'un espace hilbertien selon deux cones mutuellement polaires,, C. R. Acad. Sci, 255 (1962), 238.   Google Scholar

[33]

J. J. Moreau, Sur les lois dufrottement, de la viscosité et de la plasticité,, Comptesrendus de l'Académie des Sciences de Paris, 271 (1970), 608.   Google Scholar

[34]

J. J. Moreau, Unilateral contactand dry friction in finite freedom dynamics,, in, 302 (1988), 1.   Google Scholar

[35]

J. J. Moreau, New computation methods in granular dynamics,, Thornton, (1993), 227.   Google Scholar

[36]

J. J. Moreau, Some numerical methods in multibody dynamics: Application to granular materials,, Eur. J. Mech. A/Solids, 13 (1994), 93.   Google Scholar

[37]

M. Moussaïd, N. Perozo, S. Garnier, D. Helbing and G. Theraulaz, The walking behaviour of pedestrian social groups and its impact on crowd dynamics,, PLoS ONE, 5 (2010).   Google Scholar

[38]

S. R. Musse, C. R. Jung, J. C. S. Jacques Jr. and A. Braun, Using computer vision to simulate the motion of virtual agents,, Computer Animation and Virtual Worlds, 18 (2007), 83.  doi: 10.1002/cav.163.  Google Scholar

[39]

L. Paoli, Time discretization of vibro-impact,, Phil. Trans. R. Soc. A, 359 (2001), 2405.  doi: 10.1098/rsta.2001.0858.  Google Scholar

[40]

S. Paris, "Characterisation of Levels of Services and Modelling of Flows of People Inside Exchange Areas,", Ph. D thesis, (2007).   Google Scholar

[41]

S. Paris, J. Pettrï and S. Donikian, Pedestrian reactive navigation for crowd simulation: A predictive approach,, Computer Graphics Forum, 26 (2007), 665.  doi: 10.1111/j.1467-8659.2007.01090.x.  Google Scholar

[42]

P. Pécol, S. Dal Pont, S. Erlicher and P. Argoul, Modelling crowd-structure interaction,, Mécanique $&$ Industries, 11 (2010), 495.   Google Scholar

[43]

P. Pécol, S. Dal Pont, S. Erlicher and P. Argoul, Discrete approaches for crowd movement modelling,, European Journal of Computational Mechanics, 20 (2011), 189.   Google Scholar

[44]

P. Pécol, S. Dal Pont, S. Erlicher and P. Argoul, Smooth/non-smooth contact modeling of human crowds movement: numerical aspects and application to emergency evacuations,, Ann. Solid Struct. Mech., 2 (2011), 69.   Google Scholar

[45]

P. Pécol, "Modélisation 2D Discrète du Mouvement des Piétons - Applicationà L'évacuation des Structures du Génie Civil et Àl'interaction Foule-Passerelle,", Ph.D thesis, (2011).   Google Scholar

[46]

P. Pécol, S. Dal Pont, S. Erlicher, J. Bodgi and P. Argoul, A 2D discrete model for crowd-structure interaction,, in Proc. of the fourth international conference Footbridge 2011, (2011), 6.   Google Scholar

[47]

N. Pelechano, J. M. Addler and N. I. Badler, Controlling individual agents in high-density crowd simulation,, in Proc. of the 2007 ACM SIGGRAPH/ Eurographics Symposium on Computer Animation, (2007), 99.   Google Scholar

[48]

F. Radjai, M. Jean, J. J. Moreau and S. Roux, Force distributions in dense two-dimensional granular systems,, Phys. Rev. Lett., 77 (1996), 264.  doi: 10.1103/PhysRevLett.77.274.  Google Scholar

[49]

F. Radjai and V. Richefeu, Mechanics of Materials,, Contact Dynamics as a Nonsmooth Discrete Element Method, 41 (2009), 715.   Google Scholar

[50]

S. Reicher, The St. Pauls riotan explanation of the limits of crowd action in terms of asocial identity model,, EJSP, 14 (1984), 1.   Google Scholar

[51]

M. Renouf, "Optimisationnumérique et Calcul Parallèle Pour L'étude des Milieux Divisés Bi- ettri Dimensionnels,", Ph. D thesis, (2004).   Google Scholar

[52]

C. Reynolds, Flocks, herds, and schools: A distributed behavioral model,, Computer Graphics, 21 (1987), 25.  doi: 10.1145/37402.37406.  Google Scholar

[53]

G. Saussine, C. Cholet, P. E. Gautier, F.Dubois, C. Bohatier and J. J. Moreau, Modelling ballast behaviour under dynamic loading. Part 1: a 2D polygonal discrete element method approach,, Comput. Methods Appl. Mech. Engrg, 195 (2006), 2841.  doi: 10.1016/j.cma.2005.07.006.  Google Scholar

[54]

J. C. Simo and T. J. R. Hughes, "Elastoplasticity and Viscoplasticity Computational Aspects,", Springer, (1996).   Google Scholar

[55]

H. Singh, R. Arter, L. Dodd and J. Drury, Modelling subgroup behavior in crowd dynamics DEM simulation,, Applied Mathematical Modelling, 33 (2009), 4408.  doi: 10.1016/j.apm.2009.03.020.  Google Scholar

[56]

M. Sung, M. Gleicher and S. Chenney, Scalable behaviors for crowd simulation,, Eurographics, 23 (2004), 519.   Google Scholar

[57]

K. Teknomo, Application of microscopic pedestrian simulation model,, Transportation Research Part F, 9 (2006), 15.   Google Scholar

[58]

J. Venel, "Modélisation Mathématique des Mouvements de Foule,", Ph. D thesis, (2008).   Google Scholar

[59]

W. Yu and A. Johansson, Modelling crowd turbulence by many-particle simulations,, Physical Review E, 76 (2007).  doi: 10.1103/PhysRevE.76.046105.  Google Scholar

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Mikhail I. Belishev, Aleksei F. Vakulenko. Non-smooth unobservable states in control problem for the wave equation in $\mathbb{R}^3$. Evolution Equations & Control Theory, 2014, 3 (2) : 247-256. doi: 10.3934/eect.2014.3.247

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