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 and 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.

[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, Specialistica in Ingegneria Civile, Universitàdegli studi di Salerno, facoltà di ingegneria, 2011.

[3]

V. Blue and J. Adler, Cellular automata microsimulation of bi-directional pedestrian flows, Journal of the Transportation Research Board, 1678 (2000), 135-141. doi: 10.3141/1678-17.

[4]

J. Bodgi, S. Erlicher and P. Argoul, Lateral vibration of footbridges under crowd - loading : Continuous crowd modelling approach, Key Engineering Materials, 347 (2007), 685-690. doi: 10.4028/www.scientific.net/KEM.347.685.

[5]

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

[6]

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

[7]

C. Cholet, "Chocs de Solides Rigides," Ph. D thesis, Université Paris VI, 1998.

[8]

P. A. Cundall, A computer model for simulating progressive large scale movements of blocky rock systems, in "Proc. of the Symposium of the International Society of Rock Mechanics," 1 (1971), 132-150.

[9]

P. A. Cundall and O. D. L. Strack, A discrete numerical model for granular assemblies, Geotechnique, 29 (1979), 47-65. doi: 10.1680/geot.1979.29.1.47.

[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), 6100-6114.

[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), 1-24.

[12]

E. Dimnet, "Mouvement et Collisions de Solides Rigides ou Déformables," Ph. D thesis, Ecole Nationale des Ponts et Chaussées, 2002.

[13]

C. Ericson, "Real Time Collision Detection," Morgan Haufmann Publishers, 2004.

[14]

M. Frémond, Rigid bodies collisions, Physics Letters A, 204 (1995), 33-41. doi: 10.1016/0375-9601(95)00418-3.

[15]

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

[16]

J. J. Fruin, Designing for pedestrians: A level of service concept, Highway Research Record, (1971), 1-15.

[17]

B. D. Hankin and R. A. Wright, Passenger flow in subways, Oper. Res., 9 (1958), 81-88.

[18]

D. Helbing and P. Molnàr, Social force model for pedestrian dynamics, Physical Review E, 51 (1995), 4282-4286. doi: 10.1103/PhysRevE.51.4282.

[19]

D. Helbing, I. Farkas and T. Vicsek, Simulating dynamic features of escape panic, Nature, 407 (2000), 487-490. doi: 10.1038/35035023.

[20]

D. Helbing, Traffic and related self-driven many-particle systems, Reviews of Modern Physics, 73 (2002), 1067-1141. doi: 10.1103/RevModPhys.73.1067.

[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.

[22]

D. Helbing, L. Buzna, A. Johansson and T. Werner, Self-organized pedestrian crowd dynamics: Experiments, simulations, and design solutions, Transportation Science, 39 (2005), 1-24.

[23]

L. F. Henderson, Thestatistics of crowd fluids, Nature, 229 (1971), 381-383. doi: 10.1038/229381a0.

[24]

S. P. Hoogendoorn, P. H. L. Bovy and W. Daamen, Microscopic pedestrian wayfinding and dynamics modelling, Pedestrian and Evacuation Dynamics, (2001), 123-154.

[25]

M. Jean and J. J. Moreau, Unilaterality and dry friction in the dynamics of rigid bodies collection, Contact Mechanics International Symposium, Presses Polytechniques et Universitaires Romanes, (1992), 31-48.

[26]

M. Jean, The non smooth contact dynamics method, Compt. Methods Appl. Math. Engrg., 177 (1999), 235-257. doi: 10.1016/S0045-7825(98)00383-1.

[27]

G. Keith Still, "Crowd Dynamics," Ph. D thesis, University of Warwick, Department of Mathematics, 2000.

[28]

R. Kimmel and J. A. Sethian, Fast marching methods for computing distance maps and shortest paths, Technical Report 669, CPAM, University of California, Berkeley, (1996).

[29]

Y. Kishino, Disk model analysisof granular media, Micromechanics of Granular Materials, (1988), 143-152.

[30]

H. Klüpfel, "A Cellular Automaton Model for Crowd Movement and Egress Simulation," Ph. D thesis,Universitat Duisburg - Essen, 2003.

[31]

B. Maury, A time-stepping scheme for inelastic collisions, Numerische Mathematik, 102 (2006), 649-679. doi: 10.1007/s00211-005-0666-6.

[32]

J. J. Moreau, Décomposition orthogonale d'un espace hilbertien selon deux cones mutuellement polaires, C. R. Acad. Sci, Ser. I, 255 (1962), 238-240.

[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), 608-611.

[34]

J. J. Moreau, Unilateral contactand dry friction in finite freedom dynamics, in "Non Smooth Mechanics and Applications" (eds. J. J. Moreau and P.-D. Panagiotopoulos), CISM Courses and Lectures, (Springer-Verlag, Wien, New York), 302 (1988), 1-82.

[35]

J. J. Moreau, New computation methods in granular dynamics, Thornton, editor, Powder $&$Grains, Balkema Press, (1993), 227-232.

[36]

J. J. Moreau, Some numerical methods in multibody dynamics: Application to granular materials, Eur. J. Mech. A/Solids, 13 (1994), 93-114.

[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).

[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), 83-93. doi: 10.1002/cav.163.

[39]

L. Paoli, Time discretization of vibro-impact, Phil. Trans. R. Soc. A, 359 (2001), 2405-2428. doi: 10.1098/rsta.2001.0858.

[40]

S. Paris, "Characterisation of Levels of Services and Modelling of Flows of People Inside Exchange Areas," Ph. D thesis, Université de Rennes $1$, 2007.

[41]

S. Paris, J. Pettrï and S. Donikian, Pedestrian reactive navigation for crowd simulation: A predictive approach, Computer Graphics Forum, 26 (2007), 665-674. doi: 10.1111/j.1467-8659.2007.01090.x.

[42]

P. Pécol, S. Dal Pont, S. Erlicher and P. Argoul, Modelling crowd-structure interaction, Mécanique $&$ Industries, EDP Sciences, 11 (2010), 495-504.

[43]

P. Pécol, S. Dal Pont, S. Erlicher and P. Argoul, Discrete approaches for crowd movement modelling, European Journal of Computational Mechanics, 20 (2011), 189-206.

[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), 69-85.

[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, Université Paris Est, 2011, à paraître.

[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, Wroclaw, Poland, July 6-9, (2011).

[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), 99-108.

[48]

F. Radjai, M. Jean, J. J. Moreau and S. Roux, Force distributions in dense two-dimensional granular systems, Phys. Rev. Lett., 77 (1996), 264-277. doi: 10.1103/PhysRevLett.77.274.

[49]

F. Radjai and V. Richefeu, Mechanics of Materials, Contact Dynamics as a Nonsmooth Discrete Element Method, 41 (2009), 715-728.

[50]

S. Reicher, The St. Pauls riotan explanation of the limits of crowd action in terms of asocial identity model, EJSP , 14 (1984), 1-21.

[51]

M. Renouf, "Optimisationnumérique et Calcul Parallèle Pour L'étude des Milieux Divisés Bi- ettri Dimensionnels," Ph. D thesis, Université Montpellier II -Sciences et Techniques du Languedoc -, 2004.

[52]

C. Reynolds, Flocks, herds, and schools: A distributed behavioral model, Computer Graphics, 21 (1987), 25-34. doi: 10.1145/37402.37406.

[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), 2841-2859. doi: 10.1016/j.cma.2005.07.006.

[54]

J. C. Simo and T. J. R. Hughes, "Elastoplasticity and Viscoplasticity Computational Aspects," Springer, Berlin, 1996.

[55]

H. Singh, R. Arter, L. Dodd and J. Drury, Modelling subgroup behavior in crowd dynamics DEM simulation, Applied Mathematical Modelling, 33 (2009), 4408-4423. doi: 10.1016/j.apm.2009.03.020.

[56]

M. Sung, M. Gleicher and S. Chenney, Scalable behaviors for crowd simulation, Eurographics,23 (2004), 519-528.

[57]

K. Teknomo, Application of microscopic pedestrian simulation model, Transportation Research Part F, 9 (2006), 15-27.

[58]

J. Venel, "Modélisation Mathématique des Mouvements de Foule," Ph. D thesis, Laboratoirede Mathématiques, Université Paris XI, Orsay, France, 2008.

[59]

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

show all references

References:
[1]

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

[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, Specialistica in Ingegneria Civile, Universitàdegli studi di Salerno, facoltà di ingegneria, 2011.

[3]

V. Blue and J. Adler, Cellular automata microsimulation of bi-directional pedestrian flows, Journal of the Transportation Research Board, 1678 (2000), 135-141. doi: 10.3141/1678-17.

[4]

J. Bodgi, S. Erlicher and P. Argoul, Lateral vibration of footbridges under crowd - loading : Continuous crowd modelling approach, Key Engineering Materials, 347 (2007), 685-690. doi: 10.4028/www.scientific.net/KEM.347.685.

[5]

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

[6]

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

[7]

C. Cholet, "Chocs de Solides Rigides," Ph. D thesis, Université Paris VI, 1998.

[8]

P. A. Cundall, A computer model for simulating progressive large scale movements of blocky rock systems, in "Proc. of the Symposium of the International Society of Rock Mechanics," 1 (1971), 132-150.

[9]

P. A. Cundall and O. D. L. Strack, A discrete numerical model for granular assemblies, Geotechnique, 29 (1979), 47-65. doi: 10.1680/geot.1979.29.1.47.

[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), 6100-6114.

[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), 1-24.

[12]

E. Dimnet, "Mouvement et Collisions de Solides Rigides ou Déformables," Ph. D thesis, Ecole Nationale des Ponts et Chaussées, 2002.

[13]

C. Ericson, "Real Time Collision Detection," Morgan Haufmann Publishers, 2004.

[14]

M. Frémond, Rigid bodies collisions, Physics Letters A, 204 (1995), 33-41. doi: 10.1016/0375-9601(95)00418-3.

[15]

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

[16]

J. J. Fruin, Designing for pedestrians: A level of service concept, Highway Research Record, (1971), 1-15.

[17]

B. D. Hankin and R. A. Wright, Passenger flow in subways, Oper. Res., 9 (1958), 81-88.

[18]

D. Helbing and P. Molnàr, Social force model for pedestrian dynamics, Physical Review E, 51 (1995), 4282-4286. doi: 10.1103/PhysRevE.51.4282.

[19]

D. Helbing, I. Farkas and T. Vicsek, Simulating dynamic features of escape panic, Nature, 407 (2000), 487-490. doi: 10.1038/35035023.

[20]

D. Helbing, Traffic and related self-driven many-particle systems, Reviews of Modern Physics, 73 (2002), 1067-1141. doi: 10.1103/RevModPhys.73.1067.

[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.

[22]

D. Helbing, L. Buzna, A. Johansson and T. Werner, Self-organized pedestrian crowd dynamics: Experiments, simulations, and design solutions, Transportation Science, 39 (2005), 1-24.

[23]

L. F. Henderson, Thestatistics of crowd fluids, Nature, 229 (1971), 381-383. doi: 10.1038/229381a0.

[24]

S. P. Hoogendoorn, P. H. L. Bovy and W. Daamen, Microscopic pedestrian wayfinding and dynamics modelling, Pedestrian and Evacuation Dynamics, (2001), 123-154.

[25]

M. Jean and J. J. Moreau, Unilaterality and dry friction in the dynamics of rigid bodies collection, Contact Mechanics International Symposium, Presses Polytechniques et Universitaires Romanes, (1992), 31-48.

[26]

M. Jean, The non smooth contact dynamics method, Compt. Methods Appl. Math. Engrg., 177 (1999), 235-257. doi: 10.1016/S0045-7825(98)00383-1.

[27]

G. Keith Still, "Crowd Dynamics," Ph. D thesis, University of Warwick, Department of Mathematics, 2000.

[28]

R. Kimmel and J. A. Sethian, Fast marching methods for computing distance maps and shortest paths, Technical Report 669, CPAM, University of California, Berkeley, (1996).

[29]

Y. Kishino, Disk model analysisof granular media, Micromechanics of Granular Materials, (1988), 143-152.

[30]

H. Klüpfel, "A Cellular Automaton Model for Crowd Movement and Egress Simulation," Ph. D thesis,Universitat Duisburg - Essen, 2003.

[31]

B. Maury, A time-stepping scheme for inelastic collisions, Numerische Mathematik, 102 (2006), 649-679. doi: 10.1007/s00211-005-0666-6.

[32]

J. J. Moreau, Décomposition orthogonale d'un espace hilbertien selon deux cones mutuellement polaires, C. R. Acad. Sci, Ser. I, 255 (1962), 238-240.

[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), 608-611.

[34]

J. J. Moreau, Unilateral contactand dry friction in finite freedom dynamics, in "Non Smooth Mechanics and Applications" (eds. J. J. Moreau and P.-D. Panagiotopoulos), CISM Courses and Lectures, (Springer-Verlag, Wien, New York), 302 (1988), 1-82.

[35]

J. J. Moreau, New computation methods in granular dynamics, Thornton, editor, Powder $&$Grains, Balkema Press, (1993), 227-232.

[36]

J. J. Moreau, Some numerical methods in multibody dynamics: Application to granular materials, Eur. J. Mech. A/Solids, 13 (1994), 93-114.

[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).

[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), 83-93. doi: 10.1002/cav.163.

[39]

L. Paoli, Time discretization of vibro-impact, Phil. Trans. R. Soc. A, 359 (2001), 2405-2428. doi: 10.1098/rsta.2001.0858.

[40]

S. Paris, "Characterisation of Levels of Services and Modelling of Flows of People Inside Exchange Areas," Ph. D thesis, Université de Rennes $1$, 2007.

[41]

S. Paris, J. Pettrï and S. Donikian, Pedestrian reactive navigation for crowd simulation: A predictive approach, Computer Graphics Forum, 26 (2007), 665-674. doi: 10.1111/j.1467-8659.2007.01090.x.

[42]

P. Pécol, S. Dal Pont, S. Erlicher and P. Argoul, Modelling crowd-structure interaction, Mécanique $&$ Industries, EDP Sciences, 11 (2010), 495-504.

[43]

P. Pécol, S. Dal Pont, S. Erlicher and P. Argoul, Discrete approaches for crowd movement modelling, European Journal of Computational Mechanics, 20 (2011), 189-206.

[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), 69-85.

[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, Université Paris Est, 2011, à paraître.

[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, Wroclaw, Poland, July 6-9, (2011).

[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), 99-108.

[48]

F. Radjai, M. Jean, J. J. Moreau and S. Roux, Force distributions in dense two-dimensional granular systems, Phys. Rev. Lett., 77 (1996), 264-277. doi: 10.1103/PhysRevLett.77.274.

[49]

F. Radjai and V. Richefeu, Mechanics of Materials, Contact Dynamics as a Nonsmooth Discrete Element Method, 41 (2009), 715-728.

[50]

S. Reicher, The St. Pauls riotan explanation of the limits of crowd action in terms of asocial identity model, EJSP , 14 (1984), 1-21.

[51]

M. Renouf, "Optimisationnumérique et Calcul Parallèle Pour L'étude des Milieux Divisés Bi- ettri Dimensionnels," Ph. D thesis, Université Montpellier II -Sciences et Techniques du Languedoc -, 2004.

[52]

C. Reynolds, Flocks, herds, and schools: A distributed behavioral model, Computer Graphics, 21 (1987), 25-34. doi: 10.1145/37402.37406.

[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), 2841-2859. doi: 10.1016/j.cma.2005.07.006.

[54]

J. C. Simo and T. J. R. Hughes, "Elastoplasticity and Viscoplasticity Computational Aspects," Springer, Berlin, 1996.

[55]

H. Singh, R. Arter, L. Dodd and J. Drury, Modelling subgroup behavior in crowd dynamics DEM simulation, Applied Mathematical Modelling, 33 (2009), 4408-4423. doi: 10.1016/j.apm.2009.03.020.

[56]

M. Sung, M. Gleicher and S. Chenney, Scalable behaviors for crowd simulation, Eurographics,23 (2004), 519-528.

[57]

K. Teknomo, Application of microscopic pedestrian simulation model, Transportation Research Part F, 9 (2006), 15-27.

[58]

J. Venel, "Modélisation Mathématique des Mouvements de Foule," Ph. D thesis, Laboratoirede Mathématiques, Université Paris XI, Orsay, France, 2008.

[59]

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

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