Macroscopic models Numerical tests Rigorous results Pedestrian Dynamics: Modeling, Validation and Calibration Macroscopic modeling and simulation of crowd dynamics Paola Goatin Inria Sophia Antipolis - Méditerranée paola.goatin@inria.fr ICERM, Brown University, August 21-25, 2017
Macroscopic models Numerical tests Rigorous results Outline of the talk Macroscopic models 1 Numerical tests 2 Some rigorous results 3 P. Goatin (Inria) Macroscopic models August 21-25, 2017 2 / 42
Macroscopic models Numerical tests Rigorous results Outline of the talk Macroscopic models 1 Numerical tests 2 Some rigorous results 3 P. Goatin (Inria) Macroscopic models August 21-25, 2017 3 / 42
Macroscopic models Numerical tests Rigorous results Mathematical modeling of pedestrian motion: frameworks Microscopic individual agents ODEs system many parameters low and high densities comp. cost ∼ ped. number. Macroscopic continuous fluid PDEs few parameters very high densities analytical theory comp. cost ∼ domain size P. Goatin (Inria) Macroscopic models August 21-25, 2017 4 / 42
Macroscopic models Numerical tests Rigorous results Macroscopic models Mass conservation Pedestrians as "thinking fluid" 1 � Averaged quantities: ∂ t ρ + div x ( ρ� v ) = 0 ρ ( t, x ) pedestrians density ρ (0 , x ) = ρ 0 ( x ) v ( t, x ) mean velocity � for x ∈ Ω ⊂ R 2 , t > 0 1 R.L. Hughes, Transp. Res. B, 2002 P. Goatin (Inria) Macroscopic models August 21-25, 2017 5 / 42
Macroscopic models Numerical tests Rigorous results Macroscopic models Mass conservation Pedestrians as "thinking fluid" 1 � Averaged quantities: ∂ t ρ + div x ( ρ� v ) = 0 ρ ( t, x ) pedestrians density ρ (0 , x ) = ρ 0 ( x ) v ( t, x ) mean velocity � for x ∈ Ω ⊂ R 2 , t > 0 Two classes 1st order models: velocity given by a phenomenological speed-density relation � v = V ( ρ ) � ν 2nd order models: velocity given by a momentum balance equation 1 R.L. Hughes, Transp. Res. B, 2002 P. Goatin (Inria) Macroscopic models August 21-25, 2017 5 / 42
Macroscopic models Numerical tests Rigorous results Macroscopic models Mass conservation Pedestrians as "thinking fluid" 1 � Averaged quantities: ∂ t ρ + div x ( ρ� v ) = 0 ρ ( t, x ) pedestrians density ρ (0 , x ) = ρ 0 ( x ) v ( t, x ) mean velocity � for x ∈ Ω ⊂ R 2 , t > 0 Two classes 1st order models: velocity given by a phenomenological speed-density relation � v = V ( ρ ) � ν 2nd order models: velocity given by a momentum balance equation Density must stay non-negative and bounded: 0 ≤ ρ ( t, x ) ≤ ρ max Different from fluid dynamics: preferred direction no conservation of momentum / energy n ≪ 6 · 10 23 1 R.L. Hughes, Transp. Res. B, 2002 P. Goatin (Inria) Macroscopic models August 21-25, 2017 5 / 42
Macroscopic models Numerical tests Rigorous results Continuum hypothesis n ≪ 6 · 10 23 but ... Brown University, Main Green, 08.21.2017 P. Goatin (Inria) Macroscopic models August 21-25, 2017 6 / 42
Macroscopic models Numerical tests Rigorous results Speed-density relation Speed function V ( ρ ) : decreasing function wrt density V (0) = v max free flow V ( ρ max ) ≃ 0 congestion Examples: speed V ( ρ ) flux ρV ( ρ ) P. Goatin (Inria) Macroscopic models August 21-25, 2017 7 / 42
Macroscopic models Numerical tests Rigorous results Desired direction of motion � µ Pedestrians: ν = − ∇ x φ seek the shortest route to destination � |∇ x φ | try to avoid high density regions P. Goatin (Inria) Macroscopic models August 21-25, 2017 8 / 42
Macroscopic models Numerical tests Rigorous results Desired direction of motion � µ Pedestrians: ν = − ∇ x φ seek the shortest route to destination � |∇ x φ | try to avoid high density regions The potential φ : Ω → R is given by the Eikonal equation � |∇ x φ | = C ( t, x , ρ ) in Ω φ ( t, x ) = 0 for x ∈ Γ outflow where C = C ( t, x , ρ ) ≥ 0 is the running cost = ⇒ the solution φ ( t, x ) represents the weighted distance of the position x from the target Γ outflow P. Goatin (Inria) Macroscopic models August 21-25, 2017 8 / 42
Macroscopic models Numerical tests Rigorous results Eikonal equation: level set curves for |∇ x φ | = 1 In an empty space: potential is proportional to distance to destination P. Goatin (Inria) Macroscopic models August 21-25, 2017 9 / 42
Macroscopic models Numerical tests Rigorous results The fastest route ... ... needs not to be the shortest! P. Goatin (Inria) Macroscopic models August 21-25, 2017 10 / 42
Macroscopic models Numerical tests Rigorous results First order models Hughes’ model 1 ν = − ∇ x φ 1 � s.t. |∇ x φ | = |∇ x φ | V ( ρ ) minimize travel time avoiding high densities CRITICISM: instantaneous global information on entire domain 1 R.L. Hughes, Transp. Res. B, 2002 2 Y. Xia, S.C. Wong and C.-W. Shu, Physical Review E, 2009 3 R.M. Colombo, Garavello and M. Lécureux-Mercier, M3AS, 2012 P. Goatin (Inria) Macroscopic models August 21-25, 2017 11 / 42
Macroscopic models Numerical tests Rigorous results First order models Hughes’ model 1 ν = − ∇ x φ 1 � s.t. |∇ x φ | = |∇ x φ | V ( ρ ) minimize travel time avoiding high densities CRITICISM: instantaneous global information on entire domain Dynamic model with memory effect 2 ν = − ∇ x ( φ + ωD ) 1 1 v ( ρ ) + βρ 2 � s.t. |∇ x φ | = v max , D ( ρ ) = discomfort |∇ x ( φ + ωD ) | minimize travel time based on knowledge of the walking domain temper the behavior locally to avoid high densities 1 R.L. Hughes, Transp. Res. B, 2002 2 Y. Xia, S.C. Wong and C.-W. Shu, Physical Review E, 2009 3 R.M. Colombo, Garavello and M. Lécureux-Mercier, M3AS, 2012 P. Goatin (Inria) Macroscopic models August 21-25, 2017 11 / 42
Macroscopic models Numerical tests Rigorous results First order models Hughes’ model 1 ν = − ∇ x φ 1 � s.t. |∇ x φ | = |∇ x φ | V ( ρ ) minimize travel time avoiding high densities CRITICISM: instantaneous global information on entire domain Dynamic model with memory effect 2 ν = − ∇ x ( φ + ωD ) 1 1 v ( ρ ) + βρ 2 � s.t. |∇ x φ | = v max , D ( ρ ) = discomfort |∇ x ( φ + ωD ) | minimize travel time based on knowledge of the walking domain temper the behavior locally to avoid high densities Non-local flow: 3 ∇ ( ρ ∗ η ) ν = − ∇ x φ � v = V ( ρ ) � ν − ε with � s.t. |∇ x φ | = 1 � |∇ x φ | 1 + |∇ ( ρ ∗ η ) | 2 1 R.L. Hughes, Transp. Res. B, 2002 2 Y. Xia, S.C. Wong and C.-W. Shu, Physical Review E, 2009 3 R.M. Colombo, Garavello and M. Lécureux-Mercier, M3AS, 2012 P. Goatin (Inria) Macroscopic models August 21-25, 2017 11 / 42
Macroscopic models Numerical tests Rigorous results Second order model Momentum balance equation 45 v ) + ∇ x P ( ρ ) = ρV ( ρ ) � ν − � v ∂ t ( ρ� v ) + div x ( ρ� v ⊗ � τ where � 2 � ρ − α V ( ρ ) = v max e ρ max |∇ x φ | = 1 /V ( ρ ) P ( ρ ) = p 0 ρ γ , p 0 > 0 , γ > 1 internal pressure τ response time 4 Payne-Whitham, 1971 5 Y.Q. Jiang, P. Zhang, S.C. Wong and R.X. Liu, Physica A, 2010 P. Goatin (Inria) Macroscopic models August 21-25, 2017 12 / 42
Macroscopic models Numerical tests Rigorous results Question Can macroscopic models reproduce characteristic features of crowd behavior? P. Goatin (Inria) Macroscopic models August 21-25, 2017 13 / 42
Macroscopic models Numerical tests Rigorous results Outline of the talk Macroscopic models 1 Numerical tests 2 Some rigorous results 3 P. Goatin (Inria) Macroscopic models August 21-25, 2017 14 / 42
Macroscopic models Numerical tests Rigorous results Numerical schemes used Space meshes: unstructured triangular / cartesian Eikonal equation: linear, finite element solver 6 / fast-sweeping First order models: Lax-Friedrichs Second order models: explicit time integration with advection-reaction splitting (HLL scheme) Non-local models: dimensional splitting Lax-Friedrichs 6 [Bornemann-Rasch, 2006] P. Goatin (Inria) Macroscopic models August 21-25, 2017 15 / 42
Macroscopic models Numerical tests Rigorous results Corridor evacuation with two exits Configuration at t = 0 Parameters choice: ρ 0 = 3 ped /m 2 initial density ρ max = 10 ped /m 2 maximal density v max = 2 m/s desired speed τ = 0 . 61 s relaxation time p 0 = 0 . 005 ped 1 − γ m 2+ γ /s 2 pressure coefficient γ = 2 adiabatic exponent α = 7 . 5 density-speed coefficient ε = 0 . 8 correction coefficient η = [1 − ( x/r ) 2 ] 3 [1 − ( y/r ) 2 ] 3 convolution kernel, with r = 15 m P. Goatin (Inria) Macroscopic models August 21-25, 2017 16 / 42
Macroscopic models Numerical tests Rigorous results Corridor evacuation with two exits t = 20 s |∇ x φ | = 1 ∇ x ( φ + ωD ) |∇ x φ | = 1 /v ( ρ ) second order non-local [Twarogowska-Duvigneau-Goatin, Mimault-Goatin] P. Goatin (Inria) Macroscopic models August 21-25, 2017 17 / 42
Macroscopic models Numerical tests Rigorous results Corridor evacuation with two exits t = 40 s |∇ x φ | = 1 ∇ x ( φ + ωD ) |∇ x φ | = 1 /v ( ρ ) second order non-local [Twarogowska-Duvigneau-Goatin, Mimault-Goatin] P. Goatin (Inria) Macroscopic models August 21-25, 2017 17 / 42
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