Particle methods are used to solve the macroscopic equations. Model Hierarchy. Microscopic. (Xt,Vt). Mesoscopic ft(x, v). Macroscopic. (Ït,ut) mean-field.
Particle Method for Multi-group Pedestrian Flow N.K. Mahato, A. Klar and S. Tiwari AG Technomathematics, TU Kaiserslautern, Germany
Introduction
Numerical Experiment
We consider a multi-group microscopic model for pedestrian flow describing the behavior of large groups. It is based on an interacting particle system coupled to an eikonal equation. Hydrodynamic multi-group models are derived from the underlying particle system as well as scalar multi-group models. The eikonal equation is used to compute optimal paths for the pedestrians. Particle methods are used to solve the macroscopic equations.
Initial information:
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Model Hierarchy
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Microscopic (Xt, Vt)
mean-field
averaging
Mesoscopic ft(x, v)
Macroscopic (ρt, ut)
Multi-group Microscopic Model: We consider a microscopic social force model[2, 3] for pedestrian flow including an optimal path computation. For N (k) (k) 2 pedestrians divided into M groups with locations xi ∈ R , and velocity vi ∈ R2. Here, the index i = 1, . . . N is used to number all pedestrians, the index k = 1, . . . , M denotes the group to which the pedestrian belongs. S (k) denotes the set of all i which are in group k. The equations of motion are
Initially, pedestrians are concentrated in the left boundary and they can leave at either of the two exits of 15m width on the right boundary. We choose the parameters as reaction time T = 0.001, time step ∆t = 0.00042, free flow velocity vmax = 2m/s, and maximum density ρmax = 10ped/m2.
Numerical Simulation
(k)
dxi (k) = vi dt M X (k) X dvi (k) (l) (k) (k,l) (k) (k) (k) =− ∇xi U (| xi − xj |) + G (xi , vi , ρ(xi )) dt (l)
(1)
−r/lr
where Ca, Cr are attractive and + Cr e • Morse potential, U (r) = −Cae repulsive strengths and la, lr are their respective length scales. • Acceleration in the desired direction, 1 (k) G (x, v, ρ) = T (k)
• Walking speed, V (ρ ) = vmax(1 − 1 (k) • ρ (x) = N
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Fig. 2: Single(top) and multi-group(bottom) pedestrians.
Fig. 1: A railway platform with a square obstacle
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!
(k) ∇Φ (x) (k) −V (ρ ) −v . (k) k∇Φ (x)k
Fig. 3: Distribution of grid particles in hydrodynamic single group(top) with Ca = 0, Cr = 60, lr = 2 and multi-group(bottom) with Ca = 50, Cr = 60, la = 5, lr = 2 models at different times t = 21s, 42s, 59s (from left to right).
ρ(k) ρmax ).
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V (ρ(k)(x))k∇Φ(k)k − 1 = 0
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• Φ(k) is given by the solution of the eikonal equation
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Fig. 4: Density of pedestrians for single(left) and multi-group(right) hydrodynamic model at time t = 23s.
with boundary conditions Φ(k)(x) = 0, x ∈ ∂ΩD ,
Φ(k)(x) = ∞, x ∈ ∂Ω \ ∂ΩD .
Comparison of Models
Mean Field Equation: The mean field equation[1] for the distribution function f (k) = f (k)(t, x, v) with k groups is ∂tf (k) + v · ∇xf (k) + S (k)f (k) = 0
(2)
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= ∇v · G (x, v, ρ(x))f
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M X
∇v · (∇xU
(k,l)
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? ρ )f
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hydro scalar
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Multi-group Hydrodynamic Model: By integrating the kinetic equation(2) against dv and vdv and using a mono-kinetic distribution function f (k) ∼ ρ(k)(x)δu(k)(x)(v) to close the resulting balance equations, we get system of hydrodynamic equation as ∂tρ(k) + ∇x · (ρ(k)u(k)) = 0 M (3) X ∂tu(k) + (u(k) · ∇x)u(k) = G(k)(x, u(k)(x), ρ(k)(x)) − ∇xU (k,l) ? ρ(l)
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Fig. 5: The evacuation time is larger in the case of multi-group
Fig. 6: The results coincide in both multi-group hydrodynamic and
hydrodynamic model.
scalar models.
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Ca= 50 Ca= 70
Ca=70, Cm =90
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f (k)dv and momentum, ρ(k)u(k)(x) =
R2
vf (k)dv.
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Multi-group Scalar Model: By neglecting time change in the momentum equation of (3), we obtain the scalar equation for ρ(k) as " !# M (k) X ∇Φ (x) (k) (k) (k,l) (l) (k) ∂tρ + ∇x · ρ − T ∇x U ? ρ − V (ρ (x)) = 0 (4) (k) k∇Φ (x)k l=1
Finite Pointset Method FPM is a meshfree particle method in which one has to establish the neighbor lists for every particle in every time step, then 1. express the PDE in Lagrangian form. 2. approximate the spatial derivatives on the R.H.S. of every particle position from its neighbors, then we get a system of ODEs in time. 3. solve the ODE system with an ODE solver.
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• Density, ρ(k) =
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Fig. 7: Models with weak and strong reciprocal attraction shows that
Fig. 8: Models with center of mass attraction has a similar influence
the evacuation time increases with increasing attraction.
as increasing the reciprocal attraction.
References [1] N. K. Mahato, A. Klar, and S. Tiwari. “Modelling and simulation of macroscopic multi-group pedestrian flow”. In: Applied Mathematical Modelling (Submitted 2016). [2] R. Etikyala, S. Go¨ttlich, A. Klar, and S.Tiwari. “Particle methods for pedestrian flow models: From microscopic to nonlocal continuum models”. In: Mathematical Models and Methods in Applied Sciences 20.12 (2014), pp. 2503– 2523. [3] J. A. Carrillo, A. Klar, S. Martin, and S. Tiwari. “Self-propelled interacting particle systems with roosting force”. In: Mathematical Models and Methods in Applied Sciences 20.Suppl. (2010), pp. 1533–1552. LATEX Tik Zposter
