# American Institute of Mathematical Sciences

December  2019, 12(6): 1273-1296. doi: 10.3934/krm.2019049

## A kinetic theory approach to model pedestrian dynamics in bounded domains with obstacles

 Department of Mathematics, University of Houston, 3551 Cullen Blvd, Houston, TX 77204, USA

Received  January 2019 Revised  May 2019 Published  September 2019

Fund Project: This work was partially supported by NSF grant DMS-1620384.

We consider a kinetic theory approach to model the evacuation of a crowd from bounded domains. The interactions of a person with other pedestrians and the environment, which includes walls, exits, and obstacles, are modeled by using tools of game theory and are transferred to the crowd dynamics. The model allows to weight between two competing behaviors: the search for less congested areas and the tendency to follow the stream unconsciously in a panic situation. For the numerical approximation of the solution to our model, we apply an operator splitting scheme which breaks the problem into two pure advection problems and a problem involving the interactions. We compare our numerical results against the data reported in a recent empirical study on evacuation from a room with two exits. For medium and medium-to-large groups of people we achieve good agreement between the computed average people density and flow rate and the respective measured quantities. Through a series of numerical tests we also show that our approach is capable of handling evacuation from a room with one or more exits with variable size, with and without obstacles, and can reproduce lane formation in bidirectional flow in a corridor.

Citation: Daewa Kim, Annalisa Quaini. A kinetic theory approach to model pedestrian dynamics in bounded domains with obstacles. Kinetic & Related Models, 2019, 12 (6) : 1273-1296. doi: 10.3934/krm.2019049
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##### References:
(A) Dependence of the dimensionless velocity modulus $v$ on the dimensionless density $\rho$ for different values of the parameter $\alpha$, which represents the quality of the environment. (B) Sketch of computational domain $\Omega$ with exit $E$ and a pedestrian located at ${\boldsymbol{x}}$, moving with direction $\theta_h$. The pedestrian should choose direction ${\boldsymbol u}_E$ to reach the exit, while direction ${\boldsymbol u}_W$ is to avoid collision with the wall. The distances form the exit and from the wall are $d_E$ and $d_W$, respectively
Definition of ${\boldsymbol u}_W$ and ${\boldsymbol u}_E$ with respect to the effective area
Evacuation process of 46 pedestrians grouped into two clusters with opposite initial directions $\theta_3$ and $\theta_7$ using the medium mesh and $\Delta t_{medium}$ for time $t = 0, 1.5, 3, 6, 10.5, 13.5$ s. The color refers to density
(A) Number of pedestrians inside the room over time computed with six different combinations of mesh and time step. For ease of comparison, (B) shows only the curves in (A) obtained with simultaneous refinements of mesh and time step
Computed evacuation time from the room with one exit versus the exit size: (A) our results and (B) results from [1]
Computational domain corresponding to the experimental set-up in [40] and initial density and direction (i.e., $\theta_1$) for the experiment with 138 pedestrians
Computed (A) mean density $D_{V}$ and (B) mean flow rate $F_{V}$ as defined in (15), and measured (C) mean density and (D) mean flow rate from [40]
(A) Different velocity moduli under consideration and (B) corresponding number of pedestrians in the room versus time for the 138 pedestrian case
Density (top) and velocity magnitude with selected velocity vectors (bottom) for the evacuation process of 138 pedestrians with the purple (left), orange (middle), and blue (right) velocity moduli at time $t = 15$ s
(A) Widths of exit 2 under consideration and corresponding width ratios and (B) evacuation time versus width ratios for two different scenarios
Computational domain with effective area for (A) an obstacle placed in the middle of the room, towards the exit, and (B) two obstacles placed symmetrically with respect to the exit
Configuration 1 with $\alpha = 1$ in the effective area: computed density for $t$ = 0, 3, 6, 7.5, 9, 13.5 s. The small square within the effective area represents the real obstacle
Configuration 1 with $\alpha = 0$ in the effective area: computed density for $t$ = 0, 3, 6, 7.5, 9, 13.5 s. The small rectangle within the effective area represents the real obstacle
Configuration 2 with $\alpha = 1$ in the effective area: computed density for $t$ = 0, 3, 6, 7.5, 10.5, 13.5 s. The small square within the effective area represents the real obstacle
Configuration 2 with $\alpha = 0$ in the effective area: computed density for $t$ = 0, 3, 6, 7.5, 10.5, 13.5 s s. The small rectangle within the effective area represents the real obstacle
Evacuation times for the room with no obstacles ($\alpha = 1$ everywhere in the domain), for room with one and two obstacles with $\alpha = 1$ and $\alpha = 0$ in the effective area
The movement process of 98 pedestrians grouped into four clusters with opposite initial direction $\theta_1$ and $\theta_5$ in the periodic corridor for $t = 0, 4.2, 12.3, 19.8, 33.9, 50.7, 72.3$ s, respectively
The movement process of 188 pedestrians grouped into four clusters with initial opposite direction $\theta_1$ and $\theta_5$ in the periodic corridor for $t = 0, 4.5, 12.6, 19.8, 33.9, 50.7, 89.7$ s, respectively
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