# A Study on Crowd Evacuation Model Considering Squeezing Equilibrium in Crowded Areas
## 2 Pedestrian Flow Model  
The traditional social force model is composed of three forces, namely self-driving force, acting force between pedestrians and acting force between pedestrians and obstacles. Self-driving force is as follows:  
\[F_{a}(v_{a},v_{0})=\frac{m_{a}[v_{0}e_{a}(t)-v_{a}e_{a}(t)]}{\tau} \tag{1}\]  
where \(\tau\) is the response time (s) of an individual, \(e_{a}(t)\) is the direction vector representing a pedestrian at time \(t\), \(v_{a}\) is the actual speed of a pedestrian; \(v_{0}\) is the expected speed of a pedestrian.  
The repulsive force between pedestrians is a form of converting the psychological action of a pedestrian into a force. Specific expression is as follows:  
\[F_{ab}(t)=A_{a}e^{\frac{R_{ab}-d_{ab}(t)}{n_{a}}}n_{ab}(t)+Kg[R_{ab}-d_{ab}(t )]n_{ab}(t)+kg[R_{ab}-d_{ab}]\Delta v_{ba}(t)t_{ab}(t) \tag{2}\]  
where \(\Delta v_{ba}(t)\) is the relative velocity (m/s) in the tangential direction, \(t_{ab}\) is a vector in the tangent direction, \(Kg\) is the coefficient (N/m) of human body flexibility, \(kg\) is the coefficient of sliding friction (N/m), \(R_{ab}\) is the sum of the radii (m) of individual \(a\) and individual \(b\), \(d_{ab}(t)\) is the centroid distance (m) between individual \(a\) and individual \(b\), \(A_{a}\) is the strength of social force between pedestrians (KN), \(B_{a}\) is the coefficient of action range of pedestrians (m), and \(n_{ab}(t)\) is a direction vector from \(b\) to \(a\).  
The repulsive force between pedestrians and obstacles is also a form of converting the psychological action of a pedestrian into a force. Specific expression is as follows:  
\[\sum_{w}F_{uw}(e_{at}r_{a}-r_{w})=A_{a}e^{\frac{R_{at}-d_{aw}(t)}{R_{a}}}n_{aw}(t) +Kg[R_{at}-d_{aw}(t)]n_{aw}(t)+kg[R_{at}-d_{aw}(t)]\Delta v_{wa}(t)t_{aw}(t) \tag{3}\]  
where \(R_{at}\) is the radius of pedestrians (the particle radius of the nearest obstacle is ignored) (m), \(d_{aw}(t)\) is the distance between the individual a and the nearest particle on the obstacle (m), \(\Delta v_{wa}(t)\) is the relative velocity (m/s) in the tangential direction, \(t_{aw}\) is a vector in the tangent direction, and \(n_{aw}(t)\) is the direction vector of the nearest particle \(w\) on the obstacle towards \(a\).  
The traditional social force model has a better evacuation effect in a simple and less dense scene, but the state of stagnation often occurs in a complex and dense scene. As for the evacuation of pedestrians in a complex scene, the navigation algorithm related to floor field values [41; 42] may be chosen to act on the driving force direction of pedestrians, so that pedestrians can successfully evacuate to the exit. However, there are also acting forces between pedestrians and between pedestrians and obstacles, which are particularly important for solving the stagnation problem of a traditional social force model and ensuring the effective evacuation effect. Therefore, the acting forces between pedestrians and between pedestrians and obstacles in the traditional social force model are improved to solve the phenomenon that the evacuation disorder of pedestrians is caused by excessive crowd density in the evacuation process.