# Stability Prediction of Soil Slopes Based on Digital Twinning and Deep Learning
## 3 Methods
### Limit Equilibrium Methods
#### 3.1.2 Optimization Method for Non-Circular Slip Surface  
The slope stability analysis consists of the following two steps:  
1. The safety factor for a slip surface in the landslide was calculated by LEMs;
2. For all possible slip surfaces, the above step was repeated to find the minimum safety factor and the corresponding critical slip surface. The optimization method for the non-circular slip surface was as follows:  
\[F=(x_{1},y_{1},x_{2},y_{2},\cdots x_{n},y_{n}) \tag{7}\]  
where \(F\) is the safety factor of the non-circular slip surface calculated by LEMs, \(n\) is the number of control points of the non-circular slip surface, and \(x_{i},y_{i}\) are the coordinates of the \(i\)-th control point of the non-circular slip surface. Since n control points correspond to 2n degrees of freedom, the non-circular slip surface optimization problem contains many degrees of freedom, and there may be many local extremum points in its independent variable space, making the task of finding the global extremum very difficult. If the slope section is complex, there may be many local extrema of the safety factor. To solve this problem, a simple and effective idea was to attempt an initial value close to the global extremum. Obviously, the closer the initial value was to the global extremum, the less likely it was to miss the global extremum. As shown in Equation (8), when searching for the minimum safety factor, the optimization of the circular slip surface involves only 3 degrees of freedom. Therefore, in this paper, the minimum safety factor of the circular sliding surface corresponding to the most dangerous sliding surface was calculated first, and the non-circular sliding surface was optimized based on it. \[F=f(x_{0},y_{0},R) \tag{8}\]  
where \((x_{0},y_{0})\) are the coordinates of the center of the circle and \(R\) is the radius. In this paper, the fmincon function of the Matlab R2021a Optimization Toolbox was used to optimize the safety factor of a non-circular sliding surface with a slope. The specific steps are as follows:  
(1) As shown in Figure 10, the enumeration method was used to continuously change the center coordinates \((x_{0},y_{0})\) and radius, and the Spencer method was used to calculate the safety factor of the circular slip surface of the slope. The minimum safety factor and the corresponding most dangerous slip surface were obtained by comparing the safety factors one by one. (2) As shown in Figure 11, according to the number of control points on the non-circular slip surface, the coordinates of discrete control points were uniformly selected on the most dangerous circular slip surface, and the line segments connecting these discrete points were taken as the initial slip surface in the optimization of the non-circular slip surface. The slip surface was approximately composed of \((n-1)\) line segments connecting n control points \(\{(x_{1},y_{1})\), \((x_{2},y_{2})\),... \((x_{n},y_{n})\}\). Points A and B are the upper and lower intersections of the slip surface and the slope, respectively; their coordinates and the horizontal coordinates of the intermediate control points are fixed, and the search variable of the non-circular slip surface composed of n control points was \(S=\{y_{2},\ y_{3},\ldots y_{n-1}\}\). A reasonable slip surface is generally concave upward, which means that the slip surface usually needs to satisfy the following constraints [29]:  
\[\tan(85*\pi/180)> \frac{y_{2}-y_{1}}{x_{2}-x_{1}}>\frac{y_{3}-y_{2}}{x_{3}-x_{2}}> \ldots> \frac{y_{n}-y_{n-1}}{x_{n}-x_{n-1}}>-\tan(\pi/6) \tag{9}\]  
In this paper, the safety factor of 3000 slope samples was calculated according to the above method. Figure 12 shows the optimization result of the non-circular slip surface of 2 slope samples, where the blue is the circular slip surface, the green is the initial non-circular slip surface, and the red is the optimal non-circular slip surface. Figure 11: Optimization of non-circular slip surface. Figure 10: Enumeration method: arrange slip arcs according to different center positions and radii.