## 2 Methodology  
A brief overview of the MPM is presented before discussing the material point strength discounting method. The material point method originated from the Particle in Cell [6] method and was later extended to the field of solid mechanics by [PERSON] et al., namely the MPM. The method is based on a coupled particles and grid framework to describe the simulated object. Specifically, the material points play the role of Lagrangian particles carrying all the material information, the background grid plays the role of an Eulerian grid to solve the control equations, impose boundary conditions, and so on, and the deformed grid is discarded at the end of the time step [23]. This combination of background grid and material points to jointly describe the object does not encounter problems such as grid distortion when solving large deformation problems, making the MPM an ideal numerical algorithm for large deformation problems [37]. The material point strength reduction method is similar to the finite element strength reduction method. The reduced mechanical parameters, i.e., the cohesion and the angle of internal friction angle, are brought into the calculation program, and when the displacement of the slope changes abruptly, the reduction factor at this time is taken as the stability factor of the slope. In this paper, all formula notations satisfy the [PERSON] summation convention, with the subscripts \(I\) and \(p\) denoting the background grid nodes and material points, respectively. In the classical MPM, a continuous material is described as a series of discrete material points, as shown in Figure 1, and the density of the material can be approximated as follows:  
\[\rho(x_{i})=\sum_{p}m_{p}\delta(x_{i}-x_{ip}) \tag{1}\]  
where \(m_{p}\) denotes the mass of the matter point, \(\delta\) denotes the Dirichlet function, and \(x_{ip}\) and \(x_{i}\) denote the coordinates of the matter points and grid nodes, respectively. Based on the theory of continuous medium mechanics, the MPM uses a weak form of discretization of the equations of motion [25]. During the simulation, all material information is stored in the material point, which needs to be updated with the aid of the background grid, and thus the relationship between the material point and the background grid needs to be established during the Lagrangian phase [38]. In the MPM, this relationship is established by means of a shape function built on the background grid nodes. According to the choice of shape function, the MPM can be divided into the classical MMP (using piece-wise linear shape functions), the generalized interpolation material point method (GIMP) (using generalized interpolation shape functions) [28], the B-spline material point method (using B-spline shape functions) [39; 40] and so on. In this paper, the uGIMP shape function is employed to ensure that the calculation has sufficient stability and good computational efficiency. Assuming that there is a field function \(f(x)\) distributed over the domain, the value of the function on the grid node \(I\) can be expressed as:  
\[f_{I}=\int\limits_{\Omega}f(x)S_{I}d\Omega=\sum\limits_{p}f_{p}S_{lp}V_{p} \tag{2}\]  
where \(S_{I}\) denotes the generalized interpolation shape function, \(f_{p}\), \(S_{lp}\), \(V_{p}\) denote the values of the field function, shape function, and volume at the material point p, respectively. Due to the use of generalized interpolation shape functions, the cell-crossing error [27] of the MPM is significantly reduced, which is very beneficial for the stability of the numerical method. At the beginning of each calculation step of the MPM, the information on the material points will be mapped to the grid nodes. The mass, momentum, and velocity of the grid nodes can be represented as:  
\[m_{I}=\sum\limits_{p}m_{p}S_{lp} \tag{3}\]  
\[P_{li}=\sum\limits_{p}m_{p}v_{pi}S_{lp} \tag{4}\]  
\[v_{li}=\frac{P_{li}}{m_{I}} \tag{5}\]  
where, \(m_{p}\) and \(v_{pi}\) denote the mass and velocity of the material points, respectively.