# Prediction of the Dangerous Range of Dumps Based on a Dynamic Process
## 3 Introduction of the Calculation Method
### Introduction of the MPM Method  
As a recently developed representative meshless particle method, the MPM discretizes a continuum research object into a series of particles. Each particle represents a part of the object within the corresponding range and contains all the material information of the corresponding area, including stress, strain, quality, and speed. At the same time, a background grid is established for solving the equation of motion in the calculation area, as shown in Figure 6a,b. The basic governing equations of the MPM are the mass conservation equation,  
\[\frac{d\rho}{dt}+\rho\
abla\cdot\mathbf{v}=0, \tag{1}\]  
and the momentum conservation equation  
\[\rho\frac{dv}{dt}=\
abla\cdot\mathbf{\sigma}+\rho\mathbf{b}, \tag{2}\]  
Figure 6: (**a**) Material point method calculation process diagram; (**b**) material point discrete representation object. where \(\rho\) is the density at the current moment, \(\mathbf{v}=\mathbf{v}(x,t)\) represents the material speed, \(\mathbf{\sigma}\) represents the Cauchy stress tensor, and \(\mathbf{b}\) represents the physical force. The mass of the particle does not change during calculations with the MPM, and thus the mass conservation equation is naturally satisfied in the calculation process. The calculation process is mainly used to solve the momentum conservation equation. Since the information about the object in the MPM is carried by the particle and the solution of the motion of the object is solved by the Euler background grid, the mass, momentum, stress, body force, and surface force of the particle at time t should be mapped to the background grid node. After the background grid node completes the calculation, the calculated motion parameters are mapped back to the particle, and the velocity and position of the particle are updated. The specific implementation steps are as follows:  
1. The momentum and mass of each particle are mapped to the background grid node by an interpolation shape function to calculate the momentum and mass of the corresponding background grid node. \[m_{l}^{n}=\sum_{p=1}^{np}m_{p}N_{lp}^{n},\] (3) \[p_{il}^{n-1/2}=\sum_{p=1}^{np}m_{p}v_{ip}^{n-1/2}N_{lp}^{n},\] (4)  
Among them, the subscripts \(I\) and \(p\) represent the grid node variable and the material point variable, respectively, and \(n\) is the calculation step. The operator \(\sum\) represents the sum of all material points and \(N_{lp}\) is a shape function. 2. Essential boundary conditions are imposed on the nodes on the boundary. Let the nodes on the fixed boundary nodes make \(p_{il}^{n-1/2}=0\). 3. The node velocity \(v_{il}^{n-1/2}=p_{il}^{n-1/2}/m_{p}\) is calculated by the momentum \(p_{il}^{n-1/2}\) of each grid node. The strain increment \(\Delta\varepsilon_{ijp}^{n-1/2}\) and the spin increment \(\Delta\Omega_{ijp}^{n-1}\) of each particle are calculated according to the velocity, and the stress of each particle is updated by the constitutive relationship: \[\Delta\varepsilon_{ijp}^{n-1/2}=\Delta t\sum_{l=1}^{8}\frac{1}{2}(N_{lp,j}^{n -1/2}+N_{lp,i}^{n}v_{jl}^{n-1/2}),\] (5) \[\Delta\Omega_{ijp}^{n-1}=\Delta t\sum_{l=1}^{8}\frac{1}{2}(N_{lp,j}^{n}v_{ il}^{n-1/2}+N_{lp,i}^{n}v_{jl}^{n-1/2}),\] (6) Update the density of the particle: \[\rho_{p}^{n+1}=\rho_{p}^{n}/(1+\Delta\varepsilon_{ijp}^{n-1/2}),\] (7)
4. Calculate the background grid node force internal force \(f_{il}^{\text{int},n}\), node external force \(f_{il}^{\text{ext},n}\) and total node force \(f_{il}^{n}\): \[f_{il}^{\text{int},n}=\sum_{p=1}^{np}N_{lp,j}^{n}\sigma_{ijp}\frac{m_{p}}{\rho _{p}},\] (8) \[f_{il}^{\text{ext},n}=\sum_{p=1}^{np}N_{lp}^{n}v_{ip}^{n}+\sum_{p=1}^{np}N_{ lp}^{n}t_{ip}^{-n}h^{-1}\frac{m_{p}}{\rho_{p}},\] (9) \[\rho\frac{dv}{dt}=\
abla\cdot\mathbf{\sigma}+\rho\mathbf{b}, \tag{2}\]0 (10) If the node is fixed in the i direction, then let \(f_{il}^{n}=0\).