## 3 Methods  
### Preparation of Virtual Ground Using DEM  
#### 3.1.1 DEM Governing Equation  
The discrete element method (DEM) is a numerical analysis technique developed to simulate the motion and interaction of particles. The method was first proposed in the 1970s with the aim of solving problems in the field of rock mechanics. In the contemporary era, the scope of its application has expanded beyond civil engineering to encompass mechanical engineering, chemical engineering, biomedical engineering, and numerous other disciplines. In DEM, the ground or waste landfill is not regarded as a continuum comprising soil particles or waste, but the dynamics of each constituent is simulated by solving the equations of motion for each individual component. Governing equations are represented as  
\[m_{i}\frac{d^{2}\mathbf{x}_{i}}{dt^{2}}=\sum_{j}\big{(}\mathbf{F}_{n,ij}+\mathbf{F}_{t,ij} \big{)}+m_{i}\mathbf{g} \tag{1}\]  
\[l_{i}\frac{d^{2}\mathbf{\omega}_{i}}{dt^{2}}=\sum_{j}\mathbf{M}_{ij} \tag{2}\]  
where \(i\), \(j\) are the element number, \(m_{i}\) is the mass of the element \(i\) (kg), \(\mathbf{x}_{i}\) is the position vector of the element \(i\) (m), \(\mathbf{F}_{n,ij}\) is the external force in the direction of the normal of the contact surface between elements \(i\) and \(j\) (N) and \(\mathbf{F}_{t,ij}\) is the external force in the direction of the tangential of the contact surface between elements \(i\) and \(j\) (N); \(\mathbf{g}\) is the vector of gravitational acceleration (m/s\({}^{2}\)). \(l_{i}\) is the moment of inertia of the element \(i\) (kg\(\cdot\)m\({}^{2}\)), \(\mathbf{\omega}_{i}\) is the angular velocity of the element \(i\) (rad/s), \(\mathbf{M}_{ij}\) is the moment generated by contact with elements \(i\) and \(j\) (N\(\cdot\)m) (see Figure 1). The simulation will model the pore structure of the ground and waste landfill sites. The simulation will apply [PERSON]'s law and [PERSON]'s viscous law as the most basic models. The calculation of adhesion will be ignored. The external forces in the normal direction of the contact surface, the external forces in the contact surface direction, and the moments generated due to rotation are given by the following equations [20; 21; 22]. \[\mathbf{F}_{n,ij}=\frac{4}{3}E_{i}\beta_{n,ij}^{\frac{3}{2}}\sqrt{R_{ij}}-\sqrt{ \frac{20}{3}}\cdot\frac{\ln e}{\sqrt{\left(\ln e\right)^{2}+\pi^{2}}}\Big{(}m _{ij}E_{ij}\sqrt{R_{ij}\beta_{n,ij}}\Big{)}^{1/2}v_{n,ij} \tag{3}\]  
\[\mathbf{F}_{t,ij}=-8G_{ij}\delta_{t,ij}\sqrt{R_{ij}\delta_{n,ij}}-\sqrt{\frac{80} {3}}\cdot\frac{\ln e}{\sqrt{\left(\ln e\right)^{2}+\pi^{2}}}\Big{(}m_{ij}E_{ ij}\sqrt{R_{ij}\beta_{n,ij}}\Big{)}^{\frac{1}{2}}v_{t,ij}\leq\mu_{s,ij}\mathbf{F}_{n,ij} \tag{4}\]  
\[\mathbf{M}_{ij}=r_{i}\;\mathbf{n}_{ij}\times\mathbf{F}_{t,ij}-\mu_{r}r_{i}\big{|}\mathbf{F}_{n,ij}\big{|}\frac{\mathbf{\omega}_{r,ij}}{\big{|}\mathbf{\omega}_{r,ij}\big{|}} \tag{5}\]  
where \(r_{i}\) is the radius of the element \(i\) (m), \(e\) is the coefficient of restitution, \(\mu_{s,ij}\) is the coefficient of static friction, \(\mu_{r}\) is the coefficient of rotational friction, \(\mathbf{n}_{ij}\) is the unit vector  
Figure 1: Diagram explaining the forces and parameters acting between two particles. The normal force (\(F_{n}\)) and the tangential force (\(F_{t}\)) acting between particles are given by the sum of [PERSON]’s law, which is proportional to displacement (\(\delta\)), and [PERSON]’s law of viscosity, which is proportional to velocity (\(v\)). in the direction of the normal at the contact surface with the elements \(i\) and \(j\), \(\mathbf{\omega}_{r,ij}\) is the relative angular velocity of elements \(i\) and \(j\) (rad/s), \(\delta_{n,ij}\) and \(\delta_{t,ij}\) are the amount of overlap (m) of the elements due to contact between elements \(i\) and \(j\) in the direction of the normal or tangential to the contact surface, \(v_{n,ij}\) and \(v_{t,ij}\) are the normal or tangential component of the relative velocity of elements \(i\) and \(j\) (m/s).