## 4 Immersed Moving Boundary for LBM-DEM Coupling  
Different from MMB and IB, in IMB, distribution function in the non-conforming boundary lattice is divided into two parts. The first part undergoes non-equilibrium bounce back while the second part collides like normal fluid lattice. The ratio of two partsdepends on solid fraction of interface lattice. In Figure 3, cells are divided into three types, and the curve is the particle boundary interface. Rectangle symbol lattice is particle cell, sphere symbol lattice is fluid cell, and pentagon symbol lattice is interface cell. The shaded section represents the fluid part of interface cell. Solid fraction of interface lattice cell is \(\varepsilon_{s}\), which varies from 0 to 1. This paper proposes a new \(\varepsilon_{s}\) calculation algorithm for single particle, which will be shown in Section 5. In multi particle collision simulation, solid fraction is calculated by summing all of the overlap area of every particle. For numerical stability, \(\varepsilon_{s}\) will be replaced with weight function \(B_{s}\) in Equation (19), in which the two parameters have the same physical meaning. The relationship between \(B_{s}\) and \(\varepsilon_{s}\) is shown in Figure 4, which shows that \(B_{s}\) changes more slowly than \(\varepsilon_{s}\) when the fluid cell's flag changes, providing a smoother transition of coupling force and a reduction of force fluctuation. \[B_{s}(\varepsilon_{s},\tau)=\frac{\varepsilon_{s}\left(\frac{\tau}{\Delta t}- \frac{1}{2}\right)}{\left(1-\varepsilon_{s}\right)+\left(\frac{\tau}{\Delta t} -\frac{1}{2}\right)} \tag{19}\]  
After streaming, interface lattice obtains a new set of \(f_{i}\) from its fluid neighbor lattice as the interface lattice is mixed components including fluid and solid. With the different solid fraction, the corresponding solid part distribution function \(B_{s}f_{i}\) will bounce back  
Figure 4: Polynomial \(B_{s}\) depends on \(\varepsilon_{s}\). Figure 3: Cell type of immersed moving boundary (IMB) method. directly, which satisfies the non-equilibrium bounce back rule. If the boundary velocity is \(\mathrm{u}_{s}\), and the boundary fluid velocity is \(\mathrm{u}\), define \(f_{i}^{eq}(\rho,\mathrm{u}_{s})\) and \(f_{i}^{eq}(\rho,\mathrm{u})\) as the boundary lattice equilibrium distribution function of solid and fluid parts. The non-equilibrium part of distribution function is \(f_{-i}(x,t)\ -f_{-i}^{eq}(\rho,\mathrm{u})\), which will bounce back to \(i\) direction. Considering the value change from the origin value \(f_{i}(\mathrm{x},t)\), the non-equilibrium bounce back expression is given by Equation (20). \[B_{n}f_{i}(\mathrm{x}+\mathrm{e}_{i}\Delta t,t+\Delta t)=B_{n}f_{i}(\mathrm{x },t)+B_{n}\Big{[}f_{-i}(\mathrm{x},t)-f_{i}(\mathrm{x},t)+f_{i}^{eq}(\rho, \mathrm{u}_{s})-f_{-i}^{eq}(\rho,\mathrm{u})\Big{]} \tag{20}\]  
The fluid parts distribution function \((1-B_{s})f_{i}\), which undergoes BGK collision, is shown in Equation (21). \[(1-B_{n})f_{i}(\mathrm{x}+\mathrm{e}_{i}\Delta t,t+\Delta t)=(1-B_{n})\Big{[} f_{i}(\mathrm{x},t)-\tfrac{\Delta t}{\tau}\Big{(}f_{i}(\mathrm{x},t)-f_{i}^{(0)}( \mathrm{x},t)\Big{)}\Big{]} \tag{21}\]  
Combining Equations (20) and (21), an additional term is added to collision operator, and the mix collision expression of interface lattice is given in Equation (22). \[f_{i}(\mathrm{x}+\mathrm{e}_{i}\Delta t,t+\Delta t)=f_{i}(\mathrm{x},t)-(1-B_{ n})\left[\frac{\Delta t}{\tau}\Big{(}f_{i}(\mathrm{x},t)-f_{i}^{(0)}(\mathrm{x},t) \Big{)}\right]+B_{n}\Omega_{i}^{s} \tag{22}\]  
\[\Omega_{i}^{s}=f_{-i}(\mathrm{x},t)-f_{i}(\mathrm{x},t)+f_{i}^{eq}(\rho, \mathrm{u}_{s})-f_{-i}^{eq}(\rho,\mathrm{u}) \tag{23}\]  
As \(\sum_{i}\tfrac{\Delta t}{\tau}\Big{(}f_{i}(\mathrm{x},t)-f_{i}^{eq}(\mathrm{ x},t)\Big{)}\mathrm{e}_{i}=0\), conventional collision operator has no effect on momentum change. New term \(\Omega_{i}^{s}\) introduces the momentum change of fluid in the interface lattice cell.