# A Scalable Moving Boundary Treatment in the Lattice Boltzmann Method
## 3 The Proposed Solid Boundary Condition  
One of the advantages of solving flow problems through the Boltzmann equation is that it can easily solve complex solid boundary problems. For stationary boundary problems, the nonequilibrium state extrapolation format is used for processing when the boundary is straight; the unified boundary processing method proposed by [PERSON] et al. [15] in 2003 is used when the boundary is curved. The method considers the problem of moving boundaries and can be directly applied to boundary processing when the grid point type does not change. When the grid point type changes, this method does not work and needs to be modified. As shown in Figure 1a, a straight boundary is used instead of a curved boundary for ease of presentation. It should be noted that to clearly describe the problem of boundary movement, resulting in the loss of the new fluid point distribution function, the solid line here is designed as a straight line. The solid line can be any irregular curve approximated by an irregular polyline connected by a series of points. In Figure 1a, the solid black line is formed by connecting the red solid points, and the intersection of the broken line and the grid line is a hollow five-corner point. The relative distance between the intersection and the grid node can approximate the influence of the actual boundary. The relative distance \(\Delta=\left|x_{f}-x_{w}\right|/\left|x_{f}-x_{b}\right|\). The grid point types of different grid points according to the position of the boundary are judged. Various signs are given in Figure 1a. Figure 1b shows the eight discrete velocity directions in the D2G9 model. Figure 1: (**a**) Schematic diagram of the movement boundary; \(\odot\), fluid grid points; \(\bullet\), solid grid points; \(\bullet\), the points that make up the boundary; \(\
star\), the intersection points of the solid boundary and the intersecting grid; the solid black line is the solid boundary curve; 1-1, 2-2, 3-3, and 4-4 are fixed positions respectively; (**b**) Schematic diagram of discrete velocity of D2G9 model. Taking the point \(x_{f}\) in Figure 1a as an example, the distribution function of direction \(\overline{\alpha}=3\) cannot be obtained by migration in the unified boundary processing format [15], which is given by Formula (6) below. \[f_{\overline{\alpha}}(\mathbf{x}_{f})=\frac{\mathbf{1}}{\mathbf{1}+\mathbf{\Delta}}\bigg{[} \Delta f_{\overline{\alpha}}\big{(}\mathbf{x}_{f}\big{)}+\Delta f_{\mathbf{\alpha}} \big{(}\mathbf{x}_{f}\big{)}+(\mathbf{1}-\mathbf{\Delta})f_{\mathbf{\alpha}}\big{(}\mathbf{x}_{f} \big{)}+2\omega_{\mathbf{\alpha}}\mathbf{\rho}_{w}\frac{3}{c^{2}}\mathbf{e}_{\mathbf{\alpha}} \mathbf{u}_{w}\bigg{]} \tag{6}\]  
where \(\mathbf{e}_{\mathbf{\alpha}}\) is the discrete velocity and \(\mathbf{\omega}_{\mathbf{\alpha}}\) is the weight coefficient. The right side of the equation is the distribution function before migration after a collision. The unknown distribution function of the fluid boundary point can be solved directly by Formula (6) when the grid point type has not changed. When the type changes, for example, the boundary in Figure 1a moves from position 1-1 to position 4-4. For the point \(\mathbf{x}_{b}\), since \(f_{\overline{\alpha}}(\mathbf{x}_{b})\) in Formula (6) has not been calculated before the boundary movement, the specific value is not given. A simple processing method is to perform a second-order extrapolation of the solid boundary point distribution function along a specific discrete velocity direction before the type is changed [30]. The distribution function of the point \(\mathbf{x}_{b}\) is constructed by Formula (7). \[f_{\overline{\alpha}}(\mathbf{x}_{b})=3f_{\overline{\alpha}}(\mathbf{x}_{f})-3f_{ \overline{\alpha}}(\mathbf{x}_{f})+f_{\overline{\alpha}}\big{(}\mathbf{x}_{f\overline {\alpha}}+\mathbf{e}_{3}\big{)} \tag{7}\]  
This method maintains the continuity of the fluid before and after the exercise.