## 4 Boundary Treatments
### Additive Particles Combined with Dummy Particles  
There are two types of particles applied on the boundary, including normal boundary particles and additive boundary particles. The solid boundary is modeled with normal particles spaced the same as inner fluid particles to ensure an initial uniform particle distribution. Then, the gap between two boundary particles is filled by additive particles, as shown in Figure 3. As a result, the boundary is actually modeled with denser particles geometrically, while the additive particles are not involved in the computation of neighboring summations, and the uniform particle distribution is thus not affected. Additive particles are only used to help create enough dummy particles for the near-wall boundary particles, as shown in Figure 4. On the contrary, dummy particles do not really exist, so they cannot be visually seen and no actual space is taken up. Only dummy particles within the kernel domain will be identified and taken into account in the computation. For fluid particle \(i,j_{1}\) is a normal boundary particle and \(j_{2}\) is an additive particle. These two particles are inside the kernel domain of particle \(i\) with distances as \(\mathrm{r}_{1}\) and \(\mathrm{r}_{2}\), respectively.  
The distance between two particles is calculated as the following in 2D cases:  
\[\mathrm{r}=\sqrt{(x_{i}-x_{j})^{2}+(y_{i}-y_{j})^{2}},\mathrm{d}x=x_{i}-x_{j},\mathrm{d}y=y_{i}-y_{j}\]  
For normal boundary particle \(j_{1}\), in addition to its contribution to the kernel summation of particle \(i\), a dummy particle \(j_{1}\)\({}^{\prime}\) with twice the distance \(2\mathrm{r}\) (\(\mathrm{d}x^{\prime}=2\mathrm{d}x,\mathrm{d}y^{\prime}=2\mathrm{d}y\)) away from \(i\), as shown in Figure 4, will be checked if it is inside of the kernel domain. If this dummy particle is inside the kernel domain, it will be treated as neighboring particles contributing to the calculations of all the physical properties of particle \(i\). Then, the next dummy particle with three times the distance \(3\mathrm{r}\) (\(\mathrm{d}x^{\prime}=3\mathrm{d}x,\mathrm{d}y^{\prime}=3\mathrm{d}y\)) away from particle  
Figure 4: Dummy particles.  
Figure 3: Sketch of two types of particles on the boundary.  
\(i\) will be checked and the same process will be carried on for other particles on the boundary within the smoothing domain of particle \(i\).  
Additive boundary particles such as particle \(j_{2}\) are in the list of neighboring particles but not involved in the computation directly. Similar to the normal boundary particles, a dummy particle \(j_{2}\)' with twice the distance away from particle \(i\) is checked first. It will contribute to the kernel summation on the condition of being within the kernel domain of \(i\). Then, dummy particles with three times or even four times distance away from particle \(i\) will be checked. Similar processes will be carried out to identify all the dummy particles of the additive boundary particles within the kernel domain and to include their contribution in the computation.  
The number of the additive particles can be decided practically. Denser additive particles will create more dummy particles. The pressure of a dummy particle is obtained by smoothing over the neighboring particles as  
\[\mathrm{P_{j}}=\frac{\sum\limits_{i}p_{i}W_{ij}}{\sum\limits_{i}W_{ij}} \tag{24}\]  
Dummy particles have the same density and mass as fluid particles. Velocities are assigned to be the same as those of the normal boundary particles, which should be ensured specifically to satisfy the fixed no-slip or slip boundary conditions, or the moving flexible boundary condition, according to the case in consideration.