# Boundary SPH for Robust Particle-Mesh Interaction in Three Dimensions
## 3 Results
### Concave Mesh and Fluid Flow Setups  
A key value of our BSPH method is sourced in its ability to handle concave boundary manifolds. To demonstrate this capability for BSPH, we established three fluid tests with three different types of mesh objects: a convex sphere, a concave elongated torus, and a concave bowl. Particles were parameterized with a support kernel radius of 0.2 m and initialized in a 125 \(\times\) 25 \(\times\) 5 grid arrangement at the start of the simulation, totaling to 31,250 particles. Particles would flow rightward at an acceleration of 9.8 m/s, similar to that of gravity. The sphere, torus, or concave bowl was placed along the particle path to interact with the particles. The simulation parameters for all three fluid flow tests are provided in Table 5.  
A first setup was designed to depict the resulting flow field around a concave sphere of 7.5 m radius. It was expected that particles would flow around the curvature of the sphere and not intersect or be submerged within the sphere's mesh manifold. Figure 14 shows the resulting flow fields at approximately 3 s after the start of the simulation, when particles are interacting with the sphere. As expected, the particles appear to flow around the sphere and form two distinct flow fields as they proceed past the sphere. No penetration of the mesh boundary was visually observed. A second setup uses an elongated torus with a tunnel formed by the elongated inner cavity of the torus. We established the torus with a depth of 1.32 m while being set to 0.35 m \(\times\) 0.35 m in width and height. In addition, the torus was rotated off-axis by 45 degrees around the y-axis, creating a diagonal path that particles would be expected to flow through, while particles outside the torus flow around the outer curvature of the mesh. Figure 15 shows how particles flow through the cavity formed by the inner ring of  
Figure 14: Screen captures of the flow field simulation with a concave sphere with a radius of 7.5 m. This visual was generated approximately 3 s after the start of the simulation. Panel (**a**) visualizes the flow field as a cone field, while Panel (**b**) is a screen capture of the _Unity3D_ simulation. The _Unity3D_ simulation uses a transparent sphere to highlight possible submerged particles. Both visualizations’ coloration represent particle speeds within a range of 0–30 m/s. \begin{table}
\begin{tabular}{|l|l|l|l|l|} \hline Flow Field & \multicolumn{1}{c|}{**Number of Particles**} & 31,250 & **Ideal Density**\(\rho_{0}\) & 600 \\ \cline{2-5} \cline{2-5} \multicolumn{1}{c|}{(Sphere,} & **Initial Spacing**\({}^{ab}\) & 0.3 & **Viscosity Coeff.** & 0.001 \\ \cline{2-5} \multicolumn{1}{c|}{} & **Render Size**\({}^{b}\) & 0.1 & **Bulk Modulus (k)** & 250 \\ \cline{2-5} \cline{2-5} \multicolumn{1}{c|}{} & **Particle Mass** & 1 & **Friction Coeff. (\(c_{friction}\))** & 0.0 \\ \cline{2-5} \cline{2-5} \cline{2-5} \multicolumn{1}{c|}{} & **Kernel Radius**\({}^{b}\) & 0.2 & **Restitution Coeff. (\(c_{friction}\))** & 0.25 \\ \hline \end{tabular}
\end{table} 
Table 5: Fluid flow setups—simulation parameters. the torus, and visual observation shows that particles collide strongly with the lip of the torus near the tunnel entrance and proceed to flow through the mesh, as expected. Due to the arrangement of the torus, particles outside of the tunnel follow the diagonal orientation of the torus and reach a choking point near the end of the torus' tunnel. Visual analysis indicates that no particles appear to intersect or be submerged within the mesh boundary manifold of the torus. A third setup uses a bowl mesh meant to appear as a cooking pot. The opening of the bowl was approximately 5 m in radius, and the bowl reached a depth of 3 m. Particles were expected to initially hit the interior of the bowl, scatter, and remain in the bowl as particles outside the bowl radius flowed past it. The results are shown visually in Figure 16; particles hit the bowl and consequently bounce around in it.