# Boundary SPH for Robust Particle-Mesh Interaction in Three Dimensions
## 3 Results
### Water Tank Setups  
Our BSPH method, despite its efficacy in the plane-based trials, still does not account for pressure irregularities in particles close to the boundary manifold. To verify this, two water tank simulations were conducted: one with an empty tank, and another with a 90-degree wedge placed on the floor in the middle of the tank. A third water tank setup was also created to simulate a dam break scenario with the wedge acting as a slope, for which we measured whether the mesh boundary method could handle fluid movement from dam breaks. The tank itself was represented by an inverted mesh cube stretched to fit a 10 m \(\times\) 7.5 m \(\times\) 2.5 m area. This means that the limits of the experiment space were contained by mesh triangles, and therefore, all particle interactions with the water tank were effectively particle-mesh boundary interactions. The tank and wedge mesh boundaries both had a friction coefficient of 0 and restitution coefficient of 0.25. Simulation parameters for all water tank simulations are provided in Table 4.  
Figure 10: The inclined plane simulation run with a 5 \(\times\) 5 grid of particles placed slightly further than the smoothing kernel radius and forced to fall onto the plane inclined at 10 degrees with a friction coefficient of 1. Panels (**a–f**) depict the pathlines of five of particles as they fall and “stick” onto the inclined plane. Panel (**g**) shows the screenshot of the particles at the end of the simulation run, where they still remain uphill and in formation. Figure 9: The inclined plane simulation run with a 5 \(\times\) 5 grid of particles placed slightly further than the smoothing kernel radius and forced to fall onto the plane inclined at 10 degrees and with a friction coefficient of 0. Panels (**a–f**) depict the pathlines of five of particles as they fall and “roll” down the incline path, while panel (**g**) shows the screenshot of the particles during movement in Unity3D. In the first two water tank tests, particles had a kernel support radius of 0.2 m and were initialized with 0.2 m of space between particles. The resulting pressure gradients were measured at the end of the simulation, when the particles no longer exhibited any movement and the fluid was still. The pressure gradients visualized here are calculated based on the original pressure equation:  
\[p=k\rho \tag{9}\]  
and are expressed in Pascal units, though the SPH implementation in _Unity3D_ still uses [PERSON]'s modified variant. The empty tank scenario (Figure 11) was initialized with 15,750 particles, while the tank with the wedge scenario (Figure 12) was initialized with 13,500 particles. Visual analysis of both results show that irregular pressure values are most prominent along the edges of the simulation area, where particles would interact with the inverted cube and wedge meshes. Irregular pressure values can also be seen along the bottom of the simulation space as well. This falls within expectation, as the mesh boundary method does not account for the loss in pressure due to empty kernel space for particles touching the boundary. Nonetheless, the simulation is stable enough to be at a standstill, and particles show no sign of escaping the boundary manifold. The wedged tank simulation shows that the particles do not penetrate the boundary formed by the wedge and instead follow its contour as expected. The final tank-based setup that we explored was a traditional dam break scenario with the same incline wedge placed in the middle of the scene. Rather than identifying pressure gradients along the boundary, the focus of this evaluation was to observe whether particles penetrated the boundary due to excessive pressure and viscosity forces generated from the fluid motion. In this setup, 20,445 particles were placed on one side of the tank and were placed at intervals of 0.15 m. This means that the particles, upon initialization of the simulation, were expected to expand outward and thus generate the necessary fluid motion to simulate a dam break scenario.