# Grid-Stamping on a Polygon Model for Implementing Arbitrary-Shaped Boundary Conditions in a Moving Particle Semi-Implicit Method

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## Abstract

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## 1. Introduction

- Fixed dummy model: This model places several dummy particles near the boundary and fixes their velocities at zero. It is relatively simple and has a low computational load. However, assigning accurate boundary conditions to the boundary is challenging. Additionally, the analytical results may have been inaccurate. In some cases, fluid particles penetrate the wall surface. In particular, this model has limitations when used for extremely thin objects. Koshizuka and Oka [3] adopted the fixed dummy model in the original MPS method. Marrone et al. [4] and Adami et al. [5] used a similar model in the SPH method.
- Mirror model: This model arranges virtual particles in symmetrical positions with respect to the positions of fluid particles near the boundary. The model satisfies the slip or no-slip condition of an object surface, prevents wall penetration, and enables a relatively accurate boundary surface placement. However, if the boundary curvature is discontinuous or complex, it can be difficult to place the corresponding virtual particles. Moreover, it has limitations in representing objects with extremely thin thicknesses. Akimoto [6] adopted the mirror model to predict the free surface of complex shapes in the MPS method, and Liu et al. [7] discussed it as a method for applying boundary conditions in the ISPH method.
- Repulsive force model: This model places wall particles on the boundary surface and imposes an artificial repulsive force on fluid particles approaching the wall particles according to their distance. It is relatively easy to apply and can forcibly prevent fluid particles from penetrating the wall surface. However, given that satisfying the slip or no-slip condition at the interface is challenging and the magnitude of the repulsive force must be determined using empirical constants, the reliability of the results can be rather low. The repulsive force model was proposed by Monaghan [8]. It has since been extended to arbitrary boundary shapes by Monaghan and Kajtar [9].

## 2. Enhanced PNU-MPS Method

#### 2.1. Governing Equations

#### 2.2. Kernel Function

#### 2.3. Gradient Model

#### 2.4. Laplacian Model

#### 2.5. Incompressible Model

#### 2.6. Free-Surface Boundary Condition

#### 2.7. Collision Model

#### 2.8. Artificial Viscosity Model

#### 2.9. Divergence-Free Model

## 3. Wall Boundary Condition

#### 3.1. Establishment of Interface for Coupled Simulation with MBD Solver

#### 3.2. Explicitly Represented Polygon Wall Boundary Model

#### 3.2.1. Pressure Gradient Term

#### 3.2.2. Viscosity Term

#### 3.3. Grid-Stamping on Polygon (G-StoP) Model

#### 3.3.1. Concept of G-StoP Model

- In Figure 5a, the vectors ${\overrightarrow{n}}_{x}$, ${\overrightarrow{n}}_{y}$, and ${\overrightarrow{n}}_{z}$ are the basis vectors in the Cartesian coordinate system. ${\overrightarrow{n}}_{i}$ is the distance vector to the wall of particle $i$ and is obtained from the polygonal CAD information corresponding to particle $i$.
- To establish parallel vectors to generate temporarily local grids on polygons, ${\overrightarrow{n}}_{x}$ is rotated along the axis of ${\overrightarrow{n}}_{\perp}$ through an angle $\theta $ in the direction of ${\overrightarrow{n}}_{i}$. At this time, Rodrigues’ rotation formula [31], which is expressed in the form of a rotational matrix, is used to rotate a vector by providing rotational angle and axis information. In other words, ${\overrightarrow{n}}_{{x}^{\prime}}$ projected in the direction of ${\overrightarrow{n}}_{i}$ as well as ${\overrightarrow{n}}_{{y}^{\prime}}$ and ${\overrightarrow{n}}_{{z}^{\prime}}$ in directions parallel to the polygon can be obtained using Equation (31), as shown in Figure 5b.
- To generate grid vectors parallel to the wall with the generated polygon vectors ${\overrightarrow{n}}_{{y}^{\prime}}$ and ${\overrightarrow{n}}_{{z}^{\prime}}$, the coordinate information ${P}_{k}\left(x\right)$ and ${P}_{k}\left(y\right)$ of the 2D local grid system in Figure 5c is used. Therefore, the lattice vector can be expressed as ${P}_{k}\left(x\right){\overrightarrow{n}}_{{z}^{\prime}}+{P}_{k}\left(y\right){\overrightarrow{n}}_{{y}^{\prime}}$, and vectors parallel to the wall and directed to each lattice point can be obtained, as shown in Figure 5d. In this study, nine lattice points were used because the effective radius of the surrounding particles was set to 2.1 times the particle size. Depending on the nature of the problem, the number of grid points could be increased or decreased.
- The final expression is simply derived in Equation (32). The distance information between the virtual and fluid particles can be obtained using the grid vectors ${P}_{k}\left(x\right){\overrightarrow{n}}_{{z}^{\prime}}+{P}_{k}\left(y\right){\overrightarrow{n}}_{{y}^{\prime}}$ and ${\overrightarrow{n}}_{i}$ parallel to the wall, and ${\overrightarrow{r}}_{k}^{wall}$ enables the assignment of the boundary conditions at the wall.

#### 3.3.2. Pressure Gradient Term

#### 3.3.3. Viscosity Term

## 4. Verification and Validation (V&V)

#### 4.1. Hydrostatic Pressure Problem with Various Corner Angles

#### 4.2. Dam Breaking Problem

#### 4.3. Subaerial Landslide Tsunami Generation Problem

#### 4.4. Wine Sloshing Problem

## 5. Conclusions

- Hydrostatic pressure simulations were conducted in a rectangular tank with various corner angles. Although the conventional ERP model exhibited a large numerical fluctuation and was unstable when the corner angle was reduced, the proposed model exhibited excellent robustness and safety without any special treatment. For a more quantitative comparison, the hydrostatic pressure was compared with the results of the ERP model and the analytic solution. The result shows that, when the corner angle was 30°, the maximum errors of the ERP and G-StoP models were 11.3% and 1.8%, respectively.
- In the dam breaking simulations, the time series of the pressure in front of the obstacle was compared with other simulations and experimental results. The rise time, peak value, and decay trend of the impact pressure matched the results of the other simulations relatively well. In addition, the flow development process was expressed qualitatively and compared reasonably with experimental results.
- In the subaerial landslide tsunami generation, the slide velocity, position, pressure, and wave height results were consistent with the experimental data. However, relative errors of approximately 4% in the experiment occurred at the position at which the slide finally settled. This may be attributed to the repulsive force from the fluid acting on the slide being stronger in this simulation than in the experiment during a collision with the bottom. Meanwhile, currently, the effect of cavitation associated with air bubbles at the top and back of the slide was not simulated. Moreover, a model that can consider this type of multiphase phenomenon should be adopted in the future.
- Finally, for the wine sloshing problem, a simple visualization experiment was performed on an in-line periodic motion with a wine glass partially filled with wine. Notably, the simulation represented qualitatively similar behavior to the free surface of wine in a glass with a curved boundary compared with the experiment.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Wall boundary treatment models in particle method. (

**a**) Fixed dummy, (

**b**) Mirror and (

**c**) Repulsive force.

**Figure 5.**Concept of G-StoP model. (

**a**) Axial vector rotation, (

**b**) Polygon parallel vector generation, (

**c**) Grid generation, (

**d**) Particle arrangement.

**Figure 7.**Side view of water tanks with various corner angles. (

**a**) Corner angle = 90°, (

**b**) Corner angle = 45°, (

**c**) Corner angle = 30°.

**Figure 9.**Distribution of pressure at $t=5\mathrm{s}$. (

**a**) ERP at corner angle = 90°, (

**b**) G-StoP at corner angle = 90°, (

**c**) ERP at corner angle = 45°, (

**d**) G-StoP at corner angle = 45°, (

**e**) ERP at corner angle = 30°, (

**f**) G-StoP at corner angle = 30°.

**Figure 10.**Time history of bottom pressure. (

**a**) Corner angle = 90°, (

**b**) Corner angle = 45°, (

**c**) Corner angle = 30°.

**Figure 11.**Depth-distributed pressure profiles of all particles. (

**a**) ERP at corner angle = 90°, (

**b**) G-StoP at corner angle = 90°, (

**c**) ERP at corner angle = 45°, (

**d**) G-StoP at corner angle = 45°, (

**e**) ERP at corner angle = 30°, (

**f**) G-StoP at corner angle = 30°.

**Figure 13.**Comparison of water surfaces interacting with obstacle. (

**a**) t = 0.2 s, (

**b**) t = 0.4 s, (

**c**) t = 0.6 s, (

**d**) t = 0.84 s.

**Figure 17.**Comparison of pressure in time series with experiment and other simulation. (

**a**) at P3, δ

^{+}−LES–SPH (Zhang et al. [35]), (

**b**) at P4.

**Figure 19.**Comparison of water surfaces with experiment and pressure distributions. (

**a**) t = 0 s, (

**b**) t = 0.4 s, (

**c**) t = 0.6 s, (

**d**) t = 1.6 s.

**Figure 21.**Comparison of experimental wine sloshing surfaces and numerical pressure distributions. (

**a**) t = 0 s, (

**b**) t = 0.73 s, (

**c**) t = 0.94 s, (

**d**) t = 2.42 s.

Condition | Value | ||
---|---|---|---|

Tilted angle ($\xb0)$ | $90$ | $45$ | $30$ |

Computational time $\left(\mathrm{s}\right)$ | $5.0$ | ||

Number of particles | $384,000$ | $387,600$ | $586,480$ |

Particle length $\left(\mathrm{m}\right)$ | $5.0\times {10}^{-3}$ | ||

Density of fluid $\left(\mathrm{kg}/{\mathrm{m}}^{3}\right)$ | $1000.0$ | ||

Kinematic viscosity of fluid $\left({\mathrm{m}}^{2}/\mathrm{s}\right)$ | $1.0\times {10}^{-6}$ |

Condition | Value |
---|---|

Computational time $\left(\mathrm{s}\right)$ | $4.0$ |

Number of particles | $5,461,200\left(246\times 111\times 200\right)$ |

Particle length $\left(\mathrm{m}\right)$ | $5.0\times {10}^{-3}$ |

Density of fluid $\left(\mathrm{kg}/{\mathrm{m}}^{3}\right)$ | $1000.0$ |

Kinematic viscosity of fluid $\left({\mathrm{m}}^{2}/\mathrm{s}\right)$ | $1.0\times {10}^{-6}$ |

Condition | Value | |||
---|---|---|---|---|

Computational time $\left(\mathrm{s}\right)$ | $4.0$ | |||

Number of particles | $3,733,956$ | |||

Particle length $\left(\mathrm{m}\right)$ | $5.0\times {10}^{-3}$ | |||

Density of fluid $\left(\mathrm{kg}/{\mathrm{m}}^{3}\right)$ | $1000.0$ | |||

Kinematic viscosity of fluid $\left({\mathrm{m}}^{2}/\mathrm{s}\right)$ | $1.0\times {10}^{-6}$ | |||

Slide mass ($\mathrm{kg}$) | $60.14$ | |||

Slide volume (${\mathrm{m}}^{3}$) | 0.038 | |||

Inertia tensor | Ixx | 2.493 | Ixy | 0.72 |

Iyy | 2.422 | Iyz | −1.314 | |

Izz | 1.583 | Izx | −2.685 |

NEMEA Wine (Greece) | |
---|---|

Dynamic viscosity | $1.75\left(20\xb0\mathrm{C}\right)\mathrm{mPa}\cdot \mathrm{s}$ |

Density | $0.992\pm 0.001\left(\mathrm{g}/\mathrm{mL}\right)$ |

Alcohol content | $12.4\pm 0.1$(%, v/v) |

Condition | Value |
---|---|

Computational time | $3.5\left(\mathrm{s}\right)$ |

Number of particles | $2,099,658$ |

Particle length | $3.0\times {10}^{-4}\left(\mathrm{m}\right)$ |

Density of fluid | $992.0\left(\mathrm{kg}/{\mathrm{m}}^{3}\right)$ |

Kinematic viscosity of fluid | $1.75\times {10}^{-6}$ |

Amplitude | $0.01\mathrm{m}$ |

Oscillation period | $0.45\left(\mathrm{s}\right)$ |

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## Share and Cite

**MDPI and ACS Style**

Shin, H.-S.; Park, J.-C.
Grid-Stamping on a Polygon Model for Implementing Arbitrary-Shaped Boundary Conditions in a Moving Particle Semi-Implicit Method. *J. Mar. Sci. Eng.* **2023**, *11*, 742.
https://doi.org/10.3390/jmse11040742

**AMA Style**

Shin H-S, Park J-C.
Grid-Stamping on a Polygon Model for Implementing Arbitrary-Shaped Boundary Conditions in a Moving Particle Semi-Implicit Method. *Journal of Marine Science and Engineering*. 2023; 11(4):742.
https://doi.org/10.3390/jmse11040742

**Chicago/Turabian Style**

Shin, Hee-Sung, and Jong-Chun Park.
2023. "Grid-Stamping on a Polygon Model for Implementing Arbitrary-Shaped Boundary Conditions in a Moving Particle Semi-Implicit Method" *Journal of Marine Science and Engineering* 11, no. 4: 742.
https://doi.org/10.3390/jmse11040742