# A New Boundary Condition Framework for Particle Method by Using Local Regular-Distributed Background Particles—The Special Case for Poisson Equation

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## Abstract

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## 1. Introduction

## 2. Methodology

#### 2.1. Governing Equations

#### 2.2. Enhancing the Boundary Condition Accuracy by Local Regular-Distributed Particles

#### 2.3. Weak Form Poisson Equation and Imposing Boundary Condition

#### 2.4. Closure Equations for the Background Particles

#### 2.4.1. Closure Equations for Different Parts of Background Particles

#### 2.4.2. Gauss–Legendre Quadrature Formula

#### 2.5. Different Interpolation Techniques

#### 2.5.1. Least Square Type Interpolation Based on Taylor Series Expansion

#### 2.5.2. Moving Particle Semi-Implicit Method (MPS)

## 3. Numerical Results and Discussion

#### 3.1. Validation of Boundary Local Background Particles Method

#### 3.2. Different Choice of Interpolation Methods for Inner Areas

#### 3.3. Different Shapes for the Boundary

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Illustration of the method for applying boundary conditions based on local regular-distributed particles: (

**a**) Closure equations for background particles outside the boundary; (

**b**) Closure equations for background particles inside the computational domain and closest to the boundary; (

**c**) Closure equations for background particles closest to the fluid particles; (

**d**) Integration region for boundary; (

**e**) Integration region within the interior.

**Figure 2.**Construction method of shape functions on 9−point cell, where the three different colors represent three different groups of construction units.

**Figure 3.**Taylor–Green vortex field: solution of the pressure field in $\mathrm{\Omega}=\left[0,2\pi \right]\times \left[0,2\pi \right]$ at $t={10}^{-4}\mathrm{s}$.

**Figure 4.**Regular discretization of domain. The black dot, yellow dot, and blue dot, respectively, represent background particles, boundary particles, and fluid particles.

**Figure 5.**Convergence of the pressure field for the Neumann boundary condition: (

**a**) Least square-type interpolation; (

**b**) MPS−LRBP (the combination of MPS method and LRBP method); (

**c**) MPS−VPM (the combination of MPS method and VPM method).

**Figure 6.**Irregular discretization of domain. The black dot, yellow dot, and blue dot, respectively, represent background particles, boundary particles, and fluid particles.

**Figure 8.**Pressure cloud map and error cloud map of a given particle distribution: (

**a**) MPS−VPM; (

**b**) MPS−LRBP.

**Figure 10.**Numerical and theoretical results of triangular boundary (

**above**) and a quarter−circle boundary (

**below**).

**Figure 11.**Error cloud map of triangular boundary (

**above**) and a quarter-circle boundary (

**below**), where the left is MPS-VPM and the right is MPS-LRBP.

Grid | Nx | Ny | Order of Convergence | |||||
---|---|---|---|---|---|---|---|---|

LS | MPS–LRBP | MPS–VPM | ||||||

$\mathit{E}\mathit{r}{\mathit{r}}_{\mathit{L}2}$ | $\mathit{E}\mathit{r}{\mathit{r}}_{\mathit{L}\mathit{\infty}}$ | $\mathit{E}\mathit{r}{\mathit{r}}_{\mathit{L}2}$ | $\mathit{E}\mathit{r}{\mathit{r}}_{\mathit{L}\mathit{\infty}}$ | $\mathit{E}\mathit{r}{\mathit{r}}_{\mathit{L}2}$ | $\mathit{E}\mathit{r}{\mathit{r}}_{\mathit{L}\mathit{\infty}}$ | |||

1st | 21 | 21 | ||||||

2.5714 | 2.5700 | 2.4922 | 2.4855 | 1.2605 | 0.6855 | |||

2nd | 41 | 41 | ||||||

2.6201 | 2.5897 | 2.3156 | 2.2794 | 1.0774 | 0.6919 | |||

3rd | 61 | 61 | ||||||

2.6250 | 2.5774 | 1.5935 | 1.5686 | 0.9137 | 0.5662 | |||

4th | 81 | 81 | ||||||

2.6196 | 2.5599 | 3.9045 | 3.7851 | 1.0827 | 1.5220 | |||

5th | 101 | 101 | ||||||

2.6106 | 2.5416 | 0.3337 | 0.8071 | 1.2138 | 2.4241 | |||

6th | 121 | 121 | ||||||

2.6000 | 2.5238 | 3.3926 | 3.1212 | 1.2980 | 1.9567 | |||

7th | 141 | 141 | ||||||

2.5889 | 2.5069 | 4.8021 | 4.0767 | 1.3334 | 0.3329 | |||

8th | 161 | 161 | ||||||

2.5776 | 2.4910 | 3.0669 | 2.8146 | 1.3413 | 0.3075 | |||

9th | 181 | 181 | ||||||

2.5665 | 2.4761 | 1.3129 | 1.6320 | 1.5538 | 0.1371 | |||

10th | 201 | 201 | ||||||

Mean value | 2.5977 | 2.5374 | 2.5793 | 2.5078 | 1.2305 | 0.9582 | ||

Global value | 2.6035 | 2.5590 | 2.4752 | 2.4301 | 1.1550 | 1.0167 |

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## Share and Cite

**MDPI and ACS Style**

Sun, Z.; Dou, L.; Mu, Z.; Tan, S.; Zong, Z.; Djidjeli, K.; Zhang, G.
A New Boundary Condition Framework for Particle Method by Using Local Regular-Distributed Background Particles—The Special Case for Poisson Equation. *J. Mar. Sci. Eng.* **2023**, *11*, 2183.
https://doi.org/10.3390/jmse11112183

**AMA Style**

Sun Z, Dou L, Mu Z, Tan S, Zong Z, Djidjeli K, Zhang G.
A New Boundary Condition Framework for Particle Method by Using Local Regular-Distributed Background Particles—The Special Case for Poisson Equation. *Journal of Marine Science and Engineering*. 2023; 11(11):2183.
https://doi.org/10.3390/jmse11112183

**Chicago/Turabian Style**

Sun, Zhe, Liyuan Dou, Zongbao Mu, Siyuan Tan, Zhi Zong, Kamal Djidjeli, and Guiyong Zhang.
2023. "A New Boundary Condition Framework for Particle Method by Using Local Regular-Distributed Background Particles—The Special Case for Poisson Equation" *Journal of Marine Science and Engineering* 11, no. 11: 2183.
https://doi.org/10.3390/jmse11112183