# Application Research of an Efficient and Stable Boundary Processing Method for the SPH Method

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. SPH Methodology and Fixed Particle Interpolation Method

#### 2.1. SPH Methodology

**u**is particle velocity vector, t is time, p is pressure, and

**f**is the body force field.

_{0}is the reference density which is equal to 1000 (kg/m

^{3}) for fresh water and c

_{0}is the speed of sound evaluated in absence of compression, i.e., with ρ = ρ

_{0}.

_{0}is the reference speed of sound, γ is the specific heat ratio of water and is equal to 7.0. Different choices can be made for EOS, generally with a very weak influence on the results (Molteni, 2007, [30]).

#### 2.2. Fixed Ghost Boundary Method

- (1)
- Arrange the fixed ghost particles on the normal unit vector (out of the fluid domain) and distribute mirror particles (i.e., interpolation points) on the opposite direction according to the shape of the surface,
- (2)
- The physical properties of mirror particles are evaluated through performing interpolation among the fluid particles,
- (3)
- The physical properties of fixed ghost particles are duplicated from the mirror particles according to the laws of mirror.

## 3. Boundary Interpolation Method

#### 3.1. Moving Least Squares (MLS) Method

^{MLS}is given by:

^{MLS}is very complex which includes a variety of operations like the tensor product, sum of matrices, and inverse of the matrix. The MLS method has the following shortcomings. First, if the sample size is less than the characteristic number, the fitting equation is under-determined and the equations with an inverse matrix do not exist. For the linear basis function, the characteristic number is three, and for the quadratic basis function, the characteristic number is six, that is, when the number of particles in the interpolating domain is less than the characteristic number, the solution will fail. Secondarily, the numerical results are unstable because of the singular or morbid state of the matrix in the process of matrix inversion. Thirdly, this method uses a large amount of calculations and has a low computational efficiency, and the calculated amount shows a quadratic relationship with the number of adjacent particles.

#### 3.2. Simplified Finite Difference Interpolation (SFDI) Method

^{SFDI}contains many summation parameters, the computational complexity of this method is a quadratic relationship with the number of neighboring particles.

#### 3.3. Normalized Interpolation Method

#### 3.4. Improved Shepard Interpolation Method

^{shepard}(x

_{i}) are at least C

^{0}continuous (Lodha, 1997, [35]) and the power parameter is set to two because this makes the Shepard interpolation be optimal in a certain sense among all stable rational interpolation schemes (Farwig, 1986, [36]). In order to ensure the continuity of the weighting functions and their derivatives, and the continuity of the physical properties of the mirror particle, μ = 4 was used in this paper.

^{shepard}.

## 4. Numerical Results

#### 4.1. Hydrostatic Tank Test

^{−4}s. The simulation ended at the physical time 35 s. The model is given by Figure 2, and four pressure detection points P

_{0}(0, 0), P

_{1}(0, 0.1), P

_{2}(0, 0.2), and P

_{3}(0, 0.25) are set on the left vertical wall.

_{0}, P

_{1}, P

_{2}, and P

_{3}using the different interpolation methods and the analytical solution. Influenced by the accuracy of interpolation methods, the stability of the pressure numerical results obtained by several interpolation methods were different. The normalized method had a lower interpolation precision, and the initial moment pressure showed obvious oscillation, while the pressure field always had slight fluctuations. In the SFDI method, the pressure results also showed significant oscillations at the initial moment. In comparison, the MLS method and the improved Shepard method had relatively stable pressure results, and the numerical values of the improved method were basically consistent with the numerical results of the MLS quadratic method, indicating that the improved Shepard method has good numerical accuracy.

#### 4.2. Dam-Breaking Problem

_{1}was chosen for the right wall according to the experimental results of Buchner (2002, [39]).

_{w}= 5.366H were used. The initial distance between particles was Δx = Δy = 0.004 m and the total particle number was 45,000. The time step equaled 10

^{−4}s and the speed of sound C

_{0}was equal to $20\sqrt{gH}$.

_{1}from experimental data and different numerical methods. All of the interpolation methods recorded the pressure change during the free-surface climb and the plunging wave closure. The MLS method, the SFDI method (Chen, 2017, [5]), and the improved Shepard method have similar calculation results, while the normal method provided significantly larger numerical values than other methods. In fact, due to the low precision of the normal method, the pressure value of the particle point had deviations at the wall surface, and the flow field pressure resulted in unstable oscillation. As shown in Figure 8, the pressure field of the improved Shepard method and the MLS method showed better continuity and stability than the normalized method. In addition, according to Ren (2016, [40]) and Philip (1999, [41]), that all the numerical results take longer to reach the right side of the straight than the experimental results may be due to that the door is gradually opened during the experiment, while the numerical simulation is instantaneous.

#### 4.3. Solitary Wave Impact on Fixed Seawalls

_{0}) and y = 0.15 m (P

_{1}) to monitor the pressure time history for comparison with the numerical results of Chen (2017, [5]) and the experimental results of Zheng (2015, [43]). Besides, in order to compare the effect of different interpolation methods on the stability of the pressure results, the MLS method (both linear-based and quadratic-based) and SFDI method were also used in this part.

_{0}and P

_{1}. From the results in both figures, it shows that the improved Shepard method can achieve good results compared with the experimental data in the process of wave peak transmission.

#### 4.4. Solitary Wave Impact on Movable Seawalls

_{threshold}= 1.10 × F

_{static}. The acceleration, velocity, and position of the seawall were calculated as follows:

_{seawall}the mass of seawall. Superscripts n and (n + 1) denote the current and future time steps, respectively.

_{static}denotes the hydrostatic force on the wall.

_{seawall}= 250 kg/m and spring stiffness coefficient k = 10 N/m. The initial distance between particles was Δx = Δy = 0.008 m and the total particle number was 8025. The time step equaled 5.0 × 10

^{−4}s and speed of sound C

_{0}was equal to $10\sqrt{gH}$.

**X**denotes the seawall displacement.

## 5. Conclusions and Discussions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**(

**a**) Arrangement of mirror particles. (

**b**) Interpolation of mirror particles. (

**c**) Assignment of fixed particles.

**Figure 2.**Initial setup of the rectangular hydrostatic test. (

**a**) Computational model on the left and (

**b**) initial pressure field on the right.

**Figure 3.**Time history comparison of static water pressure. Comparison between the MLS, SFDI, normalized, and improved Shepard methods at (

**a**) P

_{0}, (

**b**) P

_{1}, (

**c**) P

_{2}, and (

**d**) P

_{3}.

**Figure 4.**Comparison of the computational efficiency between the improved Shepard and other methods. Average time spent in a single iteration (

**a**) and a sped-up comparison (

**b**).

**Figure 7.**Dam-break flow against a vertical wall. Comparison between the pressure loads measured experimentally and predicted by the numerical model at P

_{1}.

**Figure 8.**An example of corner point pressure distribution. Comparison between the (

**a**) normalized method, (

**b**) improved Shepard, and (

**c**) MLS method (quadratic based).

**Figure 9.**Comparison of computational efficiency between the improved Shepard and other methods for dam-breaking. (

**a**) Average time spent in a single iteration, (

**b**) a sped-up comparison.

**Figure 11.**Comparisons of the numerical wave impact pressure time histories with experimental data at P

_{0}(

**a**) and P

_{1}(

**b**).

**Figure 12.**Comparison of the computational efficiency between the improved Shepard and other methods for solitary wave impact on fixed seawalls. (

**a**) Average time spent in a single iteration, (

**b**) a sped-up comparison.

**Figure 15.**Comparison of the non-dimensional impact force (

**left**) and seawall displacement (

**right**) for the seawall at rest and while in motion with numerical data from Liang et al. (2017, [45]).

Interpolation Method | Calculated Amount (Relationship with Neighboring Particles) |
---|---|

Moving Least Squares method | Quadratic |

Simplified Finite Difference Interpolation method | Quadratic |

Normalized interpolation method | Linear |

Improved Shepard method | Linear |

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**MDPI and ACS Style**

Huang, X.; Chen, W.; Hu, Z.; Zheng, X.; Jin, S.; Zhang, X.
Application Research of an Efficient and Stable Boundary Processing Method for the SPH Method. *Water* **2019**, *11*, 1110.
https://doi.org/10.3390/w11051110

**AMA Style**

Huang X, Chen W, Hu Z, Zheng X, Jin S, Zhang X.
Application Research of an Efficient and Stable Boundary Processing Method for the SPH Method. *Water*. 2019; 11(5):1110.
https://doi.org/10.3390/w11051110

**Chicago/Turabian Style**

Huang, Xing, Wu Chen, Zhe Hu, Xing Zheng, Shanqin Jin, and Xiaoying Zhang.
2019. "Application Research of an Efficient and Stable Boundary Processing Method for the SPH Method" *Water* 11, no. 5: 1110.
https://doi.org/10.3390/w11051110