# Research on the Water Entry of the Fuselage Cylindrical Structure Based on the Improved SPH Model
## 2 Improved SPH Model
### TL-SFPM (Total Lagrangian Smoothed Particle Hydrodynamics) Method for Solids  
When a solid particle is in a tensile state, the attraction between particles may be unstable, leading to tensile instability. This tensile instability will lead to particle aggregation and the numerical fracture phenomenon, which introduces great obstacles into the application of SPH in solid mechanics [50]. The TL-SFPM method effectively addresses tensile instability, which leads to particle clustering and numerical fracture phenomena in the simulation of solids. [PERSON] et al. [46] identified that the cause of tensile instability in SPH lies in using the Euler kernel function and proposed that the kernel function can improve this issue. The TL-SFPM method is a high-order formulation based on the Lagrangian kernel function [51]. It considers the incomplete supported domain at the boundary and enhances the high-order formulation's computational efficiency by simplifying the selection mode of neighbor particles. Figure 2 illustrates the selection mode of neighbor particles in the internal, boundary, and corner regions, where the red particles represent the estimated particle, \(i\), the blue particles represent the neighbor particles, and the numbers 1, 2, 3, and 4 correspond to the identifiers of the blue particles. Four neighbor particles for particle \(i\) in the interior region are closest to particle \(i\), while for particle \(i\) in the boundary region, four neighbor particles are particle \(i\) itself, and the other three particles closest to particle \(i\). The gradient calculation formulas for particle \(i\) in the different regions are given by Equations (6)-(8). \[ \begin{bmatrix}f_{X}(\mathbf{X}_{i})\\ f_{Y}(\mathbf{X}_{i})\end{bmatrix} = \begin{bmatrix}\frac{f_{2}-f_{1}}{2d}\\ \frac{f_{4}-f_{3}}{2d}\end{bmatrix}  \tag{6}\]  
\[ \begin{bmatrix}f_{X}(\mathbf{X}_{i})\\ f_{Y}(\mathbf{X}_{i})\end{bmatrix} = \begin{bmatrix}-\frac{f_{1}-f_{1}}{d}\\ \frac{f_{4}-f_{3}}{2d}\end{bmatrix}  \tag{7}\]  
\[ \begin{bmatrix}f_{X}(\mathbf{X}_{i})\\ f_{Y}(\mathbf{X}_{i})\end{bmatrix} = \begin{bmatrix}\frac{1}{d}\left(\begin{array}[ ]{c}-0.8(f_{1}-f_{i})-0.2(f_{3}-f_{i})\\ +0.2(f_{4}-f_{i})\\ -0.2(f_{1}-f_{i})+0.2(f_{3}-f_{i})\\ +0.8(f_{4}-f_{i})\end{array}\right)\end{bmatrix}  \tag{8}\]  
Figure 2: Particles’ selected mode for different regions (the red particles represent the estimated particle, \(i\), the blue particles represent the neighbor particles). where \(f_{\mathbf{X}}(\mathbf{X}_{i})=\partial f(\mathbf{X}_{i})/\partial\mathbf{X}\), \(f_{Y}(\mathbf{X}_{i})=\partial f(\mathbf{X}_{i})/\partial Y\), \(f_{j}=f\big{(}\mathbf{X}_{j}\big{)}\big{|}j=1,2,3,4\) and \(f_{i}=f(\mathbf{X}_{i})\). \(d\) is the particle spacing. In the total Lagrangian SPH method, the discretized governing equations for the solid are as follows [52]:  
\[\left\{ \begin{array}{l}\rho_{i}\text{det}(\mathbf{F}_{i})-\rho_{i}^{0}=0\\ \frac{dv_{i}}{dt}=\frac{1}{\rho_{i}^{0}}\sum\limits_{j}\left(\mathbf{P}_{i}+\mathbf{P} _{j}+\Pi_{ij}^{0}\right)\cdot\
abla^{0}W_{ij}V_{j}^{0}\\ \rho_{i}^{0}\frac{d\mathbf{x}_{i}}{dt}=\sum\limits_{j}\left[\left(\mathbf{v}_{j}-\mathbf{v }_{i}\right)\otimes\
abla^{0}W_{ij}V_{j}^{0}\right]^{T}:\mathbf{P}_{i}\\ p_{i}=(1-\gamma\eta)p_{H}+\gamma\rho_{i}\mathbf{e}_{i}\end{array} \right. \tag{9}\]  
where \(\rho_{i}\), \(\mathbf{v}_{i}\), \(e_{i}\) and \(p_{i}\) represent the density, velocity, internal energy, and pressure of the particle \(i\), respectively. \(\mathbf{F}_{i}\) represents the deformation gradient of particle \(i\), and det (\(\mathbf{F}_{i}\)) denotes the determinant of \(\mathbf{F}_{i}\).