## 2 Characteristics, Principles, and a Case Study of SPH
### Principles of SPH
#### 2.2.2 Construction of Smooth Functions  
The smoothing function \(W\) plays a pivotal role in the SPH approximation method, significantly influencing both the accuracy and computational efficiency of the resulting function expression. Commonly used kernel functions include the following:  
1. Gaussian smooth function. The Gaussian smoothing function is renowned for its exceptional smoothness, stability, and high precision. It is particularly effective in scenarios involving non-uniform or irregular particle distributions. Although, from a mathematical perspective, the function does not converge to zero in absolute terms (unless the radius \(R\) approaches infinity), its numerical attenuation is remarkably rapid. As a result, it can be treated as having tight support in practice, effectively mitigating edge effects. \(d\) is the dimension of the space. \[G(R,h)=\frac{1}{\left(\sqrt{\pi}h\right)^{d}}e^{-R^{2}}\] (19)  
However, the application of the Gaussian smoothing function comes with the challenge of computational complexity, particularly when higher-order derivatives need to be calculated. Its wide support domain necessitates the consideration of more particles when approximating interactions, which directly increases the computational burden and data processing time. To optimize this process, more efficient numerical methods or algorithm optimization techniques--such as parallel computing, approximation algorithms, or hardware-accelerated solutions--can be employed to reduce computational costs while maintaining accuracy. 2. Cubic spline curve. The cubic spline smoothing function is widely recognized in modern scientific literature for its comparable performance to the Gaussian smoothing function within a narrow, compact support domain. However, its piecewise linear second derivative leads to a slight reduction in stability when compared to smoother kernel functions. To address this issue, researchers are actively investigating methods to enhance the cubic spline by combining it with other advanced smoothing techniques, aiming to create more stable and adaptable smoothing functions. This strategy seeks to overcome the inherent limitations of cubic splines while preserving their advantages for broader applications. \[W(R,h)=\alpha_{d}\times \begin{cases}\frac{2}{3}-R^{2}+\frac{1}{2}R^{2},&0\leq R \leq 1\\ \frac{1}{6}\times\left(2-R\right)^{2},&1\leq R\leq 2\\ 0,&R\geq 2\end{cases}  \tag{20}\]  
In the formula, \(\alpha_{d}\) is related to the dimension space. The value of one-dimensional space is \(\frac{1}{h}\), the value of two-dimensional space is \(\frac{15}{7\pi h^{2}}\), and the value of three-dimensional space is \(\frac{3}{2\pi h^{2}}\). 3. High-order spline curve. Building on the cubic spline function, the spline interpolation method expands by introducing higher-order spline functions, specifically quartic and quintic spline functions. These higher-order spline functions are not only numerically more stable, but also morphologically closer to the Gaussian smoothing function, thereby enhancing their applicability and accuracy in addressing complex data and curve-fitting tasks. The quartic spline function is  
\[W(R,h)=\alpha_{d}\times \begin{cases}\left(2.5+R\right)^{4}-5\times\left(1.5+R \right)^{4}+10\times\left(0.5+R\right)^{4},&0\leq R<0.5\\ \left(2.5-R\right)^{4}-5\times\left(1.5-R\right)^{4},&0.5\leq R<1.5\\ \left(2.5-R\right)^{4},&1.5\leq R<2.5\\ 0,&R\geq 2.5\end{cases}  \tag{21}\]In the above formula, the value of \(\alpha_{d}\) in one-dimensional space is \(\frac{1}{24h}\). The quintic spline function is  
\[W(R,h)=\alpha_{d}\times\left\{ \begin{array}{c}\left(3-R\right)^{5}-6\times \left(2-R\right)^{5}+15\times\left(1-R\right)^{5},\ \ 0\leq R<1\\ \left(3-R\right)^{5}-6\times\left(2-R\right)^{5},\ \ 1\leq R<2\\ \left(3-R\right)^{5},\ \ 2\leq R<3\\ 0,\ \ R\geq 3\end{array} \right.