## 2 Characteristics, Principles, and a Case Study of SPH
### Principles of SPH
#### 2.2.2 Construction of Smooth Functions
\tag{22}\]  
In the above formula, the value of \(\alpha_{d}\) in one-dimensional space is \(\frac{1}{120h}\), the value in two-dimensional space is \(\frac{7}{478h^{2}}\), and the value in three-dimensional space is \(\frac{3}{359h^{3}}\). The main features of the smooth function are summarized as follows:  
The initial requirement is the normalization condition:  
\[\int\Omega\ W\big{(}x-x^{\prime},h\big{)}dx^{\prime}=1 \tag{23}\]  
The second criterion involves fulfilling the conditions of compactness:  
\[W\big{(}x-x^{\prime},h\big{)}=0;\big{|}x-x^{\prime}\big{|}>\kappa h \tag{24}\]  
This feature effectively transforms smoothed particle hydrodynamics (SPH) from global coordinates to local coordinates. This conversion results in a series of sparse matrices, which is essential for computational efficiency. The third: The non-negative smooth function \(W(x-x^{\prime},h)>0\), evaluated at any point \(x\) within the support domain centered at \(x^{\prime}\), plays a crucial role in accurately capturing the physical implications of certain phenomena. Ensuring its non-negativity is paramount. The fourth: As the distance between particles increases, the smooth function value associated with these particles should monotonically decrease. As the distance between the two interaction particles increases, their interaction is reduced. The fifth: As the smoothing length \(h\) approaches zero, the smooth function should satisfy the conditions of the Dirac \(\delta\) function. \[\lim_{h\to 0}W\big{(}x-x^{\prime},h\big{)}=\delta\big{(}x-x^{\prime}\big{)} \tag{25}\]  
This inherent property ensures that, as the smoothing length approaches zero, the approximated value asymptotically converges towards the genuine function value. The sixth: The smooth function should be a puppet function (symmetry). This property implies that particles located at an equivalent distance from the specified particle, regardless of their specific position, should exert an identical influence on that particle. The seventh: The smooth function should be fully smooth. To achieve optimal results for a function and its derivatives, the smooth function must possess full continuity. The function and its first two derivatives are approximated with an accuracy up to the specified order of \(n\), utilizing the smooth function to fulfill the conditions required for such an approximation. Assuming that the smooth kernel function solely depends on a polynomial function related to the relative distance between particles, within the support domain delimited by the influence width \(\kappa h\), the following form can be assumed:  
\[W\big{(}x-x^{\prime},h\big{)}=W(R)=a_{0}+a_{1}R+a_{2}R^{2}+a_{3}R^{3}+\cdots a _{n}R^{n} \tag{26}\]  
By substituting \(W\) into the conditional equations pertaining to the smooth function and solving the resulting system of linear equations for the parameters \(a_{0},a_{1}a_{2},a_{3}\ldots a_{n}\), one can determine the specific form of the smooth function. Hence, we can conclude that the derived expression for the smooth function serves as a versatile weight function in the integral representation of the function and its derivatives.