## 2 Characteristics, Principles, and a Case Study of SPH
### Principles of SPH
#### 2.2.1 Construction of SPH Basic Equation  
The integral formulation of smoothed particle hydrodynamics (SPH) for calculating the derivative of a field function enables the computation of the spatial gradient using both the field function values and the derivative of the smoothing kernel W. This approach eliminates the need to depend exclusively on the derivative of the field function. Consequently, it reduces the requirement for the field function to be continuously differentiable, thereby enhancing both stability and robustness in simulations. 1. Kernel approximation method. The field function core is as follows: \[\left\langle f(x)\right\rangle=\int\limits_{\Omega}f\big{(}x^{\prime}\big{)}W \big{(}x-x^{\prime},h\big{)}dx^{\prime}\] (1)  
\(W\) represents the smoothing function, which defines the value of the function at point \(x\) in relation to point \(x^{\prime}\). It must adhere to the following three essential conditions:  
1. The initial condition pertains to regularization, ensuring that the function adheres to a specific set of standards or norms. \[\int\
olimits_{\Omega}\ W\big{(}x-x^{\prime},h\big{)}dx^{\prime}=1\] (2)
2. The second is that when the degree of smoothness tends to zero, the properties of the Dirac function are as follows: \[\underset{h\to 0}{lim}\ W\big{(}x-x^{\prime},h\big{)}=\delta\big{(}x-x^{\prime}\big{)}\] (3)
3. The third condition aims to strengthen the constraints, ensuring a tighter and more rigorous adherence to the specified requirements. \[W\big{(}x-x^{\prime},h\big{)}=0;\big{|}x-x^{\prime}\big{|}>kh\] (4)  
\(f\) is a function of a three-dimensional coordinate \(x\), \(\Omega\) represents the integral volume that encloses \(x\), and h denotes the smoothing length. \(\kappa\), a constant associated with the smoothing function at point \(x\), governs the extent (non-zero) of the function's influence, defining its effective region, known as the support domain, centered at point \(x\). Using the aforementioned formula, [PERSON] et al. (2018) developed an innovative numerical approach leveraging meshless smoothed particle hydrodynamics (SPH) to model the failure of brittle inhomogeneous materials (tunnel lining structures). This method tracks the propagation of microcracks and the resultant macroscopic mechanical behavior. Figure 4 plots the failure process and the final failure pattern of the specimen H03-01. The failure process corresponds in sequence to the points marked as 'a' to 'f' on the stress-strain curve. The predicted cracks are indicated by the damaged particles in dark black. When the specimen fails, these damaged particles can be separated according to the governing equations, and fragments will form subsequently. The final failure pattern is obtained by plotting the particles according to their locations. Formula (1) can be written as a standard function of any SPH frame:  
\[\left\langle f(x)\right\rangle=f^{\prime}(x)\int\
olimits_{\Omega}\ W\big{(}x -x^{\prime},h\big{)}\mathrm{d}x^{\prime} \tag{5}\]  
In the above formula, \(f(x)\) represents the kernel function approximation of the function \(f(x)\). The estimation of the error in the kernel function approximation can be derived through the Taylor series expansion centered at the function's argument \(x\), yielding the subsequent outcomes:  
\[ \begin{array}{c}\left\langle f(x)\right\rangle=f(x)\int\
olimits_{\Omega} \ W(x-x^{\prime},h)\mathrm{d}x^{\prime}\\ +f^{\prime}(x)\int\
olimits_{\Omega}\ (x^{\prime}-x)W(x-x^{\prime},h) \mathrm{d}x^{\prime}+r\big{(}h^{2}\big{)}\end{array}  \tag{6}\]Here, r is the error term after the Taylor series expansion, and since \(W\) is an even function related to \(x\), \((x^{\prime}-x)W(x-x^{\prime},h)\) is an odd function. \[\int\Omega\ \left(x^{\prime}-x\right)W\big{(}x-x^{\prime},h\big{)}\mathrm{d}x^{ \prime}=0 \tag{7}\]  
By combining the above formulas, the error expression for the kernel function approximation can be derived as follows:  
\[\langle f(x)\rangle=f(x)+r\Big{(}h^{2}\Big{)} \tag{8}\]  
It is evident that the kernel approximation method is second-order accurate.