## 2 Characteristics, Principles, and a Case Study of SPH
### Principles of SPH
#### 2.2.1 Construction of SPH Basic Equation
However, if the smoothing function does not satisfy the normalization condition or is not an even function, the kernel approximation method does not inherently ensure second-order accuracy. By replacing \(f(x)\) with \(\langle\
abla\cdot f(x)\rangle\rangle\), we can obtain  
\[\langle\
abla\cdot f(x)\rangle=\int\Omega\ \big{[}\
abla\cdot f(x^{\prime}) \big{]}f\big{(}x^{\prime}\big{)}W\big{(}x-x^{\prime},h\big{)}\mathrm{d}x^{\prime} \tag{9}\]  
Figure 4: Predicted failure process and final failure pattern for specimen H03-01. (**a**) One can observe the isolated microcracks are randomly distributed throughout the specimen (**b**) The density of microcracks increases but the specimen still maintainsintegrity and stability. (**c**) A few clusters of the microcracks form isolated fractures or cracks. (**d**) These isolated fractures grow rapidly At stage (**e,f**), more fractures coalesce and interact with each other and further develop into inclined shear faults across the specimen. By applying the divergence theorem, the kernel approximation of the derivative of the field function can be derived as follows:  
\[ \begin{array}{c}\left\langle\
abla\cdot f(x)\right\rangle=\int\
olimits_{\Omega }\
abla\cdot[f(x^{\prime})W(x-x^{\prime},h)]\mathrm{d}x^{\prime}\\ -\int\
olimits_{\Omega}f(x^{\prime})\cdot\
abla W(x-x^{\prime},h)\mathrm{d}x^ {\prime}\end{array}  \tag{10}\]  
(2) Particle approximation method. In the SPH method, a finite set of particles, each with distinct mass and occupying unique spatial positions, is used to represent the overall state of the system. The process of discretization, which involves the superposition and summation of these particles, is referred to as the particle approximation method or particle approximation. By using the volume \(\Delta V_{j}\) of the particle to represent the infinitesimal volume element \(\mathrm{d}x^{\prime}\) at particle \(j\), the mass of the particle can be expressed as  
\[m_{j}=\Delta V_{j}\rho_{j} \tag{11}\]  
Convert the continuous integral expression for the SPH kernel approximation method into a discrete form to facilitate the summation of contributions from all particles within the specified domain. [PERSON] et al. (2018) introduced a computationally efficient three-point integration scheme that significantly reduces computational costs for field-scale applications. The effectiveness of the proposed stabilization method was validated, demonstrating its ability to manage coupled flow and deformation in saturated porous media. This approach can be utilized to predict issues related to large deformation and retrogressive failure in such materials. For example, there is the multi-field coupling problem caused by tunnel seepage water. Seepage, which involves the movement of water through the surrounding rock mass into a tunnel, is another key factor that affects tunnel safety and durability. Uncontrolled seepage into tunnels can lead to significant problems such as flooding, instability of the surrounding rock, and increased maintenance costs. Water seepage can lead to the erosion of fine particles in the surrounding soil or rock, resulting in a reduction in the shear strength of the ground and potentially causing subsidence or collapse. Figure 5 illustrates the observed behavior of pore water pressure and effective stress in sensitive clay, with similar patterns evident throughout the simulation. The failure process recurs for subsequent sliding blocks until the debris materials accumulating in front of the crater provide enough lateral support to stabilize the remaining slope, resulting in a final stable back scrap at t = 40 s. Six rotational slides are noted, with each sliding mass being of comparable size, mainly due to the assumed homogeneous material properties. The final run-out distance reaches 192 m. It is important to note that these results can only qualitatively predict the formation and evolution of shear bands.