## 2 Characteristics, Principles, and a Case Study of SPH  
### Characteristics of SPH  
The smoothed particle hydrodynamics (SPH) method is built upon three core components: smoothing, particles, and fluid dynamics. In this framework, \"smoothing\" involves the stable and continuous approximations derived from the weighted averaging of adjacent particles, enabling a fluid-like behavior in simulations. The term \"particles\" signifies a meshless representation system, wherein each particle embodies a discrete element of the simulated domain. \"Fluid dynamics\" refers to the primary application for which the SPH method has been specifically developed and refined. Compared to conventional grid-based techniques, the SPH method presents three unique advantages:  
1. Meshless: The problem domain is entirely represented by particles, which act as the fundamental computational units for estimating field variables. This particle-based approach removes the necessity for grid-based partitioning, thereby sidestepping the challenges related to mesh distortion and reconstruction.
2. Adaptive: The kernel approximation of the field function is recalculated at each time step using all particles within the continuous domain. This adaptability ensures that  
Figure 3: Seepage process simulation diagram.  
the SPH method maintains robustness and consistency, even with arbitrary variations in particle distribution over time.
3. Lagrangian Integration: The SPH method seamlessly integrates Lagrangian properties with particle characteristics. Through the application of particle approximation techniques to discrete partial differential equations formulated in the Lagrangian framework--encompassing essential variables like velocity, density, and energy--the method establishes a coherent and precise system for capturing fluid dynamics.  
The unique integration of meshless, adaptive, and Lagrangian characteristics renders the smoothed particle hydrodynamics (SPH) method a compelling alternative to conventional mesh-based approaches, especially in contexts characterized by significant deformation. This method's versatility and robustness allow it to excel in simulating intricate fluid dynamics scenarios with both accuracy and efficiency. In contrast to other mesh-free techniques, where nodes serve merely as interpolation points, SPH particles intrinsically embody both material properties and computational elements. Functioning as both approximation locations and material constituents, these particles endow the SPH method with exceptional flexibility, facilitating the precise modeling and simulation of fluid dynamics. As a result, the SPH method finds widespread application across diverse domains of scientific research and engineering practice. The benefits of the SPH method encompass the following:  
1. It is a mesh-free framework that demonstrates resilience to distortion and deformation.
2. It has high adaptability, making it suitable for scenarios with evolving boundaries.
3. The elimination of extensive element divisions effectively mitigates errors associated with local approximations found in finite element methods (FEMs).
4. There is continuity of the field function and its gradient within the solution domain, achieved through kernel function interpolation.
5. The resolution of governing equations is simplified, thereby easing the integration of chemical and thermodynamic effects.