# Error Characteristic Analysis of Fluid Momentum Law Test Apparatus
## 2 Experimental Device and Numerical Model
### The Principle of SPH-FEM Coupling Algorithm
#### 2.2.1 The Principle of SPH Algorithm  
The Smoothing Particle Hydrodynamics (SPH) method is a meshless, pure Lagrange method. The SPH method was first applied to fluid free surface flow by Monaghan in 1994. This method is currently widely used in the fields of fluid free motion, impact collision, explosion, soil mechanics, etc. [17; 18; 19; 20; 21; 22].  
The basic logic of the SPH algorithm is to replace the continuum fluid with a series of randomly distributed particles, and estimate the space function and its derivative of the N-S equation through the particle set and interpolation kernel function, thereby transforming the original N-S equation containing both time and space derivatives into an equation only containing the time derivative [17]. The SPH method includes two parts: integral representation and particle approximation. The kernel function is the core and basis of the integral representation part of the SPH algorithm. The kernel function is approximated by the integral field function, and its expression is shown in Formula (1). On this basis, the kernel approximation equation is used to re-approximate by particle by applying the values corresponding to adjacent particles in the local area to superimpose and replace the function and its derivative integral form, named as particle approximation, as shown in Formula (2). Finally, the particle approximation process of fluid mechanics equations is realized [23; 24].  
\[\{f(x)\}\ \approx\int\limits_{\Omega}f\big{(}x^{\prime}\big{)}\,W\big{(}x-x^{ \prime},h\big{)}\,dx^{\prime} \tag{1}\]  
\[\langle f(x_{i})\rangle=\sum\limits_{j=1}^{N}\frac{m_{j}}{\rho_{j}}f\big{(}x_{ j}\big{)}\,W\big{(}x_{i}-x_{j},h\big{)} \tag{2}\]  
In Formulas (1) and (2), \(f\) is the field function, \(\Omega\) is the calculation area, \(x\) is the coordinate vector, \(h\) is the smooth length, \(W\) is the kernel function, \(\rho\) is the density, and \(m\) is the mass.  
The SPH method is applied to the N-S equation, specifically as follows: mass conservation equation (Formula (3)), momentum conservation equation (Formula (4)) and energy conservation equation (Formula (5)); where \(P_{i}\) represents the pressure of particle \(i\), \(v_{i}\)\({}^{\beta}\) represents the velocity of particle \(i\) in the \(\beta\) direction, \(x_{i}\)\({}^{\beta}\) represents the coordinate of particle \(i\) in the \(\beta\) direction, \(\sigma_{i}\)\({}^{\beta}\), \(e_{i}\)\({}^{\beta}\) represent the stress and strain tensor of  
\[\frac{de_{i}}{dt}=\frac{1}{2}\sum_{j=1}^{N}m_{j}\left(\frac{p_{i}}{\rho_{i}^{2}}+ \frac{p_{j}}{\rho_{j}^{2}}\right)\rho_{ij}^{\beta}\frac{\partial W_{ij}}{\partial x _{i}^{\beta}}+\frac{\mu_{i}}{2\rho_{i}}\varepsilon_{i}^{a\beta}\varepsilon_{j}^{ a\beta} \tag{5}\]  
The advantage of SPH algorithm to deal with the problem of fluid-solid impact is that its convection term is directly simulated by the movement of particles, which completely eliminates the problem of numerical divergence on the free interface, and ensures the clear and accurate tracking of the free interface; the mesh-free feature also avoids the net problems caused by lattice distortion and reconstruction.