# Numerical Analysis of the Installation Process of Screw Piles Based on the FEM-SPH Coupling Method
## 2 Basic Theory of SPH
### Solution of the SPH  
The core of the SPH method is interpolation, and an important step in the computation is the domain search. As is shown in Figure 1, this search is carried out on the spherical region with a radius of 2 \(h\) (smoothing length) in the influence domain of each particle, and all particles in this region are listed for comparison at each time step. At the same time, the smooth length \(h\) is used to ensure that there are sufficient influence domains within the critical domain of a particle to avoid problems with material compression and expansion. When the SPH particles are separated from each other, the smooth length will increase. Further, when particles are moved closer to each other, the smooth length decreases. This can ensure that the same number of particles are contained in the critical domain to complete the calculation. For the solution of the SPH method, it is usually divided into two steps. The first step is to obtain the field function in the continuous form as an integral representation (Kernel function). The second step is to approximate the SPH particles to achieve the particle discretization of the integral expression. In the derivation of an SPH formulation, the coordinate integral form of the particle is usually expressed as:  
\[F(x)=\int_{v}F\big{(}x^{\prime}\big{)}\partial\big{(}x-x^{\prime}\big{)}dx^{\prime} \tag{1}\]  
where \(v\) is volume domain, \(x\) is coordinate vectors of particle, \(x^{\prime}\) is coordinate vectors of any particle in the volume domain and \(\partial(x-x^{\prime})\) is the Dirac delta function [29], and it can be given as:  
\[\partial\big{(}x-x^{\prime}\big{)}= \begin{cases}1,\ x=x^{\prime}\\ 0,\ x\
eq x^{\prime}\end{cases}  \tag{2}\]  
Figure 1: The domain search of the SPH method. In fact, the Dirac delta function is difficult to realize because it has only one non-zero value. It is usually replaced by the smoothing function \(W(\mathbf{x}-\mathbf{x}^{\prime},\ h)\). Therefore, the coordinate integral form of the particle can also be expressed as:  
\[F(x)=\int_{\mathcal{D}}F\big{(}\mathbf{x}^{\prime}\big{)}W\big{(}\mathbf{x}-\mathbf{x}^{ \prime},\ h\big{)}\mathbf{dx}^{\prime} \tag{3}\]  
where \(h\) is the smoothing length. The smoothing function is usually expressed by \(B\) spline smooth functions. It can be expressed as:  
\[W\big{(}\mathbf{x}-\mathbf{x}^{\prime},\ h\big{)}=W(q,\ h)=\alpha_{q}\times\left\{ \begin{array}{c}\frac{2}{8}-q^{2}+\frac{q^{3}}{2}\ 0\leq q<1\\ \frac{(2-q)^{2}}{6}\ 1\leq q<2\\ 0\ q\geq 2\end{array} \right. \tag{4}\]  
where \(q=|\mathbf{x}-\mathbf{x}^{\prime}|\) is the distance between particles and \(\alpha_{q}\) is \(\frac{15}{7\pi h^{2}}\) and \(\frac{3}{2\pi h^{2}}\) in two-dimensional space and three-dimensional space. The smoothing function \(W\) is related to the smooth length \(h\) of the particle influence domain and the corresponding position between the particles. According to the definition of the SPH method, the continuum, when is discretized by the SPH method, is a number of particles with a certain mass and volume. Thus, for the second step of the solution of the SPH, the physical quantities of the particle can be summed up by the physical quantities of all particles in the particle's influence area. This is also an important reason for the discrete SPH particles. The SPH method is based on a quadrature formula for moving particles, so the particle approximation of \(f(x)\) can be expressed as:  
\[f(x)\approx\sum_{j=1}^{M}\frac{m_{j}}{\rho_{j}}f\big{(}\mathbf{x}_{j}\big{)}W(\bm {x}-\mathbf{x}_{j},h) \tag{5}\]  
Therefore, the function of the particle \(i\) is approximated as:  
\[f(x_{i})\approx\sum_{j=1}^{M}\frac{m_{j}}{\rho_{j}}f\big{(}\mathbf{x}_{j}\big{)}W( \mathbf{x}_{i}-\mathbf{x}_{j},h) \tag{6}\]  
where \(\rho_{j}\) is the density of particle \(j\) and \(m_{j}\) is the mass of particle \(j\). \(M\) is total number of particles in the influence area at \(\mathbf{x}\).