# Analysis of Movement Law and Influencing Factors of Hill-Drop Fertilizer Based on SPH Algorithm
## 2 Methodology
### Fertilizer Motion Simulation  
Soil is an unsaturated multiphase system with a complicated constitutive relationship, and soil disturbance is large when soil is furrowed by agricultural machinery. The finite element method is prone to cause mesh distortion and lower accuracy. SPH is a meshless algorithm that has been gradually developed in the last 20 years. Because the SPH algorithm does not need to define mesh, it can avoid mesh distortion and accuracy damage caused by large deformation, and it can effectively deal with problems such as soil multiphase, looseness and dispersed physical properties [16; 17]. Previous studies have proven that SPH can solve fluent problems effectively, such as free surface flow [18; 19] and bubble rising [20; 21]. The process of disturbing soil can be seen as the flow of soil particles, and, therefore, SPH should be an effective method for simulating disturbance processes. The kernel approximation and particle approximation are two very important steps in the SPH method [22]. To correctly characterize particle motion, an approximate function is needed to characterize particle motion information. The integral expression of the SPH function \(f(x)\) can be expressed as a discretized form that supports the summation of all particles in the domain [23; 24]:  
Figure 1: Corn hill-drop fertilizer planter. \[\prod f(x_{i})=\sum_{j=1}^{n}\frac{m_{j}}{p_{j}}f\left(\vec{x}_{i}\right)W\left( \vec{x}-\vec{x}_{j},h\right) \tag{1}\]  
The function expression of the i-th particle is:  
\[f\left(x_{i}\right)=\sum_{j=1}^{n}\frac{m_{j}}{p_{j}}f\left(\vec{x}-\vec{x}_{i }\right)W\left(\vec{x}-\vec{x}_{j},h\right) \tag{2}\]  
In the formula, \(h\) is the particle smooth length in the \(W\) influence domain; \(m_{j}\) is the particle mass; \(p_{j}\) is the particle density; and \(n\) is the total number of particles in the support domain. In this work, we chose the cubic spline function [25], which closely resembles a Gaussian function [26], as our smoothing function. It is defined using an auxiliary function \(\theta\left(x\right)\):  
\[W(x,h)=\frac{1}{h(x)^{v}}\theta(x) \tag{3}\]  
In the formula, \(W\) is the smooth kernel function (interpolation kernel), \(v\) is the space dimension, and \(H\) is the particle smooth length. The expression of the cubic B-spline kernel function is:  
\[\theta(u)=\left\{ \begin{array}{cl}\left(1-1.5u^{2}+0.75u^{3}\right)C&0<|u| \leq 1\\ 0.25C(2-u)^{3}&1<|u|<2\\ 0&2\leq|u|\end{array} \right. \tag{4}\]  
In the formula, \(C\) is a constant and is determined by the space dimension. The particle smooth length has a significant impact on efficiency and calculation accuracy [27]. The fixed smooth length can cause problems, such as numerical distortion and compression instability. The variable smooth length can keep the same number of particles in the domain. The range of variation is:  
\[h_{\min}h_{0}<h<h_{\max}h_{0} \tag{5}\]  
In the formula, \(h_{0}\) is the initial smooth length. The value of the smoothing length is chosen based on actual problems. The smooth length is between 0.2 and 2 times the initial smooth length, which can meet most requirements [22]. In this study, we chose 1.5 times particle spacing as the smoothing length. The SPH algorithm discretizes a continuum into an interactive particles, and applies speed, displacement and other physical quantities to each particle. The mechanical behavior of the overall system can be obtained by tracking the motion trail of each particle [28; 29; 30]. In view of problems such as easy concentration of fertilizer and insufficient fertilization depth due to fertilizer movement, the designed hill-drop fertilizer device was modeled, and the disturbance process of fertilizer in the soil was simulated by the cover. The simulation model mainly included opener, cover, fertilizer, and soil. Because of the complex structure of the hill-drop fertilizer device, the indirect modeling method was chosen in ANSYS/LS-DYNA.