# The Influence of Bit Edge Shape Parameters on Bone Drilling Force Based on Finite Element Analysis
## 4 Results and Analysis
### Particle Swarm Optimization  
Particle Swarm Optimization (PSO), proposed by Prof. [PERSON] and Prof. [PERSON] in 1995, is a stochastic parallel optimization algorithm that is simple and easy to implement. PSO algorithm has been paid much attention since it was introduced. PSO has been widely \begin{table}
\begin{tabular}{c c c} \hline \hline
**Coefficient of Regression** & **Estimated Value** & **Confidence Interval of Regression** \\  & & **Coefficient** \\ \hline \(\beta_{0}\) & 5.3306 & [2.9511, 7.7101] \\ \(\beta_{1}\) & 0.1777 & [\(-\)0.2789, 0.6343] \\ \(\beta_{2}\) & \(-\)0.0183 & [\(-\)0.2631, 0.2266] \\ \(\beta_{3}\) & 0.6808 & [0.5008, 0.8609] \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Regression results of drilling force prediction model. \begin{table}
\begin{tabular}{c c c} \hline \hline
**R\({}^{2}\)** & **F** & **P** \\ \hline
0.9503 & 31.8356 & 0.0011 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Model statistic scale. Figure 7: Residual diagram. used in single-objective optimization, constrained optimization, dynamic optimization, multi-objective optimization, and dynamic multi-objective optimization [39]. Particle swarm optimization is used to optimize the structure of the twist drill bit as follows:  
1. Particle swarm initialization  
Assuming that the optimal problem is in the \(D\) dimension when the initial number of particles is \(N\), the position of each particle can be represented by the \(D\) dimension vector and expressed as follows:  
\[X_{i}=\left[x_{i1},x_{i2},\cdots,x_{iD}\right]^{T}\qquad i=1,2,\cdots,N \tag{4}\]  
Similarly, the velocity of a particle is represented by a \(D\) vector:  
\[V_{i}=\left[v_{i1},v_{i2},\cdots,v_{iD}\right]^{T}\qquad i=1,2,\cdots,N \tag{5}\]  
In the particle swarm, the initial position and velocity of each individual must be random, and the number of groups must be reasonable. Otherwise, the search range will be too extensive, and the calculation will be too large; in this paper, three parameters are used to describe the position and velocity of particles to meet the search requirements. At the same time, it can avoid the position of particles beyond the predetermined range or cause high-speed flight problems. Firstly, the position of a single particle is confirmed according to the selected factor range. Based on the influence range's size, the particle's velocity is defined differently. Finally, the equation can realize the initialization of the particle swarm. \[ \begin{array}{l}x=\text{limit}(i,1)+(\text{limit}(i,2)-\text{limit}(i,1) )\cdot rand(N,D)\\ v=\text{limit}(i,1)+(\text{limit}(i,2)-\text{limit}(i,1))\cdot rand(N,D) \end{array}  \tag{6}\]
2. Calculation of individual and group fitness values  
Particles in the new position will be judged by the size of their fitness function and best if the value is more significant. Each particle updates the extremum of the individual according to the corresponding function value of the current position and compares it with the global extremum. 3. Update particle velocity and position  
In any iteration, the formula is derived from previous individual particles and population extremes. Update self-location and speed:  
\[ \begin{array}{l}v_{ij}(t+1)=wv_{ij}(t)+c_{1}r_{1}\big{(}P_{\text{best}}-x_{ ij}(t)\big{)}+c_{2}r_{2}\big{(}G_{best}-x_{ij}(t)\big{)}\\ x_{ij}(t+1)=x_{ij}(t)+v_{ij}(t+1)\end{array}  \tag{7}\]  
where, \(V_{i}\left(t\right)\) is the velocity of the particle \(i\) at the time of \(t\), \(w\) is the inertial weight, \(X_{i}\left(t\right)\) is the position of the particle at the time of \(t\), \(r_{1}\), \(r_{2}\) is a random number between \((0,1)\), \(c_{1}\), \(c_{2}\) is the learning factor, \(P_{best}\) is the optimal value for an individual, and \(G_{best}\) is the optimal value for a population. From the above formula, we can see that the particle renewal rate is affected by the inertia, self-factor and population factor, which represents the speed of the last iteration and the extreme value propensity to individuals and groups, respectively.