## 4 Results  
In the section, in order to illustrate the efficiency and accuracy of the present ES-FEM algorithm for conical pick problems, the numerical results of EF-FEM are listed and are compared with the solutions of other S-FEM models and FEM in stress, strain energy, and displacement.  
The geometrical parameters of the practical conical pick used in our experiment are given in Figure 5. The material parameters in the experiment used are: \(E=650,000\) MPa and \(v=0.3\). At the same time, the normal force of \(F_{z}=5.5\) kN, the cutting force of \(F_{y}=0.6\) kN, and the lateral force of \(F_{x}=0.002\) kN are imposed on the top surface of the conical pick, and the force area is three-quarters of the top area shown in Figure 6. First, the numbers and area of the forced elements are obtained based on the ES-FEM mesh; and then the forces of one element are imposed on three nodes through the face normal algorithm. Besides, we fully fix the bottom of the conical pick.  
For practical problems, it is very difficult to find the analytical solution. Hence, we calculate the reference solutions of the problem using the standard FEM with the mesh of 65,686 T4 elements, and 197,058 degrees of freedom.  
In Figure 7a, the conical pick is divided by tetrahedron elements as the background mesh. We use the connectivity of the background mesh listed above the tables and the algorithm mentioned in Section 3.1 to create an ES mesh for our numerical simulation (Figure 7b). From this figure, we can observe the smoothing domains are surrounding the edges in the background mesh. Seven background meshes are applied to different methods such as FEM, ES-FEM, FS-FEM, NS-FEM, which have the same numbers of nodes, elements, and degrees of freedom. The detailed information of these seven meshes is shown in Table 3.  
Figure 5: Geometrical model of the conical pick.  
reference value and FEM analysis results. Von-Mises stress is computed using the stress components in the form of:  
\[\sigma_{v}=\sqrt{\frac{1}{2}\big{[}\big{(}\sigma_{11}-\sigma_{22}\big{)}^{2}+ \big{(}\sigma_{22}-\sigma_{33}\big{)}^{2}+\big{(}\sigma_{33}-\sigma_{11}\big{)}^ {2}+6\big{(}\sigma_{23}^{2}+\sigma_{31}^{2}+\sigma_{12}^{2}\big{)}\big{]}} \tag{11}\]  
Figure 8 is the reference value of Von-Mises stress \(\sigma_{v}\) using the standard FEM with 197,058 degrees of freedom. Then we plot the distribution of von-Mises stress using the present ES-FEM and FEM with the mesh having 6413 T4 elements in Figure 9. It is easily seen that the Von-Mises stress \(\sigma_{v}\) distributions of the ES-FEM model are similar to that of FEM.  
Further, we compares the maximum Von-Mises stress of ES-FEM with FEM, NS-FEM, and FS-FEM against the degrees of freedom in Figure 10. It is obviously observed that FEM is the stiffest, and ES-FEM is softened by the smoothing technique. Besides, NS-FEM still keeps the property of upper bounds. Among the results of S-FEMs and the FEM, the ES-FEM solution is the closest one to the reference solution.  
Figure 8: The reference value of Von-Mises stress \(\sigma_{v}\).  
Figure 9: **(a)** The Von-Mises stress \(\sigma_{v}\) distribution using ES-FEM; **(b)** The Von-Mises stress \(\sigma_{v}\) distribution using FEM.  
Next, the distribution of Von-Mises stress along the generator of the tip which includes the max Von-Mises stress using different methods for the conical pick are computed and shown in Figure 11. We first find that the maximum von-Mises in the Figure 11 was about 7.5 mm away from the apex of conical pick, and the von-Mises would be steadily reducing with the distance increasing from the apex. In addition, it is shown that the ES-FEM Von-Mises are closer to the reference solutions than that of FEM. It was further proof that our ES-FEM for the practical conical pick is more accurate and efficient among other S-FEM methods and FEM.