## 4 Results
### The Nodes Displacement Solutions  
In this part, we will examine the displacement solution for the conical pick. The reference value of nodes' displacements along y-axis \(u_{2}\) and z-axis \(u_{3}\) are plotted in Figure 13 and we draw the approximate displacement solutions along the y-axis \(u_{2}\) and z-axis \(u_{3}\) using ES-FEM in Figure 14. From these figures it is distinctly observed that the displacement solutions of ES-FEM are reasonable and accurate.  
Table 6 shows the comparison of displacements of ES-FEM with FEM, NS-FEM, and FS-FEM. Figure 15 is plotted by the distance from the tip of the conical pick on the horizontal axis and the node's displacement on the vertical. The comparison is fair and rigorous as long as the same distribution of nodes is used. Overall, the displacements of ES-FEM are closer to the reference solutions than those of FEM. This also verifies that, regardless of strain and displacement, ES-FEM is the most accurate and stable method compared with FEM and other S-FEMs.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Type of Mesh** & **FEM** & **ES-FEM** & **FS-FEM** & **NS-FEM** \\ \hline Mesh 1 & 0.230063 & 0.05698 & 0.18135 & 0.3587 \\ Mesh 2 & 0.175547 & 0.09385 & 0.12717 & 0.2911 \\ Mesh 3 & 0.17543 & 0.03804 & 0.13738 & 0.2458 \\ Mesh 4 & 0.16957 & 0.04057 & 0.13147 & 0.2390 \\ Mesh 5 & 0.143046 & 0.048381 & 0.10709 & 0.2019 \\ Mesh 6 & 0.136554 & 0.04841 & 0.10297 & 0.1914 \\ Mesh 7 & 0.111434 & 0.052456 & 0.07997 & 0.1655 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Error in strain energy norm obtained using different methods for the conical picks.  
Figure 13: The reference value of nodes’ displacements along (**a**) the y-axis \(u_{2}\) and (**b**) z-axis \(u_{3}\).  
In order to further verify the accuracy and stability, the error of displacement of the ES-FEM model was conducted. Displacement errors and convergence rates of these numerical methods are analyzed using displacement norms. The displacement norm is defined as:  
\[e_{d}=\left[\int\limits_{\Omega}\frac{1}{2}(\mathbf{u}-\widetilde{\mathbf{u}})^ {T}\mathbf{c}(\mathbf{u}-\widetilde{\mathbf{u}})d\Omega\right]^{1/2}=\left[ \sum\limits_{i=1}^{N_{e}}\int\limits_{\Omega_{i}^{2}}\frac{1}{2}(\mathbf{u}- \widetilde{\mathbf{u}})^{T}\mathbf{c}(\mathbf{u}-\widetilde{\mathbf{u}})d \Omega\right]^{1/2} \tag{13}\]  
where \(\mathbf{u}\) is the reference solution of the displacements, and \(\widetilde{\mathbf{u}}\) is the numerical solution of the displacements using a numerical model. Table 7 compares the solution errors in the displacement norm obtained using FEM, ES-FEM, FS-FEM, and NS-FEM. The error of ES-FEM is about 0.216 times that of FEM, 0.8 times that of FS-FEM, and 0.213 times that of NS-FEM. In terms of convergence rate, compared with other methods, the ES-FEM performs the best.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**The Distance from the Tip** & **FEM** & **ES-FEM** & **FS-FEM** & **NS-FEM** & **Reference Solution** \\ \hline
2 & 9.294 & 9.5071 & 9.4433 & 9.6623 & 9.483 \\
4 & 9.157 & 9.363 & 9.3193 & 9.565 & 9.355 \\
5 & 8.95 & 9.0689 & 9.0429 & 9.2355 & 9.0555 \\
6 & 8.33 & 8.5585 & 8.5044 & 8.7291 & 8.5391 \\
7 & 7.6 & 7.7885 & 7.7633 & 8.03 & 7.784 \\
8 & 6.011 & 6.0869 & 6.055 & 6.2914 & 6.0714 \\
9 & 4.506 & 4.5986 & 4.5898 & 4.706 & 4.596 \\
10 & 3.613 & 3.6663 & 3.664 & 3.8142 & 3.6642 \\
11 & 2.907 & 2.979 & 2.9783 & 2.999 & 2.979 \\
12 & 2.39 & 2.4642 & 2.4541 & 2.686 & 2.4596 \\
14 & 1.881 & 1.981 & 1.9506 & 2.11 & 1.9783 \\ \hline \hline \end{tabular}
\end{table}
Table 6: The displacement along the tip using FEM, ES-FEM, FS-FEM, and NS-FEM for the conical pick (\(10^{-3}\) mm).  
Figure 14: The approximate displacement solutions along (**a**) the y-axis \(u_{2}\) and (**b**) z-axis \(u_{3}\) using ES-FEM.