## 3 The Formulation of the Edge-Based Smoothed Finite Element Method (ES-FEM) for Conical Picks  
### Create the Edge-Based Smoothing Domains for Conical Picks  
Before constructing the ES-FEM model, we uses the typical linear element T4 for 3D problems as the background mesh, which can automatically mesh and is very suitable for complex problem domains. Based on the T4 mesh, the edge-based smoothing domain can be created following the algorithm introduced below. In order to illustrate the process clearly, a simple ES smoothing domain is plotted in Figure 4 and the descriptions of elements and faces are listed in Table 2. Nodes n\({}_{1}\), n\({}_{2}\), n\({}_{3}\), n\({}_{4}\) made up the element 1 (\(e_{1}\)); and nodes n\({}_{1}\), n\({}_{2}\), n\({}_{3}\), n\({}_{5}\) made up the element 2 (\(e_{2}\)). Face1 (\(f_{1}\)) contained nodes n\({}_{1}\), n\({}_{3}\), n\({}_{4}\); Face2 (\(f_{2}\)) contained nodes n\({}_{1}\), n\({}_{2}\), n\({}_{3}\); Face3 (\(f_{3}\)) contained nodes n\({}_{1}\), n\({}_{3}\), n\({}_{5}\). In fact, ES smoothing domain relies on each edge so the number of smoothing domains equals the number of edges in the background. The smoothed domain will adopt the edges on the element as the indexes. Take the element in the Figure 4, for example, there are two elements \(e_{1}\) and \(e_{2}\). We will discuss how to build the smoothing domain of edge n\({}_{1}\)n\({}_{3}\). The edge of n\({}_{1}\)n\({}_{3}\) is an internal edge which is related to the different elements. Let us focus on the element \(e_{1}\), because the process is the same as that in the element \(e_{2}\). In this element, the edge locates on the faces \(f_{1},f_{2}\), and \(f_{3}\). Then connect the center of one face, the centroid of this element, and one vertex of the edge to form a surface-segment of the smoothing domain. Repeat the process and obtain six-surface segments for one element, n\({}_{3}\)cf\({}_{1}\)ce\({}_{1}\), n\({}_{1}\)cf\({}_{1}\)ce\({}_{1}\), n\({}_{3}\)cf\({}_{2}\)ce\({}_{1}\), n\({}_{1}\)cf\({}_{2}\)ce\({}_{1}\). Similarly, we can build other six-surface segments for the element \(e_{2}\). It is clearly seen that the 3D-edge-based smoothed domain is enclosed by these surfaces. Hence, the ID of the smoothing domain is the same as the ID of the edge, which is surrounded by the smoothing domain. In Figure 4, one T4 element is separated by six smoothing domains according to the edges of this element.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Cutting Frequency** & **The Normal Force (kN)** & **The Cutting Force (kN)** & **The Lateral Force (kN)** \\ \hline
1 & 5.15 & 0.635 & 0.0019 \\
2 & 5.78 & 0.66 & 0.0021 \\
3 & 6.15 & 0.68 & 0.0027 \\
4 & 4.88 & 0.55 & 0.0015 \\
5 & 6.01 & 0.605 & 0.0014 \\
6 & 5.58 & 0.59 & 0.0022 \\
7 & 5.5 & 0.59 & 0.0024 \\
8 & 5.47 & 0.595 & 0.0026 \\
9 & 4.98 & 0.495 & 0.0012 \\ The average force & 5.5 & 0.6 & 0.002 \\ Standard deviation & 0.43955 & 0.0559 & \(5.385\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: The first nine cutting test data of the conical picks.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Element/Face** & **The Nodes of Element/Face** & **The Centroid/The Center** \\ \hline Element 1 (\(e_{1}\)) & n\({}_{1}\)n\({}_{2}\)n\({}_{3}\)n\({}_{4}\) & \(C_{e1}\) \\ Element 2 (\(e_{2}\)) & n\({}_{1}\)n\({}_{2}\)n\({}_{3}\)n\({}_{5}\) & \(C_{e2}\) \\ Face1 (\(f_{1}\)) & n\({}_{1}\)n\({}_{3}\)n\({}_{4}\) & \(CThrough the above introduction, the constructing algorithm needs the connectivity of element, face, edge, and node. Therefore, for creating a smoothing domain quickly, we first need to get the connectivity of nodes separately with elements, faces, and edges; the connectivity of edges separately with elements, faces, and nodes; the connectivity of faces separately with elements, edges, and nodes; and the connectivity of elements separately with faces, edges, and nodes.