## 4 First and Conventional Quadratic Edge Elements
### A New Type of Quadratic Edge Element  
There are two edges on the side of a conventional quadratic hexahedron element. Within an edge element, because the curls of the edge shape functions are linearly dependent, many edge variables are redundant. The computation time and storage requirements of these elements has increased significantly. Therefore, eliminating these redundant variables helps improve efficiency at no cost to computational accuracy. In this section, we adopt a new method to eliminate redundant variables from the conventional quadratic element.  
Let us first consider a one-sided conventional quadratic edge element, as shown in Figures 3 and 4. We named the two edges on this side edge1 and edge2. \(N_{1}\) and \(N_{2}\) are the shape functions of edge1 and edge2, and their corresponding edge variables are \(A_{1}\) and \(A_{2}\). Because one of these is redundant, we decided to eliminate \(A_{2}\) and consider \(A_{1}=A_{2}\). Assume \(A_{\text{side}}\) is the vector field interpolation contribution of the two edges:  
\[\begin{split} A_{\text{side}}=N_{1}A_{1}+N_{2}A_{2}\\ =(N_{1}+N_{2})A_{1}\end{split} \tag{47}\]  
Make a line integration of \(A\) along this side and name it as \(A_{\text{unif}}\). Note that the following integration is orthogonal:  
\[\int_{eKL}N_{IL}\text{d}s=\delta_{(IL)(KL)} \tag{48}\]  
Here, d\(s\) is the differential vector of the line element along edge \(eKL\), and \(\delta_{(IL)(KL)}\) represents the Kronecker delta. Thus, the integration \(A_{\text{unif}}\) is written as:  
\[A_{\text{unif}}=\int\limits_{\text{edge1}}A\cdot\text{d}s+\int\limits_{\text{ edge2}}A\cdot\text{d}s=A_{1}+A_{2}=2A_{1} \tag{49}\]  
Then Equation (47) is rewritten as:  
\[A_{\text{side}}=(N_{1}+N_{2})A_{1}=(N_{1}+N_{2})\frac{A_{\text{unif}}}{2}=N_{ \text{unif}}A_{\text{unif}} \tag{50}\]  
In Equation (50), we obtain a new shape function \(N_{\text{unif}}=(N_{1}+N_{2})/2\) for the edge, as shown in Figure 5b. According to this, \(N_{\text{unif}}\) can be written in detail as follows:  
Figure 5: Two-edge unification: (**a**) edge before unification; (**b**) edge after unification.  
For the serendipity quadratic hexahedron element, the edges on the sides along the \(\xi\)-axis direction:  
\[N_{k}=\frac{1}{2}(1+4\zeta_{k}\xi)\left(1-\eta^{2}\right)\left(1-\xi^{2}\right)\Delta\zeta \tag{51}\]  
The edges on the sides along the \(\eta\)-axis direction:  
\[N_{j}=\frac{1}{8}\left(1+\xi_{j}\xi\right)\left(1+\zeta_{j}\xi\right)\left( \xi_{j}\xi+\zeta_{j}\xi-1\right)\Delta\eta \tag{52}\]  
The edges on the sides along the \(\zeta\)-axis direction:  
\[N_{k}=\frac{1}{8}(1+\xi_{k}\xi)(1+\eta_{k}\eta)(\xi_{k}\xi+\eta_{k}\eta-1)\Delta\zeta \tag{53}\]  
For the Lagrange quadratic hexahedron element, the edges on the sides along the \(\xi\)-axis direction:  
\[N_{i}=\frac{1}{8}\eta\zeta(\eta_{i}+\eta)(\zeta_{i}+\zeta)\Delta\xi \tag{54}\]  
The edges on the sides along the \(\eta\)-axis direction:  
\[N_{j}=\frac{1}{8}\xi\zeta\left(\xi_{j}+\xi\right)\left(\zeta_{j}+\zeta\right)\Delta\eta \tag{55}\]  
The edges on the sides along the \(\zeta\)-axis direction:  
\[N_{k}=\frac{1}{8}\xi\eta(\xi_{k}+\xi)(\eta_{k}+\eta)\Delta\zeta \tag{56}\]  
The shape functions of the edges in the element and the faces remained intact. Therefore, we obtained two new quadratic edge elements (Figure 6). There is only one edge on the sides of the two new quadratic edge elements (Figure 6); therefore, the redundant variable can be effectively eliminated. Table 1 lists various types of elements and the distribution of their basis functions.