## 6 Elimination of Degrees of Freedom  
Because the \(\vec{A},\vec{T}-\vec{A}\) formulation uses two vector potentials in \(\overline{\Omega}_{\rm c}\), the total number of degrees of freedom in \(\overline{\Omega}_{\rm c}\) is \(2\cdot n_{e}\), where \(n_{e}\) represents the total number of edges in the finite element mesh in \(\overline{\Omega}_{\rm c}\). This is significantly more degrees of freedom compared to the formulations that use one vector and one scalar potential in \(\overline{\Omega}_{\rm c}\) and have a total of \(n_{e}+n_{n}\) degrees of freedom in \(\overline{\Omega}_{\rm c}\), where \(n_{n}\) is the number of nodes in the finite element mesh in \(\overline{\Omega}_{\rm c}\). A higher number of degrees of freedom adversely affects the working memory and duration of the simulation. Because \(\vec{J}_{\alpha\beta\gamma}\) is neglected when calculating \(\vec{J}_{\alpha\beta}\), the current vector potential \(\vec{T}\) should only define \(\vec{J}_{\alpha\beta}\) in (15). Thus, in an orthogonal \(\alpha\beta\gamma\)-coordinate system defined locally within a lamination sheet, where the \(\gamma\)-direction represents the normal direction of the lamination sheet, (15) can be written as  
\[\vec{J}_{\alpha\beta}=\
abla\times\vec{T}=\partial_{\beta}T_{\gamma}\vec{a}_ {\alpha}-\partial_{\alpha}T_{\gamma}\vec{a}_{\beta}. \tag{31}\]  
As can be seen from (31), only \(\vec{T}_{\gamma}\) is needed, whereas the tangential components \(\vec{T}_{\alpha}\) and \(\vec{T}_{\beta}\) are zero, and therefore redundant. If the right prismatic finite elements (wedge or hexahedron) with the base parallel to \(\alpha\beta\) plane and the height aligned with the \(\gamma\)-direction are used, then it is possible to geometrically decouple the degrees of freedom associated with the \(\vec{T}_{\gamma}\) component from the degrees of freedom associated with the \(\vec{T}_{\alpha}\) and \(\vec{T}_{\beta}\) components, as shown in Figure 5. Because \(\vec{T}\) is interpolated with edge basis functions, the degrees of freedom associated with the edges lying in the \(\gamma\)-direction then approximate only the \(\vec{T}_{\gamma}\) component, and the degrees of freedom associated with the edges belonging to the base of the element approximate \(\vec{T}_{\alpha}\) and \(\vec{T}_{\beta}\) components. In Figure 5, \(T_{i}\) represents the degree of freedom associated with edge \(e_{i}\), and is calculated as a line integral of the current vector potential \(\vec{T}\) along \(e_{i}\). Because each edge of a finite element is assumed to be either aligned with the \(\gamma\)-direction or parallel to the \(\alpha\beta\) plane, it is associated with either \(T_{i\gamma}\) or \(T_{i\alpha\beta}\). If the \(e_{i}\) edge is aligned with the \(\gamma\)-direction, then it is associated with \(T_{i\gamma}\) degree of freedom:  
\[T_{i\gamma}=\int\limits_{e_{i}}\vec{T}d\vec{t}=\int\limits_{e_{i}}\left(T_{ \alpha}\vec{a}_{\alpha}+T_{\beta}\vec{a}_{\beta}+T_{\gamma}\vec{a}_{\gamma} \right)\vec{a}_{\gamma}dl=\int\limits_{e_{i}}T_{\gamma}dl. \tag{32}\]  
On the other hand, if the \(e_{i}\) edge is parallel to the \(\alpha\beta\) plane, then \(T_{i\alpha\beta}\) follows as:  
\[T_{i\alpha\beta}=\int\limits_{e_{i}}\vec{T}d\vec{t}=\int\limits_{e_{i}}\left(T _{\alpha}\vec{a}_{\alpha}+T_{\beta}\vec{a}_{\beta}+T_{\gamma}\vec{a}_{\gamma }\right)(\vec{a}_{\alpha}dl_{\alpha}+\vec{a}_{\beta}dl_{\beta})=\int\limits_{e _{i}}T_{\alpha}dl_{\alpha}+\int\limits_{e_{i}}T_{\beta}dl_{\beta}. \tag{33}\]  
Because the \(\vec{T}_{\alpha}\) and \(\vec{T}_{\beta}\) components are equal to zero, each \(T_{i\alpha\beta}\) degree of freedom should be equal to zero. In order to achieve this, surface condition \(\vec{T}\times\vec{n}=0\), which implies that \(\vec{T}_{\alpha}=0\) and \(\vec{T}_{\beta}=0\), can be given on the basis of the prismatic finite elements. Figure 6a shows a sequence of the prismatic (wedge) finite elements extruded in the \(\gamma\)-direction in three layers between the two interface surfaces \(\Gamma_{\text{nc}}\).