# On Modeling Antennas Using MoM-Based Algorithms: Wire-Grid versus Surface Triangulation
## 2 Materials and Methods
#### Triangle-Grid: Algorithm  
Depiction of the object geometry in the simulation using Triangle-Grid is executed by representing it as a set of vertexes arranged in three-dimensional space, which are then linked to form a surface triangular mesh. To construct such meshes, the Delaunay criterion is frequently used. According to this criterion, the circumcircle of every triangle in the mesh should not contain any other vertex [336]. This maximizes the minimum angle within each triangle of the mesh, making it closer to uniform, which can increase the accuracy of the solution [11]. Each RWG function is defined over each pair of triangles \({T_{n}}^{+}\) and \({T_{n}}^{-}\) that shares an edge with length \(l_{n}\)  
\[{\bf f}_{n}({\bf r})=\pm\frac{l_{n}}{2A_{n}^{(\pm)}}\rho_{n}^{(\pm)},{\bf r} \in T_{n}^{(\pm)} \tag{42}\]  
where \(A_{n}^{\pm}\) and \(\rho_{n}^{\pm}\) are the area and the vector of source point in the local triangle \(T_{n}^{\pm}\), respectively. Approximately, the basis function of the common edge \(n\) corresponds to a small finite electric dipole characterized by length \(d_{n}=1\,{\bf r}_{n}{}^{c-}-{\bf r}_{n}{}^{c+}\) (Figure 6). The current density \(\mathbf{J}\) over the entire surface of the modeling structure, which consists of \(M\) common edges of the surface mesh, can be represented as a superposition of basic functions as follows  
\[\mathbf{J}=\sum_{m=1}^{M}I_{m}\mathbf{f}_{m}(\mathbf{r}) \tag{43}\]  
where \(I_{m}\) are the normal components of the current densities. The elements of the impedance matrix (15) when using the RWG functions are rewritten as follows  
\[z_{mn}=l_{m}\big{(}j\omega\big{(}\mathbf{A}_{mn}^{+}\cdot\mathbf{\rho}_{m}^{c +}/2+\mathbf{A}_{mn}^{-}\cdot\mathbf{\rho}_{m}^{c-}/2\big{)}+\Phi_{mn}^{-}- \Phi_{mn}^{+}\big{)}, \tag{44}\]  
\[\mathbf{A}_{mn}^{\pm}=\frac{\upmu}{4\pi}\left[\frac{l_{n}}{2A_{n}^{+}}\int \limits_{T_{n}^{+}}\rho_{n}^{+}\left(\mathbf{r}^{\prime}\right)G_{m}^{\pm} \big{(}\mathbf{r}^{\prime}\big{)}dS^{\prime}+\frac{l_{n}}{2A_{n}^{-}}\int \limits_{T_{n}^{+}}\rho_{n}^{-}\left(\mathbf{r}^{\prime}\right)G_{m}^{\pm} \big{(}\mathbf{r}^{\prime}\big{)}dS^{\prime}\right], \tag{45}\]  
\[\[PERSON]_{mn}^{\pm}=-\frac{1}{4\pi j\omega\epsilon}\left[\frac{l_{n}}{A_{n}^{+}} \int\limits_{T_{n}^{+}}G_{m}^{\pm}\big{(}\mathbf{r}^{\prime}\big{)}dS^{\prime} -\frac{l_{n}}{A_{n}^{-}}\int\limits_{T_{n}^{-}}G_{m}^{\pm}\big{(}\mathbf{r}^{ \prime}\big{)}dS^{\prime}\right] \tag{46}\]  
where  
\[G_{m}^{\pm}\big{(}\mathbf{r}^{\prime}\big{)}=\frac{e^{-jk|\mathbf{r}_{m}^{\pm }-\mathbf{r}^{\prime}|}}{\left|\mathbf{r}_{m}^{\pm}-\mathbf{r}^{\prime}\right|}. \tag{47}\]  
At the stage of calculating the diagonal elements of the impedance matrix, singularity arises. This singularity is eliminated by dividing the original triangles of mesh into 9 equal small sub-triangles by the use of the \"one-third\" rule [337; 338]. Assuming that the integrand is constant within each of the sub-triangles, the original integral of the Green's function is replaced by a sum of the following form  
\[\int\limits_{T_{m}}G(\mathbf{r})dS^{\prime}=\frac{A_{m}}{9}\sum\limits_{i=1}^ {9}G(\mathbf{r}_{i}^{c}) \tag{48}\]  
Figure 6: RWG basis function defined for a pair of adjacent triangles. where \(\mathbf{r}_{i}^{c}\), \(i=1,\,\ldots,\,9\) are the midpoints of nine sub-triangles and \(A_{m}\) is the area of the primary triangle. To inject the excitation, the voltage \(V_{in}\) is set in an infinitesimal gap \(\Delta\) between two adjacent triangles with a common edge \(n\) (Figure 7). The electric field strength in the gap is defined as  
\[\mathbf{E}=-\
abla\varphi=\frac{V_{in}}{\Delta}\mathbf{n}_{y} \tag{49}\]  
where \(V\) is the electric potential and \(\mathbf{n}_{y}\) is the unit normal to the adjacent edge of the triangles.