# On Modeling Antennas Using MoM-Based Algorithms: Wire-Grid versus Surface Triangulation
## 2 Materials and Methods
### Wire-Grid
#### 2.2.2 Wire-Grid: Algorithm  
In this section, the algorithm of the wire-grid approach on which the Wire-Grid code is based is presented. Among the existing approaches to antenna modeling, the most promising from our point of view is the one based on the thin-wire approximation of metal component surfaces [92; 179]. In this approach, the conductor is assumed to be ideal in the form of a cylinder located along one of the coordinate axes (one-dimensional problem), with a cross-sectional radius much smaller than the wavelength of the radiated signal and the conductor physical length. This simplification allows one to use a scalar function instead of the vector function of surface current density, which greatly simplifies the problem to be solved [180; 181]. This approach can be applied to represent continuous body surfaces by a grid of wires (wire-grid or thin-wire-grid). The wire-grid approach in combination with the formulations of the EFIE and MoM is widely used in the well-known program NEC. Compared to the surface triangular approximation and the use of RWG functions, this approach demonstrates its inherent weaknesses in modeling fields in the near zone, but it has proved its efficiency on a large number of scattering and radiation problems in the far zone, associated with conductive bodies of arbitrary shape [182; 183; 184]. Therefore, wire-grid approximation is still widely used in computational electrodynamics [185]. The advantage of the wire-grid over the triangle-grid is the necessity to calculate linear integrals, and the disadvantage is the necessity to use a large number of wires to represent curvilinear boundaries. For three-dimensional electrodynamic problems, the use of MoM requires the following solution steps. First, the conductive parts of a given structure are replaced by equivalent surface electric currents, and then, the problem of the environment excitation by these currents is solved. The corresponding boundary conditions are imposed on the obtained solution, which are further used to calculate the equivalent currents. An important aspect of this solution process is the discretization of conductive surfaces into elementary areas and the approximation of the current within each of them. Therefore, in order to explain the work of the algorithm used here, we chose a simple well-known example of the current distribution on the surface of thin wire, usually used for these purposes. Consider the distribution of current on the surface of an arbitrarily oriented thin wire of length \(L\) and radius \(a\) (\(a<<\lambda\)). Figure 1 shows an example of the orientation of a wire along the \(x\)-axis. The surface area is denoted by \(S\). The applied field is assumed to be arbitrary, so the algorithm described below is applicable to both the antenna problem and the scattering problem. The current and charge densities are considered to be approximated by current and charge filaments on the conductor axis, and the current flows only in the direction of the conductor axis and is zero at its ends. Condition (1) is applied only to the axial component of the electric field strength [9]. Figure 1: General view (**a**) and segmentation of a thin wire (**b**). The solution of the scattering problem is based on the following steps:  
1. Excite the wire by an external electric field (\(\mathbf{E}^{\mathrm{inc}}\)). 2. Assume that the tangential component of the electric field strength vector on the surface of the wire is equal to zero. Then, for an arbitrarily oriented in space wire you get \[E_{l}=0.\] (16)
3. Determine the relationship between the incident and scattered electromagnetic waves using (1). At that time \[E_{l}^{inc}=-E_{l}^{sat}.\] (17)
4. Write (2), (5), and (6) on the wire surface \(S\) taking into account (4) and (17). As a result, the following is obtained \[-E_{l}^{\mathrm{inc}}=-j\omega A_{l}-\frac{\partial\Phi_{l}}{\partial l}\] (18) where \[A_{l}=\mu\int\limits_{ \begin{subarray}{c}along\\ the\ axis\end{subarray} }\frac{e^{-jkR}}{4\pi R}\,I(l)dl,\] (19) \[\Phi_{l}=\frac{1}{\epsilon}\int\limits_{