# Wire-Grid and Sparse MoM Antennas: Past Evolution, Present Implementation, and Future Possibilities
## Why MoM?
### Overview of Computional Methods
Nevertheless, in the case of using FDTD to solve the wave equation directly in the time domain, the computational complexity is the same as the MCG (\(N^{1.5}\) for the two-dimensional problem and \(N^{1.33}\) for the three-dimensional problem) [38], except that FDTD generates a solution for the entire time domain, and therefore for all frequencies at once. This algorithm is optimal because it generates O(\(N^{x}\)) solutions using O(\(N^{x}\)) operations. The Finite Element Method (FEM) is widely used in the mechanical analysis of structures. Despite the fact that the mathematical treatment of the method was proposed in 1943 by [PERSON] [39] for solving electromagnetic problems, this method was not applied until 1968. Since then it has been used in the analysis of waveguides, electrical machines, semiconductor devices, multiconductor lines, electromagnetic radiation from biological objects, etc. [40; 41; 42; 43; 44; 45; 46; 47]. The general nature of the FEM allows for the building of universal computer programs based on it in order to solve a wide range of problems. Therefore, programs developed to solve problems from other disciplines can be and have been successfully applied to solve problems from another discipline with or without minor modifications [48]. The main idea of FEM is that any continuous value (temperature, pressure, etc.) is approximated by a discrete model that is constructed using a set of piecewise continuous functions defined on a finite number of subdomains [49]. The algorithm for constructing a discrete model of a continuous quantity is as follows. A finite number of points are fixed in the area under consideration. These points are further called nodes. Assume that the studied continuous value at each node is a variable that is to be determined in the process of solving the problem. The region of the continuous value change is divided into elements. These elements have common nodes between them and, together, approximate the shape of the entire region. The continuous value is approximated within each element by a polynomial, the coefficients of which are calculated according to the quantities of this value in the nodes. Each element is approximated by its own polynomial, and the coefficients of the polynomials are chosen so that the continuity of the value along the boundaries of neighboring elements is preserved. Then, the finite elements are combined into an assembly. In this assembly, the nodal quantities of the unknown functions are chosen in such a way that a sufficient approximation of a continuous distribution is achieved. This step leads to a SLAE with respect to the nodal quantities. After that, the resulting system has to be solved, i.e., the nodal quantities must be found. To approximate a continuous region, when solving the formulated boundary problems, a system of pyramidal elements is used that allows writing down a system of equations for E (or H). Their solution determines the interpolation polynomial coefficients. Then the finite elements are combined into an assembly, and the interpolation functions are expressed through global nodal quantities and global coordinates (related to the model as a whole). A sufficiently complete presentation of the method is given in [50; 51]. The main advantages of FEM are the clear physical meaning at all stages of the calculation, which allows one to easily control the results, and the simplicity of calculating the characteristics of combined structures that include multiple elements of various sizes. The disadvantages of FEM lie in the need for an efficient finite element generator; the complexity of the equations; the discretization of the occupied region with a variable step, which leads to the increase in the computational cost while forming a volumetric model; and the increase in the computational cost in multicomponent environments. If the FEM nodes are ordered correctly, the dense matrix will appear only in the lower right corner of the system matrix [52; 53].