# Wire-Grid and Sparse MoM Antennas: Past Evolution, Present Implementation, and Future Possibilities
## Why MoM? ### Overview of Computional Methods  
In the past, methods were developed to efficiently solve general electromagnetic (EM) problems, which were based on either differential or integral equations. The resulting systems of linear equations are solved either by direct methods or iteratively. The development of computer technology, combined with the development of fast algorithms with reduced computational complexity and memory requirements, has made it possible to accurately solve problems of electrically large objects numerically. These numerical methods are based on solving partial differential equations (e.g., using the finite difference method [FDM] or finite element method [FEM]) that are reduced to sparse matrices, or solving integral equations that are converted to dense matrix equations (e.g., using the method of moments [MoM]). Solving EM problem with \(N\) unknown by a direct solver (Gauss elimination) requires computational complexity O(\(N^{3}\)) and memory requirements O(\(N^{2}\)). However, by correctly ordering the matrix elements, it is possible to effectively compress and solve the corresponding band matrix equation with computational complexity O(\(N\)). The method of conjugate gradients (MCG) [32] is used to solve the Poisson equation. Thus, it converges in O(\(N^{0.5}\)) iterations for two-dimensional problems and in O(\(N^{0.33}\)) for three-dimensional problems, respectively. When a multigrid method is used to solve the same equation, the number of iterations does not depend on the order of the matrix [33]. As a consequence, the total computational cost associated with the MCG for such problems scales as O(\(N^{1,5}\)) for two-dimensional problems and O(\(N^{1,33}\)) for three-dimensional problems. For multigrid methods, the total computational cost scales as \(N\). The finite difference time domain (FDTD) method is one of the most popular methods for solving electrodynamic problems numerically and was first described in [34]. However, the abbreviation of the method's name was suggested by the author of [35]. The method is based on a procedure deployed in the time domain during which continuous electromagnetic waves are replaced by their discrete analogues [36]. The discrete representation is realized for the space and time included in [PERSON]'s equations. This method is a simple and straightforward approach to solving [PERSON]'s equations and has an explicit solution scheme that does not require linear algebraic equations (SLAEs) to be stored, as well as an implicit scheme that reduces to a system of SLAEs [37]. Besides, the FDTD method quite simply describes the inhomogeneity of the medium. Working in the time domain allows for results to be obtained in a wide range of frequencies. This method allows the researcher to specify the material at each point of the volume and choose not only from a wide range of metals and dielectrics, but also materials with nonlinear properties. It allows one to directly simulate the effects on the holes as well as the shielding effects, while the fields inside and outside the shield can be calculated either, directly or not. However, this method, like all methods, has disadvantages. First, the counting volume must be divided into a very large number of cells (the subarea [cell] size is small compared to the smallest wavelength), which means large memory and simulation time costs. This makes it difficult to simulate problems with long, thin spatial structures, such as the fields obtained from currents along conductors. Second, the calculation of the fields is performed at each point of the counting volume, so if you want to find the field at a certain distance from the source, it probably means that the counting volume will be excessively large. But it must be finite to fit in the computer's memory, which in most cases is achieved by setting artificial boundary conditions on the counting volume, the use of which can lead to data distortion.