# Wire-Grid and Sparse MoM Antennas: Past Evolution, Present Implementation, and Future Possibilities
## Why MoM?
### MoM Acceleration  
Computation time and memory requirements are greatly reduced by efficient moment matrix filling, the use of closed-form Green's functions, and symmetry in the problem formulation; thus, a circuit of moderate electrical size can be analyzed in an acceptable period of time on a personal computer. The solvers of integral equations usually use fewer unknowns than the solvers of differential equations, because in the first case only the induced sources are unknowns, whereas in the latter case the unknown is the field. However, the use of integral equation solvers results in dense matrices [83]. If the matrix equation is solved using the LU-decomposition (Gaussian elimination) or the iterative method (CG or similar methods), the computational cost can be significant [84]. Performing the LU-decomposition requires O(\(N^{3}\)) operations and O(\(N^{2}\)) memory and provides a solution for all scatterer excitations. The CG method requires O(\(N^{2}\)) operations per iteration for dense matrices, since the costliest step in the CG iteration is the multiplication of the matrix by a vector. As a rule, the number of iterations increases as the electrical size of the object increases. Many researchers have tried to reduce the complexity of the traditional MoM algorithm by reducing the computational cost of the corresponding matrix-vector multiplication. Methods that use matrix operations, such as the order recursive method of moments (ORMoM) proposed in [85] or the fast multipole method (FMM) proposed in [86], reduce the computational cost of matrix multiplication. The impedance matrix localization (IML) method developed by [87] is based on the use of basic functions that lead to a sparse MoM matrix, which, in turn, speeds up the operation of matrix multiplication by a vector. It is worth noting that IML is only suitable for smooth surfaces. Likewise, the complex multipole beam (CMB) method [88] was introduced, which is also applicable only to smooth surfaces. The pseudo-grid algorithm described in [89] has been used to develop a mixed potential integral equation method in the spatial domain. This method, which may be used to analyze microstrip inhomogeneities and antennas of arbitrary shape, is derived from applying MoM to the mixed potential integral equation (MPIE) in the spatial domain to find both current and charge distribution on the microstrip surface. The ordered recursive Gaussian elimination (ORGE) method [90] effectively exploits the data overlap that occurs when designing and optimizing millimeter-wave integrated circuits (MMICs) using MoM, because multiple subtasks involving dimensional changes in different parts of the circuit must be iteratively modeled, which leads to significant data overlap among these subtasks since each subtask is solved independently of the others, without regard to data overlap. Later, researchers proposed the direct matrix manipulation (DMM) in [91] and methods based on wavelets [92], then in [93] they proposed a multistructure MoM technique based on simultaneous analysis of different structures. In practice, all structures under study are derived from the same global structure, and solutions of derived problems are obtained by simple analytical manipulations performed on the solution of the global structure. This concept avoids a large number of repetitive calculations and therefore dramatically reduces the computation time. Unlike most other fast methods, this method does not rely on simplifying assumptions but simply reuses the available information for further calculations. As a result, the proposed simultaneous simulation provides exactly the same accuracy as the equivalent individual simulation. Meanwhile, the computer requirements are significantly reduced. The work on reducing the modeling time is mainly related to obtaining an approximation of the original matrix with acceptable accuracy. This approximation is used to speed up the iterative solution by substituting the original matrix with the approximated one.