Solution box: SOLBOX-09, SOLBOX-10

Periodicity is a major property of many structures in electromagnetic applications, such as antenna arrays [1], frequency-selective surfaces, metamaterials [2], and photonic crystals [3]. Array elements (unit cells, antennas, etc.) are periodically arranged in order to enhance the response of a single element or to gain new capabilities due to interactions between them. Naturally, electromagnetic simulations of periodic structures have a long history in the literature. This is still an active area as the structures become more complicated and require more sophisticated (accurate, efficient, stable) tools for precise analysis. When the structure is infinitely large, and particularly when the periodicity is sub-wavelength, there are excellent tools based on Fourier series and finite-diff erence models, e.g., see [4] and [5]. Similarly, when the elements can be represented pointwise with or without interactions, there are analytical (e.g., using array factor) and semi-analytical (e.g., using lumped elements and/or transmission-line models) techniques. On the other hand, challenges arise when each element is a complex electromagnetic problem itself while it needs to interact with other complex elements. In addition, if the finiteness is an important geometric parameter of a periodic structure, its full-wave models may lead to large-scale problems that can be difficult to solve numerically. In such cases, it is tempting to substitute high-order mathematical models for the elements [6], specifically by isolating their inner dynamics to derive the required equivalence [7] within the decomposed models. A well-known method in this path is called the Equivalence-Principle Algorithm (EPA), which has been used and improved (e.g., hybridized with or used in other methods) by numerous researchers [7–11].