## 1 Introduction  
Radomes are weatherproof structures that protect an antenna from several external factors such as ice, rain, or debris, as well as serving in some cases as a protective barrier for personnel who work near fast-rotating elements. In some applications, the radome contributes to the aerodynamic properties of the platform (e.g., aircrafts). Radomes are used with many common industrial purposes [1, 2, 3], such as telecommunications, vehicular, maritime or automotive applications, defense systems, air traffic control, satellite communications, aircrafts, and more. It is a part of the antenna system [4, 5, 6, 7, 8, 9], which can seriously affect and degrade the transmitted and received signals, and is a key element on the radar cross section (RCS) [10] of the platform on which it is mounted. Different types of radomes can be applied with either monolithic [11] or multilayered configurations [12, 13] (e.g., A-, B-, or C-type sandwich radomes). More advanced designs include frequency selective surfaces (FSSs) embedded within the inner, outer, or intermediate interfaces of the radome [14, 15, 16], where the term interface refers to a surface that separates two different media. Some factors introduced by the radome can affect the behavior of the antenna, such as dissipative losses within a dielectric material, electrical phase shifts introduced by the radome, or internal reflections between the antenna and radome or even between different parts of the radome itself. Some performance parameters to be considered with the introduction of the radome structure are the insertion losses; effect on the antenna sidelobe levels; boresight error; and, for some applications, the effect on the RCS. Complex radomes commonly have long and costly development cycles. Prototyping, testing, and production are a large percentage of this cost. Therefore, simulation can significantly reduce development time and costs, although the analysis of complex antennas with the corresponding covering radomes can pose a formidable computational challenge [17, 18] owing to the electrical size of the problem as well as convergence difficulties arising from the presence of the dielectric layers [19] and, especially, from small mesh elements corresponding to the FSS embedded within one or more interfaces. It is common to find works that combine asymptotic approaches such as physical optics (PO) [20] with rigorous techniques for the analysis of this type of structure. The iterative physical optics-boundary integral-finite element method (IPO-BI-FEM) has been used to study a sandwich tangent-ogive radome [21]. The dyadic Green's function technique combined with PO is applied to obtain the radiation pattern of a hemispherical radome fed from a circular aperture in [22]. A large single-layer FSS radome is analyzed using PO in [23]. Multilayered radomes including FSS are addressed in [24] using PO in combination with a ray-tracing technique. The full-wave analysis of complex multilayered radomes with FSS seems, however, to be lacking in the literature owing to the computational challenge involved. There are a number of efficient variations of the method of moments [25] used to reduce the computational requirements to analyze large problems. The multilevel fast multipole algorithm (MLFMA) [26] is a well-known approach that allows fast matrix-vector products in the iterative solution process, avoiding storing the far-field coupling matrix coefficients. The MLFMA is combined with the characteristic basis function method (CBFM) in [27], which reduces the number of unknowns of the problem about one order of magnitude. The definition of macro-basis functions, which, in the context of the CBFM, are called characteristic basis functions (CBFs), requires additional CPU-time, but makes the solution process faster.