# A Subcell Finite-Difference Time-Domain Implementation for Narrow Slots on Conductive Panels
## 1 Introduction  
Modern on-board equipment is subject to aggressive electromagnetic environmental effects (E3) that impacts it directly and indirectly. As a paradigm, electromagnetic interference (EMI) and lightning indirect effects (LIE) are the main indirect threats to an aircraft from the EMC point-of-view [1]. EMC protection relies on shielded enclosures to safeguard sensitive equipment from disruptive E3 [2]. The shielding effectiveness (SE) parameter evaluates the protective capabilities of these enclosures by assessing their ability to shield the interior from external electromagnetic fields. Two main mechanisms affect enclosure shielding performance. Firstly, the conductive properties of enclosure walls play a vital role in mitigating electromagnetic wave penetration, reflecting or absorbing external electromagnetic radiation. Secondly, discontinuities in the enclosure walls, such as apertures (windows, slots, gaskets, junctions), can potentially facilitate electromagnetic wave propagation, posing vulnerabilities. This work proposes and evaluates three numerical models of slots on conductive panels to address both mechanisms, aiming to assess the overall enclosure shielding capabilities. Numerical computational tools, including the finite-difference time-domain (FDTD) method, have become reliable tools for assessing EMC at all levels and scenarios. FDTD is formulated as an explicit method on a staggered time and space mesh known as Yee's cube [3]. The FDTD method is conditionally stable and its analytical expression is determined by the causality Courant-Friedrichs-Lewy (CFL) criterion [4]. The CFL number (CFLN) bound the maximum allowable time step as a function of the minimum space step. Consequently, when dealing with small features, a refined space step is enforced in the usual FDTD, leading to a drastic decrease in the time step and, in turn, an increase in computer memory and CPU time required for the simulation. An alternative approach to address these cases, is the Dispersive Magnetic Material Approximation (DMMA) method. These are a type of subcell model integrated into the FDTD method. Originally, it was proposed by [PERSON] [5], and since then, numerous improvements have been done, e.g., [6]. The DMMA represents a robust method based on a capacitive approximation, wherein the narrow aperture is substituted with effective permittivity and permeability. Usually, these capacity models are frequency-dependent, but many authors use a static approximation to simplify the model and mitigate computational costs. Nonetheless, despite its strengths, DMMA exhibits certain drawbacks, which include:  
* The finite-difference time-domain (FDTD) method typically uses a mesh size smaller than one-tenth of the wavelength, \(\lambda>10\Delta\); therefore, the width of a narrow slot is assumed to be \(w\ll\Delta<\lambda\) (where \(\lambda\) is the wavelength, \(\Delta\) is the cell size of the mesh, and \(w\) is the width of the slot). Consequently, DMMA [7] is inherently applicable only when assuming a uniform field distribution on the slot;
* It assumes waves are perpendicularly impinging on the slot, with the electric field polarized in the direction across it;
* The magnetic condition must be placed on the dual grid, not matching the primary grid alignment used for PEC and dielectrics, thus making it tricky to implement in FDTD meshes. Though DMMA proves to be robust even if these conditions are relaxed, in this paper, we show two full-wave computer-affordable alternatives allowing us to gain generality over the hypothesis of DMMA. They do not make any assumption either in the angle of incidence or in the electric polarization. The first one is the conformal approximation (CA) generalized from [8] and extended in ref. [9], which uses a locally conformal treatment embedded in the usual structured FDTD mesh. CA models the aperture and its bendings naturally and only assumes its width is smaller or almost equal to the FDTD cell size (\(w\lesssim\Delta\)).