**3.1. Description of 3D TLM method**

where *n* and *m* are integers describing the *nm*-th mode, *λ* is the free-space wavelength, *w* and *h* the waveguide dimensions, and ¯ is the normalized complex permittivity ¯ = *<sup>r</sup>* − *jσ*/ (*ω*0), where *σ* is the wall conductivity. Finally, a study of modal propagation in curved

Ray models use high-frequency expansions of Maxwell's equations and are based on optical laws to model the fields. They constitute the most commonly used deterministic techniques to analyze confined environments [32]. In ray-based methods, the waves are treated as rays propagating perpendicular to the wave fronts. This approximation is valid at positions sufficiently distant from the source. The resulting spherical waves can be approximated by a locally plane wave on a small portion of the sphere. Two methods are widely used: ray launching and ray tracing [8, 10]. Curved tunnels, arched cross sections, or bends of a tunnel can be treated by ray methods provided that radii of curvature of the surface be large

Current tunnel cross sections are getting smaller, and antennas have to operate close to the tunnel walls, such that these interfere with their near-fields. As a result, antenna performances cannot be guaranteed by using asymptotic methods. For this reason, full-wave

In electromagnetism, two main families of numerical methods exist: frequency and time-domain methods. Nowadays, time-domain methods are becoming increasingly used [11, 31]. They are attractive because of their relative simplicity and their ability to account for arbitrary geometries, obstacles like vehicles, nonlinearities, and different materials and to determine transient response and perform wideband characterization [6, 32, 35]. Thus, full-wave methods attract more attention for the purpose of deterministic radio coverage predictions, near-field considerations, simplifications, and practical implementations [13]. However, efficient and adequate models are still needed to describe the wave propagation in tunnel environments. A modal approach is preferred because it allows one to describe the propagation by modes [1, 2, 29]. An efficient time-domain modal approach based on the

transmission-line matrix (TLM) method will be introduced in the next section.

**3. The Transmission-Line Matrix (TLM) method for guiding structures**

The transmission-line matrix (TLM) method was developed by P. B. Johns and his co-workers in the 1970s [17]. Its theoretical foundations are based on Huygens's model of wave propagation. The TLM method is very attractive and flexible to analyze electromagnetic field problems and has received increased recognition for full-wave analysis of arbitrary-shaped guiding structures [15]. Through the years, several nodes have been developed in TLM for structured and unstructured meshes in two and three dimensions. We will focus on the symmetrical condensed node (SCN), which is the most widely used for 3D structures.

rectangular tunnels can be found in [20].

compared to the wavelength [22, 33].

**2.4. Full-wave models**

models accounting for near-field effects become more relevant.

**2.3. Asymptotic models**

242 Advanced Electromagnetic Waves

The symmetrical condensed node (SCN) consists of a network of interconnected multi-port devices where the electromagnetic wave propagation is simulated by the propagation of traveling pulses; a unit cell is presented in (Figure 1). On each arm, two orthogonal ports are employed to account for any polarization.

**Figure 1.** Scheme of the SCN node

To solve an electromagnetic problem, the solution region has to be divided into a number of these elementary nodes.

A rule of thumb consists of taking the maximum cubic cell size as ∆*l* = *λ*/10, where *λ* correspond to the medium wavelength of the highest frequency of interest. Then, excitation is imposed. The objective of the source terms is to simulate a desired phenomenon in the structure. For instance, in guiding structures, a given mode can be selectively excited by an adequate field distribution that corresponds to the mode transverse configuration (mode template). Some other problems require to analyze the structure over a wide frequency range. This is achieved by exciting the structure with a wideband time signal such as Dirac, Gaussian, etc. Thus, in general, the excitation corresponds to a space-time distribution. Nodes are interconnected by these virtual transmission lines, and the excitation propagates from the source nodes to the adjacent nodes at each time step.

The method is carried out basically through two processes: scattering and connection. In the scattering process, voltage pulses *nV<sup>i</sup>* are incident upon the node from each of the link-lines (halfway between two nodes) at each time step *n*∆*t*. These pulses are then scattered to produce a set of scattered voltages, *kV<sup>r</sup>*, which become incident on adjacent nodes at the next time step (*n* + 1)∆*t*. In the connection process, pulses are simply exchanged among immediate neighbors.

Volumic methods such as TLM full-wave methods require a limitation of the computational domain for open problems in which fields exist at large distances. Thus, somehow the problem has to be bounded. The boundary conditions are the set of conditions which specifies the behavior of fields at the boundaries of the computational domain.

### *3.1.1. Boundary conditions in TLM*

The boundary conditions link the electromagnetic fields through the tangential or normal field values. Since TLM is based on the equivalence between Maxwell's equations and equations for voltages and currents that travel in a mesh of interconnected transmission lines, a relationship of the involved voltages at the boundary can be found. The scattered voltages *V<sup>r</sup>* are always known values, and the incident voltages *V<sup>i</sup>* are unknown. Any resistive load at a boundary may be simulated by introducing a reflection coefficient Γ as shown in (3), [7].

$$
\Gamma\_{k+1} V\_{armj}^{i} (\mathbf{x}, y, z) = \Gamma\_k V\_{armj}^{r} (\mathbf{x}, y, z) \tag{3}
$$

This formalism allows us to represent a variety of boundary conditions as long as a reflection coefficient Γ can be defined. For instance, for a perfectly electric conductor (PEC), boundary is simulated by choosing Γ = −1; a perfectly magnetic conductor (PMC) is implemented by choosing Γ = 1. The reflection coefficient for lossy boundaries [15], relating the incident and reflected voltages, can be expressed in Laplace domain by the (4).

$$\Gamma\left(\mathbf{s}\right) = \frac{V^r\left(\mathbf{s}\right)}{V^i\left(\mathbf{s}\right)} = \frac{\left[\Delta y \mathbf{Z}\_s\left(\mathbf{s}\right) - \mathbf{Z}\_{zy}\Delta z\right]}{\left[\Delta y \mathbf{Z}\_s\left(\mathbf{s}\right) + \mathbf{Z}\_{zy}\Delta z\right]}\tag{4}$$

where <sup>∆</sup>*<sup>y</sup>* and <sup>∆</sup>*<sup>z</sup>* are the cell dimensions; *Zzy* the impedance of the arms, equal to *<sup>Z</sup>*<sup>0</sup> <sup>=</sup> *<sup>µ</sup>*0/<sup>0</sup> for the standard SCN node; and *Zs* is the surface impedance (SI). In the case of a good conductor, an approximation by a real number for Γ is presented in [28]. However, Γ is in general complex and would alter the shape of the excitation pulses, which cannot be accounted for in the TLM method [7].
