**1. Introduction**

In electrostatic and electromagnetic studies of highly elongated cylinders, and ensembles or arrays of such cylinders, it is well-known [1] that the most important effects can be treated by replacing the three-dimensional structure by the corresponding cross-sectional two-dimen‐ sional (2D) profile. Such cylinders are described by *a* < < *L* , where *L* is the length of the cylinder, and *a* is a parameter characterizing its cross-section. This condition ensures that for real objects such as charged conducting cylinders, the field induced by their ends will have no impact on electric field distribution well away from the ends, or this impact will be vanishingly small as *L* →*∞*. Such idealization in 2D electrostatics causes nonphysical solutions: the arbitrary profiled charged cylinder, for instance, generates a potential that logarithmically grows instead of vanishing as expected at distant observation points. This means that the potential is unbounded at infinity. The nonphysical nature of the solution is commented by many authors. However, for systems of charged conductors this difficulty can be avoided in one of two ways: by setting the net charge of the system to be equal to zero or by introducing earthed infinite planes or closed earthed shields. This issue is discussed in [2]. In spite of the seeming limitations that the above conditions enforce on 2D electrostatic modelling, they naturally occur in almost all problems arising in practice. In particular, 2D boundary value problems for the Laplace and Helmholtz equation describe many problems of practical interest arising aero- and hydrodynamics (potential fluid flow), electrostatics, electromagnetic scattering studies, acoustics, elasticity theory, etc.

The long-standing interest in the investigation of the electrostatic field in periodic structures continues because of numerous applications. One example is the analysis of the propagation of the transverse electromagnetic (TEM) wave in open and shielded multi-conductor trans‐ mission lines [3, 4]. When the contour of a conductor coincides with the coordinate surface of one of the coordinate systems in which the Laplace's equation is separable, the Fourier method (method of separation of variables) is used. More generally, a variety of potential problems have been solved by the conformal mapping method. These results are described in many classical handbooks and monographs. The number of such solved problems is highly restrict‐ ed. Nowadays, the need for simulation of devices used in practice requires development of more universal methods to tackle problems with objects of various finite-width shapes. One of such numerous examples is the capacitance calculation for thick electrodes [5] where a physically reasonable meaning of 'edge capacitance' arises only because an accurate charge distribution over the whole electrode could not be accurately calculated. Though solutions obtained for single objects may adequately describe the real situation, most practical problems deal with a finite number of objects (say, conductors). Even when a conductor is of canonical shape (circular or elliptic cylinder), the solution of an electrostatic multi-conductor problem for an assembly of cylinders of different radii is a very bulky and lengthy procedure. Solving this problem as a classical boundary value problem for Laplace's equation and enforcing the pre-assigned boundary conditions at the surface of each conductor, it is necessary to make multiple re-expansions of the eigen functions of the Laplace's equation in each local coordinate system in terms of that chosen to satisfy the boundary conditions.

Electromagnetic and acoustic problems described by the Helmholtz equation can also be considered in two dimensions. A great variety of publications consider the problem involving infinite gratings. They are often used in antenna applications as polarizers and filters. In [6], a vector diffraction formulation for the analysis of perfectly conducting gratings of finite width and thickness is presented. The grating is assumed to have a finite number of infinitely long arbitrarily shaped rods, and is illuminated by an arbitrary plane wave. Electric and magnetic field integral equations are used to numerically solve the corresponding TM and TE electro‐ magnetic problems. Periodic structures in the millimetre wave range are considered in [7]: this paper studies single and double periodic devices using a semi-analytical mode-matching technique. Diffraction of the TM-polarized Gaussian beam by *N* equally spaced slits (finite grating in a planar perfectly conducting thick screen) is investigated in [8]. Numerous publications consider the different types of arrays. A general approach was presented in [9] for solving the 2D scattering of a plane wave by an arbitrary configuration of perfectly conducting circular cylinders in front of a plane surface with general reflection properties. Acoustic scattering by a cluster of small sound-soft obstacles was considered in [10]. The 2D scattering of a Gaussian beam by a periodic array of circular cylinders is studied in [11]. A study of the electromagnetic scattering from multi-layered periodic arrays of parallel circular cylinders is presented in [12]. The electromagnetic scattering by multiple perfectly conducting arbitrary polygonal cylinders is analysed in [13].

**1. Introduction**

178 Advanced Electromagnetic Waves

scattering studies, acoustics, elasticity theory, etc.

system in terms of that chosen to satisfy the boundary conditions.

In electrostatic and electromagnetic studies of highly elongated cylinders, and ensembles or arrays of such cylinders, it is well-known [1] that the most important effects can be treated by replacing the three-dimensional structure by the corresponding cross-sectional two-dimen‐ sional (2D) profile. Such cylinders are described by *a* < < *L* , where *L* is the length of the cylinder, and *a* is a parameter characterizing its cross-section. This condition ensures that for real objects such as charged conducting cylinders, the field induced by their ends will have no impact on electric field distribution well away from the ends, or this impact will be vanishingly small as *L* →*∞*. Such idealization in 2D electrostatics causes nonphysical solutions: the arbitrary profiled charged cylinder, for instance, generates a potential that logarithmically grows instead of vanishing as expected at distant observation points. This means that the potential is unbounded at infinity. The nonphysical nature of the solution is commented by many authors. However, for systems of charged conductors this difficulty can be avoided in one of two ways: by setting the net charge of the system to be equal to zero or by introducing earthed infinite planes or closed earthed shields. This issue is discussed in [2]. In spite of the seeming limitations that the above conditions enforce on 2D electrostatic modelling, they naturally occur in almost all problems arising in practice. In particular, 2D boundary value problems for the Laplace and Helmholtz equation describe many problems of practical interest arising aero- and hydrodynamics (potential fluid flow), electrostatics, electromagnetic

The long-standing interest in the investigation of the electrostatic field in periodic structures continues because of numerous applications. One example is the analysis of the propagation of the transverse electromagnetic (TEM) wave in open and shielded multi-conductor trans‐ mission lines [3, 4]. When the contour of a conductor coincides with the coordinate surface of one of the coordinate systems in which the Laplace's equation is separable, the Fourier method (method of separation of variables) is used. More generally, a variety of potential problems have been solved by the conformal mapping method. These results are described in many classical handbooks and monographs. The number of such solved problems is highly restrict‐ ed. Nowadays, the need for simulation of devices used in practice requires development of more universal methods to tackle problems with objects of various finite-width shapes. One of such numerous examples is the capacitance calculation for thick electrodes [5] where a physically reasonable meaning of 'edge capacitance' arises only because an accurate charge distribution over the whole electrode could not be accurately calculated. Though solutions obtained for single objects may adequately describe the real situation, most practical problems deal with a finite number of objects (say, conductors). Even when a conductor is of canonical shape (circular or elliptic cylinder), the solution of an electrostatic multi-conductor problem for an assembly of cylinders of different radii is a very bulky and lengthy procedure. Solving this problem as a classical boundary value problem for Laplace's equation and enforcing the pre-assigned boundary conditions at the surface of each conductor, it is necessary to make multiple re-expansions of the eigen functions of the Laplace's equation in each local coordinate

It should be noted that the long history of solving the Laplace and Helmholtz equations is marked by the development of many numerical methods which are useful in simulation of practical devices. Such methods include the finite difference technique, extrapolation [14], point-matching method [15], boundary element method [16], spectral-space domain method [17], finite element method [18-20], transverse modal analysis [21] and mode-matching method [22]. A numerical integral equation approach is used in [23] to explore plane-wave scattering from a nonplanar surface with a sinusoidal height profile for the case of the magnetic field parallel to the surface ridges (TM polarization). In spite of effectiveness of these methods in many cases and flexibility in geometrical representation of the structures, modelling of ridges still have some substantial drawbacks. Most of the methods require large resources in terms of computational time and storage. Often the solutions obtained with such purely numerical methods need to be verified through comparison to other results: accuracy generally cannot be guaranteed for a greater number of iterations or larger-scale computations. This problem becomes pronounced in some topologically complex configurations. In electromagnetics, the corresponding class of numerical solutions is applicable in the low to intermediate frequency range. Resonant systems behaviour cannot be reliably analysed with the numerical techniques (see [24]). Analytical-numerical methods such as those based on the method of analytical regularization (MAR) are designed to overcome these drawbacks in the resonant regime. A comparative analysis of the MAR and other methods was conducted in [25], and in [26] the distinctive features of each of the discussed methods are clearly described. The abovementioned methods are mostly suited for analysis of a single or very few conductors. In the case of a significant number of conductors with individual profiles, the effectiveness of such purely numerical methods is highly problematic because of the rapidly growing scale of computations.

In order to address these difficulties, we present here a semi-analytical approach to the analysis of 2D electrostatic and electrodynamic field problems for multi-conductor systems. The problems to be solved are treated as the classical Dirichlet boundary value problems for the Laplace and Helmholtz equations. It is well-known [27, 28] that solutions to the Laplace and Helmholtz equations can be represented as a single-layer potential at points exterior to the body of a single conductor with contour *S* is given by

$$\mathcal{L}\mathcal{L}(q) = \bigcup\_{\mathcal{S}} \mathcal{G}(p, q)\mathcal{Z}(p)dS\_p \tag{1}$$

where *Z*(*p*) is related to the linear charge distribution on the contour *S* in the case of the Laplace equation, and to the linear current density in case of the Helmholtz equation. *<sup>G</sup>*(*p*, *<sup>q</sup>*)= <sup>−</sup> <sup>1</sup> <sup>2</sup>*<sup>π</sup>* log| *<sup>p</sup>* <sup>−</sup>*<sup>q</sup>* | is the Green's function for Laplace's equation in 2D space. If the contour *S* is charged to some prescribed potential value *V*0, then *Z*(*p*) may be found by solving the equation:

$$-\frac{1}{2\pi} \int\_{s} \log|p - q| Z(p) dS\_p = V\_{0'} \qquad q \in S\_p \tag{2}$$

This equation may be classified as a first kind Fredholm equation with a singular kernel; it is ill-posed [29]. Nevertheless, this problem has been tackled by many authors who used direct numerical schemes for solving its discrete analogue in a form of a first kind algebraic equation. Theoretically, any numerical method applied to solve this equation is unable to guarantee uniform convergence, or pre-determined computational accuracy.

The only way to avoid these shortcomings is to transform the initial equation into a second kind Fredholm equation, discretization of which guarantees uniform convergence and any pre-determined accuracy of the numerical solution depending on truncation number. We employ the MAR, in particular, described in [30, 31]. An accurate solution to wave scattering by a single infinitely long cylinder of arbitrary cross-section by the MAR was obtained in [32]. The details of the algorithm for cylinders of closed arbitrary profile are presented in [26, 31]. In this chapter, we generalize the MAR for a multi-conductor potential problem where each body is an arbitrary profiled cylinder.
