**2. Regularization of the electrostatic problem: MAR**

### **2.1. Problem statement**

Consider (*N*-1) arbitrary profiled charged perfect electric conductor (PEC) cylinders embedded into a homogeneous dielectric medium with relative permittivity *ε<sup>r</sup>* (Figure 1). The finite dielectric medium is bounded by the infinitesimally thin, grounded cylindrical shell.

Rigorous Approach to Analysis of Two-Dimensional Potential Problems, Wave Propagation and Scattering... http://dx.doi.org/10.5772/61287 181

**Figure 1.** Problem geometry.

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

() (,) () *<sup>p</sup>*

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.

contour *S* is charged to some prescribed potential value *V*0, then *Z*(*p*) may be found by solving

*p q Z p dS V q S*

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

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

Consider (*N*-1) arbitrary profiled charged perfect electric conductor (PEC) cylinders embedded into a homogeneous dielectric medium with relative permittivity *ε<sup>r</sup>* (Figure 1). The finite

dielectric medium is bounded by the infinitesimally thin, grounded cylindrical shell.

<sup>1</sup> log ( ) , <sup>2</sup> *p p*

<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

0


*U q G p q Z p dS* = ò (1)

*S*

body of a single conductor with contour *S* is given by

*s*

uniform convergence, or pre-determined computational accuracy.

**2. Regularization of the electrostatic problem: MAR**

p

body is an arbitrary profiled cylinder.

**2.1. Problem statement**

*<sup>G</sup>*(*p*, *<sup>q</sup>*)= <sup>−</sup> <sup>1</sup>

180 Advanced Electromagnetic Waves

the equation:

The problem is to find electrostatic potential *U* elsewhere inside the shielded region. This electrostatic problem is fully described by the Dirichlet boundary value problem for the Laplace equation:

$$
\Delta \mathcal{U} = 0 \tag{3}
$$

with boundary conditions of the potentials *Vn* given at the surface of each of *N* cylinders:

$$\left. \left. \mathcal{U} \mathcal{U} \right|\_{\mathbb{S}\_n} = V\_n, n = 1, \dots, N - 1; \left. \mathcal{U} \right|\_{\mathbb{S}\_N} = V\_N = 0. \tag{4}$$

To employ the regularization procedure, all contours *Sn* must be smooth enough and non-self crossing to provide their continuous parameterization and twice differentiation at each point of *Sn*.

### **2.2. Problem solution**

The main challenge of this problem is that all the conductors are arbitrary-shaped and the classical separation of variables method is not applicable here. We use a more general approach based on an integral representation. Using the superposition principle, we seek the solution for the total field potential *U* as the sum of the single-layer potentials contributed by each cylinder:

$$\text{MI}\left(q\right) = \sum\_{j=1}^{N} \int G\left(\left|p - q\right|\right) Z\_{j}\left(p\right) dS\_{p};\tag{5}$$

where *Zj* is the unknown line charge density of the *j* th conductor scaled by 4*π* / *ε*, *Sj* is the boundary contour of the *j* th conductor and points *q* lie in the area between the contours. The Kernel *G* of the integral equation (5) is the 2D free space Green's function:

$$G = -\frac{1}{2\pi} \log\left(|p - q|\right) \tag{6}$$

where | *p* −*q* | is the distance between points *q* and *p*.

Applying boundary conditions to (5), one can arrive at the coupled system of integral equations for the unknowns *Zj* :

$$-\sum\_{j=1}^{N} \int\_{S\_j} G\_{ij}\left(\left|p-q\right|\right) Z\_j\left(p\right) dl\_p = V\_{u'} \quad i = 1, \ldots, N \tag{7}$$

Equation (7) represents a system of first kind Fredholm integral equations that is generally illposed.

The contours of the conductors' cross-sections should be smooth. Thus for the analysis of the rectangular and square conductors, corners should be smoothed. The two most common types of parameterization are by angle and by arc length. Here, we use parameterization by angle. After parameterization of the contours *η*(*θ*)≡(*x*(*θ*), *y*(*θ*)) and introducing some new notations:

$$\begin{aligned} \left| z\_{\prec} = l\_{\succ}(\theta) \mathbb{Z}\_{\succ} \big( \eta\_{\prec}(\theta) \big) \right\rangle \quad l(\theta) &= \left| \left[ \mathbf{x}'(\theta) \right]^2 + \left[ \mathbf{y}'(\theta) \right]^2 \right|^{1/2} \\ \left| R\_{\prec}(\theta, \boldsymbol{\tau}) = \left| \boldsymbol{p} - \boldsymbol{q} \right| &= \left| \eta\_{\prec}(\theta) - \eta\_{\succ}(\boldsymbol{\tau}) \right| = \left| \left[ \mathbf{x}\_{\prec}(\theta) - \mathbf{x}\_{\succ}(\boldsymbol{\tau}) \right]^2 + \left[ \mathbf{y}\_{\succ}(\theta) - \mathbf{y}\_{\succ}(\boldsymbol{\tau}) \right]^2 \right|^{1/2} \end{aligned} \tag{8}$$

we obtain the system of *N* integral equations:

$$-\sum\_{j=1}^{N} \int\_{-\pi}^{\pi} G\left(R\_{s\_j}(\theta,\tau)\right) z\_j\left(\tau\right) d\tau = V\_s\left(\theta\right), s = 1, 2...N. \tag{9}$$

The described approach permits us to consider a broader set of possible boundary conditions than simply a constant, though in the application to be described, a constant is deployed on the RHS of (9).

For the kernels *G*(*Rsj* (*θ*, *τ*)) such that *s* ≠ *j*, points corresponding to *θ* and *τ* belong to different contours and so *Rsj* (*θ*, *τ*)≠0 everywhere; hence, the corresponding integral terms do not contain singularities. For *Gss*(*θ*, *τ*) the corresponding integral contains a singularity of loga‐ rithmic type at the points *θ* =*τ*. In this case, we analytically separate the Green's function into the singular part and a remainder *L sj* that does not contain any singularity:

$$\begin{aligned} -2\pi G\left(R\_{\boldsymbol{\cdot}\boldsymbol{\cdot}}(\theta,\boldsymbol{\tau})\right) &= \log\left(R\_{\boldsymbol{\cdot}\boldsymbol{\cdot}}(\theta,\boldsymbol{\tau})\right) = L^{\boldsymbol{\cdot}}(\theta,\boldsymbol{\tau}), \quad \boldsymbol{s} \neq \boldsymbol{j}, \\ -2\pi G\left(R\_{\boldsymbol{\cdot}\boldsymbol{\cdot}}(\theta,\boldsymbol{\tau})\right) &= \log\left(R\_{\boldsymbol{\cdot}\boldsymbol{\cdot}}(\theta,\boldsymbol{\tau})\right) = L^{\boldsymbol{\cdot}}(\theta,\boldsymbol{\tau}) + \log\left(2\sin\left|\frac{\theta-\boldsymbol{\tau}}{2}\right|\right), \quad \boldsymbol{s} = \boldsymbol{j}. \end{aligned} \tag{10}$$

Now we can determine *L sj* from (10) as follows:

where *Zj*

182 Advanced Electromagnetic Waves

for the unknowns *Zj*

*j j jj*

 hq

q

qt

the RHS of (9).

For the kernels *G*(*Rsj*

contours and so *Rsj*

posed.

is the unknown line charge density of the *j* th conductor scaled by 4*π* / *ε*, *Sj* is the

= - - (6)

1/2 2 2

 t (8)

 q

boundary contour of the *j* th conductor and points *q* lie in the area between the contours. The

( ) <sup>1</sup> log <sup>2</sup> *G p q* p

Applying boundary conditions to (5), one can arrive at the coupled system of integral equations

*G Z p dl V i N p q*

Equation (7) represents a system of first kind Fredholm integral equations that is generally ill-

The contours of the conductors' cross-sections should be smooth. Thus for the analysis of the rectangular and square conductors, corners should be smoothed. The two most common types of parameterization are by angle and by arc length. Here, we use parameterization by angle. After parameterization of the contours *η*(*θ*)≡(*x*(*θ*), *y*(*θ*)) and introducing some new notations:

1/2 2 2

 q

, , 1,2... .

 q

 t

ë ûë û


(*θ*, *τ*)) such that *s* ≠ *j*, points corresponding to *θ* and *τ* belong to different

(*θ*, *τ*)≠0 everywhere; hence, the corresponding integral terms do not

( ) { }

== - = - + -

( ( )) ( ) ( )

 t t

*sj j s*

*sj <sup>s</sup> <sup>j</sup> <sup>s</sup> <sup>j</sup> <sup>s</sup> <sup>j</sup> p q*

 q

*R xx yy*

, () () () () () () ,


 q

*GR z d V s N*

The described approach permits us to consider a broader set of possible boundary conditions than simply a constant, though in the application to be described, a constant is deployed on

contain singularities. For *Gss*(*θ*, *τ*) the corresponding integral contains a singularity of loga‐

é ùé ù ë ûë û

, '( ) '( ) ,

, 1,...,


Kernel *G* of the integral equation (5) is the 2D free space Green's function:

( ) ( )

( ) ( ( )) ( ) { }

qt

*zl Z l x y*

= =+

 h q h t

we obtain the system of *N* integral equations:

*N*

p

p

1

= -

*j*

 q

*ij j pn <sup>j</sup> <sup>S</sup>*

where | *p* −*q* | is the distance between points *q* and *p*.

1

*N*

=

*j*

:

$$\begin{aligned} L^s(\theta, \tau) &= \log \left( R\_{\prec}(\theta, \tau) \right) - \log \left( 2 \sin \left| \frac{\theta - \tau}{2} \right| \right), \quad s = j; \\ L^s(\theta, \tau) &= \log \left( R\_{\prec}(\theta, \tau) \right), \quad s \neq j. \end{aligned} \tag{11}$$

The function *L sj* , *s* = *j* is a regular function, defined everywhere except at points *θ* =*τ* ; the function *L sj* , *s* ≠ *j* is defined everywhere. It can be shown that for the Laplace's equation this regular function has the same degree of smoothness as the contour parameterization. An exact expression for *L sj* , *s* = *j* at the points of singularity where *θ* =*τ* was obtained analytically:

$$L^{\psi}(\theta,\pi) = \log\left(l(\theta)\right),\tag{12}$$

where *l*(*θ*)= *x*(*θ*) <sup>2</sup> + *y*(*θ*) 2 is an arc length in the point *θ*.

Now we can redefine function *L sj* , *s* = *j* everywhere by the formula:

$$L^{\psi} = \begin{cases} \log(R\_{\circ}(\theta, \tau)) - \log\left(2\sin\left|\frac{\theta - \tau}{2}\right|\right), \theta \neq \tau, \\\log\left(l(\theta)\right), \theta = \tau. \end{cases} \tag{13}$$

Using the well-known Fourier expansion, we can formulate an expression for the singular part of the Green's function:

0

¹

*n*

$$\log\left(2\sin\left|\frac{\theta-\pi}{2}\right|\right) = \frac{1}{2}\sum\_{n=-\alpha}^{\alpha} \frac{e^{in(\theta-\pi)}}{|n|},\tag{14}$$

As the function *L sj* is regular, we can expand it into double Fourier series:

$$L^{s\mid}\left(\theta,\tau\right) = \sum\_{n = -\infty}^{\infty} \sum\_{m = -\infty}^{\infty} l\_{nm}^{s\mid} e^{l\left(n\theta \circ m\tau\right)}.\tag{15}$$

Also the unknown function *zj* and the given potential function are represented by their Fourier series:

$$\varphi\_{/}\left(\pi\right) = \sum\_{-\infty}^{\alpha} \xi\_{n}^{j} e^{in\tau}, \qquad V\_{/}(\theta) = \sum\_{-\alpha}^{\alpha} \nu\_{n}^{j} e^{in\theta} \tag{16}$$

After substitution of all expansions into (9), one can arrive at the system of *N* integral equations:

$$\sum\_{n=-\alpha}^{\alpha} \frac{\xi\_n^{\kappa}}{|n|} e^{\kappa n \theta} - 2 \sum\_{j=1}^{N} \sum\_{n=-\alpha}^{\alpha} e^{in\theta} \left( \sum\_{m=-\alpha}^{\alpha} I\_{n,-m}^{j} \xi\_m^{j} \right) = \sum\_{n=-\alpha}^{\alpha} \nu\_n^{j} e^{\kappa n \theta}, \ \theta \in [-\pi, \pi], \ \ s = 1, 2, \ldots, N. \tag{17}$$

Using orthogonal properties and completeness of the functions {*einφ*}*n*=−*<sup>∞</sup> <sup>n</sup>*=*<sup>∞</sup>* and defining the rescaled unknown Fourier coefficients of charge density function *ξ<sup>n</sup> <sup>s</sup>* as follows: *ξ*˜ *<sup>n</sup> <sup>s</sup>* <sup>=</sup> *<sup>ξ</sup><sup>n</sup> s σn* , *<sup>σ</sup><sup>n</sup>* <sup>=</sup> <sup>|</sup>*<sup>n</sup>* <sup>|</sup> 1/2 when *n* ≠0 and *σ*<sup>0</sup> =1, we obtain the following infinite system of linear algebraic equations:

$$
\tilde{\xi}\_n^{\tilde{\varepsilon}^s} (1 - \delta\_{n0}) + \sum\_{j=1}^N \sum\_{n=-\alpha}^n \sigma\_n \sigma\_n \mathbf{l}\_{n,-m}^{s\vert} \tilde{\xi}\_n^{\vert} = \sigma\_n \mathbf{v}\_{n\prime}^s \cdot n = 0, \pm 1, \pm 2...; \ s = 1, 2, ..., N. \tag{18}
$$

Following the steps suggested in [33], it can be shown that coefficient matrix in (18) is square summable:

$$\sum\_{j=1}^{N} \sum\_{m=-\alpha}^{\alpha} \|\sigma\_n \sigma\_m l\_{n,-m}^{s^j}\|^2 < \infty, \ n = 0, \pm 1, \pm 2...; \ s = 1, 2,...,N. \tag{19}$$

Thus the infinite system (18) is of a second Fredholm kind and can now be effectively solved by a truncation method. The solution of the truncated system monotonically and rapidly converges to the exact solution. The above solution automatically incorporates the reciprocal influence of all charged cylinders, allowing accurate calculation of the line charge densities on the boundaries and then the field potentials at any point of the space between the conductors. Fourier expansions in (18) are calculated numerically as all functions are regular.
