**4. Numerical results**

The numerical code was validated by comparing obtained results with known analytical solutions for the coaxial line with a centred inner conductor [35] and the coaxial line with a shifted inner conductor [36]. Results obtained coincide for up to 16 decimal places with the published solutions starting with *Ntr* =16 for the centred inner conductor and *Ntr* =128 for the inner conductor located close to the shield. Our results are also in a good agreement with other semi-analytical and numerical techniques (for example, presented in [37]).

As an illustration of the effectiveness of the obtained solution, we calculate the capacitance matrix for the assembly of arbitrary profiled cylinders located inside the grounded shield. There are no limitations on the number of cylinders with arbitrary smooth cross-sections. The high efficiency of the code is also the result of employment of the discrete Fast Fourier Transform. This makes filling of the matrix very fast routine procedure. For example, the computation time for a problem with the four inner cylinders and truncation number *Ntr* =256 does not exceed 4.5 s on a standard PC.

Efficiency of the developed method is also illustrated by the behaviour of the normalized truncation error versus truncation number (see [30]) calculated in the maximum norm sense as:

$$e\left(\mathbf{N}\_{tr}\right) = \frac{\max\_{n \in \mathcal{N}\_{tr}} \left| \mathbf{x}\_n^{\mathcal{N}\_{tr} \circ \mathbf{1}} - \mathbf{x}\_n^{\mathcal{N}\_{tr}} \right|}{\max\_{n \in \mathcal{N}\_{tr}} \left| \mathbf{x}\_n^{\mathcal{N}\_{tr} \circ \mathbf{1}} \right|},\tag{31}$$

where | *xn Ntr*+1 <sup>|</sup> *<sup>n</sup>*=0 *Ntr* and {*xn Ntr*+1 }*n*=0 *Ntr*+1 denote the solutions to the systems (19) or (29) truncated to *Ntr* and *Ntr* + 1 equations, respectively. The results of the calculations of truncation error for the infinite system (18) defining the solution to the Laplace equation are shown in the Figure 2. The considered structure is a circular shield of radius 1 with elliptic conductor with major semi-axis *b*<sup>1</sup> = 0.5 and various values of the minor semi-axis *b*2, embedded in the centre of the shield.

Figure 3 shows the condition number behaviour in the same case. The results are quite accurate and stable: for a simple structure like this, the condition number has reached a stable value even for small values of the truncation number.

In these examples, the ellipse is parameterized by the angle as a parameter. Fewer number of points on the sides of a slender ellipse results in decreasing accuracy for smaller *b*2. Arc length parameterization is one way to overcome this drawback. Other parameterizations could be even more effective, but they often require some adjustments for each shape.

Various shapes of conductor will be considered in this chapter. For all system configurations here and below, the inner conductors' potentials are set to be 1; the shield is grounded. The profile of each interior conductor is described by the super-ellipse equation (32), where function *ρ*(*φ*) and its derivative are continuous [38], with

$$\varphi(\varphi) = \left[ \left| \frac{1}{a} \cos \left( m \frac{\varphi}{4} \right) \right|^{u\_2} + \left| \frac{1}{b} \sin \left( m \frac{\varphi}{4} \right) \right|^{u\_3} \right]^{-1/u\_1} \tag{32}$$

**Figure 2.** Normalized truncation error versus *Ntr* : major semi-axis of the inner ellipse *b*1 = 0.5 and various values of the minor semi-axis *b*2.

**Figure 3.** Condition number versus *Ntr*.

inner conductor located close to the shield. Our results are also in a good agreement with other

As an illustration of the effectiveness of the obtained solution, we calculate the capacitance matrix for the assembly of arbitrary profiled cylinders located inside the grounded shield. There are no limitations on the number of cylinders with arbitrary smooth cross-sections. The high efficiency of the code is also the result of employment of the discrete Fast Fourier Transform. This makes filling of the matrix very fast routine procedure. For example, the computation time for a problem with the four inner cylinders and truncation number *Ntr* =256

Efficiency of the developed method is also illustrated by the behaviour of the normalized truncation error versus truncation number (see [30]) calculated in the maximum norm sense

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Figure 3 shows the condition number behaviour in the same case. The results are quite accurate and stable: for a simple structure like this, the condition number has reached a stable value

In these examples, the ellipse is parameterized by the angle as a parameter. Fewer number of points on the sides of a slender ellipse results in decreasing accuracy for smaller *b*2. Arc length parameterization is one way to overcome this drawback. Other parameterizations could be

Various shapes of conductor will be considered in this chapter. For all system configurations here and below, the inner conductors' potentials are set to be 1; the shield is grounded. The profile of each interior conductor is described by the super-ellipse equation (32), where

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r j In this equation, *a* and *b* are the figure size parameters, *n*1, *n*2, *n*<sup>3</sup> define corner sharpness and *m* represents the symmetry. This formula allows us to model a great variety of shapes such as an ellipse (*n* =*n*<sup>1</sup> =*n*<sup>2</sup> =*n*<sup>3</sup> =2, *m*=4), a rectangle with rounded-off corners (*n* =*n*<sup>1</sup> =*n*<sup>2</sup> =*n*<sup>3</sup> >2, *m*=4), a star with the smooth rays (*n*<sup>1</sup> =2, *n*<sup>2</sup> =*n*<sup>3</sup> >2, *m* is equal to the number of rays) and many others.

This parameterization is infinitely differentiable which gives us a great advantage in accuracy. To demonstrate this property, comparison of two different parameterizations used in the solution to the Helmholtz equation for a single rectangle with rounded-off corners is presented in Figure 4.

**Figure 4.** Comparison of super-ellipse and smoothed rectangular parameterizations.

Parameterization 1 stands for a super-ellipse formula; straight lines with a combination of quarter circles are used for the Parameterization 2. The super-ellipse parameterization uses *n* =*n*<sup>1</sup> =*n*<sup>2</sup> =*n*3, (see (32)); the greater the *n* is, the sharper the corners of the rectangle are. Sharper corners require higher truncation number to get the same level of the accuracy due to the parameterization by the angle. In Parameterization 2, *r* is a radius of curvature used to smoothen the corner, *h* is rectangle height. In all cases, rectangle height/width ratio is equal to 0.5. Parameterization 2 is not twice differentiable - there is a discontinuity in the second derivative at the joining of the straight line and the quarter circle. This account for slow convergence of the second parameterization as *Ntr* →*∞*.
