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ScienceDirect Procedia Computer Science 103 (2017) 378 – 383

XIIth International Symposium «Intelligent Systems», INTELS’16, 5-7 October 2016, Moscow, Russia

The use of genetic algorithm for construction objects with necessary average values scattering characteristics I.Ya. Lvovicha, Ya.E. Lvovichb, A.P. Preobrazhenskiyb*, O.N. Choporovb, Yu.S. Sakharovc b

a Paneuropean university, Bratislava, Slovakia Voronezh institute of high technologies, Voronezh, Russia c International University “Dubna”, Voronezh, Russia

Abstract In the paper we consider the possibility of constructing models of objects that have the maximum average values of the characteristics of scattering at a certain sector of angles. For optimization of these characteristics we use genetic algorithm. We developed algorithm on the base of the dependencies of the characteristic dimensions of a hollow structure with a maximum average values of the characteristics of scattering were calculated. © 2017 The TheAuthors. Authors.Published Publishedbyby Elsevier B.V. © 2017 Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the scientific committee of the XIIth International Symposium «Intelligent Systems». Peer-review under responsibility of the scientific committee of the XIIth International Symposium “Intelligent Systems” Keywords: hollow structure; optimization; genetic algorithm; integral equation; characteristics of scattering.

1. Introduction In some cases it is necessary to know not the angular dependence of the scattering parameters and their average values in certain sectors of the viewing angles. The literature provides data for the average radar cross section (RCS) values of some objects. Based on the analytical formulas for bodies of simple shape it is possible to obtain analytical expressions of the average values of RCS from the size of the object. In the general case the calculation of the scattering parameters is performed with the use of numerical methods1,2,3.

* Corresponding author. E-mail address: [email protected]

1877-0509 © 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the scientific committee of the XIIth International Symposium “Intelligent Systems” doi:10.1016/j.procs.2017.01.124

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I. Ya. Lvovich et al. / Procedia Computer Science 103 (2017) 378 – 383

It is of practical interest to construct a fairly simple algorithm for calculating the sizes of objects with maximal average values of scattering characteristics of a specific sector of observation angles 4,5,6. For bodies of simple shape dependence of the maximum of the average values of the characteristics from the dimensions of bodies can be written analytically. This paper addresses the problem of determining the maximum of the average values of the scattering parameters for the two-dimensional model of the hollow structure7,8,9. The analytical dependence of the maximum of the average values of the scattering parameters from the dimensions of the object in this case cannot be derived, however, there is the possibility of constructing the approximate function (interpolating polynomial), allowing to obtain an approximation of this dependence at sufficiently small error 11-16. 2. The model Illustration of our proposed approach will be conducted in the framework of the two-dimensional model. It is known that two-dimensional model of the hollow structure can be used to estimate the scattering characteristics of hollow structures with rectangular cross section 17,18,19. Let the dimensions of the hollow structure L1, L2, L3 (Fig. 1). Then, the total value of the loop structure L= L1+L2+L3. As the characteristic size, select the size of L1+L3. It is necessary to find L1+L3, and L, for which the average RCS at specified sectors of the angles 'T reaches the maximum values. The angle T measured from the normal to the aperture of the cavity.

Fig. 1 Scheme of the scattering of electromagnetic waves on a hollow structure with the size L1, L2, L3, the load has a slope at the angle of M.

The sector angle was varied in wide range: 5q d 'T d 90q, that is considered the area of the front hemisphere. The calculation of the scattering parameters was based on the method of integral equations. A strict calculation method was performed due to the following reasons: 1) the size of the object was varied over a wide range, including in the low frequency region; 2) On the basis of approximate analytical methods to obtain an acceptable estimate only for values of RCS in the area of local highs chart backscattering. The lows and back scattering error can be many tens of dB. Fredholm equation of the first kind for the density of the unknown electric current in the case of E-polarization the following: For Fredholm equations of the first kind that contains the unknown density of the electric current in the Epolarization 20, we can write the following expression: Z P 4

E

³ j (t ) H

2 0 [k

L0 (W , t )] [ '2 (t ) K '2 (t ) dt

E z0 (W ) , D d W d E ,

(1)

D

where L0 (W , t )

[[ (W ) [ (t )]2 [K (W ) K (t )]2 - represents the distance between the observation point and the

point of integration. E z0 (W ) – denotes the longitudinal component of the tension of the primary electric field for points on the contour. The contour is specified in a parametric form: x [ (t ), y K (t ), D d t d E and are the first derivatives of the corresponding functions, k 2 S / O , O – length of the incident electromagnetic wave.

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Equation (1) is solved based on the method of moments we find the longitudinal electric currents, having a density of j

z j (t ) , D d t d E ,

(2)

Two-dimensional RCS of a hollow structure can be found based on the following expression 2

V (I )

(60 S ) 2 k D (I ) ,

(3)

where E

³ j (t )

D (I )

[ '2 (t ) K '2 (t ) exp(i k d (t , I )) dt , d (t , I )

[ (t ) cos(I ) K (t ) sin(I ) .

D

The average RCS, we calculated, based on the following expression N

V

V (T i )

¦ N 1 ,

(4)

i 0

where V (T i ) - the value of RCS for angle ob observation T i . The problem of determining L1+L3 and L giving the maximum average RCS in the sector of angles was solved in the following way. The value of the sector angle 'T was chosen. For different values of L1+L3 the values of L are determined. The function is multiextremal, so when calculating L, we used the genetic method with successive narrowing of the field determined values. For each plot mesh was used as the local optimization method of golden section. In Fig. 2 the dependences of the length of the contour L from the aperture size L1+L3 cavity, which maximizes the average RCS for the various sectors of the viewing angles. The approximation of the received dependences L1+L3 from the L polynomial dependencies in the framework of the method of least squares was carried out. In Fig. 3 shows the dependence of relative error of linear approximation from the sector angle for polynomial degree n from 1 to 4, respectively. It is seen that the maximum error of the approximation will be when the sector angle is in the range of 0qd d15q. Acceptable approximation (i.e., relative error less than 5%) is achieved for 30qd d90q, when the degree of the approximating polynomial nt3. In the genetic algorithm21,22,23, we first set the initial population. Then, a selection is carried out. For stage selection it is required from the entire population to make a specific choice of her share, which will remain "alive" at this stage of evolution. We assume that the survival probability of the individuals of p should depend on the values of the fitness function Fitness(p). The proportion of survivors m is usually a parameter of the genetic algorithm, and its just asking in advance. On the basis of the selection of the M individuals of the population P should remain mM specimens that will be included in the final population P'. The remaining individuals die. We reviewed roulette method in which the probability of selecting individuals more likely than the better value of the fitness function f i

di

, where f i is the probability of selecting i-th individual, d i is the value of the fitness

M

¦d

i

i 1

function for the i-th individual, M is the number of individuals in the population. Selection of parents was based on the procedures of inbreeding, when the first parent is selected randomly, and the second is chosen such that is most similar to the first parent.


When the crossing was considered pertsevidny crossing-over (crossover Shuffler). In this algorithm, individuals are selected for crossover, random sharing genes. Then select a point for one-point crossing-over and carry out exchange of parts of chromosomes. After crossing the created then key re-shuffled. Thus, at each crossing-over created not only the new offspring, but also modifierade parents (old parents removed), which allows to reduce the number of operations compared to uniform crossing-over. 3. Results In Fig. 4 the values of the coefficients of the approximating polynomial of the third degree b0 b1 x b2 x 2 b3 x 3 for different sectors of the viewing angles. A similar approach can be used for bodies with a different shape: parallelepipeds, cylinders, etc. Note that the results of calculations of the dimensions of the reflector with the maximum average RCS can be used in the design of an object consisting of several elementary reflectors. Thus, the algorithm has the following stages: 1. Sets the aperture size of the sector angle 'T; 2. To determine the coefficients of the polynomial for a given 'T; 3. Calculation of L polynomial basis. y ( x)

Fig. 2 The dependence of the length of contour for hollow structure from L 1 +L3 with maximal average RCS in the sector of angles 'T =5q (curve 1), 15q (curve 2)

Fig. 3. The dependence of relative error of polynomial approximation of the size of the aperture a of the length of the contour of th e hollow structure of L for various degrees n of the approximating polynomial

381

382


Fig. 4. The dependence of the coefficients b i of the approximating polynomial of the third degree from the sector of viewing angles 'T

Thus, the above formulated approach can lead to the algorithm for calculating the scattering characteristics of electromagnetic wave two-dimensional hollow structures. The main advantages of this algorithm is as follows: 1) On the base of the considered algorithm the hollow structure can be considered as a body with the complex shape, so we do not use the waveguide approach (eg, boundary-integral modal method19, or mixed method), we can account the reflections from the outer region of this structure, which allows us to analyze the scattered electromagnetic field around the corners of the flat sector electromagnetic wave fall. The algorithm is better to use in the calculation of the RCS for hollow structures with relatively small dimensions (for wavelengths - the resonance region), in which poorly work ray methods; 2) The universality of the algorithm due to the fact that for the calculation of the scattering characteristics of the perfectly conducting hollow structures of various forms need to change the circuit structure in programming with regard to the necessary steps of the partition - the main steps of the algorithm are the same.

4. Conclusion On the basis of the considered model, the paper shows the possibility of determining the characteristic dimensions of the object that has the maximum average values of the scattering characteristics on the example of a hollow structure with use genetic algorithm. On the basis of obtained results it is possible to design objects with a low level of secondary electromagnetic radiation.

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