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Advanced Methods for Solving Nonlinear Eigenvalue Problems of Generalized Phase Optimization

Written By

Mykhaylo Andriychuk

Submitted: November 9th, 2021 Reviewed: February 25th, 2022 Published: April 18th, 2022

DOI: 10.5772/intechopen.103948

IntechOpen
Matrix Theory - Classics and Advances Edited by Mykhaylo Andriychuk

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Matrix Theory - Classics and Advances [Working Title]

Dr. Mykhaylo I. Andriychuk

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Abstract

In the process of solving the problems of generalized phase optimization the necessity to apply an eigenvalue approach often appears. The practical statement of the optimization problems consists of using the amplitude characteristics of functions that are sought. The usual way of optimization is deriving the Euler equation of the functional, which is used as criterion of optimization. As a rule, such equation is an integral one. It is worth pointing out that the integral equations of the generalized phase optimization are nonlinear ones. The characteristic property of such equations is non-uniqueness of solutions and their branching or bifurcation. The determination of branching solutions leads to the investigation of the corresponding homogeneous equations and the respective eigenvalue problem. This problem is nonlinear because of specificity of the statement of the optimization problem. The study of the above problem allows us to determine a set of points, in which the respective eigenvalues are equal to unity that determines the branching points of solutions. The data of calculations testify to the ability of the approach proposed to determine the solutions of nonlinear equations numerically with not large computations.

Keywords

  • nonlinear optimization
  • variational approach
  • radiation characteristic
  • nonlinear eigenvalue problem
  • bifurcation of solutions
  • computational modeling

1. Introduction

The nonlinear eigenvalue approach is used in this chapter for the study of the properties to solutions of the generalized phase problem related to the synthesis of radiation systems through the incomplete data. Such incompleteness is considered here in the example of an indeterminate phase characteristic of function, which characterizes the radiation of the plane antenna arrays.

The problems with an indeterminate phase of the wave field arise in various applications and are widely described in the literature. The most well-known of these is the so-called phase problem (see, for example, [1, 2, 3, 4]). It consists in restoring the phase distribution (argument) of the Fourier transform of a finite function by its amplitude (module) given (measured) along the entire real axis. This problem belongs to the classical problems of recovery (identification) and requires the conditions of existence of a unique solution.

In this chapter, another class of inverse problems is considered, and it can be termed optimization (design) problems. In sense of the Fourier transform, this can be, for example, the problem of finding such a finite complex function, the modulus of its Fourier transform satisfies a certain requirement (e.g., is close to a given positive function). As a rule, such requirements are formulated in the variational form, as the minimization of certain functionals. Obviously, such a formulation does not require a uniform solution. On the contrary, the existence of many solutions is often desirable because the above allows many degrees of freedom to determine an appropriate solution. The characteristic applications of such phase optimization problems include the theory of power transmission lines, field converters, antennas and resonators. The first works dealing with nonlinear inverse problems of such type appeared in the second part of the last century (see, for instance [5, 6, 7, 8, 9, 10]).

In mathematical terms, problems of this type are reduced to the nonlinear integral equations of Hammerstein type [11, 12, 13]. They contain a linear kernel and a nonlinear multiplier that depends on a complex unknown function as an integrand. As a rule, the argument (phase) of this function appears there separately from the module. Similar equations are found in the literature in the context of the mentioned phase problem [14, 15]. They have different solutions, and the study of their structure and process of branching or bifurcation is an interesting mathematical problem [16].

Due to their nonlinearity, the problems under consideration require the development and application of special analytical and numerical methods for their solving. Along with the iterative methods that simulate the physical processes of field formation, the various modifications of Newton’s method could be the most promising in this direction [17]. One such modification, which uses solving the nonlinear eigenvalue problems and searching for the zero curves of respective determinants, is proposed in this Chapter. It allows simultaneously with the finding of the branch of solutions to detect the presence of branching points on it and to determine them approximately, provided by this the initial approximations for more accurate calculation.

The nonlinear eigenvalue problems arise in pure and applied mathematics, as well as in the different areas of science that investigate the nonlinear phenomena [18, 19]. A variety of analytical-numerical methods have been elaborated till now for solving the nonlinear problems in acoustics, electrodynamics, fluid dynamics and other areas of applied science [20, 21]. The methods, developed until that time, were focused mainly on solving one-dimensional problems. The difficulties of analytical and computational nature appear if to apply them to a multidimensional problem. The method of implicit function is one of effective tools that been applied for solving the two- and three-dimensional nonlinear eigenvalue problem in the last two decades [22, 23, 24]. The extension of this method, which leads to solving the Cauchy problem (21) and (22), we apply in Section 3 to solve the nonlinear two-dimensional eigenvalue problem.

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2. The operators of direct electrodynamics problem

In the physical relation, the radiation system represents the plane array with the rectangular or hexagonal placement of radiators. Firstly, we consider the array with the rectangular ordering of separate elements (Figure 1a).

Figure 1.

The placement of elements for the considered arrays. a) rectangular. b) hexagonal.

Consider a plane rectangular array consisting of N2M2=2N+12M+1identical elements (radiators), which are located in the xOyplane of the Cartesian coordinate system equidistantly for each of the coordinates. Since the radiators are identical, it is possible to formulate the synthesis problem not for the whole three-dimensional vector directivity pattern (DP), but only for some complex scalar function fx1x2that is termed as the array multiplier. This function for a rectangular equidistant array has the form [25]:

fx1x2=AIn=NNm=MMInmeic1nx1+c2mx2E1

where I=InmNnNMmMis a set of excitations (currents) in the array’s elements, x1=sinθcosφ/sinα1, x2=sinθsinφ/sinα2are the generalized angular coordinates, c1=kd1sinα1, c2=kd2sinα2, k=2π/λis wave number, d1and d2are the distances between radiators along the Oxaxis and Oyaxis respectively, α1and α2are the angular coordinates, within which the desired power DP Px1x2is not equal to zero (Px1x20outside these angles). The function fx1x2possesses 2π/c1periodicity with respect to x1and 2π/c2periodicity with respect to x2. Let us denote the region of change of coordinates x1and x2on one period as Ω=x1x2:x1π/c1x2π/c2. Below, the function fx1x2is termed as the DP of array.

A similar formula can be derived for the array with the hexagonal placement of separate elements (Figure 1b)

fx1x2=AIm=M2M2n=N1mN1mInmeic1nx1+c2mx2E2

where M=2M2+1is quantity of the linear subarrays, then N=2N1m+1is the number of elements in the mth subarray.

Eqs. (1) and (2) for DP fx1x2represent the result of a linear operator A, which acts on a complex-valued space HI=CN2×M2(rectangular case) or HI=CN0×M(hexagonal case) to the space of complex functions of two variables defined in the domain Ω. The value N0determines the number of elements in the central linear subarray in the hexagonal case.

Assume that the desired power DP Ps1s2is not equal to zero in some regions G¯Ω, and it is equal to zero outside. The optimization problem is formulated as the minimization problem of the functional

σαI=PAI2f2+αII2E3

where fand Idetermine the norms in the space of DPs and space of currents respectively, which are defined by the inner products

ff2=f1f2f=Ωf1x1x2f¯2x1x2dx1dx2E4
II2=I1I2I=4π2c1c2n=NNm=MMI1nmI¯2nmE5

here the values f¯2x1x2and I¯2nmare conjugated to f2x1x2and I2nm.

The nonlinear integral equation for the complex vector Iof currents in space HI, which is derived using the necessary condition of the minimum of functional (3), has the form [26].

αI+2AAIAI2APAI=0E6

Here Ais the operator adjoint to A, its form is defined by equality AIff=IAfI. Using the inner products (4), (5) and Eq. (1) we obtain.

Afnm=c1c24π2Ωfx1x2eic1nx1+c2mx2dx1dx2,n=N,N+1,N1,N,m=M,M+1,,M1,M.E7

If to act by operator Aon both the parts of (6), we get a nonlinear integral equation of Hammerstein type for the function f

αf+2AAff2AAPf=0E8

The kernel of the AAoperator for the rectangular array is defined as

Kc1c2x1x1'x2x2'=K1c1x1x1'K2c2x2x2',E9

where

K1x1x1'c1=c1πsinN2c1x1x1'/2sinc1x1x1'/2E10
K2x2x2'c2=c2πsinM2c2x2x2')/2sinc2x2x2')/2E11

The kernel of the AAoperator for the hexagonal array is more complicated because we can not to present it in the form of two multipliers

Kc1,c2,x1,x1',x2x12'=sinc1N101/2x1x1'sin1/2c1x1x1'++2m=1M2cosmc2x2x2'{sinc1N1m1/2x1x1'sin1/2c1x1x1',N1misodd,2n=1N1mcosc1n1/2x1x1',N1mis even.E12

The kernels (9) and (12) of the integral Eq. (8) are real and degenerate. Since Eqs. (6) and (8) are nonlinear ones, both may have a non-unique solution. The number of solutions and their properties is studied according to the method proposed in [16, 27]. In the practical applications, the solution of Eqs. (6) and (8) is performed by the method of successive approximations. The convergence of the method depends on the parameter α, desired DP Px1x2, as well as the parameters c1and c2contained in the kernels (9) and (12).

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3. Search for the bifurcation curves

We should use the linear integral equation to define the bifurcation curves according to [16]. Based on this equation, we pass to the respective eigenvalue problems, solutions of which allow us to find the characteristic values of parameters c1and c2in the kernel of equation, at which the bifurcation appears.

3.1 Description of procedure

The linear equation

αf=2AAPfE13

is used to study the properties of Eq. (8).

In contrast to a similar equation for the amplitude DP synthesis problem [25], Eq. (8) does not have a trivial nonzero initial solution f0for all parameters c1and c2; the trivial solution f0is zero for it, so in contrast to the problem of synthesis by amplitude DP, we are not talking about the branching of solutions, but about their bifurcation.

The problem of finding bifurcation curves is reduced to the corresponding eigenvalue problem. The equation for eigenfunctions and corresponding eigenvalues, which refers to (13), is

gx1x2=2λα1Ωgx1'x2'K1c1x1x1'K2c2x2x2'dx1'dx2'E14

As stated by the branching theory of solutions of the nonlinear equations [16], the bifurcation points can be those values of c1and c2at which Eq. (14) has nonzero solutions.

Using the properties of the degeneracy of the kernel AA, we reduce Eq. (14) to the equivalent system of the linear algebraic equations (SLAE). The coefficients of matrix of this equation depend on the parameters c1and c2analytically. To this end, the equations for eigenfunctions corresponding to (13) are written as

gx1x2=n=NNm=MMxnmeic1nx1+c2mx2E15

where

xnm=c1c24π2ΩPx1'x2'gx1'x2'eic1nx1'+c2mx2'dx1'dx2'E16

Multiplying both the parts of (15) on Px1'x2'eic1kx1'+c2lx2'at k=N,N+1,,N1,N,l=M,M+1,,M1,Mand integrating over the domain Ω, we obtain a system of linear algebraic equations to determine the quantities xnm

xkl=n=NNm=MManmklc1c2xnm,k=N,N+1,,N1,N,l=M,M+1,,M1,M,E17

where

anmkl=c1c24π2ΩPx1x2ei[c1nkx1+c2mlx2dx1dx2E18

and matrix of the coefficients anmklis self-adjoint and Hermitian.

Thus, we obtained a two-parameter nonlinear spectral problem corresponding to a homogeneous SLAE (17). This problem can be given as

EMAMc1c2x=0E19

where AMis the matrix of coefficients anmkl, EMis a unit matrix of dimension N2M2.

For the system (19), the equality

Ψc1c2=detEMAMc1c2=0E20

must be met to have a non-zero solution.

One can easy to make sure that the function Ψc1c2is real. Moreover, since AMc1c2is the Hermitian matrix, then EMAMc1c2is Hermitian too. The determinant of the Hermitian matrix is a real number [28]. Thus, Ψc1c2is a real function of real arguments c1and c2.

Consequently, the problem to find the eigenvalues of Eq. (14) or to determine the solution of the equivalent SLAE (19) is reduced to finding zeros of function Ψc1c2.

If to consider the equation Ψc1c2=0as a problem of determining an implicit function c2=c2c1in the vicinity of some point c1, we get Cauchy problem [29].

dc2dc1=Ψc1'c1c2Ψc2'c1c2E21
c2c10=c20E22

To retrieve the initial conditions (22) we pass to an auxiliary one-dimensional nonlinear spectral problem if to substitute c2by c2=γc1in Eq. (20) with some real parameter γ. As a result, we get the one-dimensional eigenvalue problem

EMAMc1γc1x˜EMA˜Mc1x˜=0E23

Eq. (20), which corresponds to Eq. (23), is

Ψc1γc1=detEMA˜Mc1=0E24

Let c10be the solution of the Eq. (24), then c10c20=c10γc10is the point that corresponds to eigenvalue λ01of Eq. (15). By solving Eqs. (21) and (22) in a small vicinity of point c10c20, we find the spectral curve of the matrix-function AMc1c2, which is the curve c2c1defining a set of the bifurcation points.

The eigenfunctions of Eq. (14) are defined as the eigenvectors of matrix AMc1c2using the resulting solution of the Cauchy problem with the sought solutions Ψc1c2. In this procedure, a four-dimensional matrix AMc1c2is reduced to a two-dimensional one by the relevant renouncement of its elements.

3.2 Defining the area of nonzero solutions

Due to the peculiarity of the problem statement according to desired power DP Px1x2, Eq. (8) has zero solution at arbitrary values of the parameters c1,c2,α. From an engineering point of view, this is a significant drawback, but for some desired DPs Px1x2it is possible to fix an area of parameters c1,c2,αat which a nonzero solution exists. At the small c1and c2, the kernel (9) is given approximately in the form

Kc1c2x1x1'x2x2'M2N2c1c2π2E25

Assuming that fx1x2is constant, the integral Eq. (8) can be rewritten as (usually for small c1and c2fx1x2const).

π2α2M2N2c1c2=1111Px1x2dx1dx24f(x1x2)2E26

The area of integration Ωin Eq. (8) is reduced in the last formula to the area 11×11because of definition of both the arguments x1,x2and parameters c1,c2.

Taking into account that fx1x22is positive, we get the following relationship between the function Px1x2and the parameters c1, c2, and α:

1111Px1x2dx1dx2π2α2M2N2c1c2>0E27

Finally, considering the case Px1x21, we obtain:

c1c2>π2α8M2N2E28

In fact; inequality (28) determines the area of parameters c1,c2,α, where nonzero solutions exist. In Figure 2, the dependence curves c2=c2c1for three different values M2and N2are shown. The results are given for array with the number of elements M2=N2=3(curve 1), M2=N2=5(curve 2) and M2=N2=11(curve 3). The area of values c1and c2, where the existence of zero solutions is possible, according to the estimate (28) is located below and to the left of the presented curves. As can be seen, the area of zero values decreases significantly with increasing N2and M2. The obtained results testify that the zero solutions of Eq. (8) for a given constant power DP can exist either at a small value c1c2corresponding to low frequencies (at a given size of array), or at the values of c1that significantly exceeding c2and vice versa. The last case corresponds to arrays with a large difference in distance of elements along the coordinate axes. Such arrays are usually rarely used in practice.

Figure 2.

The curvesc2=c2c1at the differentM2andN2.

3.3 Determination of bifurcation lines

3.3.1 The case of rectangular array

The finding of bifurcation lines of the nonlinear Eq. (8) was performed for the array containing N2M2=1111=121radiators for the desired power DP Px1x2=1at Λc=c1c20<c1c22for the different values of the parameter αin (3).

The search for bifurcation lines can be performed directly by investigating the properties of the determinant (20) as a function of the parameters c1and c2. In addition, the function (20) depends on the parameter α; so the set of its eigenvalues also depends on this parameter, i.e. the set of spectral curves that separate the areas of zero and nonzero solutions.

The behavior of the corresponding curves when changing the parameter αis shown in Figure 3. The behavior of the determinant (24) depending on the parameters c1and c2at α=0.5is given in Figure 3a; and in Figure 3bd, the intersection of this function with a plane Ψc1γc1=0is illustrated at the different α. This results in a set of curves that correspond to a set of spectral lines separating the area of zero and nonzero solutions. At a fixed size of array, the area where zero solutions can exist expands if the parameter αincreases, this area is located below the left of the first curve.

Figure 3.

The spectral curves ofEq. (19)at the differentα.

The curves marked by number 1 correspond to the solutions with constant (zero or even) phase DP; curves with number 2 correspond to the solutions with phase DP that is even with respect to the Ox1axis, and odd with respect to the Ox2axis, and curves numbered by 3 correspond to the solutions with a phase DP odd with respect to two coordinate axes. The proposed procedure is quite approximate, it does not allow to separate the curves that correspond to different types of solutions and thus identify the areas where there is a nonzero solution for the synthesized power DP with the specified phase property.

The method of implicit function proposed in [23] and developed for plane array in [30] is devoid of this drawback.

At the first step of this method, a series of one-dimensional eigenvalue problems is solved, by this the different values of parameter γare prescribed by the relation c2=γc1and a one-dimensional problem is solved with respect to c1. In Figures 4 and 5, the first four eigenvaluesof the problem at γ=1.0and γ=0.2are shown. The values c1ic2i=c1i, i=1,2,3,4, at which λi=1, are the bifurcation points in the plane c1Oc2. By this, the set of points c1c2at which the eigenvalue λi=1is determined approximately from the graphical data.

Figure 4.

The first eigenvalues at the rayc1=c2,α=0.5.

Figure 5.

The first eigenvalues at the rayc2=0.2c1,α=0.5.

The next step is to refine the values c1ic2iby solving the transcendental Eq. (20), and the point c1ic2i, which is considered as the initial approximation.

In the final step, the bifurcation curve in the plane c1c2is determined by solving Eqs. (21) and (22), after specification of the values c1ic2i. In Figure 6, the bifurcation curves c11c21c14c24that correspond to the first four eigenvalues are shown. The curve with number 1 corresponds to the solution with the zero (even) phase of the created DP. This curve corresponds to that is marked by 1 in Figure 3b.

Figure 6.

The bifurcation curves corresponding to setc1ic2i,i=1,,4,N2=M2=11.

There are no nonzero solutions with such a phase property for the values c1and c2above and to the right of this curve. Curves 2 and 2′ correspond to solutions in which the phase DP is symmetric about one axis and asymmetric about the other axis (obviously, for a plane array, there are two such curves and they are antisymmetrical). Curve 2 corresponds that is marked by 2 in Figure 3b. The curve with number 3 corresponds to a solution with a phase DP antisymmetrical (odd) with respect to both the axes. The location of the areas of zero and non-zero solutions is the same as in Figure 3b. It should be noted that the problem of refining the roots of Eq. (20) is the most time-consuming in computational relation because refining the roots of this equation requires a series of computational experiments with different values c1,0ic2,0iof initial parameters close to approximate values.

3.3.2 The case of hexagonal array

Firstly, we consider the procedure of determination of bifurcation curves by finding zero lines of determinant (24). The results, similar to those are presented in Figure 3a and b for the rectangular array, are shown in Figures 7 and 8. One can see that the behavior of function Ψc1c2is more complex than in the case of rectangular array. The obtained graphs testify that the solutions with other different behavior of phase argfx1x2of the DP appear additionally. One such solution is marked by number 4. Other solutions appear when parameters c1and c2increase at the fixed α.

Figure 7.

The surface of determinant(24)values for the hexagonal array,α=0.5.

Figure 8.

Zero lines of determinant(24)for the hexagonal array,α=0.5.

Search of the bifurcation curves is carried out similarly to the case of rectangular array. The numerical results are presented for the array with Ntot=61elements for the desired power DP N0x1x2=1at Λc=c1c20<c1c22.0for the different values of αin the functional (3). At the first step, the one-dimensional eigenvalue problems were solved at the different values of parameter γ. In Figure 9, the first four eigenvalues are shown at γ=1.0, and in Figure 10, they are shown at γ=0.2. Similar to the case of rectangular array, the points, in which λi=1are moved to right and the distance between them increases at γ=0.2. The values c1ic2i=γc1i, where i=1,2,3,4, are the bifurcation points in the plane c1c2. The points c1ic2i, for which the eigenvalues λi=1are determined approximately in this step.

Figure 9.

The first eigenvalues for the rayc1=c2atα=0.5.

Figure 10.

The first eigenvalues for the rayc2=0.2c1,α=0.5.

The specification of values c1ic2iby solving Eq. (20) is carried out in the next step, and the points c1ic2iof the graph data from the Figures 9 and 10 are used as initial approximations. The usual numerical half-division method is used for this goal.

The bifurcation points c1ic2i, i=1,2,3,4for the first four eigenvalues in the rays, c2=γc1are shown in Figure 11. The respective curves of bifurcations, which are obtained by solving Eqs. (21) and (22), are shown in Figure 12. As in the case of a rectangular array, to obtain the necessary data, we should carry out precise computations.

Figure 11.

The bifurcation points at the raysc2=γc1.

Figure 12.

The bifurcation curves in the planec1c2.

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4. The engineering applications

The results presented in this Section demonstrate how the knowledge about the point of bifurcation obtained as the solutions of the nonlinear eigenvalue problems allows us to understand better the process of bifurcation and how to get the solutions, which are the most optimal in sense of the used criterion of optimization.

4.1 The method of successive approximations

The properties of solutions to Eq. (8) obtained by using the method of successive approximation are related directly with the properties of phase characteristic of the eigenfunctions, which are determined at solving the eigenvalue problem. Prescribing the initial approximation f0for the iterative process for solving Eq. (8) with the specified property of the phase argf0, we could receive the solution of Eq. (8) with the same phase property in the wide range of characteristic parameters c1and c2. This is important for the engineering design of arrays having the fixed phase characteristics of radiation in the defined range of frequencies.

The method of successive approximations

fn+1βfn+1βBfn=0,n=0,1,2,E29

is used for solving Eq. (8) with a set of specific physical parameters of array. In the last formula,

Bf=2αAAPff2fE30

Parameter β01in (29) is used to accelerate the convergence of iterative process. To substantiate the condition of convergence of the iterative process (29), we apply Theorem 2.6.2 [26] (p. 133), which states that the operator AAbe contraction one. This requirement is met when the inequality

α>2AAfN0f2E31

met. The results of numerical calculations show that condition (31) is overestimated and for some values of the problem parameters the iterative process (29) converges for values αthat do not satisfy the estimate (31).

4.2 The case of rectangular array

In Figure 13, the dependence of the convergence of the iterative process (29) on the value of parameter αfor a desired power DP Px1x2=1at the fixed values β=0.1, c1=c2=2.0, the number of radiators M2N2=1111=121is shown. The required accuracy ε=103.

Figure 13.

The character of convergence of iterative process(29)at the differentα,M2=N2=11.

The results of solving the optimizing problem for this desired power DP at α=0.5are shown in Figure 14. The approximation quality to a desired DP Psignificantly depends on the parameters c1and c2at both fixed N2and M2. The mean-square deviation (MSD) (the first term in (3)) is equal to 0.0847 for c1=c2=1.0, and it is equal to 0.0075 for c1=c2=3.14.

Figure 14.

The created power RPf(a) and optimal distribution of currentsInm(b) atc1=c2=1.0.

The synthesized DP ffor larger c1,c2has not only a more optimal mean-square approximation, but it is also closer to the shape of the desired DP P. The optimal amplitudes Inmof currents in the array’s elements are close to constant at such parameters c1and c2.

When solving the optimizing problem for desired power DP of a more complex form, the quality of the approximation significantly depends on both the parameter αand the type of initial approximation for the phase of a given DP. The results are shown for the desired power DP

Px1x2=sinπx1sinπx2,1x11,1x21E32

at c1=c2=3.14and α=0.2in Figure 15. Despite the fact that the shape of desired DP Px1x2is more complex that in the previous example, decrease of α(from 0.5 to 0.2) and simultaneous increase of c1and c2(from 1.0 to 3.14) allows us to get the amplitude fx1x2of created DP, which is very close to the Px1x2. The optimal distribution of currents’ amplitudes Inm(Figure 15b) approaches the shape of the created DP.

Figure 15.

The created power RPf2atc1=c2=3.14.

We have used an additional optimization parameter βin Eq. (29), which, as shown by the results of numerical calculations, accelerates the convergence of iterative process significantly. In Figure 16, the results of the study of the influence of this parameter on the rate of convergence at a fixed value of the parameter α=0.5are shown. The results are given for the desired power DP Px1x2=1at c=1c2=2.0. In order to achieve the accuracy 103of calculations, one needs 157 iterations at β=0.01. If parameter βincreases to a certain value, the number of iterations decreases significantly, so at β=0.05, β=0.10, β=0.15, β=0.20, and β=0.25one requires 62 iterations, 37 iterations, 28 iterations, 20 iterations, and 16 iterations, respectively. At the subsequent increase, the number of required iterations begins to increase and already at β=0.30the iterative process begins to diverge. Numerical calculations show that the limit valueof β, at which the iterative process (29) begins to diverge, significantly depends on the value of the parameter α. So, if this parameter decreases, the threshold value of βincreases. The dependence of the convergence on the array’s parameters (c1,c2,M2,N2,d1,d2) is not so significant.

Figure 16.

The convergence of iterative process(29)for the differentβ.

The values of the functional (3) for the created DP with different phases are shown in Figure 17. The solid curve corresponds to the phase DP even with respect to two axes, the dotted curve corresponds to phase DP even with respect to one axis and odd with respect to the other, the dashed curve corresponds to the phase DP odd with respect to both the axes.

Figure 17.

The values of functional(3)versus the phase of created DP.

One can see that the values of functional at the fixed c(frequency) significantly depend on the phase of the created DP. The value σ=0.2is achieved for the “even-even” solution at c=0.79, for the “even-odd” solution at c=0.71and for the “odd-odd” solution at c=0.625. That is, within the used criterion, the latter type of solution is 21% better than the first one. From this fact, it follows that at a fixed distance between the radiators for the desired DP Px1x2=1, the number of array’s elements can be reduced by 21% with the same value of MSD. A similar situation is observed for the characteristics of DP at σ=0.1, i.e. “odd-odd” solution is better on 19.4% than “even-even”.

4.3 The case of hexagonal array

The results of solution of the optimization problem for two given power DPs P1x1x21and

P2x1x2=2x12+x221x12x22,x12+x221,0,x12+x22>1,E33

in the form of body of rotation are shown in Figures 18 and 19 at α=0.5.

Figure 18.

The amplitude of created DPf2forP1atc1=2.0,c2=2.236.

Figure 19.

The amplitudef2of created DP forP2atc1=2.0,c2=2.236.

As previously, the optimization problem consists of solving Eq. (8) by the method of successive approximation (29). The MSD (the value of the first term in (3)) for the first desired DP is equal to 0.3774, and it is equal to 0.2218for the second desired DP.

Similar to the case of rectangular array, the approximation quality to the desired DP Pdepends on both the parameters c1, c2, and α. The characteristic of MSD of DPs for αat the different c1on the ray c2=1.118c1is shown in Figures 20 and 21. The chosen relation between c1and c2provides the regularity of the array’s geometry, and as the numerical computations have shown, gives the ability to get the close characteristics of radiation in the planes x1and x2.

Figure 20.

The MSD versus the parameterαfor the DPP1.

Figure 21.

The MSD versus parameterαfor DPP2.

The largest MSD for the P1is achieved at α=1.0for c1=0.5, and it is equal to 1.96443, it diminishes almost linearly if parameter αdecreases. The largest MSD for DP P2is equal to 1.43685. One should note that the value of MSD diminishes if αdecreases, but the norm IHIof current growth that is unacceptable from the engineering point of view.

The approximation quality to the desired DP depends also on the type of initial data, which are prescribed for the iterative process (29). The dependence of the values of functional (3) on the parity of phase of the initial approximation f0for the DP P1is shown in Figure 22. The results are shown for four types of initial approximation: odd with respect to both the axes (dotted curve), even with respect to the Ox1axis and odd with respect to the Ox2axis (dashed curve), odd with respect to the Ox1axis and even with respect to the Ox2axis (dash-dot curve), and even with respect to both the axis (solid curve). The initial approximation f0, corresponding to the even phase with respect to both the axis, is optimal for this DP, moreover the values of σαfor the small values of parameters c1and c2differ significantly, but starting from c1=0.8this difference does not exceed 10%.

Figure 22.

The values ofσαversus the initial approximation of initial approximation for the iterative process(29).

The dependence of convergence of the iterative process (29) on the parameter αat the fixed β=0.1is shown in Figure 23. As in the case of a rectangular array, the iterative process converges most slowly at α=1.5, one needs 67 iterations to achieve the accuracy that is equal to 103. The number of iterations decreases if αdiminishes. For example, one needs 30 iterations to achieve the same accuracy at α=0.2.

Figure 23.

The convergence of iterative process(29)versusα.

The dependence of convergence on the parameter βis studied too. The necessary number of iterations that needs to achieve the accuracy 103at β=0.01, β=0.05, β=0.10, β=0.15, β=0.20, and β=0.30(curves 1–6 respectively) is shown in Figure 24. It is substantiated that the iterative process converges most slowly at β=0.01, it is necessary 152 iterations to achieve the prescribed accuracy. The most optimal among the considered βis β=0.20, one needs 20 iterations only to achieve the above accuracy. The iterative process becomes slow at the subsequent growth of β. For example, one needs 37 iterations to achieve this accuracy; the iterative process (29) becomes convergent at β>0.40. This testifies that in the process of computations one should to limit by non-large values of β(β0.20) that guarantees the convergence and considerably grows its speed on the contrast with small β(β0.01).

Figure 24.

The convergence of iterative process(29)versus number of iteration,α=0.5.

More information about the problem under investigation one can find in [30, 31, 32, 33, 34].

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5. Conclusions

The problem of finding the solutions to the nonlinear integral equations and their properties is reduced to nonlinear two-dimensional eigenvalue problems that lead to the subsequent application of an implicit function method for solving the Cauchy problem for the linear differential equation. The area of non-zero solutions to the above equations is determined by involving the solving transcendental equation, which is got by equating to zero of determinant related to the eigenvalue problem. The results of solving the nonlinear eigenvalue problems are applied subsequently for specification of the bifurcation points and obtaining the bifurcation curves. The approach does not depend on the form of operator determining the radiation properties of physical system (plane rectangular and hexagonal arrays). The obtained results are the constructive basis on which a series of practical engineering problems of optimization was solved numerically.

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Conflict of interest

The author declares no conflict of interest.

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Written By

Mykhaylo Andriychuk

Submitted: November 9th, 2021 Reviewed: February 25th, 2022 Published: April 18th, 2022