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DIPED-2008 Proceedings

APPLICATION OF MODIFIED NEWTON METHOD TO NONLINEAR INTEGRAL EQUATION OF CIRCULAR ANTENNA SYNTHESIS PROBLEM M. I. Andriychuk, O. O. Bulatsyk, N. N. Voitovich Pidstryhach Institute of Applied Problems of Mechanics and Mathematics, NASU 3 ”B” Naukova St., Lviv, 79601, Ukraine Abstract. An equation arisen in the synthesis problem of antenna systems according to prescribed amplitude radiation pattern is investigated. The proposed modification of Newton method allows finding the branched solutions depending on the value characteristic physical parameter. The algorithm is described and numerical results presented.

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I. Introduction The synthesis problems according to the prescribed amplitude pattern belong to the problems with free phase, those are reducible to nonlinear integral equations of the Hammerstein type. Such problems have non-unique solutions branching as a parameter varies, and their investigation has both the theoretical and practical interests. Recently a class of equations of such a type was found having semi-analytical solutions expressed by finite-order polynomials [1]–[3]. In general case, such equations can apparently be solved only by numerical methods. Several methods were proposed and successfully applied to concrete problems [4]–[7], permitting, in particular, to investigate partially the branching process. However, the problem of developing numerical methods those would allow finding all the solutions and detecting their branching points is till topical. Different modifications of the Newton method have been used for solving nonlinear equations with branching solutions (see, e.g. [8]). The most suitable of them for this purpose is likely the one based on the singular decomposition of the Jacobi matrix, because the behavior of the singular values near the branching points can be used for their identification. This approach was successfully applied to certain problems of the considered type. Significant difficulties arise when there are several small singular values caused in particular by the geometry degeneration or the branching point multiplicity. Just such a problem is considered in this paper. II. Description of the method In general case, the regularized amplitude-phase optimization problem based on the functional 2 2 σα (u ) = F − | Au | + α u (1)

with a compact operator A , given positive function F and regularization parameter α , is reduced to the nonlinear equation of the form [4]

Φ ≡ α f + B( f − F exp(i arg f )) = 0 , (2) where f = Au , B = AA* . In the case of two-dimension circular antenna of the radius a considered here, u is the current on the antenna, f is the pattern created by u , the operator B acts as follows


135 2π

Bg =

∫ Ω(ϕ, ϕ ) g (ϕ )d ϕ , 1

1

1

(3)

0

where

Ω(ϕ, ϕ1 ) = J 0 (2c sin((ϕ − ϕ1 ) / 2)) ,

(4)

ϕ, ϕ1 are the angular coordinates of the points in the far-field zone, J 0 is the Bessel function of the first kind, c = ka is the variable parameter of the problem; a nonessential factor is omitted in (3). Analogous problem for the linear antenna was considered in [3] using the finite-polynomial presentation. Aiming to find a branch of solutions f (ϕ; c) depending on the parameter c , we consider (2) as the equation Φ( f , c) = 0 with respect to a point (c, f ) on this branch. Following the general idea of the Newton method, we assume that (c0 , f 0 ) is the current approximation to this point and find the next one from the demand that equation (1) is satisfied in the first order of the increments ε = c − c0 , δ (ϕ ) = f (ϕ ) − f 0 (ϕ ) . This demand leads to the equation Φc ⋅ ε + Φ f ( f 0 , c0 , δ ) = −Φ0 ,

(5)

where Φc =

∂B [ f 0 − F exp(i arg f 0 )] , ∂c c=c0

δ ], f0 , Φ0 = Φ( f 0 , c0 ) . Here it is accounted that in our approximation Φ f ( f 0 , c0 , δ ) = αδ + B0 [δ − iF exp(i arg f 0 ) Im

B0 = B c=c

0

(6) (7)

δ . (8) f0 Since the unknown complex function δ is involved under the operator B0 only in the form Im(δ / f 0 ) , it is convenient to introduce two real functions y and z , as follows: δ = f 0 ⋅ ( y + iz ) . Then equation (5) obtains the form exp(i arg f ) = exp(i arg f 0 ) + i Im

Φc ⋅ ε + (α I + B0 )[ f 0 ⋅ y ] + iα ( f 0 ⋅ z ) + iB0 [( f 0 − F exp(i arg f 0 )) z ] = −Φ0 ,

(9)

where I is the unit operator. Acting on (9) by the operator D = (α I + B0 )−1 , we obtain D[Φc ] ⋅ ε + f 0 ⋅ y + if 0 ⋅ z − iDB0 [ F exp(i arg f 0 ) ⋅ z ] = −D[Φ0 ] . (10) Finally, eliminating the unknown function y by multiplying the both sides of (10) by f 0 and separating the imaginary part we arrive at the linear equation Im { f 0 ⋅ D[Φc ]} ⋅ ε + f 0 ⋅ z − Re { f 0 ⋅ DB0 [ F exp(i arg f 0 ) ⋅ z ]} = − Im { f 0 ⋅ D[Φ0 ]} , (11) 2

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with respect to the unknown constant ε and function z . After ε and z are determined, the sought function δ is straightforwardly calculated from (10) and thereby determining the next approximation to (c, f ) is completed. The limiting point on the branch (if found) is determined not uniquely. To provide a uniform distribution of such points on the branch we should choose the suitable initial approximations. III. Numerical results The linear equation (11) to be solved on each step of the Newton method, is nonstandard. First, it is underdetermined because it contains not only the unknown function f but also the constant c . Second, if f 0 is a solution to (2), then the lefthand side of (11) becomes zero at ε = 0, z = 1 . Finally, at the branching point this side vanishes also in the function describing the direction of the solution branched off there. These troubles can be overcame when solving the equation approximately, using the singular decomposition of its matrix. The singular vectors corresponding to the zero and very small singular values must not participate in the solution. Moreover, the points where more than one singular values are very small should be detected as those suspected for the branching points. The algorithm was tested on the problem with F (ϕ ) = sin 2 (ϕ / 2) + 0.1 , α = 1 . In Fig. 1 the branching process is schematically shown. For identification of a solution we took its phase increment Δ = max arg( f (ϕ1 )) − arg( f (ϕ2 )) when moving ϕ1 ,ϕ2 ∈(0,2 π )

around the circle. The curves describing different solutions f j , j = 0,1, 2,3 are marked by their indices. The real solution f 0 exists at all c (such type of solutions can exist more than one, with different points on the sign alternation; here we consider only one of them). With c increasing the sign alternations appear in this solution; we suppose that its phase increment grows jump-wise. At c1 = 1.43 two new solutions f1 , f 2 branch off from f 0 , with different evenness of their phase: arg( f1 ) is even, arg( f 2 ) is odd. In the first case the increment is a smooth function of ϕ , in the second one it also has unexpected jumps. The solution f3 starts at c2 = 2.56 from f 0 having already two sign alternations. For comparison, the amplitude and phase distributions of different solutions are shown in Fig. 2. It is significant that the amplitude distribution of the real solution approaches the desired function F (ϕ ) noticeably worse than the complex ones. On the other hand, the phase distribution of the solution f 2 nearest to F (ϕ ) in the modulus, is smooth but has the large increment, about 3π . As it was expected, this solution branches off from f 0 at the first branching point and has odd phase distribution. REFERENCES

1. N. N. Voitovich, O. O. Reshnyak. Solution of nonlinear integral equation of synthesis of the linear antenna arrays. Journ of Applied Electromagnetism. 1999, Vol. II, No 1, pp. 43-52. 2. O. O. Bulatsyk, N. N. Voitovich. Analytical solutions to a class of non-linear integral equations connected with modified phase problem. Information Extraction


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and Processing . – 2003, Vol. 19(95), pp. 33-39. (In Ukrainian). 3. O. O. Bulatsyk, N. N. Voitovich. Comparison of two optimization criteria in antenna problems with free phase pattern. Proc of Xth Intern. Seminar/Workshop on Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory (DIPED-2005), Lviv, 2005, pp. 204–207. 4. M. I. Andriychuk, N. N. Voitovich, P. A. Savenko, V. P. Tkachuk. The Antenna Synthesis according to Amplitude Radiation Pattern. Numerical Methods and Algorithms. Kiev, Naukova Dumka, 1993 (In Russian). 5. P. A. Savenko. Numerical solution of one class of nonlinear problems of the radiating system synthesis theory. Journ of Computational Mathematics and Mathematical Physics. 2000, Vol. 40, No 6, pp. 929-939. (In Russian). 6. P. O. Savenko. Nonlinear Problems of Radiating Systems Synthesis. Lviv, IAPMM NASU, 2002. (In Ukrainian). 7. Yu. P. Topolyuk. Properties of gradient method for free phase problem with compact operator. Proc. of XIIth International Seminar/Workshop on Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory (DIPED-2007), September 17-20, 2007, Lviv, Ukraine, pp. 86-88. 8. P. Deuflhard: Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms. Series Computational Mathematics 35, Springer, 2004.

Fig. 1. Phase increment of different solutions. F = sin 2 (ϕ / 2) + 0.1 ; α = 1 .

a)

b)

Fig. 2. Amplitude (a) and phase (b) distributions of different solutions at c = 3 .