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Comparison of modified method of auxiliary currents and modified null-field method applied to the problem of diffraction by an impedance body


A.G. Kyurkchan, S.A. Manenkov, E.S. Negorozhina

The paper considers the scalar diffraction problem of a plane wave by an impedance body of revolution. To solve the problem, we used two methods: a modified method of auxiliary currents (MMAC) and a modified null-field method (MNFM). In this paper we derive the integral equations of the first kind to which the initial boundary problem amounts. We reduce these equations to an infinite system of one-dimensional integral equations by expanding the kernels and unknown functions into the Fourier series. The latter is solved by the collocation method, and the corresponding integrals are replaced by the Riemann sums. Thus, the simplest version of MMAC and MNFM as a method of discrete sources is used. As is known, in MMAC and MNFM some condition is set on the auxiliary surface of revolution which located inside the scatterer. To obtain efficient numerical algorithms it is essential to choose the auxiliary surface, using the analytic continuation of the boundary of the body. Besides this surface should cover all the singularities of continuation of the secondary wave field inside the surface of the body. Thus, the main point of MMAC and MNFM is correct choice of the auxiliary surface. Significant difference of this work is the use of a suitable coordinate system, in which the problem is solved. We used elongated spheroidal and oblate spheroidal as well as toroidal coordinates. This fact allowed us to obtain efficient algorithms for solving problems of diffraction on highly elongated and highly flattened scatterers, and on the bodies of the toroidal shape. This paper describes a method of constructing of the auxiliary surface in these coordinate systems based on the introduction of the corresponding complex variable. In solving the problems of diffraction by scatterers as the bodies of revolution it is difficult to calculate the Fourier coefficients of the Green function of free space, that is, so-called Vasiliev’s S-functions. The efficient algorithm allowing to calculate S-functions for all the angular harmonics is developed in the paper. This algorithm is based on the extraction of the static part of the integrand of S-functions and their derivatives. The regular part of Vasiliev’s S-functions is found with the rectangular method, and the singular is found with the help of the recurrence formula given in the paper. The developed algorithms are applied to the problem of diffraction of a plane wave on elongated and oblate spheroids. It is shown that both techniques allow us to find a pattern for elongated and oblate spheroids with semi-axis ratio 50. The modification of MMAC considered in the paper is compared with standard version of MMAC using the spherical coordinates. Significant gain in the volume of calculations for spheroidal coordinates when choosing the auxiliary surface is shown. Besides the absolute value of the pattern for these spheroids are obtained with MMAC and MNFM. It is shown the coincidence of the numerical results. In the paper we consider the problem of diffraction on multi-lobes of revolution for various shapes, in particular Chebyshev and toroidal particles. A numerical algorithm similar to the method of bisection allowing to find the maximum degree of deformation of the border of the scatterer for the choice of the auxiliary surface is developed. With the use of this algorithm we get the angular dependence of the pattern and the distribution of the residual on the initial (for MMAC) and auxiliary (for MNFM) contour of the axial cross-section of the body. The optical theorem accuracy check up is carried out in the paper. It is shown that the optical theorem is fulfilled with high accuracy for both methods.

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