Geodesic
Application of number-theoretic grids in problems of sound diffraction by elastic bodies
Čebyševskij sbornik, Tome 23 (2022) no. 5, pp. 206-226.

Voir la notice de l'article provenant de la source Math-Net.Ru

The article considers the problem of a plane harmonic sound wave diffraction by an elastic ellipsoid. To represent the scattered field, a representation in the form of a Kirchhoff integral is used. This leads to the need to solve the Fredholm integral equation of the second kind to determine the displacement potential in the scattered wave on the surface of the scatterer. It is shown that the use of quadrature formulas based on number-theoretic grids allows you to reduce the number of calculations for the approximate calculation of integrals, when solving the integral equation and when calculating the scattered acoustic pressure in near field. This method is compared with the calculation of integrals by the simple cell method, which has the same order of accuracy. The time of solving the problem is compared with the calculation of pressure in the vicinity of the ellipsoid based on the solution of an integral equation by two methods for calculating integrals.
Keywords: diffraction, sound waves, linear integral equations, quadrature formulas, periodization, Smolyak grids, parallelepiped grids. 
@article{CHEB_2022_23_5_a17,
     author = {N. N. Dobrovol'skii and S. A. Skobel'tsyn and L. A. Tolokonnikov and N. V. Larin},
     title = {Application of number-theoretic grids in problems of sound diffraction by elastic bodies},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {206--226},
     publisher = {mathdoc},
     volume = {23},
     number = {5},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2022_23_5_a17/}
}
TY  - JOUR
AU  - N. N. Dobrovol'skii
AU  - S. A. Skobel'tsyn
AU  - L. A. Tolokonnikov
AU  - N. V. Larin
TI  - Application of number-theoretic grids in problems of sound diffraction by elastic bodies
JO  - Čebyševskij sbornik
PY  - 2022
SP  - 206
EP  - 226
VL  - 23
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CHEB_2022_23_5_a17/
LA  - ru
ID  - CHEB_2022_23_5_a17
ER  - 
%0 Journal Article
%A N. N. Dobrovol'skii
%A S. A. Skobel'tsyn
%A L. A. Tolokonnikov
%A N. V. Larin
%T Application of number-theoretic grids in problems of sound diffraction by elastic bodies
%J Čebyševskij sbornik
%D 2022
%P 206-226
%V 23
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CHEB_2022_23_5_a17/
%G ru
%F CHEB_2022_23_5_a17
N. N. Dobrovol'skii; S. A. Skobel'tsyn; L. A. Tolokonnikov; N. V. Larin. Application of number-theoretic grids in problems of sound diffraction by elastic bodies. Čebyševskij sbornik, Tome 23 (2022) no. 5, pp. 206-226. http://geodesic.mathdoc.fr/item/CHEB_2022_23_5_a17/

[1] D. Kolton, R. Kress, Metody integralnykh uravnenii v teorii rasseyaniya, Mir, M., 1987, 311 pp.

[2] S. Kirkup, “The Boundary Element Method in Acoustics: A Survey”, Appl. Sci., 9 (2019), 1642

[3] S. M. Rao, “An iterative method to solve acoustic scattering problems using a boundary integral equation”, J. Acoust. Soc. Am., 130:4 (2011), 1792–1798

[4] J. A. Fawcett, “Scattering from a finite cylinder near an interface”, J. Acoust. Soc. Am., 136:2 (2014), 485–493

[5] A. M. A. Alsnayyan, J. Li, S. Hughey, A. Diaz, B. Shanker, “Efficient isogeometric boundary element method for analysis of acoustic scattering from rigid bodies”, J. Acoust. Soc. Am., 147:5 (2020), 3275–3284

[6] K. Brebbiya, Zh. Telles, L. Vroubel, Metody granichnykh elementov, Mir, M., 1987, 524 pp.

[7] E. L. Shenderov, Izluchenie i rasseyanie zvuka, Sudostroenie, L., 1989, 304 pp.

[8] N. N. Dobrovolskii, S. A. Skobeltsyn, L. A. Tolokonnikov, N. V. Larin, “O primenenii teoretiko-chislovykh setok v zadachakh akustiki”, Chebyshevskii sbornik, 22:3 (2021), 368–382

[9] I. Yu. Rebrova, V. N. Chubarikov, N. N. Dobrovolskii, M. N. Dobrovolskii, N. M. Dobrovolskii, “O klassicheskikh teoretiko-chislovykh setkakh”, Chebyshevckii sbornik, 19:4 (2018), 118–176

[10] N. M. Korobov, “Primenenie teoretiko-chislovykh setok v integralnykh uravneniyakh i interpolyatsionnykh formulakh”, Sbornik statei. Posvyaschaetsya akademiku Mikhailu Alekseevichu Lavrentevu k ego shestidesyatiletiyu, Tr. MIAN SSSR, 60, 1961, 195–210

[11] Yu. N. Shakhov, “O priblizhennom reshenii uravnenii Volterra II roda metodom iteratsii”, Dokl. AN SSSR, 136:6 (1961), 1302–1305

[12] M. Z. Gecmen, E. Celik, “Numerical solution of Volterra-Fredholm integral equations with Hosoya polynomials”, Math. Meth. Appl. Sci., 44 (2021), 11166–11173

[13] W. Shatanawi, N. Mlaiki, D. Rizk, et al., “Fredholm-type integral equation in controlled metric-like spaces”, Adv. Differ. Equ., 358 (2021)

[14] S. C. Buranay, M. A. Ozarslan, S. S. Falahhesar, “Numerical Solution of the Fredholm and Volterra Integral Equations by Using Modified Bernstein-Kantorovich Operators”, Mathematics, 9 (2021), 1193

[15] V. A. Bykovskii, “Diskretnoe preobrazovanie Fure i tsiklicheskaya svertka na tselochislennykh reshetkakh”, Dokl. AN SSSR, 302:1 (1988), 11–13

[16] Y. Kolomoitsev, J. Prestin, “Approximation properties of periodic multivariate quasi-interpolation operators”, Journal of Approximation Theory, 270 (2021), 105631

[17] S. C. Buranay, M. A. Ozarslan, S. S. Falahhesar, “Numerical Solution of the Fredholm and Volterra Integral Equations by Using Modified Bernstein-Kantorovich Operators”, Mathematics, 9 (2021), 1193

[18] N. N. Dobrovolskii, “Otklonenie dvumernykh setok Smolyaka”, Chebyshevskii sbornik, 8:1 (2007), 110–152

[19] N. M. Korobov, Teoretiko-chislovye metody v priblizhennom analize, vtoroe izdanie, MTsNMO, M., 2004, 288 pp.

[20] N. M. Dobrovolskii, A. R. Esayan, O. V. Andreeva, N. V. Zaitseva, “Mnogomernaya teoretiko-chislovaya Fure interpolyatsiya”, Chebyshevskii sbornik, 5:1 (2004), 122–143

[21] M. A. Isakovich, Obschaya akustika, Nauka, M., 1973, 496 pp.

[22] V. Novatskii, Teoriya uprugosti, Mir, M., 1975, 872 pp.

[23] E. Skuchik, Osnovy akustiki, v. 1, Mir, M., 1976, 520 pp.

[24] N. N. Kalitkin, Chislennye metody, Nauka, M., 1978, 512 pp.

[25] F. Mors, G. Feshbakh, Metody teoreticheskoi fiziki, v. 2, IL, M., 1960, 886 pp.

[26] P. Benerdzhi, R. Batterfild, Metod granichnykh elementov v prikladnykh naukakh, Mir, M., 1984, 494 pp.

[27] V. D. Kupradze, Metody potentsiala v teorii uprugosti, Fizmatgiz, M., 1963, 473 pp.

[28] G. A. Korn, T. M. Korn, Spravochnik po matematike dlya nauchnykh rabotnikov i inzhenerov, Nauka, M., 1978, 832 pp.

[29] N. M. Korobov, “Vychislenie kratnykh integralov metodom optimalnykh koeffitsientov”, Vestn. Mosk. un-ta, 1959, no. 4, 19–25

[30] J. J. Faran, “Sound scattering by solid cylinders and spheres”, J. Acoust. Soc. Amer., 23:4 (1951), 405–418

[31] L. Flax, G. C. Gaunaurd, H. Überall, “Theory of resonance scattering”, Physical Acoustics, 15, eds. Mason W.P., Thurson R.N., Academic, New York, 1981, 191–294

[32] L. Flax, G. C. Gaunaurd, H. Überall, “Theory of resonance scattering”, Physical Acoustics, 15, eds. Mason W.P., Thurson R.N., Academic, New York, 1981, 191–294