UDC 519.6, 539.3
1 O.S. Popov National Academy of Telecommunications, Odessa, Ukraine
 pr-bob@ukr.net
|
2 O.S. Popov National Academy of Telecommunications, Odessa, Ukraine
kovalev@ukr.net
|
3 V.M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine
igor.sayko1988@gmail.com
|
|
NUMERICAL ANALYSIS OF SYSTEMS OF SINGULAR INTEGRAL EQUATIONS
OF THE FIRST KIND WITH AN INDEFINABLE INDEX IN THE PROBLEM
OF THE DIFFRACTION OF PLANE WAVES ON A RIGID INCLUSION
Abstract. By reducing the systems of singular integral equations (SIE) to two types, we have carried out a numerical analysis of the problem of mathematical physics about the interaction of stationary plane strain waves with a rigid inclusion (cavity with a clamped contour) located in an infinite isotropic elastic medium. The problem is solved using the systems of SIEs of the 1st and 2nd kinds, where the latter has an indefinable index. The conditionality of the models is analyzed using cluster high-precision computational schemes.
Keywords: singular integral equations, equation index, condition number, numerical experiment, plane wave diffraction, rigid stiff inclusion (clamped cavity).
FULL TEXT
REFERENCES
- Selezov I.T., Kryvonos Yu.G. Mathematical methods in the problems of wave propagation and diffraction [in Russian]. Kyiv: Nauk. dumka, 2012. 204 p.
- Guz A.N., Kubenko V.D., Cherevko M.A. Elastic wave diffraction [in Russian]. Kyiv: Nauk. dumka, 1978. 307 p.
- Shibahara M., Tateno S., Kuroyanagi O. Diffraction of steady stress waves by arbitrary shaped discontinuities in elastic medium. Bulletin of the JSME. 1980. Vol. 23, Iss. 178. P. 493–500.
- Colton D., Kress R. Integral equation methods in scattering theory. Philadelphia: SIAM, 2013, 271 p.
- Jain D.L., Kanwal R.P. Scattering of elastic waves by circular cylindrical flaws and inclusion. J. Appl. Phys. 1979. Vol. 50, N 6. P. 4067–4109. https://doi.org/10.1063/1.326489.
- Bostrm A. Scattering by a smooth elastic obstacle. J. Acoust. Soc. Amer. 1980. Vol. 67, Iss. 6. P. 1904–1913.
- Mow C.C., Pao Y.-H. The diffraction of elastic waves and dynamic stress concentration. Santa Monica: Rand Corporation, 1971. 681 p.
- Pao Y.H. Elastic waves in solids. J. Appl. Mech. 1983. Vol. 50, Iss. 4b, P. 1152–1164.
- Aleksander M.B., Balaban S.M., Karpinski M., Rajba S.A., Chyzh V.M. Information security environment in wireless sensor networks. Ternopil: Ternopil Ivan Pul’uj National Technical University, 2016. 224 p.
- Lifanov I.K. The method of singular integral equations and a numerical experiment [in Russian]. Moscow: Janus LLP, 1995. 520 p.
- Panasyuk V.V., Savruk M.P., Nazarchuk Z.T. The method of singular integral equations in two-dimensional diffraction problems. Kyiv: Nauk. dumka, 1984. 344 p.
- Muskhelishvili N.I. Singular integral equations [in Russian]. 3rd ed. Moscow: Nauka, 1968. 513 p.
- Panchenko B.E., Saiko I.N. High-precision maximum stresses in the problem of the interaction of elastic waves with a system of cylindrical cavities under conditions of plane deformation. Kibernetika i sistemnyj analiz. 2015. Vol. 51, N 5. P. 139–148.
- Panchenko B.E. The solution of two-dimensional problems of diffraction of elastic waves by cylindrical inhomogeneities: Diss. ... cand. Phys.-Math. sciences. Sumy state un-ty Sumy, 1996. 125 p.
- Nazarenko A.M., Panchenko B.E., Lozhkin A.M. The method of singular integral equations in problems of diffraction of elastic waves by cylindrical inclusions. Visnik Sumy state un-ty. Ser. Physics, mathematics, mechanics. 2004. N 8. P. 144–150.
- Filshtinsky L.A. Periodic solutions of the theory of elasticity for a cylinder in R3. Theor. and appl. Mechanics. Kharkov: Osnova, 1990. Iss. 21. P. 13–20.
- Abapolova E.A., Soldatov A.P. On the theory of singular integral equations on a smooth contour. Scientific Bull. Ser. Mathematics. Physics. 2010. N 5 (76), Iss. 18. P. 6–20.
- Kiyasov S.N. Study of solvability and estimation of the number of solutions for one class of singular integral equations. Siberian Mathematical Journal. 2000. Vol. 41, N 6. P. 1357–1362.
- Sheshko M.A., Shulyaev D.S., Rasolko G.A., Mastyanitsa V.S. On the conditionality of a matrix of a linear algebraic system arising from the approximation of a singular integral equation with a Cauchy kernel. Differents. equations. 1999. Vol. 35, N 9. P. 1278–1285.
- Khimich A.N., Molchanov I.N., Popov A.V. Numerical software of the intelligent MIMD computer Inpark [in Russian]. Kyiv: Nauk. dumka, 2007. 220 p.
- Khimich A.N., Molchanov I.N., Popov A.V., Chistyakova T.V., Yakovlev M.F. Parallel algorithms for solving problems of computational mathematics [in Russian]. Kyiv: Nauk. dumka, 2008. 247 p.
- Khimich A.N., Popov A.V., Polyanko V.V. Algorithms of parallel computations for linear algebra problems with irregularly structured matrices. Cybernetic and Systems Analysis. 2011. Vol 47, N 6. P. 973–985.
- Pao Y.H., Mow С.C. Dynamic stress concentration in an elastic plate with rigid circular inclusion. Proc. Fourth U.S. National Congress of Applied Mechanics, ASME. Vol. 1. Oxford; London; New York; Paris: Pergamon Press., 1962. P. 335–345.
