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Home » Issues » Archive of Issues » 2023 - 3 issue » pp. 434-451

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Russia Moscow
Year
2023
Volume
33
Issue
3
Pages
434-451
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Section Mathematics
Title On one semi-analytical approximation of the normal derivative of the simple layer potential near the boundary of a two-dimensional domain
Author(-s) Ivanov D.Yu.a
Affiliations Russian University of Transporta
Abstract On the basis of piecewise quadratic interpolation, semi-analytical approximations of the normal derivative of the simple layer potential near and on the boundary of a two-dimensional domain are obtained. To calculate the integrals formed after the interpolation of the density function, exact integration over the variable $\rho=(r^{2}-d^{2})^{1/2} $ is used, where $d$ and $r$ are the distances from the observed point to the boundary of the domain and to the boundary point of integration, respectively. The study proves the stable convergence of such approximations with cubic velocity uniformly near the boundary of the class $C^{5}$, as well as on the boundary itself. It is also proved that, by analogy with the exact function, the approximations suffer a discontinuity at the boundary, the magnitude of which is proportional to the values of the interpolated density function, but they can be extended on the boundary to functions that are continuous either on a closed internal border domain or on a closed external one. Theoretical conclusions about uniform convergence are confirmed by the results of calculating the normal derivative near the boundary of a unit circle.
Keywords quadrature formula, normal derivative of simple layer potential, boundary element method, near singular integral, boundary layer effect, uniform convergence
UDC 519.644.5
MSC 31-08, 31A10
DOI 10.35634/vm230304
Received 26 February 2023
Language Russian
Citation Ivanov D.Yu. On one semi-analytical approximation of the normal derivative of the simple layer potential near the boundary of a two-dimensional domain, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2023, vol. 33, issue 3, pp. 434-451.
References
  1. Brebbia C.A., Telles J.C.F., Wrobel L.C. Boundary element techniques, Berlin: Springer, 1984. https://doi.org/10.1007/978-3-642-48860-3
    Translated under the title Metody granichnykh elementov, Moscow: Mir, 1987.
  2. Zhang Yaoming, Li Xiaochao, Sladek V., Sladek J., Gao Xiaowei. A new method for numerical evaluation of nearly singular integrals over high-order geometry elements in 3D BEM, Journal of Computational and Applied Mathematics, 2015, vol. 277, pp. 57-72. https://doi.org/10.1016/j.cam.2014.08.027
  3. Krutitskii P.A., Fedotova A.D., Kolybasova V.V. Quadrature formula for the simple layer potential, Differential Equations, 2019, vol. 55, no. 9, pp. 1226-1241. https://doi.org/10.1134/S0012266119090106
  4. Krutitskii P.A., Reznichenko I.O. Quadrature formula for the harmonic double layer potential, Differential Equations, 2021, vol. 57, no. 7, pp. 901-920. https://doi.org/10.1134/S0012266121070077
  5. Lv Jiahe, Miao Yu, Zhu Hongping. The distance sinh transformation for the numerical evaluation of nearly singular integrals over curved surface elements, Computational Mechanics, 2014, vol. 53, issue 2, pp. 359-367. https://doi.org/10.1007/s00466-013-0913-0
  6. Niu Zhongrong, Cheng Changzheng, Zhou Huanlin, Hu Zongjun. Analytic formulations for calculating nearly singular integrals in two-dimensional BEM, Engineering Analysis with Boundary Elements, 2007, vol. 31, issue 12, pp. 949-964. https://doi.org/10.1016/j.enganabound.2007.05.001
  7. Gao X.W., Yang K., Wang J. An adaptive element subdivision technique for evaluation of various 2D singular boundary integrals, Engineering Analysis with Boundary Elements, 2008, vol. 32, issue 8, pp. 692-696. https://doi.org/10.1016/j.enganabound.2007.12.004
  8. Zhang Yao-Ming, Gu Yan, Chen Jeng-Tzong. Stress analysis for multilayered coating systems using semi-analytical BEM with geometric non-linearities, Computational Mechanics, 2011, vol. 47, issue 5, pp. 493-504. https://doi.org/10.1007/s00466-010-0559-0
  9. Gu Yan, Chen Wen, Zhang Bo, Qu Wenzhen. Two general algorithms for nearly singular integrals in two dimensional anisotropic boundary element method, Computational Mechanics, 2014, vol. 53, issue 6, pp. 1223-1234. https://doi.org/10.1007/s00466-013-0965-1
  10. Niu Zhongrong, Hu Zongjun, Cheng Changzheng, Zhou Huanlin. A novel semi-analytical algorithm of nearly singular integrals on higher order elements in two dimensional BEM, Engineering Analysis with Boundary Elements, 2015, vol. 61, pp. 42-51. https://doi.org/10.1016/j.enganabound.2015.06.007
  11. Cheng Changzheng, Pan Dong, Han Zhilin, Wu Meng, Niu Zhongrong. A state space boundary element method with analytical formulas for nearly singular integrals, Acta Mechanica Solida Sinica, 2018, vol. 31, issue 4, pp. 433-444. https://doi.org/10.1007/s10338-018-0040-8
  12. Krutitskii P.A., Kolybasova V.V. Numerical method for the solution of integral equations in a problem with directional derivative for the Laplace equation outside open curves, Differential Equations, 2016, vol. 52, issue 9, pp. 1219-1233. https://doi.org/10.1134/S0012266116090135
  13. Krutitskii P.A., Reznichenko I.O. Quadrature formula for the double layer potential with differentiable density, Differential Equations, 2022, vol. 58, issue 8, pp. 1114-1125. https://doi.org/10.1134/S0012266122080122
  14. Krutitskii P.A., Reznichenko I.O. Quadrature formula for a double layer potential in the case of the Helmholtz equation, Computational Mathematics and Mathematical Physics, 2022, vol. 62, issue 3, pp. 411-426. https://doi.org/10.1134/S0965542522030095
  15. Gao Xiao-Wei, Zhang Jin-Bo, Zheng Bao-Jing, Zhang Chuanzeng. Element-subdivision method for evaluation of singular integrals over narrow strip boundary elements of super thin and slender structures, Engineering Analysis with Boundary Elements, 2016, vol. 66, pp. 145-154. https://doi.org/10.1016/j.enganabound.2016.02.002
  16. Gong Yanpeng, Dong Chunying, Qin Fei, Hattori G., Trevelyan J. Hybrid nearly singular integration for isogeometric boundary element analysis of coatings and other thin 2D structures, Computer Methods in Applied Mechanics and Engineering, 2020, vol. 367, 113099. https://doi.org/10.1016/j.cma.2020.113099
  17. Gu Yan, Zhang Chuanzeng. Fracture analysis of ultra-thin coating/substrate structures with interface cracks, International Journal of Solids and Structures, 2021, vol. 225, 111074. https://doi.org/10.1016/j.ijsolstr.2021.111074
  18. Gong Yanpeng, Dong Chunying, Bai Yang. Evaluation of nearly singular integrals in isogeometric boundary element method, Engineering Analysis with Boundary Elements, 2017, vol. 75, pp. 21-35. https://doi.org/10.1016/j.enganabound.2016.11.005
  19. Ivanov D.Yu. A refinement of the boundary element collocation method near the boundary of domain in the case of two-dimensional problems of non-stationary heat conduction with boundary conditions of the second and third kind, Vestnik Tomskogo Gosudarstvennogo Universiteta. Matematika i Mekhanika, 2019, no. 57, pp. 5-25 (in Russian). https://doi.org/10.17223/19988621/57/1
  20. Ivanov D.Yu. A refinement of the boundary element collocation method near the boundary of a two-dimensional domain using semianalytic approximation of the double layer heat potential, Vestnik Tomskogo Gosudarstvennogo Universiteta. Matematika i Mekhanika, 2020, no. 65, pp. 30-52 (in Russian). https://doi.org/10.17223/19988621/65/3
  21. Ivanov D.Yu. On uniform convergence of approximations of the double layer potential near the boundary of a two-dimensional domain, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2022, vol. 32, issue 1, pp. 26-43 (in Russian). https://doi.org/10.35634/vm220103
  22. Dunford N., Schwartz J.T. Linear operators. Part 1: General theory, Hoboken: John Wiley and Sons, 1988.
    Translated under the title Lineinye operatory. Obshchaya teoriya, Moscow: Inostrannaya Literatura, 1962.
  23. Smirnov V.I. Kurs vysshei matematiki. Tom 4. Chast' 2 (A course of higher mathematics. Vol. 4. Part 2), Moscow: Nauka, 1981.
  24. Ivanov D.Yu. Stable solvability in spaces of differentiable functions of some two-dimensional integral equations of heat conduction with an operator-semigroup kernel, Vestnik Tomskogo Gosudarstvennogo Universiteta. Matematika i Mekhanika, 2015, no. 6 (38), pp. 33-45 (in Russian). https://doi.org/10.17223/19988621/38/4
  25. Berezin I.S., Zhidkov N.P. Metody vychislenii. Tom 1 (Methods of computation. Vol. 1), Moscow: Fizmatgiz, 1962.
  26. Fikhtengol'ts G.M. Kurs differentsial'nogo i integral'nogo ischisleniya. Tom 2 (Differential and integral calculus course. Vol. 2), Moscow: Fizmatlit, 2003.
  27. Zorich V.A. Mathematical analysis. II, Berlin: Springer, 2016. https://doi.org/10.1007/978-3-662-48993-2
    Original Russian text published in Zorich V.A. Matematicheskii analiz, Chast' 1, Moscow: Moscow Center for Continuous Mathematical Education, 2012.
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