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Numerical Construction of the Transform of the Kernel of the Integral Representation of the Poincaré–Steklov Operator for an Elastic Strip

  • NUMERICAL METHODS
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Abstract

For an isotropic stratified elastic strip we consider the Poincaré–Steklov operator that maps normal stresses into normal displacements on part of the boundary. A new approach is proposed for constructing the transform of the kernel of the integral representation of this operator. A variational formulation of the boundary value problem for the transforms of displacements is obtained. A definition is given and the existence and uniqueness are proved for a generalized solution of the problem. An iteration method for solving variational equations is constructed, and conditions for its convergence are obtained based on the contraction mapping principle. The variational equations are approximated by the finite element method. As a result, at each step of the iteration method, it is required to solve two independent systems of linear algebraic equations, which are solved using the tridiagonal matrix algorithm. A heuristic algorithm is proposed for choosing the sequence of parameters of the iteration method that ensures its convergence. Verification of the developed computational algorithm is carried out, and recommendations on the use of adaptive finite element grids are given.

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REFERENCES

  1. Lebedev, V.I. and Agoshkov, V.I., Operatory Puankare–Steklova i ikh prilozheniya v analize (Poincaré–Steklov Operators and Their Applications in Analysis), Moscow: Akad. Nauk SSSR, 1983.

    MATH  Google Scholar 

  2. Lebedev, V.I., Funktsional’nyi analiz i vychislitel’naya matematika (Functional Analysis and Computational Mathematics), Moscow: Fizmatlit, 2000.

    MATH  Google Scholar 

  3. Bobylev, A.A., Application of the conjugate gradient method to solving discrete contact problems for an elastic half-plane, Mech. Solids, 2022, vol. 57, no. 2, pp. 317–332.

    Article  Google Scholar 

  4. Bobylev, A.A., Algorithm for solving discrete contact problems for an elastic strip, Mech. Solids, 2022, vol. 57, no. 7, pp. 1766–1780.

    Article  Google Scholar 

  5. Uflyand, Ya.S., Integral’nye preobrazovaniya v zadachakh teorii uprugosti (Integral Transforms in Problems of Elasticity Theory), Leningrad: Nauka, 1967.

    Google Scholar 

  6. Vatul’yan, A.O. and Plotnikov, D.K., On a study of the contact problem for an inhomogeneous elastic strip, Mech. Solids, 2021, vol. 56, no. 7, pp. 1379–1387.

    Article  MATH  Google Scholar 

  7. Vorovich, I.I., Aleksandrov, V.M., and Babeshko, V.A., Neklassicheskie smeshannye zadachi teorii uprugosti (Nonclassical Mixed Problems of Elasticity Theory), Moscow: Nauka, 1974.

    Google Scholar 

  8. Barber, J.R., Contact Mechanics, Cham: Springer, 2016.

    MATH  Google Scholar 

  9. Nikishin, V.S., Static contact problems for multilayer elastic bodies, in Mekhanika kontaktnykh vzaimodeistvii (Mechanics of Contact Interactions), Moscow, 2001, pp. 212–233.

  10. Aizikovich, S.M., Aleksandrov, V.M., Belokon’, A.V., Krenev, L.I., and Trubchik, I.S., Kontaktnye zadachi teorii uprugosti dlya neodnorodnykh sred (Contact Problems of Elasticity Theory for Inhomogeneous Media), Moscow: Fizmatlit, 2006.

  11. Aizikovich, S.M., Galybin, A.N., and Krenev, L.I., Semi-analytical solution for mode I penny-shaped crack in a soft inhomogeneous layer, Int. J. Solid Struct., 2015, vol. 53, pp. 129–137.

    Article  Google Scholar 

  12. Trubchik, I., Evich, L., and Ladosha, E., Computational model of the deformation of thin gradient coating lying on nondeformable foundation, AIP Conf. Proc., 2018, vol. 1922, article ID 120011.

  13. Babeshko, V.A., Glushkov, E.V., and Glushkova, N.V., Methods of constructing Green’s matrix of a stratified elastic half-space, USSR Comput. Math. Math. Phys., 1987, vol. 27, no. 1, pp. 60–65.

    Article  MATH  Google Scholar 

  14. Bakhvalov, N.S., Zhidkov, N.P., and Kobel’kov, G.M., Chislennye metody (Numerical Methods), Moscow: Binom, 2011.

    MATH  Google Scholar 

  15. Koltunov, M.A., Kravchuk, A.S., and Maiboroda, V.P., Prikladnaya mekhanika deformiruemogo tverdogo tela (Applied Mechanics of Deformable Solids), Moscow: Vyssh. Shkola, 1983.

  16. Trenogin, V.A., Funktsional’nyi analiz (Functional Analysis), Moscow: Fizmatlit, 2002.

    Google Scholar 

  17. Godunov, S.K., Sovremennye aspekty lineinoi algebry (Modern Aspects of Linear Algebra), Novosibirsk: Nauchn. Kniga, 1997.

  18. Samarskii, A.A. and Nikolaev, E.S., Metody resheniya setochnykh uravnenii (Methods for Solving Grid Equations), Moscow: Nauka, 1978.

  19. Voevodin, V.V. and Kuznetsov, Yu.A., Matritsy i vychisleniya (Matrices and Calculations), Moscow: Nauka, 1984.

  20. Parlett, B.N., The Symmetric Eigenvalue Problem, Englewood Cliffs, NJ: Prentice-Hall, 1980. Translated under the title: Simmetrichnaya problema sobstvennykh znachenii. Chislennye metody, Moscow: Mir, 1983.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, 119991, Russia

    A. A. Bobylev

  2. Moscow Center for Fundamental and Applied Mathematics, Moscow, 119991, Russia

    A. A. Bobylev

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  1. A. A. Bobylev
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Correspondence to A. A. Bobylev.

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Translated by V. Potapchouck

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Bobylev, A.A. Numerical Construction of the Transform of the Kernel of the Integral Representation of the Poincaré–Steklov Operator for an Elastic Strip. Diff Equat 59, 119–134 (2023). https://doi.org/10.1134/S0012266123010093

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  • DOI: https://doi.org/10.1134/S0012266123010093

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