Cybernetics And Systems Analysis logo
Editorial Board Announcements Abstracts Authors Archive
Cybernetics And Systems Analysis
International Theoretical Science Journal
Volume 56 >>> № 6 NOVEMBER — DECEMBER 2020
UDC 517.95:519.63
N.A. Vareniuk1, E.F. Galba2, I.V. Sergienko3


1 V.M. Glushkov Institute of Cybernetics, National Academy
of Sciences of Ukraine, Kyiv, Ukraine

 nvareniuk@ukr.net

2 V.M. Glushkov Institute of Cybernetics, National Academy
of Sciences of Ukraine, Kyiv, Ukraine

 e.f.galba@ukr.net

3 V.M. Glushkov Institute of Cybernetics, National Academy
of Sciences of Ukraine, Kyiv, Ukraine

 incyb@incyb.kiev.ua

VARIATIONAL STATEMENTS AND DISCRETIZATION OF THE BOUNDARY-VALUE
PROBLEM OF THE ELASTICITY THEORY WHEN TENSION ON THE BOUNDARY
OF THE DOMAIN IS KNOWN

Abstract. The equations of elastic equilibrium of bodies in displacements with the stresses set on the surface of the body are considered. Under the conditions that ensure the solution of this boundary-value problem, its solution will be unique in the whole space of vector functions where it exists. Two variational problems for the considered static problem of the theory of elasticity with a unique solution in the whole space are proposed and investigated. The mathematical apparatus of the study is one of the variants of the Korn inequality that is proved in the article. Discretization of these variational problems by the finite-element method and convergence of discrete solutions is considered.

Keywords: elasticity theory problem, variational statements, existence of a unique solution in function spaces, discrete problems, methods for solving discrete problems.



FULL TEXT

REFERENCES

  1. Galba E.F., Sergienko I.V. Methods for calculating weighted pseudoinverse matrices and weighted normal pseudosolutions with degenerate weights. Kibernetika i sistemnyj analiz. 2018. Vol. 54, N 3. P. 65–93.

  2. Sergienko I.V., Galba E.F., Deineka V.S. Representations and decompositions of weighted pseudoinverse matrices, iterative methods and regularization of problems. 1. Positive-definite weights. Kibernetika i sistemnyj analiz. 2008. Vol. 44, N 1. P. 47–73.

  3. Khimich A.N. Perturbation estimates for solving least squares problems. Kibernetika i sistemnyj analiz. 1996. Vol. 32, N 3. P. 142–145.

  4. Molchanov I.N., Yakovlev M.F. Iterative processes for solving one class of incompatible systems of linear algebraic equations. Zh. Vychisl. Mat. Mat. Fiziki. 1975. Vol. 15, N 3. P. 547–558.

  5. Neumann M., Plemmons R.J. Convergent nonnegative matrices and iterative methods for consistent linear systems. Numer. Math. 1978. Vol.31, N 3. P. 265–279.

  6. Marchuk G.I., Kuznetsov Yu.A. Iterative methods and quadratic functionals [in Russian]. Novosibirsk: Nauka, 1972. 205 p.

  7. Kuznetsov Yu.A. Computational methods in subspaces [in Russian]. In col.: Computational processes and systems. Issue 2. G.I. Marchuk (Ed.). Moscow: Nauka, 1985. P. 265–350.

  8. Molchanov I.N., Yakovlev M.F. On a class of iterative processes for solving inconsistent systems of linear algebraic equations. Dokl. AN SSSR. 1973. Vol. 209, N 4. P. 782–784.

  9. Molchanov I.N., Yakovlev M.F. On two-step iterative methods for solving one class of incompatible systems of linear algebraic equations. Dokl. AN SSSR. 1974. Vol. 214, N 6. P. 1265–1268.

  10. Keller H.B. On the solution of singular and semidefinite linear systems by iterations. SIAM J. Numer. Anal. 1965. N 2. P. 281290.

  11. Meyer C.D., Plemmons R.J. Convergent powers of a matrix with applications to iterative methods for singular linear systems. SIAM J. Numer. Anal. 1977. Vol. 14, N 4.P. 699–705.

  12. Douglas J., Pearcy C. On convergence of alternating direction procedures in the presence of singular operators. Numer. Math. 1963. N 5. P. 175–184.

  13. O’Carrol M.J. Inconsistencies and S.O.R. convergence for the discrete Neumann problem for a rectangle. J. Inst. Math. Appl. 1973. N 11. P. 343–350.

  14. Barret J.W., Elliott C.M. A practical finite element approximation of a semi-definite Neumann problem on a curved domain. Numer. Math. 1987. Vol. 51, N 1. P. 23–36.

  15. Shieh A.S.L. On convergence of the conjugate gradient methods for singular capacitance matrix equations from the Neumann problem of the Poisson equation. Numer. Math. 1977/78. Vol. 29, N 3. P. 307–327.

  16. Bakhvalov N.S., Bogachev K.Yu., Master J.F. An efficient algorithm for solving stiff elliptic problems with applications to the fictitious domain method. Zh. Vychisl. Mat. Mat. Fiziki. 1999. Vol. 39, N 6. P. 919–931.

  17. Molchanov I.N. Numerical methods for solving some problems in the theory of elasticity [in Russian]. Kiev: Nauk. dumka, 1979. 316 p.

  18. Dyakonov E.G. On the solution of systems of equations of the projection-difference method for non-negative operators. In collection: Computational methods of linear algebra. Proceedings of the All-Union Conference. G.I. Marchuk (ed.). Novosibirsk: VTS SO AN SSSR. 1977. P. 51–60.

  19. Samarskiy A.A., Nikolaev E.S. Methods for solving grid equations [in Russian]. Moscow: Nauka, 1978. 588 p.

  20. Kuznetsov Yu.A. Matrix iterative methods in subspaces: Dis. ... Dr. Phys.-Math. Sciences: Computing Center of the USSR Academy of Sciences. Moscow, 1982. 197 p.

  21. Giese J.H. On the truncation error in a numerical solution of the Neumann problem for a rectangle. J. Math. Phys. 1958. Vol. 37, N 2. P. 169177.

  22. Volkov E.A. On the grid method for a boundary value problem with oblique and normal derivative. Zh. Vychisl. Mat. Mat. Fiziki. 1961. Vol. 1, N 4. P. 607–621.

  23. Molchanov I.N., Galba E.F. On the convergence of difference schemes for the second boundary value problem of elasticity theory. Dokl. AN UkrSSR. 1982. Ser. А, N 11. P. 11–14.

  24. Molchanov I.N., Galba E.F. Variational formulations of the second boundary value problem of elasticity theory [in Russian]. Dokl. AN UkrSSR. 1986. Ser. А, N 8. P. 17–20.

  25. Korneev V.G. Finite element method schemes of high orders of accuracy [in Russian]. Leningrad: Izd-vo Leningrad. un-ta, 1977. 207 p.

  26. Oganesyan L.A., Rukhovets L.A. Variational-difference methods for solving elliptic equations. Yerevan: Izd-vo AN Arm. SSR, 1979. 335 p.

  27. Aliev B. Difference scheme for solving the second static boundary value problem of the theory of elasticity. Zh. Vychisl. Mat. Mat. Fiziki. 1970. Vol. 10, N 6. P. 1481–1490.

  28. Morozov V.A. Regularization methods for unstable problems [in Russian]. Moscow: Izd-vo Moskovskogo un-ta, 1987. 217 p.

  29. Khimich A.N., Popov A.V., Polyanko V.V. Parallel computation algorithms for linear algebra problems with irregular matrices. Kibernetika i sistemnyj analiz. 2011. Vol. 47, N 6. P. 159–174.

  30. Mikhlin S.G. The problem of the minimum of a quadratic functional [in Russian]. Moscow; Leningrad: Gos. izd-vo tekhn.-teoret. lit., 1952. 216 p.

  31. Friedrich K.O. On the boundary-value problems of the theory of elastisity and Korn’s inequality. Ann. Math. 1947. Vol. 48, N 2. P. 441–471.

  32. Duveau G., Lyons J.-L. Inequalities in mechanics and physics [Russian translation]. Moscow: Nauka, 1980. 384 p.

  33. Fichera G. Existence theorems in elasticity theory [Russian translation]. Moscow: Mir, 1974. 160 p.

  34. Sjarle F. The finite element method for elliptic problems [Russian translation]. Moscow: Mir, 1980. 512 p.

  35. Strang G., Fix J. Theory of the finite element method [Russian translation]. Moscow: Mir, 1977. 352 p.

  36. Molchanov I.N., Galba E.F. On finite element methods for the Neumann problem. Numer. Math. 1985. Vol. 46, N 4. Р. 587–598.

  37. George A., Liu J. Numerical solution of large sparse systems of equations [Russian translation]. Moscow: Mir, 1984. 333 p.
Editorial Board Announcements Abstracts Authors Archive
about scope office editorial board instructions for authors subscriptions contacts
© 2020 Kibernetika.org. All rights reserved.