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International Theoretical Science Journal
Volume 59 >>> № 5 SEPTEMBER — OCTOBER 2023
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UDC 519.6
A.N. Khimich1, A.V. Popov2


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

 khimich505@gmail.com

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

 alex50popov@gmail.com

SOLVING Ill-POSED PROBLEMS OF THE THEORY OF ELASTICITY
USING HIGH-PERFORMANCE COMPUTER SYSTEMS

Abstract. A method of efficient analysis and solution of conditionally correct problems, which have a unique solution in the subspace, is proposed. The use of a discrete finite-element model on the entire space to obtain a unique solution on the subspace of the initial variational problem is justified. To find a normal pseudo-solution (or a weighted normal pseudo-solution) of a discrete problem (a system of linear algebraic equations with a sparse symmetric semidefinite matrix), a three-stage regularization method is proposed. This method makes it possible to obtain approximations of these solutions with a predetermined accuracy. Efficient adaptive high-performance algorithms of the specified method have been developed for solving systems of linear algebraic equations with sparse symmetric semi-definite matrices using modern computers with parallel computing.

Keywords: high-performance computing, finite element method, three-stage regularization method, sparse symmetric semidefinite matrix, system of linear algebraic equations, first fundamental problem of elasticity theory, variable computer environment.


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REFERENCES

  1. Strang G., Fix G.J. An analysis of the finite element method. New York: Prentice-Hall, 1973. 306 p.

  2. Zienkiewicz O.C., Morgan K. Finite Elements and Approximation. New York: A Wiley-Interscience Publication, 1983. 320 p.

  3. Gorodetsky A.S., Evzerov I.D. Computer models of structures [in Russian]. Kyiv: FACT, 2007. 394 с.

  4. Molchanov I.N., Galba E.F. Variational Statements of the Static Problem of the Theory of Elasticity for Given External Forces. Ukr. math. journal. 1991. Vol. 43, N 2. P. 158–161.

  5. Molchanov I.N., Levchenko I.S., Fedonyuk N.N., Khimich A.N., Chistyakova T.V. Numerical simulation of stress concentration in an elastic half-space with a two-layer inclusion. Prikladnaya mekhanika. 2002. Vol. 38, N 3. P. 65–71.

  6. Khimich A.N., Yakovlev M.F. On the solution of systems with matrices of incomplete rank. Computer mathematics. Coll. scientific works. 2003. Iss. 1. P. 1–15.

  7. Popov A.V., Khimich A.N. Research and solution of the first main problem of the theory of elasticity. Computer mathematics. Coll. scientific works.. 2003.Iss. 2. P. 105–114.

  8. Popov O.V. On an effective method for solving incorrect problems with sparse matrices. Theory of optimal solutions. Coll. of science works. 2013. N 12. P. 77–81.

  9. 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. 248 p.

  10. Khimich A.N., Popov A.V., Polyanko V.V. Algorithms of parallel computations for linear algebra problems with irregularly structured matrices. Cybernetics and Systems Analysis. 2011. Vol. 47, Iss. 6. P. 973–985. https://doi.org/10.1007/s10559-011-9377-4.

  11. Popov O.V. On parallel algorithms for factorization of sparse matrices. Kompʹyuternaya matematyka. Sb. nauch. trudov. 2013. Iss. 2. P. 115–124. URI: http://dspace.nbuv. gov.ua/handle/123456789/84755" .

  12. Timoshenko S.P., Goodier J.N. Theory of Elasticity. New York: McGraw-Hill, 1970.

  13. Mikhlin S.G. Variational methods in mathematical physics [in Russian]. Moscow: Nauka, 1970. 454 p.

  14. Galba E.F., Gladkiy A.V., Khimich A.N., Yakovlev M.F. On the well-posedness of the first main problem of elasticity theory on a subspace. Computer mathematics. Coll. scientific works. 2002. Iss. 1. P. 54–62.

  15. Siarlet P.G. The Finite Element Method for Ellipticp Problems. North Holland, 1978. 529 p.

  16. Khimich A.N. Perturbation bounds for the least squares problem. Cybernetics and Systems Analysis. 1996. Vol. 32, Iss. 3. P. 434–436. https://doi.org/10.1007/BF02366509.

  17. Khimich A.N. Estimates of the total error in solving systems of linear algebraic equations for matrices of arbitrary rank. Computer mathematics. Coll. scientific works. 2002. Iss. 2. P. 41–49.

  18. Morozov V.A. Regularization Methods for Unstable Problems [in Russian]. Moscow: Moscow University Publishing House, 1987. 217 p.

  19. Voevodin V.V., Kuznetsov Yu.A. Matrices and calculations [in Russian]. Moscow: Nauka, 1984. 318 с.

  20. Popov A.V., Chistyakov O.V. On the effectiveness of algorithms with multilevel parallelism. Fizyko-matematychne modelyuvannya ta informatsiyni tekhnolohii. 2021. Iss. 33. С. 133–137. https://doi.org/10.15407/fmmit2021.33.133.

  21. Khimich O.M., Sydoruk V.A. The use of mixed bit rate in mathematical modeling. Mathematical and computer modeling. Series: Physical and mathematical sciences. 2019. Iss. 19. P. 180–187.

  22. Khimich О.М., Chistyakova T.V, Sidoruk V.A., Ershov P.S. Adaptive computer technologies for solving problems of computational and applied mathematics. Cybernetics and Systems Analysis. 2021. Vol. 57, N 6. P. 990–997. https://doi.org/10.1007/s10559-021-00424-z .

  23. Khimich O.M., Sydoruk V.A. A tiling hybrid algorithm for the factorization of structurally symmetric matrices. Theory of optimal solutions. Coll. of science works. 2017. N 2017. P. 125–132.

  24. George A., J. Liu. Numerical solution of large sparse systems of equations [Russian translation]. Moscow: Mir, 1984. 334 p.




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