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Khimich, O. M.; Popov, A. V.

Solving ill-posed problems of the theory of elasticity using high-performance computing systems. (English. Ukrainian original) Zbl 07765735

Cybern. Syst. Anal. 59, No. 5, 743-752 (2023); translation from Kibern. Sist. Anal. 59, No. 5, 66-77 (2023).
Summary: A method for the efficient analysis and solution of conditionally well-posed problems with a unique solution in the subspace is proposed. The use of the discrete finite-element model in the entire space to obtain a unique solution on the subspace of the original variational problem is substantiated. To find a normal pseudo-solution to the discrete problem (a system of linear algebraic equations with a sparse symmetric semi-definite 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 method have been developed to solve systems of linear algebraic equations with sparse symmetric semi-definite matrices using modern computers with parallel computing.

MSC:

65Nxx Numerical methods for partial differential equations, boundary value problems
74Axx Generalities, axiomatics, foundations of continuum mechanics of solids
65Fxx Numerical linear algebra

Keywords:

high-performance computing; variable computing environment; finite element method; three-stage regularization method; first fundamental problem of theory of elasticity
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References:

[1] Strang, G.; Fix, GJ, An Analysis of the Finite Element Method (1973), New York: Prentice-Hall, New York · Zbl 0356.65096
[2] Zienkiewicz, OC; Morgan, K., Finite Elements and Approximation (1983), New York: Wiley-Interscience Publ, New York · Zbl 0582.65068
[3] A. S. Horodetsky and I. D. Evzerov, Computer Models of Structures [in Russian], FACT, Kyiv (2007).
[4] Molchanov, IN; Galba, EF, Variational formulations of the static problem of elasticity theory under specified external forces, Ukr. Math. J., 43, 2, 134-137 (1991) · Zbl 0737.73097 · doi:10.1007/BF01060496
[5] Molchanov, IN; Levchenko, IS; Fedonyuk, NN; Khimich, AN; Chistyakova, TV, Numerical simulation of the stress concentration in an elastic half-space with a two-layer inclusion, Intern. Applied Mech., 38, 3, 308-314 (2002) · Zbl 1094.74532 · doi:10.1023/A:1016078010683
[6] A. N. Khimich and M. F. Yakovlev, “Solving systems with matrices of incomplete rank,” Komp. Matem., A Collection of Sci. Works, Iss. 1, 1-15 (2003).
[7] A. V. Popov and A. N. Khimich, “Investigation and solution of the first main problem of the theory of elasticity,” Komp. Matem., A Collection of Sci. Works, Iss. 2, 105-114 (2003).
[8] O. V. Popov, “An efficient method for solving ill-posed problems with sparse matrices,” Teoriya Optym. Rishen’, A Collection of Sci. Works, No. 12, 77-81 (2013).
[9] Khimich, AN; Molchanov, IN; Popov, AV; Chistyakova, TV; Yakovlev, MF, Parallel Algorithms for Solving Problems of Computational Mathematics (2008), Kyiv: Naukova Dumka, Kyiv
[10] Khimich, AN; Popov, AV; Polyanko, VV, Algorithms of parallel computations for linear algebra problems with irregularly structured matrices, Cybern. Syst. Analysis, 47, 6, 973-985 (2011) · Zbl 1298.65201 · doi:10.1007/s10559-011-9377-4
[11] O. V. Popov, “On parallel algorithms for factorization of sparse matrices,” Komp. Matem., A Collection of Sci. Works, Iss. 2, 115-124 (2013). URL: http://dspace.nbuv.gov.ua/handle/123456789/84755.
[12] Timoshenko, SP; Goodier, JN, Theory of Elasticity (1970), New York: McGraw-Hill, New York · Zbl 0266.73008
[13] Mykhlin, SG, Variational Methods in Mathematical Physics (1970), Moscow: Nauka, Moscow · Zbl 0214.07002
[14] E. F. Galba, A. V. Gladkyi, A. N. Khimich, and M. F. Yakovlev, “On the well-posedness of the first basic problem of the theory of elasticity in a subspace,” Komp. Matem., A Collection of Sci. Works, Iss. 1, 54-62 (2002).
[15] P. G. Siarlet, The Finite Element Method for Elliptic Problems, North Holland (1978). · Zbl 0383.65058
[16] Khimich, AN, Perturbation bounds for the least squares problem, Cybern. Syst. Analysis, 32, 3, 434-436 (1996) · Zbl 0877.15005 · doi:10.1007/BF02366509
[17] A. N. Khimich, “Estimates of the total error of the solution to systems of linear algebraic equations for matrices of arbitrary rank,” Komp. Matem., A Collection of Sci. Works, Iss. 2, 41-49 (2002).
[18] Morozov, VA, Regularization Methods for Unstable Problems (1987), Moscow: Izd. MGU, Moscow
[19] Voevodin, VV; Kuznetsov, YuA, Matrices and Calculations (1984), Moscow: Nauka, Moscow
[20] A. V. Popov and O. V. Chistyakov, “On the efficiency of algorithms with multilevel parallelism,” Phys.-Math. Modeling and Information Technologies, Iss. 33, 133-137 (2021). doi:10.15407/fmmit2021.33.133.
[21] O. M. Khimich and V. A. Sydoruk, “Using mixed bitness in mathematical modeling,” Mathem. and Comp. Modeling, Ser. Phys. Math. Sci., Iss. 19, 180-187 (2019).
[22] Khimich, ÎÌ; Chistyakova, TV; Sidoruk, VA; Yershov, PS, Adaptive computer technologies for solving problems of computational and applied mathematics, Cybern. Syst. Analysis, 57, 6, 990-997 (2021) · Zbl 1505.65327 · doi:10.1007/s10559-021-00424-z
[23] O. M. Khimich and V. A. Sydoruk, “A hybrid tiling algorithm for the factorization of structurally symmetric matrices,” Teoriya Optym. Rishen’, A Collection of Sci. Works, No. 2017, 125-132 (2017).
[24] George, A.; Liu, J.; Ng, E., Computer Solution of Sparse Linear Systems (1994), Orlando: Acad. Press, Orlando
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