Abstract:
The scalar Poisson equation is considered in a domain having a cut with unilateral constraints specified on its edges. An iterative method is proposed for solving the problem. The method is based on domain decomposition and the Uzawa algorithm for finding a saddle point of the Lagrangian. According to the method, the original domain is divided into two subdomains and a linear problem for Poisson’s equation is solved in each of them at every iteration step. The solution in one domain is related to that in the other by two Lagrange multipliers: one is used to match the solutions, and the other, to satisfy the unilateral constraint. Examples of the numerical solution of the problem are given.
Key words:
scalar Poisson equation, theory of cracks, unilateral constraint, domain decomposition method, Lagrange multipliers, Uzawa algorithm.
Citation:
E. M. Rudoy, “Domain decomposition method for a model crack problem with a possible contact of crack edges”, Zh. Vychisl. Mat. Mat. Fiz., 55:2 (2015), 310–321; Comput. Math. Math. Phys., 55:2 (2015), 305–316
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\by E.~M.~Rudoy
\paper Domain decomposition method for a model crack problem with a possible contact of crack edges
\jour Zh. Vychisl. Mat. Mat. Fiz.
\yr 2015
\vol 55
\issue 2
\pages 310--321
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\jour Comput. Math. Math. Phys.
\yr 2015
\vol 55
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\pages 305--316
\crossref{https://doi.org/10.1134/S0965542515020165}
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Linking options:
https://www.mathnet.ru/eng/zvmmf10160
https://www.mathnet.ru/eng/zvmmf/v55/i2/p310
This publication is cited in the following 14 articles:
T. S. Popova, “Numerical Solution of the Equilibrium Problem for a Two-dimensional Elastic Body with a Delaminated Rigid Inclusion”, Lobachevskii J Math, 45:11 (2024), 5402
E. M. Rudoy, H. Itou, N. P. Lazarev, “Asymptotic justification of the models of thin inclusions in an elastic body in the antiplane shear problem”, J. Appl. Industr. Math., 15:1 (2021), 129–140
Y. Kanno, “An accelerated Uzawa method for application to frictionless contact problem”, Optim. Lett., 14:7 (2020), 1845–1854
Nyurgun P. Lazarev, Vladimir V. Everstov, Natalya A. Romanova, “Fictitious domain method for equilibrium problems of the Kirchhoff–Love plates with nonpenetration conditions for known configurations of plate edges”, Zhurn. SFU. Ser. Matem. i fiz., 12:6 (2019), 674–686
R. V. Namm, G. I. Tsoy, G. Woo, “Modified Lagrange functional for solving elastic problem with a crack in continuum mechanics”, Commun. Korean Math. Soc., 34:4 (2019), 1353–1364
R. V. Namm, G. I. Tsoi, “Solution of a contact elasticity problem with a rigid inclusion”, Comput. Math. Math. Phys., 59:4 (2019), 659–666
Robert Namm, Georgiy Tsoy, Ellina Vikhtenko, Communications in Computer and Information Science, 974, Optimization and Applications, 2019, 35
Robert Namm, Georgiy Tsoy, Communications in Computer and Information Science, 1090, Mathematical Optimization Theory and Operations Research, 2019, 536
E. M. Rudoy, N. P. Lazarev, “Domain decomposition technique for a model of an elastic body reinforced by a Timoshenko's beam”, J. Comput. Appl. Math., 334 (2018), 18–26
N. A. Kazarinov, E. M. Rudoy, V. Yu. Slesarenko, V. V. Shcherbakov, “Mathematical and numerical simulation of equilibrium of an elastic body reinforced by a thin elastic inclusion”, Comput. Math. Math. Phys., 58:5 (2018), 761–774
E. M. Rudoy, N. A. Kazarinov, V. Yu. Slesarenko, “Numerical simulation of the equilibrium of an elastic two-layer structure with a crack”, Num. Anal. Appl., 10:1 (2017), 63–73
E. M. Rudoy, V. V. Shcherbakov, “Domain decomposition method for a membrane with a delaminated thin rigid inclusion”, Sib. elektron. matem. izv., 13 (2016), 395–410
E. M. Rudoy, “Numerical solution of an equilibrium problem for an elastic body with a delaminated thin rigid inclusion”, J. Appl. Industr. Math., 10:2 (2016), 264–276
E. M. Rudoy, “Numerical solution of an equilibrium problem for a membrane with a delaminated thin rigid inclusion”, All-Russian Conference on Nonlinear Waves: Theory and New Applications (Wave16), Journal of Physics Conference Series, 722, IOP Publishing Ltd, 2016, UNSP 012033
Citation in format AMSBIB
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