Far Eastern Mathematical Journal

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Duality method for solving model crack problem


Chervyakova M.V., Namm R.V., Vikhtenko E.M.

2016, issue 2, Đ. 137-146


Abstract
We consider the duality method based on the use of modified Lagrangian functional for solving a model of elastic problem with a crack. An article presents the theorems, allowing to use Uzawa method for search a saddle point of the modified Lagrangian functional. The results of numerical experiments are given.

Keywords:
model problem with a crack, variational inequality, modified Lagrangian functional, Uzawa method, finite element method

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References

[1] V.L. Berdichevskii, Variatsionnye printsipy mekhaniki sploshnoi sredy, Nauka, M, 1983.
[2] A.S. Kravchuk, Variatsionnye i kvazivariatsionnye neravenstva v mekhanike, MGA-PI, M, 1997.
[3] G. Diuvo, Zh.L. Lions, Neravenstva v fizike i mekhanike, Nauka, M, 1980.
[4] I. Glavachek, Ia. Gaslinger, I. Nechas, Ia. Lovishek, Reshenie variatsionnykh neravenstv v mekhanike, Mir, M, 1986.
[5] N. Kikuchi, J. T. Oden, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Method, SIAM Philadelphia, 1988.
[6] L.A. Galin, Kontaktnye zadachi teorii uprugosti i viazkouprugosti, Nauka, M, 1980.
[7] N.F. Morozov, Matematicheskie voprosy teorii treshchin, Nauka, M, 1984.
[8] A.M. Khludnev, V.A. Kovtunenko, Analysis of cracks in solids, WIT Press, Southampton; Boston, 2000.
[9] A.M. Khludnev, Zadachi teorii uprugosti v negladkikh oblastiakh, Fizmatlit, M, 2010.
[10] E.G. Gol'shtein, N.V. Tret'iakov, Modifitsirovannye funktsii Lagranzha.Teoriia i metody optimizatsii, Nauka, M, 1989.
[11] K. Grossman, A.A. Kaplan, Nelineinoe programmirovanie na osnove bezuslovnoi optimizatsii, Nauka. Sib. otd., Novosibirsk, 1981.
[12] D. Bertsekas, Uslovnaia optimizatsiia i metody mnozhitelei Lagranzha, Radio i sviaz', M, 1987.
[13] E.M. Vikhtenko, R.V. Namm, ôO metode dvoistvennosti dlia resheniia model'noi zadachi s treshchinoiö, Trudy instituta matematiki i mekhaniki UrO RAN, 22:1 (2016), 36ľ43.
[14] E.M. Vikhtenko, G. Woo, R.V. Namm, ôSensitivity functionals in contact problems of elasticity theoryö, Comp. Math. and Math. Phys., 54:7 (2014), 1190ľ1200.

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