Skip to main content
SpringerLink
Log in

Modified lagrange functionals to solve the variational and quasivariational inequalities of mechanics

  • Applications of Mathematical Programming
  • Published:
Automation and Remote Control Aims and scope Submit manuscript

Abstract

Consideration was given to the methods of solving the model problem with friction and the contact problem of the elasticity theory with a given friction which are based on the duality schemes with modified Lagrange functionals.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Fichera, G., Existence Theorems in Elasticity, Berlin: Springer, 1972. Translated under the title Teoremy sushchestvovaniya v teorii uprugosti, Moscow: Mir, 1986.

    Google Scholar 

  2. Vikhtenko, E.M. and Namm, R.V., Iterative Proximal Regularization of the Modified Lagrange Functional to Solve the Quasivariational Signorini Inequality, Zh. Vychisl. Mat. Mat. Fiz., 2008, vol. 48, no. 9, pp. 1571–1579.

    MathSciNet  Google Scholar 

  3. Namm, R.V. and Sachkov, S.A., Solution of the Quasivariational Signorini Inequality by Successive Approximations, Zh. Vychisl. Mat. Mat. Fiz., 2009, vol. 49, no. 5, pp. 805–814.

    MathSciNet  MATH  Google Scholar 

  4. Hlavacek, I., Haslinger, J., Necas, J., and Lovisek, J., Solution of Variational Inequalities in Mechanics, Berlin: Springer, 1988. Translated under the title Reshenie variatsionnykh neravenstv v mekhanike, Moscow: Mir, 1986.

    Book  MATH  Google Scholar 

  5. Namm, R.V., On Uniqueness of the Smooth Solution in the Statistical Problem with Coulomb Friction and Two-side Contact, Prikl. Mat. Mekh., 1995, vol. 59, no. 2, pp. 330–335.

    MathSciNet  Google Scholar 

  6. Bertsekas, D.P., Nonlinear Programming, Belmont: Athena Scientific, 1999.

    MATH  Google Scholar 

  7. Gol’shtein, E.G. and Tret’yakov, N.V., Modifitsirovannye funktsii Lagranzha. Teoriya i metody optimizatsii (Modified Lagrange Functions), Moscow: Nauka, 1989.

    Google Scholar 

  8. Vikhtenko, E.M. and Namm, R.V., Duality Scheme to Solve the Semicoercive Signorini Problem with Friction, Zh. Vychisl. Mat. Mat. Fiz., 2007, vol. 47, no. 12, pp. 2023–2036.

    MathSciNet  Google Scholar 

  9. Kushniruk, N.N. and Namm, R.V., Method of Lagrange Multipliers to Solve the Semicoercive Model Problem with Friction, Sib. Zh. Vychisl. Mat., 2009, vol. 12, no. 4, pp. 409–420.

    MATH  Google Scholar 

  10. Woo, G., Kim, S., Namm, R.V., and Sachkov, S.A., Method of Iterative Proximal Regularization to Seek the Saddle Point in the Semicoercive Signorini Problem, Zh. Vychisl. Mat. Mat. Fiz., 2006, vol. 46, no. 11, pp. 2024–2031.

    MathSciNet  Google Scholar 

  11. Antipin, A.S., Methods of Nonlinear Programming Based on the Direct and Dual Modifications of the Lagrange Function, Preprint of the Systems Study Inst., Moscow, 1979.

  12. Bertsekas, D., Constrained Optimization and Lagrange Multiplier Methods, New York: Academic, 1982. Translated under the title Uslovnaya optimizatsiya i metody mnozhitelei Lagranzha, Moscow: Radio i Svyaz’, 1987.

    MATH  Google Scholar 

  13. Grossman, K. and Kaplan, A.A., Nelineinoe programmirovanie na osnove bezuslovnoi minimizatsii (Nonlinear Programming Based on Unconditional Minimization), Novosibirsk: Nauka, 1981.

    Google Scholar 

  14. Rockafellar, R.T., Convex Analysis, Princeton: Princeton Univ. Press, 1970. Translated under the title Vypuklyi analiz, Moscow: Mir, 1973.

    MATH  Google Scholar 

  15. Kushniruk, N.N., Namm, R.V., and Tkachenko, A.S., On Stable Smoothing Method to Solve the Model Problem of Mechanics with friction, Zh. Vychisl. Mat. Mat. Fiz., 2011, vol. 51, no. 6, pp. 1032–1042.

    MathSciNet  Google Scholar 

  16. Vasin, V.A. and Eremin, I.I., Operatory i iteratsionnye protsessy feierovskogo tipa (teoriya i prilozheniya) (Operators and Iterative Processes of the Fejer Type (Theory and Applications)), Moscow: Inst. Komp. Issled., 2005.

    MATH  Google Scholar 

  17. Polyak, B.T., Vvedenie v optimizatsiyu, Moscow: Nauka, 1983. Translated into English under the title Introduction to Optimization, New York: Optimization Software, 1987.

    MATH  Google Scholar 

  18. Vasil’ev, F.P., Chislennye metody resheniya ekstremal’nykh zadach (Numerical Methods to Solve Extremal Problems), Moscow: Nauka, 1980.

    Google Scholar 

  19. Glowinski, R., Lions, J.-L., and Tremolieres, R., Numerical Analysis of Variational Inequalities, Amsterdam: North-Holland, 1981. Translated under the title Chislennoe issledovanie variatsionnykh neravenstv, Moscow: Mir, 1979.

    MATH  Google Scholar 

  20. Marchuk, G.I. and Agoshkov, Yu.M., Vvedenie v proektsionno-setochnye metody (Introduction to the Projection-Net Methods), Moscow: Nauka, 1981.

    MATH  Google Scholar 

  21. Ciarlet, P., The Finite Element Method for Elliptic Problems, Amsterdam: North-Holland, 1978. Translated under the title Metod konechnykh elementov dlya ellipticheskikh zadach, Moscow: Mir, 1980.

    MATH  Google Scholar 

  22. Khludnev, A.M., Zadachi teorii uprugosti v negladkikh oblastyakh (Problems of the Elasticity Theory in the Nonsmooth Domains), Moscow: Fizmatlit, 2010.

    Google Scholar 

  23. Kikuchi, N. and Oden, T., Contact Problem in Elasticity: A Study of Variational Inequalities and Finite Element Methods, Philadelphia: SIAM, 1988.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Pacific State University, Khabarovsk, Russia

    E. M. Vikhtenko & R. V. Namm

  2. Amur State University, Blagoveshchensk, Russia

    N. N. Maksimova

Authors
  1. E. M. Vikhtenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. N. Maksimova
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. R. V. Namm
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Original Russian Text © E.M. Vikhtenko, N.N. Maksimova, R.V. Namm, 2012, published in Avtomatika i Telemekhanika, 2012, No. 4, pp. 3–17.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Vikhtenko, E.M., Maksimova, N.N. & Namm, R.V. Modified lagrange functionals to solve the variational and quasivariational inequalities of mechanics. Autom Remote Control 73, 605–615 (2012). https://doi.org/10.1134/S0005117912040017

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0005117912040017

Keywords

Search

Navigation