Skip to main content
SpringerLink
Log in

Canonical Dual Approach for Contact Mechanics Problems with Friction

  • Chapter
  • First Online:
Canonical Duality Theory

Part of the book series: Advances in Mechanics and Mathematics ((AMMA,volume 37))

  • 793 Accesses

Abstract

This paper presents an application of Canonical duality theory to the solution of contact problems with Coulomb friction. The contact problem is formulated as a quasi-variational inequality which solution is found by solving its Karush–Kuhn–Tucker system of equations. The complementarity conditions are reformulated by using the Fischer–Burmeister complementarity function, obtaining a non-convex global optimization problem. Then canonical duality theory is applied to reformulate the non-convex global optimization problem and define its optimality conditions, finding a solution of the original quasi-variational inequality. We also propose a methodology for finding the solutions of the new formulation, and report the results on well-known instances from literature.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Panagiotopoulos, R.D., Fahed, A.M.: Robot hand grasping and related problems: optimal control and identification. Int. J. Robot. Res. 13, 127–136 (1994)

    Article  Google Scholar 

  2. Conry, T.F., Seireg, A.: A mathematical programming method for design for elastic bodies in contact. J. Appl. Mech. 38, 1293–1307 (1971)

    Google Scholar 

  3. Friedman, V.M., Chernina, V.S.: Iterative methods applied to the solution of contact problems between bodies. Mekh. Tverd. Tela 1, 116–120 (1967). (in Russian)

    Google Scholar 

  4. Duvaut, G., Lions, J.L.: Inequalities in Mechanics and Physics. Springer, Berlin (1976)

    Book  MATH  Google Scholar 

  5. Ferris, M.C., Pang, J.S.: Engineering and economic applications of complementarity problems. SIAM Rev. 39, 669–713 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Outrata, J., Kocvara, M., Zowe, J.: Nonsmooth approach to optimization problems with equilibrium constraints. Kluwer Academic Publishers, Dordrecht and Boston (1998)

    Book  MATH  Google Scholar 

  7. Baiocchi, C., Capelo, A.: Variational and Quasivariational Inequalities: Applications to Free Boundary Problems. Wiley, New York (1984)

    MATH  Google Scholar 

  8. Mosco, U.: Implicit variational problems and quasi variational inequalities. In: Gossez, J., Lami Dozo, E., Mawhin, J., Waelbroeck, L. (eds.) Nonlinear Operators and the Calculus of Variations. Lecture Notes in Mathematics, vol. 543, pp. 83–156. Springer, Berlin (1976)

    Google Scholar 

  9. Outrata, J., Kocvara, M.: On a class of quasi-variational inequalities. Optim. Methods Softw. 5, 275–295 (1995)

    Article  MATH  Google Scholar 

  10. Pang, J.S., Fukushima, M.: Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games. Comp. Manag. Sci. 2, 21–56 (2005). (Erratum: ibid 6, 373–375 (2009))

    Google Scholar 

  11. Chan, D., Pang, J.S.: The generalized quasi-variational inequality problem. Math. Oper. Res. 7, 211–222 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fukushima, M.: A class of gap functions for quasi-variational inequality problems. J. Ind. Manag. Optim. 3, 165–171 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Harms, N., Kanzow, C., Stein, O.: Smoothness Properties of a Regularized Gap Function for Quasi-Variational Inequalities. Preprint 313, Institute of Mathematics, University of Würzburg, Würzburg, March 2013

    Google Scholar 

  14. Nesterov, Y., Scrimali, L.: Solving strongly monotone variational and quasi-variational inequalities. CORE Discussion Paper 2006/107, Catholic University of Louvain, Center for Operations Research and Econometrics (2006)

    Google Scholar 

  15. Noor, M.A.: On general quasi-variational inequalities. J. King Saud Univ. 24, 81–88 (2012)

    Article  Google Scholar 

  16. Ryazantseva, I.P.: First-order methods for certain quasi-variational inequalities in Hilbert space. Comput. Math. Math. Phys. 47, 183–190 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Facchinei, F., Kanzow, C., Sagratella, S.: Solving quasi-variational inequalities via their KKT conditions. Math. Prog. Ser. A (2013). doi:10.1007/s10107-013-0637-0

    MATH  Google Scholar 

  18. Latorre, V., Sagratella, S.: A canonical duality approach for the solution of affine quasi-variational inequalities. J. Glob. Optim. (2014). doi:10.1007/s10898-014-0236-5

    MATH  Google Scholar 

  19. Facchinei, F., Kanzow, C., Sagratella, S., et al.: On the solution of the KKT conditions of generalized Nash equilibrium problems. SIAM J. Optim. 21, 1082–1108 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. I and II. Springer, New York (2003)

    MATH  Google Scholar 

  21. Gao, D.Y.: Duality Principles in Nonconvex Systems: Theory. Methods and Applications. Springer, New York (2000)

    Book  MATH  Google Scholar 

  22. Gao, D.Y.: Canonical dual transformation method and generalized triality theory in nonsmooth global optimization. J. Glob. Optim. 17(1/4), 127–160 (2000)

    Article  MATH  Google Scholar 

  23. Gao, D.Y.: Canonical duality theory: theory, method, and applications in global optimization. Comput. Chem. 33, 1964–1972 (2009)

    Article  Google Scholar 

  24. Wang, Z.B., Fang, S.C., Gao, D.Y., et al.: Canonical dual approach to solving the maximum cut problem. J. Glob. Optim. 54, 341–352 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Zhang, J., Gao, D.Y., Yearwood, J.: A novel canonical dual computational approach for prion AGAAAAGA amyloid fibril molecular modeling. J. Theor. Biol. 284, 149–157 (2011)

    Article  MathSciNet  Google Scholar 

  26. Gao, D.Y., Ruan, N., Pardalos, P.M.: Canonical dual solutions to sum of fourth-order polynomials minimization problems with applications to sensor network localization. In: Pardalos, P.M., Ye, Y.Y., Boginski, V., Commander, C. (eds.) Sensors: Theory, Algorithms and Applications. Springer (2010)

    Google Scholar 

  27. Ruan, N., Gao, D.Y.: Global optimal solutions to a general sensor network localization problem. To appear in Perform. Eval. 2013 published online at arXiv:654731

  28. Latorre, V.: A potential reduction method for canonical duality, with an application to the sensor network localization problem. J. Glob. Optim. (2014). arXiv:1403.5991

  29. Latorre, V., Gao, D.Y.: Canonical dual solution to nonconvex radial basis neural network optimization problem. Neurocomputing 134, 189–197 (2014)

    Article  Google Scholar 

  30. Latorre, V., Gao, D.Y.: Canonical duality for solving general nonconvex constrained problems. Opt, Lett. 10(8), 1763–1779 (2016). arXiv:1310.2014

  31. Facchinei, F., Kanzow, C., Sagratella, S.: QVILIB: a library of quasi-variational inequality test problems. Pac. J. Optim. 9, 225–250 (2013)

    MathSciNet  MATH  Google Scholar 

  32. Dirkse, S.P., Ferris, M.C.: The PATH solver: a non-monotone stabilization scheme for mixed complementarity problems. Optim. Methods Softw. 5, 123–156 (1995)

    Article  Google Scholar 

  33. Jaru\(\check{s}\)ek. J.: Contact problems with bounded friction. Coercive case. Czechoslov. Math. J. 33, 237–261 (1983)

    Google Scholar 

  34. Ne\(\check{c}\)as, J., Jaru\(\check{s}\)ek, J., Haslinger, J.: On the solution of the variational inequality to the Signorini-problem with small friction. Bolletino deal UMI 17B, 796–811 (1980)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer, Control and Management Engineering “Sapienza”, University of Rome, Via Ariosto 25, 00185, Rome, Italy

    Vittorio Latorre & Simone Sagratella

  2. Faculty of Science and Technology, Federation University Australia, Mt Helen, VIC, 3353, Australia

    David Yang Gao

Authors
  1. Vittorio Latorre
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Simone Sagratella
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. David Yang Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vittorio Latorre .

Editor information

Editors and Affiliations

  1. Faculty of Science and Technology, Federation University Australia, Mt. Helen, Victoria, Australia

    David Yang Gao

  2. Department of Computer, Control, and Managment Engineering, Sapienza University of Rome, Rome, Roma, Italy

    Vittorio Latorre

  3. Faculty of Science and Technology, Federation University Australia, Mt. Helen, Victoria, Australia

    Ning Ruan

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Latorre, V., Sagratella, S., Gao, D.Y. (2017). Canonical Dual Approach for Contact Mechanics Problems with Friction. In: Gao, D., Latorre, V., Ruan, N. (eds) Canonical Duality Theory. Advances in Mechanics and Mathematics, vol 37. Springer, Cham. https://doi.org/10.1007/978-3-319-58017-3_8

Download citation

Publish with us

Policies and ethics

Search

Navigation