Skip to main content
SpringerLink
Log in

Shape Optimization in 2D Contact Problems with Given Friction and a Solution-Dependent Coefficient of Friction

  • Published:
Set-Valued and Variational Analysis Aims and scope Submit manuscript

Abstract

The paper deals with shape optimization of elastic bodies in unilateral contact. The aim is to extend the existing results to the case of contact problems, where the coefficient of friction depends on the solution. We consider the two-dimensional Signorini problem, coupled with the physically less accurate model of given friction, but assume a solution-dependent coefficient of friction. First, we investigate the shape optimization problem in the continuous, infinite-dimensional setting, followed by a suitable finite-dimensional approximation based on the finite-element method. Convergence analysis is presented as well. Next, an algebraic form of the state problem is studied, which is obtained from the discretized problem by further approximating the frictional term by a quadrature rule. It is shown that if the coefficient of friction is Lipschitz continuous with a sufficiently small modulus, then the algebraic state problem is uniquely solvable and its solution is a Lipschitz continuous function of the control variable, describing the shape of the elastic body. For the purpose of numerical solution of the shape optimization problem via the so-called implicit programming approach we perform sensitivity analysis by using the tools from the generalized differential calculus of Mordukhovich. The paper is concluded first order optimality conditions.

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. Beremlijski, P., Haslinger, J., Kočvara, M., Outrata, J.V.: Shape optimization in contact problems with Coulomb friction. SIAM J. Optim. 13, 561–587 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Beremlijski, P., Haslinger, J., Kočvara, M., Kučera, R., Outrata, J.V.: Shape optimization in three-dimensional contact problems with Coulomb friction. SIAM J. Optim. 20, 416–444 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chenais, D.: On the existence of a solution in a domain identification problem. J. Math. Anal. Appl. 52, 189–219 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  4. Clarke, F.F.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    MATH  Google Scholar 

  5. Haslinger, J., Neittaanmäki, P.: Finite Element Approximation for Optimal Shape, Material and Topology Design, 2nd edn. Wiley, New York (1996)

    MATH  Google Scholar 

  6. Haslinger, J., Hlaváček, I., Nečas, J.: Numerical methods for unilateral problems in solid mechanics. In: Ciarlet, P.G., Lions, J.-L. (eds.) Handbook of Numerical Analysis, vol. IV, part 2, pp. 313–491. North Holland, Amsterdam (1996)

    Google Scholar 

  7. Haslinger, J., Neittaanmäki, P., Tiihonen, T.: Shape optimization in contact problems based on penalization of the state inequality. Appl. Math. 31, 54–77 (1986)

    MATH  Google Scholar 

  8. Haslinger, J., Vlach, O.: Signorini problem with a solution dependent coefficient of friction (model with given friction): approximation and numerical realization. Appl. Math. 50, 153–171 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Henrion, R., Jourani, A., Outrata, J.: On the calmness of a class of multifunctions. SIAM J. Optim. 13, 603–618 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hlaváček, I., Mäkinen, R.: On the numerical solution of axisymmetric domain optimization problems. Appl. Math. 36, 284–304 (1991)

    MathSciNet  MATH  Google Scholar 

  11. Kočvara, M., Outrata, J.V.: Optimization problems with equilibrium constraints and their numerical solution. Math. Program. B 101, 119–150 (2004)

    Article  MATH  Google Scholar 

  12. Ligurský, T.: Theoretical analysis of discrete contact problems with Coulomb friction. Appl. Math. (2011, accepted)

  13. Luo, Z.-Q., Pang, J.-S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996)

    Book  Google Scholar 

  14. Mordukhovich, B.S.: Generalized differential calculus for nonsmooth and set-valued mappings. J. Math. Anal. Appl. 183, 250–288 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  15. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory, II: Applications, Grundlehren Series (Fundamental Principles of Mathematical Sciences), vols. 330 and 331, Springer, Berlin (2006)

    Google Scholar 

  16. Nitsche, J.A.: On Korn’s second inequality. RAIRO Anal. Numer. 15, 237–248 (1981)

    MathSciNet  MATH  Google Scholar 

  17. Outrata, J.V.: Optimality conditions for a class of mathematical programs with equilibrium constraints. Math. Oper. Res. 24, 627–644 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  18. Pathó, R.: Shape Optimization in Contact Problems with Friction. Diploma Thesis, Charles University in Prague (2009). Available at: http://www.karlin.mff.cuni.cz/~patho/dipl.pdf

  19. Rockafellar, R.T., Wets, R.: Variational Analysis. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  20. Sokolowski, J., Zolesio, J.P.: Introduction to Shape Optimization: Shape Sensitivity Analysis. Springer, Berlin (1992)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Numerical Mathematics, Charles University in Prague, Sokolovská 83, 186 75, Praha 8, Czech Republic

    Jaroslav Haslinger & Róbert Pathó

  2. Institute of Information Theory and Automation, Czech Academy of Sciences, Pod Vodárenskou věží 4, 182 08, Praha 8, Czech Republic

    Jiří V. Outrata

  3. Grad. School of Information Technology and Math. Sciences, University of Ballarat, Ballarat, VIC, 3353, Australia

    Jiří V. Outrata

Authors
  1. Jaroslav Haslinger
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jiří V. Outrata
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Róbert Pathó
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Róbert Pathó.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Haslinger, J., Outrata, J.V. & Pathó, R. Shape Optimization in 2D Contact Problems with Given Friction and a Solution-Dependent Coefficient of Friction. Set-Valued Anal 20, 31–59 (2012). https://doi.org/10.1007/s11228-011-0179-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11228-011-0179-7

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation