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The Signorini Problem and More Nonsmooth BVPs and Their Boundary Integral Formulation

  • Joachim Gwinner
  • Ernst Peter Stephan
Chapter
Part of the Springer Series in Computational Mathematics book series (SSCM, volume 52)

Abstract

In this chapter we deal with unilateral and nonsmooth boundary value problems, in particular Signorini problems without and with Tresca friction and nonmontone contact problems from adhesion/delamination in the range of linear elasticity. We show how the boundary integral techniques developed in the previous chapters can be used to transform those problems to boundary variational inequalities. This opens the way to the numerical treatment of these nonlinear problems by the BEM as detailed in Chap.  11.

References

  1. 85.
    C. Carstensen, J. Gwinner, FEM and BEM coupling for a nonlinear transmission problem with Signorini contact. SIAM J. Numer. Anal. 34, 1845–1864 (1997)MathSciNetCrossRefGoogle Scholar
  2. 114.
    M. Costabel, Boundary integral operators on Lipschitz domains: elementary results. SIAM J. Math. Anal. 19, 613–626 (1988)MathSciNetCrossRefGoogle Scholar
  3. 136.
    M. Costabel, E.P. Stephan, Coupling of finite and boundary element methods for an elastoplastic interface problem. SIAM J. Numer. Anal. 27, 1212–1226 (1990)MathSciNetCrossRefGoogle Scholar
  4. 145.
    C. Eck, Existence and regularity of solutions to contact problems with friction (german), Ph.D. thesis, Universität Stuttgart, 1996Google Scholar
  5. 146.
    C. Eck, J. Jarušek, M. Krbec, Unilateral Contact Problems. Pure and Applied Mathematics, vol. 270 (Chapman & Hall/CRC, Boca Raton, FL, 2005)CrossRefGoogle Scholar
  6. 151.
    I. Ekeland, R. Témam, Convex Analysis and Variational Problems. Classics in Applied Mathematics, vol. 28 (Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999)Google Scholar
  7. 163.
    E. Ernst, M. Théra, A converse to the Lions-Stampacchia theorem. ESAIM Control Optim. Calc. Var. 15, 810–817 (2009)MathSciNetzbMATHGoogle Scholar
  8. 182.
    A. Gachechiladze, R. Gachechiladze, J. Gwinner, D. Natroshvili, A boundary variational inequality approach to unilateral contact problems with friction for micropolar hemitropic solids. Math. Methods Appl. Sci. 33, 2145–2161 (2010)MathSciNetCrossRefGoogle Scholar
  9. 184.
    R. Gachechiladze, J. Gwinner, D. Natroshvili, A boundary variational inequality approach to unilateral contact with hemitropic materials. Mem. Differ. Equ. Math. Phys. 39, 69–103 (2006)Google Scholar
  10. 201.
    D. Goeleven, Noncoercive Variational Problems and Related Results. Pitman Research Notes in Mathematics Series, vol. 357 (Longman, Harlow, 1996)Google Scholar
  11. 205.
    H. Guediri, On a boundary variational inequality of the second kind modelling a friction problem. Math. Methods Appl. Sci. 25, 93–114 (2002)MathSciNetCrossRefGoogle Scholar
  12. 212.
    J. Gwinner, On fixed points and variational inequalities—a circular tour. Nonlinear Anal. 5, 565–583 (1981)MathSciNetCrossRefGoogle Scholar
  13. 214.
    J. Gwinner, Boundary Element Convergence for Unilateral Harmonic Problems. Progress in Partial Differential Equations: Calculus of Variations, Applications (Pont-à-Mousson, 1991). Pitman Res. Notes Math. Ser., vol. 267 (Longman Sci. Tech., Harlow, 1992), pp. 200–213Google Scholar
  14. 215.
    J. Gwinner, A discretization theory for monotone semicoercive problems and finite element convergence for p-harmonic Signorini problems. Z. Angew. Math. Mech. 74, 417–427 (1994)CrossRefGoogle Scholar
  15. 221.
    J. Gwinner, E.P. Stephan, Boundary Element Convergence for a Variational Inequality of the Second Kind. Parametric Optimization and Related Topics, III (Güstrow, 1991). Approx. Optim., vol. 3 (Lang, Frankfurt am Main, 1993), pp. 227–241Google Scholar
  16. 222.
    J. Gwinner, E.P. Stephan, A boundary element procedure for contact problems in plane linear elastostatics. RAIRO M2AN 27, 457–480 (1993)MathSciNetCrossRefGoogle Scholar
  17. 227.
    H. Han, A direct boundary element method for Signorini problems. Math. Comput. 55, 115–128 (1990)MathSciNetCrossRefGoogle Scholar
  18. 229.
    F. Hartmann, The Somigliana identity on piecewise smooth surfaces. J. Elasticity 11, 403–423 (1981)MathSciNetCrossRefGoogle Scholar
  19. 232.
    J. Haslinger, C.C. Baniotopoulos, P.D. Panagiotopoulos, A boundary multivalued integral “equation” approach to the semipermeability problem. Appl. Math. 38, 39–60 (1993)Google Scholar
  20. 249.
    I. Hlaváček, J. Haslinger, J. Nečas, J. Lovíšek, Solution of Variational Inequalities in Mechanics. Applied Mathematical Sciences, vol. 66 (Springer, New York, 1988)CrossRefGoogle Scholar
  21. 256.
    G.C. Hsiao, E.P. Stephan, W.L. Wendland, On the Dirichlet problem in elasticity for a domain exterior to an arc. J. Comput. Appl. Math. 34, 1–19 (1991)MathSciNetCrossRefGoogle Scholar
  22. 257.
    G.C. Hsiao, W.L. Wendland, A finite element method for some integral equations of the first kind. J. Math. Anal. Appl. 58, 449–481 (1977)MathSciNetCrossRefGoogle Scholar
  23. 259.
    G.C. Hsiao, W.L. Wendland, Boundary Integral Equations. Applied Mathematical Sciences, vol. 164 (Springer, Berlin, 2008)CrossRefGoogle Scholar
  24. 261.
    J. Jarušek, Contact problems with bounded friction coercive case. Czechoslov. Math. J. 33(108), 237–261 (1983)Google Scholar
  25. 266.
    N. Kikuchi, J.T. Oden, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM Studies in Applied Mathematics, vol. 8 (Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988)Google Scholar
  26. 267.
    D. Kinderlehrer, G. Stampacchia, An Introduction to Variational Inequalities and Their Applications. Classics in Applied Mathematics, vol. 31 (Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000)Google Scholar
  27. 297.
    M. Maischak, E.P. Stephan, Adaptive hp-versions of BEM for Signorini problems. Appl. Numer. Math. 54, 425–449 (2005)MathSciNetCrossRefGoogle Scholar
  28. 319.
    J. Nečas, J. Jarušek, J. Haslinger, On the solution of the variational inequality to the Signorini problem with small friction. Boll. Un. Mat. Ital. B (5) 17, 796–811 (1980)Google Scholar
  29. 327.
    J. Nečas, Direct Methods in the Theory of Elliptic Equations. Springer Monographs in Mathematics (Springer, Heidelberg, 2012). Translated from the 1967 French original by G. Tronel and A. Kufner, and a contribution by Ch. G. SimaderCrossRefGoogle Scholar
  30. 332.
    N. Ovcharova, Regularization Methods and Finite Element Approximation of Hemivariational Inequalities with Applications to Nonmonotone Contact Problems, Ph.D. thesis, Universität der Bundeswehr München, 2012Google Scholar
  31. 333.
    N. Ovcharova, On the coupling of regularization techniques and the boundary element method for a hemivariational inequality modelling a delamination problem. Math. Methods Appl. Sci. 40, 60–77 (2017)MathSciNetCrossRefGoogle Scholar
  32. 335.
    N. Ovcharova, J. Gwinner, A study of regularization techniques of nondifferentiable optimization in view of application to hemivariational inequalities. J. Optim. Theory Appl. 162, 754–778 (2014)MathSciNetCrossRefGoogle Scholar
  33. 336.
    P.D. Panagiotopoulos, J. Haslinger, On the dual reciprocal variational approach to the Signorini-Fichera problem. Convex and nonconvex generalization. Z. Angew. Math. Mech. 72, 497–506 (1992)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Joachim Gwinner
    • 1
  • Ernst Peter Stephan
    • 2
  1. 1.Fakultät für Luft- und RaumfahrttechnikUniversität der Bundeswehr MünchenNeubiberg/MünchenGermany
  2. 2.Institut für Angewandte MathematikLeibniz Universität HannoverHannoverGermany

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