BEM for Contact Problems

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


In literature we find various finite element discretization schemes that tackle variational inequalities that arise from scalar unilateral Signorini problems and from contact problems without and with friction in solid mechanics, see e.g. [199, 249, 266]. Each scheme has to overcome several challenges, mainly the discretization of a cone, a primal one in variational inequalities or a dual one in mixed methods, the non-differentiability of the friction functional in the classical sense and the reduced regularity of the solution at the a priori unknown free boundary/interface from contact to non-contact and from stick to slip.


  1. 14.
    I. Babuška, A.K. Aziz, Survey Lectures on the Mathematical Foundations of the Finite Element Method. The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (Proc. Sympos., Univ. Maryland, Baltimore, MD, 1972). Academic Press, New York, 1972, With the collaboration of G. Fix and R. B. Kellogg, pp. 1–359Google Scholar
  2. 27.
    M. Bach, S.A. Nazarov, W.L. Wendland, Stable Propagation of a Mode-1 Planar Crack in an Isotropic Elastic Space. Comparison of the Irwin and the Griffith Approaches. Current Problems of Analysis and Mathematical Physics (Taormina, Italy, 1998), Aracne, Rome, 2000, pp. 167–189Google Scholar
  3. 32.
    R.E. Bank, R.K. Smith, A posteriori error estimates based on hierarchical bases. SIAM J. Numer. Anal. 30, 921–935 (1993)MathSciNetzbMATHGoogle Scholar
  4. 33.
    L. Banz, H. Gimperlein, A. Issaoui, E.P. Stephan, Stabilized mixed hp-BEM for frictional contact problems in linear elasticity. Numer. Math. 135, 217–263 (2017)MathSciNetzbMATHGoogle Scholar
  5. 35.
    L. Banz, G. Milicic, N. Ovcharova, Improved stabilization technique for frictional contact problems solved with hp-bem. Comput. Math. Appl. Mech. Eng. (2018, to appear)Google Scholar
  6. 36.
    L. Banz, A. Schröder, Biorthogonal Basis Functions in hp-adaptive fem for Elliptic Obstacle Problems, 2015, pp. 1721–1742Google Scholar
  7. 37.
    L. Banz, E.P. Stephan, On hp-adaptive BEM for frictional contact problems in linear elasticity. Comput. Math. Appl. 69, 559–581 (2015)Google Scholar
  8. 38.
    L. Banz, E.P. Stephan, Comparison of mixed hp-bem (stabilized and non-stabilized) for frictional contact problems. J. Comput. Appl. Math. 295, 92–102 (2016)Google Scholar
  9. 39.
    H.J.C. Barbosa, T.J.R. Hughes, Circumventing the Babuška-Brezzi condition in mixed finite element approximations of elliptic variational inequalities. Comput. Methods Appl. Mech. Eng. 97, 193–210 (1992)MathSciNetzbMATHGoogle Scholar
  10. 45.
    C. Bernardi, Y. Maday, A.T. Patera, Domain Decomposition by the Mortar Element Method. Asymptotic and Numerical Methods for Partial Differential Equations with Critical Parameters (Beaune, 1992), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 384 (Kluwer Acad. Publ., Dordrecht, 1993), pp. 269–286Google Scholar
  11. 46.
    Ch. Bernardi, Y. Maday, Polynomial interpolation results in Sobolev spaces. J. Comput. Appl. Math. 43, 53–80 (1992)MathSciNetzbMATHGoogle Scholar
  12. 47.
    Ch. Bernardi, Y. Maday, Spectral Methods. Handbook of Numerical Analysis, Vol. V. Handb. Numer. Anal. (North-Holland, Amsterdam, 1997), pp. 209–485Google Scholar
  13. 48.
    Ch. Bernardi, Y. Maday, Spectral, Spectral Element and Mortar Element Methods. Theory and Numerics of Differential Equations (Durham, 2000) (Springer, Berlin, 2001), pp. 1–57Google Scholar
  14. 59.
    D. Braess, A posteriori error estimators for obstacle problems—another look. Numer. Math. 101, 415–421 (2005)MathSciNetzbMATHGoogle Scholar
  15. 63.
    S.C. Brenner, L.R. Scott, The Mathematical Theory of Finite Element Methods, 3rd edn. Texts in Applied Mathematics, vol. 15 (Springer, New York, 2008)zbMATHGoogle Scholar
  16. 73.
    C. Carstensen, Interface problem in holonomic elastoplasticity. Math. Methods Appl. Sci. 16, 819–835 (1993)MathSciNetzbMATHGoogle Scholar
  17. 75.
    C. Carstensen, A posteriori error estimate for the symmetric coupling of finite elements and boundary elements. Computing 57, 301–322 (1996)MathSciNetzbMATHGoogle Scholar
  18. 90.
    C. Carstensen, O. Scherf, P. Wriggers, Adaptive finite elements for elastic bodies in contact. SIAM J. Sci. Comput. 20, 1605–1626 (1999)MathSciNetzbMATHGoogle Scholar
  19. 91.
    C. Carstensen, E.P. Stephan, Adaptive coupling of boundary elements and finite elements. RAIRO Modél. Math. Anal. Numér. 29, 779–817 (1995)MathSciNetzbMATHGoogle Scholar
  20. 101.
    A. Chernov, Nonconforming boundary elements and finite elements for interface and contact problems with friction – hp-version for mortar, penalty and Nitsche’s methods, Ph.D. thesis, Leibniz Universität Hannover, 2006Google Scholar
  21. 102.
    A. Chernov, M. Maischak, E.P. Stephan, A priori error estimates for hp penalty BEM for contact problems in elasticity. Comput. Methods Appl. Mech. Eng. 196, 3871–3880 (2007)MathSciNetzbMATHGoogle Scholar
  22. 103.
    A. Chernov, M. Maischak, E.P. Stephan, hp-mortar boundary element method for two-body contact problems with friction. Math. Methods Appl. Sci. 31, 2029–2054 (2008)MathSciNetzbMATHGoogle Scholar
  23. 111.
    A.R. Conn, N.I.M. Gould, P.L. Toint, Trust-Region Methods. MPS/SIAM Series on Optimization (Society for Industrial and Applied Mathematics (SIAM); Mathematical Programming Society (MPS), Philadelphia, PA, 2000)Google Scholar
  24. 142.
    R.A. DeVore, G.G. Lorentz, Constructive Approximation. Grundlehren der Mathematischen Wissenschaften, vol. 303 (Springer, Berlin, 1993)zbMATHGoogle Scholar
  25. 147.
    C. Eck, S.A. Nazarov, W.L. Wendland, Asymptotic analysis for a mixed boundary-value contact problem. Arch. Ration. Mech. Anal. 156, 275–316 (2001)MathSciNetzbMATHGoogle Scholar
  26. 148.
    C. Eck, H. Schulz, O. Steinbach, W.L. Wendland, An Adaptive Boundary Element Method for Contact Problems. Error Controlled Adaptive Finite Elements in Solid Mechanics (Wiley, 2003), pp. 181–209Google Scholar
  27. 149.
    C. Eck, O. Steinbach, W.L. Wendland, A symmetric boundary element method for contact problems with friction. Math. Comput. Simul. 50, 43–61 (1999)MathSciNetzbMATHGoogle Scholar
  28. 150.
    C. Eck, W.L. Wendland, A residual-based error estimator for BEM-discretizations of contact problems. Numer. Math. 95, 253–282 (2003)MathSciNetzbMATHGoogle Scholar
  29. 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
  30. 160.
    H. Engels, Numerical Quadrature and Cubature. Computational Mathematics and Applications (Academic Press, London-New York, 1980)Google Scholar
  31. 173.
    R.S. Falk, Error estimates for the approximation of a class of variational inequalities. Math. Comput. 28, 963–971 (1974)MathSciNetzbMATHGoogle Scholar
  32. 199.
    R. Glowinski, Numerical Methods for Nonlinear Variational Problems. Scientific Computation (Springer, Berlin, 2008)Google Scholar
  33. 200.
    R. Glowinski, J.-L. Lions, R. Trémolières, Numerical Analysis of Variational Inequalities. Studies in Mathematics and Its Applications, vol. 8 (North-Holland Publishing, Amsterdam-New York, 1981)zbMATHGoogle Scholar
  34. 204.
    P. Grisvard, Elliptic Problems in Nonsmooth Domains. Monographs and Studies in Mathematics, vol. 24 (Pitman (Advanced Publishing Program), Boston, MA, 1985). Reprint 2011Google Scholar
  35. 205.
    H. Guediri, On a boundary variational inequality of the second kind modelling a friction problem. Math. Methods Appl. Sci. 25, 93–114 (2002)MathSciNetzbMATHGoogle Scholar
  36. 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)zbMATHGoogle Scholar
  37. 217.
    J. Gwinner, On the p-version approximation in the boundary element method for a variational inequality of the second kind modelling unilateral contact and given friction. Appl. Numer. Math. 59, 2774–2784 (2009)MathSciNetzbMATHGoogle Scholar
  38. 218.
    J. Gwinner, hp-FEM convergence for unilateral contact problems with Tresca friction in plane linear elastostatics. J. Comput. Appl. Math. 254, 175–184 (2013)MathSciNetzbMATHGoogle Scholar
  39. 220.
    J. Gwinner, N. Ovcharova, From solvability and approximation of variational inequalities to solution of nondifferentiable optimization problems in contact mechanics. Optimization 64, 1683–1702 (2015)MathSciNetzbMATHGoogle Scholar
  40. 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
  41. 222.
    J. Gwinner, E.P. Stephan, A boundary element procedure for contact problems in plane linear elastostatics. RAIRO M2AN 27, 457–480 (1993)MathSciNetzbMATHGoogle Scholar
  42. 226.
    G. Hämmerlin, K.-H. Hoffmann, Numerische Mathematik, 4th edn. (Springer-Lehrbuch, Springer, Berlin, 1994)zbMATHGoogle Scholar
  43. 227.
    H. Han, A direct boundary element method for Signorini problems. Math. Comput. 55, 115–128 (1990)MathSciNetzbMATHGoogle Scholar
  44. 239.
    N. Heuer, M.E. Mellado, E.P. Stephan, A p-adaptive algorithm for the BEM with the hypersingular operator on the plane screen. Int. J. Numer. Methods Eng. 53, 85–104 (2002)zbMATHGoogle Scholar
  45. 249.
    I. Hlaváček, J. Haslinger, J. Nečas, J. Lovíšek, Solution of Variational Inequalities in Mechanics. Applied Mathematical Sciences, vol. 66 (Springer, New York, 1988)zbMATHGoogle Scholar
  46. 254.
    P. Houston, E. Süli, A note on the design of hp-adaptive finite element methods for elliptic partial differential equations. Comput. Methods Appl. Mech. Eng. 194, 229–243 (2005)MathSciNetzbMATHGoogle Scholar
  47. 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
  48. 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
  49. 275.
    A. Krebs, E.P. Stephan, A p-version finite element method for nonlinear elliptic variational inequalities in 2D. Numer. Math. 105, 457–480 (2007)MathSciNetzbMATHGoogle Scholar
  50. 284.
    J.-L. Lions, E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. I. Grundlehren der mathematischen Wissenschaften, vol. 181 (Springer, New York-Heidelberg, 1972). Translated from the French by P. KennethGoogle Scholar
  51. 292.
    M. Maischak, Fem/bem Methods for Signorini-Type Problems: Error Analysis, Adaptivity, Preconditioners. Habilitationsschrift (Leibniz Universität Hannover, 2004)Google Scholar
  52. 297.
    M. Maischak, E.P. Stephan, Adaptive hp-versions of BEM for Signorini problems. Appl. Numer. Math. 54, 425–449 (2005)MathSciNetzbMATHGoogle Scholar
  53. 299.
    M. Maischak, E.P. Stephan, Adaptive hp-versions of boundary element methods for elastic contact problems. Comput. Mech. 39, 597–607 (2007)MathSciNetzbMATHGoogle Scholar
  54. 326.
    L. Nesemann, E.P. Stephan, Numerical solution of an adhesion problem with FEM and BEM. Appl. Numer. Math. 62, 606–619 (2012)MathSciNetzbMATHGoogle Scholar
  55. 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
  56. 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)MathSciNetzbMATHGoogle Scholar
  57. 334.
    N. Ovcharova, L. Banz, Coupling regularization and adaptive hp-BEM for the solution of a delamination problem. Numer. Math. 137, 303–337 (2017)MathSciNetzbMATHGoogle Scholar
  58. 373.
    Ch. Schwab, p- and hp-Finite Element Methods.Theory and Applications in Solid and Fluid Mechanics. Numerical Mathematics and Scientific Computation (The Clarendon Press, Oxford University Press, New York, 1998)Google Scholar
  59. 386.
    W. Spann, On the boundary element method for the Signorini problem of the Laplacian. Numer. Math. 65, 337–356 (1993)MathSciNetzbMATHGoogle Scholar
  60. 387.
    W. Spann, Error estimates for the boundary element approximation of a contact problem in elasticity. Math. Methods Appl. Sci. 20, 205–217 (1997)MathSciNetzbMATHGoogle Scholar
  61. 391.
    O. Steinbach, Numerische Näherungsverfahren für elliptische Randwertprobleme (Springer, 2003)Google Scholar
  62. 393.
    O. Steinbach, Boundary element methods for variational inequalities. Numer. Math. 126, 173–197 (2014)MathSciNetzbMATHGoogle Scholar
  63. 405.
    E.P. Stephan, M. Suri, The h-p version of the boundary element method on polygonal domains with quasiuniform meshes. RAIRO Modél. Math. Anal. Numér. 25, 783–807 (1991)MathSciNetzbMATHGoogle 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

Personalised recommendations