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A Residual Type Error Estimate for the Static Coulomb Friction Problem with Unilateral Contact

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Recent Advances in Contact Mechanics

Part of the book series: Lecture Notes in Applied and Computational Mechanics ((LNACM,volume 56))

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Abstract

This paper is concerned with residual error estimators for finite element approximations of Coulomb frictional contact problems. A recent uniqueness result by Renard in [66] for the continuous problem allows us to perform the a posteriori error analysis.

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References

  1. Achdou, Y., Hecht, F., Pommier, D.: A posteriori error estimates for parabolic variational inequalities. J. Sci. Comput. 37(3), 336–366 (2008)

    Google Scholar 

  2. Adams, R.A.: Sobolev spaces. Academic Press (1975)

    Google Scholar 

  3. Ainsworth, M., Oden, J.T.: A posteriori error estimation in finite element analysis. Wiley, Chichester (2000)

    Book  MATH  Google Scholar 

  4. Ainsworth, M., Oden, J.T., Lee, C.-Y.: Local a posteriori error estimators for variational inequalities. Numer. Methods Partial Differential Equations 9, 23–33 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  5. Andersson, L.-E.: Existence results for quasistatic contact problems with Coulomb friction. Appl. Math. Optim. 42, 169–202 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  6. Atkinson, K., Han, W.: Theoretical numerical analysis: a functional analysis framework. Texts in Applied Mathematics, vol. 39. Springer, New-York (2001); 2nd edn. (2005)

    MATH  Google Scholar 

  7. Babuška, I., Rheinboldt, W.: Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15, 736–754 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  8. Babuška, I., Strouboulis, T.: The finite element method and its reliability. Clarendon Press, Oxford (2001)

    Google Scholar 

  9. Becker, R., Rannacher, R.: A feed-back approach to error control in finite element methods: basic analysis and examples. East-West J. Numer. Math. 4, 237–264 (1996)

    MathSciNet  MATH  Google Scholar 

  10. Belhachmi, Z., Hecht, F., Tahir, S.: Adaptive finite element for a fictitious domain formulation of some variational inequalities (in preparation)

    Google Scholar 

  11. Ben Belgacem, F., Renard, Y.: Hybrid finite element methods for the Signorini problem. Math. Comp. 72, 1117–1145 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Blum, H., Suttmeier, F.: An adaptive finite element discretization for a simplified Signorini problem. Calcolo 37, 65–77 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bostan, V., Han, W.: A posteriori error analysis for finite element solutions of a frictional contact problem. Comput. Methods Appl. Mech. Engrg. 195, 1252–1274 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bostan, V., Han, W., Reddy, B.D.: A posteriori error estimation and adaptive solution of elliptic variational inequalities of the second kind. Appl. Numer. Math. 52, 13–38 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Braess, D.: A posteriori error estimators for obstacle problems - another look. Numer. Math. 101, 415–421 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Braess, D., Carstensen, C., Hoppe, R.: Convergence analysis of a conforming adaptive finite element method for an obstacle problem. Numer. Math. 107, 455–471 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Brenner, S.C., Scott, L.R.: The mathematical theory of finite element methods. Springer (2002)

    Google Scholar 

  18. Brezzi, F., Fortin, M.: Mixed and hybrid finite element methods. Springer (1991)

    Google Scholar 

  19. Carstensen, C.: Numerical analysis of the primal problem of elastoplasticity with hardening. Numer. Math. 82, 577–597 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  20. Carstensen, C., Scherf, O., Wriggers, P.: Adaptive finite elements for elastic bodies in contact. SIAM J. Sci. Comput. 20, 1605–1626 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  21. Chen, Z., Nochetto, R.H.: Residual type a posteriori error estimates for elliptic obstacle problems. Numer. Math. 84, 527–548 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ciarlet, P.G.: The finite element method for elliptic problems. In: Ciarlet, P.G., Lions, J.-L. (eds.) Handbook of Numerical Analysis, vol. II, part 1, pp. 17–352. North Holland (1991)

    Google Scholar 

  23. Clément, P.: Approximation by finite elements functions using local regularization. RAIRO Anal. Numer. 9, 77–84 (1975)

    MATH  Google Scholar 

  24. Coorevits, P., Hild, P., Hjiaj, M.: A posteriori error control of finite element approximations for Coulomb’s frictional contact. SIAM J. Sci. Comput. 23, 976–999 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  25. Coorevits, P., Hild, P., Lhalouani, K., Sassi, T.: Mixed finite element methods for unilateral problems: convergence analysis and numerical studies. Math. Comp. 71, 1–25 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  26. Coorevits, P., Hild, P., Pelle, J.-P.: A posteriori error estimation for unilateral contact with matching and nonmatching meshes. Comput. Methods Appl. Mech. Engrg. 186, 65–83 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  27. Duvaut, G.: Problèmes unilatéraux en mécanique des milieux continus. In: Actes du Congrès International des Mathématiciens (Nice 1970), Tome 3, Gauthier-Villars, pp. 71–77 (1971)

    Google Scholar 

  28. Duvaut, G., Lions, J.L.: Les inéquations en mécanique et en physique. Dunod (1972)

    Google Scholar 

  29. Eck, C., Jaruček, J.: Existence results for the static contact problem with Coulomb friction. Math. Models Meth. Appl. Sci. 8, 445–468 (1998)

    Article  MATH  Google Scholar 

  30. Eck, C., Jaruček, J., Kozubek, M.: Unilateral contact problems: variational methods and existence theorems. Pure and Applied Mathematics. CRC Press (2005)

    Google Scholar 

  31. Eck, C., Wendland, W.: A residual-based error estimator for BEM-discretizations of contact problems. Numer. Math. 95, 253–282 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  32. Erdmann, B., Frei, M., Hoppe, R., Kornhuber, R., Wiest, U.: Adaptive finite element methods for variational inequalities. East-West J. Numer. Math. 1, 165–197 (1993)

    MathSciNet  MATH  Google Scholar 

  33. Glowinski, R.: Lectures on numerical methods for nonlinear variational problems. Notes by Vijayasundaram, M.G., Adimurthi, M. (eds.) Tata Institute of Fundamental Research Lectures on Mathematics and Physics, Tata Institute of Fundamental Research, Bombay, vol. 65. Springer, Berlin (1980)

    Google Scholar 

  34. Han, W.: A posteriori error analysis via duality theory. With applications in modeling and numerical approximations. Advances in Mechanics and Mathematics, vol. 8. Springer, New York (2005)

    MATH  Google Scholar 

  35. Han, W., Sofonea, M.: Quasistatic contact problems in viscoelasticity and viscoplasticity. American Mathematical Society (2002)

    Google Scholar 

  36. Hansbo, P., Johnson, C.: Adaptive finite element methods for elastostatic contact problems. In: Grid Generation and Adaptive Algorithms, Minneapolis, MN (1997); IMA Vol. Math. Appl., vol. 113, pp. 135–149. Springer, New York (1999)

    Google Scholar 

  37. 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. 313–485. North Holland (1996)

    Google Scholar 

  38. Hassani, R., Hild, P., Ionescu, I., Sakki, N.-D.: A mixed finite element method and solution multiplicity for Coulomb frictional contact. Comput. Methods Appl. Mech. Engrg. 192, 4517–4531 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  39. Hild, P.: Non-unique slipping in the Coulomb friction model in two-dimensional linear elasticity. Q. Jl. Mech. Appl. Math. 57, 225–235 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  40. Hild, P.: Multiple solutions of stick and separation type in the Signorini model with Coulomb friction. Z. Angew. Math. Mech. 85, 673–680 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  41. Hild, P.: Solution multiplicity and stick configurations in continuous and finite element friction problems. Comput. Methods Appl. Mech. Engrg. 196, 57–65 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  42. Hild, P., Lleras, V.: Residual estimators for Coulomb friction. SIAM J. Number. Anal. 47(5), 3550–3583 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  43. Hild, P., Nicaise, S.: A posteriori error estimations of residual type for Signorini’s problem. Numer. Math. 101, 523–549 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  44. Hild, P., Nicaise, S.: Residual a posteriori error estimators for contact problems in elasticity. Math. Model. Numer. Anal. 41, 897–923 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  45. Hild, P., Renard, Y.: An error estimate for the Signorini problem with Coulomb friction approximated by finite elements. SIAM J. Numer. Anal. 45, 2012–2031 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  46. Hoppe, R., Kornhuber, R.: Adaptive multilevel methods for obstacle problems. SIAM J. Numer. Anal. 31, 301–323 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  47. Hüeber, S., Wohlmuth, B.: An optimal error estimate for nonlinear contact problems. SIAM J. Numer. Anal. 43, 156–173 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  48. Jarušek, J.: Contact problems with bounded friction. Coercive case. Czechoslovak. Math. J. 33, 237–261 (1983)

    MathSciNet  Google Scholar 

  49. Johnson, C.: Adaptive finite element methods for the obstacle problem. Math. Models Methods Appl. Sci. 2, 483–487 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  50. Kikuchi, N., Oden, J.T.: Contact problems in elasticity. SIAM (1988)

    Google Scholar 

  51. Kinderlehrer, D., Stampacchia, G.: An introduction to variational inequalities and their applications. Pure and Applied Mathematics. Academic Press, New York-London (1980)

    MATH  Google Scholar 

  52. Kornhuber, R.: A posteriori error estimates for elliptic variational inequalities. Comput. Math. Applic. 31, 49–60 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  53. Ladevèze, P., Leguillon, D.: Error Estimate Procedure in the Finite Element Method and Applications. SIAM J. Numer. Anal. 20, 485–509 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  54. Laursen, T.: Computational contact and impact mechanics. Springer (2002)

    Google Scholar 

  55. Lee, C.Y., Oden, J.T.: A posteriori error estimation of h-p finite element approximations of frictional contact problems. Comput. Methods Appl. Mech. Engrg. 113, 11–45 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  56. Liu, W., Yan, N.: A posteriori error estimators for a class of variational inequalities. J. Sci. Comput. 15, 361–393 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  57. Louf, F., Combe, J.-P., Pelle, J.-P.: Constitutive error estimator for the control of contact problems involving friction. Comput. and Structures 81, 1759–1772 (2003)

    Article  MathSciNet  Google Scholar 

  58. Maischak, M., Stephan, E.: Adaptive hp-versions of BEM for Signorini problems. Appl. Numer. Math. 54, 425–449 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  59. Maischak, M., Stephan, E.: Adaptive hp-versions of boundary element methods for elastic contact problems. Comput. Mech. 39, 597–607 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  60. Maz’ya, V.G., Shaposhnikova, T.O.: Theory of multipliers in spaces of differentiable functions. Pitman (1985)

    Google Scholar 

  61. Nečas, J., Jarušek, J., Haslinger, J.: On the solution of the variational inequality to the Signorini problem with small friction. Bolletino U.M.I. 17(5), 796–811 (1980)

    MATH  Google Scholar 

  62. Nochetto, R., Siebert, K., Veeser, A.: Pointwise a posteriori error control for elliptic obstacle problems. Numer. Math. 95, 163–195 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  63. Nochetto, R., Siebert, K., Veeser, A.: Fully localized a posteriori error estimators and barrier sets for contact problems. SIAM J. Numer. Anal. 42, 2118–2135 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  64. Rannacher, R., Suttmeier, F.-T.: A posteriori error control in finite element methods via duality techniques: application to perfect plasticity. Comput. Mech. 21, 123–133 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  65. Rannacher, R., Suttmeier, F.-T.: A posteriori error estimation and mesh adaptation for finite element models in elasto-plasticity. Comput. Methods Appl. Mech. Engrg. 176, 333–361 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  66. Renard, Y.: A uniqueness criterion for the Signorini problem with Coulomb friction. SIAM J. Math. Anal. 38, 452–467 (2006)

    Article  MathSciNet  Google Scholar 

  67. Rocca, R., Cocou, M.: Existence and approximation of a solution to quasistatic Signorini problem with local friction. Internat. J. Engrg. Sci. 39, 1233–1255 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  68. Shillor, M., Sofonea, M., Telega, J.J.: Models and analysis of quasistatic contact. Varational methods. Springer (2004)

    Google Scholar 

  69. Suttmeier, F.-T.: General approach for a posteriori error estimates for finite element solutions of variational inequalities. Comput. Mech. 27, 317–323 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  70. Suttmeier, F.-T.: Error bounds for finite element solutions of elliptic variational inequalities of second kind. East-West J. Numer. Math. 9, 307–313 (2001)

    MathSciNet  MATH  Google Scholar 

  71. Suttmeier, F.-T.: On a direct approach to adaptive fe-discretizations for elliptic variational inequalities. J. Numer. Math. 13, 73–80 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  72. Veeser, A.: Efficient and reliable a posteriori error estimators for elliptic obstacle problems. SIAM J. Numer. Anal. 39, 146–167 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  73. Verfürth, R.: A review of a posteriori error estimation and adaptive mesh-refinement techniques. Wiley and Teubner (1996)

    Google Scholar 

  74. Verfürth, R.: A review of a posteriori error estimation techniques for elasticity problems. Comput. Methods Appl. Mech. Engrg. 176, 419–440 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  75. Weiss, A., Wohlmuth, B.: A posteriori error estimator and error control for contact problems. Math. Comp. 78(267), 1237–1267 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  76. Wohlmuth, B.: An a posteriori error estimator for two body contact problems on non-matching meshes. J. Sci. Comput. 33, 25–45 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  77. Wriggers, P.: Computational Contact Mechanics. Wiley (2002)

    Google Scholar 

  78. Wriggers, P., Scherf, O.: Different a posteriori error estimators and indicators for contact problems. Mathl. Comput. Modelling 28, 437–447 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  79. Zienkiewicz, O.C., Zhu, J.Z.: A simple error estimator and adaptive procedure for practical engineering analysis. Internat. J. Numer. Methods Engrg. 24, 337–357 (1987)

    Article  MathSciNet  MATH  Google Scholar 

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Hild, P., Lleras, V. (2013). A Residual Type Error Estimate for the Static Coulomb Friction Problem with Unilateral Contact. In: Stavroulakis, G. (eds) Recent Advances in Contact Mechanics. Lecture Notes in Applied and Computational Mechanics, vol 56. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33968-4_6

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