Mesh and model adaptivity for frictional contact problems | Numerische Mathematik Skip to main content
Springer Nature Link
Log in

Mesh and model adaptivity for frictional contact problems

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

The article focuses on adaptive finite element methods for frictional contact problems. The approach is based on a reformulation of the mixed form of the underlying Signorini problem with friction as a nonlinear variational equation using nonlinear complimentarity functions. The usual dual weighted residual framework for a posteriori error estimation is applied. However, we have to take into account the nonsmoothness of the problem formulation. Error identities for measuring the discretization as well as the model error with respect to a model hierarchy of friction laws are derived and a method for the numerical evaluation of them is proposed. The estimates are utilized in an adaptive framework, which balances the discretization and the model error. Several numerical examples substantiate the accuracy of the proposed estimates and the efficiency of the adaptive method.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Japan)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  1. Actis, R.L., Szabo, B.A., Schwab, C.: Hierarchic models for laminated plates and shells. Comp. Methods Appl. Mech. Engrg. 172, 79–107 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ainsworth, M., Oden, J., Lee, C.: Local a posteriori error estimators for variational inequalities. Numer. Methods Partial Differ. Equ. 9, 23–33 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bangerth, W., Rannacher, R.: Adaptive finite element methods for differential equations. Lectures in Mathematics, ETH Zürich. Birkhäuser, Basel (2003)

  4. Bartels, S., Carstensen, C.: Averaging techniques yield reliable a posteriori finite element error control for obstacle problems. Numer. Math. 99(2), 225–249 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Becker, R., Rannacher, R.: An optimal control approach to a posteriori error estimation in finite element methods. Acta Numerica 10, 1–102 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Betten, J.: Bemerkungen zum Versuch von Hohenemser. ZAMM 55, 149–158 (75)

  7. Beyer, F., Blum, H., Kumor, D., Rademacher, A., Willner, K., Schneider, T.: Experimental and simulative investigations of tribology in sheet-bulk-metal-forming. Key Engrg. Mat. 639, 283–290 (2015)

    Article  Google Scholar 

  8. Billade, N., Vemaganti, K.: Hierarchical models of thin elastic structures: Overview and recent adcances in error estimation and adaptivity. Comp. Methods Appl. Mech. Engrg. 196, 3508–3523 (2007)

    Article  MATH  Google Scholar 

  9. Blum, H., Braess, D., Suttmeier, F.T.: A cascadic multigrid algorithm for variational inequalities. Comput. Vis. Sci. 7(3–4), 153–157 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Blum, H., Frohne, H., Frohne, J., Rademacher, A.: Semi-smooth Newton methods for mixed FEM discretizations of higher-order for frictional, elasto-plastic two-body contact problems. Comput. Method. Appl. Mech. Eng. 309, 131–151 (2016)

    Article  MathSciNet  Google Scholar 

  11. Blum, H., Schroeder, A., Suttmeier, F.: A posteriori estimates for FE-solutions of variational inequalities. In: F. Brezzi, et al. (eds.) Numerical mathematics and advanced applications. Proceedings of ENUMATH 2001, the 4th European conference, Ischia, July 2001, pp. 669–680. Springer, Berlin (2003)

  12. Blum, H., Suttmeier, F.T.: An adaptive finite element discretisation for a simplified Signorini problem. Calcolo 37(2), 65–77 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bohinc, U.: Adaptive analysis of plate structures. Ph.D. thesis, L’Ecole Normale Supérieure de Cachan (2011)

  14. Bowden, F.P., Tabor, T.: The Friction and Lubrication of Solids. Clarendon Press, Oxford (2001)

    MATH  Google Scholar 

  15. Braack, M., Ern, A.: A posteriori control of modeling errors and discretization errors. Multiscale Model. Simul. 1(2), 221–238 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Braack, M., Taschenberger, N.: A posteriori control of modeling and discretization errors for quasi periodic solutions. J. Numer. Math. 22(2), 87–108 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Braess, D.: A posteriori error estimators for obstacle problems - another look. Numer. Math. 101(3), 415–421 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Braess, D., Carstensen, C., Hoppe, R.: Error reduction in adaptive finite element approximations of elliptic obstacle proplems. J. Comput. Math. 27, 148–169 (2009)

    MathSciNet  MATH  Google Scholar 

  19. Braess, D., Carstensen, C., Hoppe, R.H.: Convergence analysis of a conforming adaptive finite element method for an obstacle problem. Numer. Math. 107(3), 455–471 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Chen, Z., Nochetto, R.H.: Residual type a posteriori error estimates for elliptic obstacle problems. Numer. Math. 84(4), 527–548 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  21. Dörsek, P., Melenk, J.: Adaptive hp-FEM for the contact problem with Tresca friction in linear elasticity: The primal-dual formulation and a posteriori error estimation. Appl. Numer. Math. 60(7), 689–704 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Duvaut, G., Lions, J.L.: Inequalities in Mechanics and Physics. Grundlehren der mathematischen Wissenschaften. Springer, Berlin (1976)

    Google Scholar 

  23. Eck, C., Jarusek, J., Krbec, M.: Unilateral Contact Problems: Variational Methods and Existence Theorems. CRC Press, Boca Raton (2005)

    Book  MATH  Google Scholar 

  24. Frohne, H.: Finite Elemente Methoden höherer Ordnung für reibungsbehaftete, elasto-plastische Mehrkörperkontaktprobleme - Fehlerkontrolle, adaptive Methoden und effiziente Lösungsverfahren. Ph.D. thesis, Technische Universität Dortmund (2018)

  25. Große-Wöhrmann, A., Blum, H., Stiemer, M.: A posteriori control of modelling errors in linear elasticity. Proc. Appl. Math. Mech. 10, 647–648 (2010)

    Article  Google Scholar 

  26. Haslinger, J.: Mixed formulation of elliptic variational inequalities and its approximation. Appl. Math. 26, 462–475 (1981)

    MathSciNet  MATH  Google Scholar 

  27. Haslinger, J., Dostál, Z., Kučera, R.: On a splitting type algorithm for the numerical realization of contact problems with coulomb friction. Comput. Methods Appl. Mech. Eng. 191(21–22), 2261–2281 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  28. Haslinger, J., Sassi, T.: Mixed finite element approximation of 3D contact problems with given friction: error analysis and numerical realization. Math. Mod. Numer. Anal. 38, 563–578 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  29. Hauer, F.: Die elasto-plastische Einglättung rauer Oberflächen und ihr Einfluss auf die Reibung in der Umformtechnik. Ph.D. thesis, Friedrich-Alexander-Universität Erlangen-Nürnberg (2014)

  30. Hild, P., Nicaise, S.: A posteriori error estimations of residual type for Signorini’s problem. Numer. Math. 101(3), 523–549 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  31. Hintermüller, M., Ito, K., Kunisch, K.: The primal-dual active set strategy as a semi-smooth newton method. SIAM J. Optim. 13(3), 865–888 (2003)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  33. Hüeber, S.: Discretization techniques and efficient algorithms for contact problems. Ph.D. thesis, Universität Stuttgart (2008)

  34. Hüeber, S., Mair, M., Wohlmuth, B.: A priori error estimates and an inexact primal-dual active set strategy for linear and quadratic finite elements applied to multibody contact problems. Appl. Numer. Math. 54(3–4), 555–576 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  36. Johnson, K.: Contact Mechanics. Cambridge University Press, Cambridge (1985)

    Book  MATH  Google Scholar 

  37. Kikuchi, N., Oden, J.: Contact problems in elasticity: A study of variational inequalities and finite element methods. SIAM Studies in Applied Mathematics. SIAM, Society for Industrial and Applied Mathematics, Philadelphia (1988)

  38. Kleemann, H.: Adaptive FEM für Mehrkörperkontaktprobleme. Ph.D. thesis, Technische Universität Dortmund (2011)

  39. Kornhuber, R., Krause, R.: Adaptive multigrid methods for Signorini’s problem in linear elasticity. Comput. Vis. Sci. 4(1), 9–20 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  40. Krause, R., Veeser, A., Walloth, M.: An efficient and reliable residual-type a posteriori error estimator for the Signorini problem. Numer. Math. 130, 151–197 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  41. Mirabella, L., Nobile, F., Veneziani, A.: An a posteriori error estimator for model adaptivity in electrocardiology. Comp. Methods Appl. Mech. Eng. 200, 2727–2737 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  42. 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 

  43. Oden, J.T., Vemaganti, K.: Estimation of local modeling error and goal-oriented modeling of heterogeneous materials; Part I: Error estimates and adaptive algorithms. J. Comput. Phys. 164, 22–47 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  44. Oden, J.T., Vemaganti, K.: Estimation of local modeling error and goal-oriented modeling of heterogeneous materials; Part II: A computational environment for adaptive modeling of heterogeneous elastic solids. Comput. Methods Appl. Mech. Eng. 190, 3–25 (2001)

    Article  MATH  Google Scholar 

  45. Paraschivoiu, M., Peraire, J., Patera, A.T.: A posteriori finite element bounds for linear-functional outputs of elliptic partial differential equations. Comput. Methods Appl. Mech. Eng. 150(1–4), 289–312 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  46. Prudhomme, S., Oden, J.: On goal-oriented error estimation for elliptic problems: application to the control of pointwise errors. Comput. Methods Appl. Mech. Eng. 176(1–4), 313–331 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  47. Rademacher, A.: NCP function-based dual weighted residual error estimators for Signorini’s problem. SIAM J. Sci. Comput. 38, A1743–A1769 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  48. Rademacher, A., Schröder, A.: Dual weighted residual error control for frictional contact problems. Comput. Methods Appl. Math. 15, 391–413 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  49. Richter, T., Wick, T.: Variational localizations of the dual-weighted residual estimator. J. Comput. Appl. Math. 279, 192–208 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  50. Schmaltz, S., Landkammer, P., Beyer, F., Kumor, D., Rademacher, A., Blum, H., Steinmann, P., Willner, K.: Vorstellung eines simulationsbenchmarks für die blechmassivumformung. In: M. Merklein, B.A. Behrens, A.E. Tekkaya (eds.) 2. Workshop Blechmassivumformung, pp. 53–68. Meisenbach, Bamberg (2013)

  51. Schröder, A.: Error control in h- and hp-adaptive FEM for Signorini’s Problem. J. Numer. Math. 17(4), 299–318 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  52. Schröder, A., Blum, H., Rademacher, A., Kleemann, H.: Mixed FEM of higher order for contact Problems with friction. Int. J. Numer. Anal. Model. 8(2), 302–323 (2011)

    MathSciNet  MATH  Google Scholar 

  53. Schröder, A., Rademacher, A.: Goal-oriented error control in adaptive mixed FEM for Signorini’s Problem. Comput. Methods Appl. Mech. Eng. 200(1–4), 345–355 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  54. Shaw, M.C.: The role of friction in deformation processing. Wear 6, 140–158 (1963)

    Article  Google Scholar 

  55. Siebert, K., Veeser, A.: A unilaterally constrained quadratic minimization with adaptive finite elements. SIAM J. Optim. 18, 260–289 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  56. Stein, E., Ohnimus, S.: Anisotropic discretization- and model-error estimation in solid mechanics by local neumann problems. Comput. Methods Appl. Mech. Eng. 176, 363–385 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  57. Stein, E., Rüter, M., Ohnimus, S.: Implicit upper bound error estimates for combined expansive model and discretization adaptivity. Comput. Methods Appl. Mech. Eng. 200, 2626–2638 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  58. Suttmeier, F.: Numerical Solution of Variational Inequalities by Adaptive Finite Elements. Advances in Numerical Mathematics. Vieweg-Teubner, Wiesbaden (2008)

    Google Scholar 

  59. 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 

  60. Walloth, M.: Adaptive numerical simulation of contact problems: Resolving local effects at the contact boundary in space and time. Ph.D. thesis, Rheinischen Friedrich-Wilhelms-Universität Bonn (2012)

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

    Article  MathSciNet  MATH  Google Scholar 

  62. Wohlmuth, B.I.: An a posteriori error estimator for two-body contact problems on non-matching meshes. J. Sci. Comput. 33(1), 25–45 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  63. Wohlmuth, B.I.: Variationally consistent discretization schemes and numerical algorithms for contact problems. Acta Numerica 20, 569–734 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  64. Wohlmuth, B.I., Krause, R.H.: Monotone multigrid methods on nonmatching grids for nonlinear multibody contact problems. SIAM J. Sci. Comput. 25(1), 324–347 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  66. Wriggers, P.: Nonlinear Finite Element Methods. Springer, Berlin Heidelberg (2008)

    MATH  Google Scholar 

Download references

Acknowledgements

The author gratefully acknowledges the financial support by the German Research Foundation (DFG) within the subproject A5 of the transregional collaborative research centre (Transregio) 73 “Sheet-Bulk-Metal-Forming”.

Author information

Authors and Affiliations

  1. Chair of Scientific Computing, Technische Universität Dortmund, Vogelpothsweg 87, 44227, Dortmund, Germany

    Andreas Rademacher

Authors
  1. Andreas Rademacher
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Andreas Rademacher.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rademacher, A. Mesh and model adaptivity for frictional contact problems. Numer. Math. 142, 465–523 (2019). https://doi.org/10.1007/s00211-019-01044-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-019-01044-8

Mathematics Subject Classification