Skip to main content

From inexact active set strategies to nonlinear multigrid methods

  • Chapter
Analysis and Simulation of Contact Problems

Part of the book series: Lecture Notes in Applied and Computational Mechanics ((LNACM,volume 27))

Abstract

Due to their efficiency and robustness, linear multigrid methods lend themselves to be a starting point for the development of nonlinear iterative strategies for the solution of nonlinear contact problems, see, e.g., [3, 1, 10, 5]. One nonlinear strategy is to reduce the contact problem to a sequence of linear problems and to solve each of these by a linear multigrid method. This approach is often connected to active set strategies or semismooth Newton methods [6]. To avoid solving the linear problems exactly, one can use inexact active set strategies, see [7, 8]. The convergence of this inexact strategy depends on the accuracy the inner problem is solved with, see [7], as well as on algorithmic parameters [8]. A second strategy is to deal directly with the nonlinearity within the multigrid method by using, e.g., nonlinear smoothers and nonlinear interpolation operators, see [10, 9, 1]. Using the convex energy for controlling the iteration process, globally convergent nonlinear multigrid methods can be constructed which allow for solving contact problems with the speed of a linear multigrid method [10]. A third possibility is to employ a saddle point approach [3] and to solve for the primal and dual variables simultaneously using an algebraic multigrid method.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. A. Brandt and C.W. Cryer, Multigrid algorithms for the solution of linear complementarity problems arising from free boundary problems SIAM J. Sci. Stat. Comput., 4: 655–684, 1983.

    Article  MATH  MathSciNet  Google Scholar 

  2. P. Averyand G. Rebel and M. Lesoinne and C. Farhat, A numerically scalable dual-primal substructuring method for the solution of contact problems. I: The frictionless case, Comput. Methods Appl. Mech. Eng., 193:2403–2426, 2004.

    Article  Google Scholar 

  3. M. Adams, Algebraic methods for constrained linear systems with applications to contact problems in solid mechanics, Num. Lin. Alg. w Appl., 1–6, 2000

    Google Scholar 

  4. R. Kornhuber and H. Yserentant, Multilevel Methods for Elliptic Problems on Domains not Resolved by the Coarse Grid, Contemp. Math., 180:49–60, 1994

    MATH  MathSciNet  Google Scholar 

  5. J. Schöberl, Solving the Signorini problem on the basis of domain decomposition techniques, Computing, 60:323–344, 1998

    MATH  MathSciNet  Google Scholar 

  6. K. Kunisch and G. Stadler, Generalized Newton methods for the 2D-Signorini contact problem with friction in function space, M2AN, 4:827–854, 2005

    Article  MATH  MathSciNet  Google Scholar 

  7. W. Hackbusch and H.D. Mittelmann, On multi–grid methods for variational inequalities, Numer. Math., 42:65–76, 1983

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  9. Ana H. Iontcheva and Panayot S. Vassilevsky, Monotone multigrid methods based on element agglomeration coarsening away from the contact boundary for the Signorini’s problem, Num. Lin. Alg. w Appl., 11:189–204, 2004

    Article  Google Scholar 

  10. R. Kornhuber and R. H. Krause, Adaptive multigrid methods for Signorini’s problem in linear elasticity, CVS, 4:9–20, 2001, Springer

    MATH  MathSciNet  Google Scholar 

  11. Tony F. Chan and Jinchao Xu and Ludmil Zikatanov, An Agglomeration Multigrid Method for Unstructured Grids, Contemp. Math., 218:67–81, 1998

    MATH  MathSciNet  Google Scholar 

  12. R. H. Krause, Monotone Multigrid Methods for Signorini’s Problem with Friction, Freie Universität Berlin, 2001

    Google Scholar 

  13. M. Hintermüller and K. Ito and K. Kunisch, The Primal-Dual Active Set Strategy as a Semismooth Newton Method, SIAM J. Optim., 13:865–888, 2003

    Article  MATH  Google Scholar 

  14. R.H.W. Hoppe and R. Kornhuber, Adaptive multilevel–methods for obstacle problems, SIAM J. Numer. Anal., 31:301–323, 1994

    Article  MATH  MathSciNet  Google Scholar 

  15. P. Bastian and K. Birken and K. Johannsen and S. Lang and N. Neuß and H. Rentz–Reichert and C. Wieners, UG – a flexible Software toolbox for solving partial differential equations, CVS, 1:27–40, 1997, Springer

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer

About this chapter

Cite this chapter

Krause, R. (2006). From inexact active set strategies to nonlinear multigrid methods. In: Wriggers, P., Nackenhorst, U. (eds) Analysis and Simulation of Contact Problems. Lecture Notes in Applied and Computational Mechanics, vol 27. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-31761-9_2

Download citation

  • DOI: https://doi.org/10.1007/3-540-31761-9_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-31760-9

  • Online ISBN: 978-3-540-31761-6

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics