Skip to main content

On the numerical solution of contact problems

Part of the Lecture Notes in Mathematics book series (LNM,volume 878)

Abstract

We consider finite element discretizations of variational problems which correspond to quasilinear elliptic boundary value problems with linear constraints. A modified block-relaxation method and a preconditioned conjugate gradient algorithm are presented which generalize known methods for bound-constraints to more general restrictions. Global convergence proofs are given and an application to the contact problem for two membranes.

Keywords

  • Contact Problem
  • Finite Element Discretizations
  • Convergence Proof
  • Preconditioned Conjugate Gradient Method
  • Descent Property

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   44.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   59.00
Price excludes VAT (Canada)
  • Compact, lightweight 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Glowinski, R., Lions, J. L., Tremolieres, R., Approximations des inéquations variationelles. Paris, Dunod 1976

    Google Scholar 

  2. Kikuchi, N., Oden, J. T., Contact porblems in elasticity, TICOM report 79-8, University of Texas at Austin 1979

    Google Scholar 

  3. McCormick, G. P., Anti-zig-zagging by bending, Manag. Sci. 15, 315–320 (1969)

    CrossRef  Google Scholar 

  4. Mittelmann, H. D., On the approximate solution of nonlinear variational inequalities, Numer. Math. 29, 451–462 (1978)

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. Mittelmann, H. D., On the efficient solution of nonlinear finite element equations I, to appear in Numer. Math.

    Google Scholar 

  6. Mittelmann, H. D., On the efficient solution of nonlinear finite element equations II, submitted to Numer. Math.

    Google Scholar 

  7. Oden, J. T., Kikuchi, N., Finite element methods for certain free boundary value problems in mechanics, in "Moving boundary problems", D. G. Wilson, A. D. Solomon, P.T. Boggs (eds.), Academic Press, New York 1978

    Google Scholar 

  8. Oetli, W., Einzelschrittverfahren zur Lösung konvexer und dualkonvexer Minimierungsprobleme, Z. Angew. Math. Mech. 54, 334–351 (1974)

    Google Scholar 

  9. O'Leary, D. P., Conjugate gradient algorithms in the solution of optimization problems for nonlinear elliptic partial differential equations. Computing 22, 59–77 (1979)

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. Ortega, J. M., Rheinboldt, W. C., Iterative solution of nonlinear equations in several variables. Academic Press, New York 1970

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1981 Springer-Verlag

About this paper

Cite this paper

Mittelmann, H.D. (1981). On the numerical solution of contact problems. In: Allgower, E.L., Glashoff, K., Peitgen, HO. (eds) Numerical Solution of Nonlinear Equations. Lecture Notes in Mathematics, vol 878. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0090685

Download citation

  • DOI: https://doi.org/10.1007/BFb0090685

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-10871-9

  • Online ISBN: 978-3-540-38781-7

  • eBook Packages: Springer Book Archive