Skip to main content
SpringerLink
Log in

Coupled Problems and Their Numerical Solution

  • Chapter
Advances in Computational Nonlinear Mechanics

Part of the book series: International Centre for Mechanical Sciences ((CISM,volume 300))

Abstract

As discussed in a previous publication [1] coupled problems in numerical analysis context can be divided into two main categories:

  1. I)

    Those, where essentially different physics exist and in which the problem domain overlap completely or partially

  2. II)

    and those where coupling occurs only via an interface. Here the seperate domains may only differ either physically or only in the numerical technique used in each of them.

Some characteristics of the finite element discretization and steady state and transient solution technique are discussed and typical problems are illustrated. Special emphasis is given to some recent developments in transient dynamic problems including

  1. 1)

    Symmetrical direct solution techniques.

  2. 2)

    Simple Staggered solutions with unconditional stability.

  3. 3)

    Domain partition for single phase domain with Implicit—Implicit Staggered or Explict—Implicit Staggered Solution.

Stability analysis are also presented for the various solution techniques.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • 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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Zienkiewicz, O.C.: Coupled Problems and their Numerical Solution, Chapter 1 in: Numerical Methods in Coupled Systems (Eds. R.W. Lewis, P. Bettess and E. Hinton ), John Wiley and Sons Ltd, 1984, pp. 35–58.

    Google Scholar 

  2. Zienkiewicz, O.C., Onate, E. and Heinrich, J.C.:. Plastic flow in metal forming. 1. Coupled thermal behaviour in extrusion. 2. Thin sheet forming. in: Proceedings of the Winter Annual Meeting of the American Society of Mechanical Engineers on ‘Applications of Numerical Methods to Forming Processes’, San Francisco, Dec., 1978, ASME, New York, 1979, pp. 107–120.

    Google Scholar 

  3. Zienkiewicz, O.C., Onate, E. and Heinrich, J.C.: A general formulation for coupled thermal flow of metals using finite elements., Int. J. Num. Meth. Engrg., 17 (1981), 1497–1514.

    Article  MATH  Google Scholar 

  4. Zienkiewicz, O.C., Chang, C.T. and Bettess, P.: Drained, undrained, consolidating and dynamic behaviour assumptions in soil. Limits of validity. Geotechnique, 30 (1980), 385–395.

    Article  Google Scholar 

  5. Zienkiewicz, O.C., Chang, C.T., Hinton, E. and Leung, S.H.: Earth dam analysis for earthquakes. Numerical solution and constitutive relations for nonlinear (damage) analysis. Paper presented to the Conference on Design of Dams to Resist Earthquakes, Institution of Civil Engineers, London, 1–2 October 1980.

    Google Scholar 

  6. Zienkiewicz, O.C.: The coupled problems of soil-pore fluid-external fluid interaction: Basis for a general geomechanics code. in Proceedings of the Fifth International Conference on Numerical Methods in Geomechanics, Nagoya. 1–5 Apr., 1985, 1731–1740.

    Google Scholar 

  7. Zienkiewicz, O.C.:The Finite Element Method, third Edn., McGraw-Hill, New York, 1977.

    MATH  Google Scholar 

  8. Zienkiewicz, O.C. and Bettess, P.: Fluid-structure dynamic interaction and wave forces. An introduction to numerical treatment. Int. J. Num. Meth. Engrg., 13 (1978), 1–16.

    Article  MATH  Google Scholar 

  9. Zienkiewicz, O.C. and Newton, R.E.: Coupled vibrations of a structure submerged in a compressible fluid. Paper presented to the International Symposium on Finite Element Techniques, Stuttgart, 1–15 May 1969.

    Google Scholar 

  10. Gupta, B.S.: On a numerical solution of the supersonic panel flutter eigenproblem. Int. J. Num. Meth. Engrg.. 10 (1976), 637–645.

    Google Scholar 

  11. Ohayon, R. and Valid, R.: True symmetric formulation of free vibrations of fluid-structure interaction–applications and extensions. In Proceedings of the Conference on Numerical Methods for Coupled Problems (Eds. E. Hinton et al ), Pineridge Press, Swansea, 1981, 335–345.

    Google Scholar 

  12. Ohayon, R.: Symmetric variational formuations for harmonic vibration problems by coupling primal and dual variables–applications to fluid-structure coupled systems. La Recherche Aerospatiale, 3 (1979), 69–77.

    MathSciNet  Google Scholar 

  13. Zienkiewicz, O.C., Wood, W.L., Hine, N.W. and Taylor, R.L.: A unified set of sinle step algorithms Part I: General formuation and applications, Int. J. Num. Meth. Engrg., 20 (1984), 1529–1552.

    Google Scholar 

  14. Lambert, J.D.: Computational Methods in Ordinary Differential Equations. John Wiley and Sons Ltd., 1973.

    Google Scholar 

  15. Gantmacher, F.R.: Applications of the theoy of matrices, Second part of the ‘A theory of matrices’, translated and revised by J.L. Brenner et al, Interscience Publishers, Inc., 1959.

    Google Scholar 

  16. Paul, D.B.: Efficient Dynamic Solutions for single and coupled multiple field problems. Ph.D. Thesis C/Ph/64/82, University College of Swansea, 1982.

    Google Scholar 

  17. Zienkiewicz, O.C. and Taylor, R.L.: Coupled Problems–A simple time-stepping procedure. Comm. Appl. Num. Meth., 1 (1985), 233–239.

    Google Scholar 

  18. Katone, M.G. and Zienkiewicz, O.C.: A unified set of single step algorithms Part 3: The Beta-m method, a generalization of the Newmark scheme. Int. J. Num. Meth. Engrg., 21 (1985), 1345–1359.

    Article  Google Scholar 

  19. Felippa, C.A. and Park, S.C.: Staggered transient analysis procedures for coupled mechanics systems: formulation. Comp. Meth. Appl. Mech. Engrg., 24 (1980), 61–111.

    Article  MATH  Google Scholar 

  20. Park, B.C.: Partitioned transient analysis procedures for coupled field problems: stability analysis. J. Appl. Mech., 47 (1980), 370–376.

    Article  MATH  Google Scholar 

  21. Park, B.C. and Felippa, C.A.: Partitioned transient analysis Procedures for coupled field problems: accuracy analysis. J. Appl. Mech., 47 (1980), 919–926.

    Article  MATH  Google Scholar 

  22. Park, B.C.: Stablization of partitioned solution procedure for pore fluid-soil interaction analysis. Int. J. Num. Meth. Engrg., 19 (1983), 1669–1673.

    Article  MATH  Google Scholar 

  23. Leung, B.H.: Earthquake response of saturated soils and liquefaction. Ph.D. Thesis C/Ph/79/84, University College of Swansea, 1984.

    Google Scholar 

  24. Zienkiewicz, O.C., Hinton, E., Leung, B.H. and Taylor, B.L.: Staggered, Time Marching Schemes in Dynamic Soil Analysis and Selective Explicit Extrapolation Algorithms. in Procedings of Conference on Innovative Numerical Analysis for the Engineering Sciences (Eds. R. Shaw et al), University of Virginia Press, 1980.

    Google Scholar 

  25. Zienkiewicz, O.C., Paul, D.B. and Chan, A.H.C.: Unconditionally stable staggered solution procedure for soil-pore fluid interaction problems. Research Report, C/R/586/87, Aug., 1987, University College of Swansea, submitted to Comm. Appl. Num. Meth..

    Google Scholar 

  26. Paul, D.B. and.Zienkiewicz, O.C.: Earthquake and post earthquake response validity of liquefiable soils using centrifuge test. Research report, C/R/585/87, Aug., 1987, University College of Swansea.

    Google Scholar 

  27. Zienkiewicz, O.C., Paul, D.B. and Chan, A.H.C.: Numerical solution of saturated flat sand bed response when subjected to earthquake using centrifuge model test. Research Report, C/R/587/87, Aug., 1987, University College of Swansea.

    Google Scholar 

  28. Lewis, R.W. and Gerner, B.W.: A finite element solution of coupled electrokinetic and hydrodynamic flow in porous media. Int. J. Num. Meth. Engrg., 5 (1972), 41–56.

    Article  MATH  Google Scholar 

  29. Marshall, R.S., Heinrich, J.C. and Zienkiewicz, O.C.: Natural convection in a square enclosure by a finite element penalty function method using primitive fluid variables. J. Num. Meth. Heat Transfer, 1 (1978), 315–330.

    Article  Google Scholar 

  30. Venter, K.V.: KVV03 data report: Revised data report of a centrifuge model test and two triazial tests. Andrew N. Schofield and Associate, Oct., 1985.

    Google Scholar 

  31. Chan, A.H.C.: A unified Finite Element Solution to Static and Dynamic problems of Geomechanics. Ph.D. Thesis, University College of Swansea, in preparation, 1988.

    Google Scholar 

  32. Zienkiewicz, O.C., Kelly, D.W. and Bettess, P.: Marriage a la mode — the best of both worlds (finite elements and boundary integrals. Paper presented to the International Symposium on Innovative Numerical Analysis in Applied Engineering Science, Versailles, May 1977. Also In Energy Methods in Finite Element Analysis (Eds. R. Glowinski, E.Y. Rodin and O.C. Zienkiewicz ), John Wiley and Sons Ltd., Chichester, 1979, 81–106.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University College of Swansea, Swansea, UK

    O. C. Zienkiewicz & A. H. C. Chan

Authors
  1. O. C. Zienkiewicz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. H. C. Chan
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. University of Stuttgart, Germany

    I. St. Doltsinis

Rights and permissions

Reprints and permissions

Copyright information

© 1989 Springer-Verlag Wien

About this chapter

Cite this chapter

Zienkiewicz, O.C., Chan, A.H.C. (1989). Coupled Problems and Their Numerical Solution. In: Doltsinis, I.S. (eds) Advances in Computational Nonlinear Mechanics. International Centre for Mechanical Sciences, vol 300. Springer, Vienna. https://doi.org/10.1007/978-3-7091-2828-2_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-7091-2828-2_4

  • Publisher Name: Springer, Vienna

  • Print ISBN: 978-3-211-82113-8

  • Online ISBN: 978-3-7091-2828-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation