Skip to main content
SpringerLink
Log in

Simple iterative method for solving problems for plates with partial internal supports

  • Published:
Journal of Engineering Mathematics Aims and scope Submit manuscript

Abstract

We consider problems for rectangular plates on one and on two partial internal supports with two opposite edges simply supported and two other edges clamped. The problems are modeled by the biharmonic equation with strongly mixed boundary conditions in the sense that there is a change of types of boundary conditions on a side of the rectangle. These boundary value problems lead to sequences of problems for the Poisson equation with strongly mixed boundary conditions. Finally, in solving these problems a version of the domain decomposition method is applied. The convergence of the obtained approximate solution is proved. Several numerical experiments show the fast convergence of the method. The efficiency of the proposed method is clear from the comparison of it with the dual series equations method used by other authors in the case of a plate with one internal support.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

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
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20

Similar content being viewed by others

References

  1. Li Z-C, Lu TT (2000) Singularities and treatments of elliptic boundary value problems. Math Comput Model 31:97–145

    Article  MATH  Google Scholar 

  2. Yao WA, Hu XF (2011) A novel singular finite element of mixed-mode crack problems with arbitrary crack tractions. Mech Res Commun 38:170–175

    Article  MATH  Google Scholar 

  3. Elliotis M, Georgiou G, Xenophontos C (2005) Solution of the planar Newtonian stick-slip problem with the singular function boundary intergral method. Int J Numer Mech Fluids 48:1001–1021

    Article  MATH  MathSciNet  Google Scholar 

  4. Elliotis M, Georgiou G, Xenophontos C (2006) The singular function boundary integral method for a two-dimensional fracture problem. Eng Anal Bound Elem 30:100–106

    Article  MATH  Google Scholar 

  5. Elliotis M, Georgiou G, Xenophontos C (2007) The singular function boundary intergral method for biharmonic problems with crack singularities. Eng Anal Bound Elem 31:209–215

    Article  MATH  Google Scholar 

  6. Li ZC, Lu TT, Hu HY (2004) The collocation Trefftz method for biharmonic equations with crack singularities. Eng Anal Bound Elem 28:79–96

    Article  MATH  Google Scholar 

  7. Christodoulou E, Elliotis M, Georgiou G, Xenophontos C (2012) Analysis of the singular function boundary integral method for a biharmonic problem with one boundary singularity. Numer Methods Partial Differ Equat 28:749–767

    Article  MathSciNet  Google Scholar 

  8. Yang WH (1968) On an integral equation solution for a plate with internal support. Q J Mech Appl Math 21(4):503–515

    Article  MATH  Google Scholar 

  9. Guminiak M, Sygulski R (2007) The analysis of internally supported thin plates by the boundary element method. Part 1 Static analysis. Found Civil Environ Eng 9:17–41

    Google Scholar 

  10. Guminiak M, Sygulski R (2007) The analysis of internally supported thin plates by the boundary element method. Part 2 Free vibration analysis. Found Civil Environ Eng 9:43–74

    Google Scholar 

  11. Katsikadelis JT, Sapountzakis EJ, Zorba EG (1990) A BEM approach to static and dynamic analysis of plates with internal supports. Comput Mech 7:31–40

    Article  Google Scholar 

  12. Pawlak Z, Guminiak M (2008) The application of fundamental solutions in static analysis of thin plates resting on the internal elastic support. Found Civil Environ Eng 11:67–96

    Google Scholar 

  13. Wei GW, Zhao YB, Xiang Y (2002) Discrete singular convolution and its application to the analysis of plates with internal supports. Part 1: theory and algorithm. Int J Numer Methods Eng 55:913–946

    Article  MATH  MathSciNet  Google Scholar 

  14. Xiang Y, Zhao YB, Wei GW (2002) Discrete singular convolution and its application to the analysis of plates with internal supports. Part 2: applications. Int J Numer Methods Eng 55:947–971

    Article  MATH  MathSciNet  Google Scholar 

  15. Zhao YB, Wei GW, Xiang Y (2002) Plate vibration under irregular internal supports. Int J Solids Struct 39:1361–1383

    Article  MATH  Google Scholar 

  16. Civalek O (2007) Three-dimensional vibration, buckling and bending analyses of thick rectangular plates based on discrete singular convolution method. Int J Mech Sci 49:752–765

    Article  Google Scholar 

  17. Sompornjaroensuk Y, Kiattikomol K (2007) Dual-series equations formulation for static deformation of plates with a partial internal line support. Theoret Appl Mech 34:221–248

    Article  ADS  MATH  MathSciNet  Google Scholar 

  18. Sompornjaroensuk Y, Kiattikomol K (2008) Exact analytical solutions for bending of rectangular plates with a partial internal line support. J Eng Math 62:261–276

    Article  MATH  MathSciNet  Google Scholar 

  19. Dang QA (1994) Boundary operator method for approximate solution of biharmonic type equation. Vietnam J Math 22(1–2):114–120

    MATH  Google Scholar 

  20. Dang QA (1998) Mixed boundary-domain operator in approximate solution of biharmonic type equation. Vietnam J Math 26(3):243–252

    MATH  MathSciNet  Google Scholar 

  21. Dang QA (2006) Iterative method for solving the Neumann boundary value problem for biharmonic type equation. J Comput Appl Math 96:634–643

    Article  MathSciNet  Google Scholar 

  22. Dang QA, Le TS (2007) Iterative method for solving a mixed boundary value problem for biharmonic type equation. In: Chuong NM et al (eds) Advances in deterministic and stochastic analysis. World Scientific, Singapore, pp 103–113

  23. Dang QA, Le TS (2009) Iterative method for solving a problem with mixed boundary conditions for biharmonic equation. Adv Appl Math Mech 1:683–698

    MATH  MathSciNet  Google Scholar 

  24. Dang QA, Truong HH, Vu VQ (2012) Iterative method for a biharmonic problem with crack. Appl Math Sci 6:3095–3108

    MATH  MathSciNet  Google Scholar 

  25. Samarskii AA (2001) The theory of difference schemes. Marcel Dekker, New York

    Book  MATH  Google Scholar 

  26. Dang QA, Quang VuV (2006) Domain decomposition method for strongly mixed boundary value problems. Vietnam J Comput Sci Cybern 22:307–318 (in Vietnamese)

    Google Scholar 

  27. Dang QA, Vu VQ (2012) A domain decomposition method for strongly mixed boundary value problems for the Poisson equation. In : Modeling, simulation and optimization of complex processes. Proceedings of the 4th international conference on HPSC, 2009, Hanoi, Vietnam, Springer, pp 65–76

  28. Quarteroni A, Valli A (1994) Numerical approximation of partial differential equations. Springer, Berlin

    MATH  Google Scholar 

  29. Samarskii A, Nikolaev E (1989) Numerical methods for grid equations, v. 1: direct methods. Birkhäuser, Basel

    Book  Google Scholar 

  30. Chinnaboon B, Chucheepsakul S, Katsikadelis JT (2007) A BEM-based meshless method for buckling analysis of elastic plates with various boundary conditions. Int J Struct Stab Dyn 7(1):81–89

    Article  MATH  MathSciNet  Google Scholar 

  31. Katsikadelis JT, Yiotis AJ (2003) The BEM for plates of variable thickness on nonlinear biparametric elastic foundation. An analog equation solution. J Eng Math 46(3–4):313–330

    Article  MATH  Google Scholar 

  32. Civalek O (2004) Application of differential quadrature (DQ) and harmonic differential quadrature (HDQ) for buckling analysis of thin isotropic plates and elastic columns. Eng Struct 26:171–186

    Article  Google Scholar 

Download references

Acknowledgments

This work is supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant 102.99-2011.24. The authors would like to sincerely thank the anonymous reviewers for their helpful comments and remarks, which improved the original manuscript.

Author information

Authors and Affiliations

  1. Institute of Information Technology, 18 Hoang Quoc Viet, Hanoi, Vietnam

    Quang A. Dang

  2. Thainguyen University of Information and Communication Technology, Thainguyen, Vietnam

    Ha Hai Truong

Authors
  1. Quang A. Dang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ha Hai Truong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Quang A. Dang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dang, Q.A., Truong, H.H. Simple iterative method for solving problems for plates with partial internal supports. J Eng Math 86, 139–155 (2014). https://doi.org/10.1007/s10665-013-9652-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10665-013-9652-7

Keywords

Search

Navigation