Skip to main content
SpringerLink
Log in

Improved group theoretic method using graph products for the analysis of symmetric-regular structures

  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

In this paper, a new combined graph-group method is proposed for eigensolution of special graphs. Symmetric regular graphs are the subject of this study. Many structural models can be viewed as the product of two or three simple graphs. Such models are called regular, and usually have symmetric configurations. The proposed method of this paper performs the symmetry analysis of the entire structure via symmetric properties of its simple generators. Here, a graph is considered as the general model of an arbitrary structure. The Laplacian matrix, as one of the most important matrices associated with a graph, is studied in this paper. The characteristic problem of this matrix is investigated using symmetry analysis via group theory enriched by graph theory. The method is developed and decomposition of the Laplacian matrix of such graphs is studied in a step-by-step manner, based on the proposed method. This method focuses on simple paths which generate large networks, and finds the eigenvalues of the network via analysis of the simple generators. Group theory is the main tool, which is improved using the concept of graph products. As a mechanical application of the method, a benchmark problem of group theory in structural mechanics is studied in this paper. Vibration of cable nets is analyzed and the frequencies of the networks are calculated using the combined graph-group method.

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

Similar content being viewed by others

References

  1. Kaveh A., Sayarinejad M.A.: Eigensolutions for matrices of special patterns. Commun. Numer. Methods Eng. 19, 125–136 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  2. Kaveh A., Sayarinejad M.A.: Eigenvalues of factorable matrices with form IV symmetry. Commun. Numer. Methods Eng. 21, 269–278 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  3. Kaveh A., Sayarinejad M.A.: Graph symmetry in dynamic systems. Comput. Struct. 82(23–26), 2229–2240 (2004)

    Article  MathSciNet  Google Scholar 

  4. Thomas D.L.: Dynamics of rotationally periodic structures. Int. J. Numer. Methods Eng. 14, 81–102 (1979)

    Article  MATH  Google Scholar 

  5. Williams F.W.: An algorithm for exact eigenvalue calculations for rotationally periodic structures. Int. J. Numer. Methods Eng. 23, 609–622 (1986)

    Article  MATH  Google Scholar 

  6. Williams F.W.: Exact eigenvalue calculations for rotationally periodic sub-structures. Int. J. Numer. Methods Eng. 23, 695–706 (1986)

    Article  MATH  Google Scholar 

  7. Hasan M.A., Hasan J.A.K.: Block eigenvalue decomposition using nth roots of identity matrix. 41st IEEE Conference on Decision and Control 2, 2119–2124 (2002)

    MathSciNet  Google Scholar 

  8. Aghayere, A.O.: Structural systems with polar symmetry: solution by quasi-circulant matrices. M.Sc. thesis, Massachusetts Institute of Technology, March 1983

  9. Kaveh A., Rahami H.: Block diagonalization of adjacency and Laplacian matrices for graph product; applications in structural mechanics. Int. J. Numer. Methods Eng. 68, 33–63 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. Kaveh, A., Nemati, F.: Eigensolution of rotationally repetitive space structures using a canonical form. Commun. Numer. Meth. Eng. (2009, in press). doi:10.1002/cnm.1265

  11. Zloković G.M.: Group theory and G-vector spaces in engineering structures, vibration, stability and statics. Ellis Horwood Limited, Chichester (1989)

    Google Scholar 

  12. Zingoni A.: An efficient computational scheme for the vibration analysis of high-tension cable nets. J. Sound Vib. 189, 55–79 (1996)

    Article  Google Scholar 

  13. Zingoni A.: Group-theoretical applications in solid and structural mechanics: a review, chapter 12. In: Topping, B.H.V., Bittnar, Z. (eds) Computational Structures Technology, Saxe-Coburg Publications, Stirlingshire (2002)

    Google Scholar 

  14. Zingoni A.: On the symmetries and vibration modes of layered space grids. Eng. Struct. 27, 629–638 (2005)

    Article  Google Scholar 

  15. Zingoni A.: On group-theoretic computation of natural frequencies for spring-mass dynamic systems with rectilinear motion. Commun. Numer. Methods Eng. 24, 973–987 (2008)

    MATH  MathSciNet  Google Scholar 

  16. Kaveh A., Nikbakht M.: Block diagonalization of Laplacian matrices of symmetric graphs via group theory. Int. J. Numer. Methods Eng. 69, 908–947 (2007)

    Article  MathSciNet  Google Scholar 

  17. Kaveh, A., Nikbakht, M.: Improved group-theoretical method for eigenvalue problems of special symmetric structures, using graph theory. Adv. Eng. Softw. (2009, in press). doi:10.1016/j.advengsoft.2008.12.003

  18. Kaveh A., Nikbakht M.: Buckling load of symmetric plane frames using canonical forms and group theory. Acta Mech. 185, 89–128 (2006)

    Article  MATH  Google Scholar 

  19. Kaveh A., Nikbakht M.: Stability analysis of hyper symmetric skeletal structures using group theory. Acta Mech. 200, 177–197 (2008)

    Article  MATH  Google Scholar 

  20. Kaveh A., Nikbakht M.: Decomposition of symmetric mass–spring vibrating systems using groups, graphs, and linear algebra. Commun Numer. Methods Eng. 23, 639–664 (2006)

    Article  MathSciNet  Google Scholar 

  21. Kaveh, A., Nikbakht, M.: Applications of group and graph products in configuration processing of symmetric structures. In: The Ninth International Conference on Computational Structures Technology, Athens, Greece, 2–5 September 2008

  22. Kaveh A., Nikbakht M.: Improved group–theoretical method for eigenvalue problems of special symmetric structures, using graph theory. In: Topping, B.H.V., Montero, G., Montenegro, R. (eds) Proceedings of the Eighth International Conference on Computational Structures Technology, Civil-Comp Press, Stirlingshire, Scotland (2006)

    Google Scholar 

  23. Kaveh A., Rahami H.: Topology and graph products; eigenproblems in optimal structural analysis. Commun. Numer. Methods Eng. 24, 929–945 (2008)

    MATH  MathSciNet  Google Scholar 

  24. Kaveh A., Rahami H.: Tri-diagonal and penta-diagonal block matrices for efficient eigensolutions of problems in structural mechanics. Acta Mech. 192, 77–87 (2007)

    Article  MATH  Google Scholar 

  25. Kaveh A., Rahami H.: Factorization for efficient solution of eigenproblems of adjacency and Laplacian matrices for graph products. Int. J. Numer. Methods Eng. 1(75), 58–82 (2008)

    Article  MathSciNet  Google Scholar 

  26. Kaveh A., Rahami H.: An efficient method for decomposition of regular structures using graph products. Int. J. Numer. Methods Eng. 61, 1797–1808 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  27. Kangwai R.D., Guest S.D., Pellegrino S.: An introduction to the analysis of symmetric structures. Comput. Struct. 71, 671–688 (1999)

    Article  MathSciNet  Google Scholar 

  28. Albert Cotton F.: Chemical Applications of Group Theory. Wiley, New York (1990)

    Google Scholar 

  29. Armstrong M.A.: Groups and Symmetry. Springer, Berlin (1988)

    MATH  Google Scholar 

  30. Imrich, W., Klavzar, S.: Product Graphs: Structure and Recognition, Wiley Interscience Series in Discrete Mathematics, New York (2000)

  31. Sabidussi G.: Graph multiplication. Math. Z. 72, 446–457 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  32. Kaveh A.: Optimal Structural Analysis, Research Studies Press, 2nd edn. Wiley, London (2006)

    Google Scholar 

  33. Weichsel P.M.: The Kronecker product of graphs. Proc. Am. Math. Soc. 13, 47–52 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  34. Healey T.J., Treacy J.A.: Exact block diagonalization of large eigenvalue problems for structures with symmetry. Int. J. Numer. Methods Eng. 31, 265–285 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  35. Pothen A., Simon H., Liou K.P.: Partitioning sparse matrices with eigenvectors of graphs. SIAM J. Matrix Anal. Appl. 11, 430–452 (1990)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Civil Engineering, Centre of Excellence for Fundamental Studies in Structural Engineering, Iran University of Science and Technology, Narmak, Tehran-16, Iran

    A. Kaveh & M. Nikbakht

  2. Faculty of Engineering, University of Tehran, Tehran, Iran

    H. Rahami

Authors
  1. A. Kaveh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Nikbakht
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. H. Rahami
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Kaveh.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kaveh, A., Nikbakht, M. & Rahami, H. Improved group theoretic method using graph products for the analysis of symmetric-regular structures. Acta Mech 210, 265–289 (2010). https://doi.org/10.1007/s00707-009-0204-1

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-009-0204-1

Keywords

Search

Navigation