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Graph transformations for efficient structural analysis

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Abstract

In this paper, some graph theoretical (topological) transformations are presented for simplifying certain problems involved in structural analysis. For each case, the main problem is stated and the proposed topological transformation is established. Once the required topological analysis is completed, a back transformation results in the solution for the main problem. The transformations studied here employ (i) models drawn on a lower dimensional space, (ii) models embedded on higher dimensional spaces and (iii) interchange models which have simpler connectivity properties than the corresponding original structural models. All these transformations are illustrated utilizing simple examples.

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References

  1. Kaveh, A.: Structural Mechanics: Graph and Matrix Methods, 3rd edn. Research Studies Press (John Wiley), London (2004)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  3. Harary, F.: Graph Theory. Addison-Wesley, Reading (1969)

    Book  MATH  Google Scholar 

  4. Berge, C.: Graphs and Hypergraphs. North-Holland Publishing, Amsterdam (1973)

    MATH  Google Scholar 

  5. West, D.B.: Introduction to Graph Theory. Prentice-Hall, Upper Saddle River, NJ (1996)

    MATH  Google Scholar 

  6. Hammack, R., Imrich, W., Klavzar, S.: Handbook of Product Graphs, 2nd edn. CEC Press, Taylor & Francis Group, LLC, Boca Raton (2011)

    MATH  Google Scholar 

  7. Ziegler, F.: Mechanics of Solids and Fluids, 2nd edn. Springer, New York (1995)

    Book  MATH  Google Scholar 

  8. de Henderson, J.C.: Lecture note on structural analysis. Private communication (1970)

  9. Langefors, B.: Analysis of elastic structures by matrix transformation with special regard to semimonocoque structures. J. Aerosp. Sci. 19, 451–458 (1952)

    MATH  Google Scholar 

  10. Argyris, J.H., Kelsey, S.: Energy Theorems and Structural Analysis. Butterworth, London (1960)

    Book  MATH  Google Scholar 

  11. Samuelsson, A.G.: Linear analysis of frame structures by use of algebraic topology. Ph.D. thesis, Chalmer Tekniska Högskola, Göteborg (1962)

  12. Maunder, E.A.W.: Topological and linear analysis of skeletal structures. Ph.D. thesis, London University, Imperial College (1971)

  13. Kaveh, A.: Application of topology and matroid theory to the flexibility analysis of structures. Ph.D. thesis, London University, Imperial College of Science and Technology (1974)

  14. Kaveh, A.: Improved cycle bases for the flexibility analysis of structures. Comput. Methods Appl. Mech. Eng. 9, 267–272 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kaveh, A.: Recent development in the force method of structural analysis. Appl. Mech. Rev. 45, 401–418 (1992)

    Article  Google Scholar 

  16. Reid, J.K.: Large Sparse Sets of Linear Equations. Academic Press, London (1971)

    Google Scholar 

  17. Pissanetskey, S.: Sparse Matrix Technology. Academic Press, London (1984)

    Google Scholar 

  18. Duff, I.S., Erisman, A.M., Reid, J.K.: Direct Methods for Sparse Matrices. Oxford Science Publication, Clarendon Press, Oxford (1986)

    MATH  Google Scholar 

  19. Kaveh, A., Rahami, H.: Compound matrix block diagonalization for efficient solution of eigenproblems in structural matrices. Acta Mech. 188(3–4), 155–166 (2007)

    Article  MATH  Google Scholar 

  20. Kaveh, A., Rahami, H.: An efficient analysis of repetitive structures generated by graph products. Int. J. Numer. Methods Eng. 84(1), 108–126 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kaveh, A., Shojaie, I., Rahami, H.: New developments in the optimal analysis of regular and near-regular structures: decomposition, graph products, force method. Acta Mech. 226(3), 665–681 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Zingoni, A.: Group-theoretic insights on the vibration of symmetric structures in engineering. Philos. Trans. R. Soc. (A) 372, 20120037 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zingoni, A., Pavlovic, M.N., Zlokovic, G.M.: A symmetry-adapted flexibility approach for multi-storey space frames: general outline and symmetry-adapted redundants. Struct. Eng. Rev. 7, 107–119 (1995)

    Google Scholar 

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

    Article  Google Scholar 

  25. Zingoni, A.: Truss and beam finite elements revisited: a derivation based on displacement-field decomposition. Int. J. Space Struct. 11, 371–380 (1996)

    Article  Google Scholar 

  26. Kaveh, A.: Optimizing the conditioning of structural flexibility matrices. Comput. Struct. 41, 489–494 (1991)

    Article  MATH  Google Scholar 

  27. Kaveh, A., Ghaderi, I.: Conditioning of structural stiffness matrices. Comput. Struct. 63, 719–727 (1997)

    Article  Google Scholar 

  28. Kaveh, A.: Space structures and the crossing number of their graphs. Mech. Struct. Mach. 21, 151–166 (1993)

    Article  MathSciNet  Google Scholar 

  29. Euler, L.: Solutio problematic ad Geometrian situs pertinentis. Commun. Acad. Petropolitanae 8, 128–140 (1736) (Translated in: Speiser, Klassische Stücke der Mathematik, Zürich pp. 127–138 (1927))

  30. de Henderson, J.C.: Topological aspects of structural analysis. Aircr. Eng. 32, 137–141 (1960)

    Article  Google Scholar 

  31. de Henderson, J.C., Maunder, E.A.W.: A problem in applied topology. J. Inst. Math. Appl. 5, 245–269 (1969)

    Article  MATH  Google Scholar 

  32. Kaveh, A.: Computational Structural Analysis and Finite Element Methods. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-02964-1

    Book  MATH  Google Scholar 

  33. Russopoulos, A.I.: Theory of Elastic Complexes. Elsevier Publishing, Amsterdam (1965)

    Google Scholar 

  34. Moitinho de Almeida, J.P., Maunder, E.A.W.: Equilibrium Finite Element Formulations. CRC Press, Taylor & Francis Group, Broken Sound Parkway, NW (2017)

    Book  MATH  Google Scholar 

  35. Kaveh, A.: A combinatorial optimization problem; optimal generalized cycle bases. Comput. Methods Appl. Mech. Eng. 20, 39–52 (1979)

    Article  MATH  Google Scholar 

  36. Kaveh, A.: Statical bases for an efficient flexibility analysis of planar trusses. J. Mech. Struct. Mech. 14, 475–488 (1986)

    Article  Google Scholar 

  37. Kaveh, A., Roosta, G.R.: Comparative study of finite element nodal ordering methods. Eng. Struct. 20(1&2), 86–96 (1998)

    Article  Google Scholar 

  38. Laman, G.: On graphs and rigidity of plane skeletal structures. J. Eng. Math. 4, 331–340 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  39. Lovasz, L., Yemini, Y.: On generic rigidity in the plane. SIAM J. Discrete Methods 3, 91–98 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  40. Sugihara, K.: On some problems in the design of skeletal structures. SIAM J. Discrete Methods 4, 355–362 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  41. Bolker, E.D., Crapo, H.: How to brace a one storey building. Environ. Plan. B 4, 125–152 (1977)

    Article  Google Scholar 

  42. Kaveh, A.: Ordering for bandwidth reduction. Comput. Struct. 24, 413–420 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  43. Kaveh, A.: Bandwidth reduction of rectangular matrices. Commun. Numer. Methods Eng. 9, 259–267 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  44. Birn, M., Osipov, V., Sanders, P., Schulz, C., Sitchinava, N.: Efficient parallel and external matching. In: Euro-Par, vol. 8097 of LNCS, pp 659–670. Springer (2013)

  45. Kaveh, A., Rahimi Bondarabady, H.A.: A Hybrid Graph-Genetic Method for Domain Decomposition, pp. 127–134. Civil-Comp Press, Leuvan (2000)

    MATH  Google Scholar 

  46. Kaveh, A., Ghobadi, M.: A multi-stage algorithm for blood banking supply chain allocation problem. Int. J. Civ. Eng. 15, 103–112 (2017)

    Article  Google Scholar 

  47. Yavari, A., Kaveh, A., Sarkani, S., Rahimi Bondarabady, H.A.: Topological aspects of meshless methods and nodal ordering for meshless discretization. Int. J. Numer. Methods Eng. 52, 921–938 (2001)

    Article  MATH  Google Scholar 

  48. Koohestani, K.: On the analytical form-finding of tensegrities. Compos. Struct. 166, 114–119 (2017)

    Article  Google Scholar 

  49. Koohestani, K.: A computational framework for the form-finding and design of tensegrity structures. Mech. Res. Commun. 54, 41–49 (2013)

    Article  Google Scholar 

  50. Koohestani, K., Guest, S.D.: A new approach to the analytical and numerical form-finding of tensegrity structures. Int. J. Solids Struct. 50(19), 2995–3007 (2013)

    Article  Google Scholar 

  51. Kaveh, A., Daei, M.: Suboptimal cycle bases of graphs using an ant colony system algorithm. Eng. Comput. 27(4), 485–494 (2010)

    Article  MATH  Google Scholar 

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Correspondence to A. Kaveh.

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This paper is dedicated to the memory of Franz Ziegler

This paper is dedicated to late Professor Dr. F. Ziegler who had a great influence on my academic life. He encouraged me to write my first two international books, and with his excellent question, he attracted my attention to the conditioning of structural matrices, resulting in completion of a topic entitled: optimal analysis of structures. He will stay in my heart and my mind as a highly influential academician.

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Kaveh, A. Graph transformations for efficient structural analysis. Acta Mech 229, 659–675 (2018). https://doi.org/10.1007/s00707-017-1991-4

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  • DOI: https://doi.org/10.1007/s00707-017-1991-4

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