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Ordering for Optimal Patterns of Structural Matrices: Graph Theory Methods

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Computational Structural Analysis and Finite Element Methods

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Abstract

In this chapter, methods are presented for ordering to form special patterns for sparse structural matrices. Such transformation reduces the storage and the number of operations required for the solution, and leads to more accurate results. Graph theory methods are presented for different approaches to reordering equations to preserve their sparsity, leading to predefined patterns. Alternative, objective functions are considered and heuristic algorithms are presented to achieve these objectives. Three main methods for the solution of structural equations require the optimisation of bandwidth, profile and frontwidth, especially for those encountered in finite element analysis. Methods are presented for reducing the bandwidth of the flexibility matrices. Bandwidth optimisation of rectangular matrices is presented for its use in the formation of sparse flexibility matrices.

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References

  1. Rosen R (1968) Matrix bandwidth minimization. In: Proceedings of the 23rd national conference of the ACM. Brandon System Press, pp 585–595

    Google Scholar 

  2. Grooms HR (1972) Algorithm for matrix bandwidth reduction. J Struct Div ASCE 98:203–214

    Google Scholar 

  3. Hall M Jr (1956) An algorithm for distinct representatives. Amer Math Mon 63:716–717

    Article  MATH  Google Scholar 

  4. Cuthill E, McKee J (1969) Reducing the bandwidth of sparse symmetric matrices. In: Proceedings of the 24th national conference of the ACM. Brandon System Press, pp 157–172

    Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Kaveh A (1977) Topological study of the bandwidth reduction of structural matrices. J Sci Tech 1:27–36

    Google Scholar 

  8. Gibbs NE, Poole WG, Stockmeyer PK (1976) An algorithm for reducing the bandwidth and profile of a sparse matrix. SIAM J Numer Anal 12:236–250

    Article  MathSciNet  Google Scholar 

  9. Sloan SW (1989) A Fortran program for profile and wavefront reduction. Int J Numer Methods Eng 28:2651–2679

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Kaveh A (2004) Structural mechanics: graph and matrix methods, 3rd edn. Research Studies Press, Baldock

    Google Scholar 

  12. Grimes RG, Pierce DJ, Simon HD (1990) A new algorithm for finding a pseudo-peripheral node in a graph. SIAM J Anal Appl 11:323–334

    Article  MathSciNet  MATH  Google Scholar 

  13. Paulino GH, Menezes IFM, Gattass M, Mukherjee S (1994) Node and element resequencing using the Laplacian of a finite element graph: part I-general concepts and algorithms. Int J Numer Methods Eng 37:1511–1530

    Article  MATH  Google Scholar 

  14. Papademetrious CH (1976) The NP-completeness of bandwidth minimization problem. Comput J 16:177–192

    Google Scholar 

  15. Cheng KY (1973) Note on minimizing the bandwidth of sparse symmetric matrices. Comput J 11:27–30

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  17. Kaveh A, Rahimi Bondarabady HA (2001) Spectral trisection of finite element models. Int J Numer Methods Heat Fluid Flow 11:358–370

    Article  MATH  Google Scholar 

  18. Kaveh A (1991) A connectivity coordinate system for node and element ordering. Comput Struct 41:1217–1223

    Article  MathSciNet  MATH  Google Scholar 

  19. Everstine GC (1979) A comparison of three resequencing algorithms for the reduction of profile and wavefront. Int J Numer Methods Eng 14:837–853

    Article  MATH  Google Scholar 

  20. Razzaque A (1980) Automatic reduction of frontwidth for finite element analysis. Int J Numer Methods Eng 15:1315–1324

    Article  MATH  Google Scholar 

  21. Sloan SW (1986) An algorithm for profile and wavefront reduction of sparse matrices. Int J Numer Methods Eng 23:1693–1704

    Article  MathSciNet  Google Scholar 

  22. Bykut A (1977) A note on an element ordering scheme. Int J Numer Methods Eng 11:194–198

    Article  Google Scholar 

  23. Kaveh A (1984) A note on a two-step approach to finite element ordering. Int J Numer Methods Eng 19:1753–1754

    Google Scholar 

  24. Fenves SJ, Law KH (1983) A two-step approach to finite element ordering. Int J Numer Methods Eng 19:891–911

    Article  MATH  Google Scholar 

  25. Kaveh A, Ramachandran K (1984) Graph theoretical approach for bandwidth and frontwidth reductions. In: Nooshin H (ed) Proceedings of the 3rd international conference on space structures .Surrey University, pp 244–249

    Google Scholar 

  26. Livesley RK, Sabin MA (1991) Algorithms for numbering the nodes of finite element meshes. Comput Syst Eng 2:103–114

    Article  Google Scholar 

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

    Article  Google Scholar 

  28. Kaveh A, Roosta GR (1995) An efficient method for finite element nodal ordering. Asian J Struct Eng 1:229–242

    Google Scholar 

  29. Jennings A (1966) A compact storage scheme for the solution of symmetric linear simultaneous equations. Comput J 9:281–285

    Article  MATH  Google Scholar 

  30. Felippa CA (1975) Solution of linear equations with skyline-stored symmetric matrix. Comput Struct 5:13–29

    Article  MATH  Google Scholar 

  31. George A (1977) Solution of linear system of equations; direct methods for finite element problems. In: Dold A, Eckmann E (eds) Lecture notes in mathematics, vol 572. Springer-Verlag, pp 52–101

    Google Scholar 

  32. Liu WH, Sherman, AH (1975) Comparative analysis of the Cuthill-McKee and Reverse Cuthill-McKee ordering algorithms for sparse matrices. Department of Computer Science, Yale University, New Haven, Report 28

    Google Scholar 

  33. Souza LT, Murray DW (1995) A unified set of resequencing algorithms. Int J Numer Methods Eng 38:565–581

    Article  MathSciNet  MATH  Google Scholar 

  34. Irons BM (1970) A frontal solution program for finite element analysis. Int J Numer Methods Eng 2:5–32

    Article  MATH  Google Scholar 

  35. Kaveh A (2013) Optimal structural analysis by concepts of symmetry and regularity. Springer Verlag, GmbH, Wien-New York

    Book  Google Scholar 

  36. Kaneko I, Lawo M, Thierauf G (1982) On computational procedures for the force methods. Int J Numer Methods Eng 18:1469–1495

    Article  MathSciNet  MATH  Google Scholar 

  37. Behzad M (1970) A characterization of total graphs. Proc Amer Math Soc 26(1970):383–389

    Article  MathSciNet  MATH  Google Scholar 

  38. Kaveh A, Mokhtar-zadeh A (1993) A comparative study of the combinatorial and algebraic force methods. In: Proceedings of the Civil-Comp93, Edinburgh, pp 21–30

    Google Scholar 

  39. Parter SV (1961) The use of linear graphs in Gauss elimination. SIAM Rev 3:119–130

    Article  MathSciNet  MATH  Google Scholar 

  40. Tinney WF (1969) Comments on using sparsity technique for power systems problem. In: Willoughby RA (ed) Report No. RA1 (11707), IBM, pp 25–34

    Google Scholar 

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Authors and Affiliations

  1. Centre of Excellence for Fundamental Studies in Structural Engineering School of Civil Engineering, Iran University of Science and Technology, Tehran, Iran

    A. Kaveh

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  1. A. Kaveh
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Kaveh, A. (2014). Ordering for Optimal Patterns of Structural Matrices: Graph Theory Methods. In: Computational Structural Analysis and Finite Element Methods. Springer, Cham. https://doi.org/10.1007/978-3-319-02964-1_4

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  • DOI: https://doi.org/10.1007/978-3-319-02964-1_4

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-02963-4

  • Online ISBN: 978-3-319-02964-1

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