References
1. Bathe K.-J. and Wilson, E.L., Numerical Methods
in Finite Element, p. 544, Englewood Cli s, N.J.:
Prentice-Hall (1976).
2. Livesley, R.K., Matrix Methods of Structural Analysis,
Pergamon International Library of Science, Technology,
Engineering and Social Studies, 2nd Ed., Elsevier
(1975).
3. Ye, C. and Wang, Y.Q. \On the use of Chebyshev
polynomials in the Rayleigh-Ritz method for vibration
and buckling analyses of circular cylindrical threedimensional
graphene foam shells", Mechanics Based
Design of Structures and Machines, pp. 1{15 (2019).
DOI: 10.1080/15397734.2019.1704776
4. Shahgholian-Ghahfarokhi, D., Safarpour, M., and
A. Kaveh and P. Rahmani/Scientia Iranica, Transactions A: Civil Engineering 28 (2021) 192{208 207
Rahimi, A. \Torsional buckling analyses of functionally
graded porous nanocomposite cylindrical shells
reinforced with graphene platelets (GPLs)", Mechanics
Based Design of Structures and Machines, pp. 1{22
(2019). DOI: 10.1080/15397734.2019.1666723
5. Choi, K.K., Haug, E.J., and Lam, H.L. \A numerical
method for distributed parameter structural optimization
problems with repeated eigenvalues", Journal of
Structural Mechanics, 10(2), pp. 191{207 (1982).
6. Kaveh, A., Optimal Analysis of Structures by Concepts
of Symmetry and Regularity, Springer (2013).
7. Curtis, A. and Reid, J.K. \The solution of large
sparse unsymmetric systems of linear equations", IMA
Journal of Applied Mathematics, 8(3), pp. 344{353
(1971).
8. Pissanetzky, S., Sparse Matrix Technology-Electronic
Edition, Academic Press (1984).
9. Dongarra, J., Foster, I., Fox, G., et al., Sourcebook
of Parallel Computing, Morgan Kaufmann Publishers
San Francisco (2003).
10. Fiedler, M. \Algebraic connectivity of graphs",
Czechoslovak Mathematical Journal, 23(2), pp. 298{
305 (1973).
11. Paulino, G.H., Menezes, I.F., Gattass, M., et al.
\Node and element resequencing using the Laplacian
of a nite element graph: part I-general concepts
and algorithm", International Journal for Numerical
Methods in Engineering, 37(9), pp. 1511{1530 (1994).
12. Biggs, N., Algebraic Graph Theory, Cambridge University
Press (1993).
13. Cvetkovic, D.M., Doob, M., and Sachs, H., Spectra of
Graphs, Academic Press, New York (1980).
14. Seidel, J., Graphs and Their Spectra, Banach Center
Publications, 1(25), pp. 147{162 (1989).
15. Chung, F.R. and Graham, F.C., Spectral Graph Theory,
American Mathematical Soc. (1997).
16. Godsil, C. and Royle, G.F., Algebraic Graph Theory,
Springer Science & Business Media (2013).
17. Chen, Y., Fan, L., Bai, Y., et al. \Assigning mountainvalley
fold lines of
at-foldable origami patterns based
on graph theory and mixed-integer linear programming",
Computers & Structures, 239, p. 106328
(2020).
18. Chen, Y. and Feng, J. \Group-theoretic exploitations
of symmetry in novel prestressed structures", Symmetry,
10(6), p. 229 (2018).
19. Kangwai, R., Guest, S., and Pellegrino, S. \An introduction
to the analysis of symmetric structures",
Computers & Structures, 71(6), pp. 671{688 (1999).
20. Kangwai, R. and Guest, S. \Symmetry-adapted equilibrium
matrices", International Journal of Solids and
Structures, 37(11), pp. 1525{1548 (2000).
21. Harth, P. and Michelberger, P. \Determination of
loads in quasi-symmetric structure with symmetry
components", Engineering Structures, 123, pp. 395{
407 (2016).
22. Zhang, J.Y., Guest, S.D., Connelly, R., et al. \Dihedral
'star' tensegrity structures", International Journal of
Solids and Structures, 47(1), pp. 1{9 (2010).
23. Chen, Y., Sareh, P., Feng, J., et al. \A computational
method for automated detection of engineering structures
with cyclic symmetries", Computers & Structures,
191, pp. 153{164 (2017).
24. Zingoni, A. \On the best choice of symmetry group
for group-theoretic computational schemes in solid and
structural mechanics", Computers & Structures, 223,
p. 106101 (2019).
25. Chen, Y. and Feng, J. \Improved symmetry method
for the mobility of regular structures using graph
products", Journal of Structural Engineering, 142(9),
p. 04016051 (2016).
26. Chen, Y., Yan, J., Sareh, P., et al. \Feasible prestress
modes for cable-strut structures with multiple
self-stress states using particle swarm optimization",
Journal of Computing in Civil Engineering, 34(3), p.
04020003 (2020).
27. Zingoni, A. \Insights on the vibration characteristics
of double-layer cable nets of D4h symmetry", International
Journal of Solids and Structures, 135, pp. 261{
273 (2018).
28. Zingoni, A. \Group-theoretic vibration analysis of
double-layer cable nets of D4h symmetry", International
Journal of Solids and Structures, 176{177, pp.
68{85 (2019).
29. Plgnataro, M. and Rizzi, N. \The e ect of multiple
buckling modes on the postbuckling behavior of plane
elastic frames. Part I. Symmetric frames", Journal of
Structural Mechanics, 10(4), pp. 437{458 (1982).
30. Kaveh, A. and Sayarinejad, M. \Graph symmetry and
dynamic systems", Computers & Structures, 82(23{
26), pp. 2229{2240 (2004).
31. Kaveh, A. and Rahami, H. \An ecient method for
decomposition of regular structures using graph products",
International Journal for Numerical Methods in
Engineering, 61(11), pp. 1797{1808 (2004).
32. Kaveh, A. and Rahami, H. \Block diagonalization of
adjacency and Laplacian matrices for graph product;
applications in structural mechanics", International
Journal for Numerical Methods in Engineering, 68(1),
pp. 33{63 (2006).
33. Imrich, W., Hammack, R., and Klavzar, S., Handbook
of Product Graphs, CRC Press (2011).
34. Thomas, D.L. \Dynamics of rotationally periodic
structures", International Journal for Numerical
Methods in Engineering, 14(1), pp. 81{102 (1979).
35. Williams, F. \An algorithm for exact eigenvalue calculations
for rotationally periodic structures", International
Journal for Numerical Methods in Engineering,
23(4), pp. 609{622 (1986).
208 A. Kaveh and P. Rahmani/Scientia Iranica, Transactions A: Civil Engineering 28 (2021) 192{208
36. Williams, F. \Exact eigenvalue calculations for structures
with rotationally periodic substructures", International
Journal for Numerical Methods in Engineering,
23(4), pp. 695{706 (1986).
37. Kaveh, A. and Nemati, F. \Eigensolution of rotationally
repetitive space structures using a canonical
form", International Journal for Numerical Methods in
Biomedical Engineering, 26(12), pp. 1781{1796 (2010).
38. Chang, Y.-W. and Yang, C.-B. \A parallel algorithm
for circulant tridiagonal linear systems", Information
Processing Letters, 65(6), pp. 331{337 (1998).
39. Kaveh, A., Pishghadam, M., and Jafarvand, A.
\Topology optimization of repetitive near-regular shell
structures using preconditioned conjugate gradients
method", Mechanics Based Design of Structures and
Machines, pp. 1{22 (2020).
DOI: 10.1080/15397734.2020.1755682
40. Kaveh, A. and Sayarinejad, M. \Eigenvalues of factorable
matrices with form IV symmetry", Communications
in Numerical Methods in Engineering, 21(6),
pp. 269{287 (2005).
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