Profile and wavefront optimization by metaheuristic algorithms for efficient finite element analysis

Document Type : Article


1 Centre of Excellence for Fundamental Studies in Structural Engineering, Iran University of Science and Technology, Narmak, Tehran, P.O. Box 16846-13114, Iran

2 School of Civil Engineering, Iran University of Science and Technology, Narmak, Tehran, P.O. Box 16846-13114, Iran


For an efficient solution of the equations arising from finite element analysis, the stiffness matrix of the model should be structured. This can be done by reducing the profile or wavefront of the corresponding graph matrix of the structure depending on whether skyline or frontal method being used, respectively. One of the efficient methods to achieve this goal is the use of the method of King, extended by Sloan. In this paper the coefficients of the priority function utilized in the generalized Sloan’s method are optimized using the recently developed metaheuristic algorithm, so-called vibrating particles system. The results are compared to those of other metaheuristic algorithms consisting of the particle swarm optimization, colliding bodies optimization, enhanced colliding bodies optimization, and tug of war optimization. These metaheuristics, are used for optimum nodal numbering of the graph models of the finite element meshes to reduce the profile and wavefront of the corresponding sparse matrices. Comparison of the results achieved by these metaheuristic algorithms and those of the King and Sloan, demonstrates the efficiency of the new metaheuristic utilized for profile and wavefront optimization.


Main Subjects

1.Kaveh, A. Applications of topology and matroid theory to the analysis of structures", Ph.D. Thesis, Imperial College of Science and Technology, London University, UK (1974).
2. Kaveh, A., Structural Mechanics: Graph and Matrix Methods, Research Studies Press, 3rd edition, Somerset, UK (2004).
3. Kaveh, A., Optimal Structural Analysis, John Wiley, 2nd Edn., Chichester, UK (2006).
4. Papademetrious, C.H. The NP-completeness of bandwidth minimization problem", Comput. J., 16, pp. 177-192 (1976). 5. Gibbs, N.E., Poole, W.G., and Stockmeyer, P.K. An algorithm for reducing the bandwidth and pro_le of a sparse matrix", SIAM J. Numer. Anal., 12, pp. 236- 250 (1976). 6. Cuthill, E. and McKee, J. Reducing the bandwidth of sparse symmetric matrices", Proceedings of the 24th National Conference ACM, Bradon System Press, NJ, pp. 157-172 (1969). 7. Bernardes, J.A.B. and Oliveira, S.L.G.D. A systematic review of heuristics for pro_le reduction of symmetric matrices", Procd. Comput. Sci., 51, pp. 221-230 (2015). 8. King, I.P. An automatic reordering scheme for simultaneous equations derived from network systems", Int. J. Numer. Methods Eng., 2, pp. 523-533 (1970). 9. Kaveh, A. and Behzadi, A.M. An e_cient algorithm for nodal ordering of networks", Iran. J. Sci. Technol., Transactions in Civil Engineering, 11, pp. 11-18 (1987). 10. Kaveh, A. and Roosta, G.R. Comparative study of _nite element nodal ordering methods", Eng. J., 20(1&2), pp. 86-96 (1998). 11. Koohestani, B. and Poli, R. Addressing the envelope reduction of sparse matrices using a genetic programming system", Comput. Optimiz. Appl., 60, pp. 789- 814 (2014). 12. Kaveh, A., Advances in Metaheuristic Algorithms for Optimal Design of Structures, 2nd Edn., Springer International Publishing, Switzerland (2017). 13. Kaveh, A. and Mahdavi, V.R. Colliding bodies optimization: A novel meta-heuristic method", Comput. Struct., 139, pp. 18-27 (2014). 14. Kaveh, A. and Ilchi Ghazaan, M. Enhanced colliding bodies optimization for design problems with continuous and discrete variables", Adv. Eng. Softw., 77, pp. 66-75 (2014). 15. Kaveh, A. and Zolghadr, A. A novel meta-heuristic algorithm: tug of war optimization", Int. J. Optim. Civil Eng., 6, pp. 469-492 (2016). 16. Kaveh, A. and Ilchi Ghazaan, M. A new metaheuristic algorithm: vibrating particles system", Sci. Iran., 24(2), pp. 551-566 (2017). 17. Sloan, S.W. An algorithm for pro_le and wavefront reduction of sparse matrices", Int. J. Numer. Methods Eng., 23, pp. 1693-1704 (1986). 18. Kaveh, A. and Rahimi Bondarabdy, H.A. A hybrid method for _nite element ordering", Comput. Struct., 80(3-4), pp. 219-225 (2002). 19. Rahimi Bondarabady, H.A., and Kaveh, A. Nodal ordering using graph theory and a genetic algorithm", Finite Elem. Anal. Des., 40(9-10), pp. 1271-1280 (2004). 20. Kaveh, A. and Shara_, P. Optimal priority functions for pro_le reduction using ant colony optimization", Finite Elem. Anal. Des., 44, pp. 131-138 (2008). 21. Kaveh, A. and Shara_, P. Ordering for bandwidth and pro_le minimization problems via charged system search algorithm", Iran. J. Sci. Technol., Transactions in Civil Engineering, 36(C1), pp. 39-52 (2012). 22. Kaveh, A. and Bijari, Sh. Bandwidth, pro_le and wavefront optimization using PSO, CBO, ECBO and TWO algorithms", Iran. J. Sci. Technol., Transactions in Civil Engineering, 41(1), pp. 1-12 (2017). 23. Everstine, G.C. A comparison of three resequencing algorithms for the reduction of matrix pro_le and wavefront", Int. J. Numer. Methods Eng., 14(6), pp. 837-853 (1979).