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International Journal of Engineering Mathematics
Volume 2017 (2017), Article ID 2783682, 21 pages
https://doi.org/10.1155/2017/2783682
Research Article

A New Iterative Numerical Continuation Technique for Approximating the Solutions of Scalar Nonlinear Equations

IFSTTAR, Aix-Marseille Université, LBA UMR T24, 13016 Marseille, France

Correspondence should be addressed to Grégory Antoni

Received 30 June 2016; Accepted 24 October 2016; Published 16 January 2017

Academic Editor: Yurong Liu

Copyright © 2017 Grégory Antoni. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. T. Belytschko, W. K. Liu, and B. Moran, Nonlinear Finite Elements for Continua and Structures, John Wiley & Sons, 2000. View at MathSciNet
  2. A. Curnier, Méthodes Numériques en Mécanique des Solides, Presses Polytechniques et Universitaires Romandes, 2000.
  3. M. Bonnet and A. Frangi, Analyse des Solides Déformables par la Méthode des Éléments Finis, Editions de l’Ecole Polytechnique, Paris, France, 2007.
  4. J. Besson, G. Cailletaud, J.-L. Chaboche, and S. Forest, Non-Linear Mechanics of Materials, vol. 167 of Solid Mechanics and Its Applications, Springer, Berlin, Germany, 2010.
  5. R. De Borst, M. A. Crisfield, J. C. Remmers, and C. V. Verhoosel, Nonlinear Finite Element Analysis of Solids and Structures, John Wiley & Sons, 2012.
  6. C. T. Kelley, Solving Nonlinear Equations with Newton's Method. Number 1 in Fundamental Algorithms for Numerical Calculations, SIAM, Philadelphia, Pa, USA, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  7. A. Quarteroni, R. Sacco, and F. Saleri, Méthodes Numériques pour le Calcul Scientifique: Programmes en MATLAB, Springer, Berlin, Germany, 2000.
  8. P. Deuflhard, Newton Methods for Nonlinear Problems, vol. 35 of Computational Mathematics, Springer, 2005.
  9. G. Antoni, “A new accurate and efficient iterative numerical method for solving the scalar and vector nonlinear equations: approach based on geometric considerations,” International Journal of Engineering Mathematics, vol. 2016, Article ID 6390367, 18 pages, 2016. View at Publisher · View at Google Scholar
  10. G. Antoni, “An efficient and straightforward numerical technique coupled to classical Newton’s method for enhancing the accuracy of approximate solutions associated with scalar nonlinear equations,” International Journal of Engineering Mathematics, vol. 2016, Article ID 8565821, 12 pages, 2016. View at Publisher · View at Google Scholar
  11. Q. S. Nguyen, Stability and Nonlinear Solid Mechanics, John Wiley & Sons, New York, NY, USA, 2000.
  12. M. A. Crisfield, “A fast incremental/iterative solution procedure that handles ‘snap-through’,” Computers & Structures, vol. 13, no. 1–3, pp. 55–62, 1981. View at Publisher · View at Google Scholar · View at Scopus
  13. M. A. Crisfield, “An arc-length method including line searches and accelerations,” International Journal for Numerical Methods in Engineering, vol. 19, no. 9, pp. 1269–1289, 1983. View at Publisher · View at Google Scholar
  14. E. Riks, “Application of newton's method to the problem of elastic stability,” Journal of Applied Mechanics, vol. 39, no. 4, pp. 1060–1065, 1972. View at Publisher · View at Google Scholar · View at Scopus
  15. E. Riks, “An incremental approach to the solution of snapping and buckling problems,” International Journal of Solids and Structures, vol. 15, no. 7, pp. 529–551, 1979. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. E. Riks, “Some computational aspects of the stability analysis of nonlinear structures,” Computer Methods in Applied Mechanics and Engineering, vol. 47, no. 3, pp. 219–259, 1984. View at Publisher · View at Google Scholar · View at Scopus
  17. E. Riks, “On formulations of path-following techniques for structural stability analysis,” in New Advances in Computational Structural Mechanics, pp. 65–79, Elsevier, Amsterdam, The Netherlands, 1992. View at Google Scholar
  18. E. Ramm, “Strategies for tracing the nonlinear response near limit points,” in Nonlinear Finite Element Analysis in Structural Mechanics, pp. 63–89, 1981. View at Google Scholar
  19. E. Ramm, “The Riks/Wempner approach—an extension of the dis-placement control method in nonlinear analyses,” in Recent Advances in Non-linear Computational Mechanics, E. Hinton, D. R. J. Owen, and C. Taylor, Eds., chapter 3, pp. 63–86, Pineridge, Swansea, UK, 1982. View at Google Scholar
  20. G. A. Wempner, “Discrete approximations related to nonlinear theories of solids,” International Journal of Solids and Structures, vol. 7, no. 11, pp. 1581–1599, 1971. View at Publisher · View at Google Scholar · View at Scopus
  21. S. A. Ragon, Z. Gürdal, and L. T. Watson, “A comparison of three algorithms for tracing nonlinear equilibrium paths of structural systems,” International Journal of Solids and Structures, vol. 39, no. 3, pp. 689–698, 2002. View at Publisher · View at Google Scholar · View at Scopus
  22. R. Tabatabaei, H. Saffari, and M. J. Fadaee, “Application of normal flow algorithm in modal adaptive Pushover analysis,” Journal of Constructional Steel Research, vol. 65, no. 1, pp. 89–96, 2009. View at Publisher · View at Google Scholar · View at Scopus