A New Class of Vector Padé Approximants in the Asymptotic Numerical Method: Application in Nonlinear 2D Elasticity


The Asymptotic Numerical Method (ANM) is a family of algorithms for path following problems, where each step is based on the computation of truncated vector series [1]. The Vector Padé approximants were introduced in the ANM to improve the domain of validity of vector series and to reduce the number of steps needed to obtain the entire solution path [1,2]. In this paper and in the framework of the ANM, we define and build a new type of Vector Padé approximant from a truncated vector series by extending the definition of the Padé approximant of a scalar series without any orthonormalization procedure. By this way, we define a new class of Vector Padé approximants which can be used to extend the domain of validity in the ANM algorithms. There is a connection between this type of Vector Padé approximant and Vector Padé type approximant introduced in [3, 4]. We show also that the Vector Padé approximants introduced in the previous works [1,2], are special cases of this class. Applications in 2D nonlinear elasticity are presented.

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Hamdaoui, A. , Hihi, R. , Braikat, B. , Tounsi, N. and Damil, N. (2014) A New Class of Vector Padé Approximants in the Asymptotic Numerical Method: Application in Nonlinear 2D Elasticity. World Journal of Mechanics, 4, 44-53. doi: 10.4236/wjm.2014.42006.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] B. Cochelin, N. Damil and M. Potier-Ferry, “Méthode Asymptotique Numerique,” Hermes Science, Paris, 2007.
[2] B. Cochelin, N. Damil and M. Potier-Ferry, “Asymptotic Numerical Method and Padé Approximants for Nonlinear Elastic Structures,” International Journal for Numerical Methods in Engineering, Vol. 37, 1994, pp. 1187-1213.
[3] J. Van Iseghem, “Vector Padé Approximants, in Numerical Mathematics and Applications,” North Holland, Amsterdam, 1985, pp. 73-77.
[4] C. Brezinski, “Comparisons between Vector and Matrix Padé Approximants,” Journal of Nonlinear Mathematical Physics, Vol. 10, Suppl. 2, 2003, pp. 1-12.
[5] H. Padé, “Sur la Représentation Approchée D’une Fonction par des Fractions Rationnelles,” Annales de l’Ecole Normale Supérieur, Vol. 9, 1892, pp. 3-93.
[6] M. Van-Dyke, “Computed-Extended Series,” Annual Review in Fluid Mechanics, Vol. 16, 1984, pp. 287-309.
[7] C. Brezinski and V. Iseghem, “Padé Approximants,” In: P.G. Ciarlet and J. L. Lions, Eds., Handbook of Numerical Analysis, Vol. 3, North-Holland, Amsterdam, 1994.
[8] G. A. Backer Jr. and P. Graves Morris, “Padé Approximants,” Encyclopedia of Mathematics and Its Application, Vol. 2, Cambridge University Press, Cambridge, 1996.
[9] H. De Boer and F. Van Keulen, “Padé Approximants Applied to a Non-Linear Finite Element Solution Strategy,” Communications in Numerical Methods in Engineering, Vol. 13, No. 7, 1997, pp. 593-602.
[10] A. El Hage-Hussein, M. Potier-Ferry and N. Damil, “A Numerical Continuation Method Based on Padé Approximants,” International Journal of Solids and Structures, Vol. 37, No. 46-47, 2000, pp. 6981-7001.
[11] J. L. Batoz and G. Dhatt, “Modélisation des Structures par Elément Finis,” Edition Hermès, Paris, Vol. 1, 1990.

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