Vectorial Padé approximants in the Asymptotic Numerical Method

Authors

  • Noureddine Damil Laboratoire de Calcul Scientifique en Mécanique, Faculté des Sciences Ben M’Sik Université Hassan II - Mohammedia Avenue Cdt Driss El Harti, BP 7955, Casablanca, MAROC
  • Rédouane Jamai Laboratoire de Calcul Scientifique en Mécanique, Faculté des Sciences Ben M’Sik Université Hassan II - Mohammedia Avenue Cdt Driss El Harti, BP 7955, Casablanca, MAROC
  • Hassane Lahmam Laboratoire de Calcul Scientifique en Mécanique, Faculté des Sciences Ben M’Sik Université Hassan II - Mohammedia Avenue Cdt Driss El Harti, BP 7955, Casablanca, MAROC

Keywords:

vectorial Padé approximants, asymptotic numerical method, quadratic approximants, non-linear elastic shells

Abstract

In this paper, we present and discuss some techniques to define vectorial Padé approximants and quadratic approximants in the Asymptotic Numerical Method (ANM). For this purpose we have to orthonormalize the basis generated by the ANM. We shall discuss the influence of the orthonormalization procedure. We give some numerical comparisons of these techniques on non-linear elastic shells problems.

Downloads

Download data is not yet available.

References

[AZR 92] AZRAR L., COCHELIN B., DAMIL N., POTIER-FERRY., An asymptotic-numerical

method to compute bifurcating branches, In: Ladevèze, P., Zienkiewicz, O.C. (Eds), New

Advances in Computational Structural Mechanics. Elsevier, Amstredam, P. 117–131,

[BAK 96] BAKER G. A. JR., GRAVES MORRIS P., Padé approximants, Encyclopedia of

Mathematics and its application, VOL. 59, 2ND EDITION, 1996.

[BRA 95] BRAIKAT B., Méthode asymptotique numérique et fortes non linéarités, Thèse

de 3ème cycle, Université Hassan II - Mohammedia, Faculté des Sciences Ben M’Sik,

Casablanca, Maroc, 1995.

[BRA 97] BRAIKAT B., DAMIL N., POTIER-FERRY., Méthodes asymptotiques numériques

pour la plasticité, Revue Européenne des Eléments Finis, VOL. 6, P. 337–357, 1997.

[BRE 94] BREZINSKI C., ISEGHEM V., Padé approximants, In: Ciarlet, P.G., Lions, J.L.

(Eds), Handbook of Numerical Analysis, VOL. 3, NORTH-HOLLAND, AMSTERDAM,

[CAD 01] CADOU J.M., MOUSTAGHFIR N., MALLIL E., DAMIL N., POTIER-FERRY., Linear

iterative solvers based on perturbation method, Comptes Rendus de l’Académie des

Sciences, Paris, VOL. 329, SÉRIE II B, P. 457–462, 2001.

[CHA 97] CHARI R., Analyse non linéaire des structures en treillis par la M.E.F. et influence

de la procédure d’orthogonalisation et du produit scalaire sur les approximants de Padé,

Thèse de 3ème cycle, Université Hassan II - Mohammedia, Faculté des Sciences Ben M’Sik,

Casablanca, Maroc, 1997.

[COC 94A] COCHELIN B., DAMIL N., POTIER-FERRY., Asymptotic-numerical methods and

Padé approximants for non-linear elastic structures, International Journal for Numerical

Method in Engineering, VOL. 37, P. 1187–1213, 1994.

[COC 94B] COCHELIN B., DAMIL N., POTIER-FERRY M., The asymptotic-numerical

method: an efficient perturbation technique for non linear structural mechanics, Revue

Européenne des Eléments Finis, VOL. 3 (2), P. 281–297, 1994.

[COC 94C] COCHELIN B., A path following technique via an asymptotic numerical method,

Computers and Structures, VOL. 53, P. 1181–1192, 1994.

[DAM 90] DAMIL N., POTIER-FERRY., A new method to compute perturbed bifurcation,

Application to the buckling of imperfect elastic structures, International Journal of Engineering

Sciences, VOL. 28, P. 943–957, 1990.

[DAM 99] DAMIL N., POTIER-FERRY M., NAJAH A., CHARI R., LAHMAM H., An iterative

method based upon Padé approximants, Communications in Numerical Methods in

Engineering, VOL. 15, P. 701–708, 1999.

[DEB 97] DE BOER H., VAN KEULEN F., Padé approximants applied to a non-linear finite

element solution strategy, Communications in Numerical Methods in Engineering,

VOL. 13, P. 593–602, 1997.

[ELH 98] ELHAGE-HUSSEIN A., DAMIL N., POTIER-FERRY M., An asymptotic numerical

algorithm for frictionless contact problems, Revue Européenne des Eléments Finis, VOL. 7,

P. 119–130, 1998.

[ELH 00] ELHAGE-HUSSEIN A., POTIER-FERRY M., DAMIL N., A numerical continuation

method based on Padé approximants, International Journal of Solids and Structures,

VOL. 37, P. 6981–7001, 2000.

[HOF 89] HOFFMANN W., Iterative algorithm for Gram-Schmidt orthogonalization, Computing,

VOL. 41, P. 335–348, 1989.

[JAM 01] JAMAI R., DAMIL N., Quadratic approximants in the asymptotic numerical

method, Comptes Rendus de l’Académie des Sciences, Paris, VOL. 329, SÉRIE II B,

P. 809–814, 2001.

[JAMA 02] JAMAL M., BRAIKAT B., BOUTMIR S., DAMIL N., POTIER-FERRY M., A

high order implicit algorithm for solving nonlinear problems, Computational Mechanics,

VOL. 28, P. 375–390, 2002.

[JAM 02] JAMAI R., Contribution des approximants de Padé et des approximants quadratiques

dans la Méthode Asymptotique Numérique: Application en calcul des structures,

Doctorat, Université Hassan II - Mohammedia, Faculté des Sciences Ben M’Sik,

Casablanca, Maroc, 2002.

[JAM 03] JAMAI R., DAMIL N., Influence of iterated Gram-Schmidt orthonormalization

in the asymptotic numerical method, Comptes Rendus Mécanique, VOL. 331, ISSUE 5,

P. 351–356, 2003.

[LAH 02] LAHMAM H., CADOU J.M., ZAHROUNI H., DAMIL N., POTIER-FERRY M.,

High-order predictor-corrector algorithms, International Journal for Numerical Method in

Engineering, VOL. 55, P. 685–704, 2002.

[LIN 00] LINGEN F. J., Efficient Gram-Schmidt orthonormalization on parallel computers,

Communications in Numerical Methods in Engineering, VOL. 16, P. 57–66, 2000.

[MAL 99] MALLIL E., Développement d’une méthode itérative basée sur les séries et les

approximants de Padé pour le calcul non linéaire des structures, Doctorat, Université

Hassan II - Mohammedia, Faculté des Sciences Ben M’Sik, Casablanca, Maroc, 1999.

[MAL 00] MALLIL E., LAHMAM H., DAMIL N., POTIER-FERRY M., An iterative process

based on homotopy and perturbation techniques, Computer Methods in Applied Mechanics

and Engineering, VOL. 190, P. 1845–1858, 2000.

[NAJ 96] NAJAH A., Calcul non linéaire des structures par des méthodes asymptotiques

numériques et accéleration de la convergence, Thèse de 3ème cycle, Université Hassan II -

Mohammedia, Faculté des Sciences Ben M’Sik, Casablanca, Maroc, 1996.

[NAJ 98] NAJAH A., COCHELIN B., DAMIL N., POTIER-FERRY M., A critical review

of asymptotic numerical methods, Archives of Computational Methods in Engineering,

VOL. 5, N◦ 1, P. 31–50, 1998.

[PAD 92] PADÉ H., Sur la représentation approchée d’une fonction par des fractions rationnelles,

Annales de l’Ecole Normale Supérieure, série 3, VOL. 9, P. 3–93, 1892.

[SHA 74] SHAFER R.E., On quadratic approximation, SIAM J. Numer. Anal., VOL. 11 (2),

P. 447–460, 1974.

[TRI 96] TRI A., COCHELIN B., POTIER-FERRY M., Résolution des équations de Navier-

Stokes et détection des bifurcations stationnaires par une méthode asymptotique numérique,

Revue Européenne des Eléments Finis, VOL. 5 (4), P. 415–442, 1996.

[VAN 84] VAN-DYKE M., Computed-extended series, Annual Review in Fluid Mechanics,

VOL. 16, P. 287–309, 1984.

[ZAH 98] ZAHROUNI H., ELASMAR H., DAMIL N., POTIER-FERRY M., Asymptotic numerical

method for non-linear constitutive laws, Revue Européenne des Eléments Finis,

VOL. 7 (7), P. 841–869, 1998.

Downloads

Published

2004-06-11

How to Cite

Damil, N., Jamai, R. ., & Lahmam, H. . (2004). Vectorial Padé approximants in the Asymptotic Numerical Method. European Journal of Computational Mechanics, 13(1-2), 33–56. Retrieved from https://journals.riverpublishers.com/index.php/EJCM/article/view/2365

Issue

Section

Original Article