Comptes Rendus
A secure version of asymptotic numerical method via convergence acceleration
Comptes Rendus. Mécanique, Volume 348 (2020) no. 5, pp. 361-374.

The paper deals with the numerical computation of difficult problems requiring many steps and many degrees of freedom such as the finite element analysis of wrinkling of film–substrate systems. The asymptotic numerical method (ANM) is well adapted to such computations but with a progressive loss of accuracy during step chaining. Thus, correction phases are necessary, which are rarely carried out within ANM. A convergence acceleration algorithm and a step-length adaptation have been included to limit the growth of computation time and to strengthen the reliability of the procedure. This modified version of the ANM is assessed by simulating the appearance and evolution of sinusoidal wrinkles under uniaxial compression.

L’article étudie le calcul numérique de problèmes difficiles nécessitant de nombreux pas et un grand nombre de degrés de liberté, comme l’étude par éléments finis du plissement de systèmes film–substrat. La méthode asymptotique numérique (ANM) est bien adaptée à de tels calculs, mais avec une lente perte de précision lors de l’enchainement de nombreux pas de calculs. Donc des étapes de correction sont indispensables, ce qui est rarement fait lors de calculs par ANM. L’algorithme de cheminement est complété par une méthode d’accélération de convergence et par une technique d’adaptation de la longueur de pas pour minimiser le temps de calcul et pour fiabiliser la procédure. Cette version modifiée de la méthode asymptotique numérique a été évaluée par la simulation de l’apparition de plis sinusoïdaux en compression uniaxiale.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/crmeca.48
Keywords: Asymptotic numerical method, Convergence acceleration, Path-following technique, Wrinkling, Film–substrate systems
Mot clés : Méthode asymptotique numérique, Accélération de convergence, Algorithme de cheminement, Plissement, Systèmes film–substrat

Pascal Ventura 1; Michel Potier-Ferry 1; Hamid Zahrouni 1

1 Université de Lorraine, CNRS, Arts et Métiers ParisTech, LEM3, F-57000 Metz, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{CRMECA_2020__348_5_361_0,
     author = {Pascal Ventura and Michel Potier-Ferry and Hamid Zahrouni},
     title = {A secure version of asymptotic numerical method via convergence acceleration},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {361--374},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {348},
     number = {5},
     year = {2020},
     doi = {10.5802/crmeca.48},
     language = {en},
}
TY  - JOUR
AU  - Pascal Ventura
AU  - Michel Potier-Ferry
AU  - Hamid Zahrouni
TI  - A secure version of asymptotic numerical method via convergence acceleration
JO  - Comptes Rendus. Mécanique
PY  - 2020
SP  - 361
EP  - 374
VL  - 348
IS  - 5
PB  - Académie des sciences, Paris
DO  - 10.5802/crmeca.48
LA  - en
ID  - CRMECA_2020__348_5_361_0
ER  - 
%0 Journal Article
%A Pascal Ventura
%A Michel Potier-Ferry
%A Hamid Zahrouni
%T A secure version of asymptotic numerical method via convergence acceleration
%J Comptes Rendus. Mécanique
%D 2020
%P 361-374
%V 348
%N 5
%I Académie des sciences, Paris
%R 10.5802/crmeca.48
%G en
%F CRMECA_2020__348_5_361_0
Pascal Ventura; Michel Potier-Ferry; Hamid Zahrouni. A secure version of asymptotic numerical method via convergence acceleration. Comptes Rendus. Mécanique, Volume 348 (2020) no. 5, pp. 361-374. doi : 10.5802/crmeca.48. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.48/

[1] B. Cochelin; N. Damil; M. Potier-Ferry Méthode asymptotique numérique, Hermes Lavoisier, Paris, 2007 | Zbl

[2] N. Damil; M. Potier-Ferry A new method to compute perturbed bifurcation: Application to the buckling of imperfect elastic structures, Int. J. Eng. Sci., Volume 26 (1990), pp. 943-957 | DOI | MR | Zbl

[3] J. M. T. Thompson; A. C. Walker The non-linear perturbation analysis of discrete structural systems, Int. J. Solids Struct., Volume 4 (1968), pp. 757-768 | DOI | Zbl

[4] A. K. Noor; J. M. Peters Tracing post-limit-point paths with reduced basis technique, Comput. Methods Appl. Mech. Eng., Volume 28 (1981), pp. 217-240 | DOI | Zbl

[5] E. Riks Some computational aspects of the stability analysis of nonlinear structures, Comput. Methods Appl. Mech. Eng., Volume 47 (1984), pp. 219-259 | DOI | Zbl

[6] B. Cochelin; N. Damil; M. Potier-Ferry The asymptotic-numerical method: an efficient perturbation technique for non-linear structural mechanics, Revue Européenne des Éléments Finis, Volume 3 (1994), pp. 281-297 | DOI

[7] B. Cochelin; C. Vergez A high order purely frequency-based harmonic balance formulation for continuation of periodic solutions, J. Sound Vib., Volume 324 (2009), pp. 243-262 | DOI

[8] E. H. Boutyour; H. Zahrouni; M. Potier-Ferry; M. Boudi Bifurcation points and bifurcation branches by asymptotic-numerical method and Padé approximants, Int. J. Numer. Methods Eng., Volume 60 (2004), pp. 1987-2012 | DOI | Zbl

[9] W. Aggoune; H. Zahrouni; M. Potier-Ferry Asymptotic numerical methods for unilateral contact, Int. J. Numer. Methods Eng., Volume 68 (2006), pp. 605-631 | DOI | MR | Zbl

[10] B. Cochelin A path-following technique via an asymptotic-numerical method, Comput. Struct., Volume 53 (1994), pp. 1181-1192 | DOI | Zbl

[11] B. Cochelin; N. Damil; M. Potier-Ferry Asymptotic-Numerical Method and Padé approximants for non-linear elastic structures, Int. J. Numer. Methods Eng., Volume 37 (1994), pp. 1187-1213 | DOI | Zbl

[12] H. Lahmam; J. M. Cadou; H. Zahrouni; N. Damil; M. Potier-Ferry High order predictor-corrector algorithms, Int. J. Numer. Methods Eng., Volume 55 (2002), pp. 685-704 | DOI | Zbl

[13] G. A. Baker Jr; P. Graves-Morris Padé Approximants: Basic Theory, Encyclopedia of Mathematics and its Applications, vol. 13, Addison-Wesley, New York, 1996

[14] A. Najah; B. Cochelin; N. Damil; M. Potier-Ferry A critical review of asymptotic numerical methods, Arch. Comput. Methods Eng., Volume 5 (1998), pp. 3-22 | DOI | MR

[15] A. Elhage-Hussein; M. Potier-Ferry; N. Damil A numerical continuation method based on Padé approximants, Int. J. Solids Struct., Volume 37 (2000), pp. 6981-7001 | DOI | Zbl

[16] S. Nezamabadi; H. Zahrouni; J. Yvonnet Solving hyperelastic material problems by asymptotic numerical method, Comput. Mech., Volume 47 (2011), pp. 77-92 | DOI | MR | Zbl

[17] Y. Guevel; E. H. Boutyour; J. M. Cadou Automatic detection and branch switching methods for steady bifurcation in fluid mechanics, J. Comput. Phys., Volume 230 (2011), pp. 3614-3629 | DOI | MR | Zbl

[18] G. A. Baker Jr Defects and the convergence of Padé approximants, Acta Appl. Math., Volume 61 (2000), pp. 37-52 | DOI | Zbl

[19] M. Potier-Ferry; J. M. Cadou Basic ANM algorithms for path following problems, Revue Européenne des Eléments Finis, Volume 13 (2004), pp. 9-32 | DOI | Zbl

[20] C. Brezinski Convergence acceleration during the 20 th century, J. Comput. Appl. Math., Volume 122 (2000), pp. 1-21 | DOI | Zbl

[21] C. Brezinski; M. R. Zaglia Extrapolation Methods: Theory and Practice, Elsevier, Amsterdam, 2013

[22] K. Jbilou; H. Sadok Vector extrapolation methods, applications and numerical comparison, J. Comput. Appl. Math., Volume 122 (2000), pp. 149-165 | DOI | MR | Zbl

[23] N. Damil; J. M. Cadou; M. Potier-Ferry Mathematical and numerical connections between polynomial extrapolation and Padé approximants: applications in structural mechanics, Commun. Numer. Methods Eng., Volume 20 (2004), pp. 699-707 | DOI | Zbl

[24] J. M. Cadou; L. Duigou; N. Damil; M. Potier-Ferry Convergence acceleration of iterative algorithms. Applications to thin shell analysis and Navier–Stokes equations, Comput. Mech., Volume 43 (2009), pp. 253-264 | DOI | MR | Zbl

[25] E. Riks An incremental approach to the solution of snapping and buckling problems, Int. J. Solids Struct., Volume 15 (1979), pp. 529-551 | DOI | MR | Zbl

[26] X. Chen; J. W. Hutchinson Herringbone buckling patterns of compressed thin films on compliant substrates, J. Appl. Mech., Volume 71 (2004), pp. 597-603 | DOI | Zbl

[27] F. Xu; M. Potier-Ferry; S. Belouettar; Y. Cong 3D finite element modeling for instabilities in thin films on soft substrates, Int. J. Solids Struct., Volume 51 (2014), pp. 3619-3632 | DOI

[28] H. Mottaqui; B. Braikat; N. Damil Discussion about parameterization in the asymptotic numerical method: Application to nonlinear elastic shells, Comput. Methods Appl. Mech. Eng., Volume 199 (2010), pp. 1701-1709 | DOI | MR | Zbl

[29] C. Brezinski Généralisations de la transformation de Shanks, de la table de Padé et de l’ϵ-algorithme, Calcolo, Volume 12 (1975), pp. 317-360 | DOI | Zbl

[30] R. Jamai; N. Damil Influence of iterated Gram-Schmidt orthonormalization in the asymptotic numerical method, C. R. Méc., Volume 331 (2003), pp. 351-356 | DOI | Zbl

[31] Z. Y. Huang; W. Hong; Z. Suo Nonlinear analyses of wrinkles in a film bonded to a compliant substrate, J. Mech. Phys. Solids, Volume 53 (2005), pp. 2101-2118 | DOI | MR | Zbl

[32] M. Lavrenčič; B. Brank; M. Brojan Multiple wrinkling mode transitions in axially compressed cylindrical shell-substrate in dynamics, Thin-Walled Struct., Volume 150 (2020), 106700 | DOI

[33] A. Cutolo; V. Pagliarulo; F. Merola; S. Coppola; P. Ferraro; M. Fraldi Wrinkling prediction, formation and evolution in thin films adhering on polymeric substrata, Mater. Design, Volume 187 (2020), 108314

[34] F. Hecht New development in FreeFem++, J. Numer. Math., Volume 20 (2012), pp. 251-265 | DOI | MR | Zbl

[35] M. Médale; B. Cochelin A parallel computer implementation of the Asymptotic Numerical Method to study thermal convection instabilities, J. Comput. Phys., Volume 228 (2009), pp. 8249-8262 | DOI | MR | Zbl

[36] Y. Guevel; T. Allain; G. Girault; J. M. Cadou Numerical bifurcation analysis for 3-dimensional sudden expansion fluid dynamic problem, Int. J. Numer. Methods Fluids, Volume 87 (2018), pp. 1-26 | DOI | MR

[37] J. M. Cadou; M. Potier-Ferry A solver combining reduced basis and convergence acceleration with applications to non-linear elasticity, Int. J. Numer. Methods Biomed. Eng., Volume 26 (2010), pp. 1604-1617 | DOI | MR | Zbl

[38] Y. Cao; J. W. Hutchinson Wrinkling phenomena in neo-Hookean film/substrate bilayers, J. Appl. Mech., Volume 79 (2012), 031019

[39] W. Wong; S. Pellegrino Wrinkled membranes III: numerical simulations, J. Mech. Mater. Struct., Volume 1 (2006), pp. 63-95 | DOI

[40] C. G. Wang; L. Lan; H. F. Tan Secondary wrinkling analysis of rectangular membrane under shearing, Int. J. Mech. Sci., Volume 75 (2013), pp. 299-304 | DOI

Cited by Sources:

Comments - Policy