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.
Revised:
Accepted:
Published online:
Mots-clés : Méthode asymptotique numérique, Accélération de convergence, Algorithme de cheminement, Plissement, Systèmes film–substrat
Pascal Ventura 1; Michel Potier-Ferry 1; Hamid Zahrouni 1
CC-BY 4.0
@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] Méthode asymptotique numérique, Hermes Lavoisier, Paris, 2007 | Zbl
[2] 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] The non-linear perturbation analysis of discrete structural systems, Int. J. Solids Struct., Volume 4 (1968), pp. 757-768 | DOI | Zbl
[4] Tracing post-limit-point paths with reduced basis technique, Comput. Methods Appl. Mech. Eng., Volume 28 (1981), pp. 217-240 | DOI | Zbl
[5] Some computational aspects of the stability analysis of nonlinear structures, Comput. Methods Appl. Mech. Eng., Volume 47 (1984), pp. 219-259 | DOI | Zbl
[6] 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] 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] 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] Asymptotic numerical methods for unilateral contact, Int. J. Numer. Methods Eng., Volume 68 (2006), pp. 605-631 | DOI | MR | Zbl
[10] A path-following technique via an asymptotic-numerical method, Comput. Struct., Volume 53 (1994), pp. 1181-1192 | DOI | Zbl
[11] 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] High order predictor-corrector algorithms, Int. J. Numer. Methods Eng., Volume 55 (2002), pp. 685-704 | DOI | Zbl
[13] Padé Approximants: Basic Theory, Encyclopedia of Mathematics and its Applications, vol. 13, Addison-Wesley, New York, 1996
[14] A critical review of asymptotic numerical methods, Arch. Comput. Methods Eng., Volume 5 (1998), pp. 3-22 | DOI | MR
[15] A numerical continuation method based on Padé approximants, Int. J. Solids Struct., Volume 37 (2000), pp. 6981-7001 | DOI | Zbl
[16] Solving hyperelastic material problems by asymptotic numerical method, Comput. Mech., Volume 47 (2011), pp. 77-92 | DOI | MR | Zbl
[17] 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] Defects and the convergence of Padé approximants, Acta Appl. Math., Volume 61 (2000), pp. 37-52 | DOI | Zbl
[19] Basic ANM algorithms for path following problems, Revue Européenne des Eléments Finis, Volume 13 (2004), pp. 9-32 | DOI | Zbl
[20] Convergence acceleration during the 20 century, J. Comput. Appl. Math., Volume 122 (2000), pp. 1-21 | DOI | Zbl
[21] Extrapolation Methods: Theory and Practice, Elsevier, Amsterdam, 2013
[22] Vector extrapolation methods, applications and numerical comparison, J. Comput. Appl. Math., Volume 122 (2000), pp. 149-165 | DOI | MR | Zbl
[23] 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] 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] 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] Herringbone buckling patterns of compressed thin films on compliant substrates, J. Appl. Mech., Volume 71 (2004), pp. 597-603 | DOI | Zbl
[27] 3D finite element modeling for instabilities in thin films on soft substrates, Int. J. Solids Struct., Volume 51 (2014), pp. 3619-3632 | DOI
[28] 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] Généralisations de la transformation de Shanks, de la table de Padé et de l’-algorithme, Calcolo, Volume 12 (1975), pp. 317-360 | DOI | Zbl
[30] Influence of iterated Gram-Schmidt orthonormalization in the asymptotic numerical method, C. R. Méc., Volume 331 (2003), pp. 351-356 | DOI | Zbl
[31] 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] Multiple wrinkling mode transitions in axially compressed cylindrical shell-substrate in dynamics, Thin-Walled Struct., Volume 150 (2020), 106700 | DOI
[33] Wrinkling prediction, formation and evolution in thin films adhering on polymeric substrata, Mater. Design, Volume 187 (2020), 108314
[34] New development in FreeFem++, J. Numer. Math., Volume 20 (2012), pp. 251-265 | DOI | MR | Zbl
[35] 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] 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] 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] Wrinkling phenomena in neo-Hookean film/substrate bilayers, J. Appl. Mech., Volume 79 (2012), 031019
[39] Wrinkled membranes III: numerical simulations, J. Mech. Mater. Struct., Volume 1 (2006), pp. 63-95 | DOI
[40] Secondary wrinkling analysis of rectangular membrane under shearing, Int. J. Mech. Sci., Volume 75 (2013), pp. 299-304 | DOI
Cited by Sources:
Comments - Policy