Summary
Various sorts of asymptotic-numerical methods have been propsed in the literature: the reduced basis technique, direct computation of series or the use of Padé approximants. The efficiency of the method may also depend on the chosen path parameter, on the order of truncature and on alternative parameters. In this paper, we compare the three classes of asymptotic-numerical method, with a view to define the “best” numerical strategy.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Azrar, L., Cochelin, B., Damil, N. and Potier-Ferry, M. (1993). “An asymptotic-numerical method to compute the post-buckling behaviour of clastic plates and shells”,International Journal for Numerical Methods in Engineering,36, pp. 1251–1277.
Baker, G.A. and Graves Morris, P. (1981). “Basic Theory”.Encyclopaedia of Mathematics and its Applications,13, Addison-Wesley Publishing Company, New York.
Boutyour, E.H., Cochelin, B., Damil, N. and Potier-Ferry, M. (1993), “Calculs nonlinéaires par des méthodes asymptotiques-numériques: applications aux structures élastiques”.Colloque national en calcul des structures 11–14 mai 1993, Giens,1, Hermes Editions, pp. 282–289.
Braikat, B. (1997), “Méthodes asymptotiques numériques et fortes non-linéarités”, Thèse de l'Université Hassan II. Casablanca.
Braikat, B., Damil, N. and Potier-Ferry, M. (1997), “Méthodes asymptotiques numériques en plasticité”,Revue Européenne des Eléments Finis,6, pp. 337–358.
Cochelin, B. (1994), “A path following technique via an Asymptotic Numerical Method”,Computers and Structures,53, 5, pp. 1181–1192.
Cochelin, B., Damil, N. and Potier-Ferry, M. (1994a), “Asymptotic Numerical Method and Padé Approximants for non-lincar clastic structures”.International Journal for Numerical Methods in Engineering,37, pp. 1187–1213.
Cochelin, B., Damil, N. and Potier-Ferry, M. (1994b), “The Asymptotic Numerical Method, an efficient perturbation technique for non-lincar structural mechanics”.Revue Européenne des Eléments Finis,3, 2, pp. 281–297.
Connor, J.J. and Morin N. (1971). “Perturbation techniques in the analysis of geometrically nonlincar shells”Proc. of Symp on High Speed Comp. of Elastic Structures, Liège, pp. 683–706.
Damil, N. and Potier-Ferry, M. (1990), “A new method to compute perturbed bifurcations: Application to the buckling of imperfect clastic structures”.International Journal of Engineering Sciences,28,3, pp. 704–719.
Damil, N. and Potier-Ferry, M. (1994), “Une technique de perturbation pour le calcul des structures avec fortes non linéarités”.Comptes Rendus de l'Académie des Sciences,318, série II, pp. 713–719.
Gallagher, R.H. (1975). “Perturbation procedures in non-lincar finite element structural analysis”,Computational Mechanics-Lectures Notes in Mathematics,461, Springer-Verlag, Berlin.
Glaum, I.W., Belytschko, T. and Masur, E.F. (1975), “Buckling of structures with finite pre-buckling deformation-a perturbation FEA”.International Journal Solids and Structures,11, pp. 1023–1033.
Kawahara, M., Yoshimura, N., Nakagawa, K. and Ohasaka, H. (1976), “Steady and unstcady finite element analysis of incompressible viscous fluid”,International Journal for Numerical Method in Engineering,10, pp. 436–456.
Masur, E.F. and Schreyer, H.L. (1967), “A second order approximation to the problem of clastic instability”,Proc. Symp. Theory of Shells, Donnell Anniv., Univ. of Texas, Houston, pp. 231–249.
Mordane, S. (1995), “Calcul d'un problème à surface libre par une méthode asymptotiquenumérique”, Thèse de l'Université Hassan II. Casablanca.
Noor, A.K. (1981), “Recent advances in reduction methods for non-lincar problems”,Computer & Structures,13, pp. 31–44.
Noor, A.K. and Peters, J.M. (1981), “Tracing post-limit paths with reduced basis technique”,Computer Methods in Applied Mechanics and Engineering,28, pp. 217–240.
Noor, A.K. and Peters, J.M. (1983), “Recent advances in reduction methods for instability analysis of structures”,Computer & Structures,16, pp. 67–80.
Padé, H. (1892), “Sur la représentation approchée d'une fonction par des fractions rationelles”,Ann. de l'Ecole Normale Sup., 3 série,9, Supp. 3–93.
Potier-Ferry, M., Damil, N., Braikat, B., Brunclot, J., Cadou, J.M., Cao, H.L. and Elhage-Husscin, A. (1997), “Traitement des fortes non-linéartités par la méthode asymptotique-numérique”,Comptes Rendus de l'Académie des Sciences,324, pp. 171–177.
Riks, E. (1984), “Some computational aspect of the stability analysis of non-lincar structures”,Computer Methods in Applied Mechanics and Engineering,47, pp. 219–259.
Thompson, J.M.T. and Walker, A.C. (1968), “The nonlinear perturbation analysis of discrete structural systems”,Inter. Journal of Solids and Structures,4, pp. 757–768.
Tri, A., Cochelin, B. and Potier-Ferry, M. (1996), “Résolution des équations de Navier-Stokes et détection des bifurcations stationnaires par une méthode asymptotique-numérique”,Revue Européenne des Eléments Finis,5, pp. 415–442.
Vannucci, P., Cochelin, B., Damil, N. and Potier-Ferry, M. (1997), “An asymptotic numerical method to compute bifurcating branches”,International Journal for Numerical Methods in Engineering, to appcar.
Van Dyke, M. (1984), “Computer-extended series”,Ann. Review of Fluid Mechanics,16, pp. 287–309.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Najah, A., Cochelin, B., Damil, N. et al. A critical review of asymptotic numerical methods. Arch Computat Methods Eng 5, 31–50 (1998). https://doi.org/10.1007/BF02736748
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02736748