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Asymptotic-numerical methods and Padé approximants for non-linear elastic structures. (English) Zbl 0805.73076
We apply asymptotic-numerical methods for computing non-linear equilibrium paths of elastic beam, plate and shell structures. The non- linear branches are sought in the form of asymptotic expansions, and they are determined by solving numerically (FEM) several linear problems with a single stiffness matrix. A large number of terms of the series can be easily computed by using recurrence formulas. We show, with some examples, that the choice of the expansion’s parameters and the use of Padé approximants play an important role in the determination of the size of the domain of convergence.

74S30 Other numerical methods in solid mechanics (MSC2010)
74S05 Finite element methods applied to problems in solid mechanics
74G60 Bifurcation and buckling
Full Text: DOI
[1] Riks, Int. J. Solids Strut. 15 pp 529– (1979)
[2] ’Strategies for tracing the nonlinear response near limit points’, in Nonlinear Finite Element Analysis in Structural Mechanics, in , and (eds.), Springer-Verlag Berlin, 63-89 (1981) · doi:10.1007/978-3-642-81589-8_5
[3] Criesfield, Int. J. numer. methods eng. 19 pp 1269– (1983)
[4] Damil, Int. J. Eng. Sci. 28 pp 943– (1990)
[5] Azrar, Int. J. numer. methods eng. 36 pp 1251– (1993)
[6] Padé, Ann. de l’Ecale Normale Sup, 3\(\deg\) série 9 pp 3– (1892)
[7] and , Padé Approximants, Part I: Basic Theory: Encyclopedia of Mathematics and its applications, Vol. 13, Addison-Wesley, 1981.
[8] Van Dyke, Ann. Rev. Fluid Mech. 16 pp 287– (1984)
[9] Noor, AIAA J. 18 pp 455– (1980)
[10] and , ’A second order approximation to the problem of elastic instability’, Proc. Symp. Theory of Shells, Donnell Anniv. Vol., Univ. of Texas, Houston, 1967, pp. 231-249.
[11] Thompson, Int. J. Solids Struct. 4 pp 757– (1968)
[12] and , ’Perturbation techniques in the analysis of geometrically nonlinear shells’, Proc. Symp. on High Speed Comp. of Elastic Structures, Liege, 1970.
[13] ’Perturbation procedures in nonlinear finite element structural analysis’, Computational Mechanics –Lecture Notes in Mathematics, 461, 75-89, Springer Verlag, Berlin (1975).
[14] Glaum, Int. J. Solids Struct. 11 pp 1023– (1975)
[15] ’On the stability of elastic equilibrium’, Thesis, Delft 1945, English translation NASA Tech. Trans., F. 10, 883, 1967.
[16] Budiansky, Adv. Appl. Mech. 14 pp 1– (1974)
[17] ’Foundations of elastic post-buckling theory’, in Buckling and Post-buckling, Lecture Notes in Physics, Vol. 288, 1-82, Springer, Berlin, 1987, pp. 1-82.
[18] ’Sulle defonmazioni termoelastiche finite’,in Proc 3rd Int. Cong. Appl. Mech. No 2, 1930, pp. 80-89.
[19] and , ’Calcul des points de bifurcation par une méthode asymptotiquenumérique’, Proc. 1\(\deg\) Congrés National de Mécanique au Maroc, 371-378, ENIM, Rabat, 1993.
[20] Cochelin, Comput. Methods Appl. Mech. Eng. 89 pp 361– (1991)
[21] ’A path following technique via an asymptotic-numerical method’, (1994) (to be published).
[22] Elastic Stability of Circular Cylindrical Shells, North-Holland, Amsterdam, 1984.
[23] Walker, Int. J. Solids Struct. 5 pp 97– (1969)
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