Power series analysis as a major breakthrough to improve the efficiency of Asymptotic Numerical Method in the vicinity of bifurcations

Journal of Computational Physics

Volumn 236, Issue 1, 2013, Pages 594-607

  Cochelin, Bruno ^a   Medale, Marc ^b  

^a AIX MARSEILLE UNIVERSITY   (France)

^b AIX MARSEILLE UNIVERSITÉ   (France)

Author keywords

Asymptotic Numerical Method; Bifurcation; Continuation; Path following; Power series analysis

Indexed keywords

BIFURCATION (MATHEMATICS);

ASYMPTOTIC NUMERICAL METHOD; BIFURCATION; BIFURCATION POINTS; CONTINUATION; GEOMETRIC POWER; OPTIMAL STEP LENGTHS; PATH FOLLOWING; POWER SERIES; POWER SERIES ANALYSIS; SIMPLE++;

NUMERICAL METHODS;

EID: 84872688959     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2012.11.016     Document Type: Article

References (36)

1. Allery C., Cadou J.-M., Hamdouni A., Razafindralandy D. Application of the asymptotic numerical method to the Coanda effect study. Revue Eur. Eléments Finis 2004, 13:57-77.
2. Allgower E.L., Georg K.G. Numerical Continuation Methods. An introduction 1990, Springer-Verlag, New York.
3. Amestoy P.R., Duff I.S., Koster J., L'Excellent J.-Y. A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM Journal on Matrix Analysis and Applications 2001, 23(1):15-41.
4. Amestoy P.R., Duff I.S., L'Excellent J.-Y. Multifrontal parallel distributed symmetric and unsymmetric solvers. Computer Methods in Applied Mechanics and Engineering 2000, 184:501-520.
5. Amestoy P.R., Guermouche A., L'Excellent J.-Y., Pralet S. Hybrid scheduling for the parallel solution of linear systems. Parallel Computing 2006, 32(2):136-156.
6. Azrar L., Cochelin B., Damil N., Potier-Ferry M. An asymptotic numerical method to compute the post-buckling behaviour of elastic plates and shells. International Journal for Numerical Methods in Engineering 1993, 36:1251-1277.
7. Baguet S., Cochelin B. On the behaviour of the ANM continuation in the presence of bifurcation. Communication in Numerical Methods in Engineering 2003, 19:459-471.
8. Baker A., Graves-Morris P. Padé Approximants 1996, Cambridge University Press.
9. Boutyour E., Zahrouni H., Potier-Ferry M., Boudi M. Asymptotic numerical method for buckling analysis of shell structures with large rotation. Journal of Computational and Applied Mathematics 2004, 168:77-85.
10. Cadou J.-M., Cochelin B., Potier-Ferry M. A numerical method to compute bifurcation points in fluid mechanics. The European Journal of Mechanics B/Fluids 2006, 25:234-254.
11. Cadou J.-M., Potier-Ferry M., Cochelin B., Damil N. Asymptotic Numerical Method for stationary Navier-Stokes equations and with Petrov-Galerkin formulation. International Journal for Numerical Methods in Engineering 2001, 50:825-845.
12. Cochelin B. A path following technique via an asymptotic numerical method. Computers and Structures 1994, 53:1181-1192.
13. Cochelin B., Damil N., Potier-Ferry M. The asymptotic numerical method: an efficient perturbation technique for non-linear structural mechanics. Revue Eur. Eléments Finis 1994, 5:415-442.
14. Cochelin B., Damil N., Potier-Ferry M. Asymptotic Numerical Method and Padé approximants for non-linear elastic structures. International Journal for Numerical Methods in Engineering 1994, 37:1187-1213.
15. Cochelin B., Damil N., Potier-Ferry M. Méthode Asymptotique Numérique 2007, Hermès-Lavoisier, (in French).
16. Damil N., Potier-Ferry M. A new method to compute perturbed bifurcations: application to the buckling of imperfect elastic structures. International Journal of Engineering Science 1990, 28:943-957.
17. Doedel E. Nonlinear numerics. Journal of the Franklin Institute 1997, 334B:1049-1073.
18. Doedel E. Lecture notes on numerical analysis of nonlinear equations. Numerical Continuation Methods for Dynamical Systems 2007, 1-49. Springer. B. Krauskopf, H. Osinga, J. Galan-Vioque (Eds.).
19. Doedel E., Keller H.B., Kernevez J.P. Numerical analysis and control of bifurcation problems. (I) Bifurcation in finite dimensions. International Journal of Bifurcation and Chaos 1991, 3:493-520.
20. Domb C., Sykes M.F. On the susceptibility of a ferromagnetic above the Curie point. Proceedings of the Royal Society of London Series A 1957, 240:214-228.
21. Domb C., Sykes M.F. Use of series expansions for the Ising model susceptibility and excluded volume problem. Journal of Mathematical Physics 1961, 2.
22. Elhage-Hussein A., Potier-Ferry M., Damil N. A numerical continuation method based on Padé approximants. International Journal of Solids and Structures 2000, 37:6981-7001.
23. Gervais J., Sadiky H. A new step length control for continuation with the Asymptotic Numerical Method. IMA Journal of Numerical Analysis 2002, 22:207-229.
24. Guevel Y., Boutyour H., Cadou J.-M. Automatic detection and branch switching methods for steady bifurcation in fluid mechanics. Journal of Computational Physics 2011, 230:3614-3629.
25. Keller H.B. Numerical Solution of Bifurcation and Nonlinear Eigenvalue Problems 1977, Academic Press, New York.
26. Manlab. An Interactive Path-Following and Bifurcation Analysis Software, 2012, http://manlab.lma.cnrs-mrs.fr/.
27. Mattei P.O. High-order perturbation expansion for the spectral analysis of fluid-loaded vibrating structure. Acta Acustica United with Acustica 2007, 93:303-313.
28. Medale M., Cochelin B. A parallel computer implementation of the asymptotic numerical method to study thermal convection instabilities. Journal of Computational Physics 2009, 228:8249-8262.
29. Pearce C. Transformation methods in the analysis of series for critical properties. Advances in Physics 1978, 27:89-148.
30. Potier-Ferry M. Foundations of Elastic Post-Buckling Theory in Buckling and Post-Buckling. Lecture notes in Physics 1987, vol. 288. Springer-Verlag, Berlin.
31. Seydel R. Practical Bifurcation and Stability Analysis 1994, Springer-Verlag.
32. Tri A., Cochelin B., Potier-Ferry M. Résolution des équations de Navier-Stokes et détection des bifurcations stationnaires par une Méthode Asymptotique Numérique. European Journal of Finite Elements 1996, 5(4):415-442.
33. Van Dyke M. Analysis and improvement of perturbation series. Journal of Mechanical and Applied Mathematics 1974, 27:423-450.
34. Van Dyke M. Computer extension of perturbation series in fluid mechanics. SIAM Journal on Applied Mathematics 1975, 28:720-734.
35. Van Dyke M. Perturbation Methods in Fluid Mechanics 1975, The Parabolic Press, Stanford, California.
36. Vannucci P., Cochelin B., Damil N., Potier-Ferry M. An asymptotic numerical method to compute bifurcating branches. International Journal for Numerical Methods in Engineering 1998, 41:1365-1389.