Stick-slip transition capturing by using an adaptive finite element method
Roquet, Nicolas ; Saramito, Pierre
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 38 (2004), p. 249-260 / Harvested from Numdam

The numerical modeling of the fully developed Poiseuille flow of a newtonian fluid in a square section with slip yield boundary condition at the wall is presented. The stick regions in outer corners and the slip region in the center of the pipe faces are exhibited. Numerical computations cover the complete range of the dimensionless number describing the slip yield effect, from a full slip to a full stick flow regime. The resolution of variational inequalities describing the flow is based on the augmented lagrangian method and a finite element method. The localization of the stick-slip transition points is approximated by an anisotropic auto-adaptive mesh procedure. The singular behavior of the solution at the neighborhood of the stick-slip transition point is investigated.

Publié le : 2004-01-01
DOI : https://doi.org/10.1051/m2an:2004012
Classification:  65N22,  65N30,  65N50,  65Z05,  90C46
@article{M2AN_2004__38_2_249_0,
     author = {Roquet, Nicolas and Saramito, Pierre},
     title = {Stick-slip transition capturing by using an adaptive finite element method},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     volume = {38},
     year = {2004},
     pages = {249-260},
     doi = {10.1051/m2an:2004012},
     mrnumber = {2069146},
     zbl = {1130.76368},
     language = {en},
     url = {http://dml.mathdoc.fr/item/M2AN_2004__38_2_249_0}
}
Roquet, Nicolas; Saramito, Pierre. Stick-slip transition capturing by using an adaptive finite element method. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 38 (2004) pp. 249-260. doi : 10.1051/m2an:2004012. http://gdmltest.u-ga.fr/item/M2AN_2004__38_2_249_0/

[1] R.A. Adams, Sobolev spaces. Academic Press (1975). | MR 450957 | Zbl 0314.46030

[2] H. Borouchaki, P.L. George, F. Hecht, P. Laug and E. Saltel, Delaunay mesh generation governed by metric specifications. Part I: Algorithms. Finite Elem. Anal. Des. 25 (1997) 61-83. | Zbl 0897.65076

[3] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer Verlag (1991). | MR 1115205 | Zbl 0788.73002

[4] F. Brezzi, M. Fortin and R. Stenberg, Error analysis of mixed-interpolated elements for Reissner-Mindlin plates. Research Repport No. 780, Instituto di Analisi Numerica, Pavie (1991). | MR 1115287 | Zbl 0751.73053

[5] A. Fortin, D. Côté and P.A. Tanguy, On the imposition of friction boundary conditions for the numerical simulation of Bingham fluid flows. Comput. Meth. Appl. Mech. Engrg. 88 (1991) 97-109. | Zbl 0745.76067

[6] M. Fortin and R. Glowinski, Méthodes de lagrangien augmenté. Applications à la résolution numérique de problèmes aux limites. Méthodes Mathématiques de l'Informatique, Dunod (1982). | Zbl 0491.65036

[7] R. Glowinski, J.L. Lions and R. Trémolières, Numerical analysis of variational inequalities. North Holland, Amsterdam (1981). | MR 635927 | Zbl 0463.65046

[8] J. Haslinger, I. Hlavàček and J. Nečas, Numerical methods for unilateral problems in solidmechanics. P.G. Ciarlet and J.L. Lions Eds., Handb. Numer. Anal. IV (1996). | MR 1422506 | Zbl 0873.73079

[9] F. Hecht, Bidimensional anisotropic mesh generator. INRIA (1997). http://www-rocq.inra.fr/gamma/cdrom/www/bamg

[10] I.R. Ionescu and B. Vernescu, A numerical method for a viscoplastic problem. An application to the wire drawing. Int. J. Engrg. Sci. 26 (1988) 627-633. | Zbl 0637.73047

[11] N. Kikuchi and J.T. Oden, Contact problems in elasticity: A study of variational inequalities and finite element methods. SIAM Stud. Appl. Math. (1988). | MR 961258 | Zbl 0685.73002

[12] N. Roquet and P. Saramito, An adaptive finite element method for Bingham fluid flows around a cylinder. Comput. Methods Appl. Mech. Engrg. 192 (2003) 3317-3341. | Zbl 1054.76053

[13] N. Roquet, R. Michel and P. Saramito, Errors estimate for a viscoplastic fluid by using P k finite elements and adaptive meshes. C. R. Acad. Sci. Paris, Série I 331 (2000) 563-568. | Zbl 1011.76047

[14] P. Saramito and N. Roquet, An adaptive finite element method for viscoplastic fluid flows in pipes. Comput. Methods Appl. Mech. Engrg. 190 (2001) 5391-5412. | Zbl 1002.76071

[15] P. Saramito and N. Roquet, Rheolef home page. http://www-lmc.imag.fr/lmc-edp/Pierre.Saramito/rheolef/ (2002).

[16] P. Saramito and N. Roquet, Rheolef users manual. Technical report, LMC-IMAG (2002). http://www-lmc.imag.fr/lmc-edp/Pierre.Saramito/rheolef/usrman.ps.gz

[17] M.G. Vallet, Génération de maillages anisotropes adaptés. Application à la capture de couches limites. Rapport de Recherche No. 1360, INRIA (1990).