Path following methods for steady laminar Bingham flow in cylindrical pipes
ESAIM: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 1, pp. 81-117.

This paper is devoted to the numerical solution of stationary laminar Bingham fluids by path-following methods. By using duality theory, a system that characterizes the solution of the original problem is derived. Since this system is ill-posed, a family of regularized problems is obtained and the convergence of the regularized solutions to the original one is proved. For the update of the regularization parameter, a path-following method is investigated. Based on the differentiability properties of the path, a model of the value functional and a correspondent algorithm are constructed. For the solution of the systems obtained in each path-following iteration a semismooth Newton method is proposed. Numerical experiments are performed in order to investigate the behavior and efficiency of the method, and a comparison with a penalty-Newton-Uzawa-conjugate gradient method, proposed in [Dean et al., J. Non-newtonian Fluid Mech. 142 (2007) 36-62], is carried out.

DOI : 10.1051/m2an/2008039
Classification : 47J20, 76A10, 65K10, 90C33, 90C46, 90C53
Mots clés : Bingham fluids, variational inequalities of second kind, path-following methods, semi-smooth Newton methods
@article{M2AN_2009__43_1_81_0,
     author = {Juan Carlos De Los Reyes and Gonz\'alez, Sergio},
     title = {Path following methods for steady laminar {Bingham} flow in cylindrical pipes},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {81--117},
     publisher = {EDP-Sciences},
     volume = {43},
     number = {1},
     year = {2009},
     doi = {10.1051/m2an/2008039},
     mrnumber = {2494795},
     zbl = {1159.76033},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an/2008039/}
}
TY  - JOUR
AU  - Juan Carlos De Los Reyes
AU  - González, Sergio
TI  - Path following methods for steady laminar Bingham flow in cylindrical pipes
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2009
SP  - 81
EP  - 117
VL  - 43
IS  - 1
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an/2008039/
DO  - 10.1051/m2an/2008039
LA  - en
ID  - M2AN_2009__43_1_81_0
ER  - 
%0 Journal Article
%A Juan Carlos De Los Reyes
%A González, Sergio
%T Path following methods for steady laminar Bingham flow in cylindrical pipes
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2009
%P 81-117
%V 43
%N 1
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an/2008039/
%R 10.1051/m2an/2008039
%G en
%F M2AN_2009__43_1_81_0
Juan Carlos De Los Reyes; González, Sergio. Path following methods for steady laminar Bingham flow in cylindrical pipes. ESAIM: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 1, pp. 81-117. doi : 10.1051/m2an/2008039. http://archive.numdam.org/articles/10.1051/m2an/2008039/

[1] J. Alberty, C. Carstensen and S. Funken, Remarks around 50 lines of Matlab: short finite element implementation. Numer. Algorithms 20 (1999) 117-137. | MR | Zbl

[2] H.W. Alt, Lineare Funktionalanalysis. Springer-Verlag (1999). | Zbl

[3] A. Bensoussan and J. Frehse, Regularity Results for Nonlinear Elliptic Systems and Applications, Applied Mathematical Sciences 151. Springer-Verlag (2002). | MR | Zbl

[4] H. Brézis, Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations, in Contributions to Non-linear Functional Analysis, E. Zarantonello Ed., Acad. Press (1971) 101-156. | MR | Zbl

[5] J.C. De Los Reyes and K. Kunisch, A semi-smooth Newton method for control constrained boundary optimal control of the Navier-Stokes equations. Nonlinear Anal. 62 (2005) 1289-1316. | MR | Zbl

[6] E.J. Dean, R. Glowinski and G. Guidoboni, On the numerical simulation of Bingham visco-plastic flow: Old and new results. J. Non-Newtonian Fluid Mech. 142 (2007) 36-62. | Zbl

[7] G. Duvaut and J.L. Lions, Inequalities in Mechanics and Physics. Springer-Verlag, Berlin (1976). | MR | Zbl

[8] I. Ekeland and R. Temam, Convex Analysis and Variational Problems. North-Holland Publishing Company, The Netherlands (1976). | MR | Zbl

[9] M. Fuchs and G. Seregin, Some remarks on non-Newtonian fluids including nonconvex perturbations of the Bingham and Powell-Eyring model for viscoplastic fluids. Math. Models Methods Appl. Sci. 7 (1997) 405-433. | MR | Zbl

[10] M. Fuchs and G. Seregin, Regularity results for the quasi-static Bingham variational inequality in dimensions two and three. Math. Z. 227 (1998) 525-541. | MR | Zbl

[11] M. Fuchs, J.F. Grotowski and J. Reuling, On variational models for quasi-static Bingham fluids. Math. Methods Appl. Sci. 19 (1996) 991-1015. | MR | Zbl

[12] R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer Series in Computational Physics. Springer-Verlag (1984). | MR | Zbl

[13] R. Glowinski, J.L. Lions and R. Tremolieres, Analyse numérique des inéquations variationnelles1976). | Zbl

[14] M. Hintermüller and K. Kunisch, Path-following methods for a class of constrained minimization problems in function spaces. SIAM J. Optim. 17 (2006) 159-187. | MR | Zbl

[15] M. Hintermüller and K. Kunisch, Feasible and non-interior path-following in constrained minimization with low multiplier regularity. SIAM J. Contr. Opt. 45 (2006) 1198-1221. | MR | Zbl

[16] M. Hintermüller and G. Stadler, An infeasible primal-dual algorithm for TV-based inf-convolution-type image restoration. SIAM J. Sci. Comput. 28 (2006) 1-23. | MR | Zbl

[17] M. Hintermüller, K. Ito and K. Kunisch, The primal-dual active set strategy as a semi-smooth Newton method. SIAM J. Optim. 13 (2003) 865-888. | MR | Zbl

[18] R.R. Huilgol and Z. You, Application of the augmented Lagrangian method to steady pipe flows of Bingham, Casson and Herschel-Bulkley fluids. J. Non-Newtonian Fluid Mech. 128 (2005) 126-143.

[19] K. Ito and K. Kunisch, Augmented Lagrangian methods for nonsmooth, convex optimization in Hilbert spaces. Nonlinear Anal. 41 (2000) 591-616. | MR | Zbl

[20] K. Ito and K. Kunisch, Semi-smooth Newton methods for variational inequalities of the first kind. ESAIM: M2AN 37 (2003) 41-62. | Numdam | MR | Zbl

[21] J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations. Springer-Verlag (1971). | MR | Zbl

[22] P.P. Mosolov and V.P. Miasnikov, Variational methods in the theory of the fluidity of a viscous-plastic medium. J. Appl. Math. Mech. (P.M.M.) 29 (1965) 468-492. | Zbl

[23] T. Papanastasiou, Flows of materials with yield. J. Rheology 31 (1987) 385-404. | Zbl

[24] G. Stadler, Infinite-dimensional Semi-smooth Newton and Augmented Lagrangian Methods for Friction and Contact Problems in Elasticity. Ph.D. thesis, Karl-Franzens University of Graz, Graz, Austria (2004).

[25] G. Stadler, Path-following and augmented Lagrangian methods for contact problems in linear elasticity. J. Comp. Appl. Math. 203 (2007) 533-547. | MR | Zbl

[26] D. Sun and J. Han, Newton and quasi-Newton methods for a class of nonsmooth equations and related problems. SIAM J. Optim. 7 (1997) 463-480. | MR | Zbl

[27] M. Ulbrich, Nonsmooth Newton-like methods for variational inequalities and constrained optimization problems in function spaces. Habilitation thesis, Technische Universität München, Germany (2001-2002).

Cité par Sources :