Hostname: page-component-7dd5485656-wlg5v Total loading time: 0 Render date: 2025-10-31T01:40:41.227Z Has data issue: false hasContentIssue false
  • English
  • Français

Path following methods forsteady laminar Bingham flowin cylindrical pipes

Published online by Cambridge University Press:  16 October 2008

Juan Carlos De Los Reyes  and
Sergio González
Juan Carlos De Los Reyes
Affiliation:
Research Group on Optimization, Departmento de Matemática, EPN Quito, Ecuador. jcdelosreyes@math.epn.edu.ec; sgonzalez@math.epn.edu.ec
Sergio González
Affiliation:
Research Group on Optimization, Departmento de Matemática, EPN Quito, Ecuador. jcdelosreyes@math.epn.edu.ec; sgonzalez@math.epn.edu.ec
Get access

Abstract

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

Keywords

Bingham fluidsvariational inequalities of second kindpath-following methodssemi-smooth Newton methods.

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 43 , Issue 1 , January 2009 , pp. 81 - 117
Copyright
© EDP Sciences, SMAI, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Alberty, J., Carstensen, C. and Funken, S., Remarks around 50 lines of Matlab: short finite element implementation. Numer. Algorithms 20 (1999) 117–137. CrossRef
H.W. Alt, Lineare Funktionalanalysis. Springer-Verlag (1999).
A. Bensoussan and J. Frehse, Regularity Results for Nonlinear Elliptic Systems and Applications, Applied Mathematical Sciences 151. Springer-Verlag (2002).
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.
De Los Reyes, J.C. and Kunisch, K., A semi-smooth Newton method for control constrained boundary optimal control of the Navier-Stokes equations. Nonlinear Anal. 62 (2005) 1289–1316. CrossRef
Dean, E.J., Glowinski, R. and Guidoboni, G., On the numerical simulation of Bingham visco-plastic flow: Old and new results. J. Non-Newtonian Fluid Mech. 142 (2007) 36–62. CrossRef
G. Duvaut and J.L. Lions, Inequalities in Mechanics and Physics. Springer-Verlag, Berlin (1976).
I. Ekeland and R. Temam, Convex Analysis and Variational Problems. North-Holland Publishing Company, The Netherlands (1976).
Fuchs, M. and Seregin, G., 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. CrossRef
Fuchs, M. and Seregin, G., Regularity results for the quasi-static Bingham variational inequality in dimensions two and three. Math. Z. 227 (1998) 525–541. CrossRef
Fuchs, M., Grotowski, J.F. and Reuling, J., On variational models for quasi-static Bingham fluids. Math. Methods Appl. Sci. 19 (1996) 991–1015. 3.0.CO;2-R>CrossRef
R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer Series in Computational Physics. Springer-Verlag (1984).
R. Glowinski, J.L. Lions and R. Tremolieres, Analyse numérique des inéquations variationnelles. Applications aux phénomènes stationnaires et d'évolution 2, Méthodes Mathématiques de l'Informatique, No. 2. Dunod (1976).
Hintermüller, M. and Kunisch, K., Path-following methods for a class of constrained minimization problems in function spaces. SIAM J. Optim. 17 (2006) 159–187. CrossRef
Hintermüller, M. and Kunisch, K., Feasible and non-interior path-following in constrained minimization with low multiplier regularity. SIAM J. Contr. Opt. 45 (2006) 1198–1221. CrossRef
Hintermüller, M. and Stadler, G., An infeasible primal-dual algorithm for TV-based inf-convolution-type image restoration. SIAM J. Sci. Comput. 28 (2006) 1–23. CrossRef
Hintermüller, M., Ito, K. and Kunisch, K., The primal-dual active set strategy as a semi-smooth Newton method. SIAM J. Optim. 13 (2003) 865–888. CrossRef
Huilgol, R.R. and You, Z., 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. CrossRef
Ito, K. and Kunisch, K., Augmented Lagrangian methods for nonsmooth, convex optimization in Hilbert spaces. Nonlinear Anal. 41 (2000) 591–616. CrossRef
Ito, K. and Kunisch, K., Semi-smooth Newton methods for variational inequalities of the first kind. ESAIM: M2AN 37 (2003) 41–62. CrossRef
J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations. Springer-Verlag (1971).
Mosolov, P.P. and Miasnikov, V.P., Variational methods in the theory of the fluidity of a viscous-plastic medium. J. Appl. Math. Mech. (P.M.M.) 29 (1965) 468–492.
Papanastasiou, T., Flows of materials with yield. J. Rheology 31 (1987) 385–404. CrossRef
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).
Stadler, G., Path-following and augmented Lagrangian methods for contact problems in linear elasticity. J. Comp. Appl. Math. 203 (2007) 533–547. CrossRef
Sun, D. and Han, J., Newton and quasi-Newton methods for a class of nonsmooth equations and related problems. SIAM J. Optim. 7 (1997) 463–480. CrossRef
M. Ulbrich, Nonsmooth Newton-like methods for variational inequalities and constrained optimization problems in function spaces. Habilitation thesis, Technische Universität München, Germany (2001–2002).