# Path following methods for steady laminar Bingham flow in cylindrical pipes

Juan Carlos De Los Reyes; Sergio González

ESAIM: Mathematical Modelling and Numerical Analysis (2008)

- Volume: 43, Issue: 1, page 81-117
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topDe Los Reyes, Juan Carlos, and González, Sergio. "Path following methods for steady laminar Bingham flow in cylindrical pipes." ESAIM: Mathematical Modelling and Numerical Analysis 43.1 (2008): 81-117. <http://eudml.org/doc/194448>.

@article{DeLosReyes2008,

abstract = {
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.
},

author = {De Los Reyes, Juan Carlos, González, Sergio},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Bingham fluids; variational inequalities of second kind; path-following methods; semi-smooth Newton methods.; semi-smooth Newton methods; duality theory; regularization},

language = {eng},

month = {10},

number = {1},

pages = {81-117},

publisher = {EDP Sciences},

title = {Path following methods for steady laminar Bingham flow in cylindrical pipes},

url = {http://eudml.org/doc/194448},

volume = {43},

year = {2008},

}

TY - JOUR

AU - De Los Reyes, Juan Carlos

AU - González, Sergio

TI - Path following methods for steady laminar Bingham flow in cylindrical pipes

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2008/10//

PB - EDP Sciences

VL - 43

IS - 1

SP - 81

EP - 117

AB -
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.

LA - eng

KW - Bingham fluids; variational inequalities of second kind; path-following methods; semi-smooth Newton methods.; semi-smooth Newton methods; duality theory; regularization

UR - http://eudml.org/doc/194448

ER -

## References

top- J. Alberty, C. Carstensen and S. Funken, Remarks around 50 lines of Matlab: short finite element implementation. Numer. Algorithms20 (1999) 117–137.
- H.W. Alt, Lineare Funktionalanalysis. Springer-Verlag (1999).
- A. Bensoussan and J. Frehse, Regularity Results for Nonlinear Elliptic Systems and Applications, Applied Mathematical Sciences151. 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.
- 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.
- 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.
- 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).
- 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.
- 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.
- M. Fuchs, J.F. Grotowski and J. Reuling, On variational models for quasi-static Bingham fluids. Math. Methods Appl. Sci.19 (1996) 991–1015.
- 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'évolution2, Méthodes Mathématiques de l'Informatique, No. 2. Dunod (1976).
- 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.
- 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.
- 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.
- 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.
- 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.
- K. Ito and K. Kunisch, Augmented Lagrangian methods for nonsmooth, convex optimization in Hilbert spaces. Nonlinear Anal.41 (2000) 591–616.
- K. Ito and K. Kunisch, Semi-smooth Newton methods for variational inequalities of the first kind. ESAIM: M2AN37 (2003) 41–62.
- J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations. Springer-Verlag (1971).
- 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.
- T. Papanastasiou, Flows of materials with yield. J. Rheology31 (1987) 385–404.
- 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).
- G. Stadler, Path-following and augmented Lagrangian methods for contact problems in linear elasticity. J. Comp. Appl. Math.203 (2007) 533–547.
- 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.
- M. Ulbrich, Nonsmooth Newton-like methods for variational inequalities and constrained optimization problems in function spaces. Habilitation thesis, Technische Universität München, Germany (2001–2002).

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.