Skip to main content Accessibility help
×
Home

Path following methods for steady laminar Bingham flow in cylindrical pipes

  • Juan Carlos De Los Reyes (a1) and Sergio González (a1)

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.

Copyright

References

Hide All
[1] Alberty, J., Carstensen, C. and Funken, S., Remarks around 50 lines of Matlab: short finite element implementation. Numer. Algorithms 20 (1999) 117137.
[2] H.W. Alt, Lineare Funktionalanalysis. Springer-Verlag (1999).
[3] A. Bensoussan and J. Frehse, Regularity Results for Nonlinear Elliptic Systems and Applications, Applied Mathematical Sciences 151. Springer-Verlag (2002).
[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.
[5] 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) 12891316.
[6] 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) 3662.
[7] G. Duvaut and J.L. Lions, Inequalities in Mechanics and Physics. Springer-Verlag, Berlin (1976).
[8] I. Ekeland and R. Temam, Convex Analysis and Variational Problems. North-Holland Publishing Company, The Netherlands (1976).
[9] 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) 405433.
[10] Fuchs, M. and Seregin, G., Regularity results for the quasi-static Bingham variational inequality in dimensions two and three. Math. Z. 227 (1998) 525541.
[11] Fuchs, M., Grotowski, J.F. and Reuling, J., On variational models for quasi-static Bingham fluids. Math. Methods Appl. Sci. 19 (1996) 9911015.
[12] R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer Series in Computational Physics. Springer-Verlag (1984).
[13] 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).
[14] Hintermüller, M. and Kunisch, K., Path-following methods for a class of constrained minimization problems in function spaces. SIAM J. Optim. 17 (2006) 159187.
[15] Hintermüller, M. and Kunisch, K., Feasible and non-interior path-following in constrained minimization with low multiplier regularity. SIAM J. Contr. Opt. 45 (2006) 11981221.
[16] Hintermüller, M. and Stadler, G., An infeasible primal-dual algorithm for TV-based inf-convolution-type image restoration. SIAM J. Sci. Comput. 28 (2006) 123.
[17] Hintermüller, M., Ito, K. and Kunisch, K., The primal-dual active set strategy as a semi-smooth Newton method. SIAM J. Optim. 13 (2003) 865888.
[18] 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) 126143.
[19] Ito, K. and Kunisch, K., Augmented Lagrangian methods for nonsmooth, convex optimization in Hilbert spaces. Nonlinear Anal. 41 (2000) 591616.
[20] Ito, K. and Kunisch, K., Semi-smooth Newton methods for variational inequalities of the first kind. ESAIM: M2AN 37 (2003) 4162.
[21] J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations. Springer-Verlag (1971).
[22] 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) 468492.
[23] Papanastasiou, T., Flows of materials with yield. J. Rheology 31 (1987) 385404.
[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] Stadler, G., Path-following and augmented Lagrangian methods for contact problems in linear elasticity. J. Comp. Appl. Math. 203 (2007) 533547.
[26] Sun, D. and Han, J., Newton and quasi-Newton methods for a class of nonsmooth equations and related problems. SIAM J. Optim. 7 (1997) 463480.
[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).

Keywords

Path following methods for steady laminar Bingham flow in cylindrical pipes

  • Juan Carlos De Los Reyes (a1) and Sergio González (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed