An iterative method for solving the regularized Bingham problem

Authors

Keywords:

iterative method, preconditioner, viscoplasticity, Bingham problem, regularization

Abstract

The paper discusses a method for numerical solution of the regularized Bingham problem. We consider the regularized model proposed by Papanastasiou. For the linearized problem, a preconditioner is developed and several estimates for the effective condition number are derived. Further, the convergence of Krylov subspace iterative methods is analyzed. The estimates are based on the Necas inequality in weighted norms. The work was supported by the Russian Foundation for Basic Research (projects 09-01-00115 and 08-01-00159).

Author Biographies

P.P. Grinevich

Lomonosov Moscow State University,
Faculty of Mechanics and Mathematics
• PhD Student

M.A. Olshanskii

Lomonosov Moscow State University,
Faculty of Mechanics and Mathematics
• Professor

References

  1. Bercovier M., Engelman M. A finite element method for incompressible non-newtonian flows // J. Comp. Phys. 1980. 36. 313-326.
  2. Bingham E. Fluidity and plasticity. New-York: McGraw-Hill, 1922.
  3. Brezzi F., Fortin M. Mixed and hybrid finite element methods. New-York: Springer, 1991.
  4. Chatzimina M., Xenophontos C., Georgiou G.C., Argyropaidas I., Mitsoulis E. Cessation of annular Poiseuille flows of Bingham plastics // J. Non-Newtonian Fluid Mech. 2007. 142. 135-142.
  5. Dean E.J., Glowinski R. Operator-splitting methods for the simulation of Bingham visco-plastic flow // Chin. Ann. of Math. 2002. 23. 187-204.
  6. Dimakopoulos Y., Tsamopoulos J. Transient displacement of a viscoplastic material by air in straight and suddenly constricted tubes // J. Non-Newtonian Fluid Mech. 2003. 112. 43-53.
  7. Elias R.N., Martins M.A. D., Coutinho A.L. G.A. Parallel edge-based solution of viscoplastic flows with the SUPG/PSPG formulation // Comput. Mech. 2006. 38. 365-381.
  8. Elman H.C., Silvester D.J., Wathen A.J. Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics.
  9. Elman H.C., Silvester D.J., Wathen A.J. Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations // Numer. Math. 2002. 90. 665-688.
  10. Fischer B., Ramage A., Silvester D.J., Wathen A.J. Minimum residual methods for augmented systems // BIT. 1998. 38. 527-543.
  11. Grinevich P.P., Olshanskii M.A. An iterative method for the Stokes-type problem with variable viscosity // SIAM J. Sci. Comp. 2009. 32. 3959-3978.
  12. Hron J., Ouazzi A., Turek S. A computational comparison of two FEM solvers for nonlinear incompressible flow // Lecture Notes in Computational Science and Engineering. Vol. 35, pp. 87-108. New York: Springer, 2003.
  13. Mitsoulis E., Huilgol R.R. Entry flows of Bingham plastics in expansions // J. Non-Newtonian Fluid Mech. 2004. 122. 45-55.
  14. Mitsoulis E., Zisis Th. Flow of Bingham plastics in a lid-driven cavity // J. Non-Newtonian Fluid Mech. 2001. 101. 173-180.
  15. Nev c as J. Les m’ e thodes directes en th’ e orie des ’ e quations elliptiques. Paris: Masson, 1967.
  16. Nicolaides R.A. Analysis and convergence of the MAC scheme. I. The linear problem // SIAM. J. Num. Anal. 1992. 29. 1579-1591.
  17. Olshanskii M.A. Analysis of semi-staggered finite-difference method with application to Bingham flows // Comp. Meth. Appl. Mech. Eng. 2009. 198. 975-985.
  18. Olshanskii M.A., Reusken A. Analysis of a Stokes interface problem // Numer. Math. 2006. 103. 129-149.
  19. Papanastasiou T.C. Flows of materials with yield // J. Rheol. 1987. 31, N 5. 385-404.
  20. Saad Y. Iterative methods for sparse linear systems. Philadelphia: SIAM, 2003.
  21. Saad Y., Schultz M.H. GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems // SIAM J. Sci. Comp. 1986. 7. 856-869.
  22. Sanchez F.J. Application of a first-order operator splitting method to Bingham fluid flow simulation // Comput. Math. Appl. 1998. 36. 71-86.
  23. Ильюшин А.А. Деформация вязко-пластичного тела // Ученые записки МГУ. Механика. 1940. Вып. 39. 3-81.
  24. Ольшанский М.А. Лекции и упражнения по многосеточным методам. М.: Физматлит, 2005.
  25. Тыртышников Е.Е. Методы численного анализа. М.: Издательский центр «Академия», 2007.
  26. Чижонков Е.В. Релаксационные методы решения седловых задач. М.: ИВМ РАН, 2002.
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Published

02-03-2010

How to Cite

Гриневич П., Ольшанский М. An Iterative Method for Solving the Regularized Bingham Problem // Numerical Methods and Programming (Vychislitel’nye Metody i Programmirovanie). 2010. 11. 78-87

Issue

Vol. 11 (2010): Issue 1.

Section

Section 1. Numerical methods and applications