Skip to main content
SpringerLink
Log in

A monolithic finite-element formulation for magnetohydrodynamics

  • Published:
Sādhanā Aims and scope Submit manuscript

Abstract

This work develops a new monolithic strategy for magnetohydrodynamics based on a continuous velocity–pressure formulation. The magnetic field is interpolated in the same way as the velocity field, and the entire formulation is within a nodal finite-element framework. The velocity and pressure interpolations are chosen so that they satisfy the Babuska–Brezzi (BB) conditions. In most of the existing formulations, a stabilized formulation is used that requires a stabilization term, and some associated mesh-dependent parameters that need to be adjusted. In contrast, no such parameters need to be adjusted in the current formulation, making it more user-friendly and robust. Both transient and steady-state formulations are developed for two- and three-dimensional geometries. An exact linearization of the monolithic strategy ensures that rapid (quadratic) convergence is achieved within each time (or load) step, while the stable nature of the interpolations used ensures that no instabilities arise in the solution. An existing analytical solution is corrected. The coarse mesh accuracy is shown to be better compared with other existing strategies in several benchmark problems, showing that the developed formulation is both robust and efficient.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7
Figure 8
Figure 9
Figure 10
Figure 11
Figure 12
Figure 13
Figure 14

Similar content being viewed by others

Abbreviations

\(\alpha \) :

parameter in the generalized trapezoidal rule

\({\varvec{b}}\) :

body force per unit mass

\({{\varvec{B}}}\) :

magnetic induction

\({\mathbb {C}}\) :

material constitutive tensor

\({\varvec{D}}\) :

electric displacement

\({\varvec{D}}_c\) :

rate of deformation tensor

\({\varvec{E}}\) :

electric field

\(\epsilon \) :

electric permittivity

\(\epsilon _0, \epsilon _r\) :

\(\epsilon \) of vacuum, relative \(\epsilon \)

\(\Gamma \) :

boundary

\({{\mathrm{Ha}}}\) :

Hartman number

\({\varvec{H}}\) :

magnetic field

\({\varvec{j}}\) :

current density

\(\mu \) :

magnetic permeability

\(\mu _0\), \(\mu _r\) :

\(\mu \) of vacuum, relative \(\mu \)

\(\mu _{\nu }\) :

fluid dynamic viscosity

\({\varvec{n}}\) :

unit normal vector

\(\nu \) :

kinematic viscosity

\(\varOmega \) :

domain

p :

fluid pressure

\({\mathrm {Re}}\) :

Reynolds number

\({\mathrm {Re}}_{\mathrm{m}}\) :

magnetic Reynolds number

\(\rho \) :

fluid density

\(\rho _c\) :

charge density

\(\sigma \) :

conductivity

t :

time

\({\varvec{t}}\) :

traction vector acting on the surface

\(t_{\Delta }\) :

time step in the transient strategy

\({\varvec{\tau }}\) :

Cauchy stress tensor

\({\varvec{u}}\) :

fluid velocity

\(\dot{<\_>}\) :

Derivative of \(<\_>\) with respect to time

\(\hat{<\_>}\) :

Discretized nodal values of \(<\_>\)

\(<\_>_{\delta }\) :

Variation of \(<\_>\)

\(<\_>_{\Delta }\) :

Increment of \(<\_>\)

\(<\_>^{n}\) :

\(<\_>\) at time \(t_n\)

\(<\_>^{k}\) :

\(<\_>\) at \(k^{\mathrm{th}}\) iteration at time \(t_{n+1}\)

References

  1. Sterl A 1990 Numerical simulation of liquid-metal MHD flows in rectangular ducts. J. Fluid Mech. 216: 161–191

    Article  MATH  Google Scholar 

  2. Hua T Q and Walker J S 1995 MHD flow in rectangular ducts with inclined non-uniform transverse magnetic field. Fusion Eng. Des. 27: 703–710

    Article  Google Scholar 

  3. Borghi C A, Cristofolini A and Minak G 1996 Numerical methods for the solution of the electrodynamics in magnetohydrodynamic flows. IEEE Trans. Magn. 32(3): 1010–1013

    Article  Google Scholar 

  4. Farahbakhsh I and Ghassemi H 2010 Numerical investigation of the Lorentz force effect on the vortex transfiguration in a two-dimensional lid-driven cavity. Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. Sci. 224(6): 1217–1230

    Article  Google Scholar 

  5. Peterson J S 1988 On the finite element approximation of incompressible flows of an electrically conducting fluid. Numer. Methods Partial Differ. Equ. 4: 57–68

    Article  MathSciNet  MATH  Google Scholar 

  6. Morley N B , Smolentsev S, Munipalli R, Ni M J, Gao D and Abdou M 2004 Progress on the modeling of liquid metal, free surface, MHD flows for fusion liquid walls. Fusion Eng. Des. 72: 3–34

    Article  Google Scholar 

  7. Shakeri F and Dehghan M 2011 A finite volume spectral element method for solving magnetohydrodynamic (MHD) equations. Appl. Numer. Math. 61: 1–23

    Article  MathSciNet  MATH  Google Scholar 

  8. Verardi S L L and Cardoso J R 1998 A solution of two-dimensional magnetohydrodynamic flow using the finite element method. IEEE Trans. Magn. 34(5): 3134–3137

    Article  Google Scholar 

  9. Planas R, Badia S and Codina R 2011 Approximation of the inductionless MHD problem using a stabilized finite element method. J. Comput. Phys. 230: 2977–2996

    Article  MathSciNet  MATH  Google Scholar 

  10. Boffi D, Brezzi F and Fortin M 2013 Mixed finite elements and applications. Berlin: Springer

    Book  MATH  Google Scholar 

  11. Salah N B, Soulaimani A and Habashi W G 2001 A finite element method for magneto-hydrodynamics. Comput. Methods Appl. Mech. Eng. 190: 5867–5892

    Article  MATH  Google Scholar 

  12. Salah N B 1999 A finite element method for the fully coupled magneto-hydrodynamics. PhD Thesis, Concordia University, Montreal, Canada

  13. Salah N B, Soulaimani A, Habashi W G and Fortin M 1999 A conservative stabilized finite element method for the magneto-hydrodynamic equations. Int. J. Numer. Methods Fluids 29: 535–554

    Article  MATH  Google Scholar 

  14. Shadid J N, Pawlowski R P, Banks J W, Chacn L, Lin P T and Tuminaro R S 2010 Towards a scalable fully-implicit fully-coupled resistive MHD formulation with stabilized FE methods. J. Comput. Phys. 229: 7649–7671

    Article  MathSciNet  MATH  Google Scholar 

  15. Nesliturk A I and Tezer-Sezgin M 2006 Finite element method solution of electrically driven magnetohydrodynamic flow. J. Comput. Appl. Math. 192: 339–352

    Article  MathSciNet  MATH  Google Scholar 

  16. Guermond J L, Lorat J and Nore C 2003 A new finite element method for magneto-dynamical problems: two-dimensional results. Eur. J. Mech. B Fluids 22: 555–579

    Article  MathSciNet  MATH  Google Scholar 

  17. Codina R and Silva N H 2006 Stabilized finite element approximation of the stationary magneto-hydrodynamics equations. Comput. Mech. 38: 344–355

    Article  MathSciNet  MATH  Google Scholar 

  18. Gerbeau J F 2000 A stabilized finite element method for the incompressible magnetohydrodynamic equations. Numer. Math. 87: 83–111

    Article  MathSciNet  MATH  Google Scholar 

  19. Schotzau D 2004 Mixed finite element methods for stationary incompressible magnetohydrodynamics. Numer. Math. 96: 771–800

    Article  MathSciNet  MATH  Google Scholar 

  20. Zhang G, He Y and Yang D 2014 Analysis of coupling iterations based on the finite element method for stationary magnetohydrodynamics on a general domain. Comput. Math. Appl. 68: 770–788

    Article  MathSciNet  MATH  Google Scholar 

  21. Greif C, Li D, Schtzau D and Wei X 2010 A mixed finite element method with exactly divergence-free velocities for incompressible magnetohydrodynamics. Comput. Methods Appl. Mech. Eng. 199: 2840–2855

    Article  MathSciNet  MATH  Google Scholar 

  22. Jin D, Ledger P D and Gil A J 2014 An hp-fem framework for the simulation of electrostrictive and magnetostrictive materials. Comput. Struct. 133: 131–148

    Article  Google Scholar 

  23. Jin D, Ledger P D and Gil A J 2016 hp-Finite element solution of coupled stationary magnetohydrodynamics problems including magnetostrictive effects. Comput. Struct. 164: 161–180

    Article  Google Scholar 

  24. Jog C S and Kumar R 2010 Shortcomings of discontinuous-pressure finite element methods on a class of transient problems. Int. J. Numer. Methods Fluids 62: 313–326

    MathSciNet  MATH  Google Scholar 

  25. Davidson P A 1990 An introduction to magnetohydrodynamics. Cambridge: Cambridge University Press

    MATH  Google Scholar 

  26. Bardi I, Biro O and Preis K 1991 Finite element scheme for 3D cavities without spurious modes. IEEE Trans. Magn. 27: 4036–4039

    Article  Google Scholar 

  27. Jog C S and Nandy A 2014 Mixed finite elements for electromagnetic analysis. Comput. Math. Appl. 68: 887–902

    Article  MathSciNet  MATH  Google Scholar 

  28. Nandy A and Jog C S 2016 An amplitude finite element formulation for electromagnetic radiation and scattering. Comput. Math. Appl. 71: 1364–1391

    Article  MathSciNet  Google Scholar 

  29. Gupta A 2000 WSMP: Watson sparse matrix package part II—direct solution of general sparse systems. IBM Research Report RC 21888 (98472)

  30. Gupta A 2002 Recent advances in direct methods for solving unsymmetric sparse systems of linear equations. ACM Trans. Math. Softw. 28(3): 301–324

    Article  MathSciNet  MATH  Google Scholar 

  31. Moreau R 1990 Magnetohydrodynamics. Dordrecht: Springer

    Book  MATH  Google Scholar 

  32. Moffatt H K 1983 Magnetic field generation in electrically conducting fluids. Cambridge: Cambridge University Press

    Google Scholar 

  33. Hunt J C R 1965 Magnetohydrodynamic flow in rectangular ducts. J. Fluid Mech. 21(4): 577–590

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati, 781039, India

    Arup Nandy

  2. Department of Mechanical Engineering, Indian Institute of Science, Bangalore, India

    C S Jog

Authors
  1. Arup Nandy
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C S Jog
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Arup Nandy.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nandy, A., Jog, C.S. A monolithic finite-element formulation for magnetohydrodynamics. Sādhanā 43, 151 (2018). https://doi.org/10.1007/s12046-018-0905-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12046-018-0905-z

Keywords

Search

Navigation