Advances in Computational Mathematics

, Volume 44, Issue 4, pp 1119–1151 | Cite as

Atomistic-continuum multiscale modelling of magnetisation dynamics at non-zero temperature

  • Doghonay ArjmandEmail author
  • Mikhail Poluektov
  • Gunilla Kreiss
Open Access


In this article, a few problems related to multiscale modelling of magnetic materials at finite temperatures and possible ways of solving these problems are discussed. The discussion is mainly centred around two established multiscale concepts: the partitioned domain and the upscaling-based methodologies. The major challenge for both multiscale methods is to capture the correct value of magnetisation length accurately, which is affected by a random temperature-dependent force. Moreover, general limitations of these multiscale techniques in application to spin systems are discussed.


Multiscale modelling Micromagnetism Atomistic-continuum coupling Landau-Liftshitz-Gilbert equation 

Mathematics Subject Classification 2010

65C20 65M55 82C31 82D40 



The authors would like to acknowledge the support of eSSENCE. The authors would also like to acknowledge the support from the Swedish Research Council (VR) and the KAW foundation (grants 2013.0020 and 2012.0031).


  1. 1.
    Abdulle, A., Weinan, E., Engquist, B., Vanden-Eijnden, E.: The heterogeneous multiscale method. Acta Numerica 21, 1–87 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aharoni, A.: Introduction to the Theory of Ferromagnetism. Oxford University Press, London (1996)Google Scholar
  3. 3.
    Andreas, C., Kákay, A., Hertel, R.: Multiscale and multimodel simulation of Bloch-point dynamics. Phys. Rev. B 89 (13), 134,403 (2014). CrossRefGoogle Scholar
  4. 4.
    Arjmand, D., Engblom, S., Kreiss, G.: Temporal upscaling in micromagnetism via heterogeneous multiscale methods. arXiv:1603.04920 (2017)
  5. 5.
    Arjmand, D., Runborg, O.: Analysis of heterogeneous multiscale methods for long time wave propagation problems. Multiscale Model. Simul. 12(3), 1135–1166 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Arjmand, D., Runborg, O.: A time dependent approach for removing the cell boundary error in elliptic homogenization problems. J. Comput. Phys. 314, 206–227 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Atxitia, U., Hinzke, D., Chubykalo-Fesenko, O., Nowak, U., Kachkachi, H., Mryasov, O.N., Evans, R.F., Chantrell, R.W.: Multiscale modeling of magnetic materials: temperature dependence of the exchange stiffness. Phys. Rev. B 82(13), 134,440 (2010). CrossRefGoogle Scholar
  8. 8.
    Banas, L., Brzezniak, Z., Neklyudov, M., Prohl, A.: Stochastic Ferromagnetism: Analysis and Numerics. De Gruyter (2013)Google Scholar
  9. 9.
    Bergqvist, L., Taroni, A., Bergman, A., Etz, C., Eriksson, O.: Atomistic spin dynamics of low-dimensional magnets. Phys. Rev. B 87 (14), 144,401 (2013). CrossRefGoogle Scholar
  10. 10.
    Brown, W.F.: Micromagnetics. Interscience Publishers, New York (1963)zbMATHGoogle Scholar
  11. 11.
    Brown, W.F.: Thermal fluctuations of a single-domain particle. Phys. Rev. 130(5), 1677–1686 (1963). CrossRefGoogle Scholar
  12. 12.
    Cervera, C.J.G.: Numerical micromagnetics: a review. Bol. Soc. Esp. Mat. Apl. 39, 103–135 (2007)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Chubykalo-Fesenko, O., Nowak, U., Chantrell, R.W., Garanin, D.: Dynamic approach for micromagnetics close to the Curie temperature. Phys. Rev. B 74(9), 094,436 (2006). CrossRefGoogle Scholar
  14. 14.
    Cimrák, I.: A survey on the numerics and computations for the Landau-Lifshitz equation of micromagnetism. Arch. Comput. Meth. Eng. 15(3), 277–309 (2008). MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    d’Aquino, M., Serpico, C., Miano, G.: Geometrical integration of Landau-Lifshitz-Gilbert equation based on the mid-point rule. J. Comput. Phys. 209(2), 730–753 (2005). MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    De Lucia, A., Krüger, B., Tretiakov, O.A., Kläui, M.: Multiscale model approach for magnetization dynamics simulations. Phys. Rev. B 94(18), 184415 (2016). CrossRefGoogle Scholar
  17. 17.
    E, W., Engquist, B.: The heterogeneous multiscale methods. Commun. Math. Sci. 1(1), 87–132 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Eriksson, O., Bergman, A., Bergqvist, L., Hellsvik, J.: Atomistic Spin Dynamics: Foundations and Applications. Oxford University Press, London (2016)Google Scholar
  19. 19.
    Evans, R.F.L., Fan, W.J., Chureemart, P., Ostler, T.A., Ellis, M.O.A., Chantrell, R.W.: Atomistic spin model simulations of magnetic nanomaterials. J. Phys. Cond. Matter 26, 103,202 (2014). CrossRefGoogle Scholar
  20. 20.
    Garanin, D.A.: Fokker-Planck and landau-Lifshitz-Bloch equations for classical ferromagnets. Phys. Rev. B 55(5), 3050–3057 (1997). CrossRefGoogle Scholar
  21. 21.
    Garcia-Sanchez, F., Chubykalo-Fesenko, O., Mryasov, O., Chantrell, R.W., Guslienko, K.Y.: Exchange spring structures and coercivity reduction in FePt/FeRh bilayers: a comparison of multiscale and micromagnetic calculations. Appl. Phys. Lett. 87(12), 122,501 (2005). CrossRefGoogle Scholar
  22. 22.
    Hinzke, D., Kazantseva, N., Nowak, U., Mryasov, O.N., Asselin, P., Chantrell, R.W.: Domain wall properties of FePt: from bloch to linear walls. Phys. Rev. B 77(9), 094,407 (2008). CrossRefGoogle Scholar
  23. 23.
    Jourdan, T., Marty, A., Lançon, F.: Multiscale method for Heisenberg spin simulations. Phys. Rev. B 77(22), 224,428 (2008). CrossRefGoogle Scholar
  24. 24.
    Kazantseva, N., Hinzke, D., Nowak, U., Chantrell, R.W., Atxitia, U., Chubykalo-Fesenko, O.: Towards multiscale modeling of magnetic materials: simulations of FePt. Phys. Rev. B 77(18), 184,428 (2008). CrossRefGoogle Scholar
  25. 25.
    Kevrekidis, I.G., Gear, C.W., Hyman, J.M., Kevrekidis, P.G., Runborg, O., Theodoropoulos, C.: Equation-free, coarse-grained multiscale computation: enabling microscopic simulators to perform system-level analysis. Commun. Math. Sci. 1(4), 715–762 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Kirschner, M.: Coarse-Graining in Micromagnetics. Ph.D. Thesis, Vienna University of Technology (2005)Google Scholar
  27. 27.
    Landau, D.P., Binder, K.: A Guide to Monte Carlo Simulations in Statistical Physics. Cambridge University Press, Cambridge (2009)CrossRefzbMATHGoogle Scholar
  28. 28.
    Miller, R. E., Tadmor, E. B.: A unified framework and performance benchmark of fourteen multiscale atomistic/continuum coupling methods. Model. Simul. Mater. Sci. Eng. 17(5), 053,001 (2009). CrossRefGoogle Scholar
  29. 29.
    Poluektov, M., Eriksson, O., Kreiss, G.: Coupling atomistic and continuum modelling of magnetism. Comput. Methods Appl. Mech. Eng. 329, 219–253 (2018). MathSciNetCrossRefGoogle Scholar
  30. 30.
    Poluektov, M., Eriksson, O., Kreiss, G.: Scale transitions in magnetisation dynamics. Commun. Comput. Phys. 20(4), 969–988 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Qu, S., Shastry, V., Curtin, W.A., Miller, R.E.: A finite-temperature dynamic coupled atomistic/discrete dislocation method. Model. Simul. Mater. Sci. Eng. 13(7), 1101–1118 (2005). CrossRefGoogle Scholar
  32. 32.
    Scholz, W., Schrefl, T., Fidler, J.: Micromagnetic simulation of thermally activated switching in fine particles. J. Magn. Magn. Mater. 233(3), 296–304 (2001). CrossRefGoogle Scholar
  33. 33.
    Tadmor, E.B., Miller, R.E.: Modeling Materials. Cambridge University Press, Cambridge (2011)CrossRefzbMATHGoogle Scholar
  34. 34.
    Tranchida, J., Thibaudeau, P., Nicolis, S.: Closing the hierarchy for non-Markovian magnetization dynamics. Physica B 486, 57–59 (2016)CrossRefGoogle Scholar

Copyright information

© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Doghonay Arjmand
    • 1
    • 2
    Email author
  • Mikhail Poluektov
    • 1
    • 3
  • Gunilla Kreiss
    • 1
  1. 1.Department of Information TechnologyUppsala UniversityUppsalaSweden
  2. 2.ANMC, Section de MathématiquesÉcole Polytechnique Fédérale de LausanneLausanneSwitzerland
  3. 3.International Institute for Nanocomposites Manufacturing, WMGUniversity of WarwickCoventryUK

Personalised recommendations