A Fast and Accurate Numerical Method for the Computation of Unstable Micromagnetic Configurations

  • Sören Bartels
  • Mario Bebendorf
  • Michael Bratsch


We present a fast and accurate numerical method to compute unstable micromagnetic configurations. The proposed scheme, which combines various state of the art methods, is able to treat the pointwise unit-length constraint of the magnetization field and to efficiently compute the stray field energy. Furthermore, numerical results are presented which are in agreement with the expected results in simple situations and allow predictions beyond theory.


Vortex State Minimum Energy Path Fast Multipole Method String Method Block Cluster 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The author wishes to thank Felix Otto and Matthias Kurzke for stimulating discussion on the topic and Dirk Praetorius for pointing out the Ref. [11] on the employed splitting method for the computation of the stray field.


  1. 1.
    Alouges, F.: A new algorithm for computing liquid crystal stable configurations: the harmonic mapping case. SIAM J. Numer. Anal. 34, 1708–1726 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alouges, F., Conti, S., DeSimone, A., Pokern, Y.: Energetics and switching of quasi-uniform states in small ferromagnetic particles. Math. Model. Numer. Anal. 38, 235–248 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Barnes, J., Hut, P.: A hierarchical \(\mathcal{O}(N\log N)\) force calculation algorithm. Nature 324, 446–449 (1986)CrossRefGoogle Scholar
  4. 4.
    Bartels, S.: Stability and convergence of finite-element approximation schemes for harmonic maps. SIAM J. Numer. Anal. 43, 220–238 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bartels, S., Prohl, A.: Convergence of an implicit finite element method for the Landau-Lifshitz-Gilbert equation. SIAM J. Numer. Anal. 44(4), 1405–1419 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bebendorf, M.: Approximation of boundary element matrices. Numer. Math. 86(4), 565–589 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bebendorf, M.: Hierarchical Matrices: A Means to Efficiently Solve Elliptic Boundary Value Problems. Volume 63 of Lecture Notes in Computational Science and Engineering (LNCSE). Springer, Berlin (2008). ISBN:978-3-540-77146-3Google Scholar
  8. 8.
    Choi, Y.S., McKenna, P.J.: A mountain pass method for the numerical solution of semilinear elliptic problems. Nonlinear Anal. 20(4), 417–437 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    DeSimone, A., Kohn, R.V., Müller, S., Otto, F.: Magnetic microstructures – a paradigm of multiscale problems. In: ICIAM 99, Edinburgh, pp. 175–190. Oxford University Press, Oxford (2000)Google Scholar
  10. 10.
    Duffy, M.: Quadrature over a pyramid or cube of integrands with a singularity at a vertex. SIAM J. Numer. Anal. 19, 1260–1262 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    García-Cervera, J.M.: Numerical micromagnetics: a review. Bol. Soc. Esp. Matem. Apl. 39, 103–135 (2007)zbMATHGoogle Scholar
  12. 12.
    Grasedyck, L., Hackbusch, W.: Construction and arithmetics of \(\mathcal{H}\)-matrices. Computing 70, 295–334 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Greengard, L.F., Rokhlin, V.: A fast algorithm for particle simulations. J. Comput. Phys. 73(2), 325–348 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Greengard, L.F., Rokhlin, V.: A new version of the fast multipole method for the Laplace equation in three dimensions. In: Iserles, A. (ed.) Acta Numerica, 1997. Volume 6 of Acta Numerica, pp. 229–269. Cambridge University Press, Cambridge (1997)Google Scholar
  15. 15.
    Hackbusch, W.: A sparse matrix arithmetic based on \(\mathcal{H}\)-matrices. Part I: introduction to \(\mathcal{H}\)-matrices. Computing 62(2), 89–108 (1999)Google Scholar
  16. 16.
    Hackbusch, W.: Hierarchische Matrizen. Springer, Berlin/Heidelberg (2009)CrossRefzbMATHGoogle Scholar
  17. 17.
    Hackbusch, W., Khoromskij, B.N.: A sparse \(\mathcal{H}\)-matrix arithmetic. Part II: application to multi-dimensional problems. Computing 64(1), 21–47 (2000)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Hackbusch, W., Nowak, Z.P.: On the fast matrix multiplication in the boundary element method by panel clustering. Numer. Math. 54(4), 463–491 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Hertel, R., Kronmüller, H.: Finite element calculations on the single-domain limit of a ferromagnetic cube – a solution to μmag standard problem no. 3 J. Magn. Magn. Mater. 238(2–3), 185–199 (2002)Google Scholar
  20. 20.
    Hubert, A., Schäfer, R.: Magnetic Domains – The Analysis of Magnetic Microstructures. Springer, Berlin (2009)Google Scholar
  21. 21.
    Jónsson, H., Mills, G., Jacobsen, K.W.: Nudged elastic band method for finding minimum energy paths of transitions. In: Berne, B.J. et al. (eds.) Classical and Quantum Dynamics in Condensed Phase Simulations. World Scientific, Singapore (1998)Google Scholar
  22. 22.
    Kruzík, M., Prohl, A.: Recent developments in the modeling, analysis, and numerics of ferromagnetism. SIAM Rev. 48(3), 439–483 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    McMichael, R.D.: Standard problem number 3 – problem specification and reported solutions. (2008)
  24. 24.
    Popović, N., Praetorius, D.: Applications of \(\mathcal{H}\)-matrix techniques in micromagnetics. Computing 74(3), 177–204 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Sauter, S., Schwab, C.: Boundary Element Methods. Springer, Heidelberg/New York (2011)CrossRefzbMATHGoogle Scholar
  26. 26.
    Schabes, M.E., Bertram, H.N.: Magnetization processes in ferromagnetic cubes. J. Appl. Phys. 64(3), 1347–1357 (1988)CrossRefGoogle Scholar
  27. 27.
    Tyrtyshnikov, E.E.: Mosaic-skeleton approximations. Calcolo 33(1–2), 47–57 (1996/1998). Toeplitz matrices: structures, algorithms and applications (Cortona, 1996)Google Scholar
  28. 28.
    Weinan, E., Ren, W., Vanden-Eijnden, E.: String method for the study of rare events. Phys. Rev. B 66, 052301 1–4 (2002)Google Scholar
  29. 29.
    Weinan, E., Ren, W., Vanden-Eijnden, E.: Energy landscape and thermally activated switching of submicron-sized ferromagnetic elements. J. Appl. Phys. 93, 2275–2282 (2003)CrossRefGoogle Scholar
  30. 30.
    Weinan, E., Ren, W., Vanden-Eijnden, E.: Simplified and improved string method for computing the minimum energy paths in barrier-crossing events. J. Chem. Phys. 126, 164103 1–8 (2007)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Sören Bartels
    • 1
  • Mario Bebendorf
    • 2
  • Michael Bratsch
    • 2
  1. 1.Universität FreiburgFreiburgGermany
  2. 2.Institut für Numerische SimulationRheinische Friedrich-Wilhelms-Universität BonnBonnGermany

Personalised recommendations