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A Gauss-Seidel projection method with the minimal number of updates for the stray field in micromagnetics simulations
1. | School of Mathematical Sciences, Soochow University, Suzhou, 215006, China |
2. | Mathematical Center for Interdisciplinary Research, Soochow University, Suzhou, 215006, China |
3. | School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China |
4. | Suzhou Institute for Advanced Research, University of Science and Technology of China, Suzhou, Jiangsu 215123, China |
Magnetization dynamics in magnetic materials is often modeled by the Landau-Lifshitz equation, which is solved numerically in general. In micromagnetics simulations, the computational cost relies heavily on the time-marching scheme and the evaluation of the stray field. In this work, we propose a new method, dubbed as GSPM-BDF2, by combining the advantages of the Gauss-Seidel projection method (GSPM) and the second-order backward differentiation formula scheme (BDF2). Like GSPM, this method is first-order accurate in time and second-order accurate in space, and it is unconditionally stable with respect to the damping parameter. Remarkably, GSPM-BDF2 updates the stray field only once per time step, leading to an efficiency improvement of about $ 60\% $ compared with the state-of-the-art of GSPM for micromagnetics simulations. For Standard Problems #4 and #5 from National Institute of Standards and Technology, GSPM-BDF2 reduces the computational time over the popular software OOMMF by $ 82\% $ and $ 96\% $, respectively. Thus, the proposed method provides a more efficient choice for micromagnetics simulations.
References:
[1] |
C. Abert, L. Exl, G. Selke, A. Drews and T. Schrefl,
Numerical methods for the stray-field calculation: A comparison of recently developed algorithms, Journal of Magnetism and Magnetic Materials, 326 (2013), 176-185.
doi: 10.1016/j.jmmm.2012.08.041. |
[2] |
S. Bartels and A. Prohl,
Convergence of an implicit finite element method for the Landau-Lifshitz-Gilbert equation, SIAM J. Numer. Anal., 44 (2006), 1405-1419.
doi: 10.1137/050631070. |
[3] |
F. Bruckner, A. Ducevic, P. Heistracher, C. Abert and D. Suess,
Strayfield calculation for micromagnetic simulations using true periodic boundary conditions, Sci. Rep., 11 (2021), 9202.
doi: 10.1038/s41598-021-88541-9. |
[4] |
J. R. Cash and A. H. Karp,
A variable order Runge-Kutta method for initial value problems with rapidly varying right-hand sides, ACM Trans. Math. Soft., 16 (1990), 201-222.
doi: 10.1145/79505.79507. |
[5] |
J. Chen, C. Wang and C. Xie,
Convergence analysis of a second-order semi-implicit projection method for Landau-Lifshitz equation, Appl. Numer. Math., 168 (2021), 55-74.
doi: 10.1016/j.apnum.2021.05.027. |
[6] |
I. Cimrák,
Error estimates for a semi-implicit numerical scheme solving the Landau-Lifshitz equation with an exchange field, IMA J. Numer. Anal., 25 (2005), 611-634.
doi: 10.1093/imanum/dri011. |
[7] |
F. G. Di, P. Carl-Martin, P. Dirk, R. Michele and S. Bernhard,
Linear second-order IMEX-type integrator for the (eddy current) Landau-Lifshitz-Gilbert equation, IMA J. Numer. Anal., 40 (2019), 2802-2838.
doi: 10.1093/imanum/drz046. |
[8] |
M. J. Donahue and D. G. Porter, OOMMF User's Guide, 2019. http://math.nist.gov/oommf/. |
[9] |
H. Fangohr, T. Fischbacher, M. Franchin, G. Bordignon, J. Generowicz, A. Knittel, M. Walter and M. Albert, NMAG User Manual (0.2.1), 2012. |
[10] |
A. Fuwa, T. Ishiwata and M. Tsutsumi,
Finite difference scheme for the Landau-Lifshitz equation, Jpn. J. Ind. Appl. Math., 29 (2012), 83-110.
doi: 10.1007/s13160-011-0054-9. |
[11] |
C. J. García-Cervera,
Numerical micromagnetics: A review, Bol. Soc. Esp. Mat. Apl., 39 (2007), 103-135.
|
[12] |
C. J. García-Cervera and We inan E,
Improved Gauss-Seidel projection method for micromagnetics simulations, IEEE Trans. Magn., 39 (2003), 1766-1770.
|
[13] |
T. L. Gilbert,
A Lagrangian formulation of gyromagnetic equation of the magnetization field, Phys. Rev., 100 (1955), 1243-1255.
|
[14] |
D. Jeong and J. Kim,
A Crank-Nicolson scheme for the Landau-Lifshitz equation without damping, J. Comput. Appl. Math., 234 (2010), 613-623.
doi: 10.1016/j.cam.2010.01.002. |
[15] |
L. D. Landau and E. M. Lifshitz,
On the theory of the dispersion of magetic permeability in ferromagnetic bodies, Phys. Z. Sowjetunion, 8 (1935), 153-169.
|
[16] |
P. Li, J. Chen, R. Du and X.-P. Wang,
Numerical methods for antiferrimagnets, IEEE Trans. Magn., 56 (2020), 7200509.
|
[17] |
P. Li, C. Xie, R. Du, J. Chen and X.-P. Wang, Two improved Gauss-Seidel projection methods for Landau-Lifshitz-Gilbert equation, J. Comput. Phys., 401 (2020), 109046, 12 pp.
doi: 10.1016/j.jcp.2019.109046. |
[18] |
M. Najafi, B. Krüger, S. Bohlens, M. Franchin, H. Fangohr, A. Vanhaverbeke, R. Allenspach, M. Bolte, U. Merkt, D. Pfannkuche, D. P. F. Möller and G. Meier,
Proposal for a standard problem for micromagnetic simulations including spin-transfer torque, J. Appl. Phys., 105 (2009), 113914.
doi: 10.1063/1.3126702. |
[19] |
D. Praetorius, M. Ruggeri and B. Stiftner,
Convergence of an implicit-explicit midpoint scheme for computational micromagnetics, Comput. Math. Appl., 75 (2018), 1719-1738.
doi: 10.1016/j.camwa.2017.11.028. |
[20] |
A. Romeo, G. Finocchio, M. Carpentieri, L. Torres, G. Consolo and B. Azzerboni,
A numerical solution of the magnetization reversal modeling in a permalloy thin film using fifth order Runge-Kutta method with adaptive step size control, Phys. B Condens. Matter, 403 (2008), 464-468.
doi: 10.1016/j.physb.2007.08.076. |
[21] |
X.-P. Wang, C. J. García-Cervera and W. E,
A Gauss-Seidel projection method for micromagnetics simulations, J. Comput. Phys., 171 (2001), 357-372.
doi: 10.1006/jcph.2001.6793. |
[22] |
C. Xie, C. J. García-Cervera, C. Wang, Z. Zhou and J. Chen, Second-order semi-implicit projection methods for micromagnetics simulations, J. Comput. Phys., 404 (2020), 109104, 14 pp.
doi: 10.1016/j.jcp.2019.109104. |
[23] |
H. Yamada and N. Hayashi,
Implicit solution of the Landau-Lifshitz-Gilbert equation by the Crank-Nicolson method, J. Magn. Soc. Jpn., 28 (2004), 924-931.
|
[24] |
L. Yang, Current Induced Domain Wall Motion: Analysis and Simulation, Ph. D thesis, HKUST, 2008. |
[25] |
L. Yang, J. Chen and G. Hu, A framework of the finite element solution of the Landau-Lifshitz-Gilbert equation on tetrahedral meshes, J. Comput. Phys., 431 (2021), Paper No. 110142, 17 pp.
doi: 10.1016/j.jcp.2021.110142. |
[26] |
L. Yang and G. Hu,
An adaptive finite element solver for demagnetization field calculation, Adv. Appl. Math. Mech, 11 (2019), 1048-1063.
doi: 10.4208/aamm.OA-2018-0236. |
[27] |
S. Zhang and Z. Li,
Roles of nonequilibrium conduction electrons on the magnetization dynamics of ferromagnets, Phys. Rev. Lett., 93 (2004), 127204.
doi: 10.1103/PhysRevLett.93.127204. |
[28] |
Micromagnetic Modeling Activity Group, National Institute of Standards and Technology, 2020. https://www.ctcms.nist.gov/rdm/mumag.org.html. |
[29] |
I. Žutić, J. Fabian and S. Das Sarma,
Spintronics: Fundamentals and applications, Rev. Mod. Phys., 76 (2004), 323-410.
|
show all references
References:
[1] |
C. Abert, L. Exl, G. Selke, A. Drews and T. Schrefl,
Numerical methods for the stray-field calculation: A comparison of recently developed algorithms, Journal of Magnetism and Magnetic Materials, 326 (2013), 176-185.
doi: 10.1016/j.jmmm.2012.08.041. |
[2] |
S. Bartels and A. Prohl,
Convergence of an implicit finite element method for the Landau-Lifshitz-Gilbert equation, SIAM J. Numer. Anal., 44 (2006), 1405-1419.
doi: 10.1137/050631070. |
[3] |
F. Bruckner, A. Ducevic, P. Heistracher, C. Abert and D. Suess,
Strayfield calculation for micromagnetic simulations using true periodic boundary conditions, Sci. Rep., 11 (2021), 9202.
doi: 10.1038/s41598-021-88541-9. |
[4] |
J. R. Cash and A. H. Karp,
A variable order Runge-Kutta method for initial value problems with rapidly varying right-hand sides, ACM Trans. Math. Soft., 16 (1990), 201-222.
doi: 10.1145/79505.79507. |
[5] |
J. Chen, C. Wang and C. Xie,
Convergence analysis of a second-order semi-implicit projection method for Landau-Lifshitz equation, Appl. Numer. Math., 168 (2021), 55-74.
doi: 10.1016/j.apnum.2021.05.027. |
[6] |
I. Cimrák,
Error estimates for a semi-implicit numerical scheme solving the Landau-Lifshitz equation with an exchange field, IMA J. Numer. Anal., 25 (2005), 611-634.
doi: 10.1093/imanum/dri011. |
[7] |
F. G. Di, P. Carl-Martin, P. Dirk, R. Michele and S. Bernhard,
Linear second-order IMEX-type integrator for the (eddy current) Landau-Lifshitz-Gilbert equation, IMA J. Numer. Anal., 40 (2019), 2802-2838.
doi: 10.1093/imanum/drz046. |
[8] |
M. J. Donahue and D. G. Porter, OOMMF User's Guide, 2019. http://math.nist.gov/oommf/. |
[9] |
H. Fangohr, T. Fischbacher, M. Franchin, G. Bordignon, J. Generowicz, A. Knittel, M. Walter and M. Albert, NMAG User Manual (0.2.1), 2012. |
[10] |
A. Fuwa, T. Ishiwata and M. Tsutsumi,
Finite difference scheme for the Landau-Lifshitz equation, Jpn. J. Ind. Appl. Math., 29 (2012), 83-110.
doi: 10.1007/s13160-011-0054-9. |
[11] |
C. J. García-Cervera,
Numerical micromagnetics: A review, Bol. Soc. Esp. Mat. Apl., 39 (2007), 103-135.
|
[12] |
C. J. García-Cervera and We inan E,
Improved Gauss-Seidel projection method for micromagnetics simulations, IEEE Trans. Magn., 39 (2003), 1766-1770.
|
[13] |
T. L. Gilbert,
A Lagrangian formulation of gyromagnetic equation of the magnetization field, Phys. Rev., 100 (1955), 1243-1255.
|
[14] |
D. Jeong and J. Kim,
A Crank-Nicolson scheme for the Landau-Lifshitz equation without damping, J. Comput. Appl. Math., 234 (2010), 613-623.
doi: 10.1016/j.cam.2010.01.002. |
[15] |
L. D. Landau and E. M. Lifshitz,
On the theory of the dispersion of magetic permeability in ferromagnetic bodies, Phys. Z. Sowjetunion, 8 (1935), 153-169.
|
[16] |
P. Li, J. Chen, R. Du and X.-P. Wang,
Numerical methods for antiferrimagnets, IEEE Trans. Magn., 56 (2020), 7200509.
|
[17] |
P. Li, C. Xie, R. Du, J. Chen and X.-P. Wang, Two improved Gauss-Seidel projection methods for Landau-Lifshitz-Gilbert equation, J. Comput. Phys., 401 (2020), 109046, 12 pp.
doi: 10.1016/j.jcp.2019.109046. |
[18] |
M. Najafi, B. Krüger, S. Bohlens, M. Franchin, H. Fangohr, A. Vanhaverbeke, R. Allenspach, M. Bolte, U. Merkt, D. Pfannkuche, D. P. F. Möller and G. Meier,
Proposal for a standard problem for micromagnetic simulations including spin-transfer torque, J. Appl. Phys., 105 (2009), 113914.
doi: 10.1063/1.3126702. |
[19] |
D. Praetorius, M. Ruggeri and B. Stiftner,
Convergence of an implicit-explicit midpoint scheme for computational micromagnetics, Comput. Math. Appl., 75 (2018), 1719-1738.
doi: 10.1016/j.camwa.2017.11.028. |
[20] |
A. Romeo, G. Finocchio, M. Carpentieri, L. Torres, G. Consolo and B. Azzerboni,
A numerical solution of the magnetization reversal modeling in a permalloy thin film using fifth order Runge-Kutta method with adaptive step size control, Phys. B Condens. Matter, 403 (2008), 464-468.
doi: 10.1016/j.physb.2007.08.076. |
[21] |
X.-P. Wang, C. J. García-Cervera and W. E,
A Gauss-Seidel projection method for micromagnetics simulations, J. Comput. Phys., 171 (2001), 357-372.
doi: 10.1006/jcph.2001.6793. |
[22] |
C. Xie, C. J. García-Cervera, C. Wang, Z. Zhou and J. Chen, Second-order semi-implicit projection methods for micromagnetics simulations, J. Comput. Phys., 404 (2020), 109104, 14 pp.
doi: 10.1016/j.jcp.2019.109104. |
[23] |
H. Yamada and N. Hayashi,
Implicit solution of the Landau-Lifshitz-Gilbert equation by the Crank-Nicolson method, J. Magn. Soc. Jpn., 28 (2004), 924-931.
|
[24] |
L. Yang, Current Induced Domain Wall Motion: Analysis and Simulation, Ph. D thesis, HKUST, 2008. |
[25] |
L. Yang, J. Chen and G. Hu, A framework of the finite element solution of the Landau-Lifshitz-Gilbert equation on tetrahedral meshes, J. Comput. Phys., 431 (2021), Paper No. 110142, 17 pp.
doi: 10.1016/j.jcp.2021.110142. |
[26] |
L. Yang and G. Hu,
An adaptive finite element solver for demagnetization field calculation, Adv. Appl. Math. Mech, 11 (2019), 1048-1063.
doi: 10.4208/aamm.OA-2018-0236. |
[27] |
S. Zhang and Z. Li,
Roles of nonequilibrium conduction electrons on the magnetization dynamics of ferromagnets, Phys. Rev. Lett., 93 (2004), 127204.
doi: 10.1103/PhysRevLett.93.127204. |
[28] |
Micromagnetic Modeling Activity Group, National Institute of Standards and Technology, 2020. https://www.ctcms.nist.gov/rdm/mumag.org.html. |
[29] |
I. Žutić, J. Fabian and S. Das Sarma,
Spintronics: Fundamentals and applications, Rev. Mod. Phys., 76 (2004), 323-410.
|






#(linear systems of equations) (dofs) | #(stray field updates) (dofs) | |
GSPM | 5 ( |
3 ( |
GSPM-BDF2 | 5 ( |
1 ( |
#(linear systems of equations) (dofs) | #(stray field updates) (dofs) | |
GSPM | 5 ( |
3 ( |
GSPM-BDF2 | 5 ( |
1 ( |
Temporal accuracy | T/1000 | T/2000 | T/4000 | T/8000 | order | |
GSPM | 3.42e-07 | 1.71e-07 | 0.86e-07 | 0.43e-09 | 0.99 | |
GSPM-BDF2 | 3.42e-07 | 1.71e-07 | 0.86e-07 | 0.43e-09 | 0.99 | |
Spatial accuracy | 1/20 | 1/40 | 1/80 | 1/160 | order | |
GSPM | 1.29e-04 | 0.39e-04 | 0.11e-04 | 0.03e-04 | 1.82 | |
GSPM-BDF2 | 1.29e-04 | 0.39e-04 | 0.11e-04 | 0.03e-04 | 1.82 |
Temporal accuracy | T/1000 | T/2000 | T/4000 | T/8000 | order | |
GSPM | 3.42e-07 | 1.71e-07 | 0.86e-07 | 0.43e-09 | 0.99 | |
GSPM-BDF2 | 3.42e-07 | 1.71e-07 | 0.86e-07 | 0.43e-09 | 0.99 | |
Spatial accuracy | 1/20 | 1/40 | 1/80 | 1/160 | order | |
GSPM | 1.29e-04 | 0.39e-04 | 0.11e-04 | 0.03e-04 | 1.82 | |
GSPM-BDF2 | 1.29e-04 | 0.39e-04 | 0.11e-04 | 0.03e-04 | 1.82 |
Temporal accuracy | T/100 | T/200 | T/400 | T/800 | order | |
GSPM | 1.00e-07 | 5.00e-08 | 2.50e-08 | 1.25e-08 | 1.00 | |
GSPM-BDF2 | 1.00e-07 | 5.00e-08 | 2.50e-08 | 1.25e-08 | 1.00 | |
Spatial accuracy | 1/6 | 1/8 | 1/10 | 1/12 | order | |
GSPM | 2.91e-14 | 1.72e-14 | 1.13e-14 | 7.92e-15 | 1.88 | |
GSPM-BDF2 | 2.91e-14 | 1.72e-14 | 1.13e-14 | 7.92e-15 | 1.88 |
Temporal accuracy | T/100 | T/200 | T/400 | T/800 | order | |
GSPM | 1.00e-07 | 5.00e-08 | 2.50e-08 | 1.25e-08 | 1.00 | |
GSPM-BDF2 | 1.00e-07 | 5.00e-08 | 2.50e-08 | 1.25e-08 | 1.00 | |
Spatial accuracy | 1/6 | 1/8 | 1/10 | 1/12 | order | |
GSPM | 2.91e-14 | 1.72e-14 | 1.13e-14 | 7.92e-15 | 1.88 | |
GSPM-BDF2 | 2.91e-14 | 1.72e-14 | 1.13e-14 | 7.92e-15 | 1.88 |
Standard Problem #4 | GSPM-BDF2 | OOMMF | Saving |
Field Ⅰ | 20.47 | 115.32 | 82% |
Field Ⅱ | 20.33 | 116.41 | 83% |
Standard Problem #4 | GSPM-BDF2 | OOMMF | Saving |
Field Ⅰ | 20.47 | 115.32 | 82% |
Field Ⅱ | 20.33 | 116.41 | 83% |
Parameters | GSPM-BDF2 | OOMMF | Saving |
Case 1 | 97.58 | 2216.85 | 96% |
Case 2 | 97.55 | 2226.45 | 96% |
Case 3 | 93.91 | 2229.22 | 96% |
Case 4 | 95.59 | 2246.40 | 96% |
Parameters | GSPM-BDF2 | OOMMF | Saving |
Case 1 | 97.58 | 2216.85 | 96% |
Case 2 | 97.55 | 2226.45 | 96% |
Case 3 | 93.91 | 2229.22 | 96% |
Case 4 | 95.59 | 2246.40 | 96% |
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