A Gauss-Seidel projection method with the minimal number of updates for the stray field in micromagnetics simulations

Panchi Li; Zetao Ma; Rui Du; Jingrun Chen

doi:10.3934/dcdsb.2022002

Article Contents

2022, Volume 27, Issue 11: 6401-6416. Doi: 10.3934/dcdsb.2022002

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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

^*Corresponding authors: Rui Du and Jingrun Chen
^*Corresponding authors: Rui Du and Jingrun Chen

Received: May 31, 2021

Revised: August 31, 2021

Early access: January 2022

Published: October 2022

P. Li was supported by the Postgraduate Research & Practice Innovation Program of Jiangsu Province via grant KYCX20_2711, R. Du was supported by NSFC via grant 11501399, and J. Chen was supported by NSFC via grant 11971021

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Abstract

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.

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Mathematics Subject Classification: Primary: 35Q99, 65Z05, 65M06.

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Figure 1. Simulation results of the magnetization on the centered slice of the material in the $ xy $ plane. Left column: a color plot of the angle between the in-plane magnetization and the $ x $-axis; Right column: an arrow plot of the in-plane magnetization; Top row: GSPM with one update of the stray field; Second row: GSPM with three updates of the stray field; Bottom row: GSPM-BDF2

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Figure 2. Magnetization profile of the initial s-state. This is generated for a random initial state under a magnetic field $ 100\;\mathrm{mT} $ along the $ [1, 1, 1] $ direction with a successive reduction to $ 0 $

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Figure 3. Left column: dynamics of the spatially averaged magnetization on the coarse mesh under the external fields; Right column: comparison of the averaged $ \langle m_y\rangle $ on the two different meshes. Top row: Field Ⅰ; Bottom row: Field Ⅱ

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Figure 4. Magnetization profile when the averaged $ \langle m_x\rangle = 0 $ under the external fields. The color map is given by the $ z $-component. Top row: Field Ⅰ; Bottom row: Field Ⅱ

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Figure 5. Magnetization dynamics and the final state for four sets of parameters. Left column: dynamics of the spatially averaged magnetization with the result of D. G. Porter as the reference; Right column: final state colored by the $ x $-component. Top row: Case 1; Second row: Case 2; Third row: Case 3; Bottom row: Case 4

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Figure 6. Magnetization dynamics and the final state in the case of $ \xi = 0.5 $ and $ bJ = 72.45 \;\mathrm{m}/\mathrm{s} $. Left: dynamics of the spatially averaged magnetization with the result of G. Finocchio et al. as the reference; Right: final state colored by the $ x $-component. The $ x $-component of the spatially averaged magnetization $ \langle M_x\rangle $ at $ 10\;\mathrm{ns} $ is $ -1.43\times10^5\;\mathrm{A}/\mathrm{m} $

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Table 1. Main computational costs of GSPM and GSPM-BDF2 per time step

	#(linear systems of equations) (dofs)	#(stray field updates) (dofs)
GSPM	5 ($ N $)	3 ($ 3N $)
GSPM-BDF2	5 ($ N $)	1 ($ 3N $)

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Table 2. Convergence rates in terms of $ \Delta t $ and $ \Delta x $ for Example 1 (1D)

Temporal accuracy	$ \Delta t $	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	$ \Delta x $	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

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Table 3. Convergence rates in terms of $ \Delta t $ and $ h $ for Example 2 (3D)

Temporal accuracy	$ \Delta t $	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	$ h $	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

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Table 4. Computational costs (in seconds) of GSPM-BDF2 and OOMMF for Standard Problem #4 when the coarse mesh is used

Standard Problem #4	GSPM-BDF2	OOMMF	Saving
Field Ⅰ	20.47	115.32	82%
Field Ⅱ	20.33	116.41	83%

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Table 5. Computational costs (in seconds) of GSPM-BDF2 and OOMMF for Standard Problem #5

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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