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Two-scale Analysis for Multiscale Landau-Lifshitz-Gilbert Equation: Theory and Numerical Methods

Xiaofei Guan, Hang Qi, Zhiwei Sun

Abstract

This paper discusses the theory and numerical method of two-scale analysis for the multiscale Landau-Lifshitz-Gilbert equation in composite ferromagnetic materials. The novelty of this work can be summarized in three aspects: Firstly, the more realistic and complex model is considered, including the effects of the exchange field, anisotropy field, stray field, and external magnetic field. The explicit convergence orders in the $H^1$ norm between the classical solution and the two-scale solution are obtained. Secondly, we propose a robust numerical framework, which is employed in several comprehensive experiments to validate the convergence results for the Periodic and Neumann problems. Thirdly, we design an improved implicit numerical scheme to reduce the required number of iterations and relaxes the constraints on the time step size, which can significantly improve computational efficiency. Specifically, the projection and the expansion methods are given to overcome the inherent non-consistency in the initial data between the multiscale problem and homogenized problem.

Two-scale Analysis for Multiscale Landau-Lifshitz-Gilbert Equation: Theory and Numerical Methods

Abstract

This paper discusses the theory and numerical method of two-scale analysis for the multiscale Landau-Lifshitz-Gilbert equation in composite ferromagnetic materials. The novelty of this work can be summarized in three aspects: Firstly, the more realistic and complex model is considered, including the effects of the exchange field, anisotropy field, stray field, and external magnetic field. The explicit convergence orders in the norm between the classical solution and the two-scale solution are obtained. Secondly, we propose a robust numerical framework, which is employed in several comprehensive experiments to validate the convergence results for the Periodic and Neumann problems. Thirdly, we design an improved implicit numerical scheme to reduce the required number of iterations and relaxes the constraints on the time step size, which can significantly improve computational efficiency. Specifically, the projection and the expansion methods are given to overcome the inherent non-consistency in the initial data between the multiscale problem and homogenized problem.
Paper Structure (23 sections, 10 theorems, 158 equations, 6 figures, 6 tables, 1 algorithm)

This paper contains 23 sections, 10 theorems, 158 equations, 6 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Two-scale analysis and numerical discretization
  3. Two-scale method
  4. Homogenized equation in 3D case
  5. Homogenized equation in 2D case
  6. Boundary value problems
  7. Initial data
  8. Numerical scheme
  9. The flowchart of the algorithm
  10. Convergence analysis under different effective fields and boundary corrections
  11. Convergence results
  12. Periodic problem
  13. Neumann problem
  14. Proof of \ref{['thm: convergence result of periodic case']}
  15. Numerical experiments
  16. ...and 8 more sections

Key Result

Theorem 3.1

\newlabelthm: convergence result of periodic case0 Let $\boldsymbol{m}^{\varepsilon} \in L^{\infty}\left(0, T ; H^2(\Omega)\right)$ be the unique solution of the multiscale LLG equation eqn: Periodic problem and $\boldsymbol{m}_0 \in L^{\infty}\left(0, T ; H^6(\Omega)\right)$ be the unique solutio Furthermore, when $n=2$, there exists some $T^{**} \in(0, T]$ independent of $\varepsilon$, such tha

Figures (6)

  • Figure 1: Multiscale exchange coefficient $\textbf{a}^\varepsilon(\boldsymbol{x})$ defined in (a) the whole computational domain $\Omega$, and (b) the reference unit cell $Y$.
  • Figure 2: Demonstration of the auxiliary functions $\chi_1,\chi_2$.
  • Figure 3: Variation of error $e_0^j$ and $e_1^j$ relative to the cell size $\varepsilon$ across different time steps $j$ in the 2D Periodic problem.
  • Figure 4: Variation of error $e_0^j$ and $e_2^j$ relative to the cell size $\varepsilon$ across different time steps $j$ in the 2D Neumann problem.
  • Figure 5: Variation of error $e_0^j$ and $e_1^j$ relative to the cell size $\varepsilon$ across different time steps $j$ in the 2D Periodic problem with exchange field and degenerated stray field.
  • ...and 1 more figures

Theorems & Definitions (17)

  • Remark 2.1
  • Theorem 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Remark 3.4
  • Remark 3.5
  • Lemma 3.6
  • Lemma 3.7
  • Lemma 3.8
  • Remark 3.9
  • ...and 7 more