Canonical structure of the LLG equation for exponential updates in micromagnetism

TL;DR

The paper addresses accurate and structure-preserving time integration for the Landau-Lifshitz-Gilbert equation in micromagnetism by deriving a canonical form ${\partial_t{\bm{m}} = {\bm{W}} \cdot {\bm{m}}}$ with a skew-symmetric generator ${\bm{W}}$. It develops an exponential update algorithm that computes ${\bm{m}}_{n+1} = \exp[{\bm{W}}^{\Delta t}_{n+1}] \cdot {\bm{m}}_n$ within SO(3), leveraging a closed-form representation ${\exp[{\bm{W}}^{\Delta t}]} = {\bf 1} + {\alpha_1}{\bm{W}}^{\Delta t} + {\alpha_2}({\bm{W}}^{\Delta t})^2$ and stable variants for small $w$. The work compares the exponential method to Backward Euler and Midpoint rules, showing superior unit-length preservation, rotation accuracy, and robustness to large time steps in representative micromagnetic tests based on realistic material parameters. This canonical, geometric approach enables efficient, structure-preserving simulations of magnetization dynamics with potential impact on large-scale micromagnetic modeling.

Abstract

In this contribution we propose an exponential update algorithm for magnetic moments appearing in the framework of micromagnetics and the Landau-Lifshitz-Gilbert (LLG) equation. This algorithm can be interpreted as the geometric integration on spheres, that a priori satisfy the unit length constraint of the normalized magnetization vector. Even though the geometric structures for this are obvious and some works already use an exponential algorithm, to the best of the authors' knowledge, there is no canonical structure of the LLG equation for the exponential update algorithm in micromagnetism. Tensor algebraic reformulations of the LLG equation allow the canonical representation of the evolution equation for the magnetization, which serves as the basis for different integrators. Based on the specific structure of the exponential of skew symmetric matrices an efficient update scheme is derived. The excellent performance of the proposed exponential update algorithm is demonstrated in representative examples.

Figures (10)

• Figure 1: Magneto-crystalline energy surface for the cubic anisotropy case ($K_1^{cub} = 2\cdot10^4$ J/m$^3$ and $K_2^{cub} = -4.5\cdot10^4$ J/m$^3$). Parameters taken from YiXu:2014:acf.
• Figure 2: The evolution path of the magnetization vectors is shown in a) for different damping coefficients, while b) shows the evolution of the magnetization components of the magnetization vector for a damping coefficient $\alpha=0$. The length of the vector remains constant $||{{\bm{m}}}||=1$ during the simulation.
• Figure 3: a) and c) depict the precession of a magnetization vector within the x$_1$-x$_2$-plane for different time increments. The corresponding length of the magnetization vector during the rotation is presented in b).
• Figure 4: a) and c) depict the precession of a magnetization vector within the x$_1$-x$_2$-plane for different time increments. The miss-match in the start and finish locations of the magnetization vectors can be clearly seen. The preservation of the norm during the rotation of the vector is presented in b).
• Figure 5: a) depicts the precession of a magnetization vector within the x$_1$-x$_2$-plane for different time increments. The corresponding norm preservation during the rotation of the vector is presented in b).