Canonical structure of the LLG equation for exponential updates in micromagnetism

TL;DR

This paper introduces a canonical exponential update algorithm for the LLG equation in micromagnetism, ensuring unit length constraints and improving computational efficiency through geometric integration and tensor algebra reformulations.

Contribution

The paper presents the first canonical structure of the LLG equation for exponential updates, enabling efficient and geometrically consistent integration in micromagnetism.

Findings

Demonstrates excellent performance in representative examples

Provides a canonical tensor algebraic reformulation of the LLG equation

Develops an efficient exponential update scheme based on skew symmetric matrices

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.