Computation of the magnetostatic interaction between linearly magnetized polyhedrons

TL;DR

This paper introduces an accurate and efficient method for computing the magnetostatic interaction energy between linearly magnetized polyhedrons, improving numerical stability and applicability in micromagnetics simulations.

Contribution

The paper presents an analytical approach to evaluate the magnetostatic energy integral, reducing a six-fold integral to a nonsingular two-dimensional surface integral for better accuracy and efficiency.

Findings

Provides a closed-form solution for part of the integral

Enables accurate numerical evaluation of magnetostatic energy

Facilitates faster computations in micromagnetic simulations

Abstract

In this paper we present a method to accurately compute the energy of the magnetostatic interaction between linearly (or uniformly, as a special case) magnetized polyhedrons. The method has applications in finite element micromagnetics, or more generally in computing the magnetostatic interaction when the magnetization is represented using the finite element method (FEM). The magnetostatic energy is described by a six-fold integral that is singular when the interaction regions overlap, making direct numerical evaluation problematic. To resolve the singularity, we evaluate four of the six iterated integrals analytically resulting in a 2d integral over the surface of a polyhedron, which is nonsingular and can be integrated numerically. This provides a more accurate and efficient way of computing the magnetostatic energy integral compared to existing approaches. The method was developed to facilitate the evaluation of the demagnetizing interaction between neighouring elements in finite-element micromagnetics and provides a possibility to compute the demagnetizing field using efficient fast multipole or tree code algorithms.