On the Equivalence of Demagnetization Tensors as Discrete Cell Size Approaches Zero in Three-Dimensional Space

TL;DR

This paper proves that certain demagnetization tensors in 3D converge to the same field as the Cauchy principal value when cell size approaches zero, with numerical validation of convergence speed.

Contribution

It demonstrates the equivalence of specific demagnetization tensors to the Cauchy principal value in the limit of vanishing cell size in 3D.

Findings

Demagnetization tensors converge to the Cauchy principal value as cell size approaches zero.

A lower bound for convergence speed is established and validated numerically.

Different tensor approaches do not necessarily diminish contributions from neighboring cells at small scales.

Abstract

The calculation of the demagnetization field is crucial in various disciplines, including magnetic resonance imaging (MRI) and micromagnetics. A standard method involves discretizing the spatial domain into finite difference cells and using demagnetization tensors to compute the field. Different demagnetization tensors can result in contributions from adjacent cells that do not approach zero, nor do their differences, even as the cell size decreases. This work demonstrates that in three-dimensional space, a specific set of magnetization tensors produces the same total demagnetization field as the Cauchy principal value when the cell size approaches zero. Additionally, we provide a lower bound for the convergence speed, validated through numerical experiments.