A magnetostatic energy formula arising from the $L^2$-orthogonal decomposition of the stray field

TL;DR

This paper introduces a new, computationally efficient formula for magnetostatic energy that avoids complex boundary integrals, potentially improving algorithms in magnetic material simulations.

Contribution

The paper presents a novel energy formula based on $L^2$-orthogonal decomposition, eliminating the need for boundary integrals and Dirichlet problem solutions.

Findings

The new formula is numerically validated as efficient.

It provides a natural analogue via magnetic induction.

Applicable within standard discretization frameworks.

Abstract

A formula for the magnetostatic energy of a finite magnet is proven. In contrast to common approaches, the new energy identity does not rely on evaluation of a nonlocal boundary integral inside the magnet or the solution of an equivalent Dirichlet problem. The formula is therefore computationally efficient, which is also shown numerically. Algorithms for the simulation of magnetic materials could benefit from incorporating the presented representation of the energy. In addition, a natural analogue for the energy via the magnetic induction is given. Proofs are carried out within a setting which is suitable for common discretizations in computational micromagnetics.