On the single field formulation in magnetostatics

TL;DR

This paper explores the equivalence and stability of different variational formulations in magnetostatics, including magnetoelastic coupling, and discusses their mathematical properties and limitations.

Contribution

It systematically analyzes the relationship between two classical magnetostatic formulations and extends the discussion to coupled magnetoelastic models.

Findings

The two formulations are not linked by standard convex duality.

Convexity and coercivity are not necessary or always preserved in the transformations.

The link between formulations remains stable even with elastic coupling.

Abstract

We systematically discuss the equivalence of two variational formulations of magnetostatics, in terms of magnetization and magnetic field on the one hand and the single field formulation using only magnetic induction. To demonstrate that this link is stable also when the magnetic laws are coupled with other variational static models, elasticity is included in the models as well. Interestingly, despite the fact that the corresponding magnetoelastic energy densities in the material can be computed via Legendre-Fenchel transform in the magnetic state variables, the two formulations are not linked by standard convex duality on the level of the functionals. In addition, convexity and coercivity of the given functional are neither required for the transformation nor always preserved by it.