2D Fourier finite element formulation for magnetostatics in curvilinear coordinates with a symmetry direction

TL;DR

This paper introduces a Fourier finite element method for magnetostatics in curvilinear coordinates with symmetry, enabling efficient, harmonic-wise numerical solutions for problems with axial or translational symmetry.

Contribution

It develops a novel Fourier-based finite element formulation in curvilinear coordinates that handles both oscillatory and non-oscillatory harmonics in magnetostatics problems.

Findings

Efficient harmonic-wise numerical solution for symmetric magnetostatic problems.

Elimination of one vector potential component for non-zero harmonics.

Independent harmonic solutions improve memory usage and parallelization.

Abstract

We present a numerical method for the solution of linear magnetostatic problems in domains with a symmetry direction, including axial and translational symmetry. The approach uses a Fourier series decomposition of the vector potential formulation along the symmetry direction and covers both, zeroth (non-oscillatory) and non-zero (oscillatory) harmonics. For the latter it is possible to eliminate one component of the vector potential resulting in a fully transverse vector potential orthogonal to the transverse magnetic field. In addition to the Poisson-like equation for the longitudinal component of the non-oscillatory problem, a general curl-curl Helmholtz equation results for the transverse problem covering both, non-oscillatory and oscillatory case. The derivation is performed in the covariant formalism for curvilinear coordinates with a tensorial permeability and symmetry restrictions on metric and permeability tensor. The resulting variational forms are treated by the usual nodal finite element method for the longitudinal problem and by a two-dimensional edge element method for the transverse problem. The numerical solution can be computed independently for each harmonic which is favourable with regard to memory usage and parallelisation.