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

We present a numerical method for the solution of three-dimensional linear magnetostatic problems by embedding their geometry into a symmetric domain and Fourier expansion in the symmetry coordinate, leading to a set of decoupled two-dimensional problems. Special cases include axial and translational symmetry. To keep the treatment universal, a framework for the Fourier decomposition of the vector potential formulation is developed in the covariant formalism with general curvilinear coordinates. Under the restriction of homogeneous material parameters along the symmetry direction, but not on fields and current densities, this leads to a decoupled set of two-dimensional equations in 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 describes the transverse problem covering both, non-oscillatory and oscillatory case. 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 using fewer degrees of freedom than a three-dimensional simulation at comparable accuracy. These properties make the developed approach useful for the analysis of perturbation harmonics in magnetic plasma confinement devices in cylindrical coordinates or magnetic flux coordinates. The method is validated on analytical and numerical test cases and applied to a racetrack-shaped coil setup.