Electromagnetic axisymmetric finite elements based on a gauged four-potential variational principle

Abstract

Electromagnetic finite elements are derived based on a variational principle that uses the electromagnetic four-potential as primary variable. The Lorentz gauge normalization is incorporated as a constraint condition through a Lagrange multiplier field λ. This “gauged principle” is used to construct elements suitable for downstream coupling with mechanical and thermal finite elements for the analysis of high-temperature superconductor devices of potential use in aerospace applications. The main advantages of the four-potential formulation are: jump discontinuities on interfaces are naturally handled, no a priori approximations are invoked, and the number of degrees of freedom per node remains modest as the problem dimensionality increases. The new elements are tested on two magnetostatic axisymmetric problems. The results are in excellent agreement with analytical solutions and previous “ungauged” finite element solutions for the one-dimensional problem of a conducting infinite wire, in which case the multiplier field has no effect. For the two-dimensional problem of a hollow cylinder connected to an infinite cylindrical feed wire, the results make physical sense although there is no known analytical solution. In this case, the multiplier field λ couples the potentials in the radial and axial directions. The effect of full and selective integration on λ, as well as that of leaving λ out of the problem, are assessed. For materials of widely different permeability, jump conditions are found to be naturally accommodated by the present formulation.