Finite Element Formulations for Systems with High-Temperature Superconductors

In this paper, we consider finite element models for high-temperature superconductors and compare two dual formulations, either magnetic-field conforming or magnetic-flux- density conforming. The electrical resistivity of superconductors is described by a power law and is strongly nonlinear. We com- pare the accuracy and the efficiency of the dual formulations by starting from simple considerations on the concavity / convexity of the constitutive law involved in each case. We then study the numerical behavior of each formulation in 1D, 2D, and 3D problems and compare their results against benchmarks. We draw general recommendations for the choice of a formulation, an iteration scheme for treating the corresponding linearized constitutive law, and a time-stepping extrapolation scheme. This approach is extended to soft ferromagnetic materials with a saturation law. Since the outcome of our analysis shows that recommended formulations for treating ferromagnets are just the opposite of those for treating superconductors, we suggest a coupled formulation for systems where both types of materials are present. The coupled formulation is shown to be accurate and more efficient than single formulations applied indistinctly to all materials.