Finite-Element Formulations for Systems With High-Temperature Superconductors

In this article, 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 compare 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 one-, two-, and three-dimensional 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.