Standard and Mixed Finite Element Formulations for Systems with Type-II Superconductors

Type-II superconductors are of great interest in high magnetic field and electric power applications due to their unique magnetic and electric properties. The design and optimization of systems in these applications require devoted numerical modeling techniques. The main objective of this dissertation is to contribute to the development of robust and efficient finite element formulations suited for systems containing type-II superconductors, and possibly ferromagnetic materials. Type-II superconductors and ferromagnetic materials are described by nonlinear constitutive laws, that may cause distinct finite element formulations to present markedly different numerical behaviors.

In this work, we first present two standard finite element formulations for magnetodynamic problems (h-phi and a) and we analyze how the involved nonlinearities can be handled in the most efficient manner. Based on the analysis results, we then propose and present four dedicated mixed finite element formulations (h-phi-a, t-a, h-phi-b, a-j). As these mixed formulations take the form of perturbed saddle-point problems, we pay a particular attention to their discretization so as to avoid numerical instabilities.

Next, we compare the performance of the six formulations on a collection of problems of increasing complexity, with geometries ranging from 1D to 3D. We highlight the fact that the best formulation is problem-dependent and we give general recommendations for obtaining efficient time-stepping and linearization techniques. We conclude by applying the formulations on two distinct problems featuring non-trivial geometries: cables made up of twisted multi-filamentary superconducting wires, and layered magnetic shields made up of a stack of a large number of superconducting tapes with a ferromagnetic substrate.