Programming finite element method based hysteresis loss computation software using non-linear superconductor resistivity andT− ϕ formulation

Abstract

Due to the rapid development of personal computers from the beginning of the 1990s, it has become a reality to simulate current penetration, and thus hysteresis losses, in superconductors with other than very simple one-dimensional (1D) Bean model computations or Norris formulae. Even though these older approaches are still usable, they do not consider, for example, multifilamentary conductors, local critical current dependency on magnetic field or varying n-values. Currently, many numerical methods employing different formulations are available. The problem of hysteresis losses can be scrutinized via an eddy current formulation of the classical theory of electromagnetism. The difficulty of the problem lies in the non-linear resistivity of the superconducting region. The steep transition between the superconducting and the normal states often causes convergence problems for the most common finite element method based programs. The integration methods suffer from full system matrices and, thus, restrict the number of elements to a few thousands at most. The so-called T − ϕ formulation and the use of edge elements, or more precisely Whitney 1-forms, within the finite element method have proved to be a very suitable method for hysteresis loss simulations of different geometries. In this paper we consider making such finite element method software from first steps, employing differential geometry and forms.

Validation of Homogenized Filament Bundle Model in AC Loss Computations

“…According to analytic formulas presented in [1] and [2], the hysteresis loss is proportional to the filament diameter. Under self-field this is true only if the filaments are placed far enough from each other [17]. However, in these computations the current distribution in single filaments does not agree with the assumptions made when deriving the analytical formulae because the filaments are near each other.…”

“…The detailed implementation of the formulation was introduced in [17]. The computation is done in 2D and only hysteresis loss in superconducting filaments and eddy current loss in matrix metal are taken into account in straight filament structures.…”

“…These currents give a relevant contribution to the overall ac losses. Using the formulation in [17], we can only study fully coupled filaments from the external field point of view, which is the worst case considering the hysteresis losses. Therefore, we study only conductors under self-field caused by the sinusoidal transport current, where coupling is not that significant.…”

A Finite Element Simulation Tool for Predicting Hysteresis Losses in Superconductors Using an H-Oriented Formulation with Cohomology Basis Functions

“…b Close-up of conducting region c of using the tree-co-tree decomposition of the mesh, as was done, e.g. in [32]. On ∂ c ∩ ∂ nc , we ensure the tangential continuity of H by setting t i = I j on all the edges i ∈ ∂ c that belong to the support of the representation of j .…”

An Optimization of a NbTi Wire Cross Section to Attain Low AC-Losses and High Critical Current

“…We review here shortly the eddy current model presented in [10] and introduce two extensions: applied field and exploitation of discrete symmetries.…”