Finite-Element Method Modeling of Superconductors: From 2-D to 3-D

Abstract

A three-dimensional (3-D) numerical modeling technique for solving problems involving superconducting materials is presented. The model is implemented in finite-element method software and is based on a recently developed 3-D formulation for general electromagnetic problems with solid conductors. It has been adapted for modeling of superconductors with nonlinear resistivity in 3-D, characterized by a power-law relation. It has first been compared with an existing and verified two-dimensional (2-D) model: Compared are the current density distribution inside the conductors and the self-field ac losses for different applied transport currents. Second, the model has been tested for computing the current distribution with typical 3-D geometries, such as corner-shaped and twisted superconductors. Finally, it has been used with two superconducting filaments in the presence of external magnetic field for verifying the existence of coupling currents. This effect deals with the finite length of the conductors and cannot be taken into account by 2-D models.

3-D Finite-Element Modelling of a Maglev System using Bulk High-Tc Superconductor and its Application

“…Basically, the nonlinearity of the equations is eliminated by introducing a linear residual, and the corresponding linear equations can be integrated after calculating the Jacobi matrix for each element. To improve the stability of the calculation, a relaxation coefficient is also introduced as suggested in (Grilli, et al, 2005). The relaxation coefficient is assigned with an initial value before calculation.…”

3-D Transient Modeling of Bulk High-Temperature Superconducting Material in Passive Magnetic Bearing Applications

“…However, the frozen image method ignores effects related to the critical current density in the HTSC and therefore has only limited use. There has been, however, a considerable effort in the 2D and 3D modeling of the HTSC superconducting wire [8], [9] and to simulate the activation process in a bulk HTSC [9], [10]. Reference [8] gives a broad comparison of the 2D and 3D finite element methods to model HTSC using the commonly used potential formulation such as (i) the magnetic vector potential and the electric scalar potential (A-V, A method) and (ii) the magnetic scalar potential and the electric vector potential (T-Ω method).…”

“…There has been, however, a considerable effort in the 2D and 3D modeling of the HTSC superconducting wire [8], [9] and to simulate the activation process in a bulk HTSC [9], [10]. Reference [8] gives a broad comparison of the 2D and 3D finite element methods to model HTSC using the commonly used potential formulation such as (i) the magnetic vector potential and the electric scalar potential (A-V, A method) and (ii) the magnetic scalar potential and the electric vector potential (T-Ω method). Due to the highly non-linear relation (Ohm's Law) between the electric field and the current density in the superconducting material [8] cites assured convergence with the T-Ω method.…”

“…Reference [8] gives a broad comparison of the 2D and 3D finite element methods to model HTSC using the commonly used potential formulation such as (i) the magnetic vector potential and the electric scalar potential (A-V, A method) and (ii) the magnetic scalar potential and the electric vector potential (T-Ω method). Due to the highly non-linear relation (Ohm's Law) between the electric field and the current density in the superconducting material [8] cites assured convergence with the T-Ω method. This is also consistent with the authors experience with the 3D, A-V, A formulation and the T-Ω formulations.…”

Lumped-Parameter Model to Describe Dynamic Translational Interaction for High-Temperature Superconducting Bearings

“…Analysis of these cases requires 3-D Finite Element Method (FEM) models [8], [9] which can become computationally expensive. Control volume techniques may also be applicable, but these also require iterative solution steps like FEM [10], [11].…”