An optimal linear solver for the Jacobian system of the extreme type-II Ginzburg--Landau problem

This paper considers the extreme type-II Ginzburg--Landau equations, a nonlinear PDE model for describing the states of a wide range of superconductors. Based on properties of the Jacobian operator and an AMG strategy, a preconditioned Newton--Krylov method is constructed. After a finite-volume-type discretization, numerical experiments are done for representative two- and three-dimensional domains. Strong numerical evidence is provided that the number of Krylov iterations is independent of the dimension $n$ of the solution space, yielding an overall solver complexity of O(n)

Simulations of Critical Currents in Polycrystalline Superconductors Using Time-Dependent Ginzburg–Landau Theory

In this thesis, we investigate the in-field critical current density $J_\text{c} (B)$ of polycrystalline superconducting systems with grain boundaries modelled as Josephson-type planar defects, both analytically and through computational time-dependent Ginzburg--Landau (TDGL) simulations in 2D and 3D. For very narrow SNS Josephson junctions (JJs), with widths smaller than the superconducting coherence length, we derive what to our knowledge are the first analytic expressions for $J_\text{c} (B)$ across a JJ over the entire applied magnetic field range. We extend the validity of our analytic expressions to describe wider junctions and confirm them using TDGL simulations. We model superconducting systems containing grain boundaries as a network of JJs by using large-scale 3D TDGL simulations applying state-of-the-art solvers implemented on GPU architectures. These simulations of $J_\text{c} (B)$ have similar magnitudes and dependencies on applied magnetic field to those observed experimentally in optimised commercial superconductors. They provide an explanation for the $B^{-0.6}$ dependence found for $J_\text{c} (B)$ in high temperature superconductors and are the first to correctly provide the inverse power-law grain size behaviour as well as the Kramer field dependence, widely found in many low temperature superconductors

pyTDGL: Time-dependent Ginzburg-Landau in Python

Time-dependent Ginzburg-Landau (TDGL) theory is a phenomenological model for the dynamics of superconducting systems. Due to its simplicity in comparison to microscopic theories and its effectiveness in describing the observed properties of the superconducting state, TDGL is widely used to interpret or explain measurements of superconducting devices. Here, we introduce $\texttt{pyTDGL}$, a Python package that solves a generalized TDGL model for superconducting thin films of arbitrary geometry, enabling simulations of vortex and phase dynamics in mesoscopic superconducting devices. $\texttt{pyTDGL}$ can model the nonlinear magnetic response and dynamics of multiply connected films, films with multiple current bias terminals, and films with a spatially inhomogeneous critical temperature. We demonstrate these capabilities by modeling quasi-equilibrium vortex distributions in irregularly shaped films, and the dynamics and current-voltage-field characteristics of nanoscale superconducting quantum interference devices (nanoSQUIDs).Comment: 14 pages, 6 figures, GitHub repository: https://github.com/loganbvh/py-tdgl , online documentation: https://py-tdgl.readthedocs.io/en/latest