High-Temperature Superconductor Models

High-temperature superconductors have the potential to be used in a wide variety of industrial applications including generators and motors, energy storage, and magnetically levitating trains. One of the most interesting properties of these superconductors is that they allow normal and superconducting regions to coexist in the same sample. Discrete packets of external magnetic flux penetrate the sample, causing penetration regions to have the properties of a normal conductor. The rest of the superconducting sample is protected from these normal regions through the formation of vortices of superconducting electrons. It is well known that the vortices arrange themselves in a hexagonal lattice pattern to minimize the free energy in the sample when the temperature is below a critical temperature. Using the standard discretization scheme on an orthogonal mesh, we have successfully modeled vortex lattice structure in three-dimensional superconducting materials (see the figure below). On the Intel DELTA, this approach obtained sustained computational rates of 4.26 gigaflops with 512 processors.


In these simulations, an orthogonal uniform mesh was required to maintain important gauge invariant symmetries in the discretization. The physical variables of interest change most rapidly at the vortex cores and require a high density of grid points in these locations. Far from the vortex core, however, the solution changes slowly and such a high density of grid points is not required. Thus, uniform meshes are clearly inefficient because the constant density of grid points means that many superfluous grid points are used away from the vortex centers. We use the technology developed as a part of the SUMAA3d Project to adaptively refine the mesh near the vortex cores. Thus fewer total mesh points are used without compromising accuracy in the modeling of vortex core structure.

This approach uses unstructured triangular meshes and a subtle problem that now arises is that standard discretization techniques fail to maintain the gauge invariant symmetries in the numerical approximation. One solution to this problem is to incorporate a fixed gauge into the equations and use finite element analysis with polynomial interpolation functions. However, this approach fails to exactly capture the exponential terms in the boundary conditions which are critical to accurately approximating the free energy functional. To avoid this problem, we have developed a new discretization scheme that maintains gauge invariance on nonorthogonal meshes and accurately models the boundary conditions.

Using this approach we have demonstrated proof-of-principle by modeling two-dimensional superconducting materials (see the figures below for results obtained on the Intel DELTA). We now hope to study materials and phenomena that were previously considered computationally intractable. These studies could include modeling materials in the high kappa regime (where the intervortex spacing is much greater than the intravortex spacing), modeling materials with hundreds of vortices, or efficiently tracking time dependent behavior.

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freitag@mcs.anl.gov / jones@cs.utk.edu / plassman@mcs.anl.gov
Argonne National Laboratory / The University of Tennessee