Topology optimization of type-II superconductors with superconductor-dielectric/vacuum interfaces based on Ginzburg-Landau theory under Weyl gauge

TL;DR

This work addresses inverse geometry design for type-II superconductors with superconductor-dielectric/vacuum interfaces using topology optimization grounded in the time-dependent Ginzburg-Landau theory under the Weyl gauge. A density-based material distribution with a split real-valued TDGL formulation enables continuous adjoint sensitivities for efficient optimization. Key contributions include a convexity-tunable $q$-parameter material interpolation for the GL parameter $κ$, an PDE-based density filter with threshold projection to enforce clear boundaries, and an adjoint framework that minimizes the time-integrated supercurrent density $|\mathbf{j}_s|^2$ to delay flux penetration. The results show robust flux pinning and delayed or suppressed flux entry across low- and high-temperature superconductors, with anisotropy shaping distinct optimized topologies, and highlight potential applications in superconducting magnets and quantum devices.

Abstract

Geometry design is a crucial and challenging strategy for improving the performance of type-II superconductors. Topology optimization is one of the most powerful approaches used to determine structural geometries. Therefore, a topology optimization approach is presented to inversely design structural geometries of both low- and high-temperature type-II superconductors with superconductor-dielectric/vacuum interfaces. In the presented approach, the magnetic response of type-II superconductors is modeled using the Ginzburg-Landau theory, where the temporal evolution of the order parameter and vector potential is described by the time-dependent Ginzburg-Landau equations under the Weyl gauge.

Figures (18)

• Figure 1: Phase diagrams of superconductors and sketches for the states of a type-II superconductor: (a) phase diagram of a type-I superconductor, where $H$ is the externally applied magnetic field, $H_c$ is the thermodynamic critical magnetic field, $T$ is the temperature and $T_c$ is the critical value of the temperature for the phase transition, respectively; (b) phase diagram of a type-II superconductor, where $H_{c1}$ and $H_{c2}$ are the lower and upper critical values of the magnetic field for the first and second second-order phase transitions, respectively; (c) sketches for the distribution of magnetic induction lines in the Meissner, mixed and normal states, where the lines with arrows are magnetic induction lines and the superconductors are marked in purple color. In (c), the magnetic field can not penetrate into the interior of the superconductor in the Meissner state; it can penetrate completely into the interior in the normal state; and the intermediate state between the Meissner and normal states is the mixed state, where the magnetic field can penetrate into normal cores of the flux lines sketched in Fig. \ref{['fig:SketchFluxLineAndMixedState']}.
• Figure 2: Sketch for flux lines/supercurrent vortices in the mixed state of a type-II superconductor, where $\mathbf{H}$ is the applied magnetic field. In the sketch, every flux line has a normal core which can be approximated by a thin cylinder with its axis parallel to the applied magnetic field, and this normal core is surrounded by circulating supercurrents.
• Figure 3: Sketches for the Cooper-pair density and magnetic induction in the superconductor and dielectric/vacuum: (a) Cooper-pair density and magnetic induction in a flux line in the type-II superconductor; (b) distribution of the Cooper-pair density and magnetic induction in the dielectric/vacuum. In the sketches, $r$ is the radial direction in the cross-section of the flux line; $\lambda$ and $\xi$ are the penetration depth and coherence length, respectively; and $\left|\psi\right|$ and $\left|\mathbf{B}\right|$ are the Cooper-pair density and magnetic-induction modulus, respectively.
• Figure 4: Graphs of the material interpolations in Eq. \ref{['eq:MaterialInterpolation']} for different values of $q$ used to tune the convexity: (a) graphs for $\kappa^{-1}$ and $I_d$; (b) graphs for $w_p$.
• Figure 5: Sketches for the first-order nodal elements and zeroth-order discontinuous element in the hexahedron shapes: (a) first-order nodal element; (b) zeroth-order discontinuous element.