Numerical simulations of superconducting rings using a Ginzburg-Landau model
In this thesis, we will present numerical simulations of type-II superconducting rings using the time dependent Ginzburg-Landau model of superconductivity. In a type-II superconductor, discrete normal regions form called vortices. Our main objective is to better understand the behavior of these vortices as we change the inner radius of the ring and the magnetic field we apply to the ring. In chapter 1, we present some history of superconductivity. We also present some current and future applications of superconductivity to real world applications. In chapter 2, we present the time dependent Ginzburg-Landau model of superconductivity. In chapter 3, we present a weak formulation of the time dependent Ginzburg-Landau equations on an annulus domain. The weak form is the problem we will approximate to obtain our simulations. To dicretize the problem, we use a Galerkin finite element method in space and the backward Euler method in time. The finite dimensional approximation of the weak solution is presented in Chapter 4. Finally, we will report the numerical results of our experiments in chapter 5. Our numerical simulation will yield a complex order parameter [Psi symbol] = [Single bond symbol Psi symbol single bond symbol]e[Superscript iØ]. [Single bond symbol Psi symbol single bond symbol]2 is the density of superconducting charge carriers. By generating contour plots of [Single bond symbol Psi symbol single bond symbol]2 and vector plots of Ø, we will be able to determine the behavior of the vortices as we vary the applied magnetic field and inner radius of the ring. The motion of a vortex induces resistance in a superconductor, so understanding and controlling vortex behavior is important in the construction of devices that use superconductors.