Numerical simulation of the magnetization of high-temperature superconductors: 3D finite element method using a single time-step iteration

We make progress towards a 3D finite-element model for the magnetization of a high temperature superconductor (HTS): We suggest a method that takes into account demagnetisation effects and flux creep, while it neglects the effects associated with currents that are not perpendicular to the local magnetic induction. We consider samples that are subjected to a uniform magnetic field varying linearly with time. Their magnetization is calculated by means of a weak formulation in the magnetostatic approximation of the Maxwell equations (A-phi formulation). An implicit method is used for the temporal resolution (Backward Euler scheme) and is solved in the open source solver GetDP. Picard iterations are used to deal with the power law conductivity of HTS. The finite element formulation is validated for an HTS tube with large pinning strength through the comparison with results obtained with other well-established methods. We show that carrying the calculations with a single time-step (as opposed to many small time-steps) produce results with excellent accuracy in a drastically reduced simulation time. The numerical method is extended to the study of the trapped magnetization of cylinders that are drilled with different arrays of columnar holes arranged parallel to the cylinder axis

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)

A Posteriori Error Estimation for the p-curl Problem

We derive a posteriori error estimates for a semi-discrete finite element approximation of a nonlinear eddy current problem arising from applied superconductivity, known as the $p$-curl problem. In particular, we show the reliability for non-conforming N\'{e}d\'{e}lec elements based on a residual type argument and a Helmholtz-Weyl decomposition of $W^p_0(\text{curl};\Omega)$. As a consequence, we are also able to derive an a posteriori error estimate for a quantity of interest called the AC loss. The nonlinearity for this form of Maxwell's equation is an analogue of the one found in the $p$-Laplacian. It is handled without linearizing around the approximate solution. The non-conformity is dealt by adapting error decomposition techniques of Carstensen, Hu and Orlando. Geometric non-conformities also appear because the continuous problem is defined over a bounded $C^{1,1}$ domain while the discrete problem is formulated over a weaker polyhedral domain. The semi-discrete formulation studied in this paper is often encountered in commercial codes and is shown to be well-posed. The paper concludes with numerical results confirming the reliability of the a posteriori error estimate.Comment: 32 page

Electrostatic potential in a superconductor

The electrostatic potential in a superconductor is studied. To this end Bardeen's extension of the Ginzburg-Landau theory to low temperatures is used to derive three Ginzburg-Landau equations - the Maxwell equation for the vector potential, the Schroedinger equation for the wave function and the Poisson equation for the electrostatic potential. The electrostatic and the thermodynamic potential compensate each other to a great extent resulting into an effective potential acting on the superconducting condensate. For the Abrikosov vortex lattice in Niobium, numerical solutions are presented and the different contributions to the electrostatic potential and the related charge distribution are discussed.Comment: 19 pages, 11 figure

On numerical methods for diffusion of electric fields in type-II superconductors

The thesis is devoted to the study of the diffusion of the electric field in type-II superconductors in low-frequency electromagnetism. The necessity for accurate numerical methods in this research domain is increasing along with the growing number and importance of industrial applications of type-II superconductors. In Chapter 1, the basic information on superconductors is shortly summarized. The mathematical model is derived based on the power law relation between electric field E and current density J describing the nonlinear resistivity behavior of type-II superconductors. The obtained model is a nonlinear degenerate transient eddy-current problem which requires deep mathematical analysis and exact study of appropriate numerical methods. Three different versions of the problem are studied: the easiest problem involving a Lipschitz-continuous modification of the E-J relation, the non-Lipschitz but coercive model based on another modification of the power law, and the most complex problem, where the unmodified power law is considered. Different stages of the analysis are worked out for these different versions of the E-J relation. The properties of special function spaces needed for the analysis of the problem are studied in Chapter 3. Basic information on the finite element method suitable for discretization of Maxwell's equations is given in Chapter 4. In Chapter 5, the convergence of the backward Euler method is studied under the assumption that the nonlinearity is defined by the unmodified power law. We deduce the convergence of the method, carry out the error estimates and present the numerical experiments. The highlight of Chapter 5 is the generalization of the div-curl lemma. In Chapter 6 we propose a linear iteration scheme to solve a 3D stationary problem. Convergence of the method, error estimates and numerical examples are carried out. The method is stable and efficient. It is based on the fixed-point principle, which constrains its speed. The convergence of the relaxation method inspired by the articles of Jäger and Kačur is studied in Chapter 7 as preliminary to possible more extensive work in this field. In the last chapter, we propose two fully discrete methods. Their convergence is proven on the basis of the error estimates

A nonlocal parabolic model for type-I superconductors

A

Modelling nonlinear effects in high temperature superconducting magnets

In the future particle colliders, the accelerator magnets keeping the particles on their tracks are required to produce magnetic fields above 20 T. This can be achieved only by using high temperature superconductors. The technology for producing the high temperature superconducting (HTS) conductors is relatively new and only recently the number of HTS conductor manufacturers has started to increase. It was only in 2016, when a 10 kA class Roebel cable made of REBCO tapes was tested in a small study coil, Feather-M0. Followed by that, in 2017 the first Roebel cable based 5 T accelerator magnet prototype Feather-M2 was constructed and tested to examine the prospects of HTS REBCO technology in accelerator magnets. The measurement results suggested that there is still a lot to learn in modelling those magnets. This thesis begins by introducing the readers to the mathematical and physical background for understanding the research presented in the attached publications. The background is followed by the chapters reviewing and synthetizing the publications. The focus in this thesis is on the AC loss modelling and thermal stability modelling. First, AC losses and magnetic field quality are modelled in Feather-M0 using a self-implemented minimum magnetic energy variation principle based simulation tool. Then, the focus is moved on the thermal stability modelling of HTS magnets by formulating the thermal model utlized in this thesis work. Next, the thermal model is utilized for scrutinizing the behavior of Feather-M2 with an inverse problem based modelling approach. Using the Feather-M2 measurement data, the inverse problem solutions are obtained for the thermal model parameters characterizing the magnet in terms of the thermal model. Furthermore, the thermal model is utilized and an optimization problem is formulated in order to determine the maximum stable operation current of Feather-M2. Finally, an energy-extraction system (EES) design for 20 T range magnet is presented and optimized by formulating and solving an optimization problem

Scalable domain decomposition methods for finite element approximations of transient and electromagnetic problems

The main object of study of this thesis is the development of scalable and robust solvers based on domain decomposition (DD) methods for the linear systems arising from the finite element (FE) discretization of transient and electromagnetic problems. The thesis commences with a theoretical review of the curl-conforming edge (or Nédélec) FEs of the first kind and a comprehensive description of a general implementation strategy for h- and p- adaptive elements of arbitrary order on tetrahedral and hexahedral non-conforming meshes. Then, a novel balancing domain decomposition by constraints (BDDC) preconditioner that is robust for multi-material and/or heterogeneous problems posed in curl-conforming spaces is presented. The new method, in contrast to existent approaches, is based on the definition of the ingredients of the preconditioner according to the physical coefficients of the problem and does not require spectral information. The result is a robust and highly scalable preconditioner that preserves the simplicity of the original BDDC method. When dealing with transient problems, the time direction offers itself an opportunity for further parallelization. Aiming to design scalable space-time solvers, first, parallel-in-time parallel methods for linear and non-linear ordinary differential equations (ODEs) are proposed, based on (non-linear) Schur complement efficient solvers of a multilevel partition of the time interval. Then, these ideas are combined with DD concepts in order to design a two-level preconditioner as an extension to space-time of the BDDC method. The key ingredients for these new methods are defined such that they preserve the time causality, i.e., information only travels from the past to the future. The proposed schemes are weakly scalable in time and space-time, i.e., one can efficiently exploit increasing computational resources to solve more time steps in (approximately) the same time-to-solution. All the developments presented herein are motivated by the driving application of the thesis, the 3D simulation of the low-frequency electromagnetic response of High Temperature Superconductors (HTS). Throughout the document, an exhaustive set of numerical experiments, which includes the simulation of a realistic 3D HTS problem, is performed in order to validate the suitability and assess the parallel performance of the High Performance Computing (HPC) implementation of the proposed algorithms.L’objecte principal d’estudi d’aquesta tesi és el desenvolupament de solucionadors escalables i robustos basats en mètodes de descomposició de dominis (DD) per a sistemes lineals que sorgeixen en la discretització mitjançant elements finits (FE) de problemes transitoris i electromagnètics. La tesi comença amb una revisió teòrica dels FE d’eix (o de Nédélec) de la primera família i una descripció exhaustiva d’una estratègia d’implementació general per a elements h- i p-adaptatius d’ordre arbitrari en malles de tetraedres i hexaedres noconformes. Llavors, es presenta un nou precondicionador de descomposició de dominis balancejats per restricció (BDDC) que és robust per a problemes amb múltiples materials i/o heterogenis definits en espais curl-conformes. El nou mètode, en contrast amb els enfocaments existents, està basat en la definició dels ingredients del precondicionador segons els coeficients físics del problema i no requereix informació espectral. El resultat és un precondicionador robust i escalable que preserva la simplicitat del mètode original BDDC. Quan tractem amb problemes transitoris, la direcció temporal ofereix ella mateixa l’oportunitat de seguir explotant paral·lelisme. Amb l’objectiu de dissenyar precondicionadors en espai-temps, primer, proposem solucionadors paral·lels en temps per equacions diferencials lineals i no-lineals, basats en un solucionador eficient del complement de Schur d’una partició multinivell de l’interval de temps. Seguidament, aquestes idees es combinen amb conceptes de DD amb l’objectiu de dissenyar precondicionadors com a extensió a espai-temps dels mètodes de BDDC. Els ingredients clau d’aquests nous mètodes es defineixen de tal manera que preserven la causalitat del temps, on la informació només viatja de temps passats a temps futurs. Els esquemes proposats són dèbilment escalables en temps i en espai-temps, és a dir, es poden explotar eficientment recursos computacionals creixents per resoldre més passos de temps en (aproximadament) el mateix temps transcorregut de càlcul. Tots els desenvolupaments presentats aquí són motivats pel problema d’aplicació de la tesi, la simulació de la resposta electromagnètica de baixa freqüència dels superconductors d’alta temperatura (HTS) en 3D. Al llarg del document, es realitza un conjunt exhaustiu d’experiments numèrics, els quals inclouen la simulació d’un problema de HTS realista en 3D, per validar la idoneïtat i el rendiment paral·lel de la implementació per a computació d’alt rendiment dels algorismes proposatsPostprint (published version