3D modelling of twisted multi-filamentary superconducting wires 

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Critical state model : Bean model

A critical state model (CSM) is often used to represent the dynamics and different operation modes of vortices.
Critical state models are based on the macroscopic behaviour of superconducting materials, derived from experimental observations of the relationship between current density and magnetic field. The critical state occurs when an applied field exceeds a type-II superconductor’s lower critical magnetic field Hc1. Magnetic flux vortices with circulating shielding currents penetrate the material to shield the interior of the material from the applied current/field.
Where there is a current flowing in the superconductor, the magnetic field experiences a Lorentz force Fl = J B . For a large Lorentz force, the vortices become de-pinned and move in the direction of the force with a velocity vv. this vortex movement will induce an electric field E = B vv. Thus, E = (kJk)J.

Minimization of an energy functional

One can define a third family of numerical method based on the minimization of an energy functional. This approach is very intuitive, since it consists in defining a functional that relates the total energy of a system (or a variation of energy with respect to some initial conditions) with the variables that define the state of this system, e.g. potentials, field variables, source terms, etc. It allows more freedom in the way one chooses the shape functions used to approximate the solution. It also allows solving classes of problems that would be difficult to solve otherwise, namely the critical state problem, which is singular in its pure form, and therefore can only be approximated to some extent when using a classical electromagnetic formulation. Although at first sight this approach requires less mathematical formalism than strictly applying the finite element method, the
process of minimizing a functional in order to obtain a well-posed discrete equation system requires good skills in functional analysis and optimization algorithms. Also, since it is generally based on integral equations, its use in 3D is likely to be limited to relatively small problems.
In the HTS community, the use of this method was first introduced by Bossavit [55] and applied by Maslouh [44]. It was also formalized later by Prigozhin [43] as a systematic approach to solve the J distribution in HTS domains based on the critical state model. A variant of the method was later introduced by Sanchez and Navau [45] and improved and generalized by Pardo et al [46] to include, among other things, current constraints.

Finite element – Finite volume hybrid method

It is usually implemented to solve convection-diffusion partial differential equations where the diffusion equations are discretized using the finite element method and the convection equations are discretized using the finite volume method. Moreover, equivalence of the computed projection of the finite element method solution on the finite volume method mesh and vice versa allows mathematically a successful numerical implementation of this hybrid approach.
Specific operators must be implemented in order to successfully compute the equivalent projection of the solution. These operators ensure the coupling between the finite element discrete form and the finite volume discrete form. It can be applied to both structured and unstructured meshed domains of complex geometries. Based on nodes and elements, as control volumes, of the meshed domain, a volume integral formulation of the problem is discretized. According to this approach, the global solution is assumed to be piecewise constant. Kameni et al [59]-[60] implemented this approach for the specific case of modeling, based on the E-formulation, high-temperature superconductors in 2D. In the partial differential equations associated to the E-formulation, the nodal finite element method appears to be not suitable for the treatment of non-linear terms while the finite volume method does not approximate well the diffusion terms because of the gradient operator. Thus, the finite volume method will be implemented in the discrete form based on the nodal finite element method to deal efficiently with the non-linear terms. The nodal finite element method will be used for the discrete form of the finite volume method in order to approximate effectively the diffusion terms. For the coupling of both methods in order to solve the E-formulation, welldefined operators will be derived in order to successfully compute the equivalent projected solution in either finite element or finite volume meshed domain.
The developed method is robust and efficient. Unlike the finite element method, it allows the modeling of superconductors with an n-value as high as 200.

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Discrete variational formulation

The finite element discretization of the equation (2.4) will need an approximation of the unknown magnetic field H based on Nedelec elements. The approximated quantities uh and vh of the temperature T and the magnetic field H will be defined over the mesh Th. They will belong respectively to the following finite element spaces : Uh = u 2 L2( ) : ujK 2 P1(K);K 2 Th.

Straight superconducting filament embedded in a Niobium matrix carrying a sinusoidal transport current

The superconducting wire of 4 mm length is made of one single superconducting filament of 50 m radius embedded in a Niobium matrix of 0:6 mm. The wire is carry a sinusoidal transport current of amplitude Im = 500 A. It is modeled using the non-thermal coupling. The computed AC losses will be compared with the one obtained with an H-formulation finite element model implemented in GetDP only.
The superconducting behaviour of the wire is characterized by a critical electric field Ec = 10􀀀7 V/mm, a critical current density Jc = 50 A/mm2 and a power law exponent n = 20. The transport current frequency will be f = 100 Hz. Computed AC losses, fig.2.2, are equivalent for all the numerical approaches implemented.

Superconducting cube subjected to an alternating transverse magnetic field

A superconducting cube of 2mm side length, subjected to an external magnetic field Ha = Hmsin(2ft)ey, will be modeled using both the non-thermal coupling and thermal coupling formulation with the discontinuous galerkin method [72]. The computed AC losses will be compared with the ones obtained with H-formulation finite element models implemented in GetDP and Comsol Multiphysics. Computations in Comsol Multiphysics will be done for both the non-thermal coupling and thermal coupling only.

Table of contents :

General Introduction
1 Superconductivity and numerical modelling 
1.1 Introduction
1.2 Superconductors : types I and II
1.2.1 Type-I superconductors
1.2.2 Type-II superconductors
1.3 HTS: macroscopic models
1.3.1 Critical state model : Bean model
1.3.2 E-J power law
1.3.3 Kim model
1.3.4 AC losses
1.4 HTS applications
1.5 Mathematical models
1.5.1 H formulation
1.5.2 E formulation
1.5.3 A 􀀀 V formulation
1.5.4 T 􀀀 formulation
1.6 Numerical methods
1.6.1 Minimization of an energy functional
1.6.2 Integral methods
1.6.3 Finite element method
1.6.4 Finite volume method
1.6.5 Finite element – Finite volume hybrid method
1.6.6 Discontinuous Galerkin method
1.7 Conclusion
2 3D modelling of high-temperature superconductors 
2.1 Introduction
2.2 Problem formulation
2.2.1 Constitutive laws
2.2.2 Differential formulation
2.2.3 Variational formulation
2.3 Finite element method
2.3.1 Mesh definition
2.3.2 Discrete variational formulation
2.3.3 Numerical treatment of the non-linearities arising from E = (J)J
2.4 Discontinuous Galerkin method
2.4.1 Mesh definition
2.4.2 Discrete Variational formulation
2.4.3 Spatial approximation
2.4.4 Numerical fluxes terms on the faces of the mesh Th
2.4.5 Boundary conditions
2.4.6 Numerical treatment of the non-linearities arising from E = (J)J
2.5 Comparisons and validations
2.5.1 Straight superconducting filament embedded in a Niobium matrix carrying a sinusoidal transport current
2.5.2 Superconducting cube subjected to an alternating transverse magnetic field
2.6 Conclusion
3 3D modelling of twisted multi-filamentary superconducting wires 
3.1 Introduction
3.2 Influence of a transverse magnetic field on a twisted mono-filament wire
3.3 Mapping from a twisted mono-filament wire to a straight monofilament wire
3.4 Mapping validation on a twisted bi-filaments wire
3.4.1 Twisted bi-filaments superconducting wire subjected to a transverse magnetic field along the y-axis
3.4.2 Twisted bi-filaments superconducting wire subjected to an axial magnetic field along the z-axis
3.5 Mapping validation on a twisted six-filaments wire
3.5.1 Twisted six-filaments superconducting wire subjected to a transverse magnetic field along the y-axis
3.5.2 Twisted six-filaments superconducting wire subjected to an axial magnetic field along the z-axis
3.6 Study of multi-filamentary wires with multiple twist pitch
3.6.1 Influence of the number of twist pitch
3.6.2 Model-order reduction of a twisted multi-filamentary wire with multiple twist pitch
3.6.3 Analysis of the impact of elliptical fields on magnetization losses
3.7 Perspective : homogenization of multi-filamentary superconducting wire
3.8 Conclusion
General conclusion
Bibliography

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