Timo Tarhasaari

Comparison of three eddy current formulations for superconductor hysteresis loss modelling

As is well known, the superconductor hysteresis loss modelling problem may be formulated as an eddy current (EC) problem in which the resistivity of the superconducting region is modelled with a power law. We compare three EC formulations suitable for the modelling of superconductor hysteresis losses. Namely, the a?v?j-, T??- and h-formulations are discussed. We review these formulations, and through simulation results the properties of these formulations are discussed and their suitabilities for different modelling situations are compared. Special attention is paid to the h-formulation: we investigate the effects of the modelling decisions related to resistivity of the air region in an h-formulation based EC solver. According to the results, these decisions affect the energy distribution of the field solution and may even lead to seemingly contradictory behaviour.

An eddy current vector potential formulation for estimating hysteresis losses of superconductors with FEM

Many people these days employ only commercial finite element method (FEM) software when solving for the hysteresis losses of superconductors. Thus, the knowledge of a modeller is in the capability of using the black boxes of software efficiently. This has led to a relatively superficial examination of different formulations while the discussion stays mainly on the usage of the user interfaces of these programs. Also, if we stay only at the mercy of commercial software producers, we end up having less and less knowledge on the details of solvers. Then, it becomes more and more difficult to conceptually solve new kinds of problem. This may prevent us finding new kinds of method to solve old problems more efficiently, or finding a solution for a problem that was considered almost impossible earlier. In our earlier research, we presented the background of a co-tree gauged T– FEM solver for computing the hysteresis losses of superconductors. In this paper, we examine the feasibility of FEM and eddy current vector potential formulation in the same problem.

Programming finite element method based hysteresis loss computation software using non-linear superconductor resistivity and T ? phiv formulation

Due to the rapid development of personal computers from the beginning of the 1990s, it has become a reality to simulate current penetration, and thus hysteresis losses, in superconductors with other than very simple one-dimensional (1D) Bean model computations or Norris formulae. Even though these older approaches are still usable, they do not consider, for example, multifilamentary conductors, local critical current dependency on magnetic field or varying n-values. Currently, many numerical methods employing different formulations are available. The problem of hysteresis losses can be scrutinized via an eddy current formulation of the classical theory of electromagnetism. The difficulty of the problem lies in the non-linear resistivity of the superconducting region. The steep transition between the superconducting and the normal states often causes convergence problems for the most common finite element method based programs. The integration methods suffer from full system matrices and, thus, restrict the number of elements to a few thousands at most. The so-called T ? formulation and the use of edge elements, or more precisely Whitney 1-forms, within the finite element method have proved to be a very suitable method for hysteresis loss simulations of different geometries. In this paper we consider making such finite element method software from first steps, employing differential geometry and forms.

Force and torque computations with hybrid methods

Alternative methods are presented for computing the magnetic forces and torques associated with hybrid solutions of magnetostatic or eddy current problems. The challenge is to retrieve the forces not just accurately but also with as little extra work as possible. The authors analyze several methods and the errors inherent in them. The analysis shows that the equivalent currents method is the best approach for computing forces with hybrid solutions. Force values obtained with test problems by using the proposed methods, including the coupling of the computed forces to the equations of motion, are compared to analytical and measured results.

A programming interface to the Riemannian manifold in a finite element environment

In this paper we present a user-programmable interface to the Riemannian manifold. At the interface, starting from the preprocessor coordinate system (chart), one can define other charts for the manifold and determine the manifold metric properties independently of the preprocessor coordinate chart representation of the manifold. Further, the interface allows one to manipulate vector fields and differential forms as abstract mathematical entities, rather than as their coefficient representations. In contrast to interfaces that use classical vector analysis, the metric is fully isolated at the interface. As an example of finite element modeling, the interface is applied to set up a boundary value problem on a Riemannian manifold.

Hybrid formulations for eddy current problem with moving objects

In this paper hybrid formulations combining differential and integral operators for the eddy current problem with moving objects are studied. In problems including moving bodies hybrid approaches are appealing as the finite element mesh does not have to connect the stationary and moving parts to each other. Both b- and h-oriented formulations are considered for 2D and 3D problems.

Modelling of local strain and stress relaxation in bronze processed Nb3Sn wires

Recently, the need for mechanical modelling of Nb3Sn wires has increased, since these wires are used in large projects such as the International Thermonuclear Experimental Reactor (ITER). The finite element method makes it possible to solve multi-axial stress and strain distributions in complicated wire geometries. However, Nb3Sn conductors possess features that make building the mechanical model challenging and complicate the use of the finite element method. This paper focuses on two problems. First, detailed modelling of geometries with large filament amount can produce numerical problems that are too large for computers. A simple approach to circumvent this problem is presented, and its validity investigated. Second, the uncertainty in modelling thermal stress, caused by stress relaxation and other high-temperature phenomena, is treated. To investigate the credibility of the models, the computed results are compared with critical current and stress–strain data measured for a bronze processed Nb3Sn wire in an axial tension test. Computed results agree well with the measured ones, and show that the mechanical strain state of Nb3Sn in filaments is very sensitive to the degree of stress relaxation.

State variables for coupled circuit-field problems

We study systematical detection of the state variables for a quasi-static coupled circuit-field problem, which may consist of several disjoint domains for both field problems and electric networks. We translate the relevant questions about coupling into linear algebraic problems, i.e., use the homology theory. This enables us to express the state variable detection problem with readily programmable and computable concepts.

Discretization schemes for hybrid methods

In this paper several original discretization strategies for hybrid finite element and integral methods are presented, using two-dimensional magnetostatics as an example. The hybrid formulations are first derived, all making use of equivalent sources. The application of the Galerkin technique and de Rham map to these formulations is then presented, before a short comparison of their respective convergence properties.

Dimensional Reduction in Electromagnetic Boundary Value Problems

If an electromagnetic boundary value problem exhibits appropriate symmetry, it is possible to solve it in a lower dimensional domain. We discuss symmetries in general and in particular show how dimensional reduction is based on symmetry.