Manifolds in electromagnetism and superconductor modelling: Using their properties to model critical current of twisted conductors in self-field with 2-D model

Physical phenomena do not depend on coordinates or metric used in computations. Keeping this in mind, it is possible to form a more general modelling perspective than many modelling programs offer. The theory of Riemannian manifolds offers foundations for a rigorous formulation of boundary value problems that are often faced in many engineering applications. For classical electromagnetism, differential forms are natural objects to model field quantities on manifolds. Important modelling principles, such as equivalence of two problems and dimensional reduction by continuous symmetry, can be formulated clearly in this framework. Naturally, there are also good tools for implementing software packages based on these ideas. In this paper we introduce this foundation and consider how the critical current of twisted superconductor in self-field can be computed with two dimensional (2-D) model without losing any information. We begin by briefly introducing the general framework for presenting boundary value problems on manifolds. Then, we discuss about the equivalence and symmetry of boundary value problems and we present the equation system we need to solve for magnetostatics problem in 2-D domains characterized by the combination of translation and rotation symmetry in three dimensional Euclidean coordinate system. This formulation is then finally used to compute the critical current of twisted superconductors when the local critical current density – magnetic flux density relation is known.

► Manifolds offer rigorous framework for classical electromagnetism related boundary value problems.
► Differential forms are the most convenient objects to model electromagnetic fields on manifolds.
► Magnetostatic problem in a twisted conductor in self-field can be computed with a 2-D model without losing any information.
► Excellent agreement between results obtained with commercial 3-D program and self-made 2-D program was attained.
► This model can be used to study effect of twisting on self-field critical current.