Some 2-D Multiscale Finite-Element Formulations for the Eddy Current Problem in Iron Laminates

The aim of this paper is to introduce and to study the performance of some multiscale finite-element formulations for the eddy current problem in laminated iron in two dimensions. The case of the main magnetic flux that is parallel to the laminates and assumed to be perpendicular to the plane of projection is considered. Exact solutions of relevant boundary value problems for eddy currents are presented. Multiscale formulations based on the magnetic vector potential (MVP) and the single component current vector potential and on a mixed formulation of the MVP and the current density are discussed. An approach for a multiscale formulation with the MVP is constructed at the best based on an exact solution. The associated weak form of the multiscale finite-element method (MSFEM) is presented. Similar to the MVP, multiscale formulations for the single component current vector potential and for a mixed formulation with the MVP and the current density are studied. This paper does not present a mathematical analysis of the MSFEM for eddy currents in laminates. The performance of the MSFEM is studied by numerous numerical experiments of different examples. This paper covers the topics exact integration versus averaging of coefficients, stability of the MSFEM in a particular case, p-refinement of micro-shape functions and standard finite-element polynomials and edge effect. Simulations show the capability of MSFEM to approximate eddy currents in iron laminates efficiently and accurately. The simulations shall also provide a collection of numerical examples to evaluate other multiscale or homogenization methods.