The boundary element method for the eddy current problem (BEM-ECP) was proposed in a number of papers and is applicable to important tasks such as the problem of inductive heating and transmission of electromagnetic energy. BEM-ECP requires the construction of a system of linear algebraic equations in which the matrix is inherently dense and is constructed out of element matrices. For the process of the element matrix computation, two cases are normally considered: far-field interaction and near-field interaction, because the construction of element matrices requires integration of a singular function. In this article, we suggest a transform that allows computing the matrix components of the near-singular interaction part while implementing only the single and double layer potentials. The previously suggested modified double layer potential (MDLP) can be integrated by means of this transform, which simplifies the program implementation of BEM-ECP significantly. Solving model problems, we analyse the drawbacks of the previously suggested approach. This analysis includes the proof of the MDLP singularity that makes the integration of this potential a rather difficult task without the help of our transform. The previously suggested approach does not work well with surfaces that are not smooth. Our approach does consider such cases, which is its main advantage. We demonstrate this on the model problems with known analytical solutions.

This paper studies the magneto-heat coupling model which describes iron loss of conductors and energy exchange between magnetic field and Ohmic heat. The temperature influences Maxwell's equations through the variation of electric conductivity, while electric eddy current density provides the heat equation with Ohmic heat source. It is in this way that Maxwell's equations and the heat equation are coupled together. The system also incorporates the heat exchange between conductors and cooling oil which is poured into and out of the transformer. We propose a weak formulation for the coupling model and establish the well-posedness of the problem. The model is more realistic than the traditional eddy current model in numerical simulations for large power transformers. The theoretical analysis of this paper paves a way for us to design efficient numerical computation of the transformer in the future.

In this paper, hp-adaptive finite element methods are studied for time-harmonic Maxwell’s equations. We propose the parallel hp-adaptive algorithms on conforming unstructured tetrahedral meshes based on residual-based a posteriori error estimates. Extensive numerical experiments are reported to investigate the efficiency of the hp-adaptive methods for point singularities, edge singularities, and an engineering benchmark problem of Maxwell’s equations. The hp-adaptive methods show much better performance than the h-adaptive method.

We consider the time-harmonic eddy current problem in its electric formulationwhere the conductor is a polyhedral domain. By proving the convergencein energy, we justify in what sense this problem is the limit of a family of Maxwell transmission problems: Rather than a low frequency limit, this limit has to be understood in the sense of Bossavit [11].We describe the singularities of the solutions.They are related to edge and corner singularities of certain problems for the scalarLaplace operator, namely the interior Neumann problem, the exterior Dirichlet problem, and possibly, an interface problem. These singularities are the limit of the singularities of the related family of Maxwell problems.