Loops and Meshes Formulations for 3-D Eddy-Current Computation in Topologically Non-Trivial Domains with Volume Integral Equations

The volume integral equation method based on the current vector potential approximated by means edge element basis function is a well-established approach for 3-D eddy currents computation. The application of the method is straightforward when simply connected geometries and no connection with external circuits are involved. In this case, in fact, the solving system is easily obtained based on tree-cotree decomposition of the primal graph. However, when multiply connected geometries or external generators are considered, 'additional' degrees of freedom, not related to interior cotree edges of the primal graph, need to be identified and involved to assure the consistency of the numerical solution. In this article, the link between the volume integral equation method and the circuit is investigated in detail, and the circuit view is used as a guide for systematically finding the additional degrees of freedom arising in case of multiply connected geometries and/or external circuits. In particular, the dual graph is introduced as the support of the circuit and it is shown that this is the natural frame for taking the topology into account. For multiply connected geometries, the additional degrees of freedom are related to loop currents crossing one only time (or an odd number of times) any cutting surface making the domain simply connected, and are found by applying a minimum path algorithm on the dual graph forming the circuit. In case of conducting domain connected to an external generator, an extended dual graph is introduced for finding the further additional degrees of freedom. This article also researches into the possibility to replace the usual current density of the elements, obtained via facet-element shape functions and exactly matching the current of the faces, with a uniform current density obtained by means of a minimum error procedure and approximately matching the current of the faces. The use of this uniform current density, besides improving the calculation time and the accuracy of the coupling coefficients, also allows the extension of the volume integral equation method to discretizations of the problem domain made of arbitrarily shaped polyhedral elements.