Simulation Tools and Developments on Integral Formulations for the Computation of Eddy Currents

Computational electromagnetics is a discipline that since many years ago has permitted deep innovations in the study of electromagnetic problems. Even if, nowadays, commercial softwares undeniably show a certain maturity when applied to practical problems, some research work has still to be done in going beyond the theoretical limits underneath the various approaches. With respect to this, integral formulations still present some open issues. Historically, the exploitation of these formulations to study eddy currents started around the 90s with the seminal works of G. Albanese, R. Martone and R. Rubinacci together with the research activity of L. Kettunen and L. R. Turner and then with G. Meunier, who more recently rediscovered them. Lately, the contributions of L. Codecasa, R. Specogna and F. Trevisan have further increased the possibilities offered by this approach by introducing a set of new shape functions for polyhedral grids that are based on a discrete geometrical reinterpretation of the physics of electromagnetic phenomena. One of the main features characterizing integral formulations to compute eddy currents stems from the fact that they do not require any discretization of the complement of the conductor to be studied. As a drawback, they lead to fully populated matrices whose assembly results to be remarkably time consuming and whose size can sometimes saturate the memory of the calculator. In this respect, this composition presents a new volume integral code for polyhedral grids describing how a fast and efficient cohomology computation can be implemented to treat also non-simply connected domains. Then, some tools are provided for the reduction of the size, and thus of the assembly time too, of the fully populated matrix. More precisely, the attention is focused on the exploitation of cyclic symmetry and on the novel topology-related issues arising when integral formulations have to be referred only to the symmetry cell of the complete conducting domain in order not to spoil the block-circulant property of the system matrix when building the cohomology generators or the gauging tree. Furthermore, also new iterative methods are considered as additional approaches to limit the size of the system matrix to be assembled: despite being already known to the computational electromagnetics community, their convergence behaviour has not been studied yet when they are applied to integral formulations as the one here proposed. Specifically, after presenting a purely iterative scheme derived from the volume integral formulation whose convergence can be somehow problematic, we propose a new direct-iterative method based on Krylov subspace techniques and on the domain splitting into multiple conductors that exhibits a much improved behaviour. The study of these methods leads to new interesting findings to be considered in addition to matrix compression techniques.