mathematics Article The Hybrid FEM-DBCI for the Solution of Open-Boundary Low-Frequency Problems Giovanni Aiello, Salvatore Alfonzetti *, Santi Agatino Rizzo and Nunzio Salerno   Citation: Aiello, G.; Alfonzetti, S.; Rizzo, S.A.; Salerno, N. The Hybrid FEM-DBCI for the Solution of Open-Boundary Low-Frequency Problems. Mathematics 2021, 9, 1968. https://doi.org/10.3390/math9161968 Academic Editor: Jacques Lobry Received: 21 June 2021 Accepted: 10 August 2021 Published: 17 August 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Dipartimento di Ingegneria Elettrica, Elettronica e Informatica (DIEEI), Università di Catania, Viale A. Doria, 6, I-95125 Catania, Italy; giovanni.aiello@dieei.unict.it (G.A.); santi.rizzo@unict.it (S.A.R.); nunzio.salerno@unict.it (N.S.) * Correspondence: alfo@dieei.unict.it; Tel.: +39-095-738-2320; Fax: +39-095-330-793 Abstract: This paper describes a particular use of the hybrid FEM-DBCI, for the computation of low-frequency electromagnetic fields in open-boundary domains. Once the unbounded free space enclosing the system has been truncated, the FEM is applied to the bounded domain thus obtained, assuming an unknown Dirichlet condition on the truncation boundary. An integral equation is used to express this boundary condition in which the integration surface is selected in the middle of the most external layer of finite elements, very close to the truncation boundary, so that the integral equation becomes quasi-singular. The method is described for the computation of electrostatic fields in 3D and of eddy currents in 2D, but it is also applicable to the solution of other kinds of electromagnetic problems. Comparisons are made with other methods, concluding that FEM-DBCI is competitive with the well-known FEM-BEM and coordinate transformations for what concerns accuracy and computing time. Keywords: finite element method; integral equations; open-boundary problems; electrostatics; skin effect; GMRES 1. Introduction Computational electromagnetics (CEM) in industrial applications are continuously growing due to the increasing performances of electronic computers and to the devel- opment of several commercial codes based on the finite element method (FEM) [1,2], by means of which complex geometries and materials can be dealt with. However, serious difficulties appear when the device to be analyzed exhibits electro- magnetic fields extending to infinity. Unfortunately, these fields are highly common in several electromagnetic systems. In order to apply FEM, the unlimited domain must be cut by means of a closed trun- cation boundary Γ T . Once the interior bounded domain has been discretized by finite elements, a boundary condition must be imposed on such a boundary, which should be very close to the true one, which is clearly not known. The simplest way to overcome this difficulty is to place Γ T far away from the system core and impose on it a homogeneous con- dition, very often of the Dirichlet type. This approach exhibits poor accuracy-computational effort ratios, especially in the optimized design of three-dimensional electromagnetic de- vices. For this reason, several specific methods have been devised to make the FEM able to compute scalar and vector fields in unbounded domains in a more efficient way. In almost all these methods, the unbounded domain is partitioned into two parts by means of a cutting closed surface Γ T , in such a way that the interior domain is sufficiently small and encloses the core of the system, whereas the unbounded exterior domain is homogeneously constituted of free space. The interior domain is analyzed by means of the FEM, whereas for the exterior one several auxiliary methods have been devised in the literature, starting from the 1970s [3]. Mathematics 2021, 9, 1968. https://doi.org/10.3390/math9161968 https://www.mdpi.com/journal/mathematics