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This paper aims to solve inhomogeneous dielectric problems by matching boundary conditions at the interfaces among homogeneous subdomains. The capabilities of Hilbert transform computations are deeply investigated in the case of limited numbers of samples, and a refined model is presented by means of investigating accuracies in a case study with three subdomains.

The accuracies, refined by Richardson extrapolation to zero error, are compared to finite element (FEM) and finite difference methods. The boundary matching procedures can be easily applied to the results of a previous Schwarz–Christoffel (SC) conformal mapping stage in SC + BC procedures, to cope with field singularities or with open boundary problems.

The proposed field computations are of general interest both for electrostatic and magnetostatic field analysis and optimization. They can be useful as comparison tools for FEM results or when severe field singularities can impair the accuracies of other methods.

This static field methodology, of course, can be used to analyse transverse electro magnetic (TEM) or quasi-TEM propagation modes. It is possible that, in some case, these may make a contribution to the analysis of axis symmetrical problems.

The most relevant result is the possible introduction of SC + BC computations as a standard tool for solving inhomogeneous dielectric field problems.

A novel method for modelling permanent magnets is investigated based on numerical approximations with rational functions. This study aims to introduce the AAA algorithm and other recently developed, cutting-edge mathematical tools, which provide outstandingly fast and accurate numerical computation of potentials and vector fields.

First, the AAA algorithm is briefly introduced along with its main variants and other advanced mathematical tools involved in the modelling. Then, the analysis of a circular Halbach array with a one-pole pair is carried out by means of the AAA-least squares method, focusing on vector potential and flux density in the bore and validating results by means of classic finite element software. Finally, the investigation is completed by a finite difference analysis.

AAA methods for field analysis prove to be strikingly fast and accurate. Results are in excellent agreement with those provided by the finite element model, and the very good agreement with those from finite differences suggests future improvements. They are also easy programming; the MATLAB code is less than 200 lines. This indicates they can provide an effective tool for rapid analysis.

AAA methods in magnetostatics are novel, but their extension to analogous physical problems seems straightforward. Being a meshless method, it is unlikely that local non-linearities can be considered. An aspect of particular interest, left for future research, is the capability of handling inhomogeneous domains, i.e. solving general interface problems.

The authors use cutting-edge mathematical tools for the modelling of complex physical objects in magnetostatics.

Gives introductory remarks about chapter 1 of this group of 31 papers, from ISEF 1999 Proceedings, in the methodologies for field analysis, in the electromagnetic community. Observes that computer package implementation theory contributes to clarification. Discusses the areas covered by some of the papers ‐ such as artificial intelligence using fuzzy logic. Includes applications such as permanent magnets and looks at eddy current problems. States the finite element method is currently the most popular method used for field computation. Closes by pointing out the amalgam of topics.