Foundations of volume integral methods for eddy current problems

Integral methods for solving eddy current problems use Biot–Savart law to produce non-local constitutive relations that lead to fully populated generalized mass matrices, better known as inductance matrices. These formulations are appealing because—unlike standard Finite Element solutions—they avoid the generation of a mesh in the insulating regions. The aim of this paper is to alleviate the three main problems of volume integral methods. First, the computation of the inductance matrix elements is slow and also delicate because of the singularity in the integral equation. This paper introduces novel face basis functions that allow a much faster inductance matrix construction with respect to the standard one based on the Rao–Wilton–Glisson (RWG) or Raviart–Thomas (RT) basis functions. Second, our basis functions work for polyhedral elements formed by any number of faces (including prisms, hexahedra and pyramids), while producing the same results as RWG and RT basis functions for tetrahedral meshes. Third, the new basis functions allow to factorize the inductance matrix and to introduce a novel family of groundbreaking low-rank inductance matrix compression techniques that show several orders of magnitude improvement in memory occupation and computational effort than state-of-the-art alternatives, allowing to solve problems that otherwise cannot be faced.