IEEE TRANSACTIONS ON MAGNETICS, VOL. 38, NO. 2, MARCH 2002 529 Fast Computational Methods for Large-Scale Eddy-Current Computation Guglielmo Rubinacci, Senior Member, IEEE, Antonello Tamburrino, Member, IEEE, Salvatore Ventre, and Fabio Villone Abstract—In this paper, we present a technique for solving large- scale problems arising from the discretization of an integral formu- lation for three-dimensional eddy current problems in the magne- toquasi-static limit using edge-element-based shape functions. The proposed approach is in the framework of the precorrected fast Fourier transform method (PFFTM) that allows to compute the product of the full stiffness matrix with a vector in operations. A key point of standard PFFTM is the introduction of point-like sources defined onto a regular grid to approximate an ar- bitrary current density in the conductor and to compute the large distance interactions by FFT. Point-like sources are not suitable for representing solenoidal current densities as required for eddy currents problems. In this paper, edge-element-based shape func- tions onto the regular grid are introduced instead of the point-like sources. This allows us to improve the approximation (solenoidal current densities are approximated by solenoidal basis functions) and to reduce further the computational cost. Index Terms—Eddy currents, fast algorithms, finite-element methods, integral equations. I. INTRODUCTION D URING the last decade, the numerical modeling of elec- tromagnetic phenomena has reached a great level of ma- turity. With the increase of computer speed and memory, new problems arising from real-world complex physical situations can be tackled. The present paper addresses the computation for large-scale problems of the eddy currents induced in a linear conductive material by means of an integral formulation. Inte- gral formulations, as opposed to differential formulations, allow to discretize only the regions where materials and sources are present. As a consequence, the problems arising from imposing the regularity conditions of the field at infinity are avoided. However, integral formulations lead to fully populated stiffness matrices requiring operations for the direct inversion and for the storage, being the number of unknowns. For this reason, differential formulations have been preferred for solving large-scale problems. Nevertheless, in the last years, the interest in integral methods for large-scale problems is increasing thanks to algorithms such as the fast multipole method (FMM) [1] and the precor- rected fast Fourier transform method (PFFTM) [2], [3] used in conjunction with iterative inversion methods. Both approaches Manuscript received July 5, 2001; revised October 25, 2001. This work was supported in part by the Italian MURST and ASI. The authors are with the Laboratory of Computational Electromagnetics, Association EURATOM/ENEA/CREATE, DAEIMI, Università degli Studi di Cassino, I-03043 Cassino, Italy (e-mail: rubinacci@unicas.it; tambur- rino@unicas.it; ventre@unicas.it; villone@unicas.it). Publisher Item Identifier S 0018-9464(02)02539-6. share the common idea that the computational speed can be increased by reducing the time spent in the computation of the dense matrix-by-vector product. The strength of both the FMM and the PFFTM is their capability to deal with irregular meshes. The FMM has been successfully applied to static problems [1], [4] and extended to wave propagation [5] problems. The PFFTM has been successfully applied to static problems [2], wave propagation problems [3], eddy-current problems [6], [7], and even in the presence of nonlinear materials [8]. In [6] and [7], which deals with eddy-current computations, a regular three-dimensional grid containing the finite-element mesh of the conducting region is introduced. The large distance interactions are computed using equivalent point-like current densities distributed on a regular grid. The short distance inter- actions are computed without approximation. Specifically, the point-like approximation is obtained imposing a best fit on the field at large distance. Moreover, the large distance interactions are obtained by means of three FFT computations, exploiting the translation invariance of the discretized integral operator on the regular grid. We note that the projection onto the regular grid maps a solenoidal current density into a point-like current density that is nonsolenoidal. In addition, a volumetric current density is represented by a point-like current density. In this paper, we replace the point-like currents with edge-element-based shape functions defined onto the regular grid. This way to approximate the original current density leads to an improvement of the accuracy of the computation of the large distance interactions with a similar computational cost. Indeed, for a regular mesh, the product of the full stiffness matrix with a vector can be written also in this case in terms convolution products as already discussed in [9]. II. MATHEMATICAL MODEL AND NUMERICAL FORMULATION A. Integral Formulation The reference geometry with which we deal consists of a con- ductive body in the free space, where eddy currents are in- duced by a time-varying flux density due to a known source cur- rent density . Assuming, for the sake of simplicity, that the conductor is linear and nonmagnetic, the equations for time harmonic fields, in the magnetoquasi-static limit are (1) (2) 0018-9464/02$17.00 © 2002 IEEE