IEEE TRANSACTIONS ON MAGNETICS, VOL. 39, NO. 3, MAY2003 1135 3-D Magnetostatic With the Finite Formulation Maurizio Repetto, Member, IEEE, and Francesco Trevisan Abstract—The paper presents the solution of a three-dimen- sional (3-D) magnetostatic problem using the finite formulation of an electromagnetic field. Two different approaches, based on a primal-dual barycentric discretization of the 3-D space, are pre- sented, considering as unknowns either the magnetic fluxes or the circulations of the vector potential. Results on simple reference configurations are reported and discussed. Index Terms—Finite formulation, magnetic analysis. I. INTRODUCTION T HE USE of finite formulations of field problems is quite widespread in many subjects of numerical analysis and engineering. The main concept behind finite formulations is related to the use of integral variables, like for instance fluxes and line inte- grals of field quantities; these variables are then constrained by means of physical laws like balance equations or topological re- lations between them. As it is put in evidence in the introduction of the Patankar book [1], this approach is a step behind with re- spect to the differential formulations of field problems. While the differential/variational approach has been fundamental for the development of an analytical treatment of field problems, it can be said that the finite approach is more natural for their nu- merical solution. In many areas of numerical analysis of phys- ical problems the finite volume method has been largely used. In computational fluid dynamics, thermal and multiphysics en- vironments the use of this method is, in fact, very frequent. In electromagnetic analysis environment the finite approach was related, in the beginning of the use of numerical methods, to finite difference schemes. Following this research line, very important algorithms and computational procedures were devel- oped, for instance the finite difference time domain (FDTD) [2] algorithm as well as the finite integration technique [3]. Other approaches using finite formulations in electromagnetic sub- jects were naturally tied to circuit expressions of field problems; see, for instance, [4]. Unfortunately, the use of finite difference schemes always have been hindered by their natural application to structured grids. Variational methods, like the finite-element method (FEM), going along well with unstructured grids, have found a much wider application to engineering problems. The work of Tonti in the definition of finite formulation of electromagnetic field (FFEF) [5] has conceptually highlighted the basic aspects of the problems allowing the extensions of the finite approach to generic unstructured space discretizations. Manuscript received June 18, 2002. M. Repetto is with the Dipartimento di Ingegneria Elettrica Industriale, Po- litecnico di Torino, 10129 Torino, Italy (e-mail: repetto@polel1.polito.it). F. Trevisan is with the Dipartimento di Ingegneria Elettrica, Gest. e Mecc. Università di Udine, Udine, Italy (e-mail: trevisan@uniud.it). Digital Object Identifier 10.1109/TMAG.2003.810161 One of the main aspects of the work of Tonti is based on the def- inition of two space grids, which are related by duality topolog- ical constraint which fit very well with the structure of electro- magnetic equations allowing a straightforward implementation of the theoretical scheme in a computational procedure. This concept is of key importance to an efficient solution of electro- magnetic fields as it is well pointed out in [6]. Following this consideration, this work is aimed to the assess- ment of the use of FFEF in a computational procedure based on an unstructured space discretization. Two different algorithms for the solution of the magnetostatic problem are formulated and implemented in computer codes and their performances tested and compared on simple reference cases. II. FINITE FORMULATION AND SPACE DISCRETIZATION According to the finite formulation, it is possible to deduce a set of algebraic equations directly from physical laws instead of getting them from a discretization process applied to differen- tial or integral equations written in terms of the field quantities. Using global variables, like currents or magnetic fluxes, a di- rect discrete formulation of physical laws can be obtained ready for the numerical implementation. In this work, we focus on the solution of the three-dimensional (3-D) magnetostatic field analysis by reformulating the magnetostatic field laws in a di- rect discrete way. Global variables are referred to oriented ge- ometrical elements of a system like points , lines , surfaces , and volumes . These variables are continuous in the pres- ence of different materials and do not require any restriction, like field functions, in terms of derivability conditions on the material media parameters. The global variables relevant to our magnetostatic problem are reported in Table I. Their dependence on the oriented ge- ometrical elements (in bold face) is evidenced within square brackets; moreover, a tilde is used to specify the outer orien- tation of the dual entities respect to the inner orientation of the primal ones. Global variables are related to the field functions by means of an integration performed on oriented lines, sur- faces, volumes, and time intervals; therefore, they are equiva- lent to the commonly used integral variables. A further classification of the global variables, in configura- tion and source variables, is important in the finite formulation of physical laws. Configuration variables describe the pattern of the field while source variables describe its sources. The link between configuration and source variables are the constitutive equations that contain the material properties and the metrical notions. A. Cell Complexes The above classification has a great impact in the numer- ical applications of the finite formulation. Following the theo- 0018-9464/03$17.00 © 2003 IEEE