Finite formulation for quasi-magnetostatics with integral boundary conditions P. Alotto ∗ , L. Codecasa † , F. Freschi ‡ , G. Gruosso † , F. Moro ∗ and M. Repetto ‡ ∗ Dipartimento di Ingegneria Elettrica - Università di Padova, Via Gradenigo 6/A, I-35131, Padova, Italy † Dipartimento di Elettronica e Informazione - Politecnico di Milano, P.zza L. Da Vinci, 32 , I-20133 Milano, Italy ‡ Dipartimento di Ingegneria Elettrica - Politecnico di Torino, C.so Duca degli Abruzzi 24, I-10129 Torino, Italy E-mail: gruosso@elet.polimi.it Abstract: A finite formulation for eddy current problems in unbounded domains is presented. The method involves the use of integral boundary conditions to model free space. The formulation has been applied to the analysis of several significant test cases in frequency domain. Keywords: Finite Formulation, Eddy Current, Magnetic Field, Hybrid Formulation. I. I NTRODUCTION In recent years several approaches based on the Finite Formulation of Electromagnetic Fields (FFEF)[1] for the solution of eddy current problems have been developed [2],[4]. All these methods need the discretization of free space thus requiring advanced and robust mesh generators. Furthermore, such approaches may require a very large amount of elements in the air region. In this work an orig- inal method, obtained by coupling the FFEF with integral boundary conditions based on a magnetic scalar potential Boundary Element Method (BEM) [6] is presented. The main advantage of the proposed method is that only active parts of the domain need to be discretised. In the present paper the formulation will be presented and some results will be discussed. In section II the FFEF will be recalled and the basis for the quasi static formulation will be introduced by means of global variables. After- wards, in section III the coupling of FFEF formulation with BEM will be derived and the system of equations for the quasi-static formulation will be presented. At the end in section IV some results will be presented and compared with those obtained by other methods. II. FFEF FORMULATION FOR QUASI - STATIC PROBLEM In this work, the magneto-static problem is formulated resorting to FFEF [1]. According to this method, physical laws are written directly in algebraic form, ready for nu- merical implementation. FFEF description is here limited to its magneto-static part only. In the following some short remarks about the formulation are presented: A. Electromagnetic variables definition and geometrical discretization The Global Variables,(GV) are domain functions, they refer to oriented space and/or time elements, such as points, lines, surfaces and volumes. A possible classifica- tion among them is based on the role of the physical vari- ables and distinguishes between Configuration Variables, (CV) and Source Variables (SV). The first ones describe the configuration of the field or of the system. Examples of CV are the circulation of the mag- netic vector potential a and the magnetic flux b. Source variables describe the sources of the field or the forces acting on the system. Examples of source variables are the electric current I and the magneto-motive force (mmf) h. In order to define global variables space-time elements must be defined and they must be oriented. The region where physical phenomena are considered is discretized, in our case, by simplicial geometrical elements (tetrahe- dral elements), resulting in a simplicial mesh S, whose ge- ometrical elements are nodes P , edges E, surfaces S and volumes V . From the simplicial complex, a barycentric one B is derived [1];it is made of geometrical elements of the same type of primal one. The duality between the two complexes S and B assures that the geometrical ele- ments correspond as [1]: P (S) ↔ V (B) , E (S) ↔ S (B) , S (S) ↔ E (B) , V (S) ↔ P (B) . B. FFEF formulation The complexes which CV and SV are associated to are denoted as primal and dual mesh, respectively. The CV involved in magneto-quasi-statics are the magnetic fluxes b on primal faces, the integral of the magnetic vector po- tential on primal edges a, the integral of the electromotive force on primal edges e and the electrical scalar potential on primal nodes ϕ. The SV are the mmf on dual edges h, the currents on dual faces I and the mmf due to perma- nent magnet on dual edge h c . Dual relations between mesh complexes can be efficiently written by specific matrices linking the different space elements [1]. Matrix D (NT × NF ), expresses links between pri- mal volumes and their bounding surfaces while C (N F × NE) contains the relations between primal faces and their bounding edges. These matrices are consist of ±1 and 0 like the topological matrices used in network analysis. Ma- trices ˜ D and ˜ C play the same role on dual space elements. Once topological matrices have been defined, topological laws can be written in compact way.  Ch = I (1) b = Ca (2) e = -jω a - G ϕ (3)  DI = 0 (4) For the constitutive relations the situation is quite dif- ferent: in fact, since constitutive laws are defined in terms of field variables, some relation is needed to write them in terms of GV [3]. This process is quite straightforward in the case of orthogonal meshes, where a local correspon- dence between CV and SV around a couple of dual ele- ments, for instance primal edge and dual face, allows one Proceedings - 359 - 12th International IGTE Symposium