Journal of Materials Processing Technology 161 (2005) 315–319 Boundary element method for multiply connected domains F. Hantila ∗,1 , M. Vasiliu 2 , M. Maricaru, A. Della Giacomo Universitatea POLITEHNICA din Bucuresti, Bucharest 060042, Romania Abstract Two procedures in the boundary element method (BEM) for multiply connected domains are reported. The first procedure introduces cuts such that the multiply connected domain is transformed into a simply connected one, so that it allows the use of the scalar potential. Each cut introduces an additional term containing the “hall” current multiplied by the solid angle under which the cut is seen. The second procedure is based on a novel magnetic vector potential formulation. It uses edge elements and tree–cotree spanning. A zero normal component of this vector potential A and the condition for its line integral along any closed path on the boundary are imposed, such that the continuity of the normal component of the magnetic flux density be rigorously satisfied. The unknowns are the tangential components of ▽ × A and the tree edge element values. © 2004 Elsevier B.V. All rights reserved. Keywords: Boundary element method; Scalar and vector magnetic potential; Edge elements; Tree–cotree techniques 1. Introduction Boundary element method is often used for solving the magnetic field problems in free space and for obtaining the stiffness matrix for boundary conditions in both magneto- static and eddy current problems. Media in motion may also be introduced – the stiffness matrix is time dependent, the electromagnetic field equations are expressed in terms of La- grangean coordinates attached to the bodies. It is well known that the smaller number of unknowns used in scalar potential formulations constitutes an impor- tant advantage in comparison with the formulations based on vector potential ones. Unfortunately, use of scalar poten- tial is accompanied by a lower accuracy in the more general case of inhomogeneous media, when large differences in the magnetic permeability values occur. Some difficulties also appear in the case of multiply connected regions. One way to overcome these difficulties is to separate the field component having  Γ H dl = i and to use the reduced scalar potential in BEM. Another way, presented in this paper, is to use di- rectly the scalar potential, together with cut contributions. ∗ Corresponding author. E-mail address: hantila@elth.pub.ro (F. Hantila). 1 Member, IEEE. 2 Student member, IEEE. Inherently, a number of additional problems must be solved in order to obtain the stiffness matrix. Several vector potential formulations for the boundary in- tegral method have been reported [1–3]. In most problems the vector potential A and the tangential component of ▽ × A are the unknowns and the discretization is performed by the nodal elements. Involving a greater number of unknowns and dealing with singularities of the surface integrals are major disadvantages. The aim of this paper is to introduce a novel BEM algo- rithm based on magnetic vector potential particularly suitable for multiply connected domains. 2. Scalar potential formulation The integral equation used in scalar BEM formulation for a simply connected domain Ω is given by: αV (r) =  ∂Ω 1 R ∂V (r ′ ) ∂n ′ dS ′ -  ∂Ω (R · n ′ ) R 3 V (r ′ )dS ′ + V 0 (1) where V is the scalar potential; ∂Ω is the boundary of the domain Ω; α is the solid angle under which a small neigh- bourhood of Ω is seen from the observation point; r and r ′ 0924-0136/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2004.07.043