IEEE TRANSACTIONS ON MAGNETICS, VOL. 48, NO. 2, FEBRUARY 2012 463 Some Treatments of Fictitious Volume Charges in Nonlinear Magnetostatic Analysis by BIE K. Ishibashi , Z. Andjelic , Y. Takahashi , T. Takamatsu , K. Tsuzaki , S. Wakao , K. Fujiwara , and Y. Ishihara Corporate Research, ABB Switzerland Ltd., Baden CH-5405, Switzerland 2-55-12-505 Sangenjaya Setagaya, Tokyo 154-0024, Japan Department of Electrical Engineering Doshisha University Kyoto 610-0321 Japan Department of Electrical Engineering and Bioscience, Waseda University, Tokyo 169-8555, Japan The scalar potential formulation by the boundary integral equation approach is attractive for numerical analysis but has fatal draw- backs due to a multi-valued function in current excitation. We derive an all-purpose boundary integral equation with double layer charges as the state variable and apply it to nonlinear magnetostatic problems by regarding the nonlinear magnetization as fictitious volume charges. We investigate two approaches how to treat the fictitious charges. In discretization by the constant volume element, a surface loop current is introduced for the volume charge. By the linear volume element, the fictitious charges are evaluated on the condition that the divergence of the magnetic flux density is zero. We give a comparative study of these two approaches. Index Terms—Boundary integral equation, double layer charge, fictitious volume charges, iterative solutions, multi-valued function. I. INTRODUCTION T HE magnetostatic analysis by the boundary integral equa- tion (BIE) derived from the scalar potential is fundamental and well known. The most of the BIEs [1]–[7] are derived from an integral representation of scalar potential to give the mag- netic field as [8]. We have another BIE [9] derived from a potential to give the magnetic flux density . The BIE derived from is attractive in the numerical anal- ysis because it contains only one unknown of the double layer charge and has many advantages as given in [10], but only a few papers of the BIE have been reported. Since is equivalent to loop currents [8], gives directly only by the Biot–Savart law. The BIE is capable of solving even at edges and corners, and so at the vertex can be evaluated [11]. Since gives instead of , this approach is free from cancellation errors due to the weak in the case of high magnetic permeability; ac- cordingly, the BIE is applicable to the problems with high per- meability. Generally speaking, the scalar potential formulation has fatal drawbacks due to a multi-valued function of the exciting poten- tial by the source currents and its application has been restricted mostly within simply connected problems. In order to apply the BIE derived from to generic problems, we introduce a ficti- tious loop current as unknown so as to derive an all-purpose inte- gral representation for the exciting potential , and then we apply the proposed approach to a nonlinear problem with multi-mate- rials. The proposed BIE has been formulated by the volume inte- gral equation approach [12], where the magnetic materials are replaced by the magnetization . The nonlinear magnetiza- tion is approximated as follows. In the magnetic moment method (MMM) [13], is reduced to surface charges on the volume element. This treatment does not work well to get good Manuscript received July 06, 2011; revised October 07, 2011; accepted Oc- tober 29, 2011. Date of current version January 25, 2012. Corresponding author: K. Ishibashi (e-mail: yui@g03.itscom.net). Digital Object Identifier 10.1109/TMAG.2011.2174778 convergence in solving iteratively the BIE and surface loop cur- rents are introduced for the surface charges. In the BIE with the single layer charge as unknown [14], [15], is reduced to volume charges on the condition that the divergence of is zero. This treatment is adopted as it is. Concerning these two treatments in evaluation of , we present a comparative study in solving the nonlinear problems without losing BIE’s advan- tages. II. FORMULATION OF MAGNETOSTATIC FIELD A. Integral Representation of Scalar Potential In the volume integral equation approach [12], a magnetic material with the surface and volume is replaced by the magnetization . The potential due to gives originally and at an observation point is given as (1) where denotes the value at is the distance from the in- tegration point to is defined as: = with the magnetic filed and flux density and the magnetic per- meability of free space, and are called the volume and surface charges defined as: with the unit outward normal . The potential in (1) is com- posed of two potentials; the potential due to and due to . The concept of magnetic shell [8], which is com- posed of the double layer charges , suggests that could be replaced by the potential due to . Taking this concept into account, we represent the total potential for giving as (2) where is the potential produced by the exciting source, is the unit normal at , and the permeability is defined as inside outside . 0018-9464/$31.00 © 2012 IEEE