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COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Emerald Article: Nonlinear magnetostatic BEM formulation using one unknown double layer charge Kazuhisa Ishibashi, Zoran Andjelic Article information: To cite this document: Kazuhisa Ishibashi, Zoran Andjelic, (2011),"Nonlinear magnetostatic BEM formulation using one unknown double layer charge", COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, Vol. 30 Iss: 6 pp. 1870 - 1884 Permanent link to this document: http://dx.doi.org/10.1108/03321641111168165 Downloaded on: 15-05-2012 References: This document contains references to 15 other documents To copy this document: permissions@emeraldinsight.com This document has been downloaded 165 times. 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The organization is a partner of the Committee on Publication Ethics (COPE) and also works with Portico and the LOCKSS initiative for digital archive preservation. *Related content and download information correct at time of download. The current issue and full text archive of this journal is available at www.emeraldinsight.com/0332-1649.htm COMPEL 30,6 Nonlinear magnetostatic BEM formulation using one unknown double layer charge 1870 Kazuhisa Ishibashi and Zoran Andjelic Corporate Research, ABB Switzerland Ltd, Baden, Switzerland Abstract Purpose – The purpose of this paper is to solve generic magnetostatic problems by BEM, by studying how to use a boundary integral equation (BIE) with the double layer charge as unknown derived from the scalar potential. Design/methodology/approach – Since the double layer charge produces only the potential gap without disturbing the normal magnetic ﬂux density, the ﬁeld is accurately formulated even by one BIE with one unknown. Once the double layer charge is determined, Biot-Savart’s law gives easily the magnetic ﬂux density. Findings – The BIE using double layer charge is capable of treating robustly geometrical singularities at edges and corners. It is also capable of solving the problems with extremely high magnetic permeability. Originality/value – The proposed BIE contains only the double layer charge while the conventional equations derived from the scalar potential contain the single and double layer charges as unknowns. In the multiply connected problems, the excitation potential in the material is derived from the magnetomotive force to represent the circulating ﬁelds due to multiply connected exciting currents. Keywords Boundary integral equation, Double layer charge, Multiply connected problem, Nonlinear magnetostatic analysis, Scalar potential, Integral equations, Electric current Paper type Research paper 1. Introduction The magnetostatic analysis by the boundary integral equation (BIE) derived from the scalar potential given in the book by Stratton (1941) is one of the basic numerical approaches and many papers have been reported (Rucker and Richter, 1988; Koizumi et al., 1990; Sawa and Hirano, 1990; Rucker et al., 1992, Minciunescu, 1998; Buchau et al., 2003, 2007). They utilize the scalar potential wH for the magnetic ﬁeld H, and derive the BIEs with two unknowns of single and double layer charges, ss and sd as the state ^ ^ variables. Hereafter, we call the BIEs “conventional BIE”. Tozoni and Mayergoyz (1974) have derived another BIE with the double layer charge sd from an integral representation of the scalar potential wB for the magnetic ﬂux density B. It seems there are several advantages described below over the conventional BIEs, but only a few papers have been reported. In this paper, we derive a BIE with the help of the concept of wB and adopting the volume integral equation (VIE) approach (Wang, 1991). The double layer charge sd is COMPEL: The International Journal for Computation and Mathematics in deﬁned as two gapless layer charges with opposite poles of the same values, and so sd Electrical and Electronic Engineering does not produce any gap between the normal components of B but it produces a Vol. 30 No. 6, 2011 pp. 1870-1884 potential gap between the layers. Since the continuity of B is automatically fulﬁlled by q Emerald Group Publishing Limited wB, it formulates easily the magnetostatic ﬁelds. With the help of the potential gap, 0332-1649 DOI 10.1108/03321641111168165 we derive a BIE with one unknown sd by enforcing the boundary condition of the continuity of the tangential component of H, while the other boundary condition of the Nonlinear continuity of the normal component of B is automatically fulﬁlled because sd does not magnetostatic affect it. By virtue of these properties, it is expected to solve accurately wB, which is deﬁnite and smoothly distributed even if B becomes inﬁnite. Since sd given by solving BEM formulation the BIE is equivalent to the loop current, it is easy to evaluate B with the help of Biot-Savart’s law even if B approaches inﬁnity at the edge and corner. In the multiply connected problem (MCP), we introduce the excitation potential in 1871 the material derived from the magnetomotive force (MMF) so as to meet easily the requirement that the line integral of H along the toroidal core must equal the total MMF due to the exciting currents, and then we formulate the magnetostatic ﬁeld by adopting the surface integral equation approach (Wang, 1991) with the help of the scalar potential wH for H (Stratton, 1941). While the original BIEs derived from wH contain ss and sd as unknowns, utilizing the boundary conditions, we derive a BIE ^ ^ only with sd . ^ This paper also includes how to solve nonlinear magnetostatic problems. 2. Integral representation of scalar potential The magnetostatic ﬁelds are represented as: 7£H ¼ J; ð1Þ 7·B ¼ 0 ð2Þ where H, B and J are the magnetic ﬁeld, ﬂux density and the electric current density. In the VIE approach given in the book by Wang (1991), a material with the surface S and volume V is replaced to the magnetization M as follows. With the magnetic permeability of free space m0, we rewrite equation (2) as: m0 7 · H ¼ 27 · ðB 2 m0 H Þ: ð3Þ Equation (3) is for determining the magnetic ﬁeld H in the free space with the magnetic charge distribution. The right side is regarded as ﬁctitious magnetic charges due to M, which is deﬁned as: M ¼ B 2 m0 H : ð4Þ Since equation (1) is not affected by the material property, it remains as it is. We rewrite the integral representation of the potential wM due to M given in the book by Bleaney and Bleaney (1976) as: Z M ·r wMP ¼ mrio dV 4pr 3 VZ Z M · dS 7·M ¼ mrio 2 dV ð5Þ 4pr V 4pr ZS Z MS mv ¼ mrio dS þ dV S 4pr V 4pr where the subscript P denotes the value at an observation point Po, the relative permeability mrio is deﬁned as mrio ¼ m/m0 inside V, mrio ¼ 1 outside V, r is the COMPEL distance from the integral point Pi to Po, and mv and MS are called the volume and surface charges deﬁned as: 30,6 mv ¼ 27 · M ¼ 27 · ðB 2 m0 H Þ ð6Þ M S ¼ n · M ¼ n · ðB 2 m0 H Þ: with the unit outward normal n. The potential wM in equation (5) is composed of two 1872 potentials; the potential wMs due to MS and that wmv due to mv. The concept of magnetic shell (Stratton, 1941), which is composed of the double layer charges sd, suggests that wMs could be replaced by the potential due to sd. Taking this concept into account, we get the total potential wB as: Z n·r wBP ¼ wBeP þ mrioP wmvP þ sd dS ð7Þ S 4pr 3 where wBe and wmv are the potentials at Po produced, respectively, by the exciting source and mv, r is the distance from an integral point Pi on S to Po and n is the unit normal at Pi. The potential wmv is represented as: Z mv wmvP ¼ dV : ð8Þ V 4pr 3. Boundary integral equation The potentials on the material surfaces with the subscripts o and i denoting the outer and inner sides are given as: Z VsP n·r wBoP ¼ wBeP þ wmvP þ sdP þ sd dS; ð9Þ 4p S 4pr 3 Z VsP n·r wBiP ¼ wBeP þ mrP wmvP 2 1 2 sdP þ sd dS ð10Þ 4p S 4pr 3 where Vs is the solid angle subtended at the singular point on the surface S (Bladel, 1991). Applying the boundary condition of the continuous condition of the tangential magnetic ﬁeld to equations (9) and (10), we derive a BIE. The boundary condition is given as: n P £ B osP n P £ B isP ¼ ð11Þ m0 m where B isP ¼ mH isP with the subscript s denoting the value on S. Since B is given as B ¼ 2 grad(wB), equation (11) is rewritten as: wBos1 2 wBos2 wBis1 2 wBis2 ¼ ð12Þ m0 Dl mDl where the subscripts 1 and 2 denote the values at the points P1 and P2 on the surface, and Dl is the distance from P1 to P2. Since equation (12) is identical to the condition that wBos/m0 ¼ wBis/m, we get the following BIE as: Z n·r C sP sdP þ sd dS ¼ 2wBeP ð13Þ S 4pr 3 where CP is the nonlinear coefﬁcient of sdP deﬁned as: Nonlinear VsP mrs 2 VsP þ 4p magnetostatic CP ¼ ð14Þ 4pðmrs 2 1Þ BEM formulation with the relative permeability mrs deﬁned as mrs ¼ jBisj/jm0Hisj on the material surface. The other boundary condition of the continuity of normal magnetic ﬂux density is automatically fulﬁlled because sd does not affect the original normal 1873 component of B. The exciting potential wBe due to the exciting coil current Ic is given as follows. Since the potential wBe is deﬁned as the line integral of the magnetic ﬂux density Be produced by Ic from the original point P0 of the potential to Po, it is given as: Z Po wBeP ¼ B e · dL ð15Þ P0 where Be is given with the help of Biot-Savart’s law as: I I c £ rc B e ¼ m0 Lc dLc ð16Þ 4pr 3 c with the distance rc from an integral point along the coil length Lc to Po. The exciting potential wBe in equation (15) is evaluated numerically, or with the help of the integral formula as: wBeP ¼ m0 I c ðVcP þ dc Þ ð17Þ where: Z nc · r c VcP ¼ 3 dS c ð18Þ S c 4pr c with a ﬂat surface Sc surrounded by a loop line current Ic and the unit normal nc on Sc, and the value of dc is the adjusting term given as: dc ¼ 2 0.5 when VcP . 0 and dc ¼ 0.5 when VcP , 0. In the magnetostatic analysis of simply connected problems, the exciting potential must be continuous and dc works for wBe to be continuous while dc is unnecessary in MCP because the discontinuity of Vc works as the value of the cut surface. When the exciting ﬁeld is produced by the a permanent magnet, the exciting potential wBe is given as: Z M e · re wBeP ¼ 3 dV e ð19Þ V e 4pr e where Me is the magnetization of the permanent magnet, Ve is its volume and re is the distance from the point at a segmental volume dVe to Po. 4. Flux density due to double layer charge Here, we evaluate the magnetic ﬂux density Bs due to the double layer charge at any point utilizing the relation between the double layer charge and loop current (Stratton, 1941). The concept of magnetic shell is shown in Figure 1. The magnetic shell is composed of the double layer charges sd, which are replaced by the loop currents Jl when each COMPEL segmental charge is regarded as constant. This assumption leads to Bs given with the 30,6 help of Biot-Savart’s law as follows: X Ns I uJ £ r L B sP ¼ sdi 3 dLi ð20Þ i¼1 DLi 4pr L 1874 where Ns is the total number of sd that equals to the number of Jl, uJ is the direction of Jl, which circulates anticlockwise along the periphery of sd, DL is the length of Jl, and rL is the distance from the integral point along Jl to Po. When we obtain sd by adopting the constant element in discretization of equation (13), we can evaluate easily B at any point by employing equation (20) because only the periphery of the constant element is taken into account as DL in equation (20). In this case, Ns equals the number of the total elements. The value on the surface is not given by equation (20) because it is singular on the surface. How to evaluate the B on the surface is described below. Since the constant element is incapable of setting the unknown sd at the edge and corner, B at these points cannot be evaluated. In order to obtain B at the edge and corner, we adopt the linear element in discretization of equation (13). The double layer charge sd at each node of the surface elements is obtained by solving equation (13), and sde with the subscript e denoting the value within the surface element is interpolated with sd at the nodes. Figure 2 shows one of the surface elements with sdi (i ¼ 1, 2, 3 and 4) shown by the symbol W at each node. We interpolate sde at the calculating point Pe within the surface element as: σd Jl Figure 1. Magnetic ﬂux density due to magnetic shell, which is equivalent to either sd or Jl y σd4 σd3 b x b Figure 2. Interpolation of double layer charge within σd1 σd2 surface element a a X 4 Nonlinear sde ¼ N i sdi ð21Þ i¼1 magnetostatic BEM formulation where the shape function Ni is given as: 1 N1 ¼ ða 2 xÞðb 2 yÞ; 4ab 1875 1 N2 ¼ ða þ xÞðb 2 yÞ; 4ab 1 N3 ¼ ða þ xÞðb þ yÞ; 4ab 1 N4 ¼ ða 2 xÞðb þ yÞ 4ab with the local coordinates, x and y, at Pe. Adopting sde interpolated as equation (21), we evaluate Bs as follows. The surface element is divided further into the sub-elements as shown in Figure 3(a) for evaluating Bs on the ﬂat surface and Figure 3(b) for Bs at the edge and corner. The solid lines in these fugures are for the original surface element and the broken lines are for subdividing the surface element. The calculating points Po for obtaining sd1 2 sd4 have been set at the intersecting point of the solid lines shown by W. We set the point Pe shown by V at the center of the sub-element surrounded by the solid and dotted lines and evaluate sde by employing equation (21). The loop current is introduced so as to circulate anticlockwise along the periphery of the sub-element as shown by the arrows in the ﬁgure on the assumption that sde on the sub-elements are constant. When we evaluate Bs at the vertex, we set one loop current Jl on the surface elements adjoining the vertex as shown in Figure 3(b), where Jl, which is approximately equivalent to sd at the vertex, circulates along each outer periphery of three adjoining sub-elements, and others as shown in Figure 3(a). The total magnetic ﬂux density B is given as: B P ¼ B sP þ B eP þ B mvP ð22Þ where Bs is the magnetic ﬂux density due to Jl given as equation (20), Be due to the exciting sources given in equation (16), and Bmv due to the equivalent magnetic charge mv given in equation (6). We can apply equation (22) to evaluate B at any point except on the surface. Even on the surface, equation (22) is capable of evaluating the normal component of B σd4 σd3 Figure 3. (a) Subdivisions for evaluating magnetic ﬂux density; (b) subdivisions of σd1 σd2 surface elements adjoining vertex (a) (b) COMPEL by setting the calculating point Pe at the center of the sub-element, but incapable of doing the tangential component Bt. Since there is the gap DBs between the inner and 30,6 outer Bt as given in equation (11), we take DBs into account to evaluate Bt and get the following equations: ðn P £ DB sP Þ n P £ B oP ¼ n P £ B sP þ ; 1876 2 ðn P £ DB sP Þ n P £ B iP ¼ n P £ B sP 2 2 where Bsp has been evaluated by employing equation (22). Substituting these equations into equation (11), DBs is determined as: 2ðm0 2 mÞ n p £ DB sP ¼ n p £ B sP : m þ m0 Finally, we get the tangential magnetic ﬂux density as: 2m0 n P £ B oP ¼ n P £ B sP ; m þ m0 ð23Þ 2m n P £ B iP ¼ n P £ B sP : m þ m0 Since the potential wB on the surface has already been given by solving equation (13), Bt also is evaluated as: wB1 2 wB2 Bti ¼ ; Dl m0 Bti ð24Þ Bto ¼ m where wB1 and wB2 are the potentials at P1 and P2 on the surface, and Dl is the distance between these points. The magnetic ﬁeld is evaluated as HP ¼ BP/m. 5. Multiply connected problems Here, we introduce an integral representation for the exciting potential in the material so as to meet easily the circulating magnetic ﬁeld produced by the exciting coil current Ic in the MCP. We derive the exciting potential wHem for H from the MMF produced in the material by Ic. A typical MCP is shown in Figure 4 where the segmental MMF dIm is deﬁned as: dI m ¼ H m · dLm dS m ¼ H m dV ð25Þ where Hm is the magnetic ﬁeld due to Ic, dSm is the surface of the segmental MMF, dLm is the thickness of the MMF with the direction of the same as that of Hm and dV is the segmental volume of the toroidal core. The potential dwHem at an observation point Po due to dIm is given as: nm · r d wHemP ¼ dI m ð26Þ 4pr 3 Nonlinear dLm Po r Hm magnetostatic BEM formulation dϕmP dSm dIm = Hm• dLmdSm 1877 Exciting Toroidal coil core Figure 4. Segmental MMF dIm due to exciting coil current where, r is the distance from dIm to Po and nm is the unit vector perpendicular to dSm. Taking the direction of nm is the same as that of Hm into consideration, the potential wHem at Po due to the total dIm in the toroidal core is given as: Z r ·H m wHemP ¼ 3 dV : ð27Þ V 4pr The magnetic ﬁeld Hm due to Ic is evaluated with the help of Biot-Savart’s law. The concept of wHem based on the segmental MMF is similar to that based on the cut surface. The difference is as follows. The potential wHec due to the coil current is given as the right side of equation (17) divided by m0 on condition that dc ¼ 0, that is, wHecP ¼ I c VcP , which is regarded as due to the total MMF concentrated on the cut surface while equation (27) is due to the segmental MMF distributed in the entire core. Next, adopting wHem in equation (27) as the exciting potential in the material, we shall derive BIEs for the MCP with the help of the SIE approach (Wang, 1991) by using the potential wH for H (Stratton, 1941). The potentials in the outer and inner sides of material, wHo and wHi, are given as: Z Z ›wHo 1 n·r wHoP ¼ wHecP þ dS 2 wHo dS; ð28Þ S ›n 4pr S 4pr 3 Z Z ›wHi 1 n·r wHiP ¼ wHemP þ dS 2 wHi dS ð29Þ S ›n 4pr S 4pr 3 where m0(›wHoP/›n) ¼ m(›wHiP/›n) for the normal B and wHoP ¼ wHiP for the tangential H on the surface. We call wH “double layer charge sd ” and call ›wHiP/›n “single layer ^ charge ss ”. Utilizing m0(›wHoP/›n) ¼ m(›wHiP/›n) on the surface, which implies that: ^ Z Z ›wHo 1 ›wHi 1 m0 dS ¼ m dS; S ›n 4pr S ›n 4pr we get BIE for solving sd as: ^ Z sdP ^ n·r ðmr þ 1Þ þ ðmr 2 1Þ sd ^ dS ¼ mr wHemP 2 wHecP : ð30Þ 2 S 4pr 3 COMPEL Once we have determined sd by solving equation (30), we obtain ss from BIE derived by ^ ^ equating wHoP in equation (28) and wHiP in equation (29) on the surface as: 30,6 Z Z 1 1 n·r 1þ ss ^ dS ¼ 2wHemP 2 wHecP þ 2 sd ^ dS: ð31Þ mr S 4pr S 4pr 3 It is important to notice that we must adopt the constant element in discritization of 1878 equations (30) and (31) because each potential adjacent to MMF is intrinsically discontinuous even if the segmental MMF is extremely thin. 6. Iterative solution of nonlinear BIE If the magnetization property is linear, the unknowns of the BIE are sd of the surface element, but if it is nonlinear, the unknowns are sd and mv of the volume element. Since the number of volume elements are much bigger than that of surface elements, it requires large amount of computer memory to solve sd and mv. In order to solve with small amount of computer memory, we propose an iterative approach adopting Newton-Raphson method combined with a simple iterative approach (called “combined iterative approach”). A. Magnetic ﬂux density due to volume charge At ﬁrst, we examine how to evaluate mv in equation (6) and Bmv due to mv in equation (22). When we adopt the constant volume element in discritization employed for the magnetic moment method (Takahashi et al., 2007), the surface magnetic charges mvs appear on the surface of the volume element. We shall divide the inside of the magnetic material into Nv volume elements and approximate mv as: 1 mvs ¼ 1 2 n vs · B i ð32Þ mr where nvs is the unit normal on the element surface, and mr is the relative permeability deﬁned as mr ¼ jBij/jm0Hij. The potential wmvs in equation (8) due to mvs is given as: XZ 1 n vs · B i wmvsP ¼ 12 dS ð33Þ Nv DS v mr 4pr where Nv is the number of the volume elements, and DSv is the area of the surface on the volume element. Since mvs is not equivalent to mv itself but to mv including sd, BmvP in equation (22) is given as: B mvP ¼ B mvsP 2 B MsP ¼ ðmrP 2 1ÞB eP XZ 1 ðn vs · B i Þr XZ uJ £ r ð34Þ þ mrP 12 3 dS 2 sd dL Nv DS v mr 4pr Ns DL 4pr 3 where Bmvs is the magnetic ﬂux density due to mvs, BMs is that due to the surface magnetic charge Ms deﬁned in equation (6), Ns is the number of the surface elements and DL is the periphery of the surface element. B. Iterative solution of nonlinear equation Nonlinear In Newton-Raphson method, the initial value of sd obtained as the solution of equation magnetostatic (13) is improved by the correction dsd, which is given by solving the following BIE as: Z BEM formulation ›C P sdP n·r dsdP þ dsd dS ¼ 2dwBP ð35Þ ›sdP S 4pr 3 1879 where dwBP is the residual deﬁned as the difference between the right and left sides of equation (13) and given as: Z n·r dwBP ¼ wBeP þ C P sdP þ sd dS: ð36Þ S 4pr 3 The coefﬁcients of the term of Jacobian matrix in equation (35) is given as: ›C P sdP ›C P ›mrs ›BsP ¼ C P þ sdP ð37Þ ›sdP ›mrs ›BsP ›sdP with the absolute value Bs of Bs because Cp is not the explicit function of sd but of mrs, which is the function of Bs. The solution sold obtained previously is improved d by dsd as: snew ¼ sold þ dsd : d d ð38Þ In the simple iterative approach, we evaluate mvs in equation (32), which works for improving sd. The evaluation of mvs requires Bi obtained according to equation (22) with the help of sd newly obtained and also mvs previously obtained. In this way, mvs is successively improved at the end of each iteration in the combined iterative approach. The nonlinear BIE (equation (13)) is solved iteratively with the nonlinear permeability m, which are approximated according to the B-H curve as: m ¼ m0 ð1; 400 2 1; 900ðBi 2 0:8Þ2 Þ; when 0 # Bi , 1:5T ð39Þ m ¼ m0 ð1 þ 468 expð26ðBi 2 1:5ÞÞÞ; when 1:5T # Bi : The ﬂowchart to solve the nonlinear magnetostatic problems is shown in Figure 5: (1) We evaluate the source potential wBe. (2) We set the initial value of the surface relative permeability mrs. (3) We solve equation (13) to obtain the initial values of sd. (4) Employing equation (22), we evaluate Bi and obtain mr. (5) With the help of equation (23), we evaluate the magnetic ﬂux density Bs on the surface in order to obtain mrs. (6) We evaluate the residual dwBP in equation (36). (7) We solve equation (35) to obtain the correction dsd. (8) With dsd, we improve sd in equation (38). (9) We get back to (4) and repeat the procedures from (4) to (9) until the value sd converges. COMPEL START 30,6 Evaluate jBe. Set mrs. 1880 Solve (13) to obtain sd. Evaluate Bi in (22) to obtain mr. Evaluate Bs in (22) and (23) to obtain mrs. Evaluate Jacobian Matrix in (37). Evaluate djB in (36). Solve (35) to obtain dsd Improve sd in (38). Figure 5. Flowchart for nonlinear magnetostatic analysis CONVERGENCE using combined iterative approach STOP The convergence in Newton-Raphson approach is good, however in the simple iterative routine it is difﬁcult to get good convergence and adopting the deceleration coefﬁcient h we improve the evaluated Bi as follows: Bi ¼ hBOld þ ð1 2 hÞBNew i i ð40Þ where the superscripts Old and New denote the value Bi previously and newly evaluated by equation (22). When the value of h is set larger, the convergence becomes more certain but the number of iterations increases. 7. Numerical validation of proposed approach A. Cylindrical core In order to check the adequacy and effectiveness of the proposed approach, ﬁrst we solve an axisymmetric problem shown in Figure 6 (Magele et al., 1988). A cylindrical core with radius 5 cm and height 20 cm is excited by a coil with inner and outer radii 6 and 8 cm, and height 15 cm. The cylindrical axis is parallel to the Z-axis. It is conﬁrmed that the computed results of B are almost equal to those given in the paper by Magele et al. (1988). The coil current is increased to 40 kAT so as to evaluate B in more saturated case. We show the computed results of the magnetic ﬂux density B in Figure 7(a) and (b) comparing with those by using the magnetic moment method (Takahashi et al., 2007). The magnetic ﬂux density Bz2 z along Z-axis is shown in Figure 7(a), and Bz2 r along the R-axis shown in Figure 7(b), where the symbol W denotes the results by the proposed method and x denotes those by the magnetic moment method. Z 2 cm Nonlinear 6 cm magnetostatic 5 cm 1 cm BEM formulation R 1881 20 cm 15 cm Cylindrical material Coil Coil Figure 6. I = 40 kAT I = 40 kAT Axisymmetric model 2 2 Magnetic flux density Bz-z (T) Magnetic flux density Bz-r (T) Cylinder Air 1.5 1.5 Cylinder Coil Air 1 1 Air Figure 7. 0.5 0.5 (a) Computed results of magnetic ﬂux density 0 0 Bz2 z along Z-axis; (b) 0 5 10 15 0 5 10 15 computed results of magnetic ﬂux density Distance along Z-axis from center [cm] Distance along R-axis from center (cm) Bz2 r along R-axis (a) (b) B. Hollow sphere (shielding problem) Next, we shall solve a problem with a magnetic hollow sphere in the uniform magnetic ﬁeld He (2,400 A/cm) shown in Figure 8, where the direction of He is parallel to the Z-axis. We analyze linear and nonlinear cases. In the linear case, mr is supposed as mr ¼ 1,000 and computed results are compared with analytical values given in the book by Bleaney and Bleaney (1976). The computed results of B along X- and Z-axis, together with analytical ones, are shown in Figure 9(a) and (b), where the symbols W and x denote those by the proposed approach and FEM, and the solid line the Z X 3 cm 5 cm Figure 8. He = 2,400A/cm Hollow sphere in uniform magnetic ﬁeld COMPEL analytical ones. In the linear analysis, the results are almost the same as analytical 30,6 values and omitted. C. Toroidal core (MCP) The last analysis model is shown in Figure 10. In this problem, the magnetization property of toroidal core is supposed to be linear with the relative magnetic 1882 permeability 500. The dimensions of the model is as follows: core radius Rt ¼ 10 cm, core cross-section dimensions Lt ¼ 5 cm and Ht ¼ 5 cm, coil dimensions Lc ¼ 10 cm and Hc ¼ 10 cm, coil angle uc ¼ 608. The coil current is 500 AT. The computed results of the magnetic ﬁeld H in the core at the midpoint of cross-section circulating along the broken line from P are shown in Figure 11(a) and those along from Ps to Pe shown in Figure 11(b). The symbols W and x show H computed by the proposed BIE and the BIEs of the surface charge-current formulation (Ishibashi et al., 2011), respectively. Both the results are almost the same inside the core but somewhat different outside the core. It is too sensitive to obtain ss from sd by employing equation (31). And so, once ^ ^ sd and ss have been determined, ss is smoothed according to the following equation: ^ ^ ^ X Ne Z nP · r ssP ¼ 2 ssi ^ 3 dS i¼1 DS 4pr where ss is the smoothed value of ss , Ne is the number of the total constant elements. ^ 100 100 Magnetic flux density Bz (T) Magnetic flux density Bz (T) o, x : Nonlinear B-H o, x : Nonlinear B-H 10–1 : Linear mr = 1,000 10–1 : Linear mr = 1,000 Shielding Shielding Figure 9. 10–2 10–2 (a) Computed results of magnetic ﬂux density B along X-axis; (b) computed 10–3 10–3 0 1 2 3 4 5 6 0 1 2 3 4 5 6 results of magnetic ﬂux Distance from sphere center along X-axis (cm) Distance from sphere center along Z-axis (cm) density B along Z-axis (a) (b) Toroidal core Exciting coil I = 500AT Lc Core radius Rt P Lt Ps Pe Coil angle qc Hc Figure 10. Ht Analysis model with magnetic core excited by coil currents Cross-section of core and coil 10 Nonlinear magnetostatic Magnetic field H (A/cm) 8 BEM formulation 6 Coil 4 1883 2 0 0 10 20 30 40 50 60 Distance along specified line (cm) (a) 50 45 Magnetic field H (A/cm) 40 35 Toroidal core 30 25 20 Figure 11. 15 (a) Computed results of 10 magnetic ﬁeld H along 5 circulating line; (b) 0 computed results of –5 –4 –3 –2 –1 0 1 2 3 4 5 magnetic ﬁeld H along Position from Ps to Pe (cm) speciﬁed line from Ps to Pe shown in Figure 10 (b) 8. Conclusions We have proposed a BIE with one unknown of the double layer charge sd. The main difference between the conventional and proposed BIEs is as follows. The conventional BIE is derived from scalar potential wH for the magnetic ﬁeld H while the proposed BIE from wB for the magnetic ﬂux density B. This slight difference makes signiﬁcant great works. Even if the unknown of the BIE is one while the conventional BIEs contain two, the boundary conditions are fulﬁlled completely and besides B is easily and directly evaluated by Biot-Savart’s law because sd is equivalent to the loop current. Since sd at the edge/corner is easily determined by the BIE and sd gives B, we can treat robustly the geometrical singularities at the edge/corner. In order to solve the MCP, we have derived a generic integral representation to give the exciting potential produced by the MMF so as to meet arbitrarily shaped exciting coils. This makes the magnetostatic analysis generic from the CAD-modeling point of view. References Bladel, J.V. (1991), Singular Electromagnetic Fields and Sources, Oxford University Press, Oxford. Bleaney, B.I. and Bleaney, B. (1976), Electricity and Magnetism, 3rd ed., Oxford University Press, Oxford. COMPEL Buchau, A., Haﬂa, W. and Rucker, W.M. 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Tozoni, O.B. and Mayergoyz, I.D. (1974), Computation of the Three-dimensional Electromagnetic Fields, Tehnika, Kiev. Wang, J.J.H. (1991), Generalized Moment Method in Electromagnetics, Wiley, New York, NY. About the authors Kazuhisa Ishibashi graduated from Electrical Engineering Department of Tokyo Metropolitan University and received PhD degree in 1986. From 1967 to 1992, he was with R&D of Toyo Seikan. Since 1992, he had been with Mechanical Engineering of Tokai University as a Professor and retired in 2010. Now, he is currently visiting ABB Corporate Research. His main interest is computational electromagnetics. Kazuhisa Ishibashi is the corresponding author and can be contacted at: yui@g03.itscom.net Zoran Andjelic is Senior Principal Scientist at ABB Corporate Research in Baden, Switzerland. There he supervises research and software development projects related to electromagnetic and coupled problems in power engineering devices and apparatus. One of his major contributions is the synthesis of 3D simulation and automatic optimization in the design process of power transformers and switchgears. To purchase reprints of this article please e-mail: reprints@emeraldinsight.com Or visit our web site for further details: www.emeraldinsight.com/reprints