HOMOGENIZATION OF QUASI-STATIC MAXWELL S EQUATIONS

Transcription

1 HOMOGENIZATION OF QUASI-STATIC MAXWELL S EQUATIONS XUE JIANG AND WEIYING ZHENG Abstract. This paper studies the homogenization of quasi-static and nonlinear Maxwell s equations in grain-oriented (GO) silicon steel laminations. GO silicon steel laminations have multiple scales and the ratio of the largest scale to the smallest scale can be up to 1 6. Direct solution of three-dimensional nonlinear Maxwell s equations is very challenging and unrealistic for large electromagnetic devices. Based on the magnetic vector potential and the magnetic field respectively, we propose two macro-scale models for the quasi-static Maxwell s equations. We prove that the microscale solutions converge to the solutions of the macro-scale models weakly in H(curl, ) and strongly in L 2 () as the thickness of lamination tends to zero. The wellposedness of the homogenized model is established by using weighted norms. Numerical experiments are carried out for a benchmark problem from the International Compumag Society, TEAM Workshop Problem 21 c M1. The numerical results show good agreements with the experimental data and validate the homogenized model. Key words. Homogenization, Maxwell s equations, eddy current, finite element method, grain-oriented silicon steel lamination AMS subject classifications. 35K55, 65N3, 78A25 1. Introduction. Quasi-static Maxwell s equations are widely used in electric engineering, such as large power transformers and electric generators, etc. The model neglects the displacement current density in Ampere s law and approximates the Maxwell s equations at very low frequency [1]: B t + curl E = in R3, (Farady s law) (1.1a) curl H = J in R 3, (Ampere s law) (1.1b) where E is the electric field, B is the magnetic flux density and H is the magnetic field. For simplicity, we neglect conduction current and only consider source current in coils. Let c = supp(σ) be the combination of conductors. Then the current density J is defined by: J = { σe in c, J s in R 3 \ c, (1.2) where σ is the electric conductivity and J s denotes the source current density carried by coils. Clearly we have supp(j s ) c =. (1.3) We are interested in grain-oriented (GO) silicon steel laminations where B is a nonlinear vector function of H and will be specified in more detail in the next section. In fact, (1.1) is also called electromagnetic eddy current problem in the engineering community. Academy of Mathematics and System Sciences, Super-computing Center, Computer Network Information Center, Chinese Academy of Sciences, Beijing, 119, China. (jxue@lsec.cc.ac.cn) NCMIS, LSEC, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing, 119, China. The second author was supported in part by China NSF grants and , by the Funds for Creative Research Groups of China , by National 863 Project of China under the grant 212AA1A39, and by the National Magnetic Confinement Fusion Science Program 211GB153. (zwy@lsec.cc.ac.cn) 1

2 In this paper, we shall study the homogenization of (1.1) for GO silicon steel laminations which are widely used in iron cores and magnetic shields of large power transformers (see Fig. 1.1 for a laboratory model). The lamination stack consists of many steel sheets and has multiple scales. The ratio of the largest scale to the smallest scale can be up to 16. The length and width of each lamination are usually several meters and the thickness ² is about mm. Moreover, each steel sheet is coated with a thin layer of insulating film whose thickness is only 2-5µm. The coating films prevent electric currents from flowing into neighboring sheets, as seen in Figure 1.2. Full three-dimensional (3D) finite element simulation is extremely difficult due to extensive unknowns from meshing both laminations and coating films. Recently, Zheng et al proposed to compute 3D eddy currents in steel laminations by omitting coating films [19, 22, 29]. The scale ratio of the system is reduced from 16 to 14. They proved that the approximate solutions converge to the solution of (1.1) strongly in L2 (, T ; L2 (Ω)). But the small parameter ² still remains and makes the simulation of large electromagnetic devices very difficult. Fig GO silicon steel laminations. (Left) the magnetic shield for protecting the magnetic plate; (Right) the magnetic shield made of laminated steel sheets..458m.3 mm.27m 4µm Fig Geometric sizes of the silicon steel laminations. There are many works in the literature studying efficient numerical methods for linear eddy current problems. In 2, Beck et al proposed a residual-based a posteriori error estimate for edge element approximation of eddy current problems [4]. They proved the reliability and efficiency of the a posteriori error estimate with respect to the approximation error in the energy norm. In 26, Zheng et al proposed an adaptive finite element method for the H-ψ formulation of time-dependent eddy current problems with multiply-connected conductors. Both the temporal and spatial meshes are refined or coarsened under reliable and efficient a posteriori error estimates [28]. In 21, Chen et al studied the adaptive finite element method for eddy current model 2

3 with circuit/field couplings [12]. In 22, Hiptmair studied multigrid method for edge element discretization of eddy current problems [18]. In 21, Ledger and Zaglmayr computed eddy current problems on multiply-connected domains by hp-finite element method and proposed an efficient solver for the discrete problem [21]. In 23, Costabel et al studied the regularity of the solution of eddy current problems [13]. However, mathematical and numerical theories for nonlinear eddy current problems are rather rare in the literature. Here we refer to Bachinger et al [3] for the numerical analysis of nonlinear multi-harmonic eddy current problems in isotropic materials. However, in the engineering community, there are considerable efforts in developing efficient numerical methods for nonlinear eddy current problems for sillicon steel laminations [11]. Most of them resort to replace physical parameters with effective parameters in Maxwell s equations, such as permeability and electric conductivity [5, 6, 2, 25]. In [7, 8], Bottauscio et al proposed a mathematical homogenization technique based on multi-scale expansion theory to derive equivalent electric parameters and effective magnetization properties. In [14, 15], Gyselinck et al deduced effective material parameters by an orthogonal decomposition of the magnetic flux density in the perpendicular and parallel directions to the lamination plane. In [17], Napieralska-Juszczak et al established equivalent characteristics of magnetic joints of transformer cores by minimizing the magnetic energy of the system. Numerical methods based on the homogenization of material parameters provide an efficient way to simulate electromagnetic fields in steel laminations. In [23], Nédélec and Wolf proved the convergence of the homogenization method for linear time-harmonic eddy current problems in a transformer core. In [9, 27], Cao et al studied multi-scale methods for Maxwell s equations in a periodic microstructure. But for eddy current problems, the materials consist of finite number of laminations, coils, and the air surrounding them. They do not construct a periodic structure. Rigorous convergence theory is still lacked in homogenization methods for time-dependent and nonlinear eddy current problems. The theme of this work is focused on the homogenization of time-dependent eddy current problems in a variational framework. In the engineering community, there are two widely-used formulations of the eddy current model, the H-formulation and the A-formulation, which are based on the magnetic field and the magnetic vector potential respectively. We start from the micro-scale H-formulation to derive the homogenized eddy current model. The main contributions of this paper are as follows: 1. We propose a homogenized H-formulation for linear time-dependent eddy current problems. All steel laminations are viewed as a conducting block in the macro-scale problem and the small parameter ɛ is neglected. The model only involves a 3D Laplacian and a 2D Laplacian and can be solved very efficiently by nodal element methods instead of edge element methods. 2. By using weighted norms, we establish the well-posedness of the macroscale problem. We prove that the micro-scale solutions {H ɛ } ɛ> converge to the macro solution H weakly in L 2 (, T ; H(curl, )) and strongly in L 2 (, T ; L 2 ()) as ɛ. 3. The homogenized Maxwell s equations are derived by choosing proper test functions in the homogenized H-formulation. Then the homogenized A- formulation is derived easily from the Maxwell s equations. 4. For nonlinear eddy current problems, we propose the homogenized H- formulation and prove that the micro-scale solution converges to the homogenized solution as ɛ. In the nonlinear case, our theories are restricted to thin coating films which have zero thickness but can still prevent eddy 3

4 currents from flowing out each lamination. This is reasonable in practical applications since the film thickness is usually about one percent of the lamination thickness (see Fig. 1.2). 5. We extend the homogenization of the H-formulation to multiply-connected conductors. It is well-known that the H-formulation meets with discontinuous scalar potential in multiply-connected domains. This makes both theoretical analysis and real computations much complicated. Our model has simple form and insures the curl-free property of the reaction magnetic field outside conductors. 6. To validate the homogenized eddy current problem, we compute an engineering benchmark problem Team Workshop Problem 21 c M1 from the International Compumag Society by finite element method. The numerical results show good agreements with the experimental data. The layout of the paper is as follows. In Section 2, we introduce some notation and Sobolev spaces and fix the setting of multiscale eddy current problem. In Section 3, we propose the H-formulation and the A-formulation of (1.1) and study the limit of micro-scale solutions. In Section 4, we derive the homogenized H-formulation, the homogenized A-formulation, and the homogenized Maxwell s equations for linear time-dependent eddy current problems. In Section 5, we derive the homogenized H-formulation for nonlinear eddy current problems. In Section 6, we propose the homogenized H-formulation for multiply-connected conductors. In Section 7, we validate the homogenized H-formulation by finite element computations of an engineering benchmark problem TEAM Workshop Problem 21 c -M1. Without specification, C denotes the generic constant which is independent of the sensitive quantities, such as the micro-scale parameter ɛ, all through the paper. 2. Preliminaries. Let the truncation domain be a sufficiently large cube which contains all conductors and coils. Let L 2 () be the usual Hilbert space of square integrable functions equipped with the following inner product and norm: (u, v) := u(x) v(x)dx and u L 2 () := (u, u)1/2. Define H m () := {v L 2 () : D ξ v L 2 (), ξ m} where ξ represents nonnegative triple index. Let H 1 () be the subspace of H 1 () whose functions have zero traces on. Throughout the paper we denote vector-valued quantities by boldface notation, such as L 2 () := (L 2 ()) 3. We define the spaces of functions having square integrable curl by H(curl, ) := {v L 2 () : curl v L 2 ()}, H (curl, ) := {v H(curl, ) : n v = on }, which are equipped with the following inner product and norm (v, w) H(curl,) := (v, w) + (curl v, curl w), v H(curl,) := (v, v) H(curl,). Here n denotes the unit outer normal to. We shall also use the spaces of functions having square integrable divergence H(div, ) := {v L 2 () : div v L 2 ()}, H (div, ) := {v H(div, ) : n v = on }. 4

5 which are equipped with the following inner product and norm (v, w) H(div,) := (v, w) + (div v, div w), v H(div,) := (v, v) H(div,). This paper is focused on a class of engineering applications where the conducting domain consists of laminated steel sheets, such as magnetic shields and iron cores in a large power transformer. Let c denote the conducting domain, nc := \ c denote the domain without conduction current, f denote the domain occupied by the coating films surrounding steel laminations, and define c := f c. Then c = M i and i j = for i j, i=1 where i denotes each steel lamination for 1 i M. Let ɛ be the thickness of each coated lamination, ɛ f be the thickness of the coating film, and ɛ c = ɛ 2ɛ f be the thickness of each steel sheet (see Fig. 1.2). Throughout the paper, we assume that the two factors θ c = ɛ c ɛ, θ f = 2ɛ f ɛ = 1 θ c (2.1) keep fixed as ɛ. For simplicity, we choose the truncation domain such that = ( L, L) 3, L = Nɛ, N Z. Fig The truncation domain and the steel laminations. The nonlinear relationship between B and H is usually specified by the so-called BH-curves B = f BH (H). (2.2) In practice, f BH is usually generated by spline interpolations with experimental data. Figure 2.2 shows the BH-curves in the rolling and transverse directions of GO silicon 5

6 Fig BH-curves in rolling (left) and transverse (right) directions of silicon steel laminations. steel laminations [1]. We assume that the eigenvalues of the Jacobian matrix J = DB DH are real and satisfy < µ min λ i (H) L (R 3 ) µ max, i = 1, 2, 3. (2.3) Throughout the paper, we append all multi-scale functions with the subscript ɛ, and make the following assumptions on the material parameters and the source current density: (H1) Let µ be the magnetic permeability in the empty space. The constitutive relation between the flux density and the magnetic field is defined by B ɛ (H) = µ H in nc, B ɛ (H) = f BH (H) in c, (H2) The electric conductivity σ ɛ is a piecewise constant σ ɛ σ > in c and σ ɛ in nc. (H3) The source current density satisfies J s L 2 (, T ; L 2 ()), J(, ) =, div J s =. We remark that the assumptions are rather mild and usually satisfied in electrical engineering. Denote the truncation boundary by Γ :=. We impose the initial and boundary conditions for (1.1) and obtain the multi-scale eddy current problem in the bounded domain: t B ɛ(h ɛ ) + curl E ɛ = in, (2.4a) curl H ɛ = σ ɛ E ɛ + J s in, (2.4b) H ɛ (, ) = in, (2.4c) E ɛ n = on Γ. (2.4d) 3. Weak formulations of the eddy current problem. In this section, we shall study two weak formulations of the eddy current problem which are based on the magnetic field and the magnetic vector potential respectively. The close relationship between the two formulations will play an important role in the homogenization analysis for the eddy current problem. For convenience, we first consider simply-connected laminations, namely, each conductor does not have holes. In this case, we refer to [1] for a family of benchmark problems from the International Compumag Society, such 6

7 as 21 c -M1, 21 c -M2, 21 c -EM1, 21 c -EM2, and 21 d -M. The theory for multiply-connected conductors will be presented in Section 6. For simplicity, we assume that the steel sheets are laminated in the x 1 -direction and the lamination stack is defined by c = (, L 1 ) (, L2 ) (, L3 ), where L 1, L 2, L 3 are fixed as ɛ. Denote the number of laminations by M = L 1 /ɛ. Then the conducting region is defined by c = M i, i = ( ) ( ) ( ) X i + ɛ f, X i+1 ɛ f, L2, L3, i=1 where X i = (i 1)ɛ for any 1 i M. For convenience in notation, we also define nc = \ c The weak H-formulation. We first introduce the variational space for the H-formulation U ɛ := H 1 () + H (curl, c ). (3.1) Throughout the paper, we shall use the convention that all functions in H (curl, D) and H 1 (D) are extended by zero to the exterior of D for any D. Then U ɛ H(curl, ) and v 2 H(curl,) = v 2 L 2 () + curl v 2 L 2 ( c) v U ɛ. From [22, Theorem 3.1] we know that U ɛ = {v H(curl, ) : curl v = in nc }. Lemma 3.1. [22, Theorem 3.1] For any v U ɛ, there exist a unique φ H 1 ()/R and a unique v c X ɛ := {w H (curl, c ) : div w = } such that v = v c + φ, v c H(curl,) + φ H 1 () C v H(curl,), where the constant C > is independent of ɛ. Since div J s =, we can write J s into the curl of the source magnetic field by the Biot-Savart Law J s = curl H s, H s (x) = 1 4π curl R 3 J s (y) x y dy in R3. (3.2) In fact we are solving the reaction field R ɛ = H ɛ H s. By (2.4b) and (3.2), it satisfies curl R ɛ = in nc. Clearly Lemma 3.1 indicates that R ɛ U ɛ. For any v = ϕ + v c U ɛ, with ϕ H 1 ()/R and v c H (curl, c ), from (2.4a) (2.4b) we have curl R ɛ curl v = σ ɛ E ɛ curl v = σ E ɛ curl v c = σ curl E ɛ v c c = σ t B ɛ(h ɛ ) v σ curl E ɛ ϕ. 7

8 Using the boundary condition (2.4d), we find that curl E ɛ ϕ = (E ɛ n) ϕ =. A weak formulation of (2.4) reads: Find R ɛ L 2 (, T ; U ɛ ) such that R ɛ (, ) = and σ t B ɛ(r ɛ + H s ) v + curl R ɛ curl v = v U ɛ. (3.3) Here (3.3) is meant in the distributional sense in time. Theorem 3.2. [22, Theorem 3.1] Let (H1) (H3) be satisfied and suppose H s H 1 (, T ; L 2 ()). Then (3.3) has a unique solution R ɛ H 1 (, T ; U ɛ ), and there exists a constant C independent of ɛ such that R ɛ H 1 (,T ;L 2 ()) + R ɛ L 2 (,T ;H(curl,)) C H s H 1 (,T ;L 2 ()). (3.4) 3.2. The weak A-formulation. Now we study the A-formulation of the eddy current problem. Since div B ɛ = by (2.4a) and (2.4c), we can write B ɛ into the curl of a magnetic vector potential [19] B ɛ = curl A ɛ, E ɛ = A ɛ t. (3.5) Substituting the above identities into (2.4b) and using (2.4d), we obtain the initial and boundary value problem σ ɛ A ɛ t + curl H ɛ (curl A ɛ ) = J s in [, T ], (3.6a) A ɛ n = on Γ [, T ], (3.6b) A ɛ (, ) = in c, (3.6c) where H ɛ ( ) stands for the inverse of the nonlinear function B ɛ ( ). A weak formulation of (3.6) reads: Find A ɛ L 2 (, T ; H (curl, )) such that A ɛ (, ) = and σ ɛ A ɛ t v + H ɛ (curl A ɛ ) curl v = J s v v H (curl, ). (3.7) Here (3.7) is meant in the sense of distributions in time. It is obvious that the solution of (3.7) is not unique in the exterior of c. In fact, if A ɛ solves (3.7), then A ɛ + ξ p also solves (3.7) for any ξ C 1 ([, T ]) and p H 1 c () := { p H 1 () : p = Const. in c }. Clearly H 1 c () depends on the small parameter ɛ through c. To study the well-posedness of the weak A-formulation, we define V ɛ = { v H (curl, ) : (v, p) = p H 1 c () }. Let V ɛ be endowed with the following inner product and norm (v, w) V ɛ = v w + curl v curl w, v V ɛ = (v, v) V ɛ, (3.8) c 8

9 for any v, w V ɛ. Then we have the orthogonal decomposition H (curl, ) = V ɛ H 1 c (). (3.9) The following lemma is well-known (cf. e.g. [3,12,19]) and plays an important role in the existence and uniqueness of the weak solution. Lemma 3.3. There exists a constant C ɛ > depending only on c, such that v H(curl,) C ɛ v V ɛ v V ɛ. A modified weak problem is: Find a ɛ L 2 (, T ; V ɛ ) such that a ɛ (, ) = and a ɛ σ ɛ t v + H ɛ (curl a ɛ ) curl v = J s v v V ɛ. (3.1) From (3.9), it is easy to see that a ɛ also satisfies a ɛ σ ɛ t v + H ɛ (curl a ɛ ) curl v = J s v v H (curl, ). (3.11) This means that a ɛ is one solution of (3.7). Here (3.1) and (3.11) are also meant in the distributional sense in time. Although the solution of (3.7) is not unique, the current density and the magnetic flux density are unique, namely, t (σ ɛa ɛ ) = t (σ ɛa ɛ ), curl A ɛ = curl a ɛ in. (3.12) Therefore, we are only interested in σ ɛ a ɛ and curl a ɛ throughout this paper. Theorem 3.4. ( [19, Theorem 2.2]) Let (H1) (H3) be satisfied. Then (3.1) has a unique solution and there exists a constant C > independent of ɛ such that σ ɛ a ɛ L 2 (,T ;L 2 ()) + curl a ɛ L 2 (,T ;L 2 ()) C J s L 2 (,T ;L 2 ()). (3.13) 3.3. Weak limit of the micro-scale solutions. Now we are going to study the limit of the micro-scale solutions as the thickness of steel lamination tends to zero. Let R ɛ be the solution of (3.3). From Theorem 3.2, there exists a subsequence of {R ɛ } ɛ> such that lim R ɛ = R weakly in L 2 (, T ; H(curl, )). (3.14) Since H s is smooth, it follows that lim H ɛ = H := R + H s weakly in L 2 (, T ; H(curl, )). (3.15) By Lemma 3.1, the reaction field R ɛ admits a unique decomposition R ɛ = u ɛ + ψ ɛ, u ɛ H(curl,) + ψ ɛ H1 () C R ɛ H(curl,), (3.16) where u ɛ X ɛ, ψ ɛ H 1 ()/R, and the constant C is independent of ɛ. From (3.4) and (3.16) we know that, with a constant C independent of ɛ, u ɛ H 1 (,T ;L 2 ()) + u ɛ L 2 (,T ;H(curl,)) + ψ ɛ H1 (,T ;H 1 ()) C. (3.17) 9

10 Since L 2 (, T ; H(curl, )) and H 1 (, T ; H 1 ()) are self-reflective, there are two subsequences still denoted by {u ɛ } ɛ> and {ψ ɛ } ɛ> such that Clearly we have lim ɛ = u weakly in L 2 (, T ; H(curl, )), (3.18) lim ɛ = ψ weakly in H 1 (, T ; H 1 ()). (3.19) R = u + ψ. (3.2) Now we are going to show that u only has one nonzero component. Let S c, S i denote the parts of c and i respectively whose unit normals are parallel to e 1 = (1,, ), namely, S c = {x } c : n x e 1, S i = {x i : n x e 1 }, 1 i M. We introduce the following Hilbert spaces { H 1 ( c ) = v L 2 ( c ) : H 1 ( c ) = } L 2 ( c ) x 3 v, v x 2 {v H 1 ( c ) : v = on c \ S } c, which are equipped with the inner product and norm ( (u, v) H1 ( = c) uv + u v + u ) v, v x 2 x 2 x 3 x H1 ( c) = (v, v)1/2. H 1 ( 3 c) c Lemma 3.5. Let u be the limit of {u ɛ } ɛ. Then u = u e 1 where u H 1 ( c ). Proof. Since u ɛ H (curl, c ), we also have u H (curl, c ). Write u ɛ = (u ɛ,1, u ɛ,2, u ɛ,3 ) and u = (u 1, u 2, u 3 ). If u 2 = u 3 =, then ( u 1 curl u =,, u ) 1 L 2 ( c ), u n = u 1 = on x 3 x c \ S c. 2 This shows that u 1 H 1 ( c ). It is left to prove u 2 = u 3 =. From Lemma 3.1, it is easy to see u ɛ X ɛ. Since i is convex, the embedding theorem in [2] shows that u ɛ H 1 ( i ) for any 1 i M. The boundary condition u ɛ n = on S i indicates that u ɛ,2 = u ɛ,3 = on S i, u ɛ,1 = on i \ S i. (3.21) For any ϕ C ( c ), (3.21) shows that u ɛ,2 (X i + ɛ f,, ). Then x1 u ɛ,2 ϕ(x)u ɛ,2 (x)dx = ϕ(x) i i X i+ɛ f y (y, x 2, x 3 )dydx = G i,1 + G i,2, (3.22) where G i,1 = G i,2 = x1 i x1 i X i+ɛ f ϕ(x) [ uɛ,2 y (y, x 2, x 3 ) u ] ɛ,1 (y, x 2, x 3 ) dydx, x 2 X i+ɛ f ϕ(x) u ɛ,1 x 2 (y, x 2, x 3 )dydx. 1,

11 An application of Schwarz s inequality yields G i,1 ϕ i Xi+1 ɛ f X i+ɛ f curl u ɛ (y,, ) dy ɛ ϕ L2 ( i) curl u ɛ L 2 ( i). By (3.21) and the formula of integral by part, the second term is estimated as follows x1 G 2,i = ϕ u ɛ,1 (y,, )dy i x 2 X i+ɛ f = ϕ x1 u ɛ,1 (y,, )dy x 2 X i+ɛ f ɛ ϕ H1 ( i) u ɛ,1 L2 ( i). Substituting the estimates for G i,1 and G i,2 into (3.22) and summing up the estimates in 1 i M, we get Cɛ ϕ H 1 ( u c) ɛ. H(curl, c) c ϕu ɛ,2 In view of (3.17), the above inequality shows that T Cɛ ϕ L 2 (,T ;H 1 ( u c)) ɛ L 2 (,T ;H(curl, Cɛ. c)) c ϕu ɛ,2 where the constant C is independent of ɛ. This means that T T ϕu 2 = lim ϕu ɛ,2 = ϕ L 2 (, T ; C ( c )). c c Hence we have u 2 =. The proof for u 3 = is similar and omitted here. Now Lemma 3.5 shows that R = u e 1 + ψ X, where i X := H 1 ( c ) e 1 + H 1 (). (3.23) Lemma 3.6. Let R be the weak limit of R ɛ in L 2 (, T ; H(curl, )). Then R (, ) = in. Proof. From (3.4), it is easy to prove that lim R ɛ = R also weakly in H 1 (, T ; L 2 ()). Take any ϕ C ([, T ]) satisfying ϕ() = 1 and ϕ(t ) =. Then the formula of integration by parts shows that, for any v L 2 (), T { (R (, ), v) = ϕ(t) T = lim } t (R, v) + ϕ (t) (R, v) dt { ϕ(t) t (R ɛ, v) + ϕ (t) (R ɛ, v) The proof is completed upon using the initial condition R ɛ (, ) =. 11 } dt = lim (R ɛ (, ), v).

12 From Theorem 3.2 and (2.3), there exists a constant C independent of ɛ such that B ɛ (H ɛ ) L 2 (,T ;L 2 ()) + curl R ɛ L 2 (,T ;L 2 ( c)) C. Then there is a common subsequence of {B ɛ (H ɛ )} ɛ> and {curl R ɛ } ɛ> still denoted by the same notation such that lim B ɛ(h ɛ ) = B weakly in L 2 (, T ; L 2 ()), (3.24) lim curl R ɛ = J weakly in L 2 (, T ; L 2 ( c )). (3.25) The following lemma is a direct consequence of Lemma 3.5, (3.18), and (3.25). Lemma 3.7. Let u = u e 1 be the limit of {u ɛ } ɛ> in (3.18). Then J = curl u = curl(ue 1 ) in c. Remark 3.8. From Lemma 3.7, it is easy to see that the first component of the homogenized eddy current density is zero. By macro-scale models, we can not compute eddy currents flowing in the normal direction to the lamination plane. 4. Homogenization of linear eddy current problem. The purpose of this section is to derive the homogenized eddy current problem for nonmagnetic materials. In this case, we have B(H) = µ H in. For simplicity, we neglect the eddy current density in coils and assume supp(j s ) c = The homogenized H-formulation. We start by presenting the strong convergence of the micro-scale solutions in L 2 (, T ; L 2 ()). Lemma 4.1. Let R ɛ be the solution of (3.3) and R be the weak limit of R ɛ in (3.14). Then lim R ɛ R L 2 (,T ;L 2 ()) =. (4.1) Proof. The proof is similar to Lemma 5.1 for nonlinear case and omitted here. To derive the macro-scale model, we introduce the characteristic function χ ɛ defined as follows χ ɛ = { 1 in c, elsewhere. (4.2) It is known that lim χ ɛ = θ c weakly star in L ( c ), namely lim χ ɛ v = c θ c v c v L 1 ( c ). (4.3) Lemma 4.2. Let v ɛ = χ ɛ ve 1 where v H 1 ( c ). Then v ɛ H (curl, c ). Proof. For any fixed v H 1 ( c ), we have v ɛ e 1 = on I i=1 S i, v ɛ e 2 = v ɛ e 3 = v 1 = on c \ S c. 12

13 This means that v ɛ n = on c. Moreover, the definition of H 1 ( c ) indicates that v ɛ L 2 ( c) = v L 2 ( c) <, ( curl v ɛ L 2 ( = v c) x 2 2 L 2 ( c) + v x 3 2 L 2 ( c) ) 1/2 <. Thus we conclude that v ɛ H (curl, c ). Theorem 4.3. Let R ɛ be the solution of (3.3) and let R = ψ +ue 1 be the limit of R ɛ. Then R solves the following macro-scale problem: Find R L 2 (, T ; X ) such that R (, ) = and θ c σ µ t (R + H s ) v + curl R curl v = v X. (4.4) Proof. Noting that B = µ H in, the weak formulation (3.3) is equivalent to T σ µ T t (R ɛ + H s ) v ɛ + curl R ɛ curl v ɛ = v ɛ C (, T ; U ɛ ). From Lemma 4.2, we know that the above equality holds for all v ɛ = χ ɛ ve 1, v C (, T ; H 1 ( c )). Then T σ µ T t (R ɛ + H s ) (χ ɛ ve 1 ) + curl R ɛ curl(χ ɛ ve 1 ) =, Since supp(curl R ɛ ) = supp(curl u ɛ ) c, we have T σ µ v T [(R ɛ + H s ) e 1 ] χ ɛ t + curl R ɛ curl(ve 1 ) =. Using Lemma 4.1 and (4.3) and letting ɛ tend to zero, the above equality yields that, for any v C (, T ; H 1 ( c )), T θ c σ µ or equivalently, T θ c σ µ [(R + H s ) e 1 ] v T t + curl R curl(ve 1 ) =, T t (R + H s ) (ve 1 ) + curl R curl(ve 1 ) =. Moreover, for any ϕ C (, T ; H 1 ()), from (3.24) we know that T T µ t (R B T + H s ) ϕ = ϕ = lim t t B ɛ(h ɛ ) ϕ =. Combining the above two equalities, R satisfies (4.4) in a distributional sense. Theorem 4.4. The homogenized problem (4.4) has a unique solution. And there is a constant C independent of ɛ such that R L (,T ;L 2 ()) + curl R L 2 (,T ;L 2 ()) C H s H 1 (,T ;L 2 ()). 13

14 Proof. In fact, Theorem 4.3 shows that R is one solution of (4.4). It is left to prove the uniqueness. The stability can be proved by standard arguments for parabolic problems and is omitted for simplicity. The uniqueness follows directly from the stability estimate. Remark 4.5. Notice that curl R = curl(ue 1 ) = (, u, u ). x 3 x 2 Problem (4.4) can be rewritten as: Find ψ H 1 ()/R and u H 1 ( c ) such that ψ (, ) =, u(, ) =, and ( t ( ψ 1 u v + ue 1 ) ( ϕ + ve 1 ) + + u ) v (4.5) θ c σ µ c x 2 x 2 x 3 x 3 H s = ( ϕ + ve 1 ) ϕ H 1 (), v H 1 t ( c ). Clearly (4.5) only involves a 3D Laplacian for ψ in and a 2D Laplacian for u in c. It can be solved by nodal finite element method instead of edge element method. This simplifies the computations greatly. We shall validate (4.5) numerically in Section The homogenized Maxwell s equations. From (3.3), it is easy to see that B φ = lim B ɛ φ = φ H 1 (), which implies that div B = in and B n = on Γ. Then there exists a unique potential A H (curl, ) such that We define the homogenized electric field by B = curl A, div A = in. (4.6) E := A t. (4.7) To derive the homogenized Maxwell s equations for E and H, we first introduce the following lemma. Lemma 4.6. Let θ c be the effective factor in (2.1). Then θ c σ c B t v + curl H curl v = c v C ( c ). (4.8) Proof. For any v C ( c ) satisfying div(ve 2 ) =, we have c B (ve 2 ) = µ c (ue 1 + ψ + H s ) (ve 2 ) = µ 14 c H s (ve 2 ).

15 Since c is simply-connected, there exist a Φ H (curl, c ) and a φ H 1 ( c ) such that (cf. e.g. [2]) ve 2 = curl Φ + φ. Since supp(j s ) c =, (3.2) and the formula of integration by parts show that B (ve 2 ) = µ c H s (ve 2 ) = µ c (curl H s Φ div H s φ) =. c It follows that c B t (ve 2 ) =. Noting that v = div(ve 2 ) =, thus x 2 u v curl H curl(ve 2 ) = curl(ue 1 ) curl(ve 2 ) = c c c x 2 x 1 = u 2 v =. c x 1 x 2 Then we conclude that, for any v C ( c ) satisfying div(ve 2 ) =, B θ c σ (ve 2 ) + curl H curl(ve 2 ) =. c t c From div B =, we also deduce that B θ c σ (ve 2 ) + curl H curl(ve 2 ) = c t c v C ( c ). (4.9) Similarly we have θ c σ c B t (ve 3 ) + curl H curl(ve 3 ) = c v C ( c ). (4.1) Moreover, the homogenized H-formulation (4.4) shows that B θ c σ (ve 1 ) + curl H curl(ve 1 ) = c t c v C ( c ). (4.11) Then the proof is completed by combining (4.9) (4.11). Theorem 4.7. Let R be the weak limit of R ɛ in L 2 (, T ; H(curl, )) and H = R + H s. Then the homogenized Maxwell s equations hold in distributional sense B + curl E = t in, (4.12a) curl H = = σe + J s in, (4.12b) H (, ) = in, (4.12c) E n = on Γ, (4.12d) 15

16 where = σ is the homogenized anisotropic conductivity defined by σ θ = c σ in c, σ = σ elsewhere. Proof. First we note the following facts: (4.12a) is obvious by (4.6) and (4.7), (4.12c) is obvious by Lemma 3.6, and (4.12d) is obvious by (4.7) and A n = on Γ. Now it is left to prove (4.12b). For any w C ( c ), there exist a Φ H (curl, c ) and a φ H 1 ( c ) such that w = curl v + φ. Notice that div E = because of div A =. Then (curl H θ c σ E ) w = (curl H θ c σ E ) curl v c c ( = curl H curl v + θ c σ curl A t = c c ( B curl H curl v + θ c σ v t ) v ) =. Since w is arbitrary, we find that θ c σ E = curl H = curl(ue 1 ) = The above equality can also be written as (, u, u ) x 3 x 2 in c. On the other hand, it s easy to see that curl H = = σe in c. curl H = curl H s = J s in nc. Adding up the above two identities leads to (4.12b) The homogenized A-formulation. From (4.7) and (4.12b), we obtain a weak formulation: Find A L 2 (, T ; H (curl, )) such that A(, ) = and A =σ t v + µ 1 curl A curl v = J s v v H (curl, ). (4.13) Clearly the homogenized potential A solves (4.13). But similar to the micro-scale problem, the solution of (4.13) is not unique. Lemma 4.8. Define { V := v H (curl, ) : (v, p) = Ṽ := {p H 1 () : p = p(x 1 ) in } c. } p Ṽ Then V is a Hilbert space equipped with the inner product and norm (v, w) V = (v 2 w 2 + v 3 w 3 ) + curl v curl w, v V = (v, v) V, c 16,

17 Proof. We need only prove that v V = implies v = for any v V. Since is simply-connected, curl v = indicates that v = grad φ for some φ H 1 (). Then φ x i = v i = in c, i = 2, 3. This means that φ Ṽ. We conclude v = from the definition of V. A modified weak formulation of (4.12) reads: Find a L 2 (, T ; V ) such that a(, ) = in c and a =σ t v + µ 1 curl a curl v = J s v v V. (4.14) Theorem 4.9. Let A be the vector potential defined in (4.6). Then a = A is the unique solution of problem (4.14). Proof. From (4.6), it is obvious that A V. Since A solves (4.13), it also solves (4.14). The proof for the uniqueness is similar to that for parabolic problems and is omitted here. Theorem 4.1. Let a ɛ, a be the solutions of (3.1) and (4.14) respectively. Then lim σ a ɛ ɛ t = = σ a t weakly in L 2 (, T ; L 2 ()), lim curl a ɛ = curl a strongly in L 2 (, T ; L 2 ()). Proof. From (3.12), (3.5), and (2.4b), we deduce that σ ɛ a ɛ t = σ ɛ E ɛ = J s curl H ɛ = curl R ɛ. Then (3.25) and Lemma 3.7 show that lim σ a ɛ ɛ = curl u = curl R weakly in L 2 (, T ; L 2 ()). t Since curl R = J s curl H, (4.12b) and (4.7) imply that lim σ a ɛ ɛ = t = σe = = σ A = t = σ a t weakly in L 2 (, T ; L 2 ()). Now notice that From Lemma 4.1 we have curl a ɛ = curl A ɛ = µ (R ɛ + H s ), curl a = curl A = µ (R + H s ). lim curl(a ɛ a) L 2 (,T ;L 2 ()) = µ lim R ɛ R L 2 (,T ;L 2 ()) =. The proof is completed. 17

18 5. Nonlinear eddy current problems. In this section, we shall study the nonlinear eddy current problem where B ɛ ( ) satisfies (H1). Unfortunately we have not proven the strong convergence of R ɛ to R for ɛ f >. The strong convergence plays an important role in our homogenization theory. In this section, we make an additional assumption θ c = 1, θ f =. (5.1) The insulating film has zero thickness but still prevents eddy currents from passing through laminations. In this case, c has inner boundaries, namely, c = M i, i = ( ) ( ) ( ) X i, X i+1, L2, L3, i=1 c = ( X 1, X M+1 ) (, L2 ) (, L3 ). This is reasonable in electrical engineering (see Fig. 1.2). In fact, Li and Zheng proposed an approximate H-formulation by omitting coating films [22]. They proved that the approximate solution converges to the exact solution as the film thickness tends to zero. In this section, we only study the homogenized H-formulation of the nonlinear eddy current problem. Since ɛ f =, from (H1) we find that B ɛ (H) = f BH (H). The H-formulation (3.3) turns out to be: Find R ɛ L 2 (, T ; U ɛ ) such that R ɛ (, ) = and σ t f BH(R ɛ + H s ) v + curl R ɛ curl v = v U ɛ. (5.2) From [22], the theories in Section 3 are still true. We shall adopt the same notation for convenience. We start from the decomposition of the reaction field R ɛ = u ɛ + ψ ɛ. Since u ɛ H (curl, c ) H (curl, c ), there exist a ζ ɛ H 1 ( c ) and a w ɛ H (curl, c ) such that u ɛ = ζ ɛ + w ɛ, div w ɛ = in c, ζ ɛ H 1 ( c) + w ɛ H(curl, c) C u ɛ H(curl,), where the constant C depends on c but is independent of ɛ. Write ˆψ ɛ = ψ ɛ + ζ ɛ. It follows that R ɛ = ˆψ ɛ + w ɛ, w ɛ ˆψ ɛ = (div w ɛ ) ˆψ ɛ =, c ˆψ H ɛ + w ɛ ψ H(curl, c) ɛ 1 H1 () + u ɛ C R H(curl, c) ɛ H(curl,). () Since c is a convex polyhedron, the embedding theorem in [2] indicates that w ɛ H 1 ( c ) and w ɛ H 1 ( c) C w ɛ H(curl, c) C R ɛ H(curl, c). Since div w ɛ = and u ɛ H 1 (, T ; L 2 ( c )), we also have w ɛ H1 (,T ;L 2 ( c)) C u ɛ H1 (,T ;L 2 ( c)), 18 div w ɛ t = in c.

19 Introduce the Sobolev-Bochner space (cf. [24, Section 7.1]) W 1,2,2(, T ; H 1 (), L 2 () ) = which is equipped with the following norm v W 1,2,2(,T ;H 1 (),L 2 () { v L 2 (, T ; H 1 ()) : v } t L2 (, T ; L 2 ()), ( ) = v 2 L 2 (,T ;H 1 ()) + v t 2 L 2 (,T ;L 2 ())) 1/2. Next we shall examine the strong convergence of {R ɛ } ɛ>. For convenience, the same notation will also denote their subsequences without causing confusion. Lemma 5.1. Let R ɛ be the solution of problem (3.3) and let R be the weak limit of R ɛ in (3.14). Then lim R ɛ R L 2 (,T ;L 2 ()) =. (5.3) Proof. First we prove the strong convergence of the regular functions {w ɛ } ɛ>. An application of (3.17) yields w ɛ L2 (,T ;H 1 ( c)) C u ɛ L2 (,T ;H(curl, c)) C. Thus there exists a constant C independent of ɛ such that w ɛ ) W 1,2,2( wɛ,t ;H 1 ( c),l 2 ( c) L2 (,T ;H 1 ( c)) + w ɛ H1 (,T ;L 2 ( c)) C. By the compact embedding (cf. [24, Lemma 7.7]) W 1,2,2(, T ; H 1 ( c ), L 2 ( c ) ) L 2(, T ; L 2 ( c ) ), there exists a subsequence of {w ɛ } ɛ> and a limit w such that lim w strongly in L 2(, T ; L 2 ) ( c ), ɛ = w weakly in W 1,2,2(, T ; H 1 ( c ), L 2 ( c ) ). (5.4) To prove the strong convergence of R ɛ, we shall utilize the divergence-free property of the magnetic flux density. Taking v = ϕ in (3.3) shows that t f BH(H ɛ ) ϕ = ϕ H 1 (). Combining with the initial condition in (2.4), we find that div f BH (H ɛ ) = in (, T ), f BH (H ɛ ) n = on Γ (, T ). (5.5) By (2.3) and the weak convergence of R ɛ, we deduce that µ min lim R ɛ R 2 L 2 (,T ;L 2 ()) T lim [f BH (H ɛ ) f BH (H )] (H ɛ H ) T = lim T f BH (H ɛ ) (H ɛ H ) = lim f BH (H ɛ ) (w ɛ w ) =. 19

20 This completes the proof. Theorem 5.2. Let R = ψ + ue 1 be the limit of R ɛ. It solves the homogenized problem: Find R L 2 (, T ; X ) such that R (, ) = and σ t f BH(R + H s ) v + curl R curl v = v X. (5.6) Proof. Take any v C (, T ; X ). Clearly (5.2) shows that T σ From (3.18) (3.2), we know that T t f BH(R ɛ + H s ) v + curl R ɛ curl v =. T T lim curl R ɛ curl v = curl R curl v. From Lemma 5.1 and the formula of integral by part in t, we have T lim T t B(R ɛ + H s ) v = t B(R + H s ) v. Taking the limit of (5.2) and using the two equalities above, we obtain T σ T t B(R + H s ) v + curl R curl v = v C (, T ; X ). Then the density of C (, T ; X ) in L 2 (, T ; X ) indicates that R solves (5.6) in a distributional sense. The initial condition has been verified in Lemma Homogenized H-formulation for multiply-connected conductors. It is well-known that the H-formulation becomes complicated when dealing with multiply-connected conductors (see e.g. Fig. 6.1 right), since the scalar potential is discontinuous in nonconducting region. In this section, we shall propose a homogenized H-formulation for multiply-connected conductors. Fig Left: simply-connected virtual laminations. Right: multiply-connected physical laminations. First we introduce two types of laminations, virtual laminations and physical laminations, as follows: 2

21 1. Virtual laminations: the laminations are simply connected and defined as follows (see Fig. 6.1 left) D c = M D i, D i = ( ) ( ) ( ) X i + ɛ f, X i+1 ɛ f, L2, L3, i=1 D c = (, L 1 ) (, L2 ) (, L3 ). 2. Physical laminations: the laminations are multiply connected and defined by (see Fig. 6.1 right) c = M i, c = D c \, i=1 = (, L 1 ) (Y 1, Y 2 ) (Z 1, Z 2 ), i = D i \, 1 i M, where < Y 1 < Y 2 < L 2 and < Z 1 < Z 2 < L 3. We remark that the theories in this section can be extended directly to laminations with multiple holes, namely, = I i=1 j=1 J (, L 1 ) (Y 2i 1, Y 2i ) (Z 2j 1, Z 2j ), < Y 1 < < Y 2I < L 2, < Z 1 < < Z 2J < L 3. First we propose the homogenized H-formulation for virtual laminations. Define X D := H 1 ( D c ) e 1 + H 1 (). Since each D i is simply-connected, by the theories in Section 4 and 5, the homogenized H-formulation reads: Find R L 2 (, T ; X D ) such that R (, ) = and θ c σ t B(R + H s ) v + curl R curl v = v X D, (6.1) where θ c < 1, B(H) = µ H for linear problems, and θ c = 1, B(H) = f BH (H) for nonlinear problems. For physical laminations, notice that \ c is a subset of the nonconducting domain. Write R = ψ + ue 1 with ψ H 1 () and u H 1 ( D c ). Then we deduce that This yields u x 2 = u x 3 =, and thus curl(ue 1 ) = curl R = in. u = u(x 1 ) in. (6.2) Now we define X := { ϕ + ve 1 : ϕ H 1 (), v H( 1 D } c ) satisfying v = v(x 1 ) in. Clearly R X. And it is easy to verify that curl v = in, v X. 21

22 Then the homogenized H-formulation for multiply-connected conductors reads: Find R L 2 (, T ; X ) such that R (, ) = and θ c σ t B(R + H s ) v + curl R curl v = v X. (6.3) Problem (6.3) can also be written in the form of (4.5). 7. Numerical validation. In this section, we shall validate the homogenized eddy current problem by finite element computation of an engineering benchmark problem TEAM Workshop Problem 21 c -M1 [1]. Here we only study the finite element approximation of the H-formulation: Find R R (, ) = and θ c σ L 2 (, T ; X ) such that t B(R + H s ) v + curl R curl v = v X, (7.1) where θ c < 1, B(H) = µ H for linear problems, and θ c = 1, B(H) = f BH (H) for nonlinear problems Finite element approximation. Let T h be a tetrahedral triangulation of such that T h Dc, T h also construct the triangulations of D c c and c respectively. For any integer m, let P m be the space of polynomials of degree at most m, and let Q m be the space of polynomials of degree at most m in each variable. We define the Lagrange finite element space on T h by V (m, T h ) = { v H 1 () : v K P m (K), K T h }. Let M h be a hexahedral partition of D c such that M h also constructs a partition c of c. Define the finite element space on M h by { U(m, M h ) = v H( 1 D c ) : v K = v 1 (x 1 )v 2 (x 2, x 3 ), v 1 P m ([X, X 1 ]), v 2 Q m ([Y, Y 1 ] [Z, Z 1 ]), K = [X, X 1 ] [Y, Y 1 ] [Z, Z 1 ] M h }. For simply-connected laminations, we have D c = c. Then a conforming finite element space is defined by U h := V (m + 1, T h ) + U(m, M h )e 1. Based on U h, the semi-discrete approximation of (7.1) reads: Find R h L 2 (, T ; U h ) such that R h (, ) = and θ c σ t B(R h + H s ) v h + curl R h curl v h = v h U h. (7.2) c The above semi-discrete scheme involves two grids T h, M h and is inconvenient in practical computations. Therefore we come up with another choice computing u by the Lagrange finite element method on T h. Define [ V h := V (m + 1, T h ) + V ( ) ] m, T h c H 1 ( c ) e 1. (7.3) 22

23 The associated semi-discrete approximation reads: Find R h L 2 (, T ; V h ) such that R h (, ) = and θ c σ t B(R h + H s ) v h + curl R h curl v h = v h V h. (7.4) c Although V h seems too small to be a good approximation to X, (7.4) yields accurate results in numerical experiments (see Section 7.3). For multiply-connected conductors, we define Û h = {v U(m, M h ) : v = v(x 1 ) in }, ˆV h = { v V ( m, T h Dc ) : v = v(x1 ) in } H 1 ( D c ). The associated conforming finite element spaces are defined by Û h = V (m + 1, T h ) + Ûhe 1, ˆV h = V (m + 1, T h ) + ˆV h e 1. The finite element approximations for multiply-connected conductors are given by replacing U h, V h with Û h, ˆV h respectively in (7.2) and (7.4) Damped Newton method. Since we prefer to solve the problem on a single mesh, the rest of the paper will be focused on the discrete problem (7.4). In this section, we present a damped Newton method and an alternating iteration method for solving (7.4). Let = t < t 1 < < t N = T be a uniform partition of the time interval [, T ] and denote by τ = T/N the time step. The fully discrete problem reads: Given R =, find R n V h, n >, such that θ c σ B n B n 1 τ v h + curl R n curl v h = c v h V h, (7.5) where B n = B(R n + H s (, t n )). Let R n,k, k be the approximate solution of (7.5) at the k th step of damped Newton method. Let H n,k := R n,k + H s (t n ) and B n,k = B(H n,k ) be the approximate magnetic field and magnetic flux density respectively. We define the residual functional F(R n,k ): V h R by F(R n,k ), v := θ cσ τ (B n 1 B n,k, v) (curl R n,k, curl v) v V h. The differential magnetic permeability at H n,k is defined by µ n,k := DB ( ) DH (H DB n,k) with DH = Bi H j Then the linearized error equation reads: Find e n,k V h such that. 1 i 3 1 j 3 a n,k (e n,k, v h ) = F(R n,k ), v h v h V h, (7.6) where the bilinear form a n,k : V h V h R is defined by a n,k (v, w) := θ cσ τ (µ n,k v, w) + (curl v, curl w). 23

24 Given < ρ n,k 1, the damped Newton method for solving (7.5) reads: R n,k+1 = R n,k + ρ n,k e n,k, k =, 1, 2,. (7.7) Now we present a simple algorithm of damped Newton method. Algorithm 7.1 (Damped Newton Method). Given the tolerance < TOL 1 and a factor δ = Set R n, := R n 1 and k =. ( ) 2. While F Rn,k V TOL do h (a) solve the linearized problem (7.6) for an approximate solution ê n,k ; (b) set ρ = δ 1 and ˆR n,k = R n,k ; (c) while F ( ) ˆR n,k V ( ) F Rn,k V do h h reduce the step size: ρ δρ, correct the approximate solution: ˆR n,k R n,k + ρ ê n,k, end while (d) set R n,k+1 ˆR n,k and k k + 1. end While 3. Set R n R n,k. Now we study the solution of problem (7.6). For convenience, we write V := V (m + 1, T h ), V c := V ( m, T h c ) H 1 ( c ). Since V h = V +V c e 1, problem (7.6) can be reformulated as: Find φ V and θ V c such that where a n,k ( φ, v) + b µn,k (v, θ) = F(R n,k ), v v V, (7.8) a c (θ, w) + b µ T n,k (φ, w) = F(R n,k ), we 1 w V c, (7.9) b µ (v, w) := θ cσ (µ v) τ 1 w, c a c (θ, w) := c ( θw + θ x 2 w x 2 + θ x 3 w x 3 Notice that in Step 2 (b) of Algorithm 7.1, we are only solving for an approximate solution of (7.6) or (7.8) (7.9). This reduces the total computational time greatly and makes the damped Newton method much more efficient compared with that solving (7.6) accurately. Given an approximate solution ê of (7.6), we define the residual functional R(ê): V h R by ). R(ê), v := F(R n,k ), v h a n,k (ê, v) v V h. Then Step 2 (b) of Algorithm 7.1 is realized via the following alternating iteration algorithm for solving (7.8) (7.9). Algorithm 7.2 (Alternating Iteration Algorithm). Given the tolerance < TOL 1 and the maximal number of iterations N = 1. 24

25 1. Set the initial guess e =, θ =, and set l =. 2. While R(e l ) V h TOL and l N do (a) solve the following elliptic problem by 5 iterations of conjugate gradient (CG) method preconditioned by algebraic multigrid (AMG) method: Find φ l+1 V such that a n,k ( φ l+1, v) = F(R n,k ), v b µn,k (v, θ l ) v V ; (b) solve the following elliptic problem by 5 iterations of CG method preconditioned by AMG method: Find θ l+1 V c such that a c (θ l+1, w) = F (R n,k ), we 1 b µ T n,k (φ l+1, w) w V c ; (c) set e l+1 φ l+1 + θ l+1 e 1 and l l + 1. end While 3. Set ê n,k = e l. Algorithms 7.2 indicates that the error equation (7.6) only involves the solution of two elliptic problems. In practice, the elliptic problems can be solved efficiently by the Boomer-AMG method (cf. [16]) Numerical experiments. Our implementation is based on the adaptive finite element package PHG [26] and the computations are carried out on the cluster LSEC-III of Chinese Academy of Sciences. The numerical experiment is performed for the TEAM Workshop Problem 21 c M1 (see Fig. 7.1). All experimental data are measured by the R & D Center of Baoding Tianwei Group Co., LTD, China [1]. Fig Geometric sizes of the silicon steel laminations. The source currents are carried in opposite directions by two coils and are 3 Ampere/Turn at a frequency of 5Hz. The conducting region consists of a lamination 25

26 stack and a magnetic plate whose dimensions are mm 3 and mm 3 respectively, as shown in Fig The lamination stack consists of 2 simplyconnected steel sheets and placed in between the coils and the magnetic plate. The thickness of each steel sheet is.3mm. The thickness of the coating film over each sheet is 4µm. We refer to [1] for more details of the model. Suppose that the steel sheets are laminated together along the x 1 -direction. Fig. 7.2 and 7.3 illustrate a cross-section of the lamination stack in the x-z plane. The dash lines represent 2 steel sheets and the solid lines represent finite element partitions. The tetrahedral mesh of the lamination stack is constructed in two steps: c is subdivided into N layers of cuboid elements (N = 3 in Fig. 7.2 and N = 4 in Fig. 7.3); each cuboid element is subdivided into six tetrahedra. We set m = 1 in (7.3). This means that the first-order and second-order Lagrange finite element spaces are adopted for u and ψ respectively. The right figures of Fig. 7.2 and 7.3 show the numerical values and experimental values of the first component of the magnetic flux density. The curve with smaller slope represents the values of B 1 at 24 points on the line {(x, y, z) : x = 5.76 mm, y = mm}, and the other curve represents the values of B 1 at 24 points on {(x, y, z) : x = mm, y = mm}. For both figures, the number of partition layers is much less than the number of steel laminations, namely, N 2. So the homogenized eddy current model saves computations significantly. And Fig also show that refining the mesh near the coils improves the computational accuracy considerably..3.2 Numerical data Measured data Bx (T) z (m) Fig Left: the lamination stack is divided into N = 3 layers of elements in the x 1 -direction. Right: numerical and experimental values of B 1. Fig. 7.4 shows the distribution of eddy current density on the cross-section of the lamination stack at x 1 =.57 (inside the second lamination from the source side). Fig. 7.5 shows the distribution of eddy current density on two cross-sections of the 26

27 .3.2 Numerical data Measured data Bx (T) z (m) Fig Left: the lamination stack is divided into N = 4 layers of elements in the x 1 -direction. Right: numerical and experimental values of B 1. magnetic plate which are at x 1 =.1 and x 1 =.9 respectively. Since the cross-section at x 1 =.1 is close to the lamination stack, Fig. 7.5 (Left) shows a clearly shielded area by the laminations. But the magnetic flux still penetrates into the plate from the opposite side, and yields stronger eddy current density on the cross-section at x =.9 (Fig. 7.5 (Right)). Fig Distribution of eddy current density on the cross-section of the lamination stack at x 1 = Concluding remarks. We study the homogenization of multi-scale timedependent electromagnetic eddy current problems in three dimension. The homogenized H-formulation is proposed for nonlinear problems and multiply-connected conductors. For nonmagnetic materials, the homogenized Maxwell s equations are also derived by linking up the H-formulation and the A-formulation. The homogenized eddy current model is validated by finite element computing of an engineering benchmark problem from the International Compumag Society. 27

28 Fig Distribution of eddy current density on two cross-sections of the magnetic plate. Left: the side adjacent to the lamination stack. Right: the opposite side. Acknowledgement. The authors thank the anonymous referees very much for their valuable suggestions and comments on this paper, particularly for the suggestion to extend the contents of numerical computations. REFERENCES [1] H. Ammari, A. Buffa, and J. Nédélec, A justification of eddy current model for the maxwell equations, SIAM J. Appl. Math., 6 (2), pp [2] C. Amrouche, C. Bernardi, M. Dauge, and V. Girault, Vector potentials in threedimensional non-smooth domains, Math. Meth. Appl. Sci., 21 (1998), pp [3] F. Bachinger, U. Langer, J. Schöberl, Numerical Analysis of Nonlinear Multiharmonic Eddy Current Problems, Numer. Math., 1 (25), pp [4] R. Beck, R. Hiptmair, R. Hoppe and B. Wohlmuth, Residual based a posteriori error estimators for eddy current computation, Math. Model. Numer. Anal. 34 (2), pp [5] A.J. Bergqvist and S.G. Engdahl, A homogenization procedure of field quantities in laminated electric steel, IEEE Trans. Magn., 37 (21), no. 5, pp [6] A. Bermúdez, D. Gómez, and P. Salgado, Eddy-current losses in laminated cores and the computation of an equivalent conductivity, IEEE Trans. Magn., 44 (28), no. 12, pp [7] O. Bottauscio, V. Chiadopiat, M. Chiampi, M. Codegone, and A. Manzin, Nonlinear homogenization technique for saturable soft magnetic composites, IEEE Trans. Magn., 44 (28), no. 11, pp [8] O. Bottauscio, M. Chiampi, and A. Manzin, Homogenized magnetic properties of heterogeneous anisotropic structures including nonlinear media, IEEE Trans. Magn., 45 (29), no. 3, pp [9] L. Cao, Y. Zhang, W. Allegretto, and Y. Lin, Multiscale asymptotic method for Maxwell s equations in composite materials, SIAM J. Numer. Anal., 47 (21), pp [1] Z. Cheng, N. Takahashi, and B. Forghani, TEAM Problem 21 Family (V. 29), approved by the International Compumag Society Board at Compumag-29, Florianópolis, Brazil, [11] Z.Cheng, N.Takahashi, B. Forghani, and et al, Electromagnetic and Thermal field Modeling and Application in Electrical Engineering, Science Press, Beijing, 29. [12] J. Chen, Z. Chen, T. Cui and L. Zhang, An adaptive finite element method for the eddy current model with circuit/field couplings, SIAM J. Sci. Comput., 32 (21), pp [13] M. Costabel, M. Dauge, and S. Nicaise, Singularities of eddy current problems, ESAIM: Mathematical Modelling and Numerical Analysis, 37 (23), pp [14] J. Gyselinck and P. Dular, A time-domain homogenization technique for laminated iron cores in 3D finite element models, IEEE Trans. Magn., 4 (24), no. 3, pp [15] J. Gyselinck, R.V. Sabariego, and P. Dular, A nonlinear time-domain homogenization technique for laminated iron cores in three-dimensional finite-element models, IEEE Trans. Magn., 42 (26), pp [16] V. E. Henson and U. M. Yang, BoomerAMG: a parallel algebraic multigrid solver and preconditioner, Appl. Num. Math., 41 (22), pp