A Novel Finite Analytic Element Method for Solving Eddy Current Problems with Moving Conductors

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D. Z. Chen K.R. Shao Department of Electrical Engineering Department of Electrical Engineering Huazhong University of Science and Technology Huazhong University of Science and Technology Wuhan, 430074 China Wuhan, 430074 China [email protected] [email protected]

Yu Haitao J.D. Lavers Department of Electrical Engineering Department of Electrical and Computer Engineering Huazhong University of Science and Technology University of Toronto Wuhan, 430074 China Toronto M5S 1A4, Canada [email protected]

Abstract

A novel finite analytic element method (FAEM) is presented. The basic idea of the method is the incorporation of local analytic solution of the governing equation in the finite element method. A local analytical solution satisfying its nodal conditions is found in each element and is used for determining the shape functions. Then, a weighted residuals scheme is followed to yield the linear algebraic equations. The presented FAEM is applied to solve 1D and 2D eddy current problems with moving conductors. Because the problem’s analytical features have been considered, the solution in each element is approximated closely and the spurious oscillations which occur in the ordinary Galerkin solutions are avoided. High accuracy is obtained with no need of very fine meshes.

1. Introduction

When magnetic fields involving moving conductors are computed, eddy currents due to the movement of conductors should be taken into account. For the A−φ method, the governing equations are 1 ∇ × ∇ × A − σ[v × ∇ × A − jωA − ∇φ] = J (1a) µ s ∇⋅σ[v ×∇× A − jωA −∇φ] = 0 (1b)

This paper was published in IEEE Transactions on Magnetics, V.37, No.5, p.3150. 中国科技论文在线______www.paper.edu.cn

where A and φ are the magnetic vector potential and the electrical scalar potential, respectively, µ and σ are the permeability and the conductivity, respectively, and v is the velocity of the media relative to the

source. The second equation is the constrained eddy current equation ( ∇ ⋅ J e = 0 ). Equation (1) is mathematically a Convection-Diffusion equation. When it is solved by using the ordinary Galerkin finite element method, particularly when both the speed of conductors and the relative permeabilities of the moving parts are very high, the element Peclet number is typically greater than unity and the numerical solutions contain spurious oscillations [1]-[3]. The element Peclet number is defined as

Pe = vσµh / 2 (2) where h is the length of the element in the direction of the velocity. Therefore, in order to eliminate the

spurious oscillations, the mesh must be refined to insure Pe <1 . This usually greatly increases the requirement of both the computer memory and the CPU time, and sometimes makes the method unpractical. To suppress the oscillations, the upwind method has been developed [4]-[6]. The upwind scheme uses an unsymmetric weight function, with its upwind side weakened and the downwind side strengthened. This method successfully precludes the spurious oscillations but introduces excessive diffusion [7], [8]. The source of the spurious oscillations is that the ordinary Galerkin method uses linear shape functions to approximate the local solution in an element. The solutions of (1) are exponential functions. A portion of

such a solution u in the interval [xi−1, xi+1 ] is shown in Fig.1. It is obvious that a linear function only provides a good local approximation to u when the interval is sufficiently small. Fig.1 also indicates that the

derivative of u at the point xi is close to the slope of line ab, the result of a backward difference, rather than the slope of line ac, the result of a central difference. The usual statement that the central difference has the second order accuracy while the backward difference has only the first order accuracy is not always true; it

needs a presupposition that the interval [xi−1, xi+1 ] is sufficiently small. This is the real reason why the spurious oscillations appear in the ordinary Galerkin solutions of the Convection-Diffusion equation when

Pe > 1 . This also helps to understand why the upwind method, which can be considered an artificial modification of the linear finite element method, works better than the ordinary Galerkin method.

Fig. 1 A portion of exponential function u in [xi-1, xi]

2. Idea of the Finite Analytic Element Method 中国科技论文在线______www.paper.edu.cn

The ordinary Galerkin finite element method uses the uniform approach to discretization and linear element shape function to deal with generic classes of partial differential equations. The particular nature of a given problem is not considered. This greatly enhances the method’s applicability but depresses the efficiency. In order to conquer this shortcoming, a novel finite analytic element method (FAEM) is proposed in this paper.

The basic idea of the FAEM is the incorporation of the local analytic solution of the governing equation in the finite element method. A local analytical solution satisfying its nodes conditions is found in each element and is used for determining the shape functions. Then, a weighted residuals scheme is followed to yield the linear algebraic equations. Because the problem’s analytical features have been contained in the shape function, the solution in each element is approximated closely and the method can be expected to be more effective and more economical.

3. One-Dimensional FAEM

In the 1D case, the governing equation can be written as ∂ 2u ∂u − 2P + k 2u = 0 (3) ∂x2 ∂x where k 2 = −jωµσ ， 2P = µσv . For convenience sake, we have used the scalar u in (3) instead of the vector A as in (1). Assume the solution domain is in [0, 1] and the boundary conditions are u(0) = 0 and u(1) = 1.

Fig. 2 1D FAEM element

The interval [0, 1] is discretized into n elements, with each of length h =1/ n . Assume the solution values

on the two nodes of the ith element are ui−1 and ui , as shown in Fig.2. The local analytical solution in the ith element is

(P+Q)ξ (P−Q)ξ u(i) = c1e + c2e (4)

2 2 Here, a local coordinate system ξ = x − (i − 0.5)h is used and Q = P − k . The coefficients c1 and c2 are determined by using the nodal conditions

u(−h/ 2) = ui−1 , u(h/ 2) = ui (5) From (5) we have 中国科技论文在线______www.paper.edu.cn

(e−−Qh(ξξ/2) −−eQh( −/2) )eP(ξ+h/2) (eQ(ξ+h/2) e−Q(ξ+h/2) )ePh(ξ−/2) uu()i =+u (6) eeQh −−−−Qh ii−1 eeQh Qh

According to (6), we can take the shape functions as (e−Q(ξ −h / 2) − eQ(ξ −h / 2) )e P(ξ +h / 2) N = , i−1 eQh − e−Qh (7) (eQ(ξ +h / 2) − e−Q(ξ +h / 2) )e P(ξ −h / 2) N = i eQh − e−Qh

Following a weighted residuals scheme and letting the weight functions Wi−1 and Wi equal the shape

functions Ni−1 and N i , respectively (the Galerkin scheme), we obtain the element matrix ⎡ (1+ e−2Qh )Q 2e(−P−Q)hQ ⎤ ⎢P − −2Qh −2Qh ⎥ A(i) = ⎢ 1− e 1− e ⎥ (8) 2e(P−Q)hQ (1+ e−2Qh )Q ⎢ − P − ⎥ ⎣⎢ 1− e−2Qh 1− e−2Qh ⎦⎥ where the element of the matrix A(i) is derived by

h / 2 dW dN dN a(i) = (− i1 i2 − 2PW i2 + k 2W N )dξ , ( i ,i = {i −1,i}) (9) i1,i2 ∫−h / 2 dξ dξ i1 dξ i1 i2 1 2 Equation (8) leads to the final FAEM linear algebraic equations as

Ph Qh −Qh −Ph e ui−1 − (e + e )ui + e ui+1 = 0 (10)

If a linear element is used the FEM linear algebraic equations are

−(1+ Pe + β)ui−1 + (2− 4β)ui −(1− Pe + β)ui+1 = 0 (11) 2 2 where Pe = Ph = vσµh / 2 is the element Peclet number defined by (2); β = k h /6 . The solutions of (3) by FAEM and by linear FEM are compared with the exact solution, shown in Fig.3. The

linear FEM solutions contain severe spurious oscillations when Pe > 1. In contrast, because all the boundary conditions and the whole equation are satisfied in the 1D cases, the FAEM always gives the exact solution. Fig.3 also indicates that the spurious oscillations in the ordinary Galerkin solution depend not only on the

Peclet number Pe but also on the parameter β .

4. Two-Dimensional FAEM

In the 2D case, the following equation will be studied

∂2u ∂2u ∂u + − 2P + k 2u = 0 (12) ∂x2 ∂y2 ∂x

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Fig. 3 Comparison of FAEM, FEM and analytic solutions for 1D problems.

Assume that the solution domain is limited to a square domain ( 0 ≤ x ≤ 1, 0 ≤ y ≤1) and the boundary conditions are u(0, y) = 0 , u(1, y) =1 , u(x,0) = u(x,1) = x (13)

A rectangular FAEM element with length of h1 and height of h2, as shown in Fig.4, will be discussed. As in

the 1D case, a local coordinate system (ξ,η) is used: ξ = x − (x1 + h1 / 2) , η = y − (y1 + h2 / 2) . Assume the

solution values on the four nodes are u1 , u2 , u3 and u4 , respectively. Let u(ξ,η) = X (ξ)Y(η) (14) Substituting (14) into (12) we have got

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Fig. 4 2D FAEM element.

X ′′ X ′ Y′′ − 2P + + k 2 = 0 (15) X X Y Using the separation of variables method X ′′ X ′ − 2P + P2 − Q2 = 0 (16) X X Y′′ − k 2 = 0 (17) Y 2

2 2 2 2 2 where P − Q = k + k2 . k2 is the separation constant to be determined. Solutions of (16) and (17) are

respectively

(P+Q)ξ (P−Q)ξ X = C1e + C2e (18)

k2η −k2η Y = D1e + D2e (19)

Therefore the solution of (12) can be written as

()P++Qξ kP22ηξ(−Q)+kη()P+−Qξk2η(P−Q)ξ−k2η uc=+12eec +c3e+c4e (20)

Equation (20) must satisfy the nodal conditions

u(−−h12/ 2, hu/ 2) =1, u(h1/ 2, −hu2/ 2) =2, (21)

u(h12/ 2, hu/ 2) =−3, u( h1/ 2, hu2/ 2) =4 Generally speaking, there are an infinite number of solutions satisfying (21), each corresponding to a 2 2 parameter k2 . If we had some prior knowledge about the solution, we might select a good k2 . We now

choose Qh1 = k2h2 , which will make the proposed shape functions have a simple form. The four coefficients

c1 , c2 , c3 and c4 can be determined with the nodal conditions (21). We therefore have

u = u1N1 + u2 N 2 + u3 N3 + u4 N 4 (22) 中国科技论文在线______www.paper.edu.cn

where N1 , N 2 , N3 and N 4 are the shape functions:

(P+Q)h1+k2h2 +(P−Q)x−k2 y e 2 (e2Qx − eQh1 )(e2k2 y − ek2h2 ) N1 = , (−1+ e2k2h2 )(−1+ e2Qh1 )

−(P−Q)h1+k2h2 +(P−Q)x−k2 y e 2 (−eQ(h1+2x) +1)(e2k2 y − ek2h2 ) N2 = , (−1+ e2k2h2 )(−1+ e2Qh1 )

−(P−Q)h1+k2h2 +(P−Q)x−k2 y e 2 (eQ(h1+2x) −1)(ek2 (h2 +2 y) −1) N3 = , (−1+ e2k2h2 )(−1+ e2Qh1 )

(P+Q)h1+k2h2 +(P−Q)x−k2 y e 2 (−e2Qx − eQh1 )(ek2 (h2 +2 y) −1) N = 4 2k2h2 2Qh1 (−1+ e )(−1+ e ) (23)

Letting the weight functions W1 , W2 , W3 and W4 equal the shape functions and following a weighted residuals scheme, we obtain the element matrix

⎡a(e) a(e) a(e) a(e) ⎤ ⎢ 1,1 1,2 1,3 1,4 ⎥ a(e) a(e) a(e) a(e) A(e) = ⎢ 2,1 2,2 2,3 2,4 ⎥ (24) ⎢ (e) (e) (e) (e) ⎥ a3,1 a3,2 a3,3 a3,4 ⎢ (e) (e) (e) (e) ⎥ ⎣⎢a4,1 a4,2 a4,3 a4,4 ⎦⎥

where

hh21/2 /2 ddWNiiddWNii dNi aP()e =−(212−12−W2+k2WN)dξηd, ii12, ∫∫−−hh/2 /2 i11i i2 21 ddξξ ddηη dξ

(,ii12= {1,2,3,4}) (25)

The analytical expressions given by (25) are easily obtainable by using software capable of symbolic evaluation. The solutions of (12) along line y = 0.5 by FAEM and by linear FEM are compared with the exact solution, 2 shown in Fig.5. When the parameter k = 0 and Pe is very small the FAEM gives a wrong solution. This is

because the coefficients derived from (25) contain very high power of Pe and Qh1 in the denominator and 2 cannot be evaluated exactly when Pe and Qh1 are approaching to zero. When k ≠ 0 and Pe is large, the FAEM solution agrees very well with the exact solution but the FEM solution contains severe spurious

oscillations. Moreover, our calculations have also shown that while Pe has a large value, the linear algebraic equations given by FAEM converge much more quickly than those given by linear FEM when solved by using BICG method. This indicates that FAEM is applicable to the problems particularly with very high Peclet number. 中国科技论文在线______www.paper.edu.cn

5. Conclusions and Discussions

We have presented a novel finite analytic element method in this paper. The basic idea of the FAEM is the incorporation of local analytic solution of the governing equation in the finite element method. Because the problem’s analytical features have been considered in the shape function, the solution in each element is approximated closely and the method is more effective and more economical. The presented FAEM is used to solve eddy current problems with moving conductors. 1D and 2D problems have been investigated. The results are found to be significantly better than obtained by using ordinary Galerkin method that uses linear elements. The method can be easily extended to the 3D cases. It should be noted that the incorporation of a local analytic solution in the numerical solution has been proposed by Ching-Jen Chen [9]. However, his method is rather close to a finite difference method (FDM) and is different from ours. It is well known that the FEM has a much stronger ability to deal with the boundary conditions than the FDM. In addition, the method proposed in this paper to represent the local analytic solution is much more concise than that used in [9]. Despite these differences, the theories proposed in [9] are instructive in developing the FAEM presented in this paper.

Fig. 5 Comparison of FAEM, FEM and analytic solutions for 2D problems 中国科技论文在线______www.paper.edu.cn

References [1] N. Burais, N. Foggia and A. Nicolas, “Numerical solution of eddy currents problems including moving conducting parts,” IEEE Trans. Magn., vol. MAG-20, no. 5, pp.1995-1997, 1984 [2] D. Rodger, P. J. Leonard and T. Karagular, “An optimal formulation for 3D moving conductor eddy current problems with smooth rotors,” IEEE Trans. Magn., vol. 26, no. 5, pp.2359-2363, 1990 [3] H. Song and N. Ida, “Modeling of velocity terms in 3D eddy current problems,” IEEE Trans. Magn., vol. 28, no. 2, pp.1178-1181, 1992 [4] S. Y. Hahn, J. Bigeon and J.C. Sabonnadiere, “An ‘upwind’ finite element method for electromagnetic field problems in moving media”, Int. J. Num. Meth. Engng., vol. 24, pp.2071-2086, 1987 [5] M. Ito, T. Takahashi and M. Odamura, “Up-wind finite element solution of travelling magnetic field problems,” IEEE Trans. Magn. Vol. 28, no. 2, pp. 1605-1610, Mar 1992 [6] H. T. Yu, K. R. Shao, K. D. Zhou et al, “Upwind-linear edge elements for 3D moving conductor eddy current problems,” IEEE Trans. Magn., vol. 32, no. 3, pp.760-763, 1996 [7] T. J. R. Hughes, A. Brooks, “Multi-dimensional upwind scheme with no crosswind diffusion,” American Society of Mechanical Engineers, Applied Mechanics Division, AMD vol. 34, pp.979, Dec 2-7 1979 [8] E. K. C. Chan and S. Williamson, “Factors influencing the need for upwinding in two-dimensional field calculation,” IEEE Trans. Magn. Vol. 28, no. 2, pp. 1611-1614, Mar 1992 [9] C. J. Chen and H. C. Chen, “Development of finite analytic numerical method for unsteady two dimensional convective transport equation,” Proceedings of International Symposium on Refined Modeling of Flows, Paris, pp. 7-10, Sep. 1982