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

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A Novel Finite Analytic Element Method for Solving Eddy Current Problems with Moving Conductors __________________________________________________________________________www.paper.edu.cn 1 A Novel Finite Analytic Element Method for Solving Eddy Current Problems with Moving Conductors 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 2 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 3 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 4 (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 中国科技论文在线_________________________________________________________________________www.paper.edu.cn 5 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 中国科技论文在线_________________________________________________________________________www.paper.edu.cn 6 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 .
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