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Fast time-domain finite element analysis of 3D nonlinear time-periodic eddy current problems with T,Φ-Φ formulation Oszkár Bíró, Gergely Koczka, Kurt Preis IGTE, Graz University of Technology Kopernikusgasse 24, 8010 Graz, Austria [email protected] Abstract— A method is presented for obtaining the steady- state solution of three-dimensional eddy current problems with nonlinear material properties and periodic excitations using the T,Φ-Φ formulation. The problem is formulated in the time- domain and the application of a fixed-point approach ensures that there is no coupling between the time-steps within one period. Therefore, the steady-state result is obtained by stepping through one period only. Comparison with results by a transient time-stepping method is presented. I. INTRODUCTION A method has been presented in [1] for solving 3D nonlinear eddy current problems under periodic excitation by the A,v-A formulation, using a fixed point algorithm in the time-domain to get the periodic solution directly without having to couple the harmonics during the non-linear iterations. The aim of this paper is to extend the method to the T,Φ-Φ formulation [2]. II. FORMULATION Applying the Galerkin method to the differential equations of the T,Φ-Φ formulation result in an ordinary differential equation system: ( ) ( ) ( ) ( ) ( ) ( ) d t t dt dt d ρ μ ρ + = + ⎡ ⎤ ⎣ ⎦ S x M x f g μ (1) where the vector x(t) includes the coefficients of the finite element approximations of the potentials T and Φ. The stiffness matrix S and the vector f depend on the resistivity ρ (not explicitly shown in the following), i.e. they are independent of x(t). However, the mass matrix M and the vector g depend on the permeability μ, i.e. on x(t) and, hence, M is time-dependent. The excitations are time-periodic and only the steady-state solution of (1) is required. A time- stepping, brute-force method usually takes a very long time, since several periods have to be stepped trough to obtain steady-state. III. FIXED-POINT METHOD Applying the fixed point method to (1) similarly to [1], results in an iterative algorithm: ( ) ( ) ( ) () ( ) ( ) ( 1) () ( 1) () s s s FP d t t dt dt ρ μ ρ + + ⎡ ⎤ + = + ⎣ ⎦ S x M x f g s d μ ( ) () () () () s s s FP d t dt μ μ ⎡ + − ⎣ M x ⎤ ⎦ (2) where () s x is the vector of unknowns in the s-th iteration step, () s μ is the permeability calculated from the s-th solution and () s FP μ is the fixed-point permeability which is different in all sian points and its distribution can be chosen to ensure optimal convergence similarly to the way the optimal fixed- point reluctivity has been selected in [3]. IV. TIME-DOMAIN TECHN Gaus Th schem IQUE kward Euler e time-discretization of (2) with a bac e using equidistant time-steps within one period T (Δt=T/n) leads to n coupled simultaneous linear equation systems ( ) ( ) ( ) ( ) ( ) () () 1 1 ( 1) 1 1 s s s s s k F μ μ + + + + + − + g g Sx M 1 () 1 k k s k k P k t t μ + + − = + Δ Δ x x f ( ) ( () () () () () () 1 1 1 s s s s s s FP k k FP k k t μ μ μ μ + + ⎡ ⎤ + − − ⎣ ⎦ Δ M x M x (3) e x k =x(kΔt), k=0, 1, …, n-1 and x n =x 0 . block-structured RESULTS he d by the T,Φ-Φ VI. REFERENCES G. Koczka, eis, “Optimal fixed-point method for eddy current problems,” on Magnetics, vol. linear eddy current problems," IEEE ol. wher T form [1] The equations (3) result in a huge equation system with a block-circulant matrix and a block- structured right hand-side which can be solved by the block- diagonalization method of [4]. It has been shown in [3] that this is equivalent to applying the discrete Fourier (or z-) transformation and results in the steady-state solution by stepping through one period only. V. NUMERICAL first example of [1] has been solve ulation leading to 32,739 degrees of freedom using the prescribed method and also by brute force stepping through 3 periods necessary to attain steady state. The two solutions have been found to practically coincide but the number of linear equation systems solved is just 330 with the present method instead of 1,653 necessary with the brute force technique. S. Außerhofer, O. Bíró and K. Pr solving 3D nonlinear periodic COMPEL, vol. 28, no. 4, pp. 1059-1067, April 2009. [2] O. Bíró, P.Böhm, K. Preis, G. Wachutka, “Edge finite element analysis of transient skin effect problems”, IEEE Transactions 36, No. 4, pp. 835-839, July 2000. [3] G. Koczka, S. Außerhofer, O. Bíró and K. Preis, "Optimal convergence of the fixed-point method for non Transactions on Magnetics, vol. 45, no. 3, pp. 948-951, March 2009. O. Bíró and K. Preis, “An efficient time domain method for nonlinear periodic eddy current problems,” IEEE Transactions on Magnetics, v [4] 42, no. 4, pp. 695–698, Apr. 2006. 978-1-4244-7062-4/10/$26.00 ©2010 IEEE
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Fast time-domain finite element analysis of 3D nonlinear time-periodic eddy current problems with T,Φ-Φ formulation
Oszkár Bíró, Gergely Koczka, Kurt Preis IGTE, Graz University of Technology
Kopernikusgasse 24, 8010 Graz, Austria [email protected]
Abstract— A method is presented for obtaining the steady-state solution of three-dimensional eddy current problems with nonlinear material properties and periodic excitations using the T,Φ-Φ formulation. The problem is formulated in the time-domain and the application of a fixed-point approach ensures that there is no coupling between the time-steps within one period. Therefore, the steady-state result is obtained by stepping through one period only. Comparison with results by a transient time-stepping method is presented.
I. INTRODUCTION A method has been presented in [1] for solving 3D
nonlinear eddy current problems under periodic excitation by the A,v-A formulation, using a fixed point algorithm in the time-domain to get the periodic solution directly without having to couple the harmonics during the non-linear iterations. The aim of this paper is to extend the method to the T,Φ-Φ formulation [2].
II. FORMULATION Applying the Galerkin method to the differential equations
of the T,Φ-Φ formulation result in an ordinary differential equation system:
( ) ( ) ( ) ( ) ( )( )dt tdt dt
dρ μ ρ+ = +⎡ ⎤⎣ ⎦S x M x f g μ (1)
where the vector x(t) includes the coefficients of the finite element approximations of the potentials T and Φ. The stiffness matrix S and the vector f depend on the resistivity ρ (not explicitly shown in the following), i.e. they are independent of x(t). However, the mass matrix M and the vector g depend on the permeability μ, i.e. on x(t) and, hence, M is time-dependent. The excitations are time-periodic and only the steady-state solution of (1) is required. A time-stepping, brute-force method usually takes a very long time, since several periods have to be stepped trough to obtain steady-state.
III. FIXED-POINT METHOD Applying the fixed point method to (1) similarly to [1],
results in an iterative algorithm:
( ) ( ) ( ) ( ) ( ) ( )( 1) ( ) ( 1) ( )s s sFP
dt tdt dt
ρ μ ρ+ +⎡ ⎤+ = +⎣ ⎦S x M x f g sd μ
( ) ( )( ) ( ) ( )s s sFP
d tdt
μ μ⎡+ −⎣M x ⎤⎦ (2)
where ( )sx is the vector of unknowns in the s-th iteration step, ( )sμ is the permeability calculated from the s-th solution and
( )sFPμ is the fixed-point permeability which is different in all
sian points and its distribution can be chosen to ensure optimal convergence similarly to the way the optimal fixed-point reluctivity has been selected in [3].
IV. TIME-DOMAIN TECHN
Gaus
Thschem
IQUE kward Euler e time-discretization of (2) with a bac
e using equidistant time-steps within one period T (Δt=T/n) leads to n coupled simultaneous linear equation systems
( )( ) ( ) ( ) ( )( ) ( )1 1
( 1)1 1
s ss ss
k F
μ μ+ ++
+ +
−+
g gSx M 1( ) 1 k ks k k
P kt tμ ++ −
= +Δ Δ
x xf
( ) ( )( ) ( ) ( ) ( ) ( ) ( )1 1
1 s s s s s sFP k k FP k kt
μ μ μ μ+ +⎡ ⎤+ − − −⎣ ⎦ΔM x M x (3)
e xk=x(kΔt), k=0, 1, …, n-1 and xn=x0. block-structured
RESULTS he d by the T,Φ-Φ
VI. REFERENCES G. Koczka, eis, “Optimal fixed-point method for eddy current problems,”
on Magnetics, vol.
linear eddy current problems," IEEE
ol.
wher
Tform
[1]
The equations (3) result in a hugeequation system with a block-circulant matrix and a block-structured right hand-side which can be solved by the block-diagonalization method of [4]. It has been shown in [3] that this is equivalent to applying the discrete Fourier (or z-) transformation and results in the steady-state solution by stepping through one period only.
V. NUMERICAL
first example of [1] has been solveulation leading to 32,739 degrees of freedom using the
prescribed method and also by brute force stepping through 3 periods necessary to attain steady state. The two solutions have been found to practically coincide but the number of linear equation systems solved is just 330 with the present method instead of 1,653 necessary with the brute force technique.
S. Außerhofer, O. Bíró and K. Pr solving 3D nonlinear periodic
COMPEL, vol. 28, no. 4, pp. 1059-1067, April 2009. [2] O. Bíró, P.Böhm, K. Preis, G. Wachutka, “Edge finite element analysis
of transient skin effect problems”, IEEE Transactions 36, No. 4, pp. 835-839, July 2000.
[3] G. Koczka, S. Außerhofer, O. Bíró and K. Preis, "Optimal convergence of the fixed-point method for nonTransactions on Magnetics, vol. 45, no. 3, pp. 948-951, March 2009. O. Bíró and K. Preis, “An efficient time domain method for nonlinear periodic eddy current problems,” IEEE Transactions on Magnetics, v
[4]
42, no. 4, pp. 695–698, Apr. 2006.
978-1-4244-7062-4/10/$26.00 ©2010 IEEE
