IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 8, AUGUST 2010
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Effectiveness of Higher Order Time Integration in Time-Domain Finite-Element Analysis Yoshifumi Okamoto1 , Koji Fujiwara2 , and Yoshiyuki Ishihara2 Department of Electrical and Electronic System Engineering, Utsunomiya University, Utsunomiya, Tochigi 321-8585, Japan Department of Electrical Engineering, Doshisha University, Kyotanabe, Kyoto 610-0321, Japan The backward Euler (BE) method is widely applied to the magnetic field analysis of various electrical machines for the reason of unconditional stability in the time-domain analysis. However, computation accuracy occasionally deteriorated in the time domain. In this paper, the effectiveness of higher order time integration using GEAR’s backward formulas is validated by the comparison with the conventional time integration technique such as BE and Galerkin method from the viewpoint of the numerical accuracy and the effect of convergence characteristics of incomplete Cholesky conjugate gradient (ICCG) and Newton–Raphson (NR) method. As a result, it is shown that the computation accuracy can be improved by GEAR’s backward formulas with the same elapsed time as the BE method. Index Terms—Backward Euler (BE) method, coupled analysis with electrical circuits, eddy current analysis, Galerkin method, GEAR’s backward difference formulas.
I. INTRODUCTION
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HE magnetic field analysis using finite-element method is an important technique to perform the practical design of electrical machines. Especially, the temporal transient analyses with magnetic nonlinearity are indispensable to the performance evaluation of various machines such as transformer or motor. Generally, the backward difference method, which is just called backward Euler (BE) method, is widely applied to the timedomain finite-element method [1], [2] for the unconditionally stable characteristic. However, there is a case, in which the accuracy of some temporal characteristics is insufficient for the low-order approximation using single time step. Therefore, the multistep method has the possibility to improve the numerical error. For example, multistep Adams algorithm is successfully applied to the plate and transformer model [3]. In this paper, we investigated the improvement of numerical error and convergence characteristics about Newton–Raphson (NR) and incomplete Cholesky conjugate gradient (ICCG) method in using the GEAR’s multistep backward difference formulas [4]. The effectiveness of GEAR’s formulas is illustrated by the comparison with BE method and Galerkin method.
T
where is the magnetic vector potential, is the electric scalar is the edge-based vector shape function, is the potential, nodal scalar shape function, the material constants and are the magnetic reluctivity and conductivity, and is the source current density, respectively. The boundary integral terms in both equations can be omitted. Hereinafter, the assembling equation of (1) and (2) is abbreviated to the -formulation as follows: (3) where corresponds with second term in (1), and components are defined as follows: of coefficient matrices and (4) (5) Various implicit schemes are described in the following sections. Hereinafter, the time step size is not adaptive but a constant value. However, there is the possibility of a fast analysis using adaptive time step size such as embedded Runge–Kutta method [5]. A. Implicit Scheme Based on Trapezoidal Rule
II. FORMULATION IN THE TIME-DOMAIN ANALYSIS This section describes the formulation using the various implicit time integration technique. The edge-based weighted and nodal residual in the - formulation are residual given as follows:
The variables and are vectors with temporal changes. Then, these variables can be discretized in time domain as follows: (6) (7)
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, and superwhere is a constant value within the range indicates the th time step. Here, the time derivative script term in (3) is given as follows:
Manuscript received December 23, 2009; revised February 20, 2010; accepted February 23, 2010. Current version published July 21, 2010. Corresponding author: Y. Okamoto (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2010.2044771 0018-9464/$26.00 © 2010 IEEE
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where is the time step size. Substituting (6)–(8) for (3) gives the equation based on trapezoidal rule as follows:
in GEAR second, in order to keep the assembling e.g., matrix symmetry. C. Stability of GEAR’s Formula in Magnetic Field Analysis
(9) and should be attached with where superscripts because the permeability is updated for the magnetic nonlinearlity every time step. The BE procedure can be introduced when is equal to one. Next, temporal weak form of (9) is arrived by using Galerkin method with a selection of as weighted coefficient [6] as follows:
(10) As a result, (10) is transformed into
In this section, the stability of GEAR’s backward difference formula with second order is investigated in 1-D magnetic field analysis with eddy current as follows: (16) where is an 1-D component of , and the characteristics of and are linear. The left-hand side term is discretized by the first-order finite difference, and the right-hand side term is discretized by implicit GEAR second. The discretized equation can be obtained as follows: (17) where is equal to , which is assumed a positive is the element interval, and the subscript describes value, the grid number in a finite difference mesh. A numerical solution of (16) can be described as follows: (18)
(11) where the above equation corresponds with (9) on the condition , therefore (11) is called Galerkin in the following section. B. GEAR’s Multistep Formulas
expresses a component with the frequency obwhere tained from Fourier series expansion of numerical solution, and is the ratio of complex amplitude. The length should be less than 1.0 from the viewpoint of the stabilized scheme. Then, substituting (18) for (17) gives the quadratic equation with respect to (19)
Multistep backward approximation is applied to the time derivative term in (3). GEAR’s backward formula with second-order (GEAR second) is shown as follows: (12) The third-order formulation (GEAR third) is expressed as follows: (13)
The solutions of (19) are shown as follows: (20) where condition is imposed. It is . When clear that both solutions are within the range is imposed, (20) another condition can be expressed by the complex number. The lengths of both complex numbers are shown as follows: (21)
When these higher order backward differences are applied to (3), the implicit procedures are given as follows:
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Similarly, the length of complex number is within the range . Therefore, GEAR’s second-order backward formula is proved as unconditionally stable as BE or Galerkin procedure. III. VERIFICATION IN THE MAGNETIC FIELD ANALYSIS
(15) The BE method should be used in the initial step calculation of the GEAR second and third formulation. Similarly, GEAR second should be used in the second step calculation of GEAR third. When the coupled problem with the electrical circuit equations are strongly formulized, appropriate coefficient should be multiplied in the equations for electrical circuits,
Various implicit procedures are applied to the nonlinear transient problem with eddy current and strongly coupled problem with the electrical circuits in this section. The - formulation is applied to the eddy current problem. NR method with the line [7] is adopted search based on functional minimization as a nonlinear iterative method. The linear equations every NR step are solved by ICCG method under the convergence criterion , where is a -norm of the residual is set up as . of linear equation at the th iteration and Hereinafter, the NR iteration is terminated under the condition
OKAMOTO et al.: EFFECTIVENESS OF HIGHER ORDER TIME INTEGRATION IN TIME-DOMAIN FINITE-ELEMENT ANALYSIS
Fig. 1. GEAR’s implicit multistep information: (a) GEAR second; (b) GEAR third.
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Fig. 3. Eddy current losses in magnetic shielding model.
Fig. 2. Magnetic shielding model (shield thickness: 1 mm).
10
T, where
is the increment of
on the element
center. A. Magnetic Shielding Model The 3-D magnetic shielding model [7], [8], which is proposed by the Institute of Electrical Engineers of Japan (IEEJ) committee as a nonlinear magnetostatic and eddy current benchmark problem, is analyzed. Magnetic shielding model is discretized by using the hexahedral edge-based elements with first order as shown in Fig. 2. The number of elements, nodes, and unknowns are 19 869, 22 154, and 59 138, respectively. The material of magnetic shield is SS400 with magnetic nonlinearity [7], and conductivity is 7.505 10 S/m. The waveform of magnetizing current is sinusoidal curve with 50 Hz, and the amplitude is set as 8.333 10 s with 48 time steps is 2000 AT. The during the two cycles of magnetizing current. The distribution of current density is homogenized by using the electric scalar potential method with regularization [9]. The eddy current losses are compared with the fine discretized results (standard solution) that are obtained by BE 1.111 10 s (3 600 time steps) in the method with same finite-element mesh. The instantaneous eddy current loss in the th time step is given as follows: (22) where appropriate implicit formula such as (8), (12), and (13) should be substituted for the first derivative term of . Fig. 3 shows the temporal behaviors of eddy current loss. There is a difference of maximum value and phase lag between
=
Fig. 4. Contours of eddy current density (t 0.02 s): (a) Backward Euler. (b) Galerkin. (c) GEAR second. (d) GEAR third. TABLE I COMPARISON OF INSTANTANEOUS EDDY CURRENT LOSS (t
= 0.02 s)
BE and standard solution. The maximum value is slightly improved by using Galerkin method, however, the phase lag still appears. These numerical errors are remarkably removed by using GEAR’s formulas on the condition of the same number of time steps. The eddy current loss distribution nearly corresponds with the standard solution. The result can be still more improved using GEAR’s third implicit formula. Table I shows the comparison of instantaneous eddy current 0.02 s. The relative errors are evaluated against the losses in standard solution. The error of Galerkin method is similar to the one of GEAR’s second order. The error can be more decreased by using GEAR’s third order. There is little difference between various contours of instantaneous eddy current density as shown in Fig. 4. The eddy current loss is calculated by volume integral of eddy current density in total conducting region. Therefore, the loss error is highly estimated by summation of local eddy current. Table II shows the convergence characteristics. The performance of ICCG method and NR convergence has the similarity
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TABLE II CONVERGENCE CHARACTERISTICS IN MAGNETIC SHIELDING MODEL
Fig. 5. Transformer model.
phase transformer model. The numbers of elements, nodes, and unknowns are 23 712, 26 460, and 67 680, respectively. The frequency of power supply is 10 kHz with sinusoidal wave, the turn ratio is two, and the core material is 10JNEX900 made by JFE is set up as 4.167 s with 96 time Steel Corporation. The steps during the four cycles of voltage source. Fig. 6 shows the distributions of primary current and secondary current normalized by that is a maximum current in the steady state. While a little phase lag appeared in BE method against the standard solution, other distributions nearly correspond with the standard solution. Table III shows the relative errors in the extremum. The accuracy of Galerkin method is similar to the one of GEAR’s second order. Table IV shows the convergence characteristics. The performances of the ICCG and NR convergence have the similarity between these implicit procedures. IV. CONCLUSION We pointed out the insufficient accuracy of the well-known BE method, and verified the effectiveness of GEAR’s higher order backward formulas to the nonlinear transient magnetic field analysis. The GEAR’s formulas were successfully applied to the eddy current problem and strongly coupled problem with electrical circuits. Especially, GEAR’s higher order formulas were effective for the eddy current analysis to improve the numerical error with similar elapsed time against the BE method. The development of the analysis method using adaptive order and adaptive time step size using GEAR’s formulas is our future perspective. REFERENCES
Fig. 6. Current distributions: (a) primary current; (b) secondary current. TABLE III COMPARISON OF WINDING CURRENT IN TRANSFORMER MODEL (t
= 0.4 ms)
TABLE IV CONVERGENCE CHARACTERISTICS IN TRANSFORMER MODEL
in various implicit formulations, and elapsed times are almost same. B. Transformer Model The performance of GEAR’s implicit formulas is verified in the strongly coupled problem with the electrical circuit equations [10]. Fig. 5 shows the finite-element mesh for the single-
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