A precise adaptive cell method for eddy-current problems based on constitutive matrix recovery technique

Abstract

An Adaptive Cell Method (ACM) is proposed for high-precision simulation of eddy-current problems on the uniform coarse grid. The calculation error of the Cell Method (CM) is analyzed to draw the conclusion that the discretization of the constitutive equations is the main error source. The spirit of the ACM is to decrease the discretization error by using the constitutive matrix recovery technique without refining the grid. The accuracy and efficiency of the proposed approach are verified by the numerical tests.

Keywords

Cell Method adaptive cell method eddy currents quasi-static magnetic field

1. Introduction

The eddy-current skin effect results in the very uneven distribution of electromagnetic field [1, 2], accordingly, the requirement of a sufficiently fine spatial discretization inevitably leads to heavy computational burden. Therefore, how to achieve high-precision solution of eddy-current problems on the uniform coarse grid is well worth studying. In this paper, the Cell Method (CM) proposed by Enzo Tonti [3, 4] is adopted to solve the eddy-current equations. Then through a numerical example, the error analysis of the Cell Method is carried out to draw the conclusion that the discretization error of the constitutive equations is the main error source. To decrease the calculation error, the Adaptive Cell Method (ACM) is presented by using the patch recovery technique to update the constitutive relations, which are the ratio of current to voltage and the ratio of magnetic flux to magnetic pressure. Because the local variation of the field is taken into account, the more precise constitutive matrices can be obtained. The numerical results verify that the ACM exhibits fourth-order convergence and can improve the accuracy of eddy-current simulations without refining the grid.

2. The error analysis of cell method

The eddy-current equation in frequency domain can be expressed as follow

$\left[{\nabla\times\left({\frac{1}{\mu}\nabla\times}\right)+\mbox{j}\omega% \sigma}\right]{\bm{E}}=0.$ (1)

In the CM, the corresonding equation can be directly written in a discrete form [5, 6]

$\displaystyle({\rm{\bf\tilde{C}M}}^{\mu}{\rm{\bf C}}+\mbox{j}\omega{\rm{\bf M}% }^{\sigma}){\bm{e}}=0,$ (2) $\displaystyle{\rm{\bf M}}^{\mu}={\rm{\bf\tilde{D}}}_{l}{\rm{\bf M}}_{\mu}^{-1}% {\rm{\bf D}}_{s}^{-1},$ (3) $\displaystyle{\rm{\bf M}}^{\sigma}={\rm{\bf\tilde{D}}}_{s}{\rm{\bf M}}_{\sigma% }{\rm{\bf D}}_{l}^{-1},$ (4)

where $\omega$ is the angular frequency, ${\bm{e}}$ is the electric field intensity vector, ${\rm{\bf D}}_{l}$ and ${\rm{\bf D}}_{s}\ ({\rm{\bf\tilde{D}}}_{l}$ and ${\rm{\bf\tilde{D}}}_{s})$ are the diagonal matrices containing the elements of the primal(dual) edge length and the facet size, respectively, ${\rm{\bf M}}_{\mu}$ and ${\rm{\bf M}}_{\sigma}$ are the diagonal matrices with the elements of the conductivity and permeability, respectively, ${\rm{\bf C}}$ and ${\rm{\bf\tilde{C}}}$ are the topological matrices corresponding to the discrete curl-operator on the primal grid and dual grid, respectively, which only contain 0, $-$ 1 and 1, ${\rm{\bf M}}^{\sigma}$ and ${\rm{\bf M}}^{\mu}$ are the conductivity and permeability constitutive matrices , which are derived in an approximate way, such as the centroid method [7, 8].

The topological matrices are considered to be error-free, and thus the computational errors of the cell method mainly originated from the approximate constitutive relations. When the field distribution functions ${\bm{E}}$ and ${\bm{H}}$ are available, the accurate constitutive matrices can be calculated in term of an integration form as follows

$\displaystyle{\rm{\bf M}}_{i,j}^{\sigma}=\frac{\int{\sigma{\bm{E}}\cdot\mbox{d% }\tilde{\bm{s}}_{j}}}{\int{{\bm{E}}\cdot\mbox{d}{\bm{l}}_{i}}},$ (5) $\displaystyle{\rm{\bf M}}_{i,j}^{\mu}=\frac{\int{{\bm{H}}\cdot\mbox{d}\tilde{% \bm{l}}_{j}}}{\int{\mu{{\bm{H}}}\cdot\mbox{d}{\bm{s}}_{i}}}.$ (6)

Next, consider the eddy currents induced in a half-infinite conductor located in $x>$ 0, driven by a harmonically varying current $I^{\text{j}\omega\text{t}}$ applied in the yoz plane, which only flow along the $y$ direction, and the current density is equal everywhere in the yoz plane. The implementation is shown in Fig. 1.

The analytical solution of the electric field intensity and the magnetic field intensity in Cartesian coordinate system are expressed as

$\displaystyle E_{y}=E_{0}e^{-kx},$ (7) $\displaystyle H_{z}=-\frac{\mbox{j}k}{\omega\mu}E_{0}e^{-kx},$ (8)

where $k$ is given by

$k=\sqrt{\frac{\omega\mu\sigma}{2}}(1+{j}).$ (9)

The error-free constitutive matrices can be derived by substituting Eqs (7) and (8) into Eqs (5) and (6) respectively, those are

$\displaystyle{\rm{\bf M}}_{y_{i,j}}^{\sigma}=\frac{\int{\sigma E_{y}\cdot\mbox% {d}\tilde{\bm{s}}_{j}}}{\int{E_{y}\cdot\mbox{d}{\bm{l}}_{i}}}\mbox{, }i=j,$ (10) $\displaystyle{\rm{\bf M}}_{z_{i,j}}^{\mu}=\frac{\int{H_{z}\cdot d\tilde{\bm{l}% }_{j}}}{\int{\mu H_{z}\cdot d{\bm{s}}_{i}}},i=j,$ (11) $\displaystyle{\rm{\bf M}}_{y_{i,j}}^{\sigma}=0,{\rm{\bf M}}_{z_{i,j}}^{\mu}=0,% i\neq j.$ (12)

Then the accurate numerical solutions can be obtained by solving Eq. (2).

In the test case, $\sigma=1\times 10^{7}$ S/m, $\mu=4\pi\times 10^{-7}$ H/m, $f=$ 100 Hz, and the penetration depth is $d=$ 0.0159 m. The computational domain is the conducting region with an outer boundary set at $x_{0}=0$ m and $x_{1}=0.3$ m.

Table 1

Relative errors of E along x axis in conductor domain

Relative depth	3.14	6.28	9.42	12.56	15.70
Relative error	1.61 $\times$ 10 ${}^{-16}$	3.84 $\times$ 10 ${}^{-16}$	1.68 $\times$ 10 ${}^{-16}$	1.21 $\times$ 10 ${}^{-16}$	8.78 $\times$ 10 ${}^{-16}$

The relative depth is the ratio of spatial depth to penetration depth. The grid step size is $\Delta x=$ 0.05 m, and the corresponding relative grid step size is $\Delta\bar{x}=\Delta x/d=$ 3.14.

Table 1 shows that the average relative error of $E$ is about 3.35 $\times$ 10 ${}^{-16}$ , which is close to the truncation error of computer. Therefore, the conclusion can be drawn that the discrete error of the constitutive matrices is the main error source of the CM.

Figure 1.

Electromagnetic distributions in half-infinite conductor.

Figure 2.

The patch recovery.

3. The adaptive cell method

Unfortunately, the actual distribution functions ${\bm{E}}$ and ${\bm{H}}$ are generally unavailable, the ACM is therefore presented by using the patch recovery technique to update the constitutive matrices. In the proposed method, the constitutive matrices are updated for the more accurate ratio of current to voltage and that of magnetic flux to magnetic pressure.

The patch recovery technique is put forward to approximate the actual distribution functions. As shown in Fig. 2, the field quantities at the primal-nodes of the eight adjacent cells are required, and then the distribution functions on the patch $C$ can be obtained by using the bicubic spline interpolation approach [9].

The piecewise interpolation functions can be written as

$\displaystyle E(x,y)=\sum\limits_{i=0}^{i=3}{\sum\limits_{j=0}^{j=3}{\alpha^{k% }_{i,j}(x-x_{0})^{i}\cdot(y-y_{0})^{j}}},$ (13) $\displaystyle H(x,y)=\sum\limits_{i=0}^{i=3}{\sum\limits_{j=0}^{j=3}{\beta^{k}% _{i,j}(x-x_{0})^{i}\cdot(y-y_{0})^{j}}}.$ (14)

Substituting Eqs (13) and (14) into Eqs (5) and (6) repectively, the integral form of the constitutive matrices can be transformed to the algebraic form, which is fourth-order accuracy due to the use of the fourth-order recovery technology.

Figure 3.

Algorithm flow chart.

Figure 4.

Cylindrical tube excited by concentric current loops.

Figure 5.

The relative errors of the CM.

The flow chart of the ACM is shown in Fig. 3, which can be attributed to the following procedures: i) Subdivide the computational domain into primal and dual orthogonal cells through the centroid method. ii) Solve the boundary value problems on the initial grid, resulting in that the electric field intensity ${\bm{E}}$ and the magnetic field intensity ${\bm{H}}$ are both the second-order accuracy. iii) Obtain the field distribution functions of the patches via the bicubic spline interpolation. iv) Substitute the recovery results into the algebraic form of the constitutive matrices to update ${\rm{\bf M}}^{\sigma}$ and ${\rm{\bf M}}^{\mu}$ . v) Solve the governing equations with the renewed constitutive matrices, resulting in that the electric field intensity ${\bm{E}}$ and the magnetic field intensity ${\bm{H}}$ are both fourth-order accuracy.

4. Application

4.1 Linear eddy current problem

Consider the induced eddy currents in a hollow cylinder, excited by a harmonically varying current $I^{{j}\omega\mbox{t}}$ applied coaxially, as shown in Fig. 4, where $\textit{Rcoil}=$ 0.1 m, $\textit{Rcylinder}=$ 0.2 m. The computational domain is the conducting cylinder with an outer boundary set at $r_{in}=$ 0.2 m, $r_{out}=$ 0.4 m, $z_{up}=$ 0.2 m, and $z_{down}=$ $-$ 0.2 m. In our numerical implementation, $\mu=4\pi\times 10^{-7}$ H/m, $\sigma=1\times 10^{7}$ S/m, $I=$ 1000 A, $f=$ 100 Hz, and the penetration depth is $d=$ 0.0159 m.

The analytic solution to the electric field intensity is [10]

$\displaystyle E(r,z)=-jw\sigma\int_{0}^{\infty}{D(k)K_{1}(\lambda r)\cos(kz)}dk,$ (15)

where

$\displaystyle D(k)=\frac{\mu aI_{m}I_{1}(ka)}{\pi b\left[{\lambda K_{0}(% \lambda b)I_{1}(kb)+kI_{0}(kb)K_{1}(\lambda b)}\right]},$ (16)

and $\lambda^{2}=k^{2}+jp^{2}$ , $p^{2}=\omega\mu\sigma$ .

Figures 5 and 6 show the comparison between the relative error of the CM and the ACM under $\Delta\bar{r}=\Delta r/d=$ 0.052, $\Delta\bar{z}=\Delta z/d=$ 0.32, and $\Delta\bar{\phi}=\Delta\phi=\frac{\pi}{2}$ . It can be clearly seen that the average relative error of the ACM is approximately 1250 times less than that of the CM.

The relative errors of the two methods are given in Table 2. The results indicate that the CM is second-order accuracy, while the ACM is fourth-order accuracy. Compared with the CM, the ACM exhibits the accelerated convergence and higher accuracy.

Table 2

Numerical results of the linear example

Relative grid step size	Relative error of the CM	Relative error of the ACM
$\Delta\bar{r}=$ 0.415, $\Delta\bar{z}=$ 2.52	1.21 $\times$ 10 ${}^{-1}$	5.80 $\times$ 10 ${}^{-3}$
$\Delta\bar{r}=$ 0.207, $\Delta\bar{z}=$ 1.26	3.07 $\times$ 10 ${}^{-2}$	3.76 $\times$ 10 ${}^{-4}$
$\Delta\bar{r}=$ 0.104, $\Delta\bar{z}=$ 0.63	7.60 $\times$ 10 ${}^{-3}$	2.31 $\times$ 10 ${}^{-5}$
$\Delta\bar{r}=$ 0.052, $\Delta\bar{z}=$ 0.32	1.90 $\times$ 10 ${}^{-3}$	1.39 $\times$ 10 ${}^{-6}$

Figure 6.

The relative errors of the ACM.

4.2 nonlinear eddy current problem

A case of 1-D nonlinear eddy current field is introduced in this subsection. In the nonlinear numerical implementation, $\sigma=1\times 10^{6}$ S/m, $f=$ 50 Hz, the basic magnetization curve is shown in Fig. 7, where $B=10^{-4}He^{-2.5H}+1.4\times 10^{-5}H$ .

The computational domain is 0 $\sim$ 0.06 m, and the field boundary conditions are $H_{x=0}=1$ A/m and $H_{x=0.06}=0$ A/m. The numerical results through the Finite Integration Technology (FIT), in which the grid step size is refined to $dx=4\times 10^{-5}$ m, are used as the reference solutions.

In the ACM, the updated constitutive matrices are calculated in term of the integration form as follows

$\displaystyle{\rm{\bf M}}_{i,j}^{\mu}=\frac{\int{\mu\left(H\right){\bm{H}}% \cdot\mbox{d}\tilde{\bm{s}}_{i}}}{\int{{\bm{H}}\cdot\mbox{d}{\bm{l}}_{j}}}% \mbox{, }i=j.$ (17)

The relative errors of the CM and the ACM are given in the Table 3. The results demonstrate that the ACM also exhibits accelerated convergence and higher accuracy compared with the CM in the nonlinear eddy current case.

Table 3

Numerical results of the nonlinear example

Grid step size	Relative error of the CM	Relative error of the ACM
$\Delta{x}=$ 0.006	1.38 $\times$ 10 ${}^{-1}$	8.10 $\times$ 10 ${}^{-3}$
$\Delta{x}=$ 0.004	6.50 $\times$ 10 ${}^{-2}$	1.30 $\times$ 10 ${}^{-3}$
$\Delta{x}=$ 0.002	1.69 $\times$ 10 ${}^{-2}$	7.44 $\times$ 10 ${}^{-5}$

Figure 7.

The basic magnetization curve.

5. Conclusion

In the paper, a precise Adaptive Cell Method is presented to solve the eddy-current problems on the uniform coarse grid. Numerical results indicate that ACM can achieve fourth-order accuracy solutions rather than second-order accuracy solutions, by using the constitutive matrix recovery technique, the ACM yields much more accurate solutions without refining the coarse grid.

Footnotes

Acknowledgments

This project was supported by National Natural Science Foundation of China (No. 51407139).

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