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An improved method which can analyze the eddy current density in conductor materials using finite volume method is proposed on the basis of Maxwell equations and

The finite volume method (FVM) is a discretization method which is well suited for the various kinds of the conservation laws of numerical simulations. Originally, the FVM was extensively used in solving the dynamic fluid problems [

The FVM is quite different from the finite difference method (FDM) or finite element method (FEM). Roughly speaking, the FDM is the oldest numeric method to approximate the differential equations through the use of Taylor’s expansions. The FDM becomes difficult to use when the coefficients involved in the equation are discontinuous (in the case of heterogeneous media) which is not a problem for FVM if discontinuities of coefficients occur corresponding with the boundary of “control volume” [

“Robust” means that the FVM behaves well for different formulations and focuses on the physical meaning; the obtained solution does not appear without physical meaning.

“Cheap” is obtained thanks to the simple and reliable computational coding for the complex problem.

The FVM maintains the FEM’s advantages, such as the facility to deal with the complex geometry [

This paper proposes an investigation work on

To test the proposed technique, the simulated results are compared with the experienced ones which are obtained from the E’NDE benchmark problems. The aim is to evaluate the impedance change due to the presence of a crack in the center of the isotropic conductor, taking the probe movements into account. The test system is showed in Figure

The system to be solved.

A-V formulation and

To explain the formulation to calculate the eddy current in the conductor, the scheme is shown in Figure

The illustration of proposed calculation.

At the beginning, the magnetic field created by the coil is applied directly to the conductor by using Biot-Savart’s law [

To carry out such a modeling, the association of FVM to this iterative process is used. Additionally, it is necessary to control the change of the variables in the iterative process of solution. A relaxation is introduced in the following equation which is based on relaxation factor

The algorithm of the calculation for FVM

As introduced in [

Two neighbor volumes.

In fact, they belong to different volumes but should have the same conductivity. A simple approximation for the conditions of passage between two media is to make the geometric mean of the two physical volumes at the interface [

There were a lot of discussions on the E’NDE benchmark problems. Bowler et al. [

Pancake coil parameters.

For problem No. 1, the dimension of the plate is 40 mm × 40 mm. The coil is fixed over the centre of the plate. In this problem, the solution requires the determination of the impedance change of the plate. When we use Biot-Savart’s law to calculate the magnetic field in the conductor created by eddy current, it risks running “out of memory” in Matlab as the matrix is full. To reduce the full matrix to a “sparse” one, we suppose that the eddy currents far away from the coil create a small magnetic field that can be neglected. It means that the maximum distance for the computation is limited to several times the outer radius of the coil. The impedance change of the plate and the difference between the measured and simulated resulted are shown in Figure

(a) Impedance change of the plate. (b) Difference of the impedance.

From these figures, we find that the impedance change approaches the measurement from

The limitation of the matrix using Biot-Savart’s law.

The impedance changes which are going be evaluated by the proposed method are compared with those obtained from measurements at the frequencies of 150 kHz and 300 kHz (see Table

Impedance change of the plate.

Frequency | Lift-off (0.5 mm) | |||
---|---|---|---|---|

150 kHz | Experiment | 1.27 | −0.99 | 0.79 |

Simulate | 1.33 | −1.05 | 0.82 | |

300 kHz | Experiment | 3.49 | −2.22 | 2.7 |

Simulate | 3.33 | −2.36 | 2.23 |

From the table, there is, respectively, 4.72% and 6.96% differences for the two frequencies between the simulated and measured results. The fact that we use the same meshes in both cases explains why in case of 300 kHz the error is higher than that in case of 150 kHz. To solve the system, a preconditioning method developed in [

For problem No. 2, the size of the plate is

The impedance changes in function of the movement of the coil for two frequencies.

We can see that the impedance change becomes smaller when the coil is far from the centre of the hole. When the coil is at about 10 mm from the centre of the hole, the impedance remains the same as if there was not a hole in the plate. For the frequency of 150 kHz, the skin depth is a little greater than the thickness of the plate while it is smaller than the thickness for the frequency of 300 kHz. Due to the issue of skin depth, BICGSTAB used 500 iterations to inverse the matrix and the algorithm used 1500 iterations for every position on the same PC, where the plate has been divided into 34000 volumes at the frequency of 150 kHz. However, 1200 iterations to inverse the matrix, 2100 iterations for every position, and 45000 volumes are needed for 300 kHz.

The E’NDE benchmark problems No. 1 and No. 2 have been solved by the