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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

A fast crack profile reconstitution model in nondestructive testing is developed using an arrayed eddy current sensor. The inverse problem is based on an iterative solving of the direct problem using genetic algorithms. In the direct problem, assuming a current excitation, the incident field produced by all the coils of the arrayed sensor is obtained by the translation and superposition of the 2D axisymmetric finite element results obtained for one coil; the impedance variation of each coil, due to the crack, is obtained by the reciprocity principle involving the dyadic Green’s function. For the inverse problem, the surface of the crack is subdivided into rectangular cells, and the objective function is expressed only in terms of the depth of each cell. The evaluation of the dyadic Green’s function matrix is made independently of the iterative procedure, making the inversion very fast.

The use of arrayed eddy current (EC) sensors in Non Destructive Testing (NDT) provides high speed inspection and better space resolution by miniaturization of their coils. The arrayed sensors can make a measurement of large surfaces without a scan, as illustrated in the

There are several configurations of arrayed eddy current sensors [

- The synchronization of the supply and the measurement is not required for the electronic component.

- The measurement of the coils impedance is carried throw the voltage measurement.

- The incident electric field on the scan surface is uniform because the coils are connected in series, and this is independent of the work piece surface state.

The investigation is done by the measurement of the impedance variation of each coil. The purpose is to determine a crack shape and size using the measurements provided by such a sensor in a real time investigation.

The inversion method proposed is based on the iterative solving of the direct problem; it is thus important to have a fast tool to solve the latter. The use of the 3D finite element method would be very expensive in terms of memory space and CPU time. On the other hand, the analytic models lack the flexibility to handle complex geometries. In this work, we use the ideal crack model [

_{0}

The direct problem is based on the generalization of the ideal crack model to an arrayed eddy current sensor [^{T}_{k}_{k}_{k}_{sk})

Once the total normal incident field

In (4), ^{2} = _{0}

The integral _{L}_{d}_{e}

The vectors

Using the reciprocity principle, the impedance variation of a coil k of the arrayed sensor is given by the following equation:

In (6)

The reference data for the inversion are obtained by a 3D finite element computation code developed in our laboratory. The computation code is based on the AV-A formulation [

The detection of the crack is observed through the variation of the impedance matrix. In the initial step, we don’t know the exact position and orientation of the crack under the arrayed sensors. The adjustment of the position of the latter by looking for the maximum variation of the matrix impedance is necessary with the aim of getting the crack in the middle and on the main axis of the arrayed sensor. This manual operation makes the inverse problem easier and reduces it to the determination of the crack profile. It is assumed that the crack is embedded in a known rectangular area of dimensions _{L}×n_{d}_{L}_{d}

The norm used here is the absolute value which takes less computation time than the square root norm. For a better consideration of the real part in the minimization of the objective function, we separate it from the imaginary part in the impedance variation as follows:

We use the genetic algorithm for the minimization of the objective function (11). Genetic algorithms have been widely used, associated to the finite element method for the optimization of electromagnetic devices [

In the first step, we suppose that the crack occupies all the rectangle surface (_{L}_{d}_{i}_{i})_{k}_{i}_{d}

For the considered example we have chosen the discretization (_{L}_{d}_{i}_{i}_{L}

In

The part of computation done by the genetic algorithms is not time consuming according to their simple operation. The most consuming time in this inversion is the evaluation of the objective function and the inversion of the reduced matrix _{i})

We have presented a fast crack profile reconstitution procedure using arrayed eddy current sensor data. The use of the reciprocity and superposition principles allows a fast resolution of the direct problem. In the inverse problem, which is based on an iterative solving of the direct one, we adopted a coarse approximation of the crack profile which is represented by only a few discrete values; this makes the inversion procedure very fast when using genetic algorithms. This method can give a real time inspection when implemented in an embedded NDT hardware.

This work was supported by Nantes Métropole and the Algerian-French cooperation PAI-Tassili program.

An arrayed eddy current sensor above a piece with a crack.

Impedance matrix measurement.

The modeled system.

Crack shape.

Example of a crack shape defined by the discrete values _{i}

The inversion flow chart.

Inversion results

The Fixed parameter of the modeled system.

Frequency: | 300 kHz | |

Coils: | Inner radius, _{1} |
0.6 mm |

Outer radius, _{2} |
1.6 mm | |

Height, |
0.8 mm | |

Lift-off, |
0.5 mm | |

Number of turns, |
140 | |

Distance between the coils, |
4 mm | |

Plate: | Thickness, |
2 mm |

Conductivity, |
1 MS/m | |

Crack: | Length, |
12 mm |

Thickness | 0.2 mm | |

Depth | Arbitrary shape ( |

The Fixed Parameters for the Genetic Algorithm.

Population : | 64 |

Crossover rates (Uniform) : | 0.8 |

Mutation rates (Heuristic) : | 0.02 |