Reconstitution of complex defects with the method of neural networ: Application for the nondestructive evaluation

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Reconstitution of complex defects with the method of neural networ: Application for the nondestructive

evaluation

Amirouche Harouz

Numerical Modeling of Electromagnetic Phenomena and Components Laboratory

MouloudMammeri University TiziOuzou, Algeria [email protected]

Hassane Mohellebi, Meziane Hamel Numerical Modeling of Electromagnetic Phenomena and

Components Laboratory MouloudMammeri University

TiziOuzou, Algeria [email protected] Abstract—in this work, we propose a reconstitution of

complex defects starting from the results obtained during a nondestructive testing by eddy currents carried out on a conducting plate by using the neural network. We provided to the neural network the values of impedance of the differential sensor calculated using a modeling by finite elements.The values of impedance are injected at the input of the neural network and the depth of the defect is recovered in its output, we used the gradient of the error propagation algorithm for performing learning of the neural network.

Keywords—Neural network, eddy currents, impedance, complex defects; nondestruction evaluation.

I. INTRODUCTION

The knowledge of the form of the defect is a very significant parameter for the engineer, in order to enable him to decide future of the part.The methods of NDT do not inform us much about the form of the defect.In this case, one more often speaks about an END or inverse problem, generally difficult to solve.We use the neural network for their capacity to calculate linear and nonlinear model, the relations between the data presented at its input and the desired output and also their capacity to learn these relations directly starting from the data modelled in the form of examples [1], [2].

II. FORMULATION

The study of eddy current problem allows considering the harmonic hypothesis and in case of [x,y] plane, the electromagnetic equation to be considered in terms of magnetic vector potential is given as follow [3]:

− + + = (1)

with: [ / ]is the electrical conductivity, [ . ]is the magnetic permeability. = 2 , where [ ]represents the frequency of feeding, [ . ]is the source current density component along z direction.

=

In the non-destructive testing by eddy current, the sensor meets a reaction of inspected material;blow the formulation of the magnetic vectorpotential will be composed of a real part and an imaginary one.

The electric impedance is given by:

= +

The expression of the impedance thus established can apply indifferently to the sensors with double function or separate functions.However, the majority of the methods of resolution use a variable of state other than induction or the magnetic flux.Consequently, it is preferable to formulate the impedance according to the magnetic vectorpotential.

By introducing the magnetic vectorpotential, equation of electric impedancebecome as follows [3]:

( ) = . . 2 . ( ).

. (2)

( ) = . . ^.2 . ( ). (3)

with: is the number of coil winding, is the air of the whorl and is the ray of the wind.

The magnetic vectorpotential is calculated after solving the magneto-dynamic partial differential equation with the finite element method (FEM),the impedance is alsodeduced after calculation under MATLABenvironment. The values from impedances obtained will be used as a database for the inverse problem.

A database is established for the study of the direct problem, and then used for the reconstitution of supposed defects in the conductive part. The study of the direct problem was made by exploiting the finite element method. The neural network technique was used for reconstitution of defects.

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III. ALGORITHM OF TRAINING THE NEURAL NETWORK The algorithm of the retro-propagation of the gradient is an iterative algorithm designed to minimize a quadratic criterion of error between the output obtained of a multi-layer network and the desired output.This minimization is carried out by an adequate configuration of the weights.The error is the difference between the value wished for the neuron of output and its computed value by propagation. The algorithm requires a continuous, non-linear and differentiable function like transfer function of the neuron [2], [4].

The objective of the method of the retro-propagation is to adapt the parameters in order to minimize a performance index given by:

( ) = ( ) (4)

with: ( ) = ∑ −

where: ( )is the quadratic error on the level of the output layer of the network, ( )the quadratic error on the level of the output network, is the vector of the desired output of the network: «the target vector", is the vector of output worked out by the network and is the number of examples in the base of training.

IV. APPLICATION AND RESULTS

We will touch with the inverse problem, by implementing a neural network able to evaluate the data provided by a differential sensor to eddy current.The objective of this evaluation is the reconstitution of dimensions (depth) of a defect of the type crack. We will consider a perceptron multi layers containing only one layer of entry having only one neuron, a hidden layer having a number of neurons which one can vary with function of sigmoid standard activation and a layer of exit having only a neuron. Our results are obtained after simulations by using the following parameters of the network, a number of neurons in the hidden layer are 10 neurons.Rate of training is 0.01 (data that we fixed for reasons of stability of the network).A maximum number of iterations are200 iterations (given which one can vary).

Fig.1.Neuralnetwork used.

Fig. 1 represents the structure of the PMC used in this work, it is a schematization simplified to show the various layers of this network.

To show the performance of this network, we used two data bases of two inspected structures comprising of the different defects.Fig. 2 and fig. 6 represent the geometry in a two- dimensional plane.

A. First structure

Fig. 2. Geometry of the first structure.

Fig.3 and fig.7 represent the the impedanceaccording to the displacement of the transmitter provided of the non destructive testing by eddy currents (direct problem) whom we used like the values of entry of the network in order to carry out the reconstitution of the defects (inverse problem).

Fig. 3.Impedance provided to the network.

Fig.4 and fig.8 represent the average quadratic errors according to the number of the iterations obtained after simulations of the program (based on the developed algorithm of the retropropagation of the error of the gradient).

Fig. 4. Average quadtratic error

0 5 10 15 20 25 30 35 40 45

0 0.5 1 1.5 2 2.5 3

Displacement of the sensor [ mm ]

Impedance [ Ohm ]

0 20 40 60 80 100 120 140 160 180 200

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6

Number of the iterations

Error calculated by the network

Depth Impedance

Outputlayer Input layer

Hidden layer

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Fig.5. Reconstruction of the defect depth B. Second structure

Fig 6.Geometry of the second structure

Fig 7. Impedance provided to the network

.

Fig 8.Average quadtratic error

Fig. 9.Reconstruction of the defect depth^.^.

Fig.5 and fig.9 provide the results of the reconstitution made by the neuron network; we superimposed the desired output (given training of the network) with the output of the network in order to be able to carry out a comparison, the results obtained at the output of the network have a gap and this tells us about the imprecision of the network to reconstruct the defect completely. In the first defect structure was rebuilt at a rate of62%in the second defect structure was rebuilt at a rate of 77%, this shows us that reconstruction is related to the type and the fault geometry

V. CONCLUSION

In this work, we carried out a nondestructive evaluation (END) and this by carrying out the reconstitution of a defect (its depth) in a conducting part inspected via the nondestructive testing by eddy currents (NDT):the qualitative characterization.END enables us to carry out a characterization more quantitative than qualitative.

During a non-destructive testing (direct problem), we seek to detect and locate the defects (to quantify them).On the other hand, during a nondestructive evaluation (inverse problem), we use the results provided by a direct problem and using the methods of inversion (neuron network in our case), we recover parameters such as the form and dimensions of the defects.

As is seen in the figures (fig. 5 and fig. 9), when the default is more complex (which is the case of the first structure), the rate of its reconstruction decreases.

References

[1] M. BERKANE, Estimation et analyse de mouvement par approche neuronale,Thèse de Doctorat, Institut National des Sciences Appliquéesde Lyon, 2010.

[2] ^.N.OUKACINE, «Utilisation des réseaux de neurones pour la reconstitution des défauts en évaluation non destructive », Mémoire de Magister, Université Mouloud Mammeri de Tizi-Ouzou, 2012..

[3] M.HAMEL, « Etude et réalisation d’un dispositif de détection de défauts par méthodes électromagnétiques », Mémoire de Magister, Université Mouloud Mammeri de Tizi-Ouzou, 2012.

[4] L.PERSONNAZ, I.RIVALS « Réseaux de neurones formels pour la modélisation, la commande et la classification », CNRS Editions, Paris, 2003.

0 5 10 15 20 25 30 35 40

-4 -2 0 2 4 6 8 10 12 14x 10^-4

Displacement of the sensor [mm]

Depth of the defect [m]

Desired output Obtained output