Electromagnetic Non-Destructive Evaluation II,
Proceedings of the 3rd International Workshop on E'NDE,
Reggio Calabria, Italy, September 1997.
ISBN: 90 5199 375 7 - published by
IOS Press
TABLE OF CONTENTS |
This paper deals with an inverse problem solution based on the simultaneous use of two different artificial neural networks (ANN) for the localization and the shape classification of defects. In a magnetic plate specimen including different defects, a classic 2D finite element method is used for eddy current computations. Results obtained from such calculations are used as a database for the inverse problem solution. A brief overview of the ANN is presented, revealing drawbacks and advantages of the proposed approach as regards to the results.
In non-destructive testing (NDT), inverse problem solutions are difficult tasks but promising. The reverse scheme compared to the forward one offers the possibility to identify studied flaws precisely. This is done with the optimization of a suitable error functional, revealing the differences between measurements and theory. For example, in eddy current testing, the sample is analyzed with electromagnetic excitations and responses. Then, those signals take part in the inverse problem solution, expecting to reveal any defect.
The accomplishment of such work always needs the human experience for a comprehensive and rigorous approach [1]. When the experimental measurements are done, some numerical methods can be employed for solving the inverse problem in an effective way [2]. Among the most well known inversion methods, one can think about those using deterministic and stochastic optimization techniques, the reconstruction based on image processing, and the artificial neural networks.
The ANN are now widely known as good solvers for highly non-linear problems. The most used, the feed-forward scheme, is effective for the representation of an underlying physical phenomenon if a large data set is available. This is the famous generalization capacity, used in a large field of applications: optimization, recognition, and so on...[3] Nevertheless, other existing ANN present attractive characteristics [4]. The competitive neural network is one of them [5].
It is still difficult, in NDT, to get the reconstruction of a defect. Experimental measurements are always sensitive and noisy, involving hard data analysis. For these reasons, the obtained defect image appears to be fuzzy, approximate, or incomplete.
In this presentation, an inverse electromagnetic problem is solved for the defect reconstruction by eddy current testing.
Instead of searching for the absolute shape reconstruction, a neural classification is used [6][7][8] to allow shape categorization. Due to this aim, a neural competitive layer is employed in complement of the classic feed-forward scheme for the defect localization. The training data for the networks are numerically obtained [9] via a 2D finite element modelling.
The device is shown in Fig. 1. It consists of a probe, exciting and detector coils, scanning over a magnetic plate specimen including a defect with air properties.
In the present case and for simplicity, the cracks are supposed to be at the same depth so that their position is only x-dependent. They are also supposed to have similar dimensions. Considering a 3-cm specimen depth, the chosen sinusoidal exciting frequency is constant and taken equal to 1 Hz for a suitable detection sensibility. This corresponds to about 0.5-cm skin depth.
Fig 1: Experimental testing circuit |
The experimental characteristics do not have to appear restrictive. The basic idea is to reveal the effectiveness of the proposed method. Then one can easily imagine any extensions if necessary, as will be discussed at another stage.
The simulations are carried out using a classic 2D finite element method with a magneto-dynamic vector potential formulation A [10]:
(1) |
where J is the applied ac current density, and µ being respectively the material conductivity and permeability.
In the proposed problem illustrated in Fig. 1, three defect families are considered to form the database for the learning period. The three families are simple shapes: a square, an isosceles triangle and a rectangular one. Computations are run for ten possible x-positions of each crack and of the probe, involving 300 finite element computations. For example, Fig. 2 illustrates the flux distribution for given crack and probe position.
Fig 2: Computation results for a square crack in position 2 and the probe in position 8 |
The B components (Bx, By) within the detector for a complete displacement, are used for establishing the neural network data set. Fig. 3 shows the calculated induction B in the detector, for a particular crack position.
Fig 3: Induction behavior for a square crack in position 4 |
The corresponding deviation B (i.e. B compared to the induction when no defect exists) is presented in Fig. 4. This magnitude is utilized in the neural data base creation.
Fig 4: Induction deviations |
That method such as moving a probe over a magnetic testing circuit allows us to note the presence of a variation in the deviation curves. This is due to the magnetic field perturbation when passing closed to the defect [11].
ANN are expected to learn the highly non-linear relationships between the magnetic deviation within the detector, and the defect. This must include the three defect families, the ten possible defect positions, and the complete scan of the probe. Defects present the air permeability and conductivity.
Then the considered problem is to obtain the position and the shape of any crack located in the sample. This is done, as mentioned previously, with the association of two different neural networks as shown in Fig. 5.
Fig 5: Neural approach for the inverse problem resolution |
The feed-forward scheme
The feed-forward neural network contains three cell layers. This architecture is adequate to learn most of the non-linear functions. The last layer is composed of linear transfer function neurons, while the two hidden layers are made up of sigmoid ones. Then a set of input vectors (i.e. B ) and the corresponding outputs (x-axis position) are used to train the network until it can give a sufficient approximation according to the position. This approximation obviously depends on the amount of available data.
For the considered problem, the data constitute an input set of only 30 vectors representing deviation B_{x} and B_{y} for all defect combinations (shape and place). The training phase is done with the Levenberg-Marquardt method, using a second order development of the neural error criteria F at step k [12]:
(2) |
and updating the neurons weights w with the formula:
(3) |
where H is the Hessian matrix of F, the gradient of F, and a relaxation coefficient.
By calculating the inverse Hessian matrix error for updating neural weights, computation time can quickly become important. Nevertheless, the learning phase convergence is highly improved.
The feed-forward network is now expected to give any defect localization precisely. This localization is a scalar representing the x-position of the unknown defect. Then it is used for the classification in the three shape families learned by a competitive layer. A competitive layer trained beforehand with the learning vector quantization method.
The competitive layer
A neural competitive layer automatically learns to classify input vectors. The learning vector quantization is chosen for training this neural network in a supervised manner. It is based on the Kohonen rule [5] for updating neurons' weights in the learning phase. After the training period, the network is able to give the closest learned vector matched with an unknown input.
A particular shape induces a particular deviation (Bx, By) when the probe is just above the crack. This principle is presented in Fig. 6 for the training data set, representing the three possible defect families for a given x-position of the probe. We can associate a region to a defect family as shown in dotted lines. In inverse problem solutions, the classification of a defect can be done according to the shortest distance of the corresponding deviation from a region.
Fig 6: (Bx, By) deviation for the three crack families and for ten given positions of the probe just above the defect |
For an unknown defect, ten computations are performed to evaluate the corresponding Bx and By vectors. Then the feed-forward network gives the x-position of the crack. This number is now rounded and used by the competitive layer for selecting the right rows in the Bx and By vectors. The shortest distance selection to a group (the continuous arrow in the right part of Fig. 6) is applied to find the most likely shape.
After the training for both neural networks, tests have been done for unknown defects. Results are shown in table 1 and Fig. 7, it is shown that a defect position can be determined with a good minimum precision and that the shape is categorized to the nearest known one.
TESTING DEFECTS | x-Position Response | Shape Response | |
Case 1: | Square, x = 5.5 | x = 5.538 | Square |
Case 2: | Hexagon, x = 4.0 | x = 3.215 | Isosceles triangle |
Case 3: | Inverse rectangular triangle, x = 8.0 | x = 7.860 | Rectangular triangle |
Case 4: | Rectangle, x = 3.0 | x = 3.055 | Isosceles triangle |
Case 5: | Circle, x = 7.0 | x = 6.620 | Square |
Fig 7: Pattern recognition (Unknown defects in dotted lines and corresponding neural responses) |
These results show the effectiveness of the proposed ANN inverse problem solution in the shape classification of unknown defects by eddy current testing. The classification offers the advantage of getting well-defined shape results. However, it is obvious that only three roughly defined models cannot pretend to be sufficient. A database made up of several hundred accurate models, using for example an elaborate grid for the drawing, appears to be necessary. Then, an automatic numerical procedure has to be elaborated. This can appear quite fastidious as regards to the computation time, but it is the only condition for an efficient classification.
Moreover, only the localization in one dimension is studied here to stay at a low stage of time consuming. Obviously, for interesting practical considerations, 2D and 3D localization problems have to be considered. In this case, an elaborated deviation map (as shown in Fig. 6) should be set from such considerations. For example, the map could include deviations and regions corresponding to a variety of depths for the defect.
Finally, applications can require volumetric reconstruction involving 3D finite element computations. If the problem must deal with thin cracks in large materials, the classic finite element formulation may lead to numerical problem [13]. Then the electromagnetic modelling needs advanced numerical tools such as combined formulations or shell elements.
A neural network approach for solving inverse problem in eddy current testing is presented in this paper. The main idea is the introduction of categorization for the shape reconstruction using a non-familiar neural network: the competitive layer. Results are presented for a simple eddy current problem using the finite element method as experimental support.
Only 1D localization and 2D shapes have been considered to reduce the calculation time. As the available computers capacities are growing up everyday, one can think about to handle the proposed method by including: precise probe displacement in 2D or 3D, large range of defects (sizes, shapes, and positions), several exciting frequencies, magnetic non-linearity, hysteresis phenomenon, etc...Then, for a more elaborated use, some necessary improvements of the proposed method are needed.
Whatever the problem to solve, a larger data set is obviously necessary to attain higher accuracy.
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