An Efficient SQUID NDE Defect Detection Approach by Using an Adaptive Finite-Element Modeling

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1 DOI /s ORIGINAL PAPER An Efficient SQUID NDE Defect Detection Approach by Using an Adaptive Finite-Element Modeling Farrokh Sarreshtedari Sasan Razmkhah Nahid Hosseini Jurgen Schubert Marko Banzet Mehdi Fardmanesh Received: 5 September 2010 / Accepted: 6 September 2010 Springer Science+Business Media, LLC 2010 Abstract Incorporating the finite-element method for the modeling of the SQUID NDE response to a predefined defect pattern, an adaptive algorithm has been developed for the reconstruction of unknown defects using an optimization algorithm for updating of the forward problem. The defect reconstruction algorithm starts with an initial estimation for the defect pattern. Then the forward problem is solved and the obtained field pattern is compared with the measured signal from the SQUID NDE system. The result is used by an optimization algorithm to update the defect structure to be incorporated in the FEM forward problem for the next iteration. Since the mentioned model based inverse algorithm normally consumes a lot of computational resources, the number of iterations plays an important role in the determination of the total response convergence time. Consequently, different optimization algorithms have been applied and their performances are compared. In this work by incorporating an efficient forward model and using the stochastic and deterministic optimization algorithms for defect updating we have investigated their performance on the inversion of the SQUID NDE signal and also their ability to defect reconstruction in the noisy environment. Keywords SQUID NDE Defect detection algorithm Finite-element modeling F. Sarreshtedari S. Razmkhah N. Hosseini M. Fardmanesh ( ) Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran fardmanesh@sharif.edu J. Schubert M. Banzet Forschungszentrum Jülich, Institute of Bio and Nano-systems, Juelich, Germany 1 Introduction Because of the very high sensitivity and dynamic range of SQUIDs as the magnetic sensor for the NDE system, SQUID based eddy current NDE systems are among the favorable approaches for precise detection of both deep shallow and surface defects [1, 2]. Furthermore the magnetic inverse problem for the identification of the defect structure using the measurement results is another utmost important issue in this context [3, 4]. Basically there are two classes of NDE signal inversion methods, which are phenomenological or model based approaches and non-phenomenological approaches [5, 6]. Each of these methods has its own benefits and drawbacks, which are investigated in different works [5]. The block diagram of the model based approach which is considered in this work is shown in Fig. 1. This method, which is among phenomenological approaches, can be used for applications that an arbitrary defect profile needs to be identified without using a large database of defect signals for network training [5]. In this method the inversion begins by an initial guess for the structure of the defect. Incorporating a forward model for the prediction of the 2-D magnetic field distribution and an optimization approach for updating the defect structure, the size and shape of the defect can be found by iterative solution of the loop. Two essential parts of this method, which determines both the needed computational resources for the solution and the ability of the technique in signal inversion, are the incorporated forward model and the updating optimization approach. Various works have been reported on these parts of the inversion solution, e.g. references [7 9]aredevotedto developing the forward model, and in references [10, 11]the effect of optimization algorithms on the inversion process has been investigated. In this work, based on our previous

2 Fig. 1 Block diagram of the model based inversion method developed FEM defect model [12] as the basis of the forward model, we have investigated the efficiency of deterministic and stochastic optimization algorithms for the adaptive defect detection process in the presence of environmental noise. Fig. 2 The block diagram of our SQUD based NDE system 2 System Setup The incorporated experimental SQUID NDE system is based on a high-t c YBCO RF-SQUID gradiometer with a flux noise below 100 µ 0 / Hz at 100 Hz in unshielded environments [13, 14]. The excitation coil of the system is a planar double-d coil that gets aligned with the SQUID using a x, y micro-positioner for x, y with θ adjustments. Using this system, which also incorporates an automated two dimensional non-magnetic scanning robot, an aluminum plate sample with intentional defects has been tested and its scanning result is processed by the proposed defect reconstruction algorithm. The examined sample contains a distribution of eight identical holes, which form the character A [15]. Figure 2 shows the block diagram of our SQUD based NDE system and Fig. 3 shows the result of experimental SQUID magnetic scanning over the mentioned sample. The incorporated scanning system parameters like the used double-d excitation coil with its 3 cm diameter have made the insurance that the individual signals from different holes have enough overlap with each other, which is required for the evaluation of the signal inversion algorithms. 3 Forward Model The proposed forward model is based on our previous FEM numerical simulator and also the principle of superposition for the summation of the magnetic field deformation due to different defects. In reference [2], by solving the governing electromagnetic equations and using a numerical approach, the perturbation of a uniform current around a spherical hole Fig. 3 The 3-D SQUID NDE measured magnetic image for the magnetic scanning of the sample which forms the character A has been studied. Then it was shown that for distances η greater than 3η 0 the perturbation in the current distribution would be negligible. Here η and η 0 are the normalized values which are related to the characteristics of the sample and also the ratios of the defect. Satisfying this criterion means that if the distances between the holes in the sample meet the mentioned criterion, the superposition principle can be considered in formation of the forward model for the identification of the magnetic field for a desired distribution of holes. Furthermore, in reference [12] an efficient FEM simulator has been developed for the numerical simulation of the SQUID NDE scanning result of a single hole. Figure 4 shows a comparison between result of the SQUID NDE experimental scanning of a single hole in an aluminum plate and its counterpart FEM simulation result. Using the superposition principle and the mentioned simulation result, our proposed forward model can simulate the 2-D magnetic field based on the suggestion of an optimization algorithm for the defect profile, as explained in the following section.

3 Fig. 5 The 2-D SQUID NDE measured magnetic image of the sample and its considered model for the distribution of holes Fig. 4 Two dimensional magnetic scanning result of a single hole in an aluminum plate. (a) Experimental result using the SQUID scanning system. (b) FEM simulation result 4 Optimization Approaches Deterministic and stochastic optimization algorithms are the two classes of optimization approaches, which have been widely used for the model based inverse solutions. In this work we have used a gradient based algorithm as the deterministic optimization approach and particle swarm algorithm as the stochastic approach for updating the defect profile in the model based inversion solution. Figure 5 shows the 2-D SQUID NDE measured magnetic image of the sample and it s considered model for the distribution of holes. 4.1 Particle Swarm Optimization Particle Swarm Optimization (PSO) belongs to the class of direct search methods used to find an optimal solution for an objective function in a search space. The PSO is a stochastic and population-based algorithm based on swarm intelligence models [16]. In this algorithm a communication structure or social network is defined that assigns neighbors for Fig. 6 Different optimization schemes. (a) Particle swarm elements (pages). (b) Gradient based sample page each individual particle for interaction. Then a population of individual particles (which are the candidate solutions) is defined as random guess and the problem solution is initialized. The swarm is typically modeled by particles in multidimensional spaces that each has a position and a velocity parameter. These particles, which fly through hyperspace, have two essential reasoning capabilities, their memory of their own best position and the knowledge for the global or their neighborhood s best. Members of a swarm communicate good positions to each other and adjust their own position and velocity based on these good positions. In this work, as shown in Fig. 6a, each particle is a distribution of eight holes in a 4 7 page. Along the process of optimization, the distribution of the holes in these particles gradually changes for the maximization of the cost function.

4 Fig. 7 The goal result of the inversion algorithms Fig. 8 Gradient based method. Iteration no = 5, time = 620 sec 4.2 Gradient Based Algorithm Gradient based algorithms have traditionally been used to find the optimum solution to various problems [17]. These algorithms are based on the fact that, if a continuous function f(x)is defined and differentiable in a neighborhood of a point p 0, then f(x)decreases fastest if one goes from p 0 in the direction of the negative gradient of f(x)at p 0. p 1 = p 0 α f(x) x=p0. (1) The gradient based algorithms work well for a small number of continuous parameters. They are usually fast but require the computation of derivatives, and they tend to get stuck in local minima. As shown in Fig. 6b, in this work the parameter which changes and optimizes the cost function is again a 4 7 page containing a distribution of eight holes. In the implemented gradient based optimization process, the initial page moves in a multidimensional space along the steepest decent direction to find the minimum of the cost function. Fig. 9 Gradient based method. Iteration no = 3, time = 378 sec 5 Results and Discussions Incorporating the mentioned optimization algorithms, the results of the model based inversion techniques are presented in Figs As is shown in Fig. 7, for both optimization methods if enough numbers of iterations and particles (for the PSO) are considered in the solution, the complete structure of the defect can be obtained. Figures 8 and 9 show the results of using a gradient based method approach with five and three iteration numbers, which have one and three error points. On the other hand Figs. 10 and 11 show the results for incorporating particle swarm optimization with mentioned parameters. It should be noted that, for all these results, the considered initial defect profile is a random one. The processing of the inversion algorithms has been run on a Quad-core, 2.4 GHz personal computer. Table 1 shows the ability of the two optimization algorithms in defect reconstruction in the presence of excess Fig. 10 Particle swarm method. Particles no = 1000, iteration no = 10, time = 214 sec noise. In this table, the values of the detected percent of holes are obtained by considering the average result of multiple running of the mentioned method. In this investigation, the considered number of iterations for the gradient based algorithm is five and for the Particle Swarm method is 10 with 1000 particles. It can be inferred from this investigation that for all the considered noise levels the gradient based one has a much better performance in defect reconstruction.

5 Implementing two stochastic and deterministic optimization algorithms along with an efficient forward model, we have applied the model based inversion solution to the SQUID NDE magnetic image of a prepared sample. The effect of the two optimization algorithms in the resistivity of the whole inversion process has also been investigated by multiple running of each methods with different noise levels. The result was that the gradient based algorithm makes the closed loop inversion process considerably more robust to excess noise with respect to the particle swarm method. The implemented inversion solution uses small enough computational resources, which can be processed on a quite normal personal computer within reasonable time frame. Acknowledgement This work was supported in part by the National Elite Foundation, Tehran, Iran. References Fig. 11 Particle swarm method, Particles no = 100, iteration no = 10, time = 21 sec Table 1 The comparison between the ability of the two optimization algorithms in defect reconstruction in the presence of excess noise Noise level Optimization algorithm Particle swarm method (detected percent of the holes) 100% 87.5% 87.5% 500% 62.5% 87.5% 1000% 62.5% 75% 5000% 33.3% 37.5% 10000% 33.3% 33.3% 6 Conclusion Gradient based method (detected percent of the holes) 1. Kleiner, R., Koelle, D., Clarke, J.: Proc. IEEE 92, 10 (2004) 2. Sepulveda, N.G., Wikswo, J.P. Jr.: J. Appl. Phys. 79(4), 2122 (1996) 3. Chen, Z., Yusa, N., Miya, K.: Nondestruct. Test. Evaluation 24(1 2), 69 (2009) 4. Liu, X., Deng, Y., Zeng, Z., Udpa, L., Udpa, S.S.: IEEE Trans. Magn. 45(3), 1486 (2009) 5. Li, Y., Udpa, L., Udpa, S.S.: IEEE Trans. Magn. 40(12), 410 (2004) 6. Liu, G.R., Han, X.: CRC press LCC (2003) 7. Liu, X., Deng, Y., Zeng, Z., Udpa, L., Udpa, S.S.: IEEE Trans. Magn. 45(3), 1486 (2009) 8. Chen, Z., Miya, K.: J. Nondestruct. Evaluation 17, 167 (1998) 9. Badics, Z., Komatsu, H., Matsumoto, Y., Aoki, K.: J. Nondestruct. Evaluation 17, 2 (1998) 10. Ratnajeevan, S., Hoole, H., Subramaniam, S., Saldanha, R., Coulomb, J.-L., Sabonnadiere, J.-C.: IEEE Trans. Magn. 27, 3 (1991) 11. Li, Y., et al.: An adjoint equation based method for 3D eddy current NDE signal inversion. In: Electromagnetic Nondestructive Evaluation (V), pp IOS, Amsterdam (2001) 12. Sarreshtedari, F., Pourhashemi, A., Asad, N., Schubert, J., Banzet, M., Fardmanesh, M.: IEEE Trans. Superconduct. 20(2), 76 (2010) 13. Fardmanesh, M., Sarreshtedari, F., Pourhashemi, A., Ansari, E., Vesaghi, M., Schubert, J., Banzet, M., Krause, H.-J.: IEEE Trans. Appl. Superconduct. 19(3), 791 (2009) 14. Khatami, Yi, Alavi, M., Sarreshtedari, F., Vesaghi, M., Banzet, M., Schubert, J., Fardmanesh, M.: J. Phys.: Conf. Ser. 1, (2008) 15. Sarreshtedari, F.: Jahed, N.MS., Hosseini, N., Pourhashemi, A., Banzet, M., Schubert, J., Fardmanesh, M.: J. Phys.: Conf. Ser., Inst. Phys., To be published (2010) 16. Kennedy, J., Eberhart, R.C.: In: IEEE International Conference on Neural Networks (1995) 17. Jameson, A.: MAE Technical Report, Princeton University, (1995)