Numerical filters as a tool for defect ... - Semantic Scholar

J. Phys. IV France 125 (2005) 531-533 Ó EDP Sciences, Les Ulis DOI: 10.1051/jp4:2005125123

Numerical filters as a tool for defect characterization in pulsed thermography W. Karpen, M. Jöchen and U. Netzelmann Fraunhofer-Institute for Nondestructive Testing, University, Bldg. 37, 66123 Saarbrücken, Germany Abstract. A calibration approach using numerical filters to characterize defects detected by pulsed thermography in polymers with different optical transparency was developed and tested. Calibration was realised by simulation of the measurement. Seventeen measurement quantities were taken from the simulation results, that supplied the numerical filter coefficients via a multi-dimensional linear regression. Measurements using flash excited thermography on black polyoxymethylene specimens with flat bottom holes resulted in very good defect depth and width reconstruction, whereas for more transparent polyamide specimens the reconstruction was less accurate.

1. INTRODUCTION Detection and characterization of hidden defects (pores, delaminations..) is an important application of pulsed thermography. Depth and lateral size of the defect are of particular interest. Very often only criteria like the time of the begin of the defect contrast and the defect half-width at the time of maximum contrast are selected as characteristic quantities for defect depth and lateral defect dimension [1]. Complications occur, when materials are partially transparent in the visual optical or infrared spectral regime. Here, distortion quantities complicate the size and depth determination. Calibration techniques have been shown to be a useful means to circumvent the complicated mathematical inversion [2]. Prerequisite is the availability of a statistically complete set of calibration specimens covering the whole range of target values and distortion quantities. In practical industrial applications this is often difficult to realize. An alternative is a numerical filter approach, that relies on a multidimensional linear regression. The advantage is, that the numerical filtering operation is very fast. Aim of the present work is to test the numerical filter technique for a certain problem. The important steps are a): numerical forward calculations of the thermal response for all possible parameter combinations of the problem, b): selection of suitable measurement quantities and expansion functions, c): calculation of the filter coefficients and d) comparison of filtered and experimental data. 2. THEORETICAL MODEL As simple model specimens, cylindrical plates with a central flat bottom hole serving as artificial defects were considered. The theoretical description is based on the thermal diffusion equation in radial symmetry for spatially constant material parameters. On top and bottom of the specimen disk, a boundary condition of third kind was considered. The optical absorption is assumed to follow Beer´s law. For weekly absorbing specimens, multiple optical reflections between specimen surface and defect surface are included. It is also included that the infrared camera obtains radiation from deeper parts of the specimen [3].

Article published by EDP Sciences and available at http://www.edpsciences.org/jp4 or http://dx.doi.org/10.1051/jp4:2005125123

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3. NUMERICAL FILTERS The numerical filter approach assumes the following relation between target values Z (here: defect depth and defect width) and measurement quantities Xj [2]:

Z(X ) = å å α f (X ) (1) j jk k j j k This expression contains filter coefficients ajk and expansion functions fk. For the expansion functions, one selects a set of functions that describes the relation between target values and measurement quantities sufficiently well. The expansion coefficients ajk are fixed by the calibration in order to minimize the influence of the distortion quantities. The calibration has to represent the testing situation precisely and complete. Then, from eq. (1) a linear system of equations for determination of the filter coefficients ajk is obtained. In order to achieve a robust signal processing, one works with a strongly overdetermined set of equations by generation as much as possible calibration data. An approximation by minimizing least square errors within a multi-linear regression is performed:

c 2 = å ( Z n - Z(X jn )) 2

(2)

n

4. DEFINITION OF MEASUREMENT QUANTITIES AND EXPANSION FUNCTIONS The number of measurement quantities must be at least the sum of the number of target values and the number of expected distortion quantities. Beside usual measurement quantities as the full half width at maximum defect contrast, the time and the absolute value of the maximum defect contrast, also more complicated quantities as the integrated defect contrast were considered. Other measurement quantities were the time tx, where the temperature contrast is the x-fold of the maximum temperature contrast, or the laterally integrated contrast in the thermal image. In total, 17 measurement quantities were defined. As expansion functions fk, single and multiple products of the measurement quantities were chosen. From these 173 possible expansion functions, only 15 were finally selected as a result of an iterative procedure. 5. SOLUTION OF THE FORWARD PROBLEM As mentioned before, calibration was not performed by experiments, but by numerical forward simulation. For numerical realization of simulation, a finite difference technique based on quotient of differences was selected [4]. The coefficient matrix has a simple form that allows a very efficient algorithm for solution. Calculation time is crucial, as thousands of forward calculation have to be performed for calibration. Two measures helped to reduce computation times significantly: 1. Splitting the problem into a homogeneous, analytically solvable part and a defect related part to be solved numerically. 2. Constructing the mesh in a way that the Alternation Direct Implicit (ADI) algorithm can be applied [5]. 6. VERIFICATION OF THE THEORETICAL MODEL The concept of numerical filters as described here can only work if there is a good agreement between simulations and experiments. Just in order to check this point, reference specimens from polyoxymethylen (POM) and polyamide (PA) were manufactured and measured with flash thermography. For the optically strongly absorbing POM a good agreement of measured and calculated temperature-spatial profiles was obtained. The agreement was less for the optically scattering PA.

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7. TEST OF THE NUMERICAL FILTERS Three numerical filters were constructed for determining the defect depth. Filter 1 allowed only variations of the optical parameters, the thermal parameters were assumed to be known. Filter 2 allowed for a small variation of the thermal parameters in addition and filter 3 a wide range of thermal parameters to cover all polymer materials. Figure 1 show some results for the reconstructed defect depth and width in comparison to the true defect depth and width. The lines try to show the trend of the data.

Figure 1. Comparison between true and reconstructed defect depth (left part) and width (right part) for test specimens from black polyoxymethylene using different numerical filters.

All filters show a quite good estimation of the true defect parameters (typically within less than 10%). For the optically milky white PA the result was much less accurate. Imperfections of the optical model of the sample, in particular the role of optical scattering, are attributed for the latter result. In conclusion, the numerical filter approach shows promising results and can be further developed for a fast inversion of certain defect situations in industrial process control. Acknowledgments This work was supported by DFG, grant KR 1232/27-1

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