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A defect shape reconstruction algorithm for pulsed thermography
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NDT&E International 40 (2007) 220–228 www.elsevier.com/locate/ndteint
A defect shape reconstruction algorithm for pulsed thermography S. Lugin, U. Netzelmann Fraunhofer-Institute for Nondestructive Testing, University Building 37, 66123 Saarbru¨cken, Germany Received 5 May 2006; received in revised form 26 October 2006; accepted 8 November 2006 Available online 8 January 2007
Abstract An algorithm for the defect shape reconstruction from pulsed thermography data was developed. The aim was to reconstruct the defect shape in a test sample with known thermal properties. The algorithm consists of a defect shape correction unit and a simulation unit. The defect shape correction unit is designed to extract the defect shape roughly and refine it sequentially while the simulation unit models the heat conduction process in the inspected sample. The developed iterative algorithm is able to reconstruct 2D as well as 3D defect shapes. The algorithm was tested using experimental data obtained on a plate-shaped steel sample with a wall thickness profile. Robust defect shape reconstruction results were demonstrated. r 2006 Elsevier Ltd. All rights reserved. Keywords: Pulsed thermography; Defect shape reconstruction; Echo defect shape
1. Introduction Pulsed thermography (PT) is a non-destructive testing method that is based on short thermal stimulation of a sample and observation of its cooling process [1]. This testing method is very convenient due to its unique features like: contact-free operation, capability to inspect large areas simultaneously and high inspection speed. The measurement setup for PT consists of a heating source (in many cases flash lamps are used) and an infrared detector (infrared camera) that observes and records the cooling process. This kind of testing is used for the detection of subsurface defects, inclusions and delaminations. PT is also often used for the characterization of thermal material properties. The inspection usually consists of three phases: (1) thermal excitation, (2) observation of the thermal response and (3) data analysis. Once the measurement has been done, it is necessary to detect defects and to assess how critical the defects are. There are two common schemes of PT [1]: thermal reflection and thermal transmission. The last one needs access to the back surface of the sample that is not possible Corresponding author. Tel.: +49 0 6819 3023873; fax: +49 0 6819 3025920. E-mail address: [email protected] (U. Netzelmann).
0963-8695/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ndteint.2006.11.001
in many practical cases. In reflection, both the camera and heating sources are located on the front side. In this work we describe a reconstruction algorithm for the PT measurement setup based on the reflective scheme. Concerning its application fields one can note that the algorithm is applicable in almost any situations where the defect depth or shape is of interest. For example, in aerospace or power generation industry the algorithm can be used to determine the residual depth of the cooling channel of a turbine rotor blade [2]. It can also be used to determine corrosion profiles, for example, of the skin of an aircraft [3]. These techniques can, therefore, become a part of predictive maintenance, allowing to assess whether the material loss of a blade or an aircraft skin is critical or can be tolerated. This will help to reduce the maintenance costs. The problem we consider is the quantitative characterization of the defect shape. The main advantage of the algorithm presented here is its high efficiency, capability to be applied in 2D as well as 3D domain and universality that allows to carry out the reconstruction for materials having different thermal properties. In this work, we present results of the 2D corrosion shape reconstruction in a steel sample. The problem of the defect shape reconstruction has a sense of determination of the depth of the defect in all points where the defect spreads. Note that most literature
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work in PT concern a particular case where the defect is a flat bottom hole and the target values are the hole depth and the hole diameter. Hamzah et al. [4] showed that the time at the point, when the temperature contrast begins to rise, is related to the hole depth independently of the hole diameter. Balageas et al. [5] suggest to transform the cooling curve to the double-logarithmic domain. The solution of one-dimensional (1D) infinite-thickness sample is a straight line in the log-domain. Any deviations from the straight line give evidence of the presence of a defect. The defect depth is related to the time where the deviation begins. To find the hole diameter, Hamzah et al. [4] and Almond and Lau [6] suggest to measure a full-width at half-maximum (FWHM) contrast for measurements of defect image diameter. Other research work [7] offers a method that permits to eliminate the lateral diffusion scattering effect in the case of planar defects. Another approach uses multi-dimensional linear regression [8]. The problem of the defect shape extraction for a general defect shape without much a priori information has not got wide discussion. It is caused by the fact that inherent to the thermal diffusion process are strong smearing and damping which complicate the analysis of defect temperature response. Only one research work [9] describes the extraction of the defect shape and proposes a solution of the ill-posed problem for a particular case when the diffusivity of the test sample is constant. In our work, we have developed a new solution for this more general problem and show for the first time reconstruction results based on experimental data. The paper is organized as follows: in Section 2, we describe the developed algorithm structure. In Section 3, we consider some aspects of the reconstruction under experimental conditions. In Section 4, the reconstruction results of a steel sample with artificially made defect are presented. 2. Defect shape reconstruction algorithm Prior to the algorithm description, we should make two remarks about PT. The first remark is that in PT defect detectability and lateral resolution in depth are parameters strongly varying with depth. If we consider defects like flat bottom holes, an empirical rule [10] says that the radius of the smallest detectable defect should be at least one to two times larger than its depth under the surface. This rule establishes that the resolution is a variable parameter depending on the defect depth. The second remark is that the problem of the defect shape reconstruction is ill-posed. From the mathematical point of view it means that two or more similar defect shapes can produce a practically equal temperature response on the front surface. For example, we simulate two defects: in both cases the defects are the flat bottom holes, but in the second case we slightly round the sharp corners of the top-hole part. In both cases the computed temperature distributions on the front surfaces are equal within the present measurement precision. This
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fact demonstrates the strong scattering effect in PT. These remarks are related to properties of the thermal diffusion process. We describe them in order to show limitations in PT affecting the reconstruction. It is obvious that it is impossible to reconstruct the defect shape or its part if it has no influence on the temperature distribution on the front surface during the measurement. To separate PT limitations at the developing stage of the defect shape reconstruction algorithm, we assume that the defect shape satisfies the resolution criteria (the length scale of the defect shape variation is larger than the defect depth) and the defect shape is smooth (does not contain sharp edges). The algorithm we propose reconstructs the defect shape in the test sample. We assume that thermal properties (thermal conductivity, specific heat, density) and geometry (length, width, thickness) of the test sample are known. The thermal properties of the defect are also assumed to be known. The defect can be any type of inclusion, an internal destruction or corrosion. The algorithm is capable to carry out the reconstruction in two as well as in threedimensional (3D) domain. In this work we describe the algorithm for the case where the excited thermal wave suffers full reflection from the defect and the back sample surface. We do this in order to put the reconstruction principles in the foreground. Note that the algorithm in the way we describe it here can already be applied to reconstruction of corrosion shapes or any air containing defects as the reflection coefficient [1] for most metals, alloys and polymers to the air is close to 1. The developed algorithm has a structure shown in Fig. 1. It comprises a defect shape correction unit and a simulation unit. The simulation unit has a model of the test sample (as the thermal properties and geometrical sizes are known) and is able to simulate the PT experiment. The algorithm operates as follows. The PT measurement with unknown defect shape is processed by the defect shape correction unit. The unit extracts a first approximate defect shape and transmits it to the simulation unit. The simulation unit inserts the expected defect shape in the test sample model and simulates the PT measurement. Then the computed simulation is transmitted to the defect shape correction unit. The unit corrects the defect shape analysing the difference between the measurement and the simulation. Again, the defect shape is transmitted to the simulation unit. The process is repeated in the same order. The defect shape is corrected iteratively until the simulation
Fig. 1. The algorithm structure.
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converges to the measurement. After some iteration cycles the defect shape correction unit provides the reconstructed defect shape. A unique aspect in the algorithm is to find the defect shape, which produces the same temperature distribution on the front surface as the real measurement has. Even in the 1D case, there is usually no direct analytical solution of this inverse problem [11]. In two-dimensional (2D) case there is no direct analytical solution of the forward problem (the simulation of the test sample having arbitrary defect shape in the PT measurement). Only finite element or finite difference methods solve this forward problem. Therefore, we propose an iterative reconstruction algorithm where the simulation unit is used as a forward solver to simulate the internal heat conduction process in the test sample after its thermal excitation. In the following subsections the defect shape correction unit and the simulation unit are described. 2.1. Defect shape correction unit The defect shape correction unit solves two tasks. It extracts an estimate of the defect shape from the measurement at the initial step and corrects the shape after each simulation. Both tasks are solved applying the principle to extract and refine the shape at the early time phase when the defect temperature response on the front surface is not much blurred by the thermal diffusion process. We emphasize that the temperature deviations on the front test sample surface detected at the early time phase contain the most valuable information about the defect shape that is possible to get. As the time passes, due to the thermal diffusion, the defect temperature response trends toward the temperature equilibrium making the defect shape extraction more difficult. Prior to the unit description, we introduce a new definition: echo defect shape (EDS). The echo defect shape is computed in all points where the defect spreads from the front surface point of view. It has a sense to detect the defect and define its depth, as it would happen in 1D case. It does not matter whether a 2D or 3D defect shape is analysed. First, some theoretical aspects concerning heat diffusion are considered. A solution for the sample temperature T(z, t) of a 1D diffusion equation for a defect-free surface absorbing infinite-thickness sample has the following form [12]: rc Q Tðz; tÞ ¼ pffiffiffipffiffiffiffiffiffiffiffipffiffi exp z2 , (1) 4lt 2 p l rc t where z is the sample depth, t is the elapsed time after delta pulse heating, Q is the input energy density, l is the thermal conductivity, r is the density and c is the specific heat capacity. The temperature distributions in the infinitethickness sample at a certain moment is schematically shown in Fig. 2 as a dashed curve.
Fig. 2. Temperature wave in the defect-free sample with thickness d.
Now we consider the case of a plate shape sample with the thickness d assuming that the thermal pulse is fully reflected from the back surface. In this case, the reflected thermal pulse comes back to the front sample surface increasing the temperature in comparison with the infinitethickness sample (see Fig. 2). This temperature increase Tincr at the time tk corresponds to the temperature at the time tk of the infinite-thickness sample in the depth 2d as the reflected wave travels double way: from the front surface to the back surface and from the back surface to the front surface. According to the accepted assumption, the thermal wave fully reflects from the back sample surface at the depth d and goes back to the front surface. The reflected wave part is represented in the figure by the dotted curve. This reflected part additionally increases the temperature in the sample over whole thickness range. Thus, the temperature distribution in the finite-thickness sample (see the solid curve) is a sum of the original and reflected wave parts. Due to the reflected thermal wave part, the temperature on the front surface of the finitethickness sample at the time tk is about Tincr higher than in the case of the infinite-thickness sample. Strictly speaking, we simplified the reflection process. In the ideal case, we have to consider as well the wave part from 2d going to the infinite depth (see the dashed curve). This wave part reflects at the time tk from the front surface, goes to the back surface and so on (method of images). The multiple reflection process has a complex solution described in [13]. We ignore the higher-order reflections of the smallest wave part as they do not significantly change the total temperature distribution in the early time range. From the temperature increase Tincr at the time tk the thickness of the sample can be found. To derive the equation for it, we consider the followings. The front surface temperature Tinf-t(tk) of the defect-free sample with infinite thickness is at z ¼ 0: Q T inft ðtk Þ ¼ pffiffiffipffiffiffiffiffiffiffiffipffiffiffiffi . 2 p l rc tk
(2)
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As it was assumed, the temperature increase Tincr(tk) of the surface sample temperature of the thickness d is the temperature of the infinite-thickness sample in the depth 2d: Q rc 2 T incr ðtk Þ ¼ pffiffiffipffiffiffiffiffiffiffiffipffiffiffiffi exp ð2dÞ . (3) 4ltk 2 p l rc tk Now we introduce the relative temperature increase Trelincr: T relincr ðtk Þ ¼
T incr ðtk Þ T s ðtk Þ T inf t ðtk Þ ¼ , T inf t ðtk Þ T inf t ðtk Þ
(4)
where the Ts(tk) is the temperature on the front surface of the sample which thickness is extracted (see Fig. 2). Substituting the variables Tincr(tk) and Tinf-t(tk) from the equations above we have: T incr ðtk Þ T inf t ðtk Þ Q rc 2 pffiffipffiffiffiffiffiffipffiffiffi exp 4lt ð2dÞ k
T relincr ðtk Þ ¼
¼
2 p
l rc tk
Q pffiffipffiffiffiffiffiffipffiffiffi 2 p l rc tk
rc 2 ¼ exp d . ltk
Then the sample thickness is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ltk log T relincr ðtk Þ . d¼ rc
ð5Þ
(6)
Thus, knowing the relative temperature increase the sample thickness can be computed. The described method can be applied to extract the defect depth in different samples. For example, consider the case where the defect is a flat bottom hole. Over the flat bottom, the thermal wave reaches the defect and is reflected in a similar way as the wave reflects from the backside of the finite-thickness sample. At early time, the temperature increase caused by the back-drilled defect is similar to the temperature increase in the finite-thickness sample described above. So the hole depth can be computed using the Eq. (6) originally derived for evaluating the thickness. At later time, the lateral diffusion flows suppress this temperature increase on the defect area. The boundary between the early and late time periods is defined by the time when the temperature contrast between the non-defect area temperature and the temperature over the defect area reaches the maximum value. But only in the early time period Eq. (6) can be used. Now we consider in detail how the depth of the flat bottom hole is extracted. First, the relative temperature contrast curve Trel-contr(t) is computed: T relcontr ðtÞ ¼
T s ðtÞ T defectfree ðtÞ , T defectfree ðtÞ
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has the same form as the relative temperature increase (Eq. (4)) except that we use the temperature on the defectfree area instead the temperature of the infinite-thickness sample. Actually, it is very convenient and has a logical background. When the wave in the defect area is reflected, the wave in the defect-free area goes further to the back surface as in the case of the infinite-thickness sample. This fact allows us to substitute the temperatures. Second, analysing the relative temperature contrast, if there is any deviation from 0 (temperature increase), the defect depth is computed by Eq. (6). Theoretically, to get most precise results of the depth extraction, it is necessary to define the relative temperature increase Trel-incr and its time tk as early as possible when the lateral diffusion flows are weak. But the detector noise, which is present in every measurement limits the minimum value Trel-incr. For example, if the thermal noise expressed in relative temperature contrast has a range of 0.01–0.015, the recommend value of Trel-incr is about 0.025. Fig. 3 shows the extraction of the time tk in the noise temperature contrast schematically. In this way the depth of the flat bottom hole is extracted. It is necessary to consider the lateral thermal flow effect. Suppose there is a flat bottom hole in a sample with the known thickness f. If we compute the depth in all points on the surface, one can notice that in some non-defect area points the depth computed by Eq. (6) is greater than the sample thickness. This effect appears due to the lateral thermal flow coming from the defect area where the temperature is higher. In these points at a certain time moment the effect of the lateral flow has a significant contribution to Trel-incr so that the points are marked as defective and the time tk is extracted. The computed depth is greater than the thickness f because the temperature increase is entirely caused by the lateral thermal flow instead of the back surface reflected wave. Obvious that in such case, to be correct, the defect depth has to be replaced on the sample thickness f. Now we can give the precise definition of the echo defect shape of the sample with the thickness f computed for the value Trel-incr: the echo defect shape is a defect shape
(7)
where Ts(t) is the temperature on the defect area point, Tdefect-free(t) is the temperature on the defect-free area point (assumed to be preselected by a operator). This definition
Fig. 3. Extraction of the time tk in the noise temperature contrast.
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computed in every point on the front sample surface in the following way: (1) The relative temperature contrast curve is computed by the equation: T relcontr ðtÞ ¼
T s ðtÞ T defectfree ðtÞ , T defectfree ðtÞ
(8)
where Ts(t) is the temperature on the point to be analysed, Tdefect-free (t) is the temperature on the defectfree area. (2) If there exists a deviation in the computed relative contrast curve higher than the value Trel-incr then its time tk is extracted (see Fig. 3 ) and the echo defect depth DEDS is computed as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ltk (9) log T relincr ðtk Þ , DEDS ¼ rc where l; r; care the sample thermal conductivity, density and specific heat capacity. (3) If there is no deviation higher than Trel-incr or the computed defect depth in the previous step is higher than the sample thickness f, the echo defect depth DEDS in this point equals f. The defect shape correction unit uses the computed echo defect shape as the initial defect shape for the first simulation. Then, once the simulation has been computed, the unit analyses the temperature distribution from the source measurement with the unknown defect shape and the simulation with the known defect shape. Further, to refine the reconstructed defect geometry, the unit computes the difference between the echo defect shape of the measurement and the echo defect shape of the simulation, adds the difference to the defect shape used in the last simulation. The correction criterion can be written as: DSSþ1 ¼ DSS þ ðEDSM EDSS Þ,
(10)
where DSS+1 is the corrected defect shape which will be used for the simulation (S+1), DSS is the defect shape used in the simulation S, EDSM and EDSS are the EDSS extracted from the measurement M and the simulation S. Some points in the refined shape DSS+1, namely, their depths can exceed the sample thickness f. Actually, that is caused by the fact that the echo shapes contain accumulated temperature contrast due to back sample reflection and due to lateral heat flow components. The defect shape correction unit checks all points in the refined shape DSS+1 replacing the depths to the sample thickness value f if they exceed it. The main reconstruction principle used in the algorithm is comparing the echo defect shapes of the measurement and the simulation. The echo defect shape, as was mentioned above, contains accumulated information including back as well as lateral reflections. In this way, if we find the echo defect shape of a certain simulation, which is equal to the echo defect shape of the inspected sample, it
means that all internal back and lateral reflections in the simulation are also equal to reflections in the inspected sample. Consequently, the simulation has the same defect shape as the inspected sample. This concept constitutes a basis of the developed defect shape reconstruction algorithm. Notice that if the computed echo defect shapes of the measurement and the simulation are equal, the proposed correction criterion does not change the reconstructed defect shape. Now we show that the difference in echo defect shapes is an appropriate correction criterion. The proposed correction criterion, namely, the difference of the echo defect shapes is explained by followings. Consider a sample having the corrosion shape shown in Fig. 4. In this case the corrosion serves as the defect. We simulate a steel sample with thickness 5 mm in the 2D domain under PT conditions. After the simulation we computed its echo defect shape. It is shown in Fig. 4 as a dashed curve. Further, we simulate the sample with the corrosion shape represented by the dashed curve. Again, the echo defect shape is extracted (see the dotted curve). We regard the fist simulation as the measurement with unknown defect shape. The solid curve is the actual defect shape and the dashed curve is its echo defect shape. We regard the second simulation as the first iteration in the reconstruction. The dashed curve is the simulated defect shape and the dotted curve is its echo defect shape. The analysis of these results shows the following. The depth in the point C (the defect centre) on the front surface is extracted with good precision by using the echo defect shape method. But the depths in surrounding points are evaluated smaller than they really are. It is caused by a lateral thermal diffusion flow from the defect centre to the sample corners. This flow additionally increases the temperature on the sample surface. At a certain time moment in a certain defect point the total temperature due to this lateral flow and due to the wave reflected from the defect (or the back sample surface) is higher than the value Trel-incr so the echo defect shape method detects the defect and extracts its probable depth. We propose to approximate the lateral flow influence in the linear form. For example, in the point P (see Fig. 4) the echo defect depth DEDSM in the first simulation (regarded as the measurement) is DEDSM ¼ Ddefect Dl:flowðMÞ ,
(11)
where Ddefect is the actual defect depth, Dl.flow(M) is the reduced depth value due to the lateral flow. In the iterative reconstruction process, the echo defect depth DEDSS of a certain simulation is DEDSS ¼ DS Dl:flowðSÞ ,
(12)
where DS is the defect depth used in the simulation S (in the first iteration DS ¼ DEDSM), Dl.flow(S) is the reduced depth value due to the lateral flow. Further, we assume that the original defect shape and the defect shape used in the simulation are close, consequently, the lateral flow influences are also close: Dl:flowðMÞ Dl:flowðSÞ . Taking it
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Fig. 4. Scheme of a source defect shape and echo defect shapes.
into account one can combine the Eqs. (11) and (12): Ddefect DEDSM DS DEDSS .
(13)
Then, to approach to actual defect depth, we compute the depth as: Ddefect DS þ ðDEDSM DEDSS Þ.
(14)
The closer is the lateral flow influences Dl:flowðMÞ Dl:flowðSÞ , the closer to 0 is the difference of echo defect shapes ðDl:flowðMÞ Dl:flowðSÞ Þ. Applying it to the iterative reconstruction principle, the equation is transformed into the following form: DSþ1 ¼ DS þ ðDEDSM DEDSS Þ,
Remark. The presented approach allows exactly to define the time range in which the test sample model is simulated. The time boundary is derived from Eq. (6) substituting the known non-defect sample parameters (thermal properties, sample thickness) and the reconstruction parameter Trelincr. The time boundary tb of the simulation is expressed as
(15)
where DS+1 is the defect depth that will be used in the iteration (S+1). In such way, converging the echo shape of the simulation to the echo shape of the measurement, the simulation defect shape DS is approaching to the real defect shape Ddefect. We tested the reconstruction approach in many cases using the forward solver for generating the source measurement data as well as for its reconstructions. The proposed approach demonstrated robust reconstruction results. Summarizing the described background, the defect shape correction criterion proposed in Eq. (10) is proven. In the end of this section, we emphasize that the reconstruction is carried out iteratively. As the initial defect shape, the algorithm uses the echo defect shape of the measurement. In the iterative reconstruction process, the algorithm corrects the reconstructed shape using the echo defect shapes. The necessary number of iterations depends on the defect shape and lies in the range from 2 to 9. As a particular example, the sample with corrosion shown in Fig. 4 can be reconstructed in two iterations with satisfactory precision.
tb ¼
f 2 rc , l logðjT relincr jÞ
(16)
where f is the sample thickness; l; r; c are the sample conductivity, density and specific heat capacity, respectively; Trel-incr is the relative temperature increase used in the reconstruction. Actually, it is very convenient as it allows to avoid redundant computations decreasing the computation time for the simulations. 2.2. Simulation unit The unit simulates the thermal conduction process during the PT experiment. Ideally, we can also simulate radiation and convection processes in PT experiments but their influences are too small so they are neglected. Simulating the thermal conduction process, the unit uses a finite element or finite difference method (FEM or FDM, respectively). In the scope of this work there is no need to consider how the thermal conduction process is modelled by means FEM or FDM (For this work, a commercial FEM solver has been used). We only pay attention to main parameters in such simulations that are: heat excitation time and energy, thermal properties of the inspected sample and the defect, geometric sizes of the inspected sample. Prior to any reconstruction, one can recommend to verify the sample simulation model against the PT experiment. For this purpose, a sample with known defect shape can be used. One can compare its measurement and its simulation results using the relative temperature contrast. If there is
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thermal noise, uneven heating and defect camera pixels. They have unfavourable effects upon the reconstruction. In this section we describe their sources and propose methods to suppress them. 3.1. Thermal noise The thermal noise is inherent to any measurement. If the single image is considered, one can notice abrupt value changes of the temperature between neighbouring pixels. This noise is effectively suppressed by a smoothing filter. The recommended smooth filter sizes are 3 and 5 pixels. 3.2. Uneven heating
Fig. 5. Sequence of actions during the reconstruction.
any discrepancy, it is necessary to find the sources and correct them. Only if there is the qualitative and quantitative correspondence of the measurement and its simulation, the developed algorithm provides precise reconstruction results. 2.3. Sequence of actions in the reconstruction algorithm
Uneven heating is a common problem in PT. It happens when the sample surface is not illuminated uniformly by a heat source. This non-uniform heating is caused by the radiation pattern of the heat source and by the variable distance from the sample surface points to the heating source. Furthermore, when testing curved turbine blades, inhomogeneous illumination will be unavoidable. An existing method [6] of suppressing uneven heating is not efficient for quantitative computations. The method interpolates values using the smooth curve computed between highest and lowest temperatures in defect-free areas. In our case the method is not applicable since the defect and defectfree area are unknown initially. We propose an own solution that is to measure a defect-free sample extra. It works as follows. After the measurement of the sample with unknown defect shape the IR camera and the heat source stay in the same positions. Then the measurement of a reference sample without defect, made of the same material and having the same geometrical sizes is carried out. 3.3. Defect camera pixels
In order to summarize the algorithm description and show the structure in clear form, we present the diagram containing the sequence of actions during the reconstruction. In the diagram following abbreviations are used: RDS is the reconstructed defect shape, Echo_Defect_Shape(X) is the function which extracts the echo defect shape from X, M is the measurement with unknown defect shape, S is the computed simulation, Simulation(X) is the function which computes the sample simulation with the defect shape X, Correction(X) is the function which corrects the shape X where the depths exceed the sample thickness, I is a counter of executed iterations, N is the number of iterations in the reconstruction. The algorithm performs commands in the order as shown in Fig. 5.
Defect pixels (e.g. occurring during ageing of some IR detector types) provide false temperature values in observing view area. It is obvious that such pixels have negative influence on reconstruction process so they have to be eliminated. The simple and effective solution for it is a median signal filtering. Every grabbed IR camera image is filtered by a median filter, which eliminates the defect camera pixels and suppress the impulse noise. Recommended median filter sizes are also 3 and 5 pixels. In short, every measurement is prepared in the following order: (1) the median filtering; (2) the smoothing filtering; (3) the relative temperature contrast computation with uneven heating compensation. Once the measurement has been processed in such way, the reconstruction algorithm can be applied.
3. Defect shape reconstruction under practical conditions
4. Experimental defect shape reconstruction results
The reconstruction of the defect shape in real experimental conditions involves some difficulties such as strong
The developed reconstruction algorithm was tested under laboratory conditions. For this purpose, a sample
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having known, well-defined 2D corrosion shape was made of stainless steel (V2A) (see Fig. 6). A corrosion-free reference sample was produced in the same way. Both samples have following parameters: length—100 mm, width—100 mm, thickness—5 mm. Like most metals and alloys the steel reflects significant amount of energy if it is excited by flash lamps. In order to increase the optical absorption coefficient, the front sample surface was coated by a thin black layer of paint. The layer has a certain influence on the cooling process. In the time range from 0 to 15 ms after firing the flash the coating mostly determines the front sample temperature distribution revealing the thermal properties of the coating material (a polymer). Later, approximately after 15–20 ms, the temperature influence of the black coating disappears and the cooling process is dominated by the thermal properties of the sample material. In the reconstruction algorithm we neglect the information in the time range from 0 to 15 ms after the excitation, only the information after 15 ms is analysed. We should do this because of two reasons. The first one is that our sample simulation model does not contain any thin coating on the front surface. Theoretically, it is possible to simulate the thin coating too, but it will be a time consuming simulation as the coating is extremely thin
Fig. 6. Photograph of the steel sample with artificially machined corrosion shape.
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(about 5–9 mm) and, consequently, a very fine mesh has to be used in the sample model. The second reason is that we coated both samples in our laboratory manually using a typical spray ballon. If it is done manually, even with precise hand movement, the discrepancies of the coating thickness between the corroded and non-corroded sample can achieve 4 mm. Because of that, the relative temperature contrast can show oscillations in the time range from 0 to 15 ms. The easiest way to overcome the problem is to analyse the data after that time, which is what we do. After the sample preparations, the PT measurements were carried out with following parameters: the image acquisition rate was 10 ms, the number of images was 300 and the approximated heating time was 10 ms. Then the measurements were filtered by the median filter and the smoothing filter. The filter sizes were 5 pixels in both cases. Further, the relative temperature contrast was computed in all points. Note that our measurement setup has only a single flash lamp placed on one side of the IR camera so the measurements had strong uneven heating. The developed method successfully suppressed the present uneven heating effect. Then the data of the single line across the corrosion shape were extracted and used for the reconstruction. The relative temperature increase Trel-incr applied in the reconstruction is 0.05. The developed algorithm reconstructed the defect shape with satisfactory precision. The three-iteration reconstruction process is shown in Fig. 7. The solid curves show the actual shape and the dotted curves are the reconstructed shapes. The source corrosion shape consists of two rounded grooves with different slopes of the edges. The right groove is less slanted and induces weak lateral thermal flows. Its reconstruction takes one iteration. During second and third iteration the shape does not change significantly. The left groove is more slanted and, consequently, induces strong lateral flows. The reconstruction takes three iterations to
Fig. 7. The corrosion shape reconstruction process in the steel sample.
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approach to the actual groove shape. The presented reconstruction was carried out using a commercial finite element solver installed on a state-of-art PC (Pentium 3.06 GHz, 512 MB RAM). The mesh model included about 5500 finite elements. The three-iteration reconstruction took 3 min. The algorithm turned out to provide robust reconstruction results from experimental data. This fact proves the correctness of the used defect shape-refining criterion. Remark. In addition, we point out that the reconstruction algorithm can be even applied in cases where some assumptions for the defect shape (smoothness) are not fulfilled. For example, reconstructions of simulated rectangular grooves demonstrated that the half-width of the reconstructed defect shape corresponds to the actual defect width and the reconstructed depth in the defect centre equals to the actual groove depth. 5. Conclusion and further work In this work, a new algorithm of the defect shape reconstruction was developed. A unique aspect in the algorithm is to combine the approach of 1D defect depth extraction and the 2D (or 3D) simulation of a PT measurement. This combination allows to carry out the reconstruction of complex defect shapes in samples extending application fields of PT. The developed algorithm was tested using experimental data and demonstrated robust results of the reconstruction of a 2D corrosion shape in the steel sample. In the future work, we see two tasks: first, it is planned to reconstruct a 3D defect shape under real experimental conditions and, second, one should study the reconstruction in samples with unknown thermal defect properties.
References [1] Maldague X. Theory and practice of infrared technology for nondestructive testing. Willey; 2001. [2] Carl V, Becker E, Sperling A. Quantitative wallthickness measurement with impuls-video-thermography. Proceedings of the 7th ECNDT, Copenhagen, 1998. p. 1918–24. [3] Han X, Favro LD, Thomas RL. Thermal wave NDI of disbonds and corrosion in aircraft. In: Proceedings of the second joint NASA/ FAA/DOD conference on aging aircraft, Willamsburg, Virginia, August 31–September 3, 1998. [4] Hamzah AR, Delpech P, Saintey MB, Almond DP. An experimental investigation of defect sizing by transient thermography. Insight 1996;38(3):167–71. [5] Balageas DL, Krapez JC, Cielo P. Pulsed photothermal modelling of layered materials. Appl Phys 1986;59(2):348–57. [6] Almond DP, Lau SK. Edge effects and a method of defect sizing by transient thermography. Appl Phys 1993;62(25):3369–71. [7] Thomas RL, Favro LD, Crowther DJ, Kuo PK. Inversion of thermal wave infrared images. In: Balageas D, Busse G, Carlomagno GM, editors. ‘‘Quantitative infrared thermography QIRT 92’’ (Editions Europeennes Thermique et Industrie, Paris, 1992). [8] Karpen W, Jo¨chen M, Netzelmann U. Numerical filters as a tool for defect characterization in pulsed thermography. J. Phys. IV (Proc.) 2005;125:531–3. [9] Bryan K, Caudill L. Reconstruction of an unknown boundary portion from Cauchy data in N-dimension. Inverse Probl 2005;21(1):239–56. [10] Vavilov VP, Taylor R. Theoretical and practical aspects of the thermal nondestructive testing of bonded structures. In: Sharpe RS, editor. Research techniques in nondestructive testing, vol. 5, 1982. p. 238–79. [11] Mandelis A, Peralta SB, Thoen J. Photoacoustic frequency-domain depth profiling of continuously inhomogeneous condensed phases: theory and simulations for the inverse problem. J Appl Phys 1991;70(3):1761–70. [12] Almond DP, Patel PM. Photothermal science and technique. London: Chapman & Hall; 1996. [13] Carslaw HS, Jaeger JC. Conduction of heat in solids. New York: Oxford University Press; 1959.
NDT&E International 40 (2007) 220–228 www.elsevier.com/locate/ndteint
A defect shape reconstruction algorithm for pulsed thermography S. Lugin, U. Netzelmann Fraunhofer-Institute for Nondestructive Testing, University Building 37, 66123 Saarbru¨cken, Germany Received 5 May 2006; received in revised form 26 October 2006; accepted 8 November 2006 Available online 8 January 2007
Abstract An algorithm for the defect shape reconstruction from pulsed thermography data was developed. The aim was to reconstruct the defect shape in a test sample with known thermal properties. The algorithm consists of a defect shape correction unit and a simulation unit. The defect shape correction unit is designed to extract the defect shape roughly and refine it sequentially while the simulation unit models the heat conduction process in the inspected sample. The developed iterative algorithm is able to reconstruct 2D as well as 3D defect shapes. The algorithm was tested using experimental data obtained on a plate-shaped steel sample with a wall thickness profile. Robust defect shape reconstruction results were demonstrated. r 2006 Elsevier Ltd. All rights reserved. Keywords: Pulsed thermography; Defect shape reconstruction; Echo defect shape
1. Introduction Pulsed thermography (PT) is a non-destructive testing method that is based on short thermal stimulation of a sample and observation of its cooling process [1]. This testing method is very convenient due to its unique features like: contact-free operation, capability to inspect large areas simultaneously and high inspection speed. The measurement setup for PT consists of a heating source (in many cases flash lamps are used) and an infrared detector (infrared camera) that observes and records the cooling process. This kind of testing is used for the detection of subsurface defects, inclusions and delaminations. PT is also often used for the characterization of thermal material properties. The inspection usually consists of three phases: (1) thermal excitation, (2) observation of the thermal response and (3) data analysis. Once the measurement has been done, it is necessary to detect defects and to assess how critical the defects are. There are two common schemes of PT [1]: thermal reflection and thermal transmission. The last one needs access to the back surface of the sample that is not possible Corresponding author. Tel.: +49 0 6819 3023873; fax: +49 0 6819 3025920. E-mail address: [email protected] (U. Netzelmann).
0963-8695/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ndteint.2006.11.001
in many practical cases. In reflection, both the camera and heating sources are located on the front side. In this work we describe a reconstruction algorithm for the PT measurement setup based on the reflective scheme. Concerning its application fields one can note that the algorithm is applicable in almost any situations where the defect depth or shape is of interest. For example, in aerospace or power generation industry the algorithm can be used to determine the residual depth of the cooling channel of a turbine rotor blade [2]. It can also be used to determine corrosion profiles, for example, of the skin of an aircraft [3]. These techniques can, therefore, become a part of predictive maintenance, allowing to assess whether the material loss of a blade or an aircraft skin is critical or can be tolerated. This will help to reduce the maintenance costs. The problem we consider is the quantitative characterization of the defect shape. The main advantage of the algorithm presented here is its high efficiency, capability to be applied in 2D as well as 3D domain and universality that allows to carry out the reconstruction for materials having different thermal properties. In this work, we present results of the 2D corrosion shape reconstruction in a steel sample. The problem of the defect shape reconstruction has a sense of determination of the depth of the defect in all points where the defect spreads. Note that most literature
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work in PT concern a particular case where the defect is a flat bottom hole and the target values are the hole depth and the hole diameter. Hamzah et al. [4] showed that the time at the point, when the temperature contrast begins to rise, is related to the hole depth independently of the hole diameter. Balageas et al. [5] suggest to transform the cooling curve to the double-logarithmic domain. The solution of one-dimensional (1D) infinite-thickness sample is a straight line in the log-domain. Any deviations from the straight line give evidence of the presence of a defect. The defect depth is related to the time where the deviation begins. To find the hole diameter, Hamzah et al. [4] and Almond and Lau [6] suggest to measure a full-width at half-maximum (FWHM) contrast for measurements of defect image diameter. Other research work [7] offers a method that permits to eliminate the lateral diffusion scattering effect in the case of planar defects. Another approach uses multi-dimensional linear regression [8]. The problem of the defect shape extraction for a general defect shape without much a priori information has not got wide discussion. It is caused by the fact that inherent to the thermal diffusion process are strong smearing and damping which complicate the analysis of defect temperature response. Only one research work [9] describes the extraction of the defect shape and proposes a solution of the ill-posed problem for a particular case when the diffusivity of the test sample is constant. In our work, we have developed a new solution for this more general problem and show for the first time reconstruction results based on experimental data. The paper is organized as follows: in Section 2, we describe the developed algorithm structure. In Section 3, we consider some aspects of the reconstruction under experimental conditions. In Section 4, the reconstruction results of a steel sample with artificially made defect are presented. 2. Defect shape reconstruction algorithm Prior to the algorithm description, we should make two remarks about PT. The first remark is that in PT defect detectability and lateral resolution in depth are parameters strongly varying with depth. If we consider defects like flat bottom holes, an empirical rule [10] says that the radius of the smallest detectable defect should be at least one to two times larger than its depth under the surface. This rule establishes that the resolution is a variable parameter depending on the defect depth. The second remark is that the problem of the defect shape reconstruction is ill-posed. From the mathematical point of view it means that two or more similar defect shapes can produce a practically equal temperature response on the front surface. For example, we simulate two defects: in both cases the defects are the flat bottom holes, but in the second case we slightly round the sharp corners of the top-hole part. In both cases the computed temperature distributions on the front surfaces are equal within the present measurement precision. This
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fact demonstrates the strong scattering effect in PT. These remarks are related to properties of the thermal diffusion process. We describe them in order to show limitations in PT affecting the reconstruction. It is obvious that it is impossible to reconstruct the defect shape or its part if it has no influence on the temperature distribution on the front surface during the measurement. To separate PT limitations at the developing stage of the defect shape reconstruction algorithm, we assume that the defect shape satisfies the resolution criteria (the length scale of the defect shape variation is larger than the defect depth) and the defect shape is smooth (does not contain sharp edges). The algorithm we propose reconstructs the defect shape in the test sample. We assume that thermal properties (thermal conductivity, specific heat, density) and geometry (length, width, thickness) of the test sample are known. The thermal properties of the defect are also assumed to be known. The defect can be any type of inclusion, an internal destruction or corrosion. The algorithm is capable to carry out the reconstruction in two as well as in threedimensional (3D) domain. In this work we describe the algorithm for the case where the excited thermal wave suffers full reflection from the defect and the back sample surface. We do this in order to put the reconstruction principles in the foreground. Note that the algorithm in the way we describe it here can already be applied to reconstruction of corrosion shapes or any air containing defects as the reflection coefficient [1] for most metals, alloys and polymers to the air is close to 1. The developed algorithm has a structure shown in Fig. 1. It comprises a defect shape correction unit and a simulation unit. The simulation unit has a model of the test sample (as the thermal properties and geometrical sizes are known) and is able to simulate the PT experiment. The algorithm operates as follows. The PT measurement with unknown defect shape is processed by the defect shape correction unit. The unit extracts a first approximate defect shape and transmits it to the simulation unit. The simulation unit inserts the expected defect shape in the test sample model and simulates the PT measurement. Then the computed simulation is transmitted to the defect shape correction unit. The unit corrects the defect shape analysing the difference between the measurement and the simulation. Again, the defect shape is transmitted to the simulation unit. The process is repeated in the same order. The defect shape is corrected iteratively until the simulation
Fig. 1. The algorithm structure.
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converges to the measurement. After some iteration cycles the defect shape correction unit provides the reconstructed defect shape. A unique aspect in the algorithm is to find the defect shape, which produces the same temperature distribution on the front surface as the real measurement has. Even in the 1D case, there is usually no direct analytical solution of this inverse problem [11]. In two-dimensional (2D) case there is no direct analytical solution of the forward problem (the simulation of the test sample having arbitrary defect shape in the PT measurement). Only finite element or finite difference methods solve this forward problem. Therefore, we propose an iterative reconstruction algorithm where the simulation unit is used as a forward solver to simulate the internal heat conduction process in the test sample after its thermal excitation. In the following subsections the defect shape correction unit and the simulation unit are described. 2.1. Defect shape correction unit The defect shape correction unit solves two tasks. It extracts an estimate of the defect shape from the measurement at the initial step and corrects the shape after each simulation. Both tasks are solved applying the principle to extract and refine the shape at the early time phase when the defect temperature response on the front surface is not much blurred by the thermal diffusion process. We emphasize that the temperature deviations on the front test sample surface detected at the early time phase contain the most valuable information about the defect shape that is possible to get. As the time passes, due to the thermal diffusion, the defect temperature response trends toward the temperature equilibrium making the defect shape extraction more difficult. Prior to the unit description, we introduce a new definition: echo defect shape (EDS). The echo defect shape is computed in all points where the defect spreads from the front surface point of view. It has a sense to detect the defect and define its depth, as it would happen in 1D case. It does not matter whether a 2D or 3D defect shape is analysed. First, some theoretical aspects concerning heat diffusion are considered. A solution for the sample temperature T(z, t) of a 1D diffusion equation for a defect-free surface absorbing infinite-thickness sample has the following form [12]: rc Q Tðz; tÞ ¼ pffiffiffipffiffiffiffiffiffiffiffipffiffi exp z2 , (1) 4lt 2 p l rc t where z is the sample depth, t is the elapsed time after delta pulse heating, Q is the input energy density, l is the thermal conductivity, r is the density and c is the specific heat capacity. The temperature distributions in the infinitethickness sample at a certain moment is schematically shown in Fig. 2 as a dashed curve.
Fig. 2. Temperature wave in the defect-free sample with thickness d.
Now we consider the case of a plate shape sample with the thickness d assuming that the thermal pulse is fully reflected from the back surface. In this case, the reflected thermal pulse comes back to the front sample surface increasing the temperature in comparison with the infinitethickness sample (see Fig. 2). This temperature increase Tincr at the time tk corresponds to the temperature at the time tk of the infinite-thickness sample in the depth 2d as the reflected wave travels double way: from the front surface to the back surface and from the back surface to the front surface. According to the accepted assumption, the thermal wave fully reflects from the back sample surface at the depth d and goes back to the front surface. The reflected wave part is represented in the figure by the dotted curve. This reflected part additionally increases the temperature in the sample over whole thickness range. Thus, the temperature distribution in the finite-thickness sample (see the solid curve) is a sum of the original and reflected wave parts. Due to the reflected thermal wave part, the temperature on the front surface of the finitethickness sample at the time tk is about Tincr higher than in the case of the infinite-thickness sample. Strictly speaking, we simplified the reflection process. In the ideal case, we have to consider as well the wave part from 2d going to the infinite depth (see the dashed curve). This wave part reflects at the time tk from the front surface, goes to the back surface and so on (method of images). The multiple reflection process has a complex solution described in [13]. We ignore the higher-order reflections of the smallest wave part as they do not significantly change the total temperature distribution in the early time range. From the temperature increase Tincr at the time tk the thickness of the sample can be found. To derive the equation for it, we consider the followings. The front surface temperature Tinf-t(tk) of the defect-free sample with infinite thickness is at z ¼ 0: Q T inft ðtk Þ ¼ pffiffiffipffiffiffiffiffiffiffiffipffiffiffiffi . 2 p l rc tk
(2)
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As it was assumed, the temperature increase Tincr(tk) of the surface sample temperature of the thickness d is the temperature of the infinite-thickness sample in the depth 2d: Q rc 2 T incr ðtk Þ ¼ pffiffiffipffiffiffiffiffiffiffiffipffiffiffiffi exp ð2dÞ . (3) 4ltk 2 p l rc tk Now we introduce the relative temperature increase Trelincr: T relincr ðtk Þ ¼
T incr ðtk Þ T s ðtk Þ T inf t ðtk Þ ¼ , T inf t ðtk Þ T inf t ðtk Þ
(4)
where the Ts(tk) is the temperature on the front surface of the sample which thickness is extracted (see Fig. 2). Substituting the variables Tincr(tk) and Tinf-t(tk) from the equations above we have: T incr ðtk Þ T inf t ðtk Þ Q rc 2 pffiffipffiffiffiffiffiffipffiffiffi exp 4lt ð2dÞ k
T relincr ðtk Þ ¼
¼
2 p
l rc tk
Q pffiffipffiffiffiffiffiffipffiffiffi 2 p l rc tk
rc 2 ¼ exp d . ltk
Then the sample thickness is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ltk log T relincr ðtk Þ . d¼ rc
ð5Þ
(6)
Thus, knowing the relative temperature increase the sample thickness can be computed. The described method can be applied to extract the defect depth in different samples. For example, consider the case where the defect is a flat bottom hole. Over the flat bottom, the thermal wave reaches the defect and is reflected in a similar way as the wave reflects from the backside of the finite-thickness sample. At early time, the temperature increase caused by the back-drilled defect is similar to the temperature increase in the finite-thickness sample described above. So the hole depth can be computed using the Eq. (6) originally derived for evaluating the thickness. At later time, the lateral diffusion flows suppress this temperature increase on the defect area. The boundary between the early and late time periods is defined by the time when the temperature contrast between the non-defect area temperature and the temperature over the defect area reaches the maximum value. But only in the early time period Eq. (6) can be used. Now we consider in detail how the depth of the flat bottom hole is extracted. First, the relative temperature contrast curve Trel-contr(t) is computed: T relcontr ðtÞ ¼
T s ðtÞ T defectfree ðtÞ , T defectfree ðtÞ
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has the same form as the relative temperature increase (Eq. (4)) except that we use the temperature on the defectfree area instead the temperature of the infinite-thickness sample. Actually, it is very convenient and has a logical background. When the wave in the defect area is reflected, the wave in the defect-free area goes further to the back surface as in the case of the infinite-thickness sample. This fact allows us to substitute the temperatures. Second, analysing the relative temperature contrast, if there is any deviation from 0 (temperature increase), the defect depth is computed by Eq. (6). Theoretically, to get most precise results of the depth extraction, it is necessary to define the relative temperature increase Trel-incr and its time tk as early as possible when the lateral diffusion flows are weak. But the detector noise, which is present in every measurement limits the minimum value Trel-incr. For example, if the thermal noise expressed in relative temperature contrast has a range of 0.01–0.015, the recommend value of Trel-incr is about 0.025. Fig. 3 shows the extraction of the time tk in the noise temperature contrast schematically. In this way the depth of the flat bottom hole is extracted. It is necessary to consider the lateral thermal flow effect. Suppose there is a flat bottom hole in a sample with the known thickness f. If we compute the depth in all points on the surface, one can notice that in some non-defect area points the depth computed by Eq. (6) is greater than the sample thickness. This effect appears due to the lateral thermal flow coming from the defect area where the temperature is higher. In these points at a certain time moment the effect of the lateral flow has a significant contribution to Trel-incr so that the points are marked as defective and the time tk is extracted. The computed depth is greater than the thickness f because the temperature increase is entirely caused by the lateral thermal flow instead of the back surface reflected wave. Obvious that in such case, to be correct, the defect depth has to be replaced on the sample thickness f. Now we can give the precise definition of the echo defect shape of the sample with the thickness f computed for the value Trel-incr: the echo defect shape is a defect shape
(7)
where Ts(t) is the temperature on the defect area point, Tdefect-free(t) is the temperature on the defect-free area point (assumed to be preselected by a operator). This definition
Fig. 3. Extraction of the time tk in the noise temperature contrast.
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computed in every point on the front sample surface in the following way: (1) The relative temperature contrast curve is computed by the equation: T relcontr ðtÞ ¼
T s ðtÞ T defectfree ðtÞ , T defectfree ðtÞ
(8)
where Ts(t) is the temperature on the point to be analysed, Tdefect-free (t) is the temperature on the defectfree area. (2) If there exists a deviation in the computed relative contrast curve higher than the value Trel-incr then its time tk is extracted (see Fig. 3 ) and the echo defect depth DEDS is computed as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ltk (9) log T relincr ðtk Þ , DEDS ¼ rc where l; r; care the sample thermal conductivity, density and specific heat capacity. (3) If there is no deviation higher than Trel-incr or the computed defect depth in the previous step is higher than the sample thickness f, the echo defect depth DEDS in this point equals f. The defect shape correction unit uses the computed echo defect shape as the initial defect shape for the first simulation. Then, once the simulation has been computed, the unit analyses the temperature distribution from the source measurement with the unknown defect shape and the simulation with the known defect shape. Further, to refine the reconstructed defect geometry, the unit computes the difference between the echo defect shape of the measurement and the echo defect shape of the simulation, adds the difference to the defect shape used in the last simulation. The correction criterion can be written as: DSSþ1 ¼ DSS þ ðEDSM EDSS Þ,
(10)
where DSS+1 is the corrected defect shape which will be used for the simulation (S+1), DSS is the defect shape used in the simulation S, EDSM and EDSS are the EDSS extracted from the measurement M and the simulation S. Some points in the refined shape DSS+1, namely, their depths can exceed the sample thickness f. Actually, that is caused by the fact that the echo shapes contain accumulated temperature contrast due to back sample reflection and due to lateral heat flow components. The defect shape correction unit checks all points in the refined shape DSS+1 replacing the depths to the sample thickness value f if they exceed it. The main reconstruction principle used in the algorithm is comparing the echo defect shapes of the measurement and the simulation. The echo defect shape, as was mentioned above, contains accumulated information including back as well as lateral reflections. In this way, if we find the echo defect shape of a certain simulation, which is equal to the echo defect shape of the inspected sample, it
means that all internal back and lateral reflections in the simulation are also equal to reflections in the inspected sample. Consequently, the simulation has the same defect shape as the inspected sample. This concept constitutes a basis of the developed defect shape reconstruction algorithm. Notice that if the computed echo defect shapes of the measurement and the simulation are equal, the proposed correction criterion does not change the reconstructed defect shape. Now we show that the difference in echo defect shapes is an appropriate correction criterion. The proposed correction criterion, namely, the difference of the echo defect shapes is explained by followings. Consider a sample having the corrosion shape shown in Fig. 4. In this case the corrosion serves as the defect. We simulate a steel sample with thickness 5 mm in the 2D domain under PT conditions. After the simulation we computed its echo defect shape. It is shown in Fig. 4 as a dashed curve. Further, we simulate the sample with the corrosion shape represented by the dashed curve. Again, the echo defect shape is extracted (see the dotted curve). We regard the fist simulation as the measurement with unknown defect shape. The solid curve is the actual defect shape and the dashed curve is its echo defect shape. We regard the second simulation as the first iteration in the reconstruction. The dashed curve is the simulated defect shape and the dotted curve is its echo defect shape. The analysis of these results shows the following. The depth in the point C (the defect centre) on the front surface is extracted with good precision by using the echo defect shape method. But the depths in surrounding points are evaluated smaller than they really are. It is caused by a lateral thermal diffusion flow from the defect centre to the sample corners. This flow additionally increases the temperature on the sample surface. At a certain time moment in a certain defect point the total temperature due to this lateral flow and due to the wave reflected from the defect (or the back sample surface) is higher than the value Trel-incr so the echo defect shape method detects the defect and extracts its probable depth. We propose to approximate the lateral flow influence in the linear form. For example, in the point P (see Fig. 4) the echo defect depth DEDSM in the first simulation (regarded as the measurement) is DEDSM ¼ Ddefect Dl:flowðMÞ ,
(11)
where Ddefect is the actual defect depth, Dl.flow(M) is the reduced depth value due to the lateral flow. In the iterative reconstruction process, the echo defect depth DEDSS of a certain simulation is DEDSS ¼ DS Dl:flowðSÞ ,
(12)
where DS is the defect depth used in the simulation S (in the first iteration DS ¼ DEDSM), Dl.flow(S) is the reduced depth value due to the lateral flow. Further, we assume that the original defect shape and the defect shape used in the simulation are close, consequently, the lateral flow influences are also close: Dl:flowðMÞ Dl:flowðSÞ . Taking it
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Fig. 4. Scheme of a source defect shape and echo defect shapes.
into account one can combine the Eqs. (11) and (12): Ddefect DEDSM DS DEDSS .
(13)
Then, to approach to actual defect depth, we compute the depth as: Ddefect DS þ ðDEDSM DEDSS Þ.
(14)
The closer is the lateral flow influences Dl:flowðMÞ Dl:flowðSÞ , the closer to 0 is the difference of echo defect shapes ðDl:flowðMÞ Dl:flowðSÞ Þ. Applying it to the iterative reconstruction principle, the equation is transformed into the following form: DSþ1 ¼ DS þ ðDEDSM DEDSS Þ,
Remark. The presented approach allows exactly to define the time range in which the test sample model is simulated. The time boundary is derived from Eq. (6) substituting the known non-defect sample parameters (thermal properties, sample thickness) and the reconstruction parameter Trelincr. The time boundary tb of the simulation is expressed as
(15)
where DS+1 is the defect depth that will be used in the iteration (S+1). In such way, converging the echo shape of the simulation to the echo shape of the measurement, the simulation defect shape DS is approaching to the real defect shape Ddefect. We tested the reconstruction approach in many cases using the forward solver for generating the source measurement data as well as for its reconstructions. The proposed approach demonstrated robust reconstruction results. Summarizing the described background, the defect shape correction criterion proposed in Eq. (10) is proven. In the end of this section, we emphasize that the reconstruction is carried out iteratively. As the initial defect shape, the algorithm uses the echo defect shape of the measurement. In the iterative reconstruction process, the algorithm corrects the reconstructed shape using the echo defect shapes. The necessary number of iterations depends on the defect shape and lies in the range from 2 to 9. As a particular example, the sample with corrosion shown in Fig. 4 can be reconstructed in two iterations with satisfactory precision.
tb ¼
f 2 rc , l logðjT relincr jÞ
(16)
where f is the sample thickness; l; r; c are the sample conductivity, density and specific heat capacity, respectively; Trel-incr is the relative temperature increase used in the reconstruction. Actually, it is very convenient as it allows to avoid redundant computations decreasing the computation time for the simulations. 2.2. Simulation unit The unit simulates the thermal conduction process during the PT experiment. Ideally, we can also simulate radiation and convection processes in PT experiments but their influences are too small so they are neglected. Simulating the thermal conduction process, the unit uses a finite element or finite difference method (FEM or FDM, respectively). In the scope of this work there is no need to consider how the thermal conduction process is modelled by means FEM or FDM (For this work, a commercial FEM solver has been used). We only pay attention to main parameters in such simulations that are: heat excitation time and energy, thermal properties of the inspected sample and the defect, geometric sizes of the inspected sample. Prior to any reconstruction, one can recommend to verify the sample simulation model against the PT experiment. For this purpose, a sample with known defect shape can be used. One can compare its measurement and its simulation results using the relative temperature contrast. If there is
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thermal noise, uneven heating and defect camera pixels. They have unfavourable effects upon the reconstruction. In this section we describe their sources and propose methods to suppress them. 3.1. Thermal noise The thermal noise is inherent to any measurement. If the single image is considered, one can notice abrupt value changes of the temperature between neighbouring pixels. This noise is effectively suppressed by a smoothing filter. The recommended smooth filter sizes are 3 and 5 pixels. 3.2. Uneven heating
Fig. 5. Sequence of actions during the reconstruction.
any discrepancy, it is necessary to find the sources and correct them. Only if there is the qualitative and quantitative correspondence of the measurement and its simulation, the developed algorithm provides precise reconstruction results. 2.3. Sequence of actions in the reconstruction algorithm
Uneven heating is a common problem in PT. It happens when the sample surface is not illuminated uniformly by a heat source. This non-uniform heating is caused by the radiation pattern of the heat source and by the variable distance from the sample surface points to the heating source. Furthermore, when testing curved turbine blades, inhomogeneous illumination will be unavoidable. An existing method [6] of suppressing uneven heating is not efficient for quantitative computations. The method interpolates values using the smooth curve computed between highest and lowest temperatures in defect-free areas. In our case the method is not applicable since the defect and defectfree area are unknown initially. We propose an own solution that is to measure a defect-free sample extra. It works as follows. After the measurement of the sample with unknown defect shape the IR camera and the heat source stay in the same positions. Then the measurement of a reference sample without defect, made of the same material and having the same geometrical sizes is carried out. 3.3. Defect camera pixels
In order to summarize the algorithm description and show the structure in clear form, we present the diagram containing the sequence of actions during the reconstruction. In the diagram following abbreviations are used: RDS is the reconstructed defect shape, Echo_Defect_Shape(X) is the function which extracts the echo defect shape from X, M is the measurement with unknown defect shape, S is the computed simulation, Simulation(X) is the function which computes the sample simulation with the defect shape X, Correction(X) is the function which corrects the shape X where the depths exceed the sample thickness, I is a counter of executed iterations, N is the number of iterations in the reconstruction. The algorithm performs commands in the order as shown in Fig. 5.
Defect pixels (e.g. occurring during ageing of some IR detector types) provide false temperature values in observing view area. It is obvious that such pixels have negative influence on reconstruction process so they have to be eliminated. The simple and effective solution for it is a median signal filtering. Every grabbed IR camera image is filtered by a median filter, which eliminates the defect camera pixels and suppress the impulse noise. Recommended median filter sizes are also 3 and 5 pixels. In short, every measurement is prepared in the following order: (1) the median filtering; (2) the smoothing filtering; (3) the relative temperature contrast computation with uneven heating compensation. Once the measurement has been processed in such way, the reconstruction algorithm can be applied.
3. Defect shape reconstruction under practical conditions
4. Experimental defect shape reconstruction results
The reconstruction of the defect shape in real experimental conditions involves some difficulties such as strong
The developed reconstruction algorithm was tested under laboratory conditions. For this purpose, a sample
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having known, well-defined 2D corrosion shape was made of stainless steel (V2A) (see Fig. 6). A corrosion-free reference sample was produced in the same way. Both samples have following parameters: length—100 mm, width—100 mm, thickness—5 mm. Like most metals and alloys the steel reflects significant amount of energy if it is excited by flash lamps. In order to increase the optical absorption coefficient, the front sample surface was coated by a thin black layer of paint. The layer has a certain influence on the cooling process. In the time range from 0 to 15 ms after firing the flash the coating mostly determines the front sample temperature distribution revealing the thermal properties of the coating material (a polymer). Later, approximately after 15–20 ms, the temperature influence of the black coating disappears and the cooling process is dominated by the thermal properties of the sample material. In the reconstruction algorithm we neglect the information in the time range from 0 to 15 ms after the excitation, only the information after 15 ms is analysed. We should do this because of two reasons. The first one is that our sample simulation model does not contain any thin coating on the front surface. Theoretically, it is possible to simulate the thin coating too, but it will be a time consuming simulation as the coating is extremely thin
Fig. 6. Photograph of the steel sample with artificially machined corrosion shape.
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(about 5–9 mm) and, consequently, a very fine mesh has to be used in the sample model. The second reason is that we coated both samples in our laboratory manually using a typical spray ballon. If it is done manually, even with precise hand movement, the discrepancies of the coating thickness between the corroded and non-corroded sample can achieve 4 mm. Because of that, the relative temperature contrast can show oscillations in the time range from 0 to 15 ms. The easiest way to overcome the problem is to analyse the data after that time, which is what we do. After the sample preparations, the PT measurements were carried out with following parameters: the image acquisition rate was 10 ms, the number of images was 300 and the approximated heating time was 10 ms. Then the measurements were filtered by the median filter and the smoothing filter. The filter sizes were 5 pixels in both cases. Further, the relative temperature contrast was computed in all points. Note that our measurement setup has only a single flash lamp placed on one side of the IR camera so the measurements had strong uneven heating. The developed method successfully suppressed the present uneven heating effect. Then the data of the single line across the corrosion shape were extracted and used for the reconstruction. The relative temperature increase Trel-incr applied in the reconstruction is 0.05. The developed algorithm reconstructed the defect shape with satisfactory precision. The three-iteration reconstruction process is shown in Fig. 7. The solid curves show the actual shape and the dotted curves are the reconstructed shapes. The source corrosion shape consists of two rounded grooves with different slopes of the edges. The right groove is less slanted and induces weak lateral thermal flows. Its reconstruction takes one iteration. During second and third iteration the shape does not change significantly. The left groove is more slanted and, consequently, induces strong lateral flows. The reconstruction takes three iterations to
Fig. 7. The corrosion shape reconstruction process in the steel sample.
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approach to the actual groove shape. The presented reconstruction was carried out using a commercial finite element solver installed on a state-of-art PC (Pentium 3.06 GHz, 512 MB RAM). The mesh model included about 5500 finite elements. The three-iteration reconstruction took 3 min. The algorithm turned out to provide robust reconstruction results from experimental data. This fact proves the correctness of the used defect shape-refining criterion. Remark. In addition, we point out that the reconstruction algorithm can be even applied in cases where some assumptions for the defect shape (smoothness) are not fulfilled. For example, reconstructions of simulated rectangular grooves demonstrated that the half-width of the reconstructed defect shape corresponds to the actual defect width and the reconstructed depth in the defect centre equals to the actual groove depth. 5. Conclusion and further work In this work, a new algorithm of the defect shape reconstruction was developed. A unique aspect in the algorithm is to combine the approach of 1D defect depth extraction and the 2D (or 3D) simulation of a PT measurement. This combination allows to carry out the reconstruction of complex defect shapes in samples extending application fields of PT. The developed algorithm was tested using experimental data and demonstrated robust results of the reconstruction of a 2D corrosion shape in the steel sample. In the future work, we see two tasks: first, it is planned to reconstruct a 3D defect shape under real experimental conditions and, second, one should study the reconstruction in samples with unknown thermal defect properties.
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