The statistical design of experiments [5,6] are well developed methodologies that apply statistics in a way of developing a 'plan' of experiments which have minimum experimental effort for a desired results reliability. In the design of experiments, it is supposed that the system is composed by an ensemble of independent input variables (or factors/parameters) and the results for each configuration as the outputs. Usually the purpose is the evaluation of how the input variables influence the results (in a independent or combined way). Some known techniques of statistical design of experiments are the factorial analysis, Taguchi and Placktt-Burmam, but the most employed is the factorial analysis in two levels (2N). The two level factorial analysis allows the research of the influence of N variables and their interactions, by changing them in two levels (high and low). The statistical analysis of the results allows the determination of their significance and the statement of an experimental equation that associates the variables with the results. In the case of the reliability analysis of ultrasonic mapping, the input variables are the experimental parameters (see table 1) and the output are the errors from the association between the image obtained by the ultrasonic technique with the known geometry and size of the discontinuity. The application of the factorial analysis of experiments is described with more details by an example in the next section.
The sample used for the discontinuity mapping is illustrated in figure 1 and simulates a double delamination. The ultrasonic scanning was performed with the immersion technique and the system is depicted in figure 2. The pulser/receiver used is a Krautkramer USIP 12, and the received signal is digitized by an Gage Lite analog to digital card. A software developed by the authors managed the acquisition by the A/D card and the positioning system by controlling a step motor. The time domain signals were saved for further processing. The scanning area was 65 x 65mm by steps of 1.6 mm. After the acquisition of the A-scan signals for all positions, they are processed to obtain the C-Scan image. The gain was adjusted in the beginning of each scanning to have the first backwall echo at 50% of the scale.
Fig 1: Geometry and dimensions [mm] of the sample used to C-Scan mapping.
Fig 2: Experimental system used in this work. | |
Among the parameters that could have an influence in the results (table 1), three were chosen to be evaluated:
- Variable A - Transducer. Two transducers with different characteristics were used. They were the Krautkramer H5M and H10M with nominal frequencies of 5 and 10 MHz respectively and 5 mm diameter. Both transducers are broadband and focalized.
- Variable B - Filtering. One of the two configurations employed filtering. The filtering was performed by the ultrasonic pulser/receiver and configured to have the lowest noise and the best signal definition. This evaluation was done visually.
- Variable C - Evaluation Criteria. For the evaluation criteria of the A-Scan signals to obtain the C-scan image, two reference curves were used. Both have an exponential decay but the first (curve 1) had an amplitude 6 dB higher than the second (curve2).
The experiment is a factorial in two levels and three variables (23) and table 2 presents the possible configurations for a complete factorial analysis. The experiment was duplicated for each configuration for further error analysis.
| Notation
| Variable A
| Variable B
| Variable C
| Result
|
| (1)
| H5M
| Without filter
| Curve 1
| Y(1)
|
| a
| H10M
| Without filter
| Curve 1
| Ya
|
| b
| H5M
| With filter
| Curve 1
| Yb
|
| ab
| H10M
| With filter
| Curve 1
| Yab
|
| c
| H5M
| Without filter
| Curve 2
| Yc
|
| ac
| H10M
| Without filter
| Curve 2
| Yac
|
| bc
| H5M
| With filter
| Curve 2
| Ybc
|
| abc
| H10M
| With filter
| Curve 2
| Yabc
|
Table 2 - Possible combinations for the proposed complete factorial design.
The complete factorial analysis allows the evaluation of each variable (A, B and C) and also of their interactions (AB, AC, BC and ABC). To obtain the influence of each variable, a comparison is made of the results for the configuration when it has a high value with the results of the configuration when it assumes a lower value. For the variable A we will have,
| EffectA = (Ya + Yab + Yac + Yabc)/4 - (Y(1) + Yb+ Yc + Ybc)/4
| (1) |
The effect of the combinations are obtained in a similar way, as for example, the effect of the interaction AB,
| EffectAB = (Y(1) + Yab + Yc + Yabc)/4 - (Ya + Yb+ Yac + Ybc)/4
| (2) |
 |
Fig 3: C-Scan image obtained with the AB configuration, showing the round form of the discontinuities in two levels. |
The results obtained in the form of C-Scan images show clearly the simulated two level delamination, as illustrated in figure 3, by the image obtained with the configuration AB.
The numerical results were calculated by the contrast between the image obtained with each configuration and the ideal image (as the real discontinuity is known). The error is then calculated in terms of percentages of wrong pixels. Table 3 presents the obtained results with the calculations performed with the Yates algorithm (manual calculations of equations as (1) and (2)) that yields the effect for each configuration.
| Y [%]
| YR [%]
| d2
| 
| Y-1
| Y-2
| Y-3
| DM
| Significance
|
| (1)
| 2.6
| 2.5
| 0.0
| 5.1
| 27.9
| 55.1
| 187.8
| -
| -
|
| a
| 14.9
| 7.9
| 49.0
| 22.8
| 27.2
| 132.7
| 48.7
| 6.1
| S
|
| b
| 4.6
| 6.2
| 2.6
| 10.8
| 64.9
| 23.2
| 2.2
| 0.3
| N
|
| ab
| 9.0
| 7.3
| 2.9
| 16.3
| 67.8
| 25.5
| -26.1
| -3.3
| S
|
| c
| 11.0
| 11.6
| 0.4
| 22.6
| 17.7
| -0.7
| 77.6
| 9.7
| S
|
| ac
| 21.2
| 21.1
| 0.0
| 42.3
| 5.5
| 2.9
| 2.3
| 0.3
| N
|
| bc
| 15.4
| 15.6
| 0.0
| 31.0
| 19.7
| -12.2
| 3.6
| 0.5
| N
|
| abc
| 20.4
| 16.4
| 16.0
| 36.8
| 5.8
| -13.9
| -1.7
| -0.2
| N
|
Table 3 - Results for each configuration and the calculations of the effects by the Yates algorithm.
The second and the third columns (Y and YR the repetition) presents the errors in percentage for each configuration the notation presented in the first column (see also table 2). The fifth column () is the summation of the second and third, and these are values used to the Yates algorithm (columns Y-1, Y-2 and Y-3) resulting in the ninth column (DM) with the results in terms of effects for each configurations (as equations 1 and 2). The observation of effects shows, for example, that the influence of the variable C (evaluation criteria) is much more important than the variable B (filtering), which the values are respectively 9.7 and 0.3. The analysis of significance of these effects must be done by estimating the error associated with these values. This is possible because the experiment was duplicated and the squared values of these differences are shown in the fourth column (d2 ) and are used to the estimate the errors by the expression,
| EPDM2 = d2/GL.N
| (3) |
where EPDM is the standard error of the average differences, GL is the degree of freedom of the experiment and N is the number of experiments. In our case we have (N=8, GL=N(2-1)=8, d2 = 70,9)
| (4) |
The relationship between EPDM and DM should be analyzed in the context of a statistical distribution and the desired level of reliability. As the variance (associated with the standard error) was estimated from the results itself, the 'student' (or t-distribution) distribution should be applied. For the interval of reliability of 95% and degree of freedom of 8, the t-student parameter is ttab,95%,8 = 2.31, that multiplied by the EPDM results in the value calculated to be the reference.
| DMcalc = ttab .EPDM = 2.31 . 1.05 = 2.43
| (5) |
Comparing DMcalc with the values of the ninth column (DM), it could be observed that the configurations A, AB and C are significant, that is, the effect was observed to be higher than the error for a 95% interval of reliability, as indicated in the tenth column (Significance). The configurations that surely change the results are the effect of the transducer (A), the evaluation criteria (C) as well as the combined effect of the transducer and the filtering (AB). The other configurations presented have effects smaller than the statistical error level.
