Table of Contents ECNDT '98 Session: Reliability  Factorial Design of Experiments Applied to Reliability Assessment in Discontinuity Mapping by UltrasoundSilvio Elton KRUGER, Ivan Costa da SILVA and Joao Marcos Alcoforado REBELLOLABOEND/PEMM/COPPE  Universidade Federal do Rio de Janeiro Caixa Postal 68505 CEP 21945970, Rio de Janeiro  RJ Brazil Corresponding Author Contact: Silvio Kruger Av. A. Buyl, 55 B1050 Brussels, Belgium, Phone: 32 2 650 2700, Fax: 32 2 650 2786 Email: skruger@ulb.ac.be , URL: http://student.ulb.ac.be/~skruger/welcome.html 
TABLE OF CONTENTS 
Statistical design of experiments techniques are used in this paper to assess how experimental factors can influence the final results in discontinuity mapping by ultrasound. As an example, the technique known as factorial design was applied to CScan mapping in a sample that simulates delamination. This analysis allows one to determine the configuration that yields the smallest error and also indicates the experimental parameters which have the greatest influence on changes in the final results.
The advent of digital experimental systems and the signal processing in the area of non destructive testing obligates discussions about the actual ways of calibration, standardization, normalization and acceptation criteria, for example. The traditional method of evaluation of signals based in the inspector experience for interpretation of the current standards should be substituted by automatic systems for evaluation and should have different characteristics of conventional methods. But this type of discussion is just beginning [1].
The reconstruction of images from ultrasonic signals reached a sophisticated level. Often these systems are also quite inexpensive [2,3], but the industrial applications have been limited by the incompatibility with the actual standardization systems which are not updated to permit these new techniques. It can be added also that only a few works in the evaluation of the reliability of these systems are available. One example is the use of theoretical model of the interaction of the inhomogeneities with ultrasound [4]. One reason for this is the inherent difficulty in standardizing systems with very rapid obsolescence that has been typical for systems of digital technology.
In this work, the statistical design of experiments is proposed as an efficient tool for the evaluation of the experimental parameters and signal processing in image formation from ultrasonic signals. The main objective is to evaluate the accuracy of the results by the quantitative verification of how the experimental system parameters influence them and reach the smallest error configuration. The technique is presented by an example in CScan ultrasonic mapping of a known discontinuity.
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 PlackttBurmam, but the most employed is the factorial analysis in two levels (2^{N}). 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.
Subsystem  Factors/Parameters 
Transducer 

Ultrasonic equipment 

Positioning system 

Signal processing and evaluation criteria 

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 Ascan signals for all positions, they are processed to obtain the CScan 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 CScan 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:
The experiment is a factorial in two levels and three variables (2^{3}) 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  Y_{a} 
b  H5M  With filter  Curve 1  Y_{b} 
ab  H10M  With filter  Curve 1  Y_{ab} 
c  H5M  Without filter  Curve 2  Y_{c} 
ac  H10M  Without filter  Curve 2  Y_{ac} 
bc  H5M  With filter  Curve 2  Y_{bc} 
abc  H10M  With filter  Curve 2  Y_{abc} 
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,
Effect_{A} = (Y_{a} + Y_{ab} + Y_{ac} + Y_{abc})/4  (Y_{(1)} + Y_{b}+ Y_{c} + Y_{bc})/4  (1) 
The effect of the combinations are obtained in a similar way, as for example, the effect of the interaction AB,
Effect_{AB} = (Y_{(1)} + Y_{ab} + Y_{c} + Y_{abc})/4  (Y_{a} + Y_{b}+ Y_{ac} + Y_{bc})/4  (2) 
Fig 3: CScan image obtained with the AB configuration, showing the round form of the discontinuities in two levels. 
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 [%]  d^{2}  Y1  Y2  Y3  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 
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 Y1, Y2 and Y3) 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 (d^{2} ) and are used to the estimate the errors by the expression,
EPDM^{2} = d^{2}/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(21)=8, d^{2} = 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 tdistribution) distribution should be applied. For the interval of reliability of 95% and degree of freedom of 8, the tstudent parameter is t_{tab,95%,8} = 2.31, that multiplied by the EPDM results in the value calculated to be the reference.
DM_{calc} = t_{tab} .EPDM = 2.31 . 1.05 = 2.43  (5) 
Comparing DM_{calc} 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.
An analysis of the results for the described example allowed some conclusions:
The application of the statistical design of experiments proved to be very satisfactory to the evaluation of the influence of experimental parameters. The knowledge of the effect of the experimental parameters in the results in a quantitative way and statistically consistent, seems to be a very powerful tool in the certification of the system reliability for NDT methods.
The authors wish to thank CNPq and CAPES, Brazilian agencies for scientific and technological research for financial support.
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