4.2. Probabilistic FLC
Overall, the experts’ annotation lay below the SVMe results, leaving room for improvement in finding the point in time when the onset of necking begins. Taking into consideration that the experts’ annotations contain some source of error in defining the onset of necking due to variance between the experts or a high sampling frequency, as depicted by
Figure 3, the confidence of being an inlier decreases towards the end of the forming process, before being classified as outlier. This decrease is expressed in terms of quantiles of probabilities as belonging to the outlier class. The probabilistic FLC for each material is visualized in
Figure 6. To be able to assess the quality of the result, the “line-fit” method as well as the experts’ annotations are depicted for comparison. In the case of DX54D (0.75 mm), excellent agreement between the estimated probabilistic FLC, of the category < 0.01 quantile, and the experts’ annotations was observed over all geometries. This was only true for the uniaxial loading conditions in the case of AC170 and DP800. DX54D (2.00 mm) shows the largest deviation from the <0.01 quantile, which might once again be the result of the low sampling rate combined with the ductility of the material. The deviation from the plane strain to biaxial strain conditions on the right side of the curve of AC170 and DP800, are a result of the poor quality of the experts’ annotations, since in these cases the onset of necking could not be defined in consistent manner. As expected, the >0.99 quantile shows very good agreement with the “line-fit” result, in all cases except DX54D (2.00 mm). The remaining quantiles fall between the <0.01 quantile and the >0.99 quantile.
4.3. Comparison with Time-Dependent Evaluation Method
The “line-fit” method evaluates comparable information to the proposed approach to determine the onset of localized necking, and consequently, is a suitable choice for comparison. The main difference is in the definition of the evaluation area for the former, as the first derivative in the thinning direction is used to determine the evaluation area with an additional restriction to 15–20 connected pixels. This area is averaged for every stage of the forming process, and the resulting curve is used to determine the onset of necking based on the intersection of two lines that are regressed in the homogeneous and inhomogeneous area of the curve. Consequently, choice of the evaluation area is crucial for the “line-fit” method as it directly affects the shape and steepness of the curve. In contrast, the approach proposed in this study identifies a certain point in time when something abnormal starts to take place with respect to the homogeneous forming area, e.g., a rising gradient, meaning that with a certain confidence at time x, the likelihood of being in the necking phase rises with progression of the forming process. The choice of pixels that are used to determine the forming limit can be chosen independently, as long as the necking area with rising gradient is included in the evaluated patches. In order to facilitate comparisons between the proposed approach and the line fit method, a threshold value of 0.9 of the maximum thinning () was used, with additional restriction to 15–20 connected pixels. The proposed method is independent from the evaluation area, as it returns a point in time to look up the strain value pairs, and consequently the evaluation area could be defined using other constraints.
The hypothesis that the “line-fit” method overestimates the onset of localized necking is supported with the evaluation of the z-displacement differences of DX54D (2.00 mm)-S030-1 as visualized in
Figure 7a. In this trial, the “line-fit” method returns stage 370 as the onset of necking. However, this seems to be optimistic as already some reduction of sheet metal thickness along the z-axis is already visible as shown in
Figure 7a (right). Investigation of the earlier stages revealed that some reduction in sheet metal thickness was already present at stage 355. Consequently, the z-displacement of specimen is a valuable source of information that could be exploited when evaluating ductile materials. This is further emphasized by the z-displacement cross-section profiles depicted in
Figure 7b. Additionally,
Figure 7c depicts the corresponding cross-section profiles of the strain distribution.
As already mentioned, the choice of evaluation area to determine the onset of necking is critical. At the beginning of the forming procedure, the thresholded area is distributed over the whole evaluation area and as the forming process progresses, this area gets concentrated towards the center, finally resulting in a single connected particle. This is depicted in
Figure 8, which compares the change in area size (using a threshold of 0.9 on
), with the change in the size of the largest connected particle.
This process is depicted by
Figure 8b, where the relationship between the size of the thresholded evaluation area and the largest connected particle, is emphasized. The thresholded area reaches its maximum before the single connected particle. Additionally, after the connected particle reaches its maximum, the HoG feature starts rising, which indicates that the necking area is unable to spread any further. This leads to an increasing gradient and emphasizes HoG is a reasonable choice as a feature descriptor. The aforementioned observation is dependent on the material properties, especially the ductility. However comparable characteristics of this effect can be found when evaluating different materials e.g., DP800-S050-1 as depicted by
Figure 8c. The thresholded area maximum and connected component maximum show high correlation, and after some stages the HoG feature starts rising. This additionally shows the dependence on the threshold, as a different threshold e.g., 0.95 would most likely result in a maximum at a different point in time, particularly with regard to the connected component. This highlights the need to find a material dependent evaluation area rather than using a fixed size of threshold or amount of pixels to determine the onset of localized necking. For example, for DX54D (2.00 mm)-S030-1 from
Figure 7 and
Figure 8 an optimal evaluation area would include the complete necking area, which corresponds to a width of around 15 px, as defined in the cross-sections plot shown in
Figure 7b. This was inferred based on the fact that at position ≈15 and ≈27, in the strain distribution plot shown in
Figure 7c, the strain remains static. Since the dependence on a threshold always influences the evaluation area and thus the resulting onset of necking, it might be advantageous to find the necking area using image segmentation techniques. However, since the onset of necking in this study is found independently of the evaluation area, it would be possible to simply use the maximum strain value of the necking stage rather than averaging an approximated area.
Leaving aside the actual evaluation area and hence the actual major and minor strain pairs, the point in time when necking is detected, can be compared. As the present study combines three trials for each geometry to determine the probabilistic FLC candidate, the average time over three trials is depicted in
Figure 9b for each quantile. There is a probability of <1% that necking is initiated before stage 350. This increases to a 50% probability by stage 356 and is >99% at stage 362. The “line-fit” method for DX54D (2.00 mm)-S030-1 suggests stage 368 to be the onset of necking on average, whereas the majority of experts from Part 1 identified stage 364, which is reasonable as they concentrated on finding a sudden increase of strain from one image to the other based on cross-sections through the strain distributions. The multiple time points in
Figure 9b are a result of the variation in the slopes of the probability progression curves, per trial, as depicted in
Figure 9a. As the trials have different lengths, the image only contains the last 40 frames of the test set per video sequence and trial.
4.4. Comparison of Deterministic and Probabilistic FLC
The deterministic and probabilistic FLC are both capable of describing the onset of necking. The D-FLC uses the binary affiliation of patches to the outlier class without consideration of the class-membership of neighboring patches. Within the P-FLC, two Gaussians are used to model the homogeneous and inhomogeneous classes and thus allow to extract probabilities for each patch and stage. This is of particular interest in the curved region, within the transition area from the inlier to the outlier class, as it enables detection of the onset of necking, based on the change in the gradient information. Furthermore, this detection is expressed in the form of a likelihood of necking, enabling a probabilistic interpretation of the forming process. For DX54D we found that the >0.99 quantile estimate of the probabilistic FLC was in close agreement with SVMf. For DP800, the line-fit method also showed strong consistency with these two approaches as depicted in
Figure 10. Due to the different evaluation schemes, the <0.01 quantile was below the SVMe curve, as it is possible to evaluate the data, that is not yet recognized as outlier, but develops into this direction. While the difference is rather large in case of DX54D (2.00 mm), it vanishes in case of DP800 due to its lower ductility. The <0.01 quantile and SVMe must not generally agree as visualized in
Figure 10a, as measurement noise or unstable outliers may affect the modeling of the distributions as well as the lookups of SVMe. Another major difference between these two evaluation strategies, is that the deterministic FLC might be extended into an on-line evaluation method, which is capable of stopping the forming process, when being trained in an incremental manner, e.g., to allow generation of ground truth based on metallographic examinations. So far, an on-line method would only be possible with the SVMf approach, as nine patches being classified as outliers is unlikely during the homogeneous forming phase. Such an on-line evaluation method is impossible in case of the probabilistic FLC as it exploits time information.
4.5. Comparison with Metallography
A quantification of the method can be achieved with the analysis on the material behavior using metallography. In [
11], a metallographic analysis of a DP800 at several forming steps and different strain paths using Nakajima tests has been presented. This analysis of the surface and thickness pointed out that the onset of necking for this dual-phase steel starts on the surface with micro-cracks that are metallographic recognizable. The micro-crack corresponds to multiple localization in the strain distribution and thus this information can be used for the definition of the class “onset of instability” and serve as ground truth in supervised pattern recognition approaches. Once the forming step is selected, the corresponding strain level in terms of major and minor strain can be found. The outcomes for the different strain paths are compared in
Figure 11 with introduced probabilistic FLC. It can be observed, that the forming level defined by the metallographic evaluation are in good agreement with the outcomes of the unsupervised method. In particular, the average FLC values from the metallographic analysis cover the expert evaluation, with the exception of the S245 geometry. Under biaxial straining, as already discussed in Part 1, the strain distribution is homogeneous on the surface most of the time and the main straining occurs in the thickness, due to the volume constancy. Therefore, the evaluation of onset of instability for this geometry is still challenging. Nevertheless, the experimental evaluation with metallography agree with the quantile distribution, confirming that the unsupervised evaluation of the onset of instability is plausible. The LF method gets in general higher FLC results than the metallography observations. According to the quantile investigation, the LF is at the same level of high quantile, namely >0.99. That means, using the LF method, the evaluated necking phase is in advanced development stage and therefore is less conservative. Instead, the experts are conservative and detect a very early stage of localization. In general, the comparison with the metallographic analysis confirms the qualification of the unsupervised method and gives an overview of the level of development of the necking using quantiles.
4.6. Factors of Influence
As already mentioned a couple of factors influence the quality of the result. One critical part is the preparation of the specimen, as it directly affects the DIC measurements. Sub-optimal preparation reduces the traceability of blocks and thus may lead to defect pixels or incorrect measurements that lead to alternating or fluctuating strain values. While defect pixels can be interpolated using neighboring pixels or time information, fluctuating strain values are difficult to detect. Fluctuating strain values would have a large impact on the evaluation approach, as the differences between stages are used to detect the onset of necking. This effect can be attenuated to a certain degree using quantile normalization, however, outliers are still introduced as the orientation and magnitude of edges (by extension, the HoG features) are affected. An example of fluctuating pixels or wrong measurements is depicted in
Figure 12, with its effects on the difference.
Further consideration is required for the geometry S245 under biaxial straining. The symmetry of the strain condition as well as of the geometry does not permit to assign a preference in the direction and thus the crack can occur in arbitrary direction. On the one hand, the placement of the test S245 could be help. In fact, it is supposed, that the crack tends to occur in the weakest direction, in general perpendicular to the rolling direction for steels and parallel to the rolling direction for aluminum alloys. On the other hand, due to the material anisotropy as well as material structure defects, the onset of crack can still happen independently to the rolling direction. This has no effect on the result of the standard or “line-fit” method as they do not evaluate the gradient with respect to the orientation. In this study the HoG feature is used and as it is not rotation invariant, the orientation of specimen placement seems important, as the gradient develops with specific orientation during forming. This behavior is described in
Figure 13 based on the last valid stage of the necking region. Two trials have corresponding or slightly varying necking orientations, while one trial clearly deviates.
However, the orientation during training is addressed using augmentation with random rotations and flipping of the data. To address this, one improvement for the biaxial loading condition might be pose normalization with respect to the last valid stage. The effect of these factors on the probability progression and decision boundary is depicted in
Figure 14 using an insufficient HoG resolution of 180° and 10° steps. Trial 1 deviates from trial 2 and 3, that lead to the outliers in the decision boundary visible in
Figure 14b and affect the quantile-based strain values. However, the spontaneous failure of this geometry is still consistently captured at ≈76 stage of the test set of each trial, as highlighted by the steep increase in the probability progression curves.