Cargo Optimizer 4 3 Crack: How to Load More Cargo in Less Space and Time

Quantifying fatigue crack growth is of significant importance for evaluating the service life and damage tolerance of critical engineering structures and components that are subjected to non-constant service loads1. Fatigue crack propagation (fcp) data are usually derived from standard experiments under pure Mode I loadings. Therefore, a straight crack path is usually assumed, which can be monitored by experimental techniques such as the direct current potential drop method2,3. Effects like crack kinking, branching, deflection or asymmetrically growing cracks cannot be captured without further assumptions, hindering the application of classical methods for multiaxial loading conditions. Alternative methods able to capture the evolution of cracks under complex loading conditions are therefore needed.

Cargo Optimizer 4 3 Crack

Download Zip

In recent years, digital image correlation (DIC) has become instrumental for the generation of full field surface displacements and strains during fcp experiments4. Coupled to suitable material models, the DIC data can help to determine fracture mechanics parameters like stress intensity factors (SIFs)5, J-integral6 as well as local damage mechanisms around the crack tip and within the plastic zone7,8. All this requires a clear knowledge of the crack path and, especially, the crack tip position. Gradient-based algorithms like the Sobel edge-finding routine can be applied to identify the crack path9. Moreover, the characteristic strain field ahead of the crack tip can help to find the actual crack tip coordinates by fitting a truncated Williams series to the experimental data10. However, the precise and reliable detection of crack tips from DIC displacement data is still a challenging task due to inherent noise and artefacts in the DIC data11.

Convolutional neural networks (CNNs) led to enormous breakthroughs in computer vision tasks like image classification12, object detection13, or semantic segmentation14. Recently, deep learning algorithms are also finding their way into materials science15, mechanics16,17, physics18 and even fatigue crack detection: Rezaie et al.19 segmented crack paths directly from DIC grayscale images whereas Strohmann et al.20 used the physical displacement field calculated by DIC as input data to segment fatigue crack paths and crack tips. Both architectures were based on the U-Net encoder-decoder model21. Pierson et al.22 developed a CNN-based method to predict 3D crack surfaces based on microstructural and micromechanical features. Moreover, CNNs are able to segment crack features from synchrotron-tomography scans23,24 and can also detect fatigue cracks in steel box grinders of bridges25. For a detailed review on fatigue modeling and prediction using neural networks we refer to the recent review article by Chen et al.26.

In the present work, we investigate the interpretability of machine-learned fatigue crack tip detection models. For this, we introduce a novel network architecture called ParallelNets. The architecture is an extension of the classical segmentation network U-Net by Ronneberger et al.21 and its modification by Strohmann et al.20 for fatigue crack segmentation in DIC data. To this purpose, we train a parallel network for the regression and segmentation of crack tip coordinates in two-dimensional displacement field data obtained by DIC during a fcp experiment. Exemplarily, we use the Grad-CAM method to obtain neural attention heatmaps for input samples from several fcp experiments. Finally, we discuss the overall attention and the individual layer-wise attention of three trained models and find relations to their performance and robustness on unseen data.

Ground truth data for the crack tip position was obtained by manual segmentation of high-resolution optical images20. Here, we use the ground truth data from experiment \(S_160, 4.7\) for training and validation (i.e. model selection).

We can view this task as a regression problem and combine a convolutional neural feature extractor with a fully connected regressor that outputs the crack tip position41. Such architectures were already used for image orientation estimation42, pose estimation43 or, more recently, respiratory pathology detection44. This approach can be advantageous since it overcomes the class imbalance problem. However, we found that such models are not precise enough for our use case and they are useless for images without crack tips or with multiple cracks.

We introduce an architecture named ParallelNets that combines the two approaches described above and train them in a parallel network45,46. The architecture is shown in Fig. 1: a classical U-Net21 encoder-decoder model is fused with a Fully Connected Neural Network (FCNN) based at the bottleneck of the U-Net. Consequently, the network has two output blocks, i.e. a crack tip segmentation from the U-Net decoder and a crack tip position from the FCNN regressor. On the one hand, we expect that this learning redundancy can lead to improved robustness because the network encoder needs to provide good latent representations for both tasks, namely segmentation and regression. On the other hand, for the same reason ParallelNets might be harder to train than a simple U-Net and the corresponding segmentation and regression losses need to be properly balanced.

Schematic ParallelNets architecture. The classical U-Net architecture21 with four encoder blocks (Down) and four decoder blocks (Up) connected by a base block (Base) is shown in blue. Encoder and decoder blocks of the same level are connected by skip connections (gray dashed lines). The additional modules of our ParallelNets architecture are shown in orange and basically consist of a fully connected neural network (FCNN) which is trained to output the crack tip position in terms of normalized x and y coordinates.

The FCNN consists of an adaptive average pooling layer followed by two fully connected layers with ReLU activation functions and finishing with a 2-neuron linear output layer. It predicts the (normalized) crack tip position \(y=\left(y_1,y_2\right) \in [-\mathrm1,1]^2\) relative to the center of the input data.

During training, we calculate the mean squared error between the prediction and the ground truth crack tip position \(\widehaty=\left(\widehaty_1,\widehaty_2\right)\in [-\mathrm1,1]^2\), i.e.

To train ParallelNets properly, we found by trial-and-error that a loss weight of \(\omega =100\) works well, since it balances both loss terms making the whole model learn both the segmentation and regression of crack tips. Lower values of \(\omega\) pronounce the segmentation task and higher values the regression task.

In terms of hyperparameter optimization, we identified the Adam optimizer50 with a learning rate of \(5\times 10^-4\) and a batch size of 16 by trial-and-error. Moreover, we tried different dropout probabilities \(p\in [0,\frac12]\) for the bottleneck of U-Net and ParallelNets but found no substantial difference.

Figure 3 shows the displacements and von Mises equivalent strain acquired by DIC for the three samples. The results are interpolated on a 256 \(\times\) 256 pixels grid. While the samples are qualitatively similar, it has to be considered that the size of the MT-specimen for \(test_\mathrmlarge\) is six times larger than the others. The deformation field around the crack tip is best visible in the von Mises equivalent strain field in Fig. 3.

We find that U-Net-1 displays inconsistent attention heatmaps. On the one hand, for \(val\) and \(test_\mathrmsmall\) the model seems to pay attention to different parts of the crack path. On the other hand, there are no areas of high attention for \(test_\mathrmlarge\). This result indicates the confusion of U-Net-1 in the evaluation of \(test_\mathrmlarge\) which may be related to the larger specimen dimensions.

Reliability of crack detection, calculated as the number of input samples with at least one pixel segmented as crack tip over the total number of input samples (every sample contains one crack tip).

The results are shown in Table 1. We are only able to calculate the Dice coefficient and deviation for the training and validation datasets since the test datasets are unlabeled. ParallelNets-1 outperforms the other networks on all datasets except the validation dataset. Especially, it is the most reliable network on unseen data (\(test_\mathrm160,2.0\) and \(test_\mathrm950,1.6\)) and reaches a perfect reliability on the training dataset. An overall test reliability of 96.8% is reached on the unseen data. Furthermore, in terms of accuracy, it shows an overall mean deviation of the crack tip position from the ground truth of 0.54 mm (training and validation data combined) with a standard deviation (std) of 0.38 mm. The model generalizes correctly also to larger specimen sizes (\(test_\mathrm950,1.6)\), although, in contrast to Strohmann et al.20, no additional synthetic training data in form of finite element simulations was needed.

The second-best network is U-Net-2 with a deviation of the crack tip position (mean/std) of 0.61/0.74 mm and an overall test reliability of 93.9% on unseen data. U-Net-1 shows the best performance only for the Dice coefficient and deviation on the validation dataset. We remark again that the networks were selected during training using the validation Dice loss as the only selection criterion. This explains why the network U-Net-1 was chosen although it is far less reliable (70% overall test reliability on unseen data) and least accurate on the training dataset (0.88 mm mean deviation). This shows the need for improved model selection criteria during or after training.

We now compare the crack detection stability of the different models. The detected crack tip positions should result in a growing crack length, i.e. the crack length \(a\) increases between subsequent samples, i.e. \(\Delta a=a_new-a_old\) should be positive. We estimate the crack length 2ff7e9595c