# Hierarchical Tensor Decomposition of Module Partition for the Mechanical Simulation
## 4 Case Study  
In this section, the proposed HTD method is applied to a case study of the module partition of the mechanical simulation. First, as part of the proposed method, the scree plot of \(\|\hat{\textbf{\theta}}_{m}-\hat{\textbf{S}}\|_{2}\) is used to determine the layer of decomposition, as shown in Figure 9. When the raw data is divided into three layers, \(\dfrac{\|\hat{\textbf{\theta}}_{3}-\hat{\textbf{S}}\|_{2}}{\|\hat{\textbf{\theta }}_{2}-\hat{\textbf{S}}\|_{2}}=\dfrac{2.3\times 10^{7}}{2.3\times 10^{7}}=1>0.8\), which indicates the number of layers should be determined to be two. Then, the proposed method and the benchmark methods are applied to the raw data. As shown in Figure 10a, it is evident that the HTD algorithm provides a background layer, a foreground layer, a sparse spots (stress concentrations) layer, and a random noise layer. Among these layers, the background layer has low intensity variation, the foreground cluster region layer has higher intensity variation than the background layer, and the sparse spots layer has a higher intensity variation than the foreground layer. Local thresholding is sensitive to sparse spots and noise, resulting in large detection errors, as shown in Figure 10b. SSD decomposed the raw data into three components, i.e., smooth background, sparse, and noise components. It can be seen in Figure 10c, the foreground cluster regions are divided into the sparse component. The low-rank method combined the foreground layer and the background layer into one-layer \(\hat{\textbf{L}}\), which also included a partial sparse stress concentration feature, as shown in Figure 10d. Figure 10e shows the result of CP decomposition method. Though CP decomposition can separate raw data into layers, each layer is the outer product of the two one-way tensors, which makes the decomposition layer difficult to interoperate according to the physical meaning. Figure 8: The foreground cluster region layer and sparse spots layer extracted by the proposed HTD method are compared with the labeled ground truth. (**a,b**) are the foreground cluster region and sparse spots, respectively, extracted by HTD. Figure 7: The extracted sparse component based on local thresholding, SSD, and RPCA. According to Equation (3), the elements that belong to sparse spots in \(\hat{\mathbf{S}}\) are extracted and binarized. A postprocessing method based on Equation (4) is applied in \(\hat{\mathbf{B}}_{2}\) to extract the clustered foreground regions. The region to be detected in \(\hat{\mathbf{B}}_{2}\) has a higher intensity value than the background. Therefore, the element \((i,j)\) satisfies \(\hat{\mathbf{B}}_{n}(i,j)>f_{n_{i}}\) is considered as foreground cluster element. \(v_{\max}=0\), \(h=1.6\). The foreground cluster and sparse stress concentration detection result of HTD, SSD, and RPCA are labeled, after postprocessed \(\hat{\mathbf{B}}_{2}\) and \(\hat{\mathbf{S}}\), as shown in Figure 11. In Figure 11a, the foreground cluster regions are detected and marked as red color, the sparse stress concentrations are labeled as green color. As shown in Figure 11b, noises with high intensities have been identified as stress concentrations, primarily as a result of their high intensity. For SSD, the background and foreground cluster regions are divided into one layer, the sparse stress concentrations are identified as shown in Figure 11c the majority of foreground regions and sparse stress concentrations are not detectable. RPCA is sensitive to the high-intensity value noise; there are many FP elements, as shown in Figure 11c. Obviously, the proposed HTD method has a better performance than the benchmark methods. Figure 11: The detection result of the foreground cluster regions and sparse stress concentrations based on the HTD, Local thresholding, RPCA, and SSD. Figure 9: The score plot of the strain contours of the mechanical simulation sample. To improve simulation results, the stress concentration and strain cluster regions were extracted by the HTD method.