# Hierarchical Tensor Decomposition of Module Partition for the Mechanical Simulation
## 3 Simulation Study  
To evaluate the effectiveness of the proposed HTD method under various conditions, several simulation samples are used in this section. Initially, two samples are simulated with different intensity variation layers within foreground cluster regions in order to demonstrate how to determine the number of layers. As illustrated in Figures 2 and 3, these tensors contain different layers of foreground cluster regions and sparse spots. Both of these simulation samples correspond to \(401\times 401\) two-way tensor \(\mathbf{X}\) based on the following model: \(\mathbf{X}=\mathbf{B}_{n}+\mathbf{S}+\mathbf{E}\), where \(\mathbf{B}_{n}\) represents the background and foreground cluster region layers, \(n=\{1,2,\ldots,N\}\), \(\mathbf{S}\) represents the sparse spots layer, and \(\mathbf{E}\) represents random noise. Specifically, Figure 2 is composed of a background layer \((\mathbf{B}_{1})\), a layer with foreground cluster regions \((\mathbf{B}_{2})\), a layer of sparse spots \((\mathbf{S})\), and a layer of random noise \((\mathbf{E})\). Figure 3 consists of a background layer \((\mathbf{B}_{1})\), two layers with foreground cluster regions \((\mathbf{B}_{2},\mathbf{B}_{3})\), a sparse spot layer \((\mathbf{S})\), and a random noise layer \((\mathbf{E})\). Essentially, \(\mathbf{B}_{n}\) is created by merging multiple two-dimensional Gaussian distributions. The centers of these two-dimensional Gaussian distributions are randomly distributed in the area of \(401\times 401\). The intensity variation of \(\mathbf{B}_{n}\) is determined by superimposing these two-dimensional distributions. With such simulation constraints, \(\mathbf{B}_{n}\) can be guaranteed to vary smoothly and with a certain intensity variation. With the increase of \(n\), the intensity variation of \(\mathbf{B}_{n}\) increases gradually. In Figure 2, \(\mathbf{B}_{1}\) is simulated by the superposition of 8 two-dimensional Gaussian distributions; \(\mathbf{B}_{2}\) is generated by a two-dimensional Gaussian distribution. In Figure 3, \(\mathbf{B}_{1}\) is generated by the superposition of 10 two dimensional Gaussian distributions; \(\mathbf{B}_{2}\) is generated by a two-dimensional Gaussian distribution; \(\mathbf{B}_{3}\) is simulated by a two-dimensional Gaussian distribution. A sparse spots layer is simulated to model the sparseness of \(\mathbf{S}\). This layer consists of several small, connected, closed areas that are tightly connected. It is noteworthy that each area includes a square of \(5\times 5\) elements, and its intensity variation is relatively larger than that of \(\mathbf{B}_{n}\). Finally, each element in the noise layer follows a Gaussian distribution \(N(0,\sigma^{2})\), where \(\sigma=1\). To determine the number of layers of decomposition, the simulation data from Figures 2 and 3 were first decomposed using the proposed HTD method. Then, the scree plot was used to determine how many layers of decomposition there were. The decomposition of the tensor with multilayer property is accomplished by determining the number of layers based on domain knowledge and evaluating the variation of \(\|\mathbf{\hat{B}}_{n}-\mathbf{\hat{S}}\|_{2}\), as shown in Figure 4. For the simulation in Figure 4a, \(\frac{\|\mathbf{\hat{B}}_{n+1}-\mathbf{\hat{S}}\|_{2}}{\|\mathbf{\hat{B}}_{n}- \mathbf{\hat{S}}\|_{2}}>\zeta\), where \(\zeta=0.8\). Referring to the simulated data \(\mathbf{X}\), when \(n=2\), \(\frac{\|\mathbf{\hat{B}}_{n+1}-\mathbf{\hat{S}}\|_{2}}{\|\mathbf{\hat{B}}_{n}- \mathbf{\hat{S}}\|_{2}}=\frac{2.3\times 10^{4}}{2.3\times 10^{4}}=1>0.8\); therefore, the number of the smooth layer of \(\mathbf{X}\) is \(n=2\). As a result, \(\mathbf{X}\) is divided into three components, \(\mathbf{\hat{B}}_{1}\), \(\mathbf{\hat{B}}_{2}\), and \(\mathbf{\hat{S}}\). For the simulation in Figure 4b, when \(n=3\), \(\frac{\|\mathbf{\hat{B}}_{n+1}-\mathbf{\hat{S}}\|_{2}}{\|\mathbf{\hat{B}}_{n}- \mathbf{\hat{S}}\|_{2}}=\frac{123\times 10^{6}}{1.23\times 10^{6}}=1>0.8\); therefore, the number of the smooth layer of \(\mathbf{X}\) is \(n=3\).