# Hierarchical Tensor Decomposition of Module Partition for the Mechanical Simulation
## 1 Introduction
Based on a single global threshold that minimizes variations in intensity within foreground clusters within an image, global thresholding approaches organize the elements in an image into two foreground clusters, a background cluster, and a foreground cluster. There are several techniques developed to overcome this problem, for example, adaptive thresholding segmentation [28] and maximum entropy threshold segmentation [29]. By assessing the relative intensity levels of elements in the local area, local thresholding establishes the boundary between foreground and background. Using a global thresholding approach for extracting foreground cluster features and segmenting objects from an image with multiple components, may result in excessive variation of intensity within a foreground cluster, making it difficult to distinguish between the features of individual regions from the background [30]. However, if the components of different types overlap in the foreground, local thresholding approaches with multiple thresholds may be merged. Defining thresholding-based solutions as an approach has a fundamental problem in the fact that they do not explicitly take into account the smoothness and sparsity of tensor structures, as  
Figure 1: A hierarchical mechanical simulation strain contours with different layers. (**a**) is the design domain and boundary conditions; (**b**) is the strain contour for (**a**) by ANSYS; (**c**) is the three-dimensional display of (**a**). well as the characteristics of the components in the raw data when setting a threshold, and as a result, they are not able to efficiently decompose tensors. In the analysis of data and the reduction of dimensionality, low-rank-based decomposition methods [31] have become widely used. The principle is based on the assumption that an image can be decomposed into a low-rank component, which lies within a low-rank subspace, and a sparse component. In addition to repairing images and recognizing objects, this method can be used to reconstruct data, detect small targets, and identify data sources [32]. There are a number of proposals for separating low-rank and sparse components from the original data, such as Robust Principal Component Analysis (RPCA) [33]. After this, several methods are proposed based on RPCA. As one example, the researcher composed a low-rank matrix and sparse component recovery problem [34] to detect the sparse spots in the infrared image. A spatially weighted PCA formulation [35] was developed to improve the local correlation structure. A low-rank component, which contains information about the smooth background and partial foreground of the background, is a component that contains two similar points: (i) represents the span of two vectors with linear correlation; (ii) sparse spots, which represent rugged areas of high intensity that are sparsely distributed. Considering the two points cited above, the low-rank decomposition method does not explicitly distinguish between the global and local smoothness of the foreground and background, respectively. Therefore, it is inapplicable for determining the foreground cluster regions' features. Many methods can be derived from functional analysis, including kernel smoothing, SVD decomposition, Fourier transformations, wavelet transformations [36], and B-spline [37] transformations. These approaches are mostly used in order to eliminate noise from the original data. The detection of defects (sparse spots) makes use of a number of approaches, including derivative operators, Laplacian operators, and Sobel operators. The methods described above, however, may not be able to simultaneously estimate the different foreground cluster regions and sparse spots due to their lack of detection accuracy and computation time. Smooth-sparse decomposition (SSD) [37] is proposed as a method for identifying sparse anomalies (sparse spots) in images. A Smooth-Sparse Decomposition model is developed in order to explore the spatial-temporal properties of images.